!
~
'¥'<, ~ (!,~ t
"
..
Pressure Buildup and Flow Tests in Wells
C. S. Matthews
.
Manager of Exploitation Engineering ,
Shell Oil Company
D. G. Russell StafJExploitation Engineer
Shell Oil Company
..""""'~~:::"~~.:.' ! ."
'.
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1
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Henry L. Doherty Memorial Society of Petroleum New York
Fund of AIME
Engineers of AIME
1967
Dallas
" ,,'.. """
Contents 1.
Introduction
1
5. ~re
1.1 Use~of ~ressureInformation in Petroleum EngIneenng 1 1.2 Early History of PressureMeasurements 1 1.3 Types of PressureInformation 1 1.4 Early History of PressureAnalysis Methods 2 1.5 Objectivesof Monograph 2 1.6 Organizationof Monograph 2 2.
Mathematical Basis Analysis Methods
for
4 4; 5
/M:6r..ultiple-Rate
2.3 2.4 2.5 2.6
Single-PhaseLiquid Flow 6 Single-PhaseGas Flow 7 Multiphase Flow 8 Solutions for Radial Flow of Fluid of Small and ConstantCompressibility 10 2.7 Conceptsof Transient,Semi-SteadyState, and Steady-StateFlow Behavior 12 2.8 The Principle of Superpositionand Approximation of Variable-Rate PressureHistories 14 2.9 Units -Field Unit and Darcy Unit Systems 16 / ~ressure Buildup Analysis 3.1 Basic Method 3.2 Skin Effect, Skin Factor, and Flow Efficiency 3.3 BoundedReservoirs 3.4 PressureBuildup for Two- or Three3 5 PhaseFlow .Pressure Buildup in Gas Wells 3.6 Effects of Wellbore Fillup and Pha Redistribution se 3.7 Effect of Partial Penetration
'7 / IV
Superposition
to
Account
for
Production Rate Variation 3.9 Alternative Methodsof Pressure
na YSIS 0f
7.1 7.2 7.3 7.4 7.5
19 21 Y
4/
Determi!1ation
Reasonsfor InterferenceTests Equations for PressureInterference ExampleCalculation,InterferenceTest Least-Squares Methods Other Methods for Computing Interference
Pressure
Analysis
We/ls
8.2 Pr~ssureFall-O~ Af!alysis to ReservoIr ...
Fillup
49 50 51 52 53 56 58 58 60 61 62 62 67 67 67 68 69 70 72 72 73
8.3 Two-Rate Injection Test AnalysIs 8.4 Gas Injection Wells
81 81
30 ~
Drillstem
Test Pressure
Determining
Volumes
4.3
~etermining Avera~e Reservoir Pressure In Bounded (Depletion-Type) Reservoirs 39
4.4 Water-Drive Reservoirs
of
Wells
35
44
Analysis
9.1 PressureBehavioron DST's 9.2 Oper~t~onalConsiderationsin ObtaInIng
4.2
.
in Injection
29
Reservoir Pressure 35 4.1 Usesof Average ReservoirPressureData 35
~--
f erence Tests
8.1 PressureFall-Off Analysis in Unit-Mobility, Liquid-Filled Reservoirs
of Average
Drainage
Wenter II I
27
30
Buildup Analysis
Flow Test Analysis
A I .
Pnor 3.8
48
6.1 General Equations for Analysis of Flowing Well Tests with Variable Rate 6.2 Two-Rate Flow Test Analysis Method 6.3 Two-Rate Flow Test Analysis in Non-Ideal Cases 6.4 Elimination of Wellbore Effects with Two-Rate Flow Tests 6.5 Tran~ien~Analysis of Gas-Well Multi-PoInt Open-FlowPotentialTests
18 18
22 24
Analysis
5.1 PressureDrawdown Analysis for TransientConditions 5.2 PressureDrawdownAnalysis for Late TransientConditions 5.3 PressureDrawdown Analysis for Semi-SteadyState Conditions 5.4 Exampleof Application of Pressure DrawdownAnalysis Methods 5.5 OperationalConsiderationswith PressureDrawdownTests 5.6 Behaviorin Non-Ideal Cases
Pressure
2.1 Basic Assumptions 2.2 The Continuity Equation
Drawdown
9.Use 3
Good
DST
Pressure
84 Data
. Theory of PressureBuildup
on DST Data
9.na 4 A I YSIS .. of DST Flow Penod PressureData
84
86
86
87
9.5 Multiple-Rate DST's 9.6 Practical Considerationsin DST Interpretation 9.7 Wireline Formation Tests 10,
88 88 88
Effect of Reservoir Heterogeneities on Pressure Behavior
t
92
10.1 PressureBehaviorNear Faults or Other ImpermeableBarriers 10.2 Effect of Lateral Changesin Hydraulic Diffusivity on PressureBehavior , 10.3
Pressure
BehavIor
, m Layered
ReservoIrs
10 4 P B h ' . N II .ressure e aVlor m atura y Fractured Formations
, '1
DrainageAreas 10.7 Effect of Pressure-Dependent Rock Properties 10.8 ConcludingComments II,
Practical Analysis
Aspects
102
114 114
11.3
115
Wells
115 116
11.8 Well Stabilization 11.9 Other Considerationsin Well Tests 11,10 MeasuringInstruments
117 117~ 118 119 119
11.11 Qualitative Interpretation of Buildup Curves
122
12 C ,
110 110
.., ofTestsm FlowmgWells 11.1 ChoIce 11.2 Choice of Testsin Injection Wells
11.7
J ~
109
of Pressure
11.4 Required Closed-InTimes 11.5 Radiusof Investigation 11.6 Notes on Fractured and Other HeterogeneousReservoirs Correction
I
of Pressure to a Datum
'
onc us Ion
12.1 The Stateof the Art 12.2 Current Problemsand Areas for Further Investigation 12.3 Value of PressureAnalysis Methods to the Petroleum Industry 12.4 .Where Do We Go From Here?
~
.Reservoir 97
114
Tests in Pumping
Nomenclature
Constant Rate, BoundedCircular ReservoirCase ConstantRate, ConstantPressure Outer BoundaryCase Appendix B: Example Pressure Buildup
131 133
Calculations Analysis
for 134
95
10.5 PressureBehavior in Hydraulically Fractured Wells 103 10.6 PressureBehavior in Non-Symmetrical
.:
1
92
Appendix A: Solutions for Radial Flow of Fluids of Small and Constant Compressibility 130 ConstantRate,Infinite ReservoirCase 130
124 124
Above
Bubble
Point
135
ReservoirBelow Bubble Point Gas Reservoir
136 138
" " Appendix C: Example Calculation Average Pressure
for 140
Matthews-Brons-Hazebroek Method Miller-Dyes-HutchinsonMethod Append1x D: Example Calculations Pressure Drawdown Analysis
140 141 for 142
TransientAnalysis Late TransientAnalysis Semi-Steady State Analysis (Reservolr . L Iml ' . tT est)
142 144
D '.ISCUSSlon
145
145
Appendix E: Example Calculations for Multiple-Rate Flow Test Analysis Two-Rate Flow Test 0 pen- Fl ow P otentla . I T est M uI tl-. Point '
147 147 148
Appendix F: Example Calculations for Injection Well Analysis 150~ Pressure. Fall-~~ AnalY,sis, Liquid-Filled Case, UnIt MobIlity Ratio PressureFall-Off Analysis Prior to Reservoir Fillup, Unit Mobility Ratio PressureFall-Off Analysis,Non-Unit Mobility Ratio
150 152
Two-Rate Injection Test
153~
152
126
" " Appendix G: Charts and Correlations~.. for Use in Pressu~e Buildup and Flow Test AnalysIs
155
126
" hy B I' bl lograp
164
125
128
Subject-Author
168~
Index
"
~-
,
Chapter 1
Introduction
1.1
Uses of Pressure Information in Petroleum Engineering
and MacDonald gauges. By 1933 there were some 10 different kinds of instruments in use.5
Several hundred technical papers have been published over the past 35 years dealing with the important subject of pressure tests in oil and gas wells. This extensive literature has evolved because the pressure behavior of a well is both a readily measurable and a highly useful quantity. Pressuredata from wells may be used to estimate how efficiently the well is completed, the need for and successof a well stimulation treatment, the general type of well treatment desirable, the degree of connectivity to other wells and many other
One of the first field-wide applications of subsurface pressuresoccurred in the East Texas field. Information obtained from periodic surveys in key wells was used to control allowables, equalizing rate of oil off-take with rate of water influx. Another early application was made in Kansas where liquid levels were measured in wells while pumping. These measurementswere used in prorating wells. This method eliminated installation of special high-capacity pumps to "potential" wells and was an early step in analyzing well behavior.
items. Pressure data from wells are used to define local and average reservoir pressures.These data, when combined with hydrocarbon and water production data and with laboratory data on fluid and rock properties, afford the means to estimate the original oil in place and the recovery which may be expected from the reservoir under various modes of exploitation. It is the purpose of this Monograph to present the subject of pressure tests of wells as a coherent whole using published techniques as a basis and adding new information and techniques where needed.
1.3 Types of Pressure Information
1.2 Early History of Pressure Measurements Instruments for measuring maximum pressures in wells were developed and applied in the United States during the early 1920'S.1One early device was simply a Bourdon gauge with a stylus which marked on a blackened face. Other devices were developed to measure liquid levels in wells, utilizing floats or sonic echos. Sclater and Stephenson2discussed an application of pressure measurements from such early devices in a gas-oil ratio study in 1928. A year later Pierce and Rawlins3 reported on a study of a relation betweenbottom-hole pressure and potential production rate. The utility of early bottom-hole pressure instruments was greatly increased by the development of continuously recording instruments such as the Amerada,4 Humble 'Referencesgiven at end of chapter.
.Except for such liquid-level measuremelltsin pum~mg wells, the usual type of pressure measurement m ea~ly days was a so-called."static': measurement. In this type, a pressure-measunngdeVIcewas lowered to the. bottom. of a well which had been closed for a penod of time, such as 24 to 72 hours. The pressure measured ~t this time was called a "stati~" .pressure. These st~tic measurementS;sufficed ~o. Indicate ~he pressure m permeable, high-productiVIty reservoIrs. However, engineers soon recognized that in most formations the static pressure measurements were very much functions of closed-in time. The lower the permeability, the longer the time required for the pressure in a well to equalize at the prevailing reservoir pressure. Thus, engineers realized very early that the rapidity with which pressure buildup occurred when a well was closed in was a reflection of the permeability of the reservoir rock around that well. This qualitative observation was an important step in developing an understanding of well pressure behavior. This understanding led to the other basic type of measurement, called transient pressure testing. In this type, the pressure variation with time is recorded after the flow rate of the well is changed. It is this type of measurement which is used in modem pressure tests of wells and, thus, is the type with which we shall mainly be concemed in this Monograph.
2
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
A stimulus for developing a quantitative interpretation of pressure data came with the introduction of the material-balance method6 of calculating original oil in place in a reservoir. To provide meaningful data for this method, engineers began to seek answersto questions such as: "How long should I shut in a well to get the required pressuresfor this method?", and "Can I extrapolate data to a static pressure?". Development of other analytical methods of analyzing reservoir performance, such as the treatment of water drive by Moore, Schilthuis and Hurst,7 increased the need for a method for quantitatively treating pressure data.
wells and pressure fall-off in injection wells. We shall also treat pressureresponseduring multiple-rate tests in both producing wells and injection wells. We have tried to provide in this Monograph an upto-date treatment for the benefit of engineerswho want to re-educate themselves on the subject of pressure tests. We have stressedexample applications particularly. For those who are more mathematically inclined, we have also presented a rather complete treatment of the mathematical basis. Most of this treatment has been placed in appendices,however, so that the Monograph's readibility will not be impaired.
1.4
Early History of Pressure Analysis Methods
The general plan has been to present, rather completely, a preferred method for each type of pressure
The first effort to present an extrapolation theory and to relate the change in pressure with time to the parameters of the reservoir was presented in 1937 by Muskat.8 He deduced, mathematically, a method for extrapolating the measured well pressure to a true static pressure. Muskat stated at the time that his method had only a qualitative application. In a sensethis was true, since this method did not take into account the important aspectof fluid compressibility. The first comprehensive treatment of pressure behavior in oil wells to include the effects of compressibility was that of Miller, Dyes and Hutchinson9 in 1950. The following year Hornero presented a somewhat different treatment. These two papers still furnish the fundamental basis for the modem theory and analysis of oilwell pressure behavior.* Subsequentpapers have brought a multitude of refinements and a deeper understanding of this subject. In this Monograph we will attempt to incorporate these refinements into the earlier basic methods.
analysis. Alternative methods will usually be discussed and referenced, and in some casespresented also. Because of Space limitations however it is not feasible to , , presentcomplete discussionsof all methods of pressure analysis. The references should provide a guide to the alternative procedures. .. 1.6 Organization of Monograph At this point some comments on the organization of the Monograph are in order. The next chapter presentsthe mathematicalbasis for pressureanalysis methods. It is not essentialthat a reader master this chapter to be able to understand and apply the methods discussed in the remainder of the book. However, we hope that every reader -even those who are not particularly well-versed in advanced mathematics-will browse through this section to enhance his basic understanding of the various pressure analysis methods. Subsequentsections are devoted to pressure buildup, pressure drawdown, pressure fall-off, interference tests
...and We will not trace the hIStOry of pressure ~nalysis
multiple-rate tests. In each case illustrative exampIes using actual field data are presented. Discussions
further. Ref. 1: as well as the many refer~ncesm later chapt~rs of this ~onograph: sh~uld furnIsh adequate matenal for those Interested m thIS aspect.
are included on drillstem test analysis, reservoir heterogeneities and on the practical aspects of bottom-hole pressure measurement.The paper ends with a discus-
1.5 Objectives of Monograph In our treatment we shall concern ourselves almost entirely with the subject of creating and analyzing the transient pressure responsein a well. By transient pressure response, we mean the pressure response which results from a change in a well's production rate. For instance, a transient pressure is created by putting a closed-in well on production. In a well which has been producing at a constant flow rate for some period of time and has reached a pseudo steady-state behavior, a pressure transient is created by closing in the well or,
sion of problems yet unsolved in pressurebehavior. We hope that the manner of presentation will make the Monograph both readable and yet practical as a guide for day-to-day use.
alternatively, by changing the producing the types of transient pressure behavior
References 1. History of Petroleum Engineering,API (1961). 2. Sclater,K. C. and Stephenson,B. R.: "Measurements of Original Pressure,Temperature,and Gas-Oil Ratio in Oil Sands", Trans.,AIME (1928)82, 119-136.
rate. Among we shall con-
3. Pierce, H. R. and Rawlins, E. L.: "The Study of a Fundamental Basis for Controlling and Gauging Nat-
sider are pressure buildup and drawdown in producing
ural Gas Wells", RI 2929 and 2930, USBM (1929). 4. Millikan, C. V. and Sidwell,C. V.: "Bottom-holePressuresin Oil Wells", Trans.,AIME (1931)92, 194-205.
*Some different approacheshave beenused by Russian and French authors. For a review of Russian pressure buildup methods,see Ref. 11. Referencesto some of the French methods this Monograph.
will
be made in subsequent chapters
--~
---~
.".
5. Hawthor~: D.. G.. Review of S~ bsur f ace Pressure I nstruments , 011 and Gas J. (April 20, 1933) 16, 40.~iiiL~__'~
of
-
Chapter 2 "',
,
Mathematical
Basis For Pressure
-
'
:"',;~1"'\.;
Analysis Methods
The pressure analysis techniques to be discussed in this Monograph have been derived from solutions of the partial differential equations describing flow of fluids through porous media for various boundary conditions. By beginning with the underlying physical principles and considering the differential equations and the solutions of interest, one can better understand the implications of pressure analysis theories. 2.1 Basic Assumptions I d f fl 'd fl ' A h mat ematica escnption 0 UI ow m a porous medium can be obtained from the following physical principles: (1) the Law of Conservation of Mass; (2) Darcy's law (or other flow law); and (3) Equation(s) of State.
sectional area, it! is the potential, \l it! is the gradIent of the potential in the direction of flow, ,II.is the viscosity of the fluid, k is the permeability of the medium (a constant) and p is the density of the fluid. The minus sign in the above equation denotes that flow occurs in the direction of decreasingpotential. Hubbert' has studied Darcy's law and its implications quite extensively, and those who are interested in the fundamen.talconsiderations concerning this law are referred to hISwork. Hubbert showed that
In flow phenomena of any type (fluids, heat, electricity) , one of the most useful statementsis a conservation principle. This is simply a statement that some physical quantity is conserved, i.e" neither created nor destroyed. In fluid flow in a porous medium, the most significant quantity conservedis mass and the conservation statement is simply (referring to an arbitrary re-
where z is the height above and Po is the pressure in an arbitrary datum plane. The forms of Eq. 2,1 for flow in the x, y and z directions are
.
gion)
..
p it!=
J !!!!-+ po
--~ U.-
k ~ ,11.. ax '
"fJit!
.p
(amount of mass mput)-(amount put)
gz
p'
of mass out-
+ and (net sinks) amount of mass introduced by sources = (increase in mass content of the region), Darcy's law expresses the fact that the volumetric
u,,=--k"a'
,II.
u.---~
y
,II.k z~oz .
Thus, for flow in the x, y and z directions, respectively, Darcy's law can be expressedas
rate of flow per unit cross-sectionalarea at any point in a uniform porous inmedium is proportional to that the gradient in potential the direction of flow at
U. = --; k. ~op
point, The law is valid for laminar flow at low Reynolds numbers,l and its mathematical expressionis*
u,,= -~
\lit! , (2.1) ,II. where u is the volumetric rate of flow per unit crossU = -~
k
0 -:-
,II.
fJY
(2.2)
k. [-az-+ pg] OP
U.= --;
~-
I
I In these equations, Ui (i=x, y or z) denotes the volumetric rate of flow per unit cross-sectionalarea in the
*See the Nomenclatureon page 128.
'
I
MATHEMATICALBASIS FOR PRESSUREANALYSIS METHODS
i
direction i. The symbols k", ky and k" are the permeabilities of the rock in the indicated directions. For radial flow, neglecting gravity, Darcy's law becomes k 0 Ur = -~ --.!!.-. ,lI. or
2.2 The Continuity Equation In this section we will develop a mathematical statement of the continuity principle. By subsequentcombination of the continuity equation with Darcy's law and equations of state, we can derive a family of differential equations which describes various flow situations. We begin by considering a single fluid flowing through a
! t
In the case of flow at high velocities, Darcy's law is no longer valid. It has been found that a quadratic ve-
porous medium of porosity cf>.We choose an arbitrary volume element within the flow region and apply the
locity correction term can be added to modify Darcy's law. In this case the flow law becomes k, cp --~ = U + D1u2, ,lI. 0"
continuity statementpresented in the previous section. Since our primary interest in this Monograph is in ra~ial flow, we ~hall derive the continuity equatio,?apphcable for radial flow as well as the more general, three-dimensional case. We first consider the three-di-
where Dl is a constant that is a function of the pore structure of the porous medium and 0"is the direction of flow. The reader who is interested in so-called non-
mensional case and choose as our arbitrary volume the rectangular parallelepiped shown in Fig. 2.1A, Th 1 t t f fl '
I
t
Darcy
,
, IS referred
flow
.
to
'8
the
papers '
by '
Houpeurt .
or,
h h I d hi h Ramey,'T T e mat ematica consl eratlons In t s c ap-, ter are
b
ase
d
on
fl
h' h b D ' 1 ow w IC 0 eys arcy saw,
Various equations of state are used in deriving the
I .,.
1,
.~
flow equations. An equation of state specifiesthe dependence of fluid density p bn the fluid pressure p and temperature T, Thus, depending on the actual fluid(s) ti' f t t . 11 b chosen t ' t presen , an appropna e equa on 0 s a e WI e ., Throughout this Monograph isothermal flow is assumed so that the equation of state will depend only on pressure, Before presenting the differential equations for flow through porous media, we should point out that a differential equation describes only the physical law or laws which apply to a situation. To obtain a solution to a specific flow problem, one must have not only the differential equation, but also the boundary and initial conditions that characterize the particular situation of interest,
5
e vo
ume
.
nc
componen , ,
s 0
ow
In t 0
th e e 1 emen
t
In the x, y and z directions are denoted by u", UII and . u", respectively. ,
These .
are volumetnc
flow
rates
per
urnt of cross-sectional area. Thus, the mass flow rate into the element in the x-direction is
The
mass
flow
rate
,.
pU" ~y ~z , .
In the
x-direction
out
of
the
element
IS ~y ~z [pu" + ~ ~pu,,)]. w~e~e ~(pu,,) is the change in mass,flux that. occ~rs within th~ element. The net flow r~te In the x-direction (amount-In less the amount-out) IS -~y
~z ~ (pu,,) .
Similar expressions can be written for the y and z directions, Assuming no mass is generated or lost in the element, the amount of net mass change in the element
":~
t P.! "..+4(p..)/ 4y
~ ""f"
P.!
-".
poy'4(p.y) .
--
y
. )--
\
/
0..+4(0..)
r p..
P.
// /./
y
.~
//;
/
I.
~J"
+4.
/
Ie
~
/
./
./././
/././
1.//
y
A
B
Fig. 2.1 Volume elementfor derivation of continuity equation: (A) in three spacedimensions;and (B) for radial flow.
6
PRESSURE
in a time increment dt can be expressedas -dt
[d(pU,,)
dY dZ + d(pUI/)
dZ dx
+
1 0
,"aT
d(pUz)dX
-0 -ar--(i/>p)
IN WELLS
...(2.5)
I
2.3 Single-PhaseLiquid Flow
t+dt
I
sIngle-phaseflow. The most Important of these, in the context of this Monograph, is the equation for isothermal flow of fluids of small and constant compressibility. The compressibility of a fluid is defined as the relative
important class of fl?w equations results for
!::.!:.!!!!-~ ] = (i/>p)t+dt-(i/>p) t .change
+ ~~+ dY
dZ
in fluid volume per unit change in pressure, or
dt
c=
Proceeding to the limit as dx, dY, dZ and dt approach zero gives
1
--- V
oV
op
.
This may also be written
0
0
0
0
1
ax(pU,,) +ay(pUI/) +az- (pUz)= -ar(i/>p) ,
.,
.,
c = p (2.3)
This equation is the continuity equation (in Cartesian form) for flow of a fluid in a porous mediurn. The continuity equation for radial flow follows from a similar development. If we consider the elementalvolume as shown on Fig. 2.IB, then the following mass balance can be written:
{ -dt
[ 8(r+dr)
] }
h(pu..) -8rh
pU..+d(pU..)
I
I
= cpph8rdr -cpph8r.lr t+dt'
-eC(p-po) P -po , .., " (2. 6) where pois the value of p at some referencepressure Po. This particular equation of state applies rather well to most liquids. If we introduce the equation of state of Eq. 2.6 into
Eq. 2.4, assume the viscosity is constant and neglect loop gravity forces, then (since -~ = c~) p
.
( k,,~
02p
'
]
+
( -F0 ok a ok --!.- + -F- -.! ox ox
~ "P
,
,(2.3a)
Eq, 2,3a is the continuity equation for radial flow. T d ' diff ' I ' a 0 enve erentia equations f or fl UI'd fl ow In p~rous mediu~, .we must. next combine Darcy'~ law WIth the continuIty equations. For the three-dimensional case, substitution of Eq. 2,2 into Eq. 2.3 yields
.
) + ~oz [~J.I.(~oz +pg)] (~J.I.~ ) + ~oy ( .!!-!:-~ ox J.I. oy (i/>p)
2
,
.,
,
.,
,
.(2,4)
0 ok oY oY + -F--!. az oz
~.I. ,,'/'
,,(2.7)
If c is small, if the permeability is constant and isotropic, if the porosity is constant and if it is assumed that the pressure gradients involved are small so that th: gradient squared terms may be neglected, the foregOIngreduces to ~!!:!-~= ~~ ax2 + oy2 + OZ2 k
(28) .
ot .,
F
d ' I fl b" or ra Ia ow, com Ination 0f Eqs. 2,6 and 2. 5 yields (viscosity constant) ~~ +~~~+C 2
(~ )
( r~ )
r or
or
k.. or or
or
at
Eq. 2.4 represents a general form for the combination of the continuity equation and Darcy's law. The final differential equation which will result from this equa~on depends on the fluid and the equation of state of Interest. For the radial flow case we obtain in similar manner: -~
~-~-
) *
"
at
-o(pu..)/or
~(rpu..) = -~ (cpp). or ot
uX
) [ ( )
( )
~t
uX
OP 2 + k" ( ay + k,,~ OP ) 2 a2p + kz""'J"Z2 02P + C k" ax
=cpp.c-+J.I.-
.ot d(pUr)/dr~ and SInce
=~
.
If c is constant then the above relationship can be
0 + k z -Foz
[ - ( ) ] --~ ~ pu..dr rd pUr -A rdr ~ r
ap
~
integratedto yield
This reduces to
~ ox
OP)
-;-"aT
FLOW TESTS
I
~his IS S!~p!y a dIrect application of the cOntin.UItypnncIple. DIVIding the equation by dX dY dZ dt YIelds dx
( rpk..
AND
dY]
= i/>pdXdY dZ -i/>pdX dy dZ I ..An
- [~~
BUILDUP
cpp.c~ + ~ ~ = T ot k.. ct ' *To establishthis relationship we have made use of a op o.p -a-i" (.pp)= .pat + Pat op o.p = .ppCat+ Pat .
MATHEMATICAL
BASIS
FOR PRESSURE
ANALYSIS
METHODS
If we assume constant permeability and porosity, cond h ap 2. ' b' li stant and small compressl I ty, an t at -IS ar negligibly small, the above equation becomes
( )
1 0 --rr or This
(
op ) -o2p --+ or ar2
equation
is
one
of
1 op -I/>p.c ap r or
the
most
used
and
a fluid
of
small
and
d
=
-a'
an
h
were
b
th
0
th
'
e
porosl
ty
an
d
in
a2p
petroleum
constant
a2p
a2p -I/>p.
~+az+T2-T(c+Cf)a' x y
op
z
t
.(2.12)
If Eq. 2.11 is expressed for radial flow it becomes a2 1 a 1 ak a 2 a -& +r-!r+ ( c +Ta)( fr) = ~ (c + Cf)-ft.
com-
p
pressibility must be assumed to obtain this equation from the original nonlinear equation with which we began. The reader should keep these assumptions in mind since solutions to this particular equation form the foundation of pressure~~ ~hniques. --Gas Eq. 2.8 and Eq. 2,9 are called diffusivity equations ~~ ,.. and the constant,~, IS called the hydraulIc dlffu-
(2.13) 2,4 Single-Phase Gas Flow An important class of single-fluid flow equations is that describing flow of gas through a porous medium. flow equations are different than those for liquid flow in that the equations of state which are used are quite different in functional form from those for liquids. The equation of state for an ideal gas is given by the ideal gas law as m pV = MRT,
~
si~y~iStOrlCaTiy~-th1sequation first arose in the study of heat conduction. Lord Kelvin called a corresponding constant in the heat-conduction equation the thermal diffusivity. Equations similar to Eq. 2.8 also arise in the study of diffusion and electrical potential distribution. Equations of this general type are known as the diffusivity equation.
where V is the volume occupied by the mass m of gas of molecular weight M, R is the gas law constant and
!
If we wish to obtain the differential equation for flow of a fluid of small and constant compressibility,
T is the absolute temperature. Since the density, p = ~, in this case is
but for the case of pressure-dependentporosity and permeability, we can further refine Eq. 2.7. If we assume constant viscosity, isotropic permeability and neglect gravity, we obtain
V
.then ~.
[( ap a2p a2p a2p n + 32 + T2 + C a x y z x
-+
~ k
)
2
+
( op )
~
2
ay
( ap) 2 + -:e-- ] Z
[ ~ax ~ox + ~ay ~ay + ~oz ~az ] = ~k ,
~= ox
~~ cp
(and
similarly
,
gas viscosity and constant rock properties, and neglecting gravity, Eq, 2,4 becomes
.(2.10)
for
y
and
(
( ~
( ~)
3ax p ~ax) +~oy p oy) + 3cz p oz =~~. k at
z),
.,
(2,14)
ax
This equation can be rewritten as 01/> -~
02p2 c2p2 a::t2 + ~+
op
at-a-p-~' 02p F2-+n+~ x y
) (
a2p + z
[ CP] 2 ) -I/>p. + -a-z -T(C
1 ak c+Ta p
op + Cf) at
) ([ ap] a
"
x
2
02p2 21/>p. ap --aZ2 = k~"
(2,15)
r ap ] 2 + ay
In the case of radial flow Eq. 2,15 becomes 02p2 1 cp2 21/>p. op I/>p.op2 -+ --= --=~.(2.16) ar2 r or k ot kp ot Either of the two right-hand forms is often used. This
(2,11)
equation is nonlinear and has been solved mainly by numerical methods.
If we rearrange Eq. 2.10 it now becomes
( 02P
for isothermal variations in pressure, op -M ap ot RT ot . From kinetic theory, the viscosity of an ideal gas depends only upon temperature. Thus, for constant
and
x
M p = liT p ,
~ + !:-.-~. at k at
This equation can be simplified somewhat by noting that
~\,
I
.1 al/> I/> P permeability are pressure-dependent.In cases in which the gradient squared terms can be neglected, Eq. 2.11 Cf
can be reduced to
engineering-the equation for radial flow of a fluid of small and constant compressibility. It is quite important to not~ that small p~essure gradients, constant rock properties,
h
were
(29)
k at'..
often
7
8
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
In the case of flow of a non-ideal gas, the gas deviation factor z is introduced into the equation of state to give p =:r
~. z
of gas liberated from a volume of oil to the oil volum~ (all referred to standard conditions) is the gas solubility factor Rs. Similarly, a gas solubility factor for water can be defined and representedby Rsw. The use of a formation volume factor to allow for the changesin volume which occur in each phase upon
If we assume laminar flow, neglect gravity and assume constant rock properties, then Eq. 2.4 becomes, for isothermal conditions,
transition from reservoir to standard surface conditions of temperature and pressureis a well known procedure. These volume factors are defined for each phase as
(~ ) ~ox(~ILZ~ox )+ ~oy(!!.-ILZ~oY) + ow ~ (~ILZ~oW) = ..!.k ~ot Z
B = oil and dissolvedgasvolume (reservoirconditions), 0 oil volume (standardconditions)
(2.17) In Eq. 2.17 we have used the symbol W for the Z co-
gasvolume (reservoirconditions) Bg = gasvolume (standardconditions)'
ordinate to avoid confusion with the gas deviation factor z.
Bto --. watervolume (standardconditions)
For radial flow Eq. 2.17 can be expressedas 1
-T r
C r
(p
aP
-r /LZ
-:ar
)
cf> '0 kat
=
(
P z
) ...(2.18)
A version of Eq. 2.18 in which higher-order terms are neglected can also be derived. This equation is ~
+ ~
or2
~
r
or
= ~ k
~ P
~ at
(~
,
-water anddissolvedgasvolume(reservoirconditions) In addition to these quantities, the concept of relative pe~meability m.ust be introduced. When th~ee immiscible 'flUIds (e.g., 011, gas and water) flow sImultaneously through a porous medium, the permeability of the rock
to each flowing phase depends on the interfacial tensions betweenthe fluids and the contact angles between the rock and the fluids. It has been found
)
..(2.19)
Z
Russe11 et a1.ave 4 h
commonly the rock
encountered conditions the to each phase is independent
that for
permeability of bulk
of fluid
19 shown that use 0f Eq. 2.as a ..nction substitute for the more rIgorous Eq. 2.18 can lead to serIous errors...permea m gas-well performance predictions for low-permeability gas reservoIrs.
properties and of flow rate (for laminar flow), and is a fu f h fl d h 0 t e UI saturations only. T e relative perb l ti t h h d fin d mea 11 es 0 eac p ase are e e as the ratIo 0f t he b1lit y t 0 a phase at preVaI .1mg saturation conditions to the single-phase permeability of the rock.
The equations for flow of a single fluid which are essentialto this Monograph have now been developed. In reality, of course, the pore space of a reservoir is occupied by more than one fluid, and any or all of these fluids may occur at saturation levels such that simultaneous flow will take place. It is essentialto an
Thus, for oil, gas and water,
. ... .
.
..
. .
krtO= kiD(So,Sto), k ko (So,SiD) kro = k '
understanding of pressure analysis methods that some basic facts about multiphase flow be developed. The brief section which follows is devoted to this.
.
.
k rg
= kg (So,SiD) k '
M I h FI 2.5 u tip ase ow A completely rigorous formulation of the equations
where
for multiphase flow should consider the spatial distribution of each component in the hydrocarbon-water systemas a function of time.5.18The approach which we take in this section will be much less rigorous. All hydrocarbon liquid which is present at atmospheric conditions, as obtained by differential vaporization, we refer to as oil. The gas phase we refer to simply as gas, without regard to its composition, and we consider the solubility of gas in the oil and water phases. Our derivation will be for radial flow only. At any instant an element of the reservoir will contain certain volumes of oil, gas and water which, when reduced to standard conditions, will be modified as a result of gas solubility in the oil and water and the compressibility of each phase. The ratio of the volume
It is beyond the scope of this Monograph to present a definitive discussionof two- or three-phaserelative permeability. For the purposes of our derivations, we shall consider simply that these are physically meaningful quantities which can be measured on a rock sample in the laboratory. Consider a unit volume of the reservoir. In this volume there is a mass of oil given by
S + S + S = 1 . 0 to g
cf> So ~Pos , 0 and a mass of water given by cf> SiD B;;; Ptos ,
MATHEMATICAL
BASIS
FOR
PR,ESSURE ANALYSIS
METHODS
wherepo.and pw.are oil and water densitiesat standard conditions. In the same reservoir unit there is a massof free gas S ~ PUB Bu and a massof dissolvedgas .l. R
'I'
P .U'
S
B
0
+
-loR 'I'.w,.
P
S
~
PU'
B
porous mediumunder conditionsof neglectof gravity forcesandcapillarypressuredifferencesbetweenphases. They representa simultaneousset of four nonlinear equationsdescribingfour unknowns,So,Su,Swand p. This complexsystemcan be solvedonly by numerical means. Martin6 has shown that in the casewhere higherorder
tD
B' 0 tD so that the total massof gas per unit volumeof reservoir is + f/>R. PU'SO + cJ>R-p,. StD B
9
..
terms
-ko po uro ---
.-!.-~ r or
neglected
(r
~
Bo op
+~~ ~
+
~
~
Bo op
-~~
BtD
fJpw ' pw.ar
expansion
of
) = !!:!!-+ ~~ = ~ or2 r or (-k )
~ or
ce =, -~
B0 po.a r '
ktD
the
the
p.
~ ot' e
where Ct is the total systemcompressibilitygiven by
and for water PtDUroo = -~
in
"",,(2.24)
opo
p.o
op
-~
~
BtD op
+ Cf'
(2.25)**
Bu op
and the quantity (klp.)t is the sum of the mobilities
For gas,
(kip.) of the fluids;i.e., k
--u Pu uru ---;;ij;
R k
op, .0 PU' ar -PUB B:- ~
t
f 0
ti' equa
f ons
II 0
1 0
[r
ko OP] -0 -at
( ~k
k
k
+ -!- + ~ ) ,11.0 p.u p.w
e
.(2.26)
~th a pressure-dep~ndent diffus~vitycoefficient..This Impo~ant fact proVl~esthe. baSISfor pressu:e.mt~rpretati~npro~e~uresm multiphasecases.This IS discussed
ows.
m
detail
m
later
chapters
of
the
Monograph.
For the sake of completeness, the simplified forms of the precedingequationsin the case of two-phase,
Oil: Tar
p.
Comparisonof Eqs. 2,24 and 2.9 showsthat under the assumed multiphase flow in aequation porous medium can conditions, be described by the diffusivity
If we neglect capillary pressuredifferences * in the systemand neglectgravity, then a continuity equation for eachphasecanbe written as in Eq. 2.3. se
( -=k )
apo --a;;-
-PUB~Bw ~ jJ.w~or '
e
be
B'
u 0 tD By use of Darcy's law we can expressthe radial massflux of oil as
Th
can
quantitiesin Eqs. 2.20, 2.21 and 2.22, theseequations canbe combinedmathematicallyto yield
(
~ar
So ) cJ>~.
(2.20)
Gas:
gas-oil flow are ~resented. The differential become the folloWIng.
equations
Oil:
(~+~+~ )~ ] -.!-.!.r or [ r p.oBo IJ.wBw jJ.,B, or [ ( R.So R.wStD Su)] =-cJ>-++ ot Bo Bw Bu 0
~~ r
(2 21)
or
-.!-.~ r or r ~+~ p.oBo
]
~ ~ [r~ ~ =~ r or IJ.wBtD or ot
( cJ>~BtD)
,
(2.22)
So+S,+Sw=1
, , , .,
, ..(2.23)
Eqs. 2.20 through 2.23 constitutethe equationsfor simultaneousflow of oil, gas and water through a *Capillary forces are not completelyneglectedbecause effectivepermeabilityterms are affectedby capillarity.
--
)~] =~ot [ (~+~ Bo
,II.,Bu or
cJ>
, , , , ., and
where
.(2.27)
Gas..
[ (
Water:
[ r ~p.oBo ~or ]=~ot ( cJ>~Bo ) ,
,
Bu
)]
(2.28)
So+ s, = 1 ,
This set of equationshas beenstudied extensivelyby Perrine,7Wellersand West et aV4 by means of numerical solutionsobtainedon digital computers.L **The term c, was added to Martin's equationsto ac-.the count for formation compressibility.
10
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
2.6
Solutions for Radial Flow of Fluid of Small and Constant Compressibility Thus far in the presentation of the math:matical basis for pressure analysis methods, we have discussed the physical laws which govern fluid flow in a porous medium and the combination of these laws into differential equations which describe the various flow ti f Eq W h th regimes which may occur. It e excep on o. 2.8 and its form for radial flow, Eq. 2.9, all the equa-
.
.
..
Fluid of small and constant compressibility; Constant fluid viscosity; Small pressuregradients; and ... Negligible graVIty forces. Again, the equation is 02 1 0P .I. P 'f'JJ.COp ¥ + -3 = ~at . r r
tions which were developed are nonlinear and not easily solved. Eqs. 2.8 and 2.9, however, are linear and can be solved analyticany for boundary conditions of interest, as we shan see presently. Not only can they be solved, but application of these solutions to reservoir conditions has, over the years, demonstrated their practical value. Because of this utility and simp~icity, these equations have become the fundamental basIs for the commonly used pressure analysis techniques. For the development of the pressure analysis theories discussed in this Monograph, three basic solutions of Eq. 2.9 are needed. These are presented in the section which fonows. Others may be found in Carslaw and Jaeger9or in the paper by Rowan and Clegg.15 The assumptions made in the development of Eq. 2.9 are summarized as fonows: Radial flow into wen opened over entire thickness of formation; Homogeneous and isotropic porous medium; Uniform thickness of the medium; Porosity and permeability constant (independent of pressure);
Ghe solutions of this equation of interest to us in the developmentof pressure analysis methods are those for the case of flow into a centrally located wen at a constant volumetric rate of production, q) As win be mentioned later in this chapter, the basic solutions for constant rate can be combined by the principle of superposition to yield solutions for arbitrary rate histories. (Three basic casesare of interest: (1) Infinite Reservoir -the case in which the wen is assumed to be situated in a porous medium of infinite radial extent; (2) Bounded Cylindrical Reservoir-the case in which the wen is assumed to be located in the center of a cylindrical reservoir with no flow across the exterior boundary; and (3) Constant Pressure Outer Boundary -the case in which the wen is situated in the center of a cylindrical area with constant pressure along the outer boundary. The specific application of each of these caseswin become apparent in the later sections of this Monograph.) The geometry and boundary conditions for these three casesare indicated schematically on Fig. 2.2. To INFINITE
RESERVOIR
P- PiASrCONSTANT
PRESSURE
BOUNDARY P = p.
CASE
co
OUTER
CASE
AT r = r
Ie""
/
/'
0
'" BOUNDED
CIRCULAR
RESERVOIR
CASE
~or Ire
=0
"" -.J
I
I
re I-r
I
~
W
; =:j : I:=-
I
I I
Fig. 2.2 Schematicdrawingof geometryand boundaryconditionsfor radial flow, constant-ratecases.
MATHEMATICAL
BASIS
FOR PRESSURE
ANALYSIS
METHODS
11
expressthe condition for constant flow rate at the wellbore (i.e., at r = rw), we may write from Darcy's law ( r ~ .Thus, q = ~ p. or r~ Thus, if we require a constant rate at the well, then we
The symbol Y is Euler's constant and is equal to 1.78. 4kt for ~ > 100,
impose the following condition on the pressure gradient at the well: ( ~ ) = --.!!!!:-~.. (2.29)
or
)
(
qp. p( r ,)t = P. + -In 41Tkh
[ 1n~
p(r, t) = P. -41Tkhqp.
)
yf/Jp.cr2
4kt'
kt
+ 0.80907 ] .
or r~ 21Tkhrw (For no flow across an exterior boundary, r = re, we must have zero flow velocity; therefore, the pressure gradient must be zero.) OP -Pwf a -0 (2.30) r r. (In all caseswe require that at t = 0 (i.e., initially) ~h~.reservoi.r.isuniformly pressured.at a value Pi)~he ffiitial condition could also be specified as a function of radius from the well; however, for our purposes the
...(2.32) The expression for pressure at the wellbore (i.e., at r = rw) is
assumptionof initial uniform pressureis adequate. The mathematical statement of the boundary conditions and development of the mathematical solutions for each of these cases is presented in Appendix A. These solutions are, of course, quite well known and have been incorporated into this Appendix solely for the sake of completeness. As is usually the case, the exact form of the mathematical expressions for the solutions of the foregoing
The solution we have presented for the infinite reservoir case is an approximation to the actual finitewellbore infinite reservoir case, and is based on the assumption of a vanishingly small wellbore radius. However, when it is evaluated at practical values of radius and time (including normal wellbore radius values), it yields almost identical results with the lesstractable finite-wellbore solution. More information on this approximation can be found in Appendix A.
problems depends on the approach taken in the analytical treatment. In this regard, several slightly different
Bounded Circular Reservoir
( )
solutions of th~mproblems in w~ch we are interested have ap'peared the petroleum literature.. Rather than
qp. In = P. + 4:;;kh
(
yf/Jp.crw2 4kt'
)
or k
Pwf = P. -~
[ln~f/Jp.crw 41Tkh
+ 0.80907
].
(2.33)
p(r, t) = P. -2;khqp.
{ reD22-1 (4rD2 +
)
tDw -
attempting to present all of these solutions and an accompanying critique, we have chosen to utilize in each of the three casesthat solution most convenient to
reD2In rD (3reD'-4reD' In reD-2reD2-1) 00 ~-=T4(reD2 -1)2 + 1Tn=l
the needs of this Monograph. The reader who is interested in a variety of these solutions is referred to Muskat,lO van Everdingen and Hurst,S Homer,l1 or CarslawandJaeger.9 The mathematical solutions for each case are listed
e-a."tD~112(anreD)[11(an)Yo(anrD)-Y1(a,,)lo(anrD)] a,,[112(anreD)-112(an)] } (2.34) where
in the section of the text which follows. Infinite Reservoir, Line Source Well
I
qp. 21Tkh
p(r,t)=pi
1
.'t' 2 E, ( -
.I.
r rD = -,rw 2
p.C r 4 kt )
re reD= -tDw rw '
=
kt f/JpocrfD2 '
and the an values are the roots of
~
,
J1 (anreD)Y 1 (an) -J1
(an) Y 1 (anreD) = O.
(2.31) where
(2.35) For the pressure at the wellbore, Pwf, for the casewhere re > > rw, Eq. 2.34 can be written
-E,
. (-x)
For x < 0.01,
=
00
f e-U udu.
Pwf = P. -2;kh"qp.
z
( 1) -E.
(-
x) ~ -y -In ( x) = In
1nreD-4 3
e-a."tD. J 2 ( ar ~ 2 2 1 2"'eD) } .(2.36) n=l an [J1 (anreD)-J1 (an)] The an values in Eq. 2.36 take on monotonically
+ 2
--05772 x'. ~
~
00
{~+2tow
12
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
increasing values as n increases; i.e., a1 < a2 < as "". Thus, for a given value of tDw,the exponentials decrease monotonically (e-at'tD~ > e-a22tD~ > ). Also, the Bessel Function portion of the terms of the seriesbecomesless as n increases.Thus, as tDwbecomes large, the terms for large n become progressivelysmaller. For tDwsufficiently large, all the terms of the summation become negligibly small. Prior to this tDwvalue, however, there is a period of time in which all terms but the first in the summation can be neglected. This period will be referred to later in the Monograph as the "late transient" period. Thus, for sufficiently large tvw, the summation term in the solution approaches zero and Eq. 2.36 becomes 2t [~+ 11' reD
PtD/= Pi -~
In reD-4]
3
of the limited drainage area manifests itself. Until the time of boundary interference in the solutions the pressure behaviors for both casesare identical. This may be seen from the behavior of the reD = 10 curve which breaks away from infinite reservoir behavior at tvw= 16. The curves for reD= 6 and reD= 8 had broken away earlier. A comparison of the dimensionlesspressure drop for an infinite reservoir with that for the constant-pressure outer-boundary case is shown on Fig. 2.4. Here, again, the solutions are identical until the effect of the boundary is felt. Since in this casea constant pressureis being maintained at the outer boundary, the flow system reaches an equilibrium condition (steady state) and the pressure at the well becomes constant. This is in contrast to the bounded reservoir solution. In that case no fluid enters the flow system and the pressure in the
.(2.37)
A form of Eq. 2.36 which is convenient for use in pressure buildup analysis and determination of average reservoir pressure is obtained by adding and subtracting a term In( ycf>p,crw2 /4kt) to obtain 2 Pw/ = Pi +~[ln~Y (t)], .(2.36a) 11' t
well and throughout the reservoir declines with time as a result of the depletion of mass from the system. Note in Fig. 2.4 that the larger-size reservoirs follow infinite reservoir behavior for a longer time than the smaller ones. Further, the steady-statevalue of Pi -Pw/ is proportional to log reD'The specific pressure behavior during the various flow regimes will be discussedin great-
where
er detail in the section which follows.
ycf>lJ.Crw272+ 4tvw 2 (In reD -4)3 Y(t) = In 4~+ eD 00 e-a.8tD~112(a r D) + 4 ~ 2[1 2 ( ) 1 "2e ( )] n=l
a"
1
a"reD
1
2.7 Concepts of Transient, Semi-Steady State and Steady-State Flow Behavior .If
we
consider
a
hypothetical
example
in
which
the
an
2.30 Constant Pressure Outer Boundary
In this case we present only the solution for the pressure behavior at the well. This expressionis Pw/ = Pi --
2 kh qp.
{ In reD-2
11'
2.20
00 ~ n=l
e-P.'tD~102(,8"reD) ,8,,2[112(,8,,) -102 (,8"reD)]
} ,.
where
re reD = ~'
(2.38)
2.1
~
a. I ::Ll.c ~
kt tvw = cf>p.CrtD2 ,
ci- C" C\J II
and ,8" is a root of 11(,8,,) YO(,8"reD)-Y
~ <;]
1(,8,,) 10 (,8"reD) = o.
The solutions which have been presented and the forms resulting from them will be used later in the text to develop the various pressure analysis methods. On Fig. 2.3 the solutions for the dimensionlesspressure drop, ~P D, at the well for the infinite reservoir and bounded cylindrical reservoir cases are shown. Values for ~PD are obtained by evaluating the terms within the braces in Eqs. 2.31 and 2.36. The results, when plotted as in Fig. 2.3, show the deviation from the "infinite reservoir behavior" which results as the effect -~
2.0
1.90
!
1.80 20
24 tow Fig. 2.3 Solutionsof the infinite and boundedcircular reservoir cases,constantrate. (After van Everdingenand Hurst.") ~
MATHEMATICAL BASIS FOR PRESSURE ANALYSIS METHODS
4
0
IxI02
3
5
8
13
IxI03
.6.8
reo=200
3.8
-
3.6
.4
3.4
.2
a.3 ~ I~ I
~
c-
a..-t\J II
6.6
3.2
a
a. <]
r eo=400
6.0
3.0
5.8 reo=300
IxI03
2.8
3
5
8 IxI04
3
5
8 IxI05
3
5
8
56.
tow Fig.
2.4
Solutions
of
the
infinite
and
constant-pressure
(After
assump~on~ the denvation
regarding of the
fulfilled, constant
then flow
Fig. 2.5. interest,
(puring the the pressure
closely, pressure
first by behavior
p~s~e
the rate
pressure will be
behavior
pletion
fr es
comes voir
larger
regime state occur ~ steadv and that
om fluid
steady
as
The. the
-
-q
of
=
Eq.
2.40
[
~
into
Eq.
1n reD
2.37
3
-4
]
..(2.41) .
C Thu~,
the
wellbore
Since
boundary, behavior
difference
the
V
betwee? pressures
productivity
1
=
the
average
the
tional
to
rate the
of
because of (As time
time. of
names
for
the
debe-
this
as
semi-steady
this
l Eq.
reser-
When
has also been call~~dos ta t e, anu ~ even stea d y
-
it canduring be well
2.4~
(The
of
the
well
for
so-called
(Another
interesting
the average ?ressu,re is
reservoir con~tant
1S easlly within
established. the
drainage
is
defined
state
state -TRANSIENT
flow.
p:I
In
semi-steady
1S constant.
con9jr~,#.S
in
the
steady-state
flow
ITRANSITIONAL (LATE TRANSIENT) PERIOO I
natural
occurrence
flow
EQ.2.31 I-
state
decline
is
volume)
inversely This
a-w~ll.)
"reservolimiiiest?'
at
reservoir of
constant
factls
that
the
pressure and the during semi-steady The volume
volumetric of
the
fact
difference flowing state average
well
is
is
steady-
Row
rate,
SEMI-STEADY STATE FLOW
14
EQ.2.37-
P
..
LINEAR PRESSURE DECLINE d
propor-
~~t~~e
the
) d,~
prevailing
usually")vpreclude
FLOW
shown semi-
during
wf
pore
as
(2.42)
~at
index flow
systems
(2.39)
pressure
and
q
--,.
!m~lies
productiVity
flow
condition
2. 'lTre
fluid-filled
~~~~~~~~f b~is
reservolf
1S constant.) index
Pwf
Thus,
gives
EQ. 2.36
h ."C
rate.
..(2.40)
is
--.l.
constant
Pw!
some.LFromdecline Eq. at 2.37 ofby pressure the
t opw!
2.33. which
cases,
-'lTre2cf>ch..
-qp. P -Pw!
floWing
spoken
other
.
-;;o-Q
Eq. in
throughout of
commonly
literature. It st d q ua s1 -ea9Y
flow
--qt P -Pi
Substitution
t;ansient{~
decline
function
Many
state
a on
reservoir
Hurst.")
.P
pressure
it is
behavior)
trans1ent the rate
,/ V
time of practical described quite
te reservo1r case from the reservoir.
a linear
in the -:=.=' state
be
for
boundary
and
made in suitably
the well schematically
then by period
described
e 1 mass
occurs,
and this
. nfini
the
becomes
of
flow across the drainage time elapses the pressure
th
of
behavior depicted
fluid are
outer
Everdingen
is--ess~~~-same-astliatill-an
If there is no more producing
deVla.
2.31 during
is
t
as
early producing behavior can
Eq.
~~-~~
as
the formation and foregoing solutions
van
/dl
-q
--~
the
-between
wePbore flow) This
t
pressure.
. Fig.
2.5 bounded
-
Sche~tIc
plot cIrcular
of reservOIr,
prc;:ssure
decline constant-rate
at
the case.
well,
14
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
pressure
is independent
of tim~
Eq.
2.38
reduces
to
by production
at a rate
this case for large time and can be shown by numerical
t1. In other words,
evaluation
ward
to simplify
to q
Pw! =
In
solution
P, -~
.
some
reservoIr
1n reD sys tems
the
(2.43)
.
c hange
pressure
Thus,
WIt h
times warranted. ~trictly state flow can only occur at any
point
speaking, however, steadyif there is no mass depletion in the
flow
past t1 and add to it (or
for
flow
at rate
in mathematical
(q2 -q1)-
terms
q1Jl. = 2;;kii
~p(t)
system.)
~p(t)
and Approximation Histories
eral, however, a well stant rate throughout
~PD(t)
kh ~PD(t)
= ~
+ (q2 -qJp.
~PD(t-tJ
(2.45) In the second
equation
above,
the first
~PD = In
this
rate. For these situations we make use of a very powerful mathematical technique commonly referred to as the 'principle possIble the for arbitrary
of su~erposition. This prin.ciple makes generation of pressure behaVIor solutions producing-rate schedules from the basic
constant-rate
pressure
behavior
the principle
an understanding
of
its
a well
the first time
of use,
which
solutions. shall
flows
and gain
consider
at two
rates
the
' =
-~
~p(t)
t < t would 1-
hist?ry
now
. El
we have
(
case IS shown
as
where
interval
the pressure
by q1Jl. = 2;;kii~PD(t),
drop be-
= Pi -Pw!
~PD(t)
is
simply
the
tim .
e
ft
th e
pro
a more
)
complex
. rate
the pressu~e be-
on FIg. 2.7. We obtain
the
q2 q
r,"
I (q2-q,) q,
!~..~,;
\ .:
.(2.44)
dimensionless
d uction...PRODUCTION rate IS
by
PRODUCT'DN RATE
: I "
pressure
drop at the well for the applicable outer boundary condition. in the braces This isin simply Eqs. 2.31,* the sum 2.36 of orthe2.38. terms contained At
given
q,p.crw2 -4k(t-t) 1
for. which. we wish to d~rive
haVIor. This
be
( -.!~ 4kt )
(q2-q1)p. kh 1T
that
Ei 2
for
-~Ei 41Tkh -4
Suppose
superposition we
case
Ap(t)
at the well is given Ap(t)
is the
( -~~4kt. )
Pi-Pw!= qp./21Tkh
shown on Fig. 2.6. In this case the well has produced at rate q1 until time ft and was then changed to rate q2' The resulting pressure behavior is shown. havior
term
or smaller than q1' To illustrate the use of relationbi1ips such as those of Eqs. 2.45, consider ~PD as defined by Eq. 2.31. '
will not have produced at conits life. Further, some of the
pressure analysis techniques to be discussed involve the use of pressure data obtained at more than one flow
During
ft onward.
creasing the producing rate by an increment (q2-q1). These expressions are valid regardless of whether q2 is
..larger .T~e solutions to the flow problems presented earh~r In thIS chapter were for the case of a constant volumetric rate of flow at flowing bottom-hole conditions. In gen-
case of
from
pressure drop from flow at the first rate. The second term is the incremental pressure drop caused by in-
Principle of Superposition of Variable-Rate Pressure
simple
the
21Tkh
to these concepts.
develop
for-
superpose)
t < t . 1 -.
In the chapters of the Monograph which follow, further ~scussion of the various flow regimes and the flow times at which they occur will be presented. The material in this chapter was meant to be an introduction
To
at time
solution
we have
.21T
2.8
beginning
the pre-t1
O
time is so slight that it is practically undetectable. In such cases the ass~ption of steady-state flow is some-
occurring
in time
(q2 -q1)
we continue
Increased
by
pressure behavior during the period from time ft onward can be calculated by adding to the pressure drop caused by rate q1 an additional pressure dro p caused
*In Eq. 2.31 the function in the braces must be evaluated at r=r.,. Eqs. 2.36 and 2.38 are written for r=r.,.
I
PRESSURE DROP CAUSED 8Y AT RATE q "
an
amount (q2 -q1). This increase in P roduction causes an additional pressure drop as shown on Fi g.2.6. The
i! ~ : :'
Pwf
,
1 ADDITIONAL PRESSURE. DROP CAUSED 8YINCREASING PRODuCTION RATE ANAMOUNT (q2-q,l I I 'I
Fig. 2.6 Production and pressu~ history of a well which has produced at two rates.
t
MATHEMATICAL BASIS FOR PRESSURE ANALYSIS METHODS
pressure behavior again by simply superposing the basic solutions. The pressure history for an arbitrary
15
q1Jl. /::::.p(t) = ~/::::.PD(t)
rate history is the sum of the pressure histories for incremental rates of production, each of which becomes operative at the time each new rate begins. This is nothing more than reapplication at each rate change of the basic principle which we illustrat d .same: the two-rate example. For the case shown on Fi; 2~; we start as before; i.e., the pressure drop during the initial time period is
+
(q2 -ql)p. 21Tkh
/::::.PD(t-t2)
/::::.PD(t-t1) .
21Tkh. ... For each r~te change the bas~c pnnclple 1~ alv:ays the continue the old solution forward In time and add to it (or superpose on it) t~e adQitional pressure drop caused by the latest change In rate. We see, then, that for a sequence of n rates, the pressure drop during the nth period is given by
q1Jl. /::::.PD(t) ./::::.p(t) /::::.p(t) = 2";k1i"
t :5: t1 :
(qS-q2)p.
+
= -&
/::::.PD(t)+ J~~~!!:.--
/::::.PD(t-t1)
As before, for the second period t1 :5: t :5:t2 :
+
-Q1Jl. (q2-qJp. /::::.p(t) -~ /::::.PD(t) + 2 kh /::::.PD(t-t1) .(q" 1T 1T At time t2 the rate changes from Q2to qs, so we must
+
add to the solution
for the second period
an ad-
ditional pressure drop ~aused. ~y the incremental ra~e change (qs -q2). This additional pressure drop IS
(qs -Q2)p. 21Tkh /::::.PD(t-t2)
or -q1Jl. /::::.p(t) -2;kh
-q"-l)p. h 21Tk
l
+
, ~PD( t - t"-l )
/::::.PD(t)
+
;
i=2
...
,
qi -qi-l ql
given by ~~;1~o2l!:.- /::::.PD(t -t2)
../::::.PD(t-
ti-l) ] .(2.46)
Thus, during the third period
*
Eq. 2.46 is the general form of the principle of superposition for the case of generation of pressure behavior for stepwise rate histories. Although we have
t2 :5: t :5: ts :
illustrated application of this method with a monotonically increasing rate sequence, the method and-
PRODUCTIONRATE q
equations are applicable to arbitrary stepwise rate variations. It is important that the reader understand the
4
principle of superposition and its application to stepwise rate sequences. The majority of the pressure analysis techniques which are presented later in the text
q qI
employ superposition methods.. : I
: I I
tit
2
I I I t3
Eq. 2.46 is completely valid also if one or more of the producing rates is zero (well closed in). For example, if the rate during the nth period is zero, then the pressure behavior (pressure buildup) during this period is given by
t
PRESSURE
Pi -PWB = ~ qlp. 1T
[/::::.PD(t"-l +
/::::.t) + n-1 ~ i=2
/::::.PD(t"-l -ti-l
+ /::::.t)] -~
qi -qi-l ql
/::::.PD(/::::.t), (2.46a)
P.I
I
I
where tn-l is the total elapsed producing time prior to
I I : I
I I I I
tI
:2
shut-in and /::::.tis the closed-in time measured from the instant of shut-in. This equation expresses the pressure behavior of a closed-in well which has produced prior to shut-in with a variable-rate history. t 13
Fig. 2.7 Production and pressure history of a well with .." stepwise IncreasIng rate history.
t
The
principle
.
of
superposition
. .
can also be ex-
.
*8 8 t ee ec Ion. 61 f or an appI Icat Ion 0f the pr1nclp Ie 0f superposition similar to that given by Eq. 2.46. --~
16
PRESSURE
pressed in continuous formdealt as opposed to far. the Suppose discrete or stepwise form we have with thus in Eq. 2.46 that the rate and time steps are taken to be infinitesimally small. In this case the summation can be written as an integral of the form
-2q1JL kh [ 6Pn(t)
Jt dq(or)
+ --1
1T
The quantity qp./41Tkhin Darcy units is 70.6 qp.B/kh in practical units. Using these conversions as a basis, Eq. 2,33 of the text becomes, for example,
d
ql
,
0 , ,
,
6Pn(t -or )dor .
or
continuous
form
of
,
the
.
th
ti
ema
al prope
Jaeger
W
f
kn
t
9 or tl requen y
I
ung
rti .
c
,
,
,
(2.47)
162,6 qp.B pi-p1O=10-3,23.
superposItion
t 1 negra.
Ii
12 I
own
as
y e.
f es
are
th
n D
I
re
e con
h u
al
n gener,
1, ame
th
d
t
C 0
!
'
'
nncip
1 ars
f
uous P
s
orm
th or
li
d
1
' . e superposItion
.
,
f
li
Fal e
tec
diff
,.
'
hni
we have .,
made
quantities -.,.,
que
al
..to
For most well-test analysIs applications Eq. 2.46 IS
quite adequate to handle the effects caused by variable flow rates. Barenblatt18 and Chaumet, Pouille and SeguierI9 have introduced a method of allowing for bl t hi h 1 ki 'th h L I vana e ra es w c mvo ves wor ng WI t e ap ace transforms of the pressure and rate histories, and eliminates the stepwise superposition procedure This method IS difficult to apply computationally and offers
.
.
presenting
g
h t
e
p.cr
.
the
and
systems
umts
no attempt .,. theIr
systems.
]
102
' umts
to start
ave
emp
WIth the basIc
dimensIons,
Readers
h
we ."
an
. t ' al 1 IS so
1 e
n
kt
kh
I
d aw
can e app e to so utions 0 near erenti equa-. tions obtained for constant boundary conditions to generate solutions for time-varying boundary conditions b
[
pnn-
erre
tin ,
],
.
ciple and those who are interested in a discussionof its ma
qp.B
or, equIvalently, ,.,
the
[log 0.000264 kt JA.Cr1O2 + 0.351
-162,6 Pi -Pro! -kh
]
'"
IS
IS referred to .m the
kt
Eq. 2.46 becomes
This
IN WELLS
tnw = 0.000264,1. 2' 'l'JA.crw
0 where oris the variable of integration corresponding to ti-l in the discrete form.
6Pt =
FLOW TESTS
text of this Monograph as dimensionless time. When rw is substituted for r, tD is called tD,o,and when re is substituted, tD is called IDe.In practical units,
d or 6Pn(t -or )dor'
ql
AND
Th'e time group, tn = ~' kt.
Jt dq(or)
--1
BUILDUP
and
Interested
then
to
1oye
d
,
physIcal develop
m a presentation
,. f This concludes the mathematical concep~ portion 0 the Monograph. It serves as the foundation for the methods to follow. Frequent reference will be made 0
f
thi
s
ty
pe
are
re
f
it, particularly
erre
d
t
0
R
e
f
S.
5
an
d
13
.
to the basic assumptions and limita-
tions of the mathematical
solutions
upon which the
pressure analysis methods are based. References ~ I, Hubbert, M, King: "Darcy's Law and the Field Equations of the Flow of Underground Fluids", Trans., AIME (1956) 207' 222-239.
no significant advantages over the classical superposition technique,
2. Hildebrand, F. B.: Advanced Calculus for Engineers, Prentice-Hall,Inc., New York (1949).
2,10
3. val}. Everdingen,A. F. and Hurst, W.: "The Application of the Laplace Transformation to Flow Problems in Reservoirs", Trans.,AIME (1949) 186, 305324.
Units-Field
Unit and Darcy Unit Systems
Before discussing the pressure analysis methods in detail, we must set forth the system of units to be employed in the text. Two sets of units are used. For the purpose of the mathematical derivations contained herein, we employ a systemof units commonly referred to as Darc um' ts Th th ti 1 d ti.Variable y , e ma ema ca enva ons pred th f th .c sente us ar m e report employ this system. For application to field data of the various mathematical expressions, we will employ a system of practical oilfield units. This dual set-up is quite handy, It prevents the occurrence of cumbersome numerical constants in th th 1 1 . b 11 .q ~ ma ematica. mam?u ~tions, ut. a ows practical umts to be used m application. Numencal examples are given with the practical equations to make clear the correct units, Table 2.1 shows the units in the two systems
.
.
.
,
.
.
TABLE2. I-DARCY ANDPRACTICAL UNITSFOR PARAMETERS ANDVARIABLES USEDIN CHAPTER 2 Parameteror DarcyUnits PracticalUnits* vol/vol/atm vol/vol/psi rI> fraction fraction h cm ft k darcies md JL cp c~ p a/tm pSI cc sec
B/D
(subsurface (stock-tank conditions) conditions) r cm ft t sec hr * "Practical Units" are also referredto as "Practical Oil-
field
Units"
and
also I
as "Oilfield
Units".
q
I
Chapter 3
I!'ii\i\\,' ._,
.(!iMJJM
,(1J
)!"';~c::( ff'1t'{ ::"JltMl'J ::'c"!i.i;J.fIJtA
,dJ(,.(>OE ,ill:'!, ({;?(Q{'t :~.. j, a ~{'r'..4'"'
~.".,..;
Pressure Buildup Analysis
,"!,"'.'\"~~,'"
i'.\~;'~;;;!~':j':'~~::~ ...'
.e... i~'r
3.1 Basic Method In the previous chapter we developed the basic equations for describing the pressure behavior in an oil reservoir. In this chapter we will show how these are applied in analyzing pressure buildup curves. We begin with the "line source" solution (Eq. 2.31) for one well in an infinite reservoir. This equation indicates that after a well has produced at rate q for time t, the bottom-hole flowing well pressure P"'f will be given by 2 ( -~ ) , (3.1) Pwf = P. + ~Ei 41Tkh 4kt
well pressure after shut-in and Prof to designate the pressure during the production period before closing in. Eq. 3.4, which was presented by Homer1 in 1951, will be our basic equation for pressure buildup analysis. As discussedin the previous chapter, it is a solution for an infinite, homogeneous, one-well reservoir containing a fluid of small and constant compressibility. As might be expected, the equation applies quite well without modification to newly completed wells in oil reservoirs above the bubble point. Modifications necessary for application of this equation to other cases will be discussedlater in this chapter.
which at times of interest reduces to
When we express Eq. 3.4 in practical oilfield units of psi, BID, cp, md and ft, it becomes**
-q}l. Pwf -P.
(
+ _4 kh 1T
so that the pressure
In
)
yCP}l.Crro2
4k
* ,..
t
(3.2) Pro. = P. -162.6-log
drop is
( Ycf>,ucrw2)
= (pressure drop caused by rate q for time t + ~t) + (pressure drop caused by rate change -q for time ~t);
or P. -Pw.
-4;;jiJi --q,u1
2
(
)
2
) ( ycf>p.crw 4k~t
+ qp.1 ycf>p.crw n 4k(t + ~t) 4;kJi n
(3.3) and
(
(I+
kh
P. ---q,u Pwf -4;k"h 1n 4kt' ..(t If ",:,enow ~lose m our w~ll for a time ~t, after ~roducmg for ti~e ~, we obtam the. ~ressu.redrop ~t time ~t by the pnncIple of superposItion discussed m Section 2.8, as P. -Pw.
q,uB
)
~t
) ..(3.4a)
~I
This equatio~ tells us tha~ if we plot the pressure Pw-; o~~,:a:Y!!~~os~perioa vs the iOgarithni ~f + ~t) I ~t, we should obtain a straight line. Pig. 3.1 shows a plot of data from a new well in an oil iservoir. As may be seen, the theory and practice ~gree very well in this case. T 0 ana1yze the curve m PIg. 3 .,1 note th at the
. .
absolute value of the slope of the curve m is equal to the coefficient of the logarithm term in Eq. 3.4a. Therefore* * * kh =
162.6qp.B
(3.5)
m
Extrapolation of the straight-line section to an infinite shut-in time, [(t + ~t)1 ~t] ~~y~s a pressure we will call p* throughout this Monograph. In this case
Pro. = P. -4kii qp. 1n. I + ~I .(3.4). 1T ~I In these equations we have used Pw. to designatethe
**See Section2.10 for derivation of the factor 162.6; this factor is also discussedin the Nomenclature.
*Throughout this Monograph, "In" will refer to the natur.al logarithm, while "log" will refer to the base 10 logarIthm.
slope should be used in this equation. This is true in all uses of the slope throughout the Monograph. Note that the slope is also given by m = 2.303 qp/4trkh, as may be seen f1lom Eq. 3.4.
***Only the magnitude (not the + or -sign)
of the
PRESSURE BUILDUP ANALYSIS
19
ii* = pi,the-itiiti-alpressure. Determination of kh and p* in this manner forms two of the basic steps in pressure buildup analysis. The quantity p* is the pressure which would be obtained at infinite shut-in time. In the case of one well in an infinite reservoir, p* is also the initial reservoir pressure. In finite reservoirs and even in infinite reservoirs containing more than one well, p* is less than the original pressure after some depletion occurs. The difference between Pi and p* is a reflection of this depletion. As will be discussedlater, p* is approximately equal to, but usually slightly greater than, the average pressure in the drainage area around the well. Note that for values of ~t small compared with t (the usual case during a.buil~up), a plot of Pro.vs log ~t should also be a straight line, as may be seen from Eq. 3.4a. The slope of the curve will be the same (though reversed in sign) whether Pro. is plotted vs log ~t or log [(t+~t)/ ~t]. However; the plot orplO, vs ~!E=~o~ be ex_tra~la!ed to p *in a ~im~ manner so that it is usually s~. over-~11,tn.p1at Pro. vs 102 r CL-!:~bt;)-I-M.To account for the fact that the production rate of a well may vary considerably over its life, one should, theoretically, use the principle of superposition discussed in Section 2.8 to approximate the true rate history (see also, Section 3.8). However, an acceptable approXimation, as discussedbelow, is to take the rate q as the last rate before closing in and to compute the flowing time from t =
.een cumulative well production smce completion production
wellbore damage. Additional pressure buildups will usually be made to obtain values for kh and wellbore damageafter a well is completed and "cleans up". Thus, the drillstem test values usually need only be approximate. Nisle8 has shown that if the production time subsequentto a short term shut-in is at least 10 times the duration of the shut-in, the error in kh arising from use of Eq. 3.6 will be less than 10 percent. For all these reasons, Eq. 3.6 will be used throughout this Monograph. For interpreting short produ~tion tests and for obtaining accurate kh values from drillstem tests, the Odeh and Seli~- method should be used. A method similar to this has also been suggestedby Trebin and Shcherbakov.41 3.2 Skin Effect, Skin Factor, and Flow Efficiency Skin Effect In many cases it has been found that the permeability of the formation near the wellbore is reduced as a result of drilling and completion practices. Invasion by drilling fluids, dispersion of clays, presence of a mud cake and of cement, presence of a high gas saturation around the wellbore, partial well penetration, limited perforation, and plugging of perforations are some of the factors responsible for this reduction in permeability. Since the effect is close to the well, transients caused by it are of small duration and may be neglected. Hence, the effect of a reduction in permeability near the well can be taken into account as an additional pressure drop ~p proportional to the rate of production q. The zone of reduced permeability has b
rate just before closing in
ca 11ed a " s ki n "4 .an 5
d th e resu lti. ng e ff e ct a " skin
effect".
6)
Skin F'attor
Another approximation for t and q has been discussed bY~_~d._S~_:!~a~oximation-I~a better ~e for o~~~ng kh from short production tests ana-drlllstem tests. Even for these cases, the approximation of Eq. 3.6 leads to correct extrapolated pressures and to reasonably accurate values for kh and
... .Quantitatively, after van Ev~rdingen,4we define the skin ~actor as. a constan~ s w~ch relates the pressure drop m the skin to the dimensionlessrate of flow.
(
)
qp. ~P.kln = S 2:;;:kji""
(3.7)
1320
Here-s is called the skin IaC"fOf.After introducing Eq. 3.7 into Eq. 3.2, we find for the well pressure after a
130
production time t
-~ ~1280
PIO!= Pi + ~ qp.
:)
[ In ( rc/>JiCrIO2 4kt ) -2s ]
.(3.8)
~ 1260 ~
This flowing pressure Prof is lower by an amount sqp./21Tkhthan the pressure in the absence of a skin
~1240 m :: 1220 ~ 120
(see Eq. 3.2). The skin effect is illustrated in Fig. 3.2 (from Hurst5). In the idealized case shown there, the pressure should rise by an amount ~P.kln immediately after shutin. In practice, the order of magnitude of the skin effect can be estimated from the difference between the pressure before shut-in and that shortly after.
118 100
10 It+ 6t) /6t
Fig. 3.1 Pressurebuildup in a nearly ideal reservoir.
I
To calculate the skin factor, it is necessaryto measure -
20
PRESSURE
the well pressure both before and after closing in. By co~bini~g ~q. 3.8 which .give~ the pressure before cloSIng m, WIth Eq. 3.4 which gives the pressure after
PIC' -PIc! --qp. -4;kh"
,"«
SK 0 ZON DA
]
(3.9)
m ;", ,~,
m
(-~
)+
JLCrw2
FROM
(3.10)
STATIC PRESSURE
""
skin
ACROSS
SKIN
HURST5
Fig. 3.2 Pressuredistributionin a reservoirwith a skin.
In this equation we have replaced the factor qp./41Tkh by its equivalent based on Eq. 3.4, m/2.303. The pressure Pw! is that measured before closing in; the pressure PI hr is obtained from the straight-line portion of the pressure buildup curve 1 hour after closing in. The italicized statement is most important. If the pressure buildup curve is not straight at 1 hour, it is necessary to extrapolate the curve backward as shown on I
IN WELLS
"~;; f1 1m!: II JJ !i~~ FLOWING PRESSURE
]
3.23
FLOW TESTS
»illji PRESSURE IN !~ FORMATION !: ::; 11A P ..PRESSURE DROP 1;
For ~t small compared with t, we can approximate (t + ~t)/t as 1. Rearranging this equation, choosing ~t = 1 hour so that P",. = PI hr, and introducing practical oilfield units, we get for s
s = 1.151[PI hr -PIc! -log
AND
WELL BORE ,;.,.(;, W *1
closing in, we find
[ In ( 'YJLCrw2 (t + ~t» ) 4kt(~t) -2s
BUILDUP
6tt
Fig. 3.3. This is necessary because Eq. 3.4 is only applicable to the straight-line portion of the curve. Usually, at early times, the curve deviates from a straight line becauseof flow into the wellbore after the well is closed in at the surface. The basic theory does not take this into account. To compensatefor this well fillup effect, it is necessaryto extrapolate the straightline portion of the curve backward to early times. hours
10
100
~ ~ .
4600
r"I
4400
4200
4000
3
3
34
100 I
(t+6t)/6t
Fig. 3.3 Pressurebuildup showing effect of wellbore damageand afterproduction. --
~' PRESSURE BUILDUP ANALYSIS fJ -t.i:? IP
21
It would have been possible to choose any time besides 1 hour in developing Eq. 3.10. T~ wo~erel'y ~ choos~
~t ch~~e_the == 10 !!o~s~v!!!~e the of constant the cQns~~ would
(~~-==19.H.-lQl
nr2-21.-
other so
~.23.~ become
The
V
factor
8 by6
( ~-
3 = Thus,
) In 2.
1
k.
(3.11)
the permeability
in the skin is greater
if if
than that in the
formation, as from fracturing or acidizing, 8 -will be negative, Hydraulically fractured wells often show values of 8 ranging from -3 to -5. te that even I f k d k t t No , 8 an rw are nown, I IS no possible to obtain both the radius of the skin and its P ermea
b i li t y
from
Eq
3
11
0
...ne
t
.
ff
ti
may
ge
lIb
d aroun
di
thi
1n!!r ' w
=
1n!-!r
riD' = If 8 is positive,
s
+ 8
Flow
from
Eq.
.,
3.7 which,
...(3.7a)
of about
2,0 may be obtained
may reach 5.0 after a fracture
Pressure
e
ffi
buildup
...
B,
treatment.
Calculation
calculations
clency. are convemently
for
kh,
summanze~
Example
.
radius riD' is small-
..
1).
8 and
flow
on a form
sheet
are
the
Calculations
h b ld pressure UI up curve sown m FIg. ... reservoIr IS above the bubble pomt.
3.3,
for
where
or p* total
= 4,585
psig.
compressibility
for Example
1-is-obtained
laboratory
measurements
(see also TrubeIO).
pressibilities are of the order of 10 X 10-6 psi-l. compressibility of non-gas saturated water varies
If 8 is negative,
the effective
fr~2
than r w. For example,
8 values
of
-3
and
-5
corre-
spond to effective well radii of 5 and 37 ft, respectively, for rw = 3 in. This effective wellbore radius concept is especially useful in discussing results of hydraulic fracturing. FI A better
relative
ow
Eff " Ic/ency
index
than
3 X 10-6 psi-l is usually
skin effect for deciding
.
= ~
quired to obtain
J
'P. Ii! p"..,etuc/ttti(t( -7'-C"-u4
-q actual -.
33
-Pwf
Rock
compressi-
th elr . .m enter
only into a logva I ues IS not re-
.
accurate values for skin and
B d dR . Gun e eservolrs
Thus far in this chapter, we have presented tibns for only one well in an infinite reservoir. approximations
for bounded
equaThese reser-
voirs if production time is not long but become poor with additional production.'In this section we will discuss modifications of previously presented theory to enable
p*
satisfactory.
reasonably
are good
.;;
and use of a value of
damage.
-equations ~;W-~ncy .Jldw Smce
The. only
bilities may be obtained from Hallll (Fig. G.5). They vary from 3 X 10-6 to 10 X 10-6 psi-!. The compressibility of gas-saturated water varies from 15 X 10-6 psi-l at 1,000 psi to 5 X 10-6 psi-l at 5,000 psi (see Ramey12). Since compressibilities an. thmi c t erm, high accuracy
upon the efficiency with which a well has been drilled andd completed fin d this providedf by a "flow efficiency". This IS e e as e ratio 0 actual productiVIty mdex of a well to its productivity index if there were no skin (8 = 0).
.
X 10-6 to 4 X 10-6 psi-l
form from
Oil com-
must theoretically travel through addito give the required pressure drop). is larger
the
To obtain p* in Fig. 3.3, we must extrapolate from P = 4,445 psig (at right ordinate) two cycles to the right at a slope of 70 psi/cycle. Thus, p* = 4,445 +
er than r,o (fluids tiona1 formation
radius
after
of moderately high formations, the flow
using the equation shown at the bottom of the sheet. It is best to obtain the oil compressibility
wellbore
A
its
usage
solution
in bounded
for
reservoirs,
the pressure
behavior
of a well
in
and J Ideal =
q p*
-Pwf
~
we obtain
~P.kln
'
*Strictly speaking, one should use p, the average pres~ure, rather than p* in this _equation. However, since p* IS a good approximation for p, and since this quantity oc-
curs in both. numerator
caused by USIng p*.~ff~
v'F;;:;
I-tp.£
..-
--
Example
The
wellbore
*"
(3.12)
qp./41Tkh, is
efficiencies
efficiency
2(70)
w riDe-B.
the effective
IS obtained
3(;;;)1. ':~
(Appendix
"
difficulty by defin mg ." an e ec ve we ore ra us , r ' W d fin thi di th t hi h k th w. e e e s ra us as a w c ma es e calculated pressure drop in an ideal reservoir equal to that in an actual reservoir with skin. Thus,
or
m = 2.303
-~P.kln -Pwf
hydraulic fracturing in formations permeability; in low-permeability
. .
.
~P.kin
Vlrnp~.87 I~DA'U
in the skin zone is less than
that in the rest of the formation, 8 will be positive; the permeabilities are equal, 8 will be zero. Finally,
-Pwf
-p*
-Ihe flow efficiency has also_been called the~ ~ctivity ratio, the condition ratio,7 and the com~~~ tacror.8 When subtracted from unity it gives the damage factor. 9 7J.,C'
rw
if the permeability
quantity
using
The radius r. of the "skin" zone around the well and the permeability k. in this zone are related to the skin
.-p* efficIency
Flow
and denominator,
little
error
is
22
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
a bounded cylindrical reservoir was obtained in Chapter 2. Eq. 2.36a of that chapter gives the following relation for flowing pressure Prof. qp. P"" = P. +"4;kji""
)
[(In
Yc/>JLCr",2 4kt -Y(t)
] .(3.13)
13"
~j 12 PROBABLE
The factor (q/L/41Tkh) Y(t) may be thought of as a pressure drop additional to that in an infinite reservoir, caused by the fact that no fluid can flow in across the
p=1190psiQ II
outer cylindrical boundary. To obtain the shut-in pressure in this bounded reservoir, we superposethe pressure drop given by Eq. 3.13 at time (t + L;:.t) on the change at time L;:.t,obtaining
[ (t+
P",. = P. --In q/L 41Tkh
L;:.t
)
L;:.t
+ Y(t + L;:.t) -Y(L;:.t)
] .(3.14)
p* = P. --.!!!!:- Y(t). 41Tkh
...(3.15)
For a cylindrical, bounded reservoir, Y(t) is a positive function which increases with time. Thus, p* will be less than Pi and the difference will increase with increasing production time. If we substitute Eq. 3.15 into Eq. 3.13, we obtain -*
q/L
+ "4;kji""ln
.1015.!.i!!'
Qoo61 .. Fig. 3.4Observedpressurebuildupcurve In well In finite reservoir. If on Fig. 3.4 we extrapolate the straight-line portion
From Eq. 2.36a we find that for small L;:.t,Y(L;:.t) ~ 0* and Y(t+ L;:.t)~ Y(t). Then when Pro. is plotted vs In [(t+ L;:.t)/ L;:.t] and extrapolated to [(t+ L;:.t)/ L;:.t] = 1, we find the extrapolated value, p*, from Eq. 3.14 as
p",,-p
10
( Yc/>JLCr",2 ) 4kt'
.(3.13a)
of the buildup curve to ~nfinite closed-in .tim~, [(I + L;:.t)/ L;:.t] = 1, we obtain a value p* which IS
greater than the average pressure p, as shown. Usually a well will not be closed ~ long enough to obtain the flattening and to observe p. However, it is possible to estimate p from the extrapolated value of p*. This is done by using the Ei-function and other functions developed in Chapter 2 (see Ref. 13) to developequations for (p* -p) vs time for drainage areas of various shapes. For a circular drainage area, a graph of (p* -P>/(q/L/41Tkh) in oilfield units is given in Fig. 3.5 as a function of kt/c/>/LcA,also in practical oilfield units (see Nomenclature). The quantity A is the drainage area of the well; for one well in a bounded reservoir, it is the reservoir area. Values of p* -p for drainage areas of other shapes will be found in Chapter 4. Discussion of the use of graphs such as Fig. 3.5 toobtainpfromp*willbedeferredtoChapter4.
On comparing this equation for a bounded reservoir with Eq. 3.2 for an infinite reservoir, we see that p* has the same meaning in an equation written for a bounded reservoir as does Pi in an equation written for an infinite reservoir. This has an important corollary. An equation written for pressure behavior in an infinite reservoir may be immediately rewritten for the finite reservoir case by substituting p* for Pi. We will use this corollary later in the Monograph. Returning now to Eq. 3.14, we see that it differs from the case of one well in an infinite reservoir by the two Y(t) terms. On evaluating Y(t) from Eq. 2.36a, we find that the e~ect of the two Y(t) terms is to cause the pressure buIldup curve to bend over at large ..-e time, as shown by Fig. 3.4. The flattened curve ~ll ~pproach, asymptoti~ally, the average pressure p m this bounded reservoIr.
If there are other wells in a reservoir, the effect of predfJ(;tion at the other wells is to cause a well to be surrounded by a drainage boundary, as shown in Fig. 3.6. On one side of this boundary fluid flows toward that well, and on the other side toward another well. For some time after a well is closed in it can be treated as if its drainage boundary still exi~ts. Thus, a well surrounded by other wells will have a buildup curve qualitatively similar to that in Fig. 3.4. For very long closed-in times this is not true, as will be discussed later under Interference Tests. A more extended discussion of average well and reservoir pressuresis presented in Chapter 4. 3.4 Pressure Buildup For Two- or Three-Phase Flow B 1ow t he bubble pomt ' 0f the 01 .1 .m the reservoIr, . gas flow will begin. At this time the pressure buildup behavior is governed by the more complicated nonlin-
* T0 Sh ow th IS we must repIace th e term In [( Y"'I'C'F.")/ 4kt] in Y(t) by the equivalentEi-function.
ear differential equations given in Chapter 2; and since the. equations are nonlinear, strictly speaking, the foregomg methods cannot be used.
O
--
PRESSURE BUILDUP ANALYSIS
23
Practical experience has shown, however, that with modifications the above methods also apply quite well below the bubble point. To arrive at the modifications, one should first note that the pressure in the oil phase in a given pore in the reservoir will be almost the same as that in the gas phase in tQe same or an adjacent pore. The two pressures will differ by the oil-gas capillary pressure, which for most situations of interest will be less than a few pounds per square inch. Thus, for practical purposes the buildup will be identical in
each phase. If we concentrate our attention on the oil phase, we can liken the buildup in this phase to buildup in a single-phasesituation. Two differences will arise. First, the compressibility will be higher in any set of pores because of the presence of gas. Secondly, the change in pressure with distance and time will be caused by the simultaneous flow of both oil aJld gas. We might expect that we could apply, at least approximately, the single-fluid methods if we use total compressibility and
7
6
5
.c
Ia.
~ 4 m
*1
:l
a.
0lD
~ "3
2
1
)
00.01
0.1
.10 0.000264
kt
9>JLCA
Fig. 3.5 Pressurefunction for one well in centerof cylindricalreservoir. DRAINAGE
BOUNDARY
1 ~ ~ :)
WELL RELATIVE
NOTE:
WELL
PRESSURES
I
WELL
RATE
I
PRESSURES ARE
FAR
RELATIVE
ARE BELOW
NOT THE
SHOWN
FOR
BOTTOM
2 RATE
EITHER
OF THE
2
WELL. DIAGRAM
Fig 3.6 Pressuredistribution in a 2:1 rectangularreservoir. ~-
24
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
total
mobility
results
instead
of
These
authors
foregoing
modifications,
this
ential
by
in
Section
that,
single-fluid
how flow
to
the
may
give
the
either to
the
methods
showing
2.5
true.
with
additional
radial
is
differential
showed
given
The
this
partial
and
the
two-phase
discussed
that
applicable
has
approach for
quantities.
show
flow
Martiw6 to
as
the
two-phase
applicable.
single-fluid
Weller15
solved
for
equations
the
and
equations
basis
of
Perrine14
2
of
most
interest.
In
obtaining same
differential
C t
following
differ-
a:;:2 +
cp
-,:-
ar
=
op
(k/
)
~'
JL where
Ct
is
total
...(3.16)
compressibility
and
(k/
JL) t
is
total
As
this
ing
the
is it
single-fluid
course,
is
and
equation equation,
completely gives
methods.
given
by
the
analogous
a The
cited
to
justification
for
derivation
final
the
justification,
numerical
works
and
Perrine
Weller. Up
out
to
curve
this
point
the
the
slope
of
should
3.16,
we
that
at
and
be.
If
would
the
abbve the
we
need
analysis
were
to
has
two-phase to
apply
not
pressure solve
average
pressure to
wish
the
boundary
the
oil
production ing
=
~~
(
r
~
JLo
When
this
Eq.
given
is
=
C1
noted
substituted
into
the
solution
to
El
is
the
note
by
rate
the
for
obtained p
is in
been
p*
obtain
the
the
oil
the
length
zone,
effective
dividing
kh
for
to
reflects
To
value
obtained
values
have
in.
that
values
which
done
oil
by
apparent
of
in
the
cumulative
measured
and
just
by
the
Weller,
around
skin
Perrine,14
3.5
Perrine
saturation
percent
oil
before
clos-
effect.
In
the
flow
gas
saturation
Pressure
the
the the
example was
buildup
in
Gas
of
can case
efficiency
Buildup
presence
wellbore
to
worked
out
reduced near
a
lead
to
the
70
wellbore.
Wells
E . xpenencehas shown th at these pressurebuild up
by
.
(
p
i'=r..'
t
cumulative
closing
distribution) for
flowing
producing
the
obtaining
transients This
gas
by
or
condition
3.16
)
its
value
pressure
by
As high an
qo
(or a
first
the of
be
before
the
3.17.
the
also
by
Eq.
in.
condition
well
this,
purpose
use
reservoir.
for
use that
correct in
example,
just
influence
s
dividing
rate
we
single-phase be
for
may by
oil
influences
the
this
obtained
reason
only
in
3.7.
except
the
and
is
Fig.
flow,
to
P"'t
Ex-
this
is
flow
replace
which
the
not
we
pointed
Eq.
but Since
buildup
differential
be
t does
s,
as
well,
by the
p.
time
what
oil
gas,
for
to
for
B, since
two-phase
demonstrat.ed
that,
an
production
and of
of
for
understand
apply-
for
the
permeability
analyzed
single-phase
completing
the
ko
curve
for
be
recommend
kg,
(Appendix
factor as
used
used
single-fluid
skin
are
t
or
calculated
buildup
may
oil
mobility.
The
have
This
water
t
we
/ JL ) t
time
oil,
calculation
2A),
the
(k
We 1
and
equation
and
to
example
quantities.
equation. o2p
permeability
our
amples
the
combined
the
In
theoretical
be
were
ko,
gas.
tpCtr2
)
methods
-4t(k/JL)t
+
C2
,
has
may
also
presented
a
be good
applied
to
gas
discussion
of
wells.
the
Tracy17
basis
for
such
application. Such approximationis based upon work it
leads,
by
integral,
use
of
the
rule
for
differentiating
a
definite
by
to
Aronofsky
=
~
(
Ei
41Tkoh Putting
r
curve,
=
we
r",
C2.
(3.17)
4t(k/JL)t and
find
) +
-tpCtr2
superposing
that
the
to
slope
of
obtain
the
a
resultant
buildup
Jenkins18
,ho
partial p
and
discussion),
differe~tial
ideal
gas.
thod
of
the
basis
approximation, with
also
Ref.
numerical
equations
On
expressed
(see
obtained
describing of
their
the
acceptable
radial
results
buildup
q
curve
In
is
practical
m
B
=
qoJLo/41Tkoh.
P"'8
oilfield
hi y
gas
t
same
.where (psi/cycle).
can
=
reasonmg
be
by oR
Y
...(3.18)
It
can
be
shown
that,
for
Z
-
T
T(
the
8e
t
+
~t
log
~t
)
.
P
P8e +
)/2
.'
(3.21a)
P"'8
phase, = m
In
-162.6
of me-
(3.21)
..Bg s
p*
the
their
units,
q"p.oBo koh
162.6
=
the
to flow
and
( buildup
and
equation
accuracy
pressure
19
solutions
this
=
162
6 .k
equation qg
=
J.
pSI
0" qg
qgt
./
(
~~
I cyce
represents
)
(3
19)
."
the
flow
only
of
-qoR8
free
gas.
(3.20)
Eqs.
slope
of
3.18 the
and two-phase
3.19,
we buildup
see
that curve
we
may to
use
Note
that,
the
calculate
changes curve
~
with of
--
P'O8
the time
vs
log
average
to average
and
factor of
computed well
pressure
P* .
according
arithmetic p*
volume
arithmetic d
at
the
formation
at
.an
pressure From
gas
well during [(t+~t)/
Eq.
3.21a,
Bg
between pressure
the P"'8'
buildup, ~t]
As the
should
is
computed
ex~apolated this slope also
average of change
the
c1
PRESSURE BUILDUP ANALYSIS
25
slightly with time. This changewill usuallybe negligible, and it will usuallybe satisfactoryto approximate PIC.by PIC!in the equationfor Bu. Whenthe gas equationis used in this form, one can use exactly the same form sheetfor buildup analysis in gas wells as was used with oil wells. It is only necessaryto convertthe gasrate in cubic feet per day to barrels per day by dividing by 5.615. The method of obtaining Bu, Cuand /l.uis shown in Appendix B, Example3A. The restof the analysisis straightforward as shown on Example 3. The curve analyzedin this caseis Fig. 3.8. By substitutingEq. 3.21a into Eq. 3.21 and rearranging,one obtainsthe following. ~ PIC.2= p*2 -325.2 !!.~~ log( ~) kgizT.c ~t (3 21b)
product is often more nearly constant than .auand thus the PIC.plot is preferred in this range. Even at low pressure,we have obtained very satisfactoryresuItswith the "unsquared"PIC.plot and, therefore,we recommendit as the commonmethod. In derivingboth Eqs. 3.21 and 3.21b, it was necessaryto assumethat the pressuregrp.dientis small and that .a and z are constant. When these assumptions are not allowable, AI-Hussainy, Ramey and Crawfords8 have shown that one can define a pseudopressurefor a gas which leadsto a form of the pressure buildup equationanalogousto Eq. 3.21. By using this gas pseudo-pressure, one can better handle pressure buildup in very low-permeabilitygas reservoirs when these are produced at high rates. See Ref. 38 for an accountof this method. At high rates of gasproduction, an additionalpressure drop will be introduced near the wellbore due
From this it can be seenthat a plot of PIC.2vs log [(t+ ~t) / ~t] should be a straight line if .auis constant. From Eq. 3.21 it can be seenthat the plot of PIC.vs log [(t+~t)/ ~t] shouldbe straightif the product .auBu is constant.At pressuresabove2,000 psi, this
to non-DarcyftoW.2O-23 The additional pressuredrop is proportionalto the productionrate q and acts just like additional wellbore damage.The skin effect s' for this caseis written s' = s + Dq. One cannot determine both the skin factor s and the non-Darcy b.t, hours 1
10
130
12
II
100
9
r
'i 8
7 100.000
10,000
1000
10
(I-At IIAt
Fig. 3.7 Pressurebuildupin a reservoirwhenboth oil and gasare flowing. ~
26
PRESSURE
coefficient D from a single pressure buildup test. The separation may be made by conducting two flow tests or two buildup tests.21If the skin effect s' is constant on two successivebuildups in a gas well, conducted after producing the well at different rates, then the non-Darcy effect is negligible. If the skin effects differ, one should plot s' vs flow rate. Extrapolation to q = 0 will give the skin effect s, alone. For a recent discussion of non-Darcy flow, see Ramey.23,38 To obtain an idea of whether non-Darcy flow is important, one may calculate the Reynolds number for flow at the wellbore.24aIf the Reynolds number is 1 or less, non-Darcy flow effects should be small. This is often the case in the authors' experience. See also Ref. 25, page 193ff, for a more complete discussion of methods of calculating Reynolds numbers and the onset of non-Darcy flow. The subject of gas-well flow testing is so extensive that it cannot be covered completely in this Monograph. Ref. 25, Theory and Practice of Testing of Gas Wells, published by the Oil and Gas Conservation Board of Alberta in 1965, provides a comprehensive treatment of this subject. Some aspectsof gas-well flow tests will be covered in Chapter 5 of this Monograph.
BUILDUP
AND
FLOW TESTS
IN WELLS
See also Ref. 26, Handbook of Natural Gas Engineering. Although gas-well flow tests are intended to measure directly the deliverability of a gas well, these tests can be very misleading (optimistically) if conducted for only a short time in a low-permeability reservoir. The reason for this is that wells in such reservoirs often require months to reach a "stabilized" condition, where the radius of drainage of the well ceases moving outward from the well. Tests of such length are generally impractical. Rather than conducting short flow-after-flow tests of doubtful extrapolatibility into the future, it is the authors' experience that a much better procedure is: (1) to determine kh and skin from a buildup test in the gas well, and (2) to use these values in an equation which predicts gas-well performance. The buildup method furnishes more fundamentally sound data and the mathematical equation allows prediction over a much longer period of time than does the common flow-after-flow test. This is especially true in low-permeability reservoirs. An analytical prediction method for gas reservoirs has been presented by Swift and Kiel,22 and more recently by Russell et al.27 When flow-arter-flow tests are re-
i),t, hours 10
3000
...
2900
280
270
260
250
240
10 I
3
10
(t+6t)
/6t
Fig. 3.8 Pressurebuildup in a gaswell. ,~~
,
~~
PRESSURE
BUILDUP
ANALYSIS
27
quired by law, it is possible, in some cases,to interpret these in terms of kh and skin as discussed in Section 6.5, and to use these values in the fundamental prediction equations. 3.6 Effects of Wellbore Fillup and Phase Redistribution Th .d 1. d h d. th e 1 ea Ize t eory Iscussed us far assumesthat a well is closed in at the sand face and that, after closing in, no production enters the wellbore. In practice, however, a well is closed in at the surface and fluid continues to flow into the wellbore for some time. Only after sufficient fluid accumulatesis the effect of closing in at the surface transmitted to the formation. For this reason there is a lag in the buildup at early times, as shown by Fig. 3.9 (adapted from Ref. 28). When the rate of flow into the wellbore is known at all times during this fillup period, it is possible to apply the principle of superposition for the gradually changing rate and thereby make use of this portion of the curve. Methods for doing this have been developed by Gladfelter, Tracy and Wilsey7and by Russell.2v In the approach of Gladfelter et a/., it is necessary ~o measure the rate of influx into the well after closing In at the surface. This is done through sonic measurements or through measurementof tubing-head and casing-head pressure simultaneously with bottom-hole pressure. These influx rates are then used to calculate a corrected buildup pressure from t:.Pcorr =
(t:.p) meaB(
qo
)
qo -q" ,!,here qo is the rate of production prior to closing In and q" is the average influx rate at a time t:.t". This corrected pressureis plotted vs logarithm of closedin time in the usual manner and is interpreted in the usual way. This method of using early values on the pressure buildup curve is very helpful in cases where the straight-line section is short or is ill-defined.
2000 g 180 '~ 1600 ~ E 1400 ~ ~ 1200 ~ ~ 1000 In 80 600 ,0001
GAS-FILLED WELL80RE
.001 201 .000264kdt/cpJLcre
Fig. 3.9 Well fillup effect. (After Miller, Dyes and Hutchinson.-)
.1
An alternative method for using pressure buildup data at early times has been developed by Russell.29 In this method it is not necessaryto measure the rate of influx after closing in at the surface. Instead, one uses a theoretical equation which gives the form the bottom-hole pressure should have as fluid accumulates in the wellbore during buildup. This leads to the result that one should plot P1O8 1 vs log
( t+t:.t't:.t )
1 -c-Kj 2 or vs log t:.t in analyzing pressure buildup data during the early fillup period. The denominator on the left makes a correction for the gradually decreasing flow into the wellbore. The quantity C2 is obtained by trialand-error as the-value which makes the curve straight at early times. An example application of this method is given in Figs. 3.1 OA and 3.1OB. After obtaining the straight-line section, the rest of the analysis is the same as for any other pressure buildup. This method has the advantage ?r requi~ng no add.itional data over that taken routinely dunng. a s~~t-In test. A r~~ent method pr~sente.dby GadzkI~gly also uses empmcal constants In fi~tIng an equatIon to the early part of the pressure buIldup curve. In addition to distortions caused by well fillup at early times, certain wells exhibit another peculiarity during buildup-that of "humping". By this we mean that the bottom-hole pressure builds up to a maximum and then decreases.An example of this is shown in Fig. 3: 11. Stegemeier.and Matth~ws3Oinvestigated this behavIor both theoretically and In the laboratory and showed that the behavior was due to segregationof oil and gas in the tubing and casing subsequentto shut-in at the surface. The rise of gas bubbles increasesthe botto~-~ole pressure: This can increas~ so much tha~ liquId In the well WIll be forced back Into the formation, thus decreasingthe bottom-hole pressure. Wells which show the humping behavior usually have the following characteristics: (1) they are completed in moderately permeable formations with a considerable skin effect or restriction to flow near the wellbore, and (2) the annulus is packed off. The phenomenon does not occur in the tighter formations since in these the wellbore pressurebuilds up so slowly over a long time period that the formation pressure will always be higher than the pressure generated by bubble rise in the tubing. Similarly, if there is no restriction to flow near the wellbore, fluid can flow back into the formation easily, always equalizing the pressure and preventing humping. If the annulus is not packed off, bubble rise in the tubing will simply unload liquid into the casing-tubingannulus rather than
h. b k .
h f
.
pus It ac Into t e ormation. One of the ways of decreasingthese wellbore effects is to use a tool31which closes in the well at the bottom.
.
28
PRESSURE BUILDUP AND FLOW TESTS IN WELLS 3200
\
PORTION OF DATA USED -..j
3100
IN
AFTERFLOW
ANALYSIS
300 290 280 270 260 01
.~
2500 ~ ; 2400
Pwf = 1590 psig q = 157 ST8/ D
a. 230
fL = 0.3 Cp 80 = 1.6
220
.=
0.1
C=2XI0-5psi-1
210
~
rl?
200
I(iJ
"'
190
180010
100 SHUT-IN
TIME,
1000 MINUTES
10pOO
;,.: jli
Fig. 3.10A Pressurebuildup curve,ShellNo.1.
1500
CI
140
In
a. ~
'+~
a.
VALUES
-N
I<2 +-
I
In ~
a.
Of 0
2.0
to
2.05
C2
U
-I
D 2.15
.2.5
130
IZO
0.1.
0 6t,
HOURS
Fig. 3.10BAfterflow analysis,ShellNo.1. ~
PRESSURE BUILDUP ANALYSIS
29
700
600
"
cn 0. 500
0:
::> 0
~400 ::> Q)
UJ 30 Q:
::> cn cn
~ 20 Q.
10
0 I
10
100 CLOSED-IN
1000
10,000
TI ME, minutes
Fig. 3.11 Hump due to rise of gasin tubing after closingin. This tool allows interpretation of a pressure buildup curve after a much shorter closed-in time. In Ref. 30, Stegemeierand Matthews also showed how leakage through the wellbore between dually completed zones at different pressure can cause an anomalous hump in measuredpressures.An example is shown in Fig. 3.12. The packing in the separation tool apparently leaked when the pressure differential between zones became small, allowing oil to flow past and cause a hump ...ec In the pressure observed In the other zone.
340
UPPERZONEPRE5JYRE
~ III ~ g::3200
3100 10000
0
a la
ene ra Ion
The effect of partial penetration of a well into a producing formation has been studied by Nisle82 and by Brons and Marting.88 An important factor in this case is the ratio of vertical to horizontal permeability. If, due to presence of shale streaks or tight layers,
JLURE BUILDUP
.~ ..; 3300
Another type of anomaly can be caused by a tubing leak in a well where the annulus is not packed off. High-pressure gas can enter from the annulus through the leak and cause the tubing to unload into the annulus. This can affect the slope of the bottom-hole pressure buildup curve. A clue as to occurrence of such a leak is the accompanying very high buildup in tubing-head pressure. 3 7 Eft t f P rt " I P t t "
the effective vertical permeability is smaIl, then the well will tend to behave as if the formation thickness is equal to the completion thickness. On the other hand, if the vertical permeability is high, the effect of partial penetration is to introduce an extra pressure drop near the well. When the standard methods of pressure buildup analysis are used, this extra pressure
1000
100
10
I
.t
+ f.t ~ Fig. 3.12.Effectof packerfailure on pressurebuildup, Williston Basinwell (upperzonepackedoff). .~
drop will show up as a "skin" or apparent wellbore damage. The buildup curve in such cases82will have the same shape as that given by Fig. 3.3. In the work bY Brons and M arting, curves are given f rom whic h It
..
.
is possible to estimate the amount of extra pressure drop caused by not penetrating the entire formation thickness. Opening only a few holes in the casing can also "
30
,
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
cause a skin effect. This is caused by the fact that the flow has to converge as it goes through the perforations. This convergence causes an extra pressure drop which can be interpreted as a "skin". Perforating devices, however, not only open a hole in the casing, but they also leave a hole some distance in the formation. Thus, in actuality the flow does not converge at the perforations, but into this hole. If these holes remain open, the skin effect will be much less than would be ...ternatrve calculated from the purely geometncal contraction eff .ressure ect by the method of Brons and Marting33 or Mus~at.24b In .the recent publication by Harris,34 this effect IS taken Into account. Of course, if the perforation holes are plugged or partially filled, the skin effect will be even greater than calculated from the constric-
the braces in the last equation. As may be seen from this equation, the absolute value of the slope is equal to 162.6 q2pB/kh. We now determine kh, sand wellbore damage exactly as in Examples 1 through 3 of Appendix B. The reader may be interested in verifying, by superposition, that the equation for s in the case of variable production rate is indeed the same as for the constant-rate case. 3 9 AI . M et h 0d S 0 f . P B UI.Id up Ana I YSIS
tion effect. 3 8 S ..A .puperp~srtlon to c~ou.nt for roductlon Rate Variation
(p -P,o.) against the closed-in time would give a straight line. The mathematical analysis was for the case of incompressible flow, and thus is not quantitatively applicable to actual reservoirs. However, the
Muskat Method .. In 1937. Muskat .performed a mathematical analysIs on the basIs of WhICh he proposed that a plot of log
..Although gene~ally the ~etho~ of calc~lating t given m Eq. 3.6 and dIscussed m Section 3.1 IS adequate to account for a variable production rate history, there are some exceptions. One exception is caused when a drastic change in production rate is made within a few ~ays .of a pressure buildup test. T~. account f~r sU:h sItuatIons, the method of superposItion as outlIned m Section 2.8 should be used. As an example, consider the case of a well which has produced at rate q1 for time tt. It then produces at rate q2 until time t2, or for
type of plot suggested by Muskat has been foune applicable in the case of compressible flow, as will be discussed later in this chapter as the "extended Muskat method". Pollard35 has used this type of plot for pressure buildup in fractured limestone wells. Other investigators have used it for pressure fall-off behavior in injection wells, as will be discussed later.
a time (t2-tt), as illustrated by the first two rates and times in Fig. 2.7. At time t2 the well is closed in for time !::Jot. From Eq. 3.8 we see that the pressure drop at the well for a constant rate q is given in oilfield units by , ,
The basis of the Miller, Dyes and Hutchinson method for pressure buildup analysis is the mathematical solution of the differential equations of pressure behavior in a fini~e reservoir. From the solution d.eveloped, one obtaIns kh from the slope of the buIldup curve exactly as in the examples of Appendix B. For
-)
cosm g m , we 0 tam Pi-POD.
=
162.6~
~
{
_Homer q1
.- 162 6 q2~ .- k
P.
h
is
equivalent
The
d
B
.
OlDl
tt
f A -== ---:':=-ppenwx, .
0 b
es
e
I
e
. 11
examp
method does differ from the
Matthews-Brons-Hazebroek method used to obtain
D~termination?f aver.age pres~ure is ma~e by u~ing solutions to the differential equation for flUId flow m a cylindrical reserv.oir. These .solutions plott~d in dimensionless form (FIg. 3.13) gIve the theoretical pressure buildup with time on closing in. By entering the curves at an actual closed-in dimensionless time, one reads the theore~cal expected buildup to final pressure. Either the . curve for no-flow over the draInage boun d ary or f or contant pressure at this boundary can be used. The curve for no-flow over the outer boun d ary (Curve A) should give approximately the same value
{-og qt I ( q2
)
t2+ !::Jot t2-t1 + !::Jot
l' }
In the last two equations, the well pressure is designated by POD' rather than POD!because the well was closed in. To determine formation parameters, one
average
system, howpressure.
[log(t2+!::Jot-t1)+s]
A + log ( t2-t1+Ut\ !::Jo t
permeability
he t
ever,
or
=
and the
.
[log(t2+!::Jot)+s]-
-q2 [log!::Jot+S'J }
in
.
t
.' .. Miller-Dyes-Hutchinson
+(q2-q1)
POD'
I
t
WI
and
m
00,
q2
effective
lon,
qt,
to
a
rates
average
cu
for
t
b
2.8
he fl ow e ffi clency .. I h h. of
pressure
ca
this
IS
of
t
superposItion
Section
Method
.
t m
Dyes and Hutchinsow8
'
this reason an example will not be repeated here. In addItIon, the MIller, Dyes andH utc hmson d eterlDlne d ..'
s,
ratio By
)
..
as I
(
0
drop
...to discussed
og
Miller
at
-.." s IS a constant.
where
t+
(1
= 1626~ . kh
.so
P.OD! 'P
V
for the average pressure p as the Matthews-BronsHazebroek method (Chapter 4) for a circular (or square) bounded reservoir. They may differ slightly
PRESSURE
BUILDUP
ANALYSIS
31
the reservoir is at semi-steady state prior to shut in, while the Matthews-Brons-Hazebroekmethod does not. An example comparison in Appendix C shows that for the case treated the answers obtained by the two methods differ by only 2 psi. The Miller-Dyes-Hutchinson Curve B for constant pressure at the drainage boundary gives a value for p which is greater than the value of p f!;!J;no flow_at the drainage radius. In the
-..
I examp
e
A --_c==:-
companson
10
dix '~
C
-
ppen
( P
fl no
gested by Muskat can be used with late-time pressure buildup data to obtain the drainage volume of a well. To do this we use Eq. 2.36 for pressure behavior in a bounded circular reservoir. We first assume that our well has produced lOng enuug1ltO~ se-mi';:s1ea-aystate prior to shut-in. At this time all the exponential terms have died out. Next, we close in our well for a time such that all but one of the exponential terms in
) ow
Eq. was
2.36
die
out.
This
occurs
after
the
pressure
buildup
.'"
-' 4,417 psia, and P (constant pressure) was 4,461 psia.
curve deVIates from a straIght hne on semI-log paper and starts to flatten out. We now superpose Eq. 2.36
It is important that the time at which the theoretical curve in Fig. 3.13.is entered be ?n the straight portion of the pressure buIldup curve. Miller, Dyes and Hutchinson suggest that the dimensionless time 6tDe at
for these flowing and production times, and introduce the relation between p and Pi. On evaluating the Bessel functions for a large outer boundary we find ' log (p-Pw.)
= log
( 118.6~ kh )-0.00168~ "'pCre
~hich th~-~_t:Y~~~!!t~!eE_be in,th_erange Q(.O.Q1_!Q 0.1. In this interval of time, wellbore filIup effects will ~ally be small and there will be little interference from production at other wells. We will adopt similar criteria later on to estimate how long a well should be clos~d in to give a usable straight-line pressure buildup section. The MiIler-Dyes-Hutchinson pressure buildup plot for reservoir shapes other than circular is shown in Fig. 3.14, after Pitzer.s6 l .."/ c;-~ .~ Extended Muskat Method29.43
(3.22) The units in this equation are the practical oilfield units discussedin Chapter 2. We see from Eq. 3.22 that a l;'~t oi~(p-p:>-;;"Zt should be '!!pear with slope {3=0.00168 k/"',ucre2 and intercept b = 118.6 q,uB/kh. From the quantity b and from knowledge of q and B, one can obtain a value for kh. A value 'f:r wellbore damage can also be obtained by using the semi-steadystate form of Eq. 2.36 together with this
In this section it will be shown that the plot sug-
kh value. As these quantities can also be obtained
0.0
r-
L~ /
./ V 0.5
V
./
/./
/' (A) ./ E
~
;-
~
(8)
-"""
10
L
f ~
./
I'
~
:: 00
/'
/
V
V
L./ L./
0 1.5
a.
~
/
/'
/'
~
./"
NO IPiFLU~ OF FLUID O\l£R THE DRAIPiAGE RADIUS
f8\
CQljSTANT PIIESSURE AT
III
nul
./
~
TH£ DRAIPiAGE IIADIUS
./
./ 20./
fi\ ~
~ ,.
./
/' V 2 5 I 1111111111111I1I11X11111! I 1 I .10-5
I I IIIIIIIIIIIIIIII!II!
III i III I
10-2
I ; I ./10-1
dt De :0.OO0264kAf IS_2.
_cre
( ~~~kl'w~
Fig. 3.13 Theoreticalpressurebuildup curves.(Frorri~Dyes Mca.H.
-~ro"S"-\Jtc.,"2.6roe.k)
{
-t
i III I
t4Vf1 -l" f
and ~~son,
-o.OOOL'ItKt-
p
D -'1'~cA
IV"
-
I III 100
"~ , I ~
.
r~.I~ .-UOIrs
~ and Perrine.") -p~-
70-'
p
I"-B/kl.
I
)
r
. ' tI'" f I;'~.!:rvo;r--S'
32
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
from shorter tests, it is not especially important to be able to obtain them from a long closed-in test. The pore volume drained by a well cannot, however, be obtained from short tests. It is this we obtain from:- v;=
should then plot 6PD vs log /:;'t on a sheetof transparent graph paper using the same scales as in Fig. 3.13. Overlay and note 6t and 6tDe at best fit. Compute the oil drainage volume V 0 from
---=-
V = 0.024 (1-8w) q6t 0 cmuA tDe
0.111So!!!!--, reservoir bbl. Thus, by a long closed-in abc /" period the drainage volume of a well can be determined. See Ref. 29 for an example calculation.
F
I
requent
-.
.
thi
y In k
The
h t
P
e
IS
not
Thi
most d
exten
d
nown.
serious M
e
may
problem
k
us
b s
b e
0
. talne
.
d
h
at
that
d
met
" IS
0
may
rom
In
arise
in
ng
sure
ki
ma
enough
f
d Ch 8 '11 b di as WI e scusse In apter,
eno ug h f -,
using th
0
i
I jI !' ~:::~::
0:
I!111;;'i"" 1:1:::::'
.1 :,., .",.
:
' .,
:';
6
.0001
I
'...
I:"
;.:.::..
I
r~ I''I
1
.:,' :;,' "I/
: : :: ,: : ,2
I l r
"""'I~':;,' Ii ",',':,, "
,001
1
.
accurately
l
I'
:
i!"I 'j Ii; :i::' ,!, , Ij: ,.
: !
i Ii:! i :!!:
i
1 '~~ II: i:
I !
I, II" ':'
'
!!!!,':;f"'::1o'"" 4"., """,'" ;!!i!:!~...,'
1 "1
1;:i!:::::-: :,:!::,,; ' I 'C"
I,...
~!:~,~~;; i : 1: ;,' ,.
I::
""
1 1
,I!;,::,:;",
0 b serve, d the
line is be
th e
drainage
tried
v al ue
volume
,.
c
I
li!i r'l!
I 1
&~" ;11"1it:: ,;..: ~,.
!j" :::!;: !'I'; :
I
1 i;; ~
"
! i I! !!: :;'! "
:
I
I
;.': ,
:::::,:-:1 "!(I"\ !ii!~~.l1
I ~:
';,!.I II,!,
l'
l-ifii!;,:: ill;;;;;
[' I
" I , ,, , " ;,,:'-"-
'
I'"
1111"1.,,,.
""'"
I"
,
Fig. 3.14Theoreticalpressurebuildupcurves.(After Pitzer."")
i
1;I,/)j
! I I
::;!!i":: '" " I ". I I
TIME
I
; :!: ;,:' I'I! '" ",
I
I
I ~'
: i: j i i ;:
I
!!: !~i;:::: ' ,,1 ,;",
:
;
~iil!'.
I
,I
SHUT-IN
,::: , I;!:
;!::::;i,;:I!:\l::~~~:~' I, -\1'"
riTf',
.01
.~_.I !' II: ""' '; I;,,:
;: i i;;: ;~::, 1 :
~:~t~..'.2.~~:!'::,:!;!: m,! clr~' III " 1 ,1. DIMENSIONLESS
P to
'j1-I\!!!i ';":'
!
""",'
111
or
i
I
I
:::;'y" !:::/,.."" 1(:1:111:j:'I'
. \1
: : ;;:': , ,'!
be
1 1
" rill:':"
,
!:~... I...?
, , , 'I!:I;"')"
t0
!lilllIIl!! il;, !!;
, ,::: ':
,\!\;!::j~1! ~m:/'
':Ii
of
~_.~
iiii\(::': 'II:":':'
,
values
known
~~ "11",
-:-'" : '
"'"
th e b en d -over
be
~1~~"-t
:.;,,:::! ~:,
; '"
f or
P must
I". /',
;;:".::~I""""",:::',
I
I, I :;,.
,.~
/~..
:I
I I
1! i I: '::::
.I !
,~1:
,,'
i!;!::
, : :
:;i:ii::...,,;": .,!;,,"',;
£"5 ~;:
..,
' i! 'ii!:
1
i'
~4'~;::i" ~ 'i ,
::
,:",., ::::':
"
5. !:' .."t
,,: ,:; II!::::,';
:\1::::' :,./'
-"
0:
" ' '
1;:1,'::
I, Lj~; z
I ! ';:;:, . i I:il :""'
; i;;':.
'~...
!I "',. II ' ' i"!
i
i ,I
:'
~ ~;:;.::~;: Q. Q. ,:1 ~3~::;'
"
!
Irll'i;;"'I"illl:""'" ; i!: I:i; ::.
I ii!I:!1;! ~21!!:\~:::::i:
.r
different
..~
! , I: :,,;
! ! I I ~i: :::
i: I
.~.
: ,: !I!;: ',., ii,!; : : ' I " .,
I" " , I, ::'' m!:; : '" ,.. ! ' I:::;,;,.: I'
III':' .I!::::::'
allows
the straight -
ma b b e su Ject t 0 I arge error. These methods for obtaining drainage volume are based both on a radial model and on central location of a producing well in that area. Non-central location can cau.sesi.gnificantdeviation in buildup character, as shown In Figs. 3.14 and 3.15. Drainage volumes obtained for non-central wells will usually be conservative since the first bend-over from the straight line will be interpreted as a reflection of the cylindrical drainage radius, Note on Fig. 3.15 that the initial deviation from a straight line for the buildup in a well draining a 5:1
at
Drainage Volumes by Curve Fitting37 , ." It IS also possible to deterInlne the drainage volume by matching observed data with the curves in Fig. 3.13. If the average pressure jj is known, the ordinate 6PD may lIe obtained as 6PD = 1.15 (P-PW8)/m. One
~
This
from
b ., .. In 0 talmng a best fit. If the well IS not closed In long
v
...~.~
that the "bend-over"
observed. .
a
h ' f th ' .Y t e portion 0 e curve IS late enough for drainage boundary effects to be felt. As a check on this the value of closed-in dimensionlesstime 0.000264 k6t/ cf>cr 2 toward the end of the buildup curve should b: of ilie order of 0.15 (or (:J6t ~ I),
N
(3 23)
.. This method IS best applied when the buildup is long
f I . th S type 0 ana YSIS, e average pres-,
.I d I tna -an -error pot, sure
.
STB '...
..
' 10
PRESSURE BUILDUP ANALYSIS
33
rectangle is like that of a well draining a square. However, the well draining the rectangle -Trans., has a later rise and then a flattening toward p. If one based a drainage fi
.
f
li
dri
I
d
I
3. Nisle, R. G. : "The Effect of a Short Term Shut-In on a Subsequent Build-Up Test on an Oil Well", AIME Pressure (1956)207, 320-321.
h
4.
van
volume
on curve 0 a cy n ca e to t e first deviation from tting the straight line, the mo drainage vol-
Everdingen,
A.
P.:
"The
Skin
fluence on the Productive Capacity AIME (1953) 198, 171-176.
ume obtained would be conservative. Note also on Fig. 3.15 the similarity between a well draining a 5: 1 rectangle and a well betweentwo faults. As the more distant boundaries begin to be felt in the 5: 1 rectangle case,the buildup curve flattens. The other buildup curve breaks upward and heads toward Pi. Again, results obtained for the drainage volume of the 11 b tw th tw f It ld b ti.StImulatIon we e een e 0 au s wou e conserva ve. In our next chapter we shall discuss more fully drainage volumes and their change with time. We shall also indicate how the p values we obtained in this chapt~r should be combined with drainage volumes to obtain an average reservoir pressure.
Effect of W
a
and
eII" ,
Its Tr
In-
ans.,
5. Hurst, W.: "Establishmentof the Skin Effect and Its Impediment to Fluid Flow into a Wellbore", Pet. Eng. (Oct., 1953) 25, B-6. 6. Hawkins, M. P., Jr.: "A Note on the Skin Effect", Trans.,AIME (1956) 207,356-357. 7. Gladfelter,R. E., Tracy, G. W. and Wilsey,L. E.: "Sele~ting yvells Which ,;Will .Respond to ProductionTreatment, Drill. and Prod. Prac., API (1955) 117. 8. Arps, J. J.: "How Well Completion Damage Can Be DeterminedGraphically", World Oil (1955)140 No.5, 225. 9. Thomas,G. B.: "Analysis of PressureBuild-Up Data", Trans.,AIME (1953) 198,125-128. 10. Trube, A. S.: "Compressibility of Natural Gases", Trans., AIME (1957) 210, 355-357; also, "Compressi-
References
1. Horner, D. R.: "Pressure Build-Up in Wells", Proc., Third World Pet. Cong., E. J. Brill, Leiden (1951) H, 503.
bility of Undersaturated Hydrocarbon Reservoir Fluids", Trans.,AIME (1957) 210,341-344. 11. Hall, H. N.: "Compressibility of Reservoir Rocks", Trans.,AIME (1953) 198,309-311.
2. Odeh,A. S. and Selig,P.: "PressureBuild-Up Analysis, Variable-Rate Case",J. Pet. Tech. (July, 1963) 790794.
12. Ramey,H. J., Jr.: "Rapid Methods for EstimatingReservoir Compressibilities",J. Pet. Tech. (April, 1964) 447-454.. I
1
D
=
0.000264
kl
o\~'
O:pcA
~~o(
~\~E.
~E.~~
~~ \~ E.\..\.. \~
~ ~E.
1(4\\ \\:11
.
f\~\~
\
oft
1
@ WELL DRAINING A
P (@)
SQUARE BOUNDARY tD=.545
w IX:
BOuNDARY DIAGRAM:;
~ ~ (/)
@
a 2.
FAULT
o/2..t
IX:
A=o2
n.
NOTE' FOR THE 5'1 RECTANGLE
~
~.-PiOASPREDICTEDBYFIGURE4.7
o'
\oV
.TWO
U.
.FAULTS A=o2
ATID=0.32
5
@ @ .001
.01
0.1
61
tt61
Fig. 3;15Illustrative pressurebuildupcurves. ~
--
I
.II ~
RECTANGLE SQUARE I
Chapter
4 , \'
~
Determination of Average
Reservoir Pressure
4.1 Uses of Average Reservoir Pressure Data h In Chapter 3 we Pointed out a method for obtaining f t
e
average
pressure
cy li n dri c al bounded well
in
a
multi-well
th or
case
of
a
sI
' ngle
well
in
a
using a prediction method whic~ relates future production to future average reservOIr pr~ss~re. Pressure measurements
throughout
e
reservoir. bounded
the
We
also ..to noted
reservoIr
that
behaves
each m
a
roughly similar manner. In this chapter we will pursue this subject further. First of all, what are average reservoir pressures used for? Broadly speaking, they are used for characterizing a reservoir, computing its oil in place and predicting future behavior. In characterizing a reserv~ir, pressures are used to relate the amount of production in a given interval of time to the pressure drop. If the pressure drop is small per unit of production, a clue is given to the existence of a water drive or to draina~e from a large reservoir volume. If th.e pres~ure dr~p ~s large for a given amount of production, this may mdicate drainage from a small sand lens or fault block. Alternatively, if the reservoir is und:rsaturated, a large pressure drop may only be .reflecting ~e low ~ompressibility of sing1e-phas~011 along WIth assocIated rock and water. Companson of well pressures ~ay indicate whether wells are separated by fa~lts or Impermeable zones. Pressure drop, of course, IS only one element in the characterization of a reservoir. Geolo~ical conditions, oil properties, water cuts and ga.s-oIlratio variation are a few of the other relevant Items necessaryfor reservoir characterization. In addition to this semi-quantitative use, pressures find a quantitative use in volumetric-balance calculationsl of oil-in-place in a reservoir: For this pu~ose, average reservoir pressuresare requIred. If water Influx calculations are to be made in combi~~tion2. with a volumetric balance, pressures at the ongInal oIl-water boundary are needed in addition to average pressures. Measurementsof areal averagepressuresare, of course, fundamental to secondary recovery and pressure maintenanceprojects to indicate effectivenessof repressuring and degreeof reservoir connectivity. Finally, extrapolation into the future is best made by
reservoIr
life
h
..
are
thus
d"Iction
almost IS tomake relate corrections suc ~ pre to actualmandatory performanceIf one and
the
..
predIctIons. 4.2 Determining Drainage Volumes of Wells As noted in Chapter 3, each well in a bounded reservoir drains a certain volume surrounding that well. How does one find the size and shape of this drainage volume? We first note that at semi-steady state (see Chapter 2, Section2.7) it can be shown3that each well in a reservoir drains a volume proportional to its production rate. * Prior to this time, each well in a reservoir tends to drain an equal share of the reservoir. A simple example to illustrate these concepts is shown in Fig. 4.1 for a 2: 1 rectangular reservoir where one well produces at twice the rate of the other. This examplewas obtained by using Ei-functions to represent the pressure drop at the wells and imaging3to reproduce the boundary of this two-well reservoir. By summing the effect of the Ei-functions, the pressure drop was calculated cata network of grid points in this rectangle for a seriesof times. At eachvalue of time, equipressure contours and flowlines were drawn. Some of the flowlines are shown on Figs. 4.1A and 4.1B at a series of times. Note that each well initially drains one-half the reservoir and that the boundary gradually shifts until each well drains a volume (and an area, since thickness is constant) proportional to its rate. The time required in this case to reach semi-steady state where each well drains a volume proportional to its rate is kt/JLcA t ~ 0.7 (or kt/JLcA ~ 2 for Well 1 and kt/JLcA ~ 1 for Well 2). This is much greater than in a symmetrical drainage area. In the symmetrical :" *Stewart8 has presentedfield confirmati~n of this for gas wells. He also illustrates ho.wthe d~arnagevol1;lmes are altered after fracturIng, agaIn becomIngproportlonal to the new rates. ~, ---~
---
36
PRESSURE BUILDUP AND FLOW TESTSIN WELLS
""'. ", ---~- -,
" 1\""'"
I
,
I / I '
'\
/
"'" / "// ",-/ I \ ' ---,
"
\ , ',', \' I 1/ / I I \, " ", ,~" 1/ " , " " '" \ 1:,1 /" I \ , " '" \1" / .1 I " ,'~ I / ,,' I '- --"" ",' / / \i I / 1/_"" .-"" / " ---= r- ,,"" / /1" I", ,""-- ~...--r " , " ../ / 1/ , ' , / , / I ,II' , , I I // ,;'/ I J:\ " " ',:,'
~
----
I
/ /
I
/
, I I / // ' /
//
"/
...'
'/.../""
/
,I, /
,
I ~ '\ !"" T,
;1
I I
/
','
'
I ' 1/
""""
"Iy/
I
,,
, ~ I
""
"
~~--
/ /,ft', /" '11 1\\'~ '" ~ / / j 'III' \... ~',
,/ I' I / /
\ I' / '11'/ " ,II, 1 / /
' '--
,
7 /1 'DRAINAGE, .1/ / /' / / / ,/ ,'\BOUNDARY \ : I / / / / ',\ " ',\ I /, / / / I , "', "'",,-"'\'/ ~ / /1 , '" \ II/ / ~./ / ""'""', , I / /" " 2--""'"
/ II " "
""
1,
/
/
//
/, /
/I
I
--"
""""
I
"
"
,,\,\',
1\ " " I'I " '" ,,'
I
'
\ , \ I\
"
',
I
, ,
"""""--',~
DIMENSIONLESS TIME, tD =0.01 ,
""
,
,
"
,
' I
\
'
\ ' \ "
\ ",
" /
',1
1/ II
/
/
/ / I / I I
/ I' , I
1
, ,\' , I 11\ '\ ~, I' \
,,/ / i .../ ~~"
,
/
','
""
~
/""
\ I
II
I,
'/'
/ I
~
\: I / I I ' / I I I
+"', 1 " , I "",'
,
,
\ \ DRAINAGE \\ T , / / I \ , BOUNDARY, .I " / I' \ \ '" "I I / ~ 1 I " '" \ II ' / I I " \ I I /./ / / '-' ~ 11'-~2 "" ~.&./ --'" /'1\\ /"" -"""' ..,... / I" , "'" " / .4' 'I' \ ~ "\ \ / /' ,I \
\ "
---~
,
\
/I, \
'\ ~f ,I / \ I / I " , '," \1' II 1 " " , , II/, ~ ' \ * -~~--~ I -~ -~ ---~---:::: " -"" " ,I " ""..,.,.', 'I / ~ , ,r , 'I / // I ,1\ \ ,
,
,:
/ / """
I, I / / \I , / ,I
/
-
~ I 1'\ ,.I J lIf" "
/1 I I I
'"
,,\
" \' , \ \ " " ---' '\ \ ""
tD=0,03
---
I
-,
\
\ ,
, ,,"""
'
/1
1 I
\\
t r! ,
" ""-, -y ,,"'"
,/ , I
I I I ,
\',
II" II,
,
I
I" III ~/!
-,
\~
I', "
,
1\,
!\ \. 'f\~, 1\.\
""""" --"--
",,/ ,/"
,
I ~,/ "
I \' I , ,,1 I,
I'
'-
I ,
,~
,
",,"'"
I
I
DRA-INAGE \ I,f
,BOUNDARY \
~_""I
--~--II',
'
,
/
/ , ' / /~ I
\ II , II
III \1' ~ 12___~/
--r---II
/11
,II 1'\
I" .I ,L , \.. , ~ ~ " I " ~"" I '
-,
/
/
,
"
/
" , \
\ t = 0.000264kt D RJ.u cAt ,whereAt=totoloreo
t D = 0.06
Fig. 4.1A Movementof a drainageboundary:relative productionrates1:2.
J
~
DETERMINATION
~ f
OF AVERAGE
RESERVOIR
, ,-----~ ,-', \, "
'" "
,
"
/ I
'"
I
'"
",'"
I
~""'"
",,""
/I
/
--'"
--.~
-'"
-"
'
--,...
\'\',
"
\
,
" ,,
'",
I
/
'
:f"
I
/'"
",/'"
/
/
"'
/
/
//
'
"
,,,'
,"
I
I II
, ",
/
I
I
,
/ .;If'
,,
'"
\ ,'\ 1'\ I \', } ~\
-- '" ~-""'-
,
\
,,', /J
~
\
--
I
"",
I ./ ~
/ ""
'" / /' """;0.--
-'"
DRAINAGE
,,
/ '"
',
/
/
I
/
",'"
/
/
//
'" '"
\
,'
--- I, I
~
/
, //
/ I,
,,
1/
/
2 'I ---
,
~
,\,\\
, \\
.I
/
-"" t
=I
/
I
,
I
/
' \
1 '\,\\
'"
\ \ \
,
II '" I / '"
, ",' ,
-P""
\
I
,I '/ I'"
'- ""'
'., '\
\
\
,
BOUNDARY
,
",,""
\,\
""
,
/ / //
I',
tD=0.2
'"
,I
- -..A
,,
\,\
I I
1//2
",'"
'"
-'"
\
--
'"
I
+
///
--""
,
,I, f
---\\'
)
",/
,
/ / II I II , II
--
,.
/
,
III
--P"'"--"", '",--
, \
I\ I II
'DRAINAGE
BOUNDARY
/
,,
.,
I"
II
/
"
\
--"" -- ~-I /
1,
,. --, , "
'
tD = 0.1
"
""', "
"
1/
I
"
"",\,
"
...'" I
',
,.,
TIME,
/
/
-,
\'
I" , / '
, /
~
11\ 11\
II' II' 1
/1
DIMENSIONLESS
'"
11\"
'"",'"
I
'"
",/
~\\IIII''~
II
'"
I,
I
\'1 , \ ,I
~
/
"'",
I
'"
/
\ ,I
/',
/
~"I t '
BOUNDARY
"
,,
/
, I
I
\I
1.!-'-,' I
/
\
I",
"
\'
,
,.
DRAINAGE
\,\,
,
'-"""' --,
//
\
~
"",
'" _-
1
,,
'"
"
\\
37
,
-, '",
\
,
PRESSURE
\\ ' \ .D \',
'
t-
\ 0.000264
-0JJcAt
kt
where At = total area
D
Fig. 4.1HMovementof a drainageboundary:relativeproductionrates1:2.
38
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
case, the time to reach semi-steadystate* is kt/
the Fig.
time 10.29
at which for a list
the of
straight-line such times.
2
1
1
1
por-
ter 10, Section 10.3. The time required to achieve semi-steadystate in such reservoirs, however, may be 10 to 50 times as long as in homogeneousones. This again points up the necessity for developing pressureaveraging methods which are applicable prior to semisteady state. In both homogeneousand heterogeneous cases, once semi-steady state is reached, the pressure at each point in the reservoir will decline at the same rate. At such time each unit volume in the reservoir is losing fluid at the same rate, and thus the drainage volume of each well must be proportional to the rate at that well . Returning
to
.
Fig.
4.1
we
.
see
that
this
figure
t t th t li t 'd ra es e wo mI s on ramage vo Iume -one
1
1
4
1
4
3
1
illus-
limI-' t
2
Fig. 4.2 Exampledrainageboundaries(basedon flow lines obtained in model studies).Numbersindicate approximaterelative productionrates of wells. Each well drains an area proportionalto its rate.
DETERMINATION
OF AVERAGE
RESERVOIR
PRESSURE
ing volume is obtained by allocating :volumes i~ .p~oportion to rate, and the other is obtained by dividing the reservoir volume equally among all wells. Even at times prior to steady state: we normally use the first limit and assum~ that dram~ge volumes are proportional to production rate. This was s~own to be satisfactory3 in ~dealizedcases.for co~puting average pressureboth pnor to, and dunng, semi-steadys~ate. .tion, Now that the size.of each dr;inage volu~e Ishestimated, what about ItS shape? .om~ :x:mp e sha: . (obtained by the method shown me: .) ar~ s ho In FI . g 4 2 As ma y be seen man ' y vanatlons ..dunng,m s ape are possible. As a practical matter, It WIll often be sufficiently accurate to assume that the drainage areas are symmetrical and to use the .- curve in Fig. 4.3 for a S q
uare
ge area to obtaIn p.
draina
Thi
.
t
b
s IS rue ecause
pressur~s ar~ most often ~sed m re~rvolr ent~neenng to obtaIn dlfferences:- dlffer~nces etween e pressures taken at two different times and between functions calculated from these pressures. Because 0f thO IS, a small
but
constant
error
m p WIll often
be urnm-
39
A slightly different approach for obtaining p has been developed by Dietz,5 who suggests: "Instead of extrapolating to p* and then correcting to give p, why not extrapolate to p directly?" He shows that this can be done if the well is produced long enough prior to shut-in to reach semi-steady state. The paper should be consulted for details. For semi-steadystate producthe Dietz method gives exactly the same result for p as the Matthews-Brons-Hazebroek3method discussed in this chapter. As the latter (MBH) method gives
correct
results
at
times
preceding,
as
well
as
semi-steady state, It IS preferable. .. ':Jv'hatcan be done for cases m which the pressure buildup curve was not recorded, but only a spot pr~ssure
was
measured
after
a
24-hour
or
..
longer
shut-In?
Suchsituation spot pressures are arises. required some states that this frequently To m cope with this, so Brons and Miller6 have devised a useful method. They point ..
out that these spot readings are usually on the stralghtbUIld up curve. S.I nce the
.
.
1me po rti on 0 f a pressure slope of the straight-line
portion
.
is 162.6
qpB/kh,
and
portant. For cases in which it is important to make a cor-
since the quantities q, p., B, k and h are either known or can be estimated, a straight line with the correct
rection for non-symmetry becauseof very non-uniform spacing or large differences in production rates, one can use the curves for non-symmetrical drainage areas (Figs. 4.3 through 4.9). To use these curves it will be necessaryto divide the reservoir into areas drained by each well. A study of some of the shapes obtained in Ref. 4 by model experiments can be helpful in this.regard. As disc~ssed, we normally make the assumption that the draInage volumes a~e proportional to pro~uctio~ rate. Ske.tching of the draInage areas and checkIng with a plarn~eter to see whether the sketched volumes are p~oportlonal to .productio~ rate .is next. This usually re~ulres several adjustments m draInage area.and can be time-~onsuming. The authors prefer the slmpl~r .approach dlscusse.d previously except in caseswhere It IS felt that non-urnform Spacing or non-uniform production rate is unusuallyacute.
slope can be drawn through the spot pressure value. This line can be extrapolated to give p* on a plot of pressure vs log [(t + !:::ot)/ !:::of].Knowing p*, one can obtain p as before. In their paper, Brons and Miller assume semi-steady state flow prior to shut-in. Their method is thus like that of Dietz. As discussed in the previous paragraph, the authors of this Monograph prefer the more general method using p*. Both systerns are equivalent at semi-steadystate. Miller, Dyes and Hutchinson7 have also presented a method for obtaining average pressure, as discussedin Section 3.9. By using their Curve A, which assumes no flow across the drainage boundary, one should obtain very nearly the same results as with the MatthewsBrons-Hazebroek method. A difference of only 2 psi was obtained between the two for the example in
4.3
Determining Average Pressure in Bounded (Depletion- Type) Reservoirs By means of the curve for the square in Fig. 4.3, it is possible to compute the average pressure in the drainage area of each well in a bounded reservoir. This is done by obtaining p* from the flow or buildup test, obtaining p* -p from Fig. 4.3 and then obtaining p by difference. In this way it is unnecessaryto close in the wells until they show the flattened portion as on Fig. 3.4. In tight reservoirs the shut-in time required to obtain the flattening is prohibitive. An example calculation for p is given in Appendix C. By proceeding as in this example, a correction can be obtained to each value of p* and the average pressure obtained for each drainage area.
A
d.
ppen IX C. . By one of the procedures discussed above, one can obtain a pressure p for the drainage area of each well in a bounded reservoir. These, in turn, can be averaged to give the over-all average reservoir pressure. This is usually done by plotting these pressures on a map and "contouring" equipressureareas.However, this practice is not fundamentally sound in a bounded reservoir. The set of average pressures, p, are not point values existing at each well. They represent averages for various drainage volumes and, as such, need not be smoothly contourable. Wells producing at high rates will tend to have a lower average pressure in their drainage volumes than lower-rate wells and vice versa. Thus, the distribution of pressure in a bounded reservoir will depend mainly on the distribution of production rates.
40
PRESSURE BUILDUP AND FLOW ~ESTSIN WELLS~
w
I
1~
~ ~ ~
uJ
~
a
{\J
J -]
iB'
to;
"0
-~
UI
v ~ .CD
~(\JU .0
:l
.s
1
«)
0.. 8
{\J 0. .0
~ ~
-ti
!
(;
j "'! ~
w 0
t
~
!
Q
n01
~ 0
IIJilJ
.t.~
{\J
.-.q
I
-:-'::.
1'1 I. r--
W
It)
~
~~~~~ ~;~*~~~~;::: ~
{\J
Ii 1;
,--,-" -0
(3
~
.1."'-
-'
,
..3~ .~.."
~
DETERMINATION
OF AVERAGE RESERVOIR PRESSURE
41
~;;:~~:~~,:;Q ;:::
...:
:
m
. It)
'"
i
='
0 .0
e ~ 0' on
~ .s on
-<
§
().-
.\~
";: r'~
~
1D4 -
N U
=
g~"v~
~ .,.~. ! 0 ~Q) '" ~ d0
r
0
g
-.d
a
: ~ ~:
~
~: ; ;
.£ ~
..;
.t
ID
In
.It) 4)j/8rlb
'"
-.
9.OL
d-* -d
:!
42
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
0
<.D
V
~
(\J
~
.g §
0 .0
...
'"
~
-Q
~
-2
...
N
<.D
'"
.s
-a
v
~
.~
.V
1.1
<.O
~
C\J (,) ~
(\J
-.
O::l~ O-e..= 0
O .It=
~ 4) :a '0 Q 0 '.:3 u
-:
§
~
0
'" ~
<.D 0
~
..
In
~
~
v
~
0: ~ 0 (\J
0
r-
<.D
It)
v
~
(\J
-O'
0
~
DETERMINATION
OF AVERAGE
RESERVOIR
PRESSURE
43
0
(,0
V
~ N
.
i= ::I 0
.c
... '"
-~
= ..tg
,
~: ...
r
~
~'"
(,0
~-rn
.~
=
~
~
'z0
.c.D~ NU;;
g
~
.O::LQ)
O-e..!
0 ~ 0
5 ~ ... oS d 0
.z
U
-d
0 ..a
Q)
~ rn
rn
(,0 .~ 0
~ IC
..; ~
0 ~
0 N
a
In
~
~
N
-
d 0
-N I
I
~
or!;
44
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
Perhaps the best method of obtaini~g .an avera~e reservoir pressure in a bounded reservoir is to obtain pre~sure su~eys in as many ~ells as possible, alternating wells m each survey penod. The surveys should cover the range of producing rates as well as all sections of the field, especially if these have differing rock properties. By plotting p vs time for each well, it will be possible to extrapolate the pressure m .Preservolr nonsurveyed wells and interpolate pressures for surve~ed wells to the time period of interest. Often a correlation can be found between the producing rates of wells and the average pressures in their drainage areas. Such a correlation can help in deciding upon a pressure in the area around non-surveyed wells. After obtaining the average pres~ure m ~e drainage
volu~e of each well, t~e next step is to weight these to obtain average reservoir pressure. The correct method of weighting is to multiply the average well pressure by the volume fraction of the reservoir it drains and then add the results. Assuming semi-steadystate conditions where the drainage volumes are proportional to production rates, we may approximate the relative drainage volume of Well i, Vi/Vt from
~ = ~ , (4.1) Vt qt where qi is the production rate of Well i and qt is the total production rate from the reservoir of volume V to From this relation we obtain , -=
-~
Pl qt
+ P2 -~ qt +
(4 2)
p obtained in this way is averaged volumetrically over the reservoir. 4.4 Water. Drive Reservoirs Two types of pressuresare needed for conducting a material balance in a water-drive reservoir: (1) the average pressure in the original oil reservoir and (2) the pressure at the original oil-water contact. The aver-
age pressure is used in the terms involving volume changes in the original oil-bearing portion. Pressures at the original oil-water contact are used to compute water influx. A different procedure must be used to obtain the average pressure in a water-drive reservoir than in a bounded (depletion-type) reservoir. In the water-drive
5
4
.c ~
IQ.~ I ~
* co Q.,g
g:
.01
.02
..3.4..
~ 0.000264
-..
kt
.,.. cA
..Fig.
4.7 Pressurefunction for rectanglesof variousshapes.
8
10
DETERMINATION
OF AVERAGE
RESERVOIR
PRESSURE
45
0
Q
z
Q
g3
'CO
= ~
"U
I
-.. ~ ~
.~
N
-~ N
=
.~ 'Q
0::1.
~
0
...
0 .~
=
O-e.
-'"
u
0-
~ .S
§ ',= u .e
~ =
3
'"
e
~
I
~ ~ .2f ~
0
~ ~ Ht::j~
~
In
..It)
N
-0
q
To
4)j/811bg.OL d-*
d
~
46
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
4
3
2 .c
~ I
10. (D
I ~
*0. 0<.0
0 0 r--
-I
-2
-3
0.01
0.1
1.0
10
O.000264kt I#IfLCA
Fig. 4.9 Pressurefunction in a 2:1 rectangleand an equilateraltriangle.
-9-
: :~ 0
0
Fig. 4.10 Flow lines in a water-drivereservoir. -~
-
100
DETERMINATION
OF AVERAGE
RESERVOIR
PRESSURE
47
case there is a strong dependenceof pressure on distance from the original oil-water contact. In the bounded case there is not necessarily any relation between position and pressureas discussedin the previous
@
section. The reason for the dependenceof pressure on position in the water-drive case is the pressure gradient from aquifer to oil reservoir which develops as oil production begins. A sketch showing flowlines in a water-drive reservoir
RADIAL FLOW LINES
=
+
~
LINEAR FLOW
is given in Fig. 4.10. The flowlines may be approx-
LINES
imated by: (1) radial flowlines near the well and (2) linear flowlines from the original oil-water contact to the well, as shown in Fig. 4.11. The averagepressure in the radial flow area may be approximated as the pressure in a bounded circular reservoir of area slightly less (say 10 percent to allow for flowlines passing between wells) than the well spacing. 1 1 ti.. d . tl
sure ca cu a on IS ma e m exac y as that
in Appendix
C except
This th
average pres-
e same manner
that the curve
for one
well m the center of a bounded cyhndncal reservOir is used to convert p* to p (see Fig. 4.3). Those who prefer the Miller, Dyes and Hutchinson system may prefer to use the average between no-flow across and constant pressure at the outer boundary for this calculation. To obtaIn
pressures
at the ongInal
oIl-water
contact,
one should plot these average pressuresvs their lateral d f h 1 1 E 1 Istance rom t e ongma 01 -water contact. xtrapo ation of this curve to the original oil-water contact will give the pressure at this point. To be strictly accurate, the drainage boundary pressure rather than the average pressure should be used in making this extrapolation. However, the averagepressureand the drainage boundary pressure are very nearly the same. In addition' it is h d .ff
.
tel
..
erence
m
pressure
.
which
IS used
m
water
Influx
calculations, and this will be almost exactly the same for
either
case
Fig. 4.11 Approximationof flow lines. References 1. Muskat,.M.: Physical Principles of Oil Production, Graw-Hlll Book Co., Inc., N. Y. (1949) 378 if. 2. van Everdingen, A. F., Timmerman, Mahon, J. J.: "Application of the
E. H. and McMaterial Balance
Equation to a Partial Water Drive Reservoir", Trans., AIME (1953) 198, 51-60. 3. Matthews, C. S., Brons, F. and Hazebroek, P.: "A Method for Det~r,r;ninationof Average Pressure in a BoundedReservoir, Trans.,AIME (1954)201,182-191. 4. Matthews,C. S. and Lefkovits, H. C.: "Studies on Pressure Distribution in Bounded Reservoirs State", Trans., AIME (1955) 204, 182-189.
at
Steady
.
.. . .Ie, tz D ..: N " D etermmatlon 0f Average ReservOir Pressurefrom Build-Up Surveys",J. Pet. Tech. (Aug., 1965)955-959. 6. Brons, F. and Miller, W. C.: "A Simple Method for Correcting SpotPressureReadings",J. Pet. Tech. (Aug., 1961)803-805. 7. Miller, C. C., Dyes,A. B. and Hutchinson,C. A., Jr.: "The Estimation of Pressure Permeability and Reservoir Pressure from Bottom-hole Build-up Characteristics", 5 D
Trans.
AIME
'".
8. Stewart, .serves",
Mc-
(1950)
189
'
91-104
'. .
P. R.: Evaluation of Individual Pet. Eng. (May, 1966) 38, 85.
:'\'
Gas
Well
Re-
Chapter 5
Pressure Drawdown Analysis
In up
Chapter
3
analysis
however,
pressure
pressure
flow
at
to
,
at
last
a
objectives not
few
hours the
possible
reach of
a
was
static
technique
will test
When
should
drawdown
test
Ideally,
a
pressure .~ canOlaares.
be
an as
the
as
a
test, rate
test
a
should so
new
has
workover
consIderations
start
with
are
further
to
Odeh
and
real
time
of
be
for
a
the
pressure
answer.
down tests the pressure
sure
analysis.
further
in wells buildup
test
data
Generally are
However,
an
extended conveniently I'
by
tical tablished.
method
umts),
is
on
time
transIent
value
flow
conditions
An
to
periods analysis
as
are
k
are
these the
"late
indicated
technique
/
TRANSIENT
uniform
in
for
and
cases
transient" schematically
applicable
to
pres-
METHOD
AMENABLE TO ANALYSIS BY TRANSIENT METHOD
/
eco-
of
PORTION OF DRAWDOWNTEST
the
AMENABLE
for
estimates b e run
,-
drawa
pres-
(RESERVOIR
TO ANALYSIS LIMIT
TEST)
~ in
"i Co
test
.p~cr2
comparative
pressure
tests
may
drawdown
used
short
test
uncertainties that case
pressure be
drawdown
may
time
PORTION OF DRAWOOWNTEST
some if
considering are In
to
afford
es-
two
PORTION OF DRAW DOWN TEST AMENABLE TO ANALYSIS BY
test?
cloSIng-In
for
there
a
(prac-
conditions
between
referred
time
t -0.00088 flow
period
sometimes
5.1,
of
state
interim
These
Fig.
a
semi-steady The
period. of
at
units),
values
excenent
drawdown
alternative
n some
can
speaking,
simp~y
a
reason
which
( or
BY
STATE METHOD
then
where interpretation.
followed
pressure
tests
purpose.
A is
buildup
affords
tests) and
buIldup,
approx-
umts
.SEMI-STEADY well
time
! i
practical
Also,
P ressure
~
,
t~~
t~
.ppcr
2
~
drawdown
pressure
buildups.
(reservoir reservoir ' .1 pnman Y
limit
of
FLOW TIME,
t,
hrs
~
volume f
thi or
:!
Nabor,12
.
D arcy
In
flow
.-c/>p.cre2
prevail.
or
available.
a
recompletion
preclude
value
I
methods
of
., In
, nomIc
by
-O,1c/>p.CTe2. t =
-,
is
to
buildup
closed
a
These
6.
wells
been
or
it
discussion
pressure
to
a
analysis:
ranges
discussed
period
1
equa
k
rate
Chapter
recommend to
flow
j
of
In
different data.
different
As
c/>/LCre2.
time a is
our in
the
j --0.00264
the
which
sufficient
in
a drawdown.
flow
compressibility.
drawdown
during
radial
three
1
pressure
k
and
on
in
flow
methods
reservoir, .LATE If a well
Also, such
the
included
opposed
drawdown
a
during
constant
tests
as
for
present
pressure
drawdown
relationships
solutions
and
shall
applicable
pressure
flow
the
small
we for
all
during
tests
depending
variable
enmneer b'
well,
producing
to a
analysis
.--~ In
reason
prior
considers
'the
i.e., of
for
same
pres-
Drawdown
constant
are
1y
The
tests
in
time
the
chapter
throughout.
into
days,
closed
is
of
fluid
this
techniques of
well
period
pressure.
drawdown
not
pressure which
multiple-rate
several
For
the
a
begun.
bottom-
techniques
on theory,
single A
period
equalize
lowered is
maintain
well
analysis
This
or
test.
to
the
for
static is
of a
Usually
to
rate
reservoir
series
analysis
based
buildup
tests.
during
test
reach
constant
of
where
flow
are
per-
cases,
a
pressure
equipment
flow
may
simply
The
possible,
drawdown
rate.
the
the ..Imate I.e., to
measuring
then
is
build-
is
formation
some
made
allow
formation,
sure
in
producing
prior to
pressure
the
pressure
test
in
It
of
measurements
in
sufficient
art"
also,
of
constant
closed
the
presented.
and
means
drawdown
hole
the
estimates
effect
by
of
obtain
skin
volume
"state was
to
meability,
the
methods
Fig.
5.1
Schematic
tIme
ranges
plot
s. for
.'. WhICh
of
pressure varIOUS
applicable.
drawdown analysIs
test methods
showing are
PRESSURE
sure
DRAWDOWN
drawdown
presented /5.1
during
each
text
which
follows.
Drawdown
Transient
of
Analysis
these
periods
is
flow
a well
in
rate,
the
reservoir
pressure
is given
by
behavior
slope
(practical
the
P
001
=
P
i
[ -Ei
-70.6qp.B kh
(-
cpp.crW2)kt 0.00105
+
2s
solu~~n
(E
.2.31)..
the
reservOIr
dunng
the
occurrence
of
This
behaVIor early
expressl°.n
o.
a
transIent
boundary
usual
r;;;-= ~
approximations 1626 it~-
Pi
I
~x_~nation
slent
flow
~ottOm-hole illustrated product
~f
Eq.
pen~
o~n
ed
flow
penod
pnor
to
effects.
vs
schematically
kh
t-
of a pressur~
be =
I
] 3.23 + ...(5.2)
log
drawdown t .should
.on FIg.
obtained
~st,
0.87s
a pl~t
be linear.
5.2.
The
of
~f IS the
from ,...
hr
(5.3)
obtained,
The
rearrangement -log
~
+
Indicated
on
the
for
the
of Eq.
5.2.
3.23
]
cpp.Crw
IS also
from
line
formula
the
(5.4) on FIg.
schematIcally
5.2. on Fig.
exists
on
begun.
This
the
5.2
that
basic
a short
plot
period
period
loading. during
In
is usually
wells
which
without
oil stored
The ginning
end of
effects
of short
this
is easily
recognized
It
is evidenced
on
pressure
from data.
At
straight-line
this
time,
is
may
period usually of the basic
begin,
test
is
It is the
in the tubing string in effect and annulus un-
annulus
t curve.
the
duration.
packers,
in the
nonlinearity
after
of the transient period of the late transient period
the
of
immediately
result of unstable flow conditions the well during this period, skin
earlier
162.6qp.B m
-Pl
pressure once
straight
1 hour.
by
ed". The duration of this nized from the appearance
This
value
[ Pi
Pl hr value
Note
5.2~n~_c~!~~~n~~_~~-
pressu,re
can
The \
In.a
kt cp 2p.C~w.
1.15
been
the
m
IS alSO vafid
well
to
[log
is obtained
for
determined,
has
from
of
pressure-time
employed
be
curve
time
'"
As discussed in Chapter 2 following Eq. 2.31 and also at the end of that chapter, Eq. 5.1 can be simplified by the
factor
can
read
the
paper.
to that
factor
value
of
semilog
drawdown
at a flow
s =
We skin, the obtain s(qp./21Tkh), this equation to by the adding pressurethe drop pressure given drop by the in
fo~~~~
skin
the
in psi/cycle on
analogous
the
of
skin
(5.1)
ide~
slope
plotted
pressure
graph
]
kh
when
In a manner
units)
the
m is the
curve
buildups,
at a constant
an infinite
where
for
Conditions
During
I
49
tests
in the
Pressure
of
ANALYSIS
the
period
be "unload-
is easily plot.
flow, i.e., the bewhen boundary
on the pressure
~s
plot
by
section
transient
recog-
flow
vs log
a drop-off
established no
longer
vails. Physically, this means that the pressure drop to production has been felt at the drainage boundary
in from predue of
0 0 0
0
EARLY DEVIATION CAUSED BY WELLBORE EFFECTS
I/O Q.
-~
Q.
~SLOPE
BEGINNING AT END OF
=
162.6QILB kh
OF DEVIATIONPERIOD TRANSIENT AT
~
00 0
10 .FLOW
0.1
: '~e
..~~
Fig. 5.2 Schematic
10
TIME,t,hrs
transient --
drawdown
analysis
plot.
100
50
PRESSURE BUILDUP AND FLOW TESTS IN WELLS ~-, From Jahnke and Emde2 it can be verified greater than 100, -2B1 ~ 0.84,
the well and, as a result of depletion, the flow regime is in the transitioiiarperiOa:-pnor-rorea:cmiigse-mt-=Steaoy state. This time interval we refer to as the late transient pe~~~~__~i~~~ :he pre~sure behavior is neither se~isteady state nor tranSIent. A procedure for analYZIng the pressure behavior in this period which is analogous to the late-time pressure buildup analysis (extended Muskat 5.2
method)
Drawdown
Late Transient
Analysis
for reD
14.6819 -a12
=
-2
. reD
Thus,
Eq.
5.6
has been developed.
Pressure
that,
can be written
A Pw.f -P
for
Conditions
which
qp. = 084
in practical
-14.6819 kt
-e 21Tkh
units
~i'CT.2 '
is
To developthe analysistheory for pressuredraw..., conditions,l down at late transIent we recall from Chapter 2 the equation for pressure behavior at constant rate in a cylindrical bounded reservoir (Eq. 2.36),andadd the pressure drop due to the skin, S(qp./21Tkh) , to obtain
[
~21Tkh -~+ cf>p.cre2
Pi -= Pwf
In~rw
2-+ 4
~ -P A = 118.6 ~e q ,.J) kh
~
(14.6819) (0.000264) kt -~i'CT,2.
(5.7) Eq.
5.7
can also be wntten
log (PWf -p)
S
A
= log (118.~)
qIiB
-0.OO168~.
kt ."p.Cre (5.8) A
00
+2
~
Bn (an,reD) e -a.2tD.. ] ,
n= 1
From this equation we see that a plot of~g(p:; ==iJT VSt sho~e linear with slope magnitude fJ = 0.00168
where B n ( ~, r eD) =
an
/12 (a"reD) 2-[J 12-.-( anreD ) - I 1 2 ( an )]
~-and cf>p.cre2
,
intercept
b = 118.6!!
kliB h
.
will
be
A E
tDw
~Jt --:;
=
le.,f --,..
cf>p.crw2' and an is the nth root 11 (anreD) Y 1 (an) For in
production the
reservoir
--qt' P -;- Pi -.I. If
we
we
~ ~~
The
9 Ji'
1" ~
vided
of
constant
is
given
y 1 (anreD) = o.
assumed
rate,
the
straight
pressure
these
(P"'f A
A P values. line
by
-p)
vs
on
That
the
log
t
Usually
plot value
(P"'f
linear
pro-
it is not. This
must be made which
-~)
vs
yields
t plot
is
using
the
best
chosen
as
A
the correct P value. Once this problem has been settled, then kh can be determined from the intercept value by
,L 2 . 1T."Cr,re
combine
log
that a trial-and-error
(an)
average
of
the value of P is known.
means
-11
at
plot
118.6
equations
and
rearrange
them,
kh
=
liB
b~
.~.
(5.9)
obtain
Pwf
-;
=
~
The
[ -2Bn
;
21Tkh n=l
(an,reD) e -a,2tD"
],
(5.5)
r-;1 l!::,
where
( In
~ = p -~ .21Tkh
~ -2+ S rw 4
)
.Drawdown
that
the
parameter
t h at the change
P
-t with in P
is
a
constant
if
time is negligible
...A.. of Interest. PhysIcally, flow penod ing semi-steady state flow.
we
assume
during
the
P IS SImply P"'f dur-
When sufficient producing time elapses to reach the late tran.sient stage of.the drawdown, all terms except the first In the summation of Eq. 5.5 become negligibly small.
We
Pwf -~.=
can then write &
[-
to
he
a
stabilized
average
(drainage
volume)
(5.10)
.reservoIr pressure value.
of
from the slope of the. basic is given by
If
we
assume
P -A
that
this
after a to build value
is
an d ne glec t . 1t s c hange
during valuecanP be -P found is known. of this the fact test, the then skin the factor from Because ' S = 0.84
e -a,2tDw1
...(5.6)
volume
-n4 4 4:-qBl = 0.1115;IibeJ.
The pressure
2B1 (al,reD)
pore
tests are ~sually undertaken well has been shut in and the pressure allowed up
Note
contributory
the well can be determined plot. This value, in barrels,
[-;;-p -~
] -1n-;:;
.(5.11)
drop across the skin zone is given by ~p(skin)
The
r e +4 3
graphical
portion
= ~.. of this analysis
.(5.12) method,
i.e.,
PRESSURE DRAWDOWN ANALYSIS
51
the trial-and-error plot, slope and intercept determination, is shown schematically on Fig. 5.3. From the analysis of a pressure drawdown curve's late transient portion, it is possible to determine the kh product, skin factor and contributory pore volume. As is the case with the late transient analysis of pressurebuildups, this type of pressure drawdown analysis is postulated on a cylindrical reservoir shape. Deviations from this cylindrical shape or from a centrally positioned well location will affect all the calculated results. If agreement is obtained on kh and s values derived from the transient and late transient analyses, then the value found for the drainage volume should be fairly reliable. Some unpublished investigations have been made of the effect of drainage area shape and well location on the late transient results. These studies, based on theoretical drawdown behavior for various
time values for various reservoir shapes.!~~g;~ ~ou8!! 4.9. noting: the nme a~wbi~n~ttl~curv~become linear with the logarithm of time.) After semi-steady state flow is achieved, the effect of reservoir shapeis to alter the value of the constant term (3/4) in Eq. 5.13. This is discussedfurther in Section 10.6. Because of the mathematical form of Eq. 5.13, it is obvious that only in fortuitous circumstances (s and re known) is it possible to determine the kh product from semi-steadystate pressurebehavior. The one vitally important item that can be determined from semi-steady state data is the drainage volume contributing to the wells' production. From Eq. 5.13 we see that plotting Pwf vs t should yield a linear relationship with a slope q .BL= 1r
drraiffiige CtC~Ct ~hCtp~~as derived from the work of Matthews, !irons and Hazebroek,9 show that acceptable results are calculated if wells are located centrally or not far off-center in fairly regular drainage areas (circles, squares, hexagons, 2:1 rectangles, equilateral triangles, etc.). However, in the case of wells eccentrically 10-
From the slope the drainage volume can be calculated. In practical units the formula for the contributory pore space V p in reservoir barrels is B V p = 0.0418 f-, ...(5.14) LC
cated in oblong areas, the calculated results may be distorted. One should inspect the pressure drawdown plots (~~P!~ssure buildup plots,i!_,~Y~~~~ ~ey indicate;"lhe-"possffiilliY"o{ oddly shaped reser~ (See example buildup plots in Figs. 3.14 and 3.15). If odd reservoir shapes and/or well positions could possibly be inferred, the results from the late transient calculation should be viewed and used with caution. Further discussion of reconciliation of results from the different analysis theories is presented in the next section. When the end of the late transient period is reached, the period of semi-steadystate pressurebehavior begins, and for this regime a different analysis technique must
where the slope .BLis that of a linear plot of pressure (in psi) vs time (in hours). Drawdown tests run for the purpose of drainage volume determination from semi-steady state pressure data are known popularly as "reservoir limit tests". This type of test was first brought to the attention of petroleum engineers in a paper written by the late Park J. Jones.8In succeedingyears the reservoir limit test has become a popular tool for obtaining information on the size of newly discoverecAydrocarbon reservoirs. Its popularity is in no small measure due to the fact that the well can continue to produce income while the needed pressuresare being obtained. In the proper
be used. 5.3
Pressure Drawdown Analysis for Semi-Steady State Conditions
-,/'" b'IIS.6 ~q/,B
If a pressure drawdown test is run for a sufficient
o- o.,:~<:::o o ~ ~
period of time (tDe = 0.3), then semi-steady state is r~ached and the pressw'e behavior at the well is given by P. -Pw/ = 2;kh qp. ~ 2kt + In ~re -4 3 + s ..::
-
This
I ~
[
equation
]
can also be written
P. -Pw/ -qt -1r
+
qp. 2;kh
3 [1n-;:;;;-re + s.] 4
f
~
SLOPE. P.oooi6B-~ ..0/,'. 0'"00..0
:: :::::~~:~,~~-~TOOLARGE } CHOOSE
LARGEST
~
WHICH GIVES STRAIGHT LINE
(5.13)
The IDe = 0.3 value quoted above is for cylindrical reservoirs. For other reservoir shapes, especially those which are the least symmetrical, the time at which semisteady state flow ~~~es. The curvesm"M~ ~ Bfon~ ana~broek9 can be consulted to ascertain
FLOW TI .I, -Fig. 5.3 Schematiclate transientdrawdownanalysisplot.
52
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
circumstances and under controlled conditions,Is this type of test can be a valuable aid in evaluating exploitation economics. In a later section the application of these tests will be discussedin detail. 5 .xamp 4 E Ie 0f AppI.Ita t Ion " 0f Pressur e Drawdown Analysis Methods
sure vs flow time. It appears that the transient flow period lasted about 2 hours and that semi-steadystate flow began in the interval, 10 to 15 hours. Using the transient analysis equations on the straight-line portion of Fig. 5.4, we find the following (see Appendix D) " Transient:
To demonstrate the use of the techniques discussed in the previous section, a field example of the application of the three analysis techniques to an extended drawdown test has been included. The details of all the calculations for this example are contained in Appendix D. The well in this example was completed as a Muddy Sandstone oil discovery in the Denver basin.
kh = 767 md-ft,
s = -5.0 .. From applIcatIon of the late transient methods (see Fig. 5.6), we obtain the following (Appendix D). Late transient: kh = 371 md-ft, = -5 4 S .,
Before undertaking additional drilling, the engineers decided to run an extended pressure drawdown test, pnmanly to test the size of the 011accumulation. After a hydraulic fracture treatment upon completion, the well was flowed to clean up and recover load oil, and was then shut in and the pressure allowed to stabilize at the static value. Then, while continuously measuring the
VII = 146,000 reservoir bbl, 17 dr or,-., acres amage area. The reservoir limit analysis on the semi-steadystate data gave the following value (Appendix D). Reservoir limit: V = 149000 reservoir bbl II or ,:, 17 acres. '
bottom-hole pressure and production rate, the well was flowed at a rate of 800 STB/D for a period of 50 hours. Fig. 5.4 is a plot of flowing bottom-hole pressure vs log t, while Fig. 5.5 is a linear coordinate plot of pres-
On the basis of these results, further drilling was not attempted. The subsequentproduction performance of this well is shown on Fig. 5.7. The well recovered 31,000 STB of oil. Using a reservoir volume of 146,-
.
2000
1800
01
..
~
1600
!oj
~
::>
In In !oj
~
1400
!oj
..J
0 :I: I
2
0 ~ ~ 0
1200
m C>
z
i
1000
0
..J Lo.
800
60010
100
1000 TIME
IN MINUTES
Fig 5.4 Flowing pressurevs logarithm of flowing time, extendedpressuredrawdowntest, Denver Basin Muddy Sandstonewell.
10,000
PRESSURE
DRAWDOWN
ANALYSIS
53
000 bbl and a water saturation of 35 percent, this amounts to a recovery efficiency of about 30 percent. A recovery of this magnitude is quite reasonable for Denver Basin depletion-type reservoirs. The example demonstrates the economic value of a transient pressure analysis. For roughly $1,000, information was obtained which prevented an obvious loss of much greater magnitude. However, all the separate facets of the analyses are not in complete agreement. In particular, why is it that the kh and s values determined from the transient and late transient methods do not agree more closely? In this case the explanation lies in the fact that the well was fractured on completion. In fractured wells the kh and s values will depend on the flowin,g time range of the pressure data used in their calculation. At early time, flow into the well is the result of essentially linear flow into the fracture, and the pressure drop per unit of production is less than with radial flow. Thus, the calculated kh value will be too high. As flowing time increases, the more radial flow away from the fracture controls the pressure behavior and truer estimates of kh result. In both instances negative skin factors are obtained. This is discussedmore completely in Section 10.5. Because of this effect and the fact that the reservoir volumes determined from the late transient and reservoir limit analyses are in near-perfect agreement, we believe the
kh and s values calculated from the late transient method to be preferable. This example has served to illustrate the need for considering all available information when analyzing pressure data. In this case, the additional knowledge of pressure behavior in fractured wells gained from theoretical studies provided a basis for choosing between answers. There are cases in which a unique interpretation of reservoir characteristics may not be possible from pressure analyses and the available geological and petrophysical data. In such casesthe engineer must acknowledge the existence of more than one possible interpretation. If he must make recommendations concerning future exploitation of a reservoir, he should be especially aware of the economic implications of each alternative solution. ..." 5.5 Operational Considerations with Pressure Drawdown Tests The properly run pressure drawdown test can yield valuable information about the reservoir. This type of test is harder to run and control, however, becausethe well is flowing during the test. The analysis methods are based on the assumption of a constant flow rate from the well. If the well will not flow at constant rate, then the pressure drawdown behavior can sometimes be analyzed by recourse to the multiple-rate
2000
1800
".-0 ~
1600
1&1
~ ~ II) II) 1&1
f
1400
1&1 -J 0
i
~
1200
I0
m ~ z
i
0 -J
1000
I&.
800
600 0
600
1200
1800 TIME
2400
IN MINUTES
Fig. 5.5 Flowing pressurevs time, extendedpressuredrawdown test, Denver Basin Muddy Sandstonewell. :
3000
54
PRESSURE
test methods which are presented in the next chapter. In the case where the well surges or "heads" due to slug flow through the tubing string, the resulting pressure data generally will not be usable. It is advisable to have some idea of the flowing characteristics of a well before committing funds and equipment for a drawdown test. The flow rate of the well should be great enough to cause easily discernible pressurechanges on all phasesof the test.
BUILDUP
AND
FLOW TESTS
IN WELLS
If it is available, special test equipment with which the oil, gas and water produced during the test can be metered as a function of time is highly desirable. Pressure measuring equipment should be especially calibrated for the range of pressuresto be encountered on the test. If an internally recording pressure bomb is used, then an effort should be made to eliminate as much as possible the necessity for retrieving the bomb to rewind the clock. Especially in those cases in which
1000
b='320 ~"-
~
,,'-..,...
/
",.0~ 0.
0/~=1'300
-0 """
" 0'0, 100
.
0
'"
'
0\",
""""'-0 ~ = 1400
'"
\
~ -0 U)
\ \
a.. ~a.
1
\
~= slope = ~= \
,
-\
a.-
0.1'35
\ \
\ \\
10
A
p = 1460
\
\
\
\0 \ A p=1490
1
0
2
4
6
8
flowing
time -hours
Fig. 5.6 Late transient analysisplot, extendedpressuredrawdown test, Denver Basin Muddy Sandstonewell. ~
-
PRESSURE DRAWDOWN ANALYSIS
55
small changes in pressure are being observed, the disturbance in the tubing created by pulling and re-running a pressure bomb when combined with gauge hysteresis effects can render significant portions of the pressure data unusable. Surface recording bottom-hole pressure gauges8are very helpful in drawdown tesung. When properly run, the pressure drawdown test affords a method for establishing the formation permeability and skin effect which is equally as reliable as the pressure buildup procedure. From the late transient portion of a pressure buildup or drawdown, the contributory drainage volume of the well can be estimated.
The long-term pressure drawdown test offers the engineer an additional means for estimating reservoir size (reservoir limit test). The pressure buildup is operationally simpler than the drawdown test, however, because it requires no measurementof production rates during the test. We believe it absolutely necessaryto devote some discussion to precautions concerning reservoir limit tests. These tests are probably the easiestof the pressure analysis techniques to misapply and obtain erroneous results. Invariably, the question which is asked concerning reservoir limit test data is: "Did
300
15, 000
..J m m
...
-&..
0
~
m
u
-
-200 I&J
10
000 ,
0 -
~
c
U)
J
I&J
0
~
~
-
...I
U)
~ ~
100
5,000
~ "
,~ ',\" 5000 ';1
10,000 CUMU~:~:
IlL
20,000
25,000
Fig. 5.7 Productionperformance,DenverBasinMuddy Sandstonewell. -~--
---
30,000
8
...
56
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
the well really reach semi-steady state drawdown behavior?" The longer the test, the greater the certainty as to whether semi-steady state was reached. If the test data are misinterpreted and the slope of the pressure vs time plot is derived from transient or late transient pressure data, then the resulting calculated drainage volume will always be conservative. This is true because the slope of the pressure-time plot (dp/dt) decreases monotonically until semi-steady state flow is reached, and calculated reservoir size is inversely proportional to the slope value. Some simple results from theoretical fluid flow studies will help to answer whether the test time was sufficient. Transient flow endsll at a flow time of t ~
2 q"ucre (practical units), 0.00264 k
and flow at semi-steadystate begins9 at a time of t ~ 2 -cJ>JLcre -or three times the first value. (See also Odeh 0.00088 k and Nabor,12 who give approximately these same values.) Thus, if we can pick the end of the transient period from the pressurevs log t plot, then the semi-steady state beginning can be estimated at three times this value. This factor of three will increase as the drainage shape departs from circular and ~s the well location .is shifted from the center of the draInage area. In certaIn
viously, difficulties in maintaining proper test conditions and the cost of the test almost preclude reservoir limit testing in such cases. The pressure measurement costs will usually be $200 to $300 per day. In planning for such tests, the engineer should estimate the permeability and other parameters and make rough calculations of the time required to reach semisteady state for various drainage radii. In this way he can help decide on the practicality of attempting a reservoir limit test. Another problem may arise in economic handling of products produced from the well. In the case of oil this usually is not too much of a problem. However, with a gas well that has no pipeline connection, considerable volumes of gas may have to be produced and flared. This adds heavily to the cost of the test. In areas where production is prorated, permission must be obtained from the regulatory body to undertake any extended drawdown tests that would violate allowable restrictions. . ProVided t he p1anmng and economIC .. requirements indicate it to be feasible, a considerable amount of useful reservoir information can usually be obtained from a well executed reservoir limit test. Surface recording bottom-hole pressure gaugesprovide a ready check on pressure behavior as the test progresses. 5.6 Behavior in Non-Ideal Cases
.
10.3.
Section
In
eory
-
C
th
pra
d
.
UI
e-
1
SIng
fl
some
. 0
t
on
e
en
e
a
e
ca
presen
an
ave
ng
we
.
es
own
much
IS
factor
the
reservoIrs dIscussed
as
than
...or
heterogeneous
of
types larger
three
(
f d
ar
tubing or casing is proportional to the rate of change of the difference between the tubing- or casing-head pressuresand bottom-hole pressures.Thus, if one plots ~(BHP-THP) vs flow time, then stable conditions prevail in the well for times beyond the point at which ~(BHP-THP) becomes essentially a constant. Generally, it is possible to recognize this point on field data from the linearity of the p vs log t plot. Ramey5 has studied wellbore storage effects in pressure buildup and drawdown of oil and gas wells. Fortunately, the casing-tubing annulus usually is isolated by a packer in a gas well so that the only gas volume available is in the tubing string. Ramey found that the time required
raw
Extended drawdown tests have been valuable to many companies in making post-discovery decisions as to further developmentdrilling. As can be seenfrom the formula for the time at which semi-steady state 2 flow begins t ~ -~~ ), however, the time re0.00088 k quired to obtain valid reservoir limit test data is directly proportional to the compressibility and inversely proportional to the permeability. In the case of a lowpermeability gas reservoir, for example, the time required for semi-steady state flow to occur for a 640acre area may be in the order of several months. Ob-
..us
Another check on the reservoir limit information can be obtained from data derived from a pressure buildup test run immediately after the drawdown test. For a fluid of small and constant compressibility, the time required to reach semi-steadystate pressure decline is the same as the shut-in time required for the well to completely build up to the average reservoir pressure. Therefore, if a post-drawdown pressure buildup is run for a shut-in time at least as great as the length of the drawdown test, a check on whether semi-steady state was reached can be made. This is done by noting whether the buildup plot (PW8vs log [(t+ ~t) / ~t]) deviates from a straight line at large time and flattens out toward the averagepressure. Unless some flattening occurs, the flow test of equal time was too short.
.
Th d
. .
h t d th t ti d 11d tt ti ti I ca aspect s 0f the meth0ds. I n this secti on We shall point out the modifications to the basic theory which must be effected to handle gas flow, multiphase flow, etc. If the well has been closed in prior to a drawdown test, there is some fluid stored in the wellbore which unloads when the well is opened. Van Everdingen and HursPo originally studied this effect in transient production analysis. Until mass equilibrium is restored in the wellbore, the surface-measuredproduction rate will be the sum of the bottom-hole inflow and wellbore storage depletion rates. On short drawdown tests in wells with extended periods of wellbore depletion, the calculated results may be in error becauseof the use of erroneous rates. Stegemeierand Matthews4 show that the net rate of fluid accumulation or depletion into the f
PRESSURE DRAWDOWN ANALYSIS
effects
are
likely
less
than
1
of
wellbore
storage
In
case
the
li mi
reservoir point of
day.
will
case tests
must
substitute
the
and
system
fluid replace of the
total
pressures
th
of the
single-fluid
In
tests
in gas wells
formulas is for two P hase
analysIs
as dis-
drawdown
by using a modification that for pressure buildup
usually
of the analysis
can
,ugBg
product
f
mean
0
sures,
and
should
..
h
t e static the
reservoir
be
evaluated
reservoir
an
CtJ.tg product
pressure.6
If
the
the
of
stant
for
studies
all
to
the
. owmg be
pressure
is not greater CtJ.tg products the
d fl
should
being analyzed the J.tgBg and mid-point
theory analoin gas wells.
range
rigorously
the
h
we
ore
pres-
on
static
of
the
data
than, say, 400 to 500 can both be evaluated and
considered
to
there
substantiate
1
.
ate
1
transient
our
.
been
d
0
we
manner for gas wells. that pressure changes
say,
psi
to
500
are
encountered
h
of
during
wells,
the
should
of the
be
taken
account.
1
for
properties plotting
including
in
gas
gas propthe
In
transient
pressures
directly,
drawdown they
of
analysIs. first
the
Instead
transform
larg-
the
gas of pres-
sures into a pseudo-pressure variable by means of an integral involving the pressures and associated gas properti ' e Thi . ful d f .of s.
s
IS
a
use
proce
ure
or
gas
a test
to
at stat-
to
cases, next
if the
possible
difficult
the
of Pressure
Tech.
(Dec.,
main-
the multichapter
are
Build-Up
An-
1966)
F.: Tables of New York.
1624-1636.
Functions,
Fourth
213, 44-50.
Pet.
Tech.
(Feb.,
J.
Pet.
Tech.
R. and
(Jan.,
Ramey,
8. Kolb..
~.
AIME
H.:
"Two
(1960)
223-233.
G ..an E d B rus Gas Well Per-
1966)
99-108. "Application
of
Testing and Deliver(May, 1966) 637-642.
~ottom-Hole
Automatic 219,
1965)
H. J., Jr.:
Re,a.l Gas Flo~ T,~eory to Well ability Forecasting, J. Pet. Tech.
M
h
.att
C
ews,
Pressur.e
Surface
Instruments
Recordmg",
Trans.,
346-349.
for
Bounded 191. 10.
,:an
reservoirs
S
B
rons,
Reservoir",
in Reservoirs", C.
F.
Laplace
C.,
H
b
aze
of
Trans.,
A.
the
d
.an
Determination
Everdingen,
11. Miller,
F
..,
Method
than,
tests
pressure-dependence
is In these of
J. Pet.
J.
formance",
b
est pressure changes are encountered during transient tests and the smallest for the semi-steady state tests AI-Hussainy and Ramey7 have described a unique pro~ cedure
frequently
"Extensions
Wells",
7. AI-Hussainy,
no
more
Usually,
to begin
6 R 11 D G G d . P erry, .usse, .., 00 nc, h J ..,H kotter, J. F.: "Methods for Predicting
a so I e
pressure-dependent
into
it is not
H. J., !r.: "Non-Darcy Flow and Wellbore Effects m Pressure Build-Up and Drawdown
of
vanation
time
methods
G.:
(1958)
5. Ramey, Storage
..tlon
erties
cases
rate.
volume. tests
believe
Id
s ou
drainage
of flowing
e.
D.
AIME
psi, at
be con-
have
view,
th
ana YSlS me
modified in this In the event 400
it
Methods",
Gas
anthmetic
lIb
based
range
.Provldmg Although
analysIs.
at
should
3. Jones, P. an~ M.ca;hee, E,',: "a;ulf Coast Wildcat Verities Reservoir Limit Test, 011 and Gas J. (June 18, 1956) 184. 4 St . G L d M tth C S ." Anomalous Pressure Build-Up Behavior", Trans.,
9
h
t at t e
Also,
2. Jahnke, E. and Emde, Ed., Dover Publications,
be
..of The
sufficient
analysIs bl
1. Russell, alysis
the total of these
many
pro~ucing
app Ica
work
References
reservoir
apter.
Pressure handled gous to
buildup
test r
In
and
simplest
for
a constant
0 ten
compressibil-
mobility with The calculation
pressure
conditions.
Their
drawdown tests are an operasound means f or evauating 1 .
are the
are right.
in a well
drop.
parameters
tests
.egemeler,
of
pressure details.
reservoir
pIe-rate f
u
of We
the
oil
3
case
bble
e
the equations for analysis must be modified slightly.
the single-fluid flowing fluids.
Ch
the .
In d
flow
b
ow
gas
compressibility
close
tain
and
for
critical
ic (including
of
flowing to for
conditions
tests,
1 e
flow
a large
Drawdown
drawdown
tests
quantities and their use in the analysis p letel y analo gous to the modifications com
cusse
in
not be important.
b
a
string wellbore
longer-duration
two-phase
rock
flow
for
t run
with
be referred
In summary, pressure tiona 11y an d th eoretically . .
cases
drawdown
) s
reservoir,
will result. In this pressure drawdown
the
some
For
pressure
test
of
in
by
)
umts.
to be important
probably of
t
the
ity and mobility
length that
.
at
wells
f
tests
is the
gas
is given
0
Lt
in
y
storage
.
J.tcgrw2Lt ( kh
concluded
out effect,
d
formula
Ramey
die skin
St
this
to no
A
) 4 785 -practic,
ys
having
...u
In feet.
effects
and
ews,
(da
t
storage
packers,
a
wellbore
without
..an
for
57
and
AIME Hur~t,
Trans.,
AIME A.
B.
W.: to
(1949) and
P
"
.:
Pressure
(1954)
Transformation
Dyes,
k
roe,
Average
201,
"The
A
in
a
182-
Applica-
Flow
Problems
186, 305-324.
Hutchinson,
C.
A.,
Jr.: "Estimation of Permeability and Reservoir Pressure from Bottom Hole Pressure Build-up Characteristics", Trans., AIME (1950) 189, 91-104. 12. Ode~,
A.
duction acteristics
.S. and History From
Nabor,
~.
"!'.:
"The
Effec.t
on Determmatlon of Formation Flow Tests", J. Pet. Tech. (Oct.,
of
ProChar1966)
1343-1350.
13. Root, P. J., Warren, J. E. and Hartsock, J. H.: "Implications of Transient Flow Theory: The Estimation Gas 141.
Reserves",
Drill.
and
Prod.
Prac.,
API
(1965)
Chapter 6
Multiple-Rate Flow Test Analysis
The methods for analyzing flowing well behavior discussedthus far have been based on the assumption of a constant producing rate. In some cases, however, the rate will vary with time. In other cases, regulatory bodies require flow tests...0 made at a series of different ..pressure rates. Gas-well tests fall Into thIScategory. ThIS chapter IS devoted
to development
of pressure
analysIs
methods
..W for handlIng both of these cases.The methods of thIS ...era chapter are partIcularly useful In the case of a floWIng well which produces at constant rate where it is not operationally or economically feasible to shut in the well for a pressure buildup or to allow the pressure to equalize prior to a pressure drawdown test. In these
Chapter 3) can be used to forecast future deliverability. 6.1
General Equations for Analysis of Flowing Well Tests with Variable Rate T deve1op the genera1 equatIons, we dIVIde the
.
d raw d own
.
.
based
on
case
wIth
any
measurements
...esire pressure
obtaIned
...rate.
analysIs
whIle
the
well
on the obtaining of made in the discussion
good of
measurements which were pressure drawdown analysis
are equally valid here. It should be realized, of course, that the measurement of production rates is more critical in the case of multiple-rate tests than with ordinary pressure drawdown at constant rate. We shall discuss this point further in the presentation of the various multiple-rate analysis methods. As will be seen, the multiple-rate methods to be presented are applicable to gas wells as well as oil wells. The objective of these methods is to determine permeability, skin effect and reservoir pressure. Determination of gas well deliverability is not a primary objective. However, once the basic parameters of kh and skin factor are determined, the methods of Swift and Kiel (Ref. 22, Chapter 3) 0; Russel[et al. (Ref. 27,
0f
.. .. ..
.
intervals
f
rom
q = q"
,
t"-l ~ t .
may
as
small
be
d t 0 app I y t 0 th e case
Th
IS
producIng, care must be exercIsed to obtaIn good production rate and pressure measurements. The remarks
eac h
q = ql , 0 ~ t ~ t1 , =., t < t< t q q- , 1 --2 , q = qs , t2 ~ t ~ t3 ,
d
method
.
.
for estimating the kh product, the skin factor and the reservoir pressure. ...The IS the
..
t erva 1s d unng .
d t. t b h h th IC e pro uc Ion ra e can e consldered const ant. Th t ti hd I e- me sc e u e IS as f 011 ows:
instances dependable transient pressure data can often be obtained by measuring the pressure responsecaused by a change in flow rate. Analysis of these data by the interpretation methods of this chapter affords a means
As
t es t .mom t .
E
p,
.
and
as
numerous
as
0 f a con ti. nuous I y c h a n gt. ng
. ..
.
d d th ' . ti I t d e pressure rop unng e mi a Ime peno IS,
q. .-=
5 2 .,
P"'f
162.6
kh
qlJL B
[I
og
t +
-] s,
(61) .
where s = log ---~~ -3.23 + O.87s .(6.2) cf>p.cr", Applying the principle of superposition (as in Section 2.8), we find the pressure drop during the second time period to be 162.6 q1/A. B P. -P"'f = kh [log t + s] +
162.6(q2 -ql)JL B [log (t -t1) kh
+ s] . (6.3)
For the third time period,
MULTIPLE-RATE FLOW TEST ANALYSIS
59
_162.6 q1JA.B -162.6 P. -Poo! -kh [log t+s] + [log (t-t1)
-162.6 + s] +
[log (t-t2)
-rate + s] ..(6.4)
(q2 -q1) JLB kh (q -q) B ~h 2/L
down tests which are begun at stabilized pressure conditions in the reservoir and in which the production is non-constant. The basic plot requires that the initial pressure value Pi be known. If it is not known,
Thus, dunng tIme penod n the pressuredrop IS gIvenby .-= p,
162.6
q,/L
P,C!
B
[log
t +
s]
162.6(qa-q2)JL kh
...+
-on [log
kh
+
(t-t1)
+
B [log (t -t2)
This equation can also be written 162.6/LB Pi -Poo! = kh[q110g t + (q2 -q1)
Odeh
-rate + s] +
by
trial
and
be
that
which
error.
That
yields
is,
the
the
best
correct straight
Pi line
and
Jlmes!
presented
the
.. foregoIng
vana bl e-
~nalysis technique a?d also some examples of applYIng the method to ollwell and gas-well pressure
drawdowns. The basic data for the oilwell example + s] .presented in their paper are depicted graphically on Fig. 6.1. Fig. 6.2 shows the variable-rate analysis method plot for these data. Further details of the example calculation can be found in the Odeh and Jones paper.
log (t -t1)
+ + ( qa-q) + 2 log (t-t.) -I ...(q" -q"-1) og (t -t"-1)] + 162.6q"p.B -
kh
Pi should
..
the basic plot.
s]
162.6(qn -qn-1)/L B[log (t -t"-1) kh
In a laterofsection of this chapter will present the application a modification of thewe variable-rate analysis theory to the determination of the kh product and
s ,
500
or P. -Poo! -162.6/LB :: [ Aqj -"" -og I q" kh j=l q"
]
( t -t j-1)
]
rlog~- k 162.6JLB kh
+
...
value
162.6(q2 -q1)JL B "
then the utility of this type of test and analysis becomes h t d somew a ImtffiShed .ISIt pOSSI ble, 0f course, t 0 determine
kh
+
The foregoing equations and plotting technique comprise the basic method for analyzing pressure draw-
3.23 + 0.87s ,
~ m
400
> a: l1J ~
300
~
(6.5) where
a:
A.qj = qj -qj-1
.;
A.q1= q1
20
to = O. From Eq, 6,5 we see during the nth period of constant rate, i,e., t"-1 $: t, if we plot p, -P
!
to
vs
q..
n -logA.qj
~ j=1
(I
-Ij-J
,
Pi = 3000
]
p.cr 00
kh = .!~~~,. and
~r
Co
.:. ~ 15
..(6.6)
m
[ bl. k -,-log-:;:--2+ m
200 "OJ
From these values we can determine the kh product and skin factor from
S -1.151
psi
q..
we should obtain a straight line of slopem'=-~~~~and intercept bl= 1626 k'h /LB [log~- k 3.23 + 0.87s
,!
100 2500
] 3.23 ...(6.7)
100
DATA
V-
MISSING
'l'JLCroo
Damage ratio or flow efficiency can be calculated for flow tests exactly as for pressure buildups (see Chap3).
0
TIME,MINUTES Fig. 6,1 Multiple-ratetestpressureand productiondata. (Data from Odehand Jones.')
-
200
60
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
skin effect from gas-well open-flow potential test data. The developmentof an analysis theory has led to the use of multiple-rate flow tests as a means of generating transient pressure data. One of the most popular of these techniques is the two-rate flow test described in the section which follows. v .and --ssuInlng 6.2 Two-Rate Flow Test AnalysIs Method The two-rate flow test, developed by Russell,2offers a means for determining the kh product, skin effect and extrapolated pressure which overcomes many of the wellbore problems experienced with pressure buildups, and also eliminates the need for shutting in the well. The required pressure data are obtained by observation of the transient bottom-hole pressure behavior V after the stabilized production rate of the well is changed
is reduced. The period of transient pressure change caused by the change in rate (labeled A on Fig. 6.3) is followed by a period during which boundary and interference effects are felt (B), and finally the well returns to a stabilized pressure decline (C). It is necessaryto measure only the pressuresjust prior to the rate change during a portion of the transient responseinterval. A that the ra te 0f pro duct Ion q2 becomes operative immediately after the change in rate, we may combine Eq. 5.2 and the principle of superposition to yield the following expression for the flowing bottemhole pressure after the rate change.
.
.
p"" = Pi -kh
to another, higher or lower, rate. In preparation for a two-rate flow "test, a well is usually stabilized for several days at a constant producing rate, and both oil and gas production rates are measured on a daily basis. Three to four hours prior to the rate change, the bottom-hole pressure bomb is lowered into the well, and pressure measurement is
[log~-
3.23 + O.87s
[log~+~t'
~log ql
162.6 q2JLB
-162.6
kh
qIPB
k p.C'"
]
~t' ] ' (6.8)
where . ql = rate pnor to the rate change,
begun. This is necessaryin order to obtain a dependable value for the flowing pressure prior to the test. The producing rate is then changed by adjustment of the
q2 = rate after the rate change, ... t = producIng time pnor to the rate change,
choke at the wellhead, and, after a usually short period of transition, the rate stabilizes at its new value. Meanwhile, the transient pressure response caused by the. rate change is being measured. The flow test pressure and production rate behavior with time are shown schematically on Fig. 6.3 for the case in which the rate
~t' = producing time measured from the instant of rate change. It IS assumed that the well has produced at constant rate ql for time t prior to the test. (This is the same assumptionmade in pressure buildup analysis theory.) We see from Eq. 6.8 that a plot of p"" vs
/
6.0
-/
/0 /
/
[IOg¥+~log~t']
0
0
~INITIAL
~
0
..I-
I.
~
-I~~
~
U)
:
@
PORTION OF PRESSURE HISTORY USED IN FLOW TEST ANALYSIS
@
BOUNDARY
-'
Co
I
PREssuRE
@II'\
... ~
'in
Co
@
:0 '"
'-CD
-0
will yield a straight line.
0
.0 50
"f
/
2
lc
PAST PRESSuRE INOT NEEDED
~
0"
ANALYSISI
HISTORY -, FOR
i
I
~
WELl. RETURNS TO STABILIZED
z:
~
PRESSURE
'.
1
AND INTERFERENCE
EFFECTS ARE FELT
DECLINE
0;-:-6"-TIME-
'.'j£~1 ,~)t
... I-
SHORT TIME LAGUSUALLYREOUIRED BEFORE NEW STABLE RATE IS REACHED
:
~
4.
q,
-
:0 0
3.0
..6.0
7.0
t
n qj-qj-l
I
J=I
)
~
1- ~
8.0
0
f
q.
I I
, I
I
log (t-tj-l)
1--1
qn
.1-
6t'-TIME-
Fig. 6.2 Multiple-ratetestbasicplot. (Data from Odeh and Jones.') 'I
-I
IU
~l.J)
Fig. 6.3 Schematicplot of productionrate and bottom-hole pressureperformancefor two-rate flow test (q, < qJ.
MULTIPLE-RATE FLOW TEST ANALYSIS
61
From the slope m (in psi/cycle) of this plot the kh product can be determined by y -162.6 qlJLB V I__~ -m'. (6.9)
cussion of this example and the details of the calculations employed in the analysis can be found in Appendix E. 6.3 Two-Rate Flow Test Analysis
...In In a manner sImIlar to that used to denve the pressure buildup theory it can be established that the skin factor is given by
Non.ldeal Cases ... As IS the case WIth pressure buIldup and pressure drawdown analysis theories, the two-rate flow test analysis theory is based on flow of fluid of small and constant compressibility. If the pressures on a tworate flow test are below the bubble point, it is necessary to substitute the total mobility and compressibility of the system into the analysis formulas, as in Example
3 = 1.151
I
[(
ql
ql -q2
)(
PI
) -log~
hr -PID
m
f/llJ-CrtD2 (6.10)
3.23]
...+ "
where p~~win2 pressure at the time of the rate change an~ is the pressure at 1 hour after the rate change on the straight-line section of the flow test Plot. Damage
ratio
or
flow
efficiency
can
be
calculated
2A, Appendix B.
for
q2 < q I
I.J
-
a:
0000
two-rate flow tests exactly as for pressurebuildups (see Chapter 3).
~ ~
cfI\
The value of Pi (equivalent to p. in pressurebuildup: theory; see Chapter 3) is given by
RETURN TO 0' SEMI STEADY:r STATEFLOW
/ EARLY DEVIATION FROM
I
[ P.
=
PtD
+
m
kt log
] -3.23
+
2 ~
0.873
f/lp.crtD2
LINEARITY DURING RESTABILIZATION
b
(6.11)
CD ~
The pressure drop across the skin zone is b" aiven by ~p (skin) = 0.87 ms (at rate ql) or q ~p (skin) = 0.87 ~ ms (at rate q2).
0
~ ~ ~
4--INCREASINGFLOW TIME
( ) ql
The interpretation
i ,
'
L-
1+61' q + -!61 ql
log ~
theory is based on the infinite
homogeneous reservoir.
3250
.0
caseof a well producIng from a bounded homogeneous dramage voIume .IS shown on P.Ig. 6 ..n 4 I t his case,I.e., q2 < ql, the effect of a boundary is to cause the points to bend over and deviate from the straight line. This d b .k eVlation ecomes progressIvely greater as the well fina11 y reaches a semI-steady state pressure decline at rate q2. PIg. 6.sows 5 h
.
.0
.
.
a typIcal fi eld example of a two-rate
flow test from the paper by RUsseiI.2This flow test was run .m a Iow-permeab1li ty li mestone reservoIr m t he
.
..
Permian Basin region of West Texas. A complete dis-
ql * 107STB/D P. * 3118 pliO q2*46STB/D h*59fl c,*9.32.'0-5pI'-1 rw*0.2fl IJ.*0.6Cp B*I.5 ~*0.06 Np*26.400 STB 1*~ '24* 5922hr 107 BASIC DATA-WELL A
324
[
PtD!vs log t + ~t'+ !l!.log ~t' ] ~t' ql will be linear for a period of time after the rate change and will then start to deviate from linearity as boundary and (or) interference effects are reflected at the well. The times at which such effects will be felt will be of the same magnitude as would be the case if a conventional buildup were being run in the well. The general appearanceof a two-rate flow test curve in the
..
log 61'
Fig. 6.4 Appearance of two-rate flow test curve in bounded
reservoir solution of the radial flow equation for a slightly compressible fluid. In practice, however, the method will generally be applied to wells which produce from bounded drainage volumes. It may be expected h th h ten at t e plot of
.
RATE
323 0-
.~ 3220 ,.j ~ 3210 ~ ~ 3200 ~ i 3190 :: m
SLOPE *90pII
31BO
317
RESUL TS =30md
316
1*-3.6 . P* *3548psI9
3.0 3.
3150
...3.5
3.6
109~'
.38
+ .9..z. 10961'
61
ql
Fig. 6.5 Two-rate flow test plot, Permian Basinwell.
,..,,-
,£"c~
3.9
62
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
For the case of a gas well, the technique of employing "average" gas properties as outlined in the discussionsof pressure buildup and pressure drawdown analysis is applicable for two-rate flow tests. If the pressure range encountered on the test is quite great then the methods suggestedby Al-Hussainy and RameyS can be applied to account for pressure dependenceof the gas properties. The effects of partial well penetration, perforations, etc., are quantitatively the same on two-rate flow tests
in a well. Although the last few points appear to lie on a straight line, in reality the true straight-line section has not yet been reached. In this case it was necessary to extend the shut-in period of the well to 6 or 7 days to overcome compression and afterproduction effects and obtain a straight-line section on the pressure buildup plot. The plot of a two-rate flow test from this same well is shown on Fig. 6.7. The flow test of 22 hours' duration enabled g?od estimate~of .kh, sand p* without loss of production from closing In the well.
as on pressure buildups. Another point pertinent to practical use of two-rate flow tests concerns the assumption of instantaneous change in flow rate in developmentof the interpretation theory. Such an instantaneous change in flow rate is never fulfilled in practice because the adjustment in bottom-hole inflow rate results from a change in choke size at the wellhead. Thus, the flow test performance of a well after a change in choke si~e is directly related to the vertical lift performance characteristics of the well. Field experiencewith the flow test method indicates that rapid stabilization of flow conditions usually occurs within the flow string and, therefore, surface producing rate measurements are not greatly distorted. Rapid restabilization appears to be directly related to the fact that the flow string is continuously resupplied with mass during a two-rate flow test. In general it has been found from field tests that the
Fig. 6.8 shows a humping buildup curve obtained from a Wilcox Sand gas-condensateproducer in South Texas. The two-rate flow test run in this well is shown on Fig. 6.9. This is a very striking example of the utility of two-rate flow tests.
restabilization period is shorter in the case of a rate reduction than in the case of a rate increase. The reason for this seemsto be as follows. In the case of a decrease in rate, an additional amount of fluid is needed to establish new flow equilibrium in the flow string. This can be provided quickly by the entering fluid. In the case of an increase in rate, there must be a net decrease in the mass content of the flow string for stabilization. This decreaseoccurs more slowly because fluid is now entering the flow string from the formation at a higher rate than before. The method of Stegemeier and Matthews4 can be used to detect when flow through the tubing has stabilized. This technique was discussedin Section 5.6. 6.4
Elimination of Wellbore Effects with Two-Rate Flow Tests Two-rate flow tests have found their primary area of application in wells in which wellbore effects obviate the pressure analysis technique. Two principal types of conditions are involved. One is the case in which compression of gas in the wellbore and a long, low-rate afterproduction period combine to render the normal pressure buildup plot non-linear. The other is the instance in which phase redistribution in the tubing string subsequent to shut-in causes "humping" buildup behaVlor. ..30
Some
examples
from
Ref.
2
are
Included
to
illustrate graphically the cases referred to above. Fig. 6.6 shows a 72-hour pressure buildup obtained
As a final note in our discussion, we would like to emphasizethat in planning and executing two-rate flow tests, one needs to have a general idea of the flow characteristics of the well. If field personnel are not familiar with the behavior of the well, it is advisable to observethe flowing behavior of the well at two or three different flow rates to obtain a general impression of its performance characteristics. By obtaining such observations in advance, one is able to make a better choice of the flow rates to be used during the flow test. A basic requirement of the two-rate procedure is that the well flow without surging or heading at each rate. 6.5
.. Transient AnalysIs of Gas-Well Multi-Point Open-Flow Potential Tests
An important and plentiful source of multiple-rate transient pressure data, especially in low-permeability reservoirs, is the flow-after-flow type of open-flow potential test run in gas wells. The general multiple-rate analysisprocedure set forth by Odeh and Jonesl can be applied to determine the kh product and skin factor from such data. 4800 440 ~ 4000 ~ a 3600 g: ~0 3200 T ~ 2800 ~ 2
72-hr PRESSURE BUILDUP Np=4145 STB
:J
.J:f Q-"o.P' -~
2000 ,.000
10 000
1+61
At Fig. 6.6 Pressurebuildupcurve,problemwell.
10
MULTIPLE.RATE
FLOW TEST ANALYSIS
3600 ~t'=22hr ~ ~ Q Q 3500\ .~ ~ w
63
°'b
~
'
(/) ~ 3400-.0 U! SLOPE =670 pslg ~ "7
At'=5.5hr 00 0 ° °
~ BASICDATA 0' 6 UI3300-I qo.=70STB/D P-0'0.34cp q =40STB/D t=8151hr h02=8 ft Ct=2.77 x10-5psi-I
3200
3.3
In several areas of the United States and Canada, the stat.e regulat?ry bodies that control oil. and gas production requIre that open-flow potential tests (OFPT) be obtained in gas wells. Results of these tests are often used in conjunction with other parameters to determine the allowable flow rate of a well. The OFPT consists of a series (commonly from one to four) of measurementsof flowing bottom-hole pressures made at various flow rates. Generally, the well is allowed to flow several hours at each rate and then the pressure is measured. In the flow-after-flow type of OFPT, the pressures are measured at the end of a flow period at a given rate, after which the rate is changed immediately to a new value without closing in the well. The pressure-rate data from an OFPT are usually analyzed by the familiar steady-state
RESULTS k .1.1md s =-4.84 p*. 5770psig
r w2.0.13 f t 2 ~ '0.12
(~) 29.8 mdIcp II =t320~ s.
Bo' 1.~2 I
:w .3186PP:. Ihr
flow method of Rawlins and Schellhardt5to determine the open-flow potential of the well, i.e., the theoretical rate at which the well would flow if the sand-facepres. sure were reduced to atmosphenc. In permeable reser-
tt21 hr 0 0
voirs each pressure measurement usually represents essentially semi-steady state flow conditions at the re-
spective rates; however, in tight reservoirs the flowing pressures measured are usually still in the transient
I
3.4
3.~ 3.6 3.7 3.8 ~9 4.0 1°9~ + ~ 1°9 At' Fig. 6.7 Two-rate flow test plot, ~rOducingbelow bubblepoint.
stage. The idealized pressure-rate-time behavior during an OFPT of a new gas well in a tight reservoir is depicted on Fig. 6.10. The "tight reservoir" restriction is in-
3600
~ 3500 '" 0-
w
FLOWINGPRESSURE = 3255psig
a: :> In In
U! a: a.
U! .J 0 :I:
3400
I
~ 0
ff0 UI
/
3300
I
.0
6t, MINUTES Fig. 6.8 Pressurebuildup curve,Wilcox well.
64
PRESSURE BUILDUP AND FLOW TESTS IN WELLS 3200
.,.-6 t/: 22 hr 3190 SLOPE: 35 psiQ
.~ 3180
RESULTS
U)
Co
~
w
a: =>
~ 3170
k
:
5.7
s
:
5.47
md
in
p* : 3560 psig
w
a: Q.
BASIC DATA-WELL C
w
0
6 3160
ql : 8781 Mcf/O
BQ: 0.0056
:I:
q2 : 6002 Mc f /0
Ct :2.24xI0
Np : 32,254 MMcf
fLQ : 0.02 cp
h
p : 3084psiQ w Plhr:3180pSiQV
I
~ 0 I-
b m 3150
: 142 ft
rw: 0.25ft ~ : 0.15
-4
psi-I
Sw : 03 t : 88,157 hr
314 ",
1
t6t: 3130 4.5
...4.9
5.0
I
3 mln 4
5.5
,
, .I.
t +6t 6t'
log
+
q2 -q;-
' log
6t
Fig. 6.9 Two-rate flow test plot, Wilco-x well.
I
serted to insure that the method will be applied only in those cases in which the measured pressures are within the transient portion of the pressure history. In Fig. 6.1 0 a four-point OFPT is shown in which the rates increase. The analysis procedure is independent of whether the rates increase or decrease during the test.
~ ~ ~ ~
By modifying the general equations (Eqs. 6.1 through 6.7) presented earlier in the chapter for flow of gas, we find that at point n(n = 1, 2, ...) of an OFPT the following expression can be written:
~ i 3 II.. 0
p.
0 Pi -PtDf..=28,958IJ.,Bg q.. k,h
[10g-.!~-
c!>IJ.,clrtDl
I
I
I
tl
t2
t3
] 3.23 + O.87S
p wf4 1 t4
TIME w
+ 28,958IJ.,Bg ~ k,h
( q/
i=1
I-
-q/-l
)
log
(t..
-t/-1)
,
q"
(6.12)
I
:
:
Z 0
I I
t
q2
:
=>
where qo and to are identically zero. The factor 28,958 is used in this equation rather than 162.6, because throughout Section 6.5, q is expressed in units of Mcf per day.
g ql ~ 0 0
:.: tl
t2
:
: :
t3
t4
TIME
Thus
,
a plot of
Pi -PtDf" .q"
vs
Fig. 6.10. Idealized pressure-fate-time. relationship for fourpomt gas-well open-flow potential test (OFPl).
MULTIPLE.RATE
FLOW TEST ANALYSIS
) IOg (I"
.~ (~ 1=1
q"
65
-li-1)
to yield a linear relationship with slope m' = ~. 0'"
., , -28,958 should be linear with slope m -koh
/LgBg. andmter-
The .kgh product.can be determined fro~ the slope m', and If slope and mtercept values from this plot are used
cept
with Eq. 6.7, the skin factor can be calculated. 3.23 + 0.87s 1 .sho~n
b' = 28,958 /LgBg[log--~koh
~/Lgctrw2
This procedure is depicted schematically on Fig. 6.11. The kh product and skin factor are determined from kgh = 28,958 /LoBg , m' and [ b' k s = 1.151 -, -log ~ g 2 + 3.23 m 'j'/LgCtrw
.
Th e gas OFPT
th e
ana thod
d
properties I . tli
YSiS
an th me
d
.
0 th
ate h
s
ld
di
pressure. However, OFPT pressure measurements are (6.14) frequently made at the wellhead with a dead-weight d .tester. In this case one should exercise care in con-
parameters b I
ou
e .
eva
d uate
f
-4.7 determined from a subsequent pressure buildup test. The use of this technique depends heavily on the accuracy of the pressure data. If possible, it is desirable to obtain direct measurements of flowing bottom-hole
]
d
I re
d
(6.13)
use b
10 h y
t
... vertmg
these
pressures
back
and
are
to
sand-face
e
II
me ou ne m e SCUSSion0 gas-we pressure B d bUl1dup and d raw d own ana IYSiS. Th e JJ.g g pro uct should be evaluated at the mean between the static reservoir and flowing wellbore pressures, and the CI,url product should be based on static reservoir pressure. If the pressure range on the OFPT is only 400 to 500 psi, all gas properties can be evaluated at the mid-point
.
.
A field example of the application of this method is on Fig. 6.12: The pressure data are from OFPT obtamed at completion on a Morrow-Chester sandstone well in the Anadarko basin of Oklahoma. The basic data and calculations lor this analysis are presented in Appendix E. The calculated results are kg = 3.5 md and s = -4.7, as compared with k" = 3.8 md and s =
conditions.
.' If
condensate
water
present
m
the
flow
string,
... this converSion can be difficult. .' . If the pressure is not fully built up to the static reservoir pressure during the shut-in period prior to an OF~T, allowanc~ mus~ be made for the effects of the preVious produ~tion. history of the. well upon the ~esuIts. The modifications to the basic procedure which
of this range and considered to be constant for all the analysis. If the total pressure range becomes greater,
0.017
Eq. 6.12 can be divided by JJ.gBg so that
0.016
P. -Pw/" q" /Lg"Bg"
0.015
may be plotted against
~
( qi
j= 1
0.014
) log (I"
-qi-1
-li-1)
,
0.013 u ~
q"
m'=0.02904 0.012
In
a.
0
-0.011 .E ~~ cf 0.010
-::::u
a
n=I,2,'.'
~ 0 .-SLOPE "' ':"
'
=m =
k g = 3 . 5 md
28958 8
0.009
kh g
0.008
,
p.g g
RESULTS kgh =140md.fl 5 =-4.7
c
ic~ 'i'" ii: 0"
~
c., 28 958p. B INTERCEPT =b'=~~'~~-fg -g log g
k9 2 -3.23 +0.875 V ~Ctp.grw
0.007, (.., / 0
00
n
.L
)=1
( qj-qj-l ) qn
b =0.00625
0.006 0.1 n
log (In-lj-,)
Fig. 6.11 Illustration of type of plot used to determine k.h and s from OFPT data.
.}:;
jcl
(
0.2 qj-qj-l
q
n
0.3
) log (tn-lj-l)
Fig. 6.12 Calculation of k.h and s from OFPT data, Anadarko Basin well.
0.4
66
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
are necessaryin this case can be established by use of the principle of superposition. In
this
chapter
we
have
presented
the
techniques
which are necessaryto the analysis of transient pressure data from variable-rate flow tests in wells, The types of
multip le-rate
tests
which
ha
ve
been
d'scussed 1
do
d n
d ee
h ,
t
., ere
are
many
d
vanations
on
these
prace
1. Odeh, A. S. and Jones, L. G.: "Pressure Analysis, Variable-Rate Case", J. Pet. Tech.
IStlCS 1963)
which the enterprising engineer may employ to accomplish a given objective. The general approach used in the theoretical presentation in this chapter may be helpful to the engineer who wishes to analyze other multiple-rate test combinations including those involv...' 109 closed-In penods.
From Two-Rate 1347-1355,
.
3 Al H '
ures
-ussamy,
Real
Gas
Flow
R
,an
Flow
d R
Theory
arney,
Tests,
J,
Well
r.,
Testing
;
"_._"-~c
If!)
..,_'::3:;~~Jr~: ,+"
~";;,; '" ~-~" ..... '", "$,.;"..n.~~.. i [;H
a..~ (J~1-
"'!tlI;'J
"'
Tech.
A
pp and
.
Ilca
(Dec.,
t'
Ion Deliver-
0
f
ability Forecasting",J. Pet. Tech. (May, 1966)637-642. 4, Stegemeier,G. L, and Matthews,C. S.: "A Study of AnomalousPressureBuild-Up Behavior", Trans.,AI ME (1958)213, 44-50, 5. Rawlins, E, L. and Schellhardt, M, A:: "Bac~-Pr,essure Data on Natural Gas Wells and Their ApplicatIon to Production Practices",Monograph 7, USBM.
,
\
Pet.
H J J ..,
to
,~
"
Drawdown (Aug., 1965)
960-964, 2. ~~ssell,D. G.: "Determination of"F°rmation Character-
not
represent a complete library of these types of tests. I
References
Chapter 7
V
Analysis of Well Interference Tests
7.1 ,Reasons for Interference Tests W en one well is closed in and its pressure is
-Pw. = p' -162,6
[
~~~th..r~ ia. tbe~ ~ir are prooriced, the test is termed an interference test. The comes from the fact that the pressure drop causedname by the producing wells at the closed-in observation well "interferes with" the pressure at the observation well, This type
of
test
can
give
information
on
reservoir
I
which cannot be obtained from ordinary pressurebuildup or drawdown tests. First- of all, one can determine y' Is the Portion of the reservoir at reservoocoonectivit , this dlwell locationf being drained d by other wells? ? A hi 1 3How A rapl y. n Inter erence test can eterInlne t s. -n-~ ot~er I~~nt use of Interference tests ISto deterInlne dire~arrese.rvoir now patter~s. This is ,done by selectively opening wells surrounding the shut-In well. In addition to this qualitative information, it is possible to obtain a quantitative estimation of connected porosity from such a test. Porosity cannot be d f b Id I Elk' 4 estimate rom a pressure UI up test a one. ms
.
.
.
.
.
has also
used
..e Interference
tests
to determine
the
nature
and magnitude of an anisotropic directional reservoir permeability. Groundwater hydrologists5.6have made much more use of interference tests than have petroleum engineers. Muskat7 shows an example of their work. .If 7.2 Equations for Pressure Interference The mathematical basis for interference tests was first presentedby Theis" in 1935,* The following method usesthe same basic equations but differs in treatment and method of analysis. This method is based on su rpo..:'!!tion ?--~~~~~~~h of.the prod~cing v:ells at tile Slffit:m oDservl1tiunwt;il. USIng the El-function solution of Chapter 2 (Eq. 2,31), we find that the pressure at the closed-in observation well due to continued production at Wells 1, 2, 3, etc., is given by:
*In this same paper Theis also developed a pressure
buildup equation similar buildup methods never groundwater hydrologists.
N~Wq;
i=lq
-E'
properties
to Eq. 3.4, However, pressure gained great popularity with
9~log kh
(
(~ ~I ) +
70,6 ~ kh
{ E' I ( O.OO105k -~l1.ca;2 (t; +
.6.t;) )
-~I1.Cai2 )}]
O.OO105kt;
..(7.1)**
h were -h .q -t
d '. . e p.ro uctlon rat.eat our observation well before It was closed m t W II ' qi = the rate of Prod Uct'on 1 a e 1 ..i ' te rf enng we 11 prIor t 0 -1 = P roducing time of l'th m shut-in of the observation well ~ .6./; = producing time interval of the jth interfering well subsequentto shut-in of the observation well,. I NW :: n~mber of mterfe~~n~wells. ' a; -distance. of the 1 mterfenng well from the observationwell. All th t't " Ifi Id ' E 71
.
quan
lies
m
q.
.are
..m 01
e
.
Units,
The log term in Eq. 7.1 gives the effect of producing and shutting in the observation well itself, The Ei terms give the pressure drop at the observation well causedby production at Wells 1,2,3, ...at distancesat, a2,a3 ' , , a reservoir bou~~.J!:.-Sf1°se by. it may be taken into account by the method of images to be discussedin Chapter 10. The "image terms" are exactly like the Eifunction terms in Eq. 7.1, there being one such term for each "image" well. The distance a; in this case is the distance from the image well to the observation well. For use in Eq. 7,1 we obtain times t, tt, .6.4, etc., by the same type of equation as in Chapter 3. 1 = cumulative production at observation well rate of production just before closing in '
**Eqs. 7.1 through 7,3 apply to a more general case than the corresponding equations in the original monograph. The authors are indebted to Raj K. Prasad for the derivation.
68
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
cumulative l?roduction at :WeIll prior to shut-In of observatlo~ well averagerate of productIQnq1 during interference test
11 =
well, best represents these quantities in the reservoir between the interfering wells. '" . An example Interference test IS shown In FIg. 7.1. The dotted line in this figure, called "Extrapolated Buildup Pressure", was obtained by extrapolating the linear portion of the log plot a~ shown on Fig. 7.~. From Eq. 7.1 we see that the dIfference between thIs extrapolated curve and the observed pressure is the sum of the Ei functions, or
incremental production at Well 1 subsequent to shut-in of observatio~ well averagerate of production q1 during interference test
1::.11=
and similarly for 12, 1::.12, etc. These equations for 4, 1::.1],etc., apply best when the rates of production at these wells are reasonably constant during the interfer-
( P*
ence test. We have already implied that this is the case by u~ing only one Ei term to represent ~ach producing well In Eq. 7.1. If the rate at a producIng well varies considerably during the test, a series of Ei functions should be used in Eq. 7.1 to represent the rate at that well; that is, the method of superposition (see Chapter 2, Section ~.8) should be employed rat~er t.~an the a?ove equations for 4. ~nd 64. Eq. 7.1 IS written for sIngle-phaseflow condItIons above the bubble point. For two-phase flow below bubble point, total mobilities and compressibilities should be used as in pressure buildup examples 2A and 2. Using Eq. 7.1, it is possible to determine the quantity (I/>JLc/k). The value of this quantity which, by trial and error, gives the best fit between observed and calculated values of the pressure drop at the observation
-162.6
~Iog kh
= -70.6~ ., kh .~ q -Ei(-
~ ~I
)
-P""
'
[ i=1 N!
.-1/>fJ.Cai2 { Ei ( O.OOI05k (I; +
)
1::.1;)
-1/>,uca;2 --: )}. ] O.OOI05kl}
(7.2)
Since the first two terms on the left side of the equation representthe straight-line extrapolation on Fig. 7.2 and the third term, P,V,',represents the observed pressure, this may be rewritten as
(/)
0.200 W
~ 1900 (f)
~
DUP
Ct: 180
EXTRAPOLA~E'py~l~
a..
w ~ 1700
\
7
I 1600 v
v
I-
I()
I()
~
w.!.~.!...~.!...
(II
~
58 psi
~ 0
PRESSURE
150
I
I
.'1; v I
I()
I
I
I
I
r--
a>
a>
m
I
143 psi ~
I
I
I
I
I
v
v
I()
I()
I()
I()
I()
I()
.!.
.!...
I
I() I
m Q
1:
I
.'1; v I
r\ ?
I
I
=
~
I.!... -C\I
..!
!..
140
1
~1
1300
",,"'! 0
40
80 120 CLOSED-IN TIME,
160 days
Fig. 7.1 Interference test in a low-permeability reservoir.
200
,olt_! -.Ic,rIU;;
,,:ii'"
ANALYSIS OF WELL INTERFERENCE TESTS
Pext -Pcb.
[
-m 2.303
(
-Ei
I '
[
69
=
incremental production at WeIll
NW ~ -El qj
{ .(
i=l q -cpp.caj2
)
-cpp.caj2 0.00105k
(4 +
)} ]
!:1t -subsequent 1 -avera
!:1tj)
23050 = -iso
..(7.3)
0.00105ktj
Example Calculation,
Calculate
the
pressure
Interference drop
at
the
and assume tha
-270 well
~P
~aused by production at Well 1 at the time when Incremental production at Well 1 subsequent to shut-in is 23,050 bbl. There is no production prior to shut-in and thus t1 = O. 0 a ta q = 140 ~/D,
~p.C/k = 10"6.
=
2.303
[ 180
.
(
-10-6
T4(fEl
(1835)2
0.00105
(3070)-
= -117.2
[1.285 Ei (-
= -117.2 = 32 psi.
[1.285 ( -~.21)]
)J
,
1.042)], ,
one can calculate the effect of production at Similarly, other wells-Wells 2, 3, 4-as a function of time and for several values of cpp.c/k. Fig. 7.3 shows results
rate at observation well prior to
shut-m, m = 270 psi/cycle, for buildup curve in observation well (from Fig. 7.2), B = 1.1, formation volume factor,
calculated rounding producing that their
C = 6.9 X 10-6 psi-1, compressibility, qt = 180 B/D,
~
cu ation The calculated pressure drop at the observation well caused by production at Well 1 is, from Eq. 7.3,
Test observation
= 128 days = 3,070 hours,
a1 = 1,835 ft,
The terms on the nght-hand sIde of thIs equatIon represent the calculated pressure drop at the observation well due to production at Wells 1, 2, 3, This will be made clear by the following example calculation. 7.3
to shut-in ge rate at WeIll
as above for the effect of four wells surthe observation well. Wells 1 and 2 began at the time the observer was closed in. Note influence was not appreciably felt at rates of
125 to 180 B/D until 60 days had passe<;f.Wells 3 and
average rate at Well 1 during test,
4 began producing at rates of 20 to 40 B/D
80 days
2000
1900 EXTRAPOLATED
.-0' cn
BUILDUP ..""
c..
.1800 w Q:
V
::>
"
~ 170
L 1
W
-.J 160 0
t-
v
"
""
""
vv
I() I
I() I() I I
~
~
I
:I: I
~
" ""
"
,;. P
IJ1I/
~
0
~
PRESSURE"
CD
I
-N
mo-
I
.
150
5 m 140
130
100
(t+~t)/~t Fig. 7.2 Interference test in a low-permeability reservoir, log-time plot.
I
70
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
after the observer was closed in. Their influence was not felt until about 8? days after they ~e~an producing and, because of theIr low rates, theIr mfluence was
7.4 Least-Squares Methods In the above example the "best" value of cpJ1.c/kwas the value which gave the best fit to the three plotted
small. The assumed value of cpJ1.c/kwhich~ave the agree~ent shown is cpp.C/k = 3.5 X 10-7. From Eq. 3.5 we
points when judged by eye. A more precise method of obtaining cpJ1.c/kis to use the method of least squares. To use this method, first measure Pext -Paba (from
J
!:~~~jve ~=
162.6 qB =
p.
162.6
m
(140)
1.1
a plot such as that of Fig. 7.2) for each data point such as the three shown in Fig. 7.3. Call this difference ~Pobs'
= 92.6.
270
Then
Then ~
( kh ) 1
cph = kc/lJ1.C
--;-
Pext -Paba
at each
point
for
several
values of cpJ1.c/k,using Eq. 7.3 as in the above example, and summing the effects of all the producers as in Fig. 7.3. Call this total calculated pressure drop caused by
1 (92.6)6.9X10-6
7=3.5XI0-7
calculate
= 4.70.
h = 43 ft,
all wells at the observer ~Pcalc' Compute (~PObS~Pcalc)2 for each measured point such as the three in Fig. 7.3. Plot a curve of ~(~PObs-~Pcalc)2 vs c/lp.c/k. The value of CpJ1.C / k which gives a minimum in this
cp = 0.11.
curve is the least-squares choice for cpJ1.c/k.
F
Note from this example that a very long closed-in time may be required to see the effects of interference. A minimum of two months was necessary in this case where the permeability is a few millidarcies. Note also that it was not necessary to find k or J1.to find cph.
Thus far in this chapter we have assumed that a value for J1./kh was available from a buildup test in our observation well. If this is not so, it is possible to obtain both J1./kh and cp,uc/k from an interference test, as shown by Morris and Tracy.s Their paper should be consulted for details.
16
PRESSURE DROP ACTUALLY
7.5 Other Methods for Computing Interference The methods presented thus far in this chapter as-
_14 ~120 ~~ ~ ~ 100 ~: ~ ~ 80 PR
sume a homogeneous reservoir. A more sophisticated method has been developed by Jacquard and Jain9 (see also JahnsI3), which takes into account variations in permeability in the reservoir. The method requires use of a digital computer to determine, by successive approximations, the modification in permeability distribution required to give a best fit to observed
~ '5 60
pressures at wells. Single..phase flow is assumed. Tho-
ME
Wo
[ ..mere ~ 4 -from 2 0 9 .., :" .:.
and Arthurlo have d~veloped a method for computing an interference "function" and a well "function" the observed flowing and shut-in pressures in wells in a reservoir. These "functions" can be used to ..60 ,. ~
80. , ' " ; ~ ~ ;:; ELAPSED TIMEIN DAYS ---" (AFTERSHUT INOFOBSERVER)
+.
'
~
OFFSET WELLPATTERN ..well .a i ~ 08siRVER 2 I ..with .A .the ..a .Sheldon Fig. 7.3 Calculated and observed pressure drop, interference test.
predict future pressure behavior in the wells. Again, the reservoir can be non-homogeneous, but flow must b ' e smglhe p ase. . A novel method of mterference determmation by "pulse testing" has been develo ped by Johnson et 01.14 In this method a production well near the observation is alternately produced and then closed in to give series of pressuIe pulses. The pulses are detected at the observation well by a very accurate (O.OOl-psi) pressure gauge. Use of this gauge allows the interference pressure pulses to be detect~d much more rapidly than normally used helical Bourdon-tube gauges. potentially more powerful method than any of foregoing is that of general reservoir simulation on digital computer. In this technique, developed by et 01.,11 multiphase segregated flow of oil and gas or water, as well as reservoir heterogeneity, is all9wed. Pressure behavior may be computed as a function of time and location. Fagin and Stewart12
I
71
ANALYSIS OF WELL INTERFERENCE TESTS
have developed a similar approach. They adjust reservoir parameters by trial and error to achieve a reasonable fit between measured pressuresand those calculated from a mathematical model by use of a computer. They then use the adjusted parameters for future reservoIr predictIon. Furt her d evelopment 0f such methods WIll undoubtedly occur m t he future smce t he methods enable determination of recovery expected from the various sections of a reservoir. In a sense
... ..
these
.
recent
methods
are
generalizations
..
an
d
exten-
6. Jac°.b' ,
sions of early "interference" calculations.
Elkins,
L.
F.:
"Reservoir
Performance
and Pressure
Jain,
C.: Data",
"Permeability Soc.
Pet.
Distribution J.
Eng.
(Dec.,
10. Thomere, R. and Arthur, K. B.: "Analysis of the Pressure a Production Well Subject InterferenceBehavior and Its of Application to Reservoir Models",to Well
Spacing,
Hum-
and
Well
Spac-
ing -Silica Arbuckle Pool, Kansas", Drill. and Prod. Prac., API (1946) 109.
paper SPE 1189 presented at 40th Annual
Fall Meet-
Sheldon, Method
D.: Simulation
ing, Denver,
11. 2.
P.
Field
1965)281-294.
References 1. Craze, R. C. and Glanville, J. W.: ble Oil & Refining Co. (1955).
Jacquard, From
J. for
Colo.
(Oct.
W., Harris, General
3-6, 1965).
C. Reservoir
D.
and Bavly, Behavior
"A
on Digital Computers", Paper 1521-G presented at 35th SPE Fall Meeting, Denver, Colo. (Oct. 2-5, 1960).
3. Elkins, L. F.: "ReservoirPerformanceand Well Spacing, Spraberry Trend Area Field of West Texas", Trans.,AIME (1953) 198, 177-196.
12. Fagin, R. G. and Stewart,C. H., Jr.: "A New Approach to the Two-DimensionalMultiphase Reservoir Simulator", Soc. Pet. Eng. J. (June, 1966) 175-182.
4 Elkins L. F. and Skov, A. M.: "Determination of .Fract~e Orientation from Pressure Interference",
13. Jahns H. 0.: "A Rapid Method for Obtaining a TwoDimensional Reservoir Description From Well Pres-
Trans.
, AIME
(1960)
219,
301-304.
5. Theis, C. Y.: "The Relationship Betweenthe i:°wering of PiezometricSurface and Rate and DuratIon of Discharge of Wells Using Ground-Water Storage", T rans., AGU
( 1935 ) II , 519.
sure 315-327.Response
Data",
Soc.
Pet.
Eng.
J.
(Dec.,
1966)
. 14. Johnson,C; R., Greenkorn,R. A. and Wo~d~,E. G.. "P~se-Testmg: A ~ew Method for ~escnbmg Reservlor (Dec.,
Flow Properties 1966) 1599-1604.
Between
Wells,
J. Pet.
Tech.
Chapter 8
Pressure Analysis in Injection Wells
8.1 It
Pressure Fall-Off Analysis in Unit Mobility, Liquid-Filled Reservoirs
.
IS
0
f
consl
.
d
era
bl
e
. Interest
an
d
. Importance
For this case the pressure behavior is described by Eq. 2.38 of Chapter 2. As shown in Ref. 4 (for a to
b
e
able to determine the characteristics of the reservoir
similar f
case),
it
is
possible
to
rewrite
Eq.
2.38
in
the
orm
(
)
in an area surrounding a water injection well. If we can determIne early m the lIfe of an InjectIon well that there is an appreciable "skin effect", remedial measures can be started before a full-scale pattern flood begins. Similarly, if we can show that a gradual buildup of skin effect is occurring with time, we can take measures to free the water of plugging material. Determination of static ~pressurein a water injection well may show that the water is entering a thief zone and not the desired reservoir. Finally, determination of the permeability of the sand around an injection well will allow estimation of the future relation between injection pressure and rate. Morse and Ott7 present a good discussion of the use and value of pressure analysis in water injection wells. In water injection wells it is natural to attempt to determine formation properties by closing in the well and using familiar pressure buildup methods. Joers and Smith, I Nowak and Lester,2 and Groeneman and
= -1 ip. -constant. t+~t + PWB P. + 41Tkh n ~t Thus, the slope of the fall-off curve may be interpreted in terms of kh exactly as in the case of a buildup. The skin effect and well damage can also be obtained in the same way as for pressure buildup, as can be shown by utilizing the Ei-function solution given in Appendix 1 of Ref. 4. Although these items are determined in exactly the same way for buildup and fall-off curves, the flow efficiency and the averagepressure must be determined differently. The reason~ average pressure must be determined differently foll~ws from consideration of the equipressurecontours and streamlines in a waterflood. Consider, for example, the regular five-spot waterflood shown in Fig. 8.1. As may be seen,the proper boundary condition midway between injector and producer is a contour of constant pressure across which
Wright3 have applied the buildup methods for singlephase, slightly compressible flow to fall-off curves obtained in this manner. For waterfloods which have "filled up" and for which the oil mobility does not differ too much from the water mobility, the theory should apply. We will discuss this liquid-filled reservoir case first because of its similarity to conventional buildup. In the next section we will cover analysis of injection wells 1\ prior to fillup.
flow takes place. In a reservoir which contains producers only, no flow takes place across the boundary between wells (see Fig. 3.6). Since the pressure distribution differs for the two cases,the averagepressure must be determined differently. In the five-spot network shown in Fig. 8.1, the 50 percent equipressure contour which divides the injection well area from the production well area enclosesa square of area A around our injector. We may
Thp basic assumptions for this method are the same as fo~ pressure buildup theory. The reservoir is assumed\to be homogeneous,of constant thickness and to contain a single fluid of small and constant compressibility. Prior to shut-in, water is injected at a constant rate through a well which completely penetrates the formation. The pressure is assumed to be constant at a radius r e from the well, as will be discussedbelow.
closely approximate the pressure behavior in this square by finding the pressure behavior in a circle of equivalent area, as was shown in Fig. 4.3. Thus, we choose the radius of the equivalent circle from 1Tr.2= A, where A is the area inside the 50 percent equipressure contour shown on Fig. 8.1. The quantity A corresponds to the drainage area of a producing well, discussed in Chapters 3 and 4.
--
PRESSURE ANALYSIS IN INJECTION WELLS Now sure
the
pre~sure
contour
in .sure F. m Ig.
WIll
near
the
lower
We
now
ent
pressures
d
an
the
this
square
waterflood
be
of
given that
balanced
..
t
t
iS analyzed
-an Pe = p.
.h
.on
pressure
pressure
to
zero. 4
ra e a
behavior
For
t th
for
this
that
simply result
by
case
the the
sum
e we
b
an
d
oun
pressure.
Th f
di
con
-
p*
The
..
shown
h
difference
boundary
to be given
Pe -p*
-E.
i.u/41Tkh-
As
pressure
to
relate between
described
we ca:
p -p*
/kh
e
IS
From
to
tion
from
the
poor
4>.ucA
is
( -~.
E
.
using
equation
(
)
of
becomes
p -p* in
to
Fig.
0.000264
8.2.
kt/4>IJ.cA
we
we
11
11.
.
I d d
m
mJec
t
the
or wa
er
on.
p
buildups, these
pSI
for W
p*
e
I
4>IJ.CA
oilfield
)
IS
accurate
can use
to
it without
in the
example
.
I
use
n
p
approxima-
p. Note
322
t d ec e
w ell , USIng p rathe:
p*
-pSI.
I ways
g1
.. ti. mJec on
fall-offs,
=
a
ne
In
*
p
t
is a good
one
cases.
. hil
125 =
It is more
.IS
s
f
since
.
A B
.. ti.. mJec on,
gas
al I ti.
approximation
h
cor-
may
fall
be
a
treated
-0
ff
s
th
ere-
-'
the.
authors
recommend
Fall-Off
use
of
p
to
calculate
I
Analysis
Ij
Fillup
:
to Reservoir
units
Mobility
Ratio
..
j
.
P fill th 1 d t b k .to reservoir ~p'. e 01 an wa.e.r an s ~ay be Idealized as shown m Fig. 8.4. The fluid saturation,
..nor
as
.. ti.
or
f
Unit
oilfield
.on
ti
F , t 0 ma k e this
this
dimensionless In
... slDlllanty
Ismcue
A ppen di x
buildups.
as
-4>IJ.CA 0.0033 kt
1,
However,
p in
Pressure
--
plotted
to
8.2
above
relationship
kt/4>,ucA
as for cases.
(
the
e
I
efficiency.
-Prior P and
.ty urn .
flow
)
8.3
Injection
uc
p
Fig.
...
d
..
B
obtain
the
an
or a pro
t
f
Ie
ca
to p in most
recting
.u The
p*
in both
at
-4kt 4>.ucre2 --I -E.
IX
li bl f
thi
the
extrapolated
found
or
acor
app
fore,
Pe =
-I
r
so
I c ose
than is
obtain
this
x
Bf
I
E xamp
by
above,
rewrite
70.6i
-I
at
ppen
to
where
c cu a IS oncalculated or an note n that s theexamp flow eefficiency
Appendix
behavior
Ei-functions.
superposition to
.
,
F
-' s, and p. Note di F f
ppen
ti
I
a
I
circular wellbore simpler
in
pressure
and
e
curves
Appendix
d.
,
A
very to m t h ese ca I cu I a ti ons.
11
equivalent
two
e
1
s h ee,t
boundary
Aif considerably we let our
fall-off
1
ormaonvoume
f orm
..t and
analysis
xamp
xamp
and
.' obtaIn kh, I 1 A
to I
pressure
lA
equlv-
ary
outer
this
it is
of
by
pressure
boundary
t e the
injection
it is possible
extrapolated
h
at
t
ti
mJec
Ref.
by
with
E d E
eet
constant
an
pressure
is used
as in Example
by
Thus,
reservoir
case analysiswas can obtained be made, m Eq. however, 2.38.
shrink
exactly
figure
pres-
such as that. th hi h e g er
h t at
producer.
( 1Tre2--WItA)
I ar one
cons
1
.IS
s
equipres-
average
e
. clrcu
The
percent
to the
thi
WIll
near
approXImate
beIng
50
equal
...between Injector
the
..s tions
the
exactly
a unit-mobility five-spot 8 1 Th f ..e reason or
pressures
al
Pe along
be
73
time
units, shown.
kt/ This
in
turn
Fig.
may
8.5.
water
be idealized
We
have
first
the
by
the
consider
same
distribution
the
case
properties.
A
shown
where
oil
mathematical
in
I
and
i
solu-
tion for the pressurebehavior in this case was dePRODUCE R
PROOUCER
veloped the
by
Hazebroek
presence
of the
single-fluid and
method
water
difficulty
wiili
using
of finding
the
curve. 8.3
difficult
The
to
know
since
the
is
at a reduced
rate.
Until
straight-line
pressure
line
Fig
PRODUCER
.
81 E ..qulpressure network. (Numbers
..
t d tr amI fi t con ours an s e mes m a ve-spo represent percentages of total pressure drop.)
broek et al.4 ... sc hematically were
for
still this
a
falls
will finding
a new
method
and
Case the
closed-in
A,
well
surface but
continue value,
no
be observed. the
straightby
Haze-
saturations shown Two pOSSI b.I li ties
.
the
stays
time.
at the
minutes;
developed
for the fluid banks and m Figs. 8.4 an d 8.5. For
of
to a low
with was
for
period
will
section
is
reason
The
few
injection
rate
correct
of fluid
surface.
in
full,
full
fall-off
slowly
considerable
off
and
long
difficulty
considered.
decreases
at the
it
The
the
overcome portion,
be
in
reason
slope
the
of the
portion
this
used. is
will
bleed
oil
method
portion
For
been
stopped
often
conventional
correct
wellbore
that
straight-line
section
wellbore
To
PROOUCER
is
will
this
point.
have
The
injection
pressure
p*
in
use the
provided
straight-line
the
even
properties.
short
this
straight-line
after-injection. time
rather
whether
to
short
same
correct
illustrates
extrapolation the
discussed
the
fall-off Fig.
just
abOut
One is that
have
et al.4 ..They show that saturation one can still
gas
surface
filled This
.
pressure
up to the happens
,I
top
when
74
v
PRESSURE
~
I.D
2~.i4:~:-~:
It')
V
It)
=---~_:-_:-~-:~IIIIIIII
~:t~-=f::t~
N
AND
FLOW TESTS
IN WELLS
-0
0-
--:- ,- _::~- ~::,---"-
:: -,--
,
BUILDUP
---=--
--
.
I -,
,
.
-:~-
. ,
, I -1
Ii)
0
-
J
--I
-~~:.~-
0;
~-.-"~~~f~~ --1~._-
::: -r-
,
-::[: --t-H~Ti-
--t-
-:-_: 'ci-~'=::.:-:_-~cJ=.-
=-: '-
-:-- -~-~-~~--'
'8
1-
L1 -+ tl-
1 -'"
.-oX
v
~:=-::-
~ i
e 0-
f
0 -e.. '" ---~
.lib
---0
"3
--II
8-
-a
,
-8
I
0
I
~d
I
"-
0 0
~
_I--
-=
S
-;,~~=-
0
-u
IEI_B~-i=
-c=£"-
"~=~- --.'c
-'"c-
IIIIII!IIIO
-:
~'- -~~---~-~--~~~~ --;:- -~
~
"~
§
---
::}::t:-
=
,
--c::rxc
-.:-
,
---,
--::':r=-:~
--:"'--
---=-~--
-~- --:+-~::c'-
'- ~ -tm-
"-
---:
.-
i.:-::t:,,~'?--~
-::
,-,--
:c:::t
-"'"':::1::-:, ,: ":~1~
-:-
-=l:-=
---
I ,
0 -0
~ ~
L =
~J
0
cn
a) 4'>1/71'lgoOL
*d -..Q.
~
to
It')
l~l
-J I
~
-
0:
PRESSURE ANALYSIS IN INJECTION WELLS
75
6t, hr
3 00
I
10
100
0 0
200
0
0
, ~~
CALCULATED AVERAGE PRESSURE
"
.;;:
100
""
.-, CI
,, ,
~
W
," "
a
a:
"
(/)
,
:J
(/) w a: Q.
-I a
," , ~"
-t)-,,;>,
4~'o "
" ~"
'<4)::'~o
"
-20
,
,, ,, ,
-30
"
"
,
p. = -322 psig
-40
10
10
10
t + 6t
10
10
I
6t
Fig. 8.3 Example pressure fall-off curve.
the reservoir pressure is high. After-flow into the formation in this case is small since it results only from expansion of fluid in the well as the pressure decreases. For Case B, the surface pressure drops to zero a short time after closing in, after which the liquid level in the well starts to sink. In this case the volume of inflow Pf" ,P I
UNFLOOOEO
I
I : : :
REGION I : :
: : I I I I I
: : I I I I I
:
:
I I
I I
: : P
PRESSURE Pe ATRADIUS re
Fig. 8.4 Oil and water banks.
P e is the Pressure at the outer radius of the oil bank (see Fig. 8.4) and PW8is the fall-off pressure in the well at closed-in time 6t. From this equation we see that a plot of log (PW8-Pe) vs At should be linear with slope ,81/2.303 and intercept bI at At = O. From
the theoretical treatment in Ref. 4, we find that the :t -INJECTOR intercept b1 is related to kh and the injection rate i P-PROOUCER b
y
r =.!.9aD re
. kh = !J!:.-. 1 = C1 -C2 b1
I 1
into the formation at any time is equal to the volume of the wellbore column between the top of the well and the liquid level at the time of interest. For both conditions it was found in Ref. 4 that the injection well closed-in pressure is given by --.B1/lt where PW8-Pe+b1e , (8 1)
P
(1-Ca)2
f
(fJ)
'
...
( 8
2
)
where f(fJ) is plotted in Fig. 8.6, and C1, C2 and C3 are obtained from Eqs. 8.5a or 8.5b. Knowing kh and the injection pressure Pw at time of shut-in, we can find s from
76
PRESSURE
s+
1 re -0.00708 (PtD-Pe) n-;:;;i,u.lkh ,
(83)
BUILDUP
AND
FLOW TESTS
IN WELLS
Eqs. 8.2, 8.3 and 8.4 are written for the practical system of units: kin md, h in ft, i in BID, Wi in bbl,
where r e may be calculated from the cumulative water
,u.in cp, and hl in psi.
injected Wi.
The quantities Cl, C2 and Ca, when expressed in practical uni~s, become f~r C.aseA where the surface pressure persIstsafter cloSIng In,
-~ re -.,
rw
I Wi (5.615) 7r1f>(S,-S,r)h
WATER
(8.4)
BANK
OIL
SQr
RESIDUAL
S z
BANK
or
RESIDUAL
~ ~ a:
OIL
I Ir
DISPLACED
OIL INJECTED
re
UNFLOODED
REGION
1
GAS 01 L
PRIOR TO FLOODI NG
7
GAS
Q
OIL PRIOR TO FLOODING
WATER
:;)
f
~ INTERSTITIAL
WATER
JECTOR
PRODUCER
Fig. 8.5 Fluid saturationsin the reservoir. 200
180
160
140
120
f(BI 100
80
60
40
20
0
0
.04
Fig.8.6Functionfor calculating 8.kh.
S 0
PRESSURE ANALYSIS IN INJECTION WELLS
77
~ Ci = 0.0538
Fig. 8.7. When the assumed value of Pe is too high, the
dt2.8ibicw. (Pw-pt)
curve
'p
turns
down,
of Pe which
~
C2 = 0 Ca = Pw -Pe Ci
'
.(8.5a)
bi
as in Curve
gives
Curve
A, Fig.
C is the
8.7. The value
correct
one.
The
intercept at At = 0 is b1, and .81is obtained by multiplying the slope of the straight line by 2.303.
where p! 1Sthe surface pressure at time ~f closm~ m, and Pw 1Sthe bottom-hole pressure at this same time. For Case B wh~re ~he surface pressure drops to zero
For caseswhere the surface pressure persists during shut-in, one need only plot 10g(Pts-Pes) vs At, where Pts is the surface tubing pressure at any time and Pes is the surface pressure which corresponds to the outer boundary pressurePe.
shortly after closmg m, -~~) Ci -0.0538 ip
An example calculation for the unit-mobility-ratio case prior to fillup is. given in .Ap~endix F, Example 2. The curve analyzed 1Sshown m F1g. 8.8.
~
)
C2 = ~Cl
Cs = !!!!!-.-=-l!!.Ci b1 and for both Cases A and B
,
(8.5b)
In this case, oil and water are allowed to have different properties. The ratio (kw/ ILw)/ (ko/ ILo) is the mobility ratio M. The saturation distribution during injection is idealized in Fig. 8.5. Water is assumed to
,
-Ci (I-Cs) 9 -2( l-Ci-C2) .saturation ..., Here dt 1Sthe diameter of tubmg or casmg expressed ~n mc les, p m gm. cc, cw. m pS1 , P m pS1,a.n...81 m hr- .The quanuty Cw1Sthe water compress1bility, ..' not total compress1bility. The reason for this 1Sthat ..h
.
Cw occurs
/
m the
.'-1
term
for
Non-Unit Mobility Ratio
..
expans10n
d
of water
displace oil and gas down to some uniform residual in the water bank and to build up the oil saturation to a uniform value in the oil bank. Only t wa
fl er
. ows
th m
t e
wa
bank er
d an
nl 0
O
il
flows
in
the
y
oil bank. This approXimation is a good one for oils of 1ess th an about 50 -cp VlSCOS1 ty. Asere th are f ew w ate-r
..
fl 00 d s f or cases 0 f hi gh er 01 . 1 VlSCOS1 ' . ty, th e resu It s m '
m the
tubing. In Case A and very often in Case B, the quantities C1,
C2,
Cs
and
9 are
small
and
j«(})
can
be
taken
1000
equal
to j(9) = 181. This means that the effect of the after.
800
flow into the 1 formation is negligible. In that case we h ave s1mpy,
600
.
ilL
kh = 181-bi
.(1°9360-1°920)2303
Plotting Results As we have seen, in applying the theory it is neces. sary to determine Pe, b1 and .81' This may be done ."' graphically as follows. We assume some reasonable valu~ for Pe,and then plot 10g(Pws-:Pe)vs At where Pws1Sthe wellbore pressure at any time At. If the re. suIt is not a straight line at large times, the value of Pe is changed and a new curve plotted. On semi-log paper the new curve can be plotted without any additional computation by shifting the previous points a certain amount Ap in pressure at the same value of time. For example, in Fig. 8.7 Curve B shows values obtained for Pe=200 psi. By mentally subtracting 30 psi from the value of Pws-Pe for each point, we plot Curve C,
400
P,= 70 =0 0412 Iv -I
200
0.
~ '°'0
--~, ~ ~ 100 t. 80 -"\ 60
'0,«x" '0'0 x',x 'b'o.. ''x Q,o CURVE B '0'Q"Q,Pe=200 psi , O..Q ~,"b'Q X,, x\ 1
40
psi \,\ x\
20
for which Pe=230 psi. It is sometimes difficult to tell when the best straight
\'x CURVE A \ Pe=260psi \
line is obtained. For example, a straight line can also be drawn through the last few points of Curve B in Fig. 8.7. We have found it helpful in such cases to "bracket" the best curve by curves of types A and B,
100
10
20
\ 30 40 50 60 CLOSED-IN TIME,HOURS
Fig. 8.7 Pressurefall-off curve.
70
78
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
this section are believed applicable to most non-unit mobility ratio waterfloods.
V 0 -volume ~ -volume
In obtaining the results in this section, we have assumed that the flow of water into the formation after closing in is negligible. This is the same assumption made in deriving pressure buildup theory. The method of this section will apply best to the latter part of the
of oil bank -So -Sor of water bank -Sg -Sgr (8.10)
The function F is also a function of 'Y, the ratio of the total compressibility in the oil bank to total compressibility in the water bank.
pressure fall-off curve when the after-flow has decreased to a small value. Criteria for judging when this method should be applied will be discussed toward the end of this section. For conditions where the oil and water banks (see
A set of curves (Figs. 8.9, 8.10, 8.11) has been constructed giving F as a function of M for values of the parameters rODand 'Y' Each figure corresponds to a value of 'Y ('Y= 1, 2 and 4) and gives curves corre-
Fig. 8.5) have different properties, show that, like the unit-mobility-ratio
sponding.t~ .values roD=O.l, 0.2,...0.9, 1. Curves for the limIting case, rOD= 0, have been drawn for 'Y = 2 and 'Y = 4 when M > 2. These curves have been
Hazebroek et al.4 case,
PW8= Pe + ble-.8~'t.
.(8.6)
From the theoretical treatment constant b 1 is related to k..,h by k..,h = ~(2F) bl t
,
...(8.7)
.
.
F . b b ' lity ti M ence' or mo I ra 0 ,
h fu H f
h
. lions. were
it is shown that the
e
nc
-,uo koh -~M'
ti
on
IS
a
com
.
ma
ti .
f on
0
k..,h
Notice that the quantities Cl, C2 and Ca are not used in these calculations. The quantities entered into Eq.
B
1
fu
esse
8.2 nc-
(8.8)
." .be !he func~on F IS a .£unction of rOD,the rati~ of the Inner radius of the OIl bank to the outer radius (see Fig. 8.4). The quantity rOD may be obtained from
1
were,
because
of
the
afterflow
from
the
wellbore
"
into
the
..
formation which took place after closIng m at the surface. No after-flow was assumed to take place .m this non-unity mobility ratio case so that these quantities did not appear. However, it is suggested that Cl calculated (for Case A or B, as applicable) to determine whether or not this as~umption is correct. If C is small then the after-flow will be small for each unit 1drop in ~ressure. For best results we suggest that
this non-unit mobility ratio method not be applied
rOD=,
h
omitted for other values of 'Y and M because of their closeness to the curve, rOD = 0.1. The plotted value of F contains a number of conversion constants such that k..,h is and hl obtained is in psi. in md-ft when iw is in BID, ,ufOis in cp
(8.9)
_/~ + 1 ., V w t ti.. t fill ames pnor 0 up,
when Cl ~ 0.1, except at large closed-in times. An estimate of s may be obtained from the bottomhole injection pressure just before closing in, PW' MOBILITY
600
100
3
4
5
RATIO,M 6
7
8
10
80
50 40
60
3
40
5
F
20
.
..30 a-
-=
.9
a-
'.
10
1.0
~.8 20
15 3
F
10
5
8
7
6
4 100
0
0 CLOSED-IN
TIME,
HOURS
Fig. 8.8 Pressure fall-off curve.
MOBILITY
RATIO,
M
1.8 2.0
Fig. 8.9 Function for calculating kh, 'Y = 1.
PRESSURE ANALYSIS IN INJECTION WELLS
[s + In~+ PtD= ~ 27rk",h rtD As In(re/ro) = (1/2) practical units,
I
2
(M-l)
79
In~
] + Pe.
S + In~=O.OO708 (PtD~Pe)kwh-~.
ro
r,o
In[(V o/V tD) + 1], we find in
3
MOBILITY ~
4
'tDJL,o I
RATIO, 6
[V 0 n --v;:;+
2
1] .(8.11
)
M 7
8
9
10
1000
-0
.1
800
.2 .3
600
.4
~OO
.5
400
.6
f'
300
--.7 r
200
--.B
I~O
--.9
100
1.0
80
:t
~ -.1
150
.2 .3 .4 .5
100
.6 .7 .9
80
eo f' ~O 40
30
20
.
I~
f :t
t1Ji f:..;:~ t:
,: 1-f ~if[1 +-t t
10.2
.4
.e
.8
1.0 M081LITY
1.2 RATIO,
1.4 M
Fig. 8.10 Functionfor calculatingkh, 'Y = 2.
..6
1.8
2.0
80
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
We will apply this method to the same curve used in the previous example to illustrate the error one mtg
.
ht
k ma
. e
1
f
h
M e
assumes
1 =
h w
. t en
1
.
tu 1S
ac
11 a
4 y
for unit mobility ratio to a reservoir where the mobility ratio is higher than 1.0 (as in the case of viscous oil), too
Iowa
value
will
be
obtained
for
wellbore
damage.
...
...Thus Results are shown and d1scussedin Appendix P, Ex-
the engineer may dec1dethat there is no reason to stimulate the well. Use of the non-unit mobility ratio
ample 3. Results show that if one applies the methods
casewould lead to the proper recommendation.
MOBILITY
5
RATIO,
M
6
7
B
9
10
1000
0-
0 .1 .2
BO
.3
6
.4
5
.5
40 .6
F 30
-.7 20
-.8 -' +
15
-.9 10 8
00=.1 -.2
15
.3
.4
10
.5 .9
.6 .8
8
.7
6
F
5 4
3
2
I
I
1.0 MOBILITY
..1.6 RATIO,
1.8 M
Fig. 8.11 Function for calculatingkh, y = 4.
2.0
PRESSURE ANALYSIS IN INJECTION WELLS
8.3
81
Two-Rate Injection Test Analysis
PID= P + 141.2 ~(
Discussion
In~+
s) ;
thus ,
As might be expected, a procedure similar to the two-rate flow test method (Section 6.3) can also be used for analysis of fluid injection wells. This procedure has an advantage over conventional fall-off methods for cases in which the surface pressure falls to zero after cessation of injection. To obtain pressure data after closing in such wells, a bottom-hole pressure bomb must be run. With the two-rate procedure, a pressure bomb usually need not be run since surface pressures generally persist throughout the two-rate transient injection test. Johnson el al.5 have also presented a variable-rate method for determining wellbore damage.Their method, which is actually quite different from that to be presented here, appears quick and practical for determin' an ... mg skiffn e ect m mjection we.IIH owever, thfe orma-
.
a=
PID-~141.2 ~
In~. TID
(8.14)
Eq. 8.14 can be used to determine s, provided a good estimate of Teis obtainable. If the value of Teis questionable, we note that the slope of the plot of log( PiID-{p
+ (i2/i1)[PID-PJ} )vs Lll' is given by
,8 = 0.000664 ~ CP,uCTe
...(8.15)
Thus, Te2=
0.000664 k ,8 ~, 'l'Jl.C
(8.16)
tion permeability and average pressure cannot be determined by this method, and we will therefore describe the more comprehensiveprocedure. Variable injection rate data may also be analyzed by the method of Odeh and Jones.8
and substitution of Eqs. 8.13 and 8.16 into Eq. 8.14 yields 1.151 log 0.00066~ k S = 1.283 ( PID-P ) ~~b 11 ,8cPp,CTID (8.17)
Theory We begin by making the same assumptionsas for the unit mobility ratio cylindrical case. From the results of Hazebroek, Rainbow and Matthews4 and Muskat,6 it can be shown that the pressure behavior of the well at time Lll' after the change in injection rate is given by ( { . } ) 181 2 (. -.) log PiID- P+t[PID-PJ = log .;h 12,u
Eq. 8.17 is an alternative formula for determination of the skin factor. For i2 = 0 the inter~retation formulas for t~o-rate tes~sbecome those whIch are ~ommon1y used m conventional pressure fall-off analysIs. The averagepressure must be determined by a trialand-error procedure as noted above. The procedure is to try various values of p until one is found which yields the best straight line on the plot of
-0.000664~,
vsLll'.
This procedure can usually be employed with a good degree of accuracy.
p = averagepressure in area between injector and producer. As discussed in Section 8.1, P = Po, the mid-
An example application of this method is given in Appendix F, Example 4.
point pressure between injector and producer. We see from Eq. 8.12 that if we plot
8.4 Gas Injection Wells For mjection wells m gas-cycling projects, the meth-
10g
( .-
P'ID
{ -~
-- })
ods discussedin Section 8.1 of this chapter, and given as Example 1, Appendix F, may be applied with slight
P + ;1 [PID p]
l' th ..modification. vs Ll e plot should be linear; and from the mtercept value b at Lll' = 0 ' we find -181.2 kh -b
(4 -;2) ,u
log (PiID-{P+(i2/i1)[PID-PJ
})
..(8.12) 'I',uCT e where PiID= injection well pressure after change in rate, PID= injection well pressure at time of rate change, and
(8.13)
An example of this type of plot IS shown m FIg. 8.12. Trial-and-error values of p are used until the best straight line is obtained, as shown for p = 3,600 psig. To determine the value of s, the skin factor, we recall that at time of rate change
The modification consists in determining and using the formation volume factor B, a quantity which was neg1ected m the case 0f water ... mjection be-
.
cause it was so close to unity. The value of B is determined at the arithmetic average of the pressures p* and PID'as in Example 3A, Appendix B. Example 3A may be used as a guide in applying the form sheet given in Example 1, Appendix F, to a gas well. For injection wells in miscible floods, or in pressure maintenance projects in oil reservoirs, the non-unitL mobility ratio case treated above would at first sight
82
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
seem most applicable. However, a useful method was obtained in that case only where the wellhead pressure falls to zero shortly after closing in. This does not happen in gas injection wells, Further, the assumption of cylindrical
banks where only one fluid is flowing
in each bank is not nearly so applicable for nonmiscible gas injection as for water injection, The best recourse for gas reservoirs is probably as f 11 0 ows. .., 1. For gas lOjection lO O1lsclbleprojects, apply the unIt-mobIlity-ratio
method
...
of
, Section
8.1
.gas this chapter
of
d
h
(Appendix F, Example 1). In USlOgthis metho ,c oose b nk d .e A as the area of the lOjected gas a .In so olng you are neglecting the expansion of the solvent and oil outside the gas. This is justifiable because of the much
..
higher compressibility of gas. 2. For gas injection into an oil reservoir, nonmiscible case, apply the same method modified for two-phase (oil-gas) flow. In applying this case, it will be necessaryto calculate total mobility and total compressibility, Total compressibility Ct may be calculated
as in Example 2A, Appendix B. Total mobility is given by (k/p.)t = ko/p.o+ kg/p.g, ,
,
The quantity kg/ p.gmay be obtalOed from the slope of th~ pressure fall-off ~urve in ~e injector, ~~ for a b~tldup wel~. To .0btalO ko, r~lative per~eability data WIll be required slOce.the relative flow ratio of gas and oil cannot be directly measured in the area around the , . ti' 11 F th 1 ti b lOjeC on we. rom e re a ve permea I lity t0 . 1 (k ) d t (k ) t th tu ti 01 0.. an 0 gas g.. a e average sa ra on In th ty f th ' , VlClm 0 e gas lOjeCtor,
.
.
...
..
k = k .~ 0 9 kg... From this and from data on p.o, (k/ p.)t may be calculated. The quantities Ct and (k/ p.)t are used in calculating the skin factor s. The form sheet used in Appendix F, Example 1, should also be used for this case of gas injection with the above modification for Ct and (k/ p.),.
10.000
DATA .~
il=2563B/D
on
.
C1.
12=742
~ 1C1.
I
.4> ~ N -rw .-1.+- 1000 1C1. '-0-' 1 .! Co
BID
IJ.w=0.37 cp Ct= 7.0XI0-6 psi-I Pw = 6777 psig =0.244 h = 31 ft = 0.3 'I
0 00 00
0 00 0 00 0 0
00 00 0
0 000
00 0
000000
00000 000
OOOOOO~e-e-o-e-e=~
=-~
-e
0000000 00
000
00 0
P=3600psig 00000
0000 000
00 0000
P=3400psig e--
00
0G-f>..e-e-e-
~-e-e
P=3800psig &---
00G-e-e e-e-o-e-e.9-6-G--
0 0 0 00
00
--0--.
6- -P
= 4000 psig --
~~ -G$..e~
--
-9-9 90s
100 0
--9-
-36
~t:HOURS
Fig. 8.17.Two-rate injection test,~
P = 4200 psig 40
44
48
50
PRESSURE ANALYSIS IN INJECTION WELLS
This is the same modification as was made in Appendix B Exam ple 2' and therefore an example will not be d ,
repeate.
83
4. Hazebroek,P., Rainbow,H. and Matthews,C. S.: "Pressure Fall-Off in Water Injection Wells", Trans.,AIME (1958) 213, 250-260.
1. J.oers,J. C. .and Smith, ~: .V.: "Determin.ationof.Eff~ctlve Formation Perme~,bllitles and OperationEfficiencies of Water Input Wells, Prod. Monthly (Oct., 1954).
5. Johnson,C. R., Greenkorn,R. A. and Widner, G. W.: "A Variable-Rate Procedure for Appraising Wellbore Damagein Waterflood Input Wells", J. Pet. Tech. (Jan., 1963)85-89. 6. Muskat,M.: The Flow of HomogeneousFluids Through Porous Media, McGraw Hill Book Co., Inc., New York (1937)Section10.10.
2. Nowak, T. J. and Le~ter, <;1.W.: "Ana!ysi~ of Pressure Fall-off. Curv~s ~btamed ~n Water Injec.ti°n Wells ~? Determme Injective Capacity and Formation Damage, Trans., AIME (1955) 204, 96-102.
7. Morse, J. V. and Ott, F.: "Field Application of Unsteady State Pressure Analyses in Reservoir Diagnosis", paper SPE 1514 presented at 41st Annual Fall Meeting, Dallas, Tex. (Oct. 2-5, 1966).
3. Groeneman,A. R. and Wright, F. F.: "Analysis of Fluid Input Wells by Shut-In Pressures",J. Pet. Tech. (July, 1956) 21-24.
8. Odeh, A. S. and Jones,L. G.: "Pressure Drawdown Analysis,Variable-RateCase",J. Pet. Tech. (Aug., 1965) 960-964.
References
F. -.'~--, " "u..w .~
"~Vtin7' "J! ,fjifs
!!Ht5ii1
m(:Ml;:~
ii!,5':~~""lha.~tfJ li')w r'i :oi:'; if}(f,~~ l';;>ri '-
: ')'111 ...,1:
;
;,
...:~ ~,'. '..'~
0;
i'os , :
.;
; ;; , r
,:
;-r ~
;~:
;;.
,,'
-',"
)'
'"
j'
'"
;f'
,,; ,,':' '1.1
:\.,;
,
..,:, ".';'!'~;(; c:
:'i
;'
f,;,,:
j
1
';'1 ~,1,;
,1:;\.;)1 !.J(!](;
1\11(';"O".!i
.;,.;li,
;
.',
'll"t'.,
'.');""~"
.;;'..£,0;
"...;;r,1 ;;(':10:";
rl!~')
": ;:tl; ,I:, "';)-~o\1;'J,)1!~~,~
,,~!1:
:IIo~,"~c~:;;j~,"j.
, :>10, ",J .I~ i""'m~,.", cT,.' ,
i)~~
c, ,
,.
; ;
: ,;,L,')
~"a hP ,pi,,! :-{?I';5iiRii/;' 1/':;' i,~ S~11l":;:; .lit-A' 0: ,- 'J\):I';;()l<; t'J.r\(""':;(1')
J-4/
,
,:?lU 1(; ""~;",iq,l!t;i ) !t" 'I'oj' 'I, ';"' 1.1~"I f, ,
,
!
["'" ;, .,0';\'d " i~ir:i;1
ri OLS: W1A}
,
.Iot~,
f)f1L;
"
r~
flt '
dJ f)'..'
t;!'Jf; Z1$X~q
,;,,~ at ~u ~,~OO .h
(WOu
Chapter 9
Any discussion of transient pressure behavior in wells would be incomplete without considering pressure behavior under drillstem test conditions. The measurement and analysjs of drillstem test (DST) pressure behavior affords the engineera practical and economical means for estimating important formation parameters prior to well completion. A properly run and interpreted DST probably yields more valuable information per dollar spent than any other evaluation tool. Hole conditions do not always permit the use of the DST as an evaluation tool. However, in those cases amenable to the technique, the fluid production and pressure information obtained are many times invaluable. Often, the only good estimates of the initial reservoir pressure are obtained by DST in the first wells in the reservoir. By utilizing some of the transient pressure methods previously discussed, the kh product and skin effect can be estimated and thus be used to help plan effective well treatments. Essentially, a DST is a temporary well completion effected for the purposes of sampling the formation fluid and establishing the probability of commercial production. Pressures were first recorded on DST's to ascertain proper test tool operation. As a result, the early pressure recording devices were rather insensitive. Recognition of the potential value of interpretation of DST pressure behavior led quickly to the development and use of accurate and sensitive pressure recorders. The recorders presently in use are generally of good quality. 9.1
Pressure Behavior on DST's
The reader who is unfamiliar with DST equipment and operational procedures is referred to the paper of van Poollen1 or the more recent presentation of McAlister, Nutter and Lebourg.2 A DST is run by lowering into the borehole on the drill pipe an arrangement of packers and surface-actuated valves. The packers are used to seal off the mud-filled annulus from the interval to be tested, and the valves allow the formation fluids to flow into the drill pipe. By closing the valves a pressure
Fig. 9.1 Diagramof currently operationalDST tool. (After McAlister, Nutter and Lebourg.")
~
85
DRILLSTEM TEST PRESSURE ANALYSIS
buildup can be obtained. A pressurerecord of the entire flow and shut-in sequence is obtained. Fig 9.1 is a schematic diagram of a currently operational DST tool. Fig. 9.2 shows the sequenceof operations of this tool from running-in the hole to retrieval. The appearanceof a pressurerecord from a drillstem test is shown schematically on Fig. 9.3. The section labeled A shows the increase in hydrostatic mud pressure as the tool is lowered into the hole. When the tool is on bottom, the maximum mud-column pressure is obtained. Setting of the packers causes compressionof the mud in the annulus in the test interval, and a cor-
respondingincreasein pressureis noted at Point B. When the test tool is opened and inflow from the formation occurs, the pressure behavior is as shown on Section C. After the test tool is closed, a period of pressure buildup labeled D results. The first flow and shut-in period is usually followed by a subsequentflow and buildup period, as shown. Finally, the test is ended and the packers are pulled loose, causing a return to hydrostatic mud pressure (Point E), and then the tool is pulled (Section F). Fluid recovery from the test may be estimated from the contents of the drill pipe or from the amount recovered at the surface if a flowing DST is obtained. The double shut-in method of testing is the most
common test procedure in use. The eventsinvolved are referred to as the initial flow and initial shut-in periods and the final flow and final shut-in periods. The initial flow period is usually of 5 to 10 minutes
t
... ~ ~ (I) (I)
... ~ Go
TIME --Fig. 9.3 SchematicDST pressurerecord. duration and is primarily for the purpose of allowing the equalization back to static reservoir pressure of the fluid in the filtrate-invaded zone near the wellbore. Both the static mud-column pressure and the setting of the packer cause mud filtrate to be squeezedinto .the formation. The brief initial flow period is designed to relieve this over-pressured condition and restore the formation to a near-original state. The initial flow period is followed by an initial shut-in period of about 30 to 60 minutes. This initial flow and shut-in sequence enables a good estimate of the static reservoir pressure to be made. At the beginning of the second flow period, the formation is hopefully restored to initial conditions and the natural flow behavior of the test zone can be obtained. This second flow period generally runs from 30 minutes to 2 hours or so. The final closed-in pressure buildup is usually slightly longer than, or at least equal
-M.F:E.DUAL CONTROL VALVE CLOSED '\
ru ~'I '\, (I \;'
SAMPLE (2750CC) RETRIEVED UNDER PRESSURE BY- PASS VALVE OPEN
w) Il-.J
WELL
FLOWING
WELL
SHUT-IN
OUT
/SAFETY SEAL PACKER DEACTIVATED (a PULLED LOOSE
OF
HOLE
Fig. 9.2 Sequenceof operationsfor MFE tool. (After McAlister, Nutter and Lebourg,2)
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
to, the second flow period. It is common in low-permeability reservoirs to employ even longer final buildups in order to obtain interpretable pressure buildup data. In addition to the common double shut-in tests, it is now possible to run drillstem tests with an arbitrary number of flow and shut-in periods. In a later section we will discuss this type of test. 9.2
Operational Considerations in Obtaining Good DST Pressure Data
Various factors govern the quality of DST pressure data. Not only must the reservoir parameters directly affecting pressure behavior be considered, but also care In the measurement of fluid recoveries and flow rates must be exercised since these quantities must be known for pressure analysis purposes. Many of the points to be mentioned in this section are discussedin greater detail by Maier,3 Dolan et al.4, and van Poollen.1 The engineershould consult, as freely as he can, with his colleagues or with DST service companies to gain any knowledge of a general nature concerning DST behavior in a particular formation. Frequently, many of the operational factors may be ascertained from previous experience in the same geologic province. Perhaps the primary consideration in planning a test is the maximum time in the hole with the test tool that can be tolerated by borehole conditions. If it is estimated that the on-bottom time during the test should be, say, 2 hours, then the test must be planned accordingly. Generally speaking, the first flow period on a DST should be at least 5 minutes and the initial closed-in period at least 30 minutes. This normally will allow expansion of mud trapped below the packers and pressure equalization in the filtrate-invaded zone so that a good estimate of static reservoir pressure can be obtained. The length of the second flow period (on a double shut-in test) is generally dictated by experience and prevailing conditions. The weaker the "blow" at the surface, the slower the rate of formation fluid influx and the longer the second flow period should be. If the drill pipe fluid load increases to the point that the hydrostatic pressure of the fluid column kills the inflow, then the final buildup should be started immediately. For tests with a weak surface blow throughout the duration of the flow period, the tool must be left open longer to sample the formation effectively. The final shut-in time should be at least equal to the flowing time if an accurate extrapolated pressure is to be obtained and if permeability changes nearby are to be detected. The lower the formation permeability, the longer the desired final pressure buildup. For a kh product of less than 10 md-ft, shut-in times of at least 2 hours are recommended. For higher kh values, times of 30 minutes to 1 hour may be sufficient. For accurate pressurereadings, service company personnel should be apprised of expected conditions (including estimated reservoir temperature and pressure
range) and the over-all test objectives so that the proper clocks and pressureelements can be selected.The most recent date the pressure bomb was calibrated at the expected conditions should be of interest. Prior to pressureinterpretation one should evaluate the accuracy of the pressure gauges by comparing their recorded pressuresat severalkey points. The hydrostatic pressure of the liquid recovery should be calculated and compared with the final flow pressure. The volume of liquid recovery should be carefully measured.The liquids recovered, both separatelyand in contaminated mixtures, should be adequately described and density measurementstaken. Gas flow on drillstem tests should be measured as accurately as possible at several equally spaced time intervals throughout the flow periods. The rate of liquid recovery can be estimated at any time by converting the rate of change of hydrostatic pressure in the drill pipe to a liquid production rate. 9.3
Use of Pressure Buildup Theory on DST Data
Pressure buildup analysis theory, as presented in Chapter 3, has been found to be applicable to analysis of DST pressure buildup data. The basic assumptions of pressure buildup theory-radial flow, infinite reservoir, single compressible fluid-are fairly well suited to DST conditions. On flowing DST's the assumption of a constant producing rate is sometimes even fulfilled. However, on a non-flowing liquid recovery test, the flow rate usually decreasesthroughout the flow period. Dolan et al.4 have shown that as long as the difference in the initial and final production rates in the flow period prior to the pressure buildup is not extreme, the average production rate can be used as a good approximation in pressure buildup analyses.This is especially true if the rate of change of the production rate with time is constant. On non-flowing liquid recovery tests this is frequently the case. The average rate of production is determined, of course, by dividing the fluid recovery by the length of the flow period. This applies for both the initial and final pressure buildups on a double shut-in test. The conclusion of Dolan et al.4 is usually acceptable as a practical matter. Odeh and SeliglOhave presented a means for calculating the proper production rate and flowing time values for use in pressure buildup analyses in instances in which the shut-in period is preceded by a short, variable-rate flow period. Use of their method yields greater accuracy in kh and s values for variablerate cases. The static formation pressureis estimated from extrapolation of the plot of P,o.vs log[ (t + ~t) / ~t] where ~t is shut-in time and t is the flowing time prior to shut-in. The appearance of the pressure buildup curves for a typical field case is shown on Fig. 9.4, from the paper by Maier.s If the initial flow period was sufficient to
DRILLSTEM TEST PRESSURE ANALYSIS
87
relieve mud compression effects and to allow the formation to expel most of the filtrate invasion, the initial buildup should extrapolate to the true formation static pressure. If it was not, then a higher value may result. The extrapolated pressure value from the second buildup curve should be fairly close to that from the initial buildup. If it is appreciably lower, then one could conclude that a very small accumulation had been tested and that significant depletion had occurred on the test. Since the inference of a small reservoir is based on comparison of extrapolated pressures, the importance of careful determination of these quantities cannot be over-emphasized. To calculate the kh product of the formation, the familiar transient pressure technique which utilizes the slope of the buildup plot is used, kh = 162.6 qllB
m
'
sure. MaierS presented a convenient simplification of Eq. 9.1. He assumedtypical values of cf>= 0.15 and rID= 0.333 ft. In that case the skin effect formula becomes
(9.2) The transient drainage radius during a DST is also of interest. As discussed in Chapter 11, the approximate relationship is
0.000264.1..kl 2 = , 0.25.
where m is the slope of the buildup plot in psi/cycle. The B and /A.values must be estimated from some type of correlation.5 A correlation for /A.is given in Fig. 9.9. The skin factor cannot be determined by the method of Chapter 3 because flow time and closed-in time are of the same order in drillstem tests. The skin factor is determined through use of the equation for the flowing pressure immediately prior to shut-in. From Eq. 5.2, Pwf = Pi -kh 162.6q/A.B[ log~ kt
where m is the slope of the buildup curve used to determine the kh product, and t is the total flow time. In the case of a flowing test in which the rate is fairly constant, Pavgis replaced by the true, final flowing pres-
-3.23
+ O.87s1.
If the rate q had been constant during the flow period, then Pwf would be the true value for the final flowing pressure. If the rate is not constant, a better approximation for this value is the average flowing pressure during the flow period, which we shall call Pavg.Rearrangement of the above equation yields the following expression for the skin factor.
.,..ucr8
A reservoir boundary at a distance r e from the well will be reflected in the pressure behavior of the well at a time " estimated from the above relationship. Thus, the drainage radius corresponding to a time t is estimated by
In the case of multiphase flow the total compressibility and total mobility of the reservoir fluid system must be substituted for the corresponding single-fluid quantities as in other transient pressure analysis techniques. In the case of DST's of extended duration, it is sometimes possible to infer the presence of reservoir heterogeneities within the radius of drainage affected by the test. The spe~ific behavioral attributes of such heterogeneities as faults, permeability pinchouts, etc., will be discussed in Chapter 10. The papers on DST pressure behavior which have appeared in the petroleum literature are well illustrated with field examples. An outstanding collection of well documented field examples can be found in the paper of Ammann.6 Pressure buildup behavior from one of Ammann's examples is shown on Fig. 9.5. 9.4
Fig. 9.4Field exampleof DST pressurebuildupcurves, (Mter Maier,')
Analysis of DST Flow Period Pressure Data
For DST's which flow at constant rate, such as gas tests and some oil tests, it is possible to analyze the flowing pressure behavior by means of the transient pressuredrawdown analysismethods presentedin Chapter 5. For the-more common low-permeability cases in which the pressure behavior during the main flow period is essentiallya record of the buildup in fluid head in the drill pipe and the flow rate is not constant, it is possible to analyze the flow period pressure behavior using the multiple-rate techniques of Chapter 6. To use these methods it is necessaryto convert the pressure
88
rise in the drill pipe to equivalent fluid production rates. This is accomplished, knowing the density of the produced fluids and the internal diameter of the drill pipe, by converting fluid head to cumulative inflow as a function of time. The slope of this cumulative inflow vs time curve is the production rate. The method of Section6.1 may be used to analyze these data. 9.5 Multiple-Rate DST's Recent developmentsin testing equipment2have produced test tools which can be opened or closed an arbitrary number of times without disturbing the packer seat. Also, by regulation of choke sizes it is possible to vary production rates on flowing DST's. This clearly opens a myriad of possibilities when one must design a DST. Fig. 9.6 shows a multiple-flow period DST pressure behavior record, taken from the paper by McAlister et 01.2These authors showed that with only slight error one may disregard prior flow and buildup periods and simply aRalyze each pressure buildup by the familiar Prosvs log[ (t+ ~t) / ~t] plot. Thus, the analysis of data from this type of test is of the same order of difficulty as that encountered with, say, the ordinary double shutin test. When should one choose to run a multiple-flow
period DST as opposedto the conventionaldouble shut-in DST? We shall offer some simple guides that can be used to answer this question from the reservoir analysis point of view. The multiple-flow DST can help substantiate reservoir depletion implied by comparison of the extrapolated initial buildup and second buildup
pressures.If the extrapolatedpressureson subsequent buildups confirnl a pronounced downward trend, then the presenceof a small reservoir can be inferred without the need for retesting. Multiple-flow DST's obviously lend confidence to the values calculated for the kh product and skin effect by providing additional calculation results for com-
PRESSURE
~UILDUP
FLOW TESTS
IN WELLS
parison. Also, change in the skin effect on successive buildups may give a clue as to whether the well might clean up when put on permanent production. 9.6
Practical Considerations in OST Interpretation
Test interpretation, in addition to estimation of liquid recovery, requires preliminary interpretation of the pressure charts. The charts should be examined carefully, first to ascertain that the tool operated properly, and second to verify that the pressures during the test were measured accurately. The accuracy of the gaugesmust be judged by comparison of key pressures against the computed mud pressure. On Fig. 9.7, taken from a paper by Black,ll the typical pressure chart configurations for successfulsingle-flow period DST's and for common types of failures are shown. Pressurecharts should always be inspected to ascertain proper tool operation. Pressurechart configuration will also vary, depending on the productive capacity of the zone being tested. In high-permeability zones, "critical" flow effects may cause the flow of fluid into the drill pipe through the bottom choke to be independent of the pressure inside the drill pipe. Critical flow will produce a nearly constant pressure throughout the flow period. Low-permeability formations are normally revealed on the pressurecharts by extremely low flowing pressures. On Fig. 9.8, typical pressure chart configurations for various reservoir flow conditions are shown. To aid further in DST interpretation, an additional figure from the paper by Black11 has been inclu~ed. A correlation of API gravity with viscosity is presented on Fig. 9.9. This figure can be used to estimate the viscosity needed in DST pressure analysis. 9.7 Wireline Formation Tests In addition to the commonly used DST methods, the wireline formation testeris often used to test formations in sand-shalestratigraphic sequences.Oftentimes, lack of a packer seat and sloughing shales prevent use of the conventional DST technique. Essentially, a wireline tester consists of a sampling chamber or chambers of several gallons capacity. These chambers are connected to an opening in a pad that is forced againstthe wall of the hole to effect a seal. Firing a shapedcharge or hydraulically forcing a tube from the center of the
0 5
51
66
-TIME,
Fig. 9.5 DST pressurebuildups from Arbuckle formation. (After Ammann.')
AND
112
156
246
MINUTES-
Fig. 9.6 Multiple-flowDST pressurerecord. (After McAlister, Nutter and Lebourg.")
DRILLSTEM TEST PRESSURE ANALYSIS
89
pad establishes communication between the chamber and the formation. The tool is run on an electrical logging cable and the valves which open and close the sample chamber are controlled from the surface. The pressure behavior during sampling, as well as a final pressure buildup, is recorded. More detailed descrip-
tions of the wireline tester can be found in the literature.7 A pressure analysis theory for the wireline tester has appeared in the literature.s The basic features which distinguish wireline tester pressure behavior from that of a conventional DST are the small fluid sample
-CRITICAL ---NON-CRITICAL
FLOW FLOW
TOP CHART
4,
I
,I
BOTTOM CHART
1,
",
JL
I
j'
I
I
1
TIMI PRESSURES:
":'
I--I
,.)
~':'-I-,
.TiMI
.
A-INITIALMUO 8-~CKER SQUEEZE C -AVE. FLOWING O-SHUTIN E -fiNAL MUO o-c -~AWOOWN
EVENTS:
1-lat~1N ~-LAST~* c-~~TTOM 4-~ SET I -~ ~.O I-~ Cl.DKO ,-aaD~~T[ II -tWAUZlNG VALVE O~[N[O I -NCR£R IM[AT[O
J -..~
WT , -LASTTIR8L[ OOT
I-~_*
1- FLOWp[~
TYPICAL CHARTS FROM A SATISFACTORY TEST
(TOP
GAUGE
1- SI«JT* P[RaI 4-TPlA.UMG OUT
NOT
BLANKED
lEGEND:
OFF;
-TOP
BOTTOM
GAUGE
BLANKED
OFF)
CHART (OR BOTH CHARTS)
---BOTTOM
CHART
(CRITICAL FLOW)
"-'
TOOL FAILED TO OPEN
PACKER FAILED
NO St«JT-IN PRESSURE
\
\ \
\ \ \ \ \ \ CHOKE PLUGGING
ANCHOR PLUGGED
TOP CLOCK STOPPED 8 STARTED
Fig. 9.7 Pressurechart interpretation:typical chartsfrom a satisfactorytestand charts from commontypes of mis-runs. (After Black.")
90
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
obtained and the fact that the flow is into a single perforation and is thus essentially spherical, as opposed to being radial in a conventional DST. Milburn and Howell9 studied results from 560 wireline tests and concluded that quantitative interpretation of this type of test is often difficult. Their study showed that the magnitude of the recorded pressureswas usually correct, but calculation of permeability from the results was not dependable. We conclude this chapter on DST pressure behavior
by reminding the reader that a properly run and analyzed DST is a very valuable evaluation tool. Like most other formation evaluation tools, it must be carefully applied and the results interpreted with the benefit of a large dose of engineering judgment and experience. In the references which we have cited on drillstem testing and also in the literature of the testing service comparnes, a large number of well-documented examples exist of DST behavior in various reservoir and operational situations. From these examples it is possible
2+3 NO
,
PERMEABILITY
1-2
VERY
LOW PERMEABILITY
SAND FACE POSSIBLY
PLUGGED
3
i
HK;H PERMEABILITY ON 3/16" BOTTOM CHOKE (CRITICAL
HIGH PERMEABILITY ON 1/4" BOTTOM CHOKE
FLOW)
(CRITICAL
HIGH PERMEABILITY WITH NO BOTTOM CHa
FLOW)
(NON-CRITICAL
,-J.l.
2
GAS TEST -UNLOADED .-WATER b- -TER
FLOW)
WATER CUSHION
EXCESSIVE
CUSHION RISING TO SURFACE CUSHION BEING PROOUCEO
FLUID HEAD INSIDE PIPE
FLOWING PRES~ ~TREAM OF CHOKE ~NED CONSTANT UNTIL THE 8ACI<
c- FLOWING DRY GAS d- TO«. SHUT IN
PRESSURE ~ INSIDE PIPE
TO LIQUID AC~TIDN BE~ EXCESSIVE, RESlA.TING
IN A DIMINISHING FLOW RATE.
NOTE:
TIE RtNIING A«
N AN) ~LING
OUT PERIODS
SHOWN CC»IPR£SSED
t:W TIAIE SCALE
t:W THESE CHARTS FM
CLARITY.
Fig. 9.8 Pressurechart interpretation:varioustestingconditions.(After Black!') ~
DRILLSTEM
TEST
PRESSURE
ANALYSIS
91
so
References
\oj ~ 4~ ~ ~ ~ : c 40 ..0 ~ ~ ~
1. van Poollen, H. K.: "Status of Drill-Stem Testing Techniques and Analysis", J. Pet. Tech. (April, 1961) 333-339. 2. McAlister, J. A., Nutter, B. P. and Lebourg, M.: "A New System of Tools for Better Control and Interpretation of Drill-Stem Tests", J. Pet. Tech. (Feb., 1965) 207-214.
0
; ~ 0 0 '" ~ 0 00
~ 35 ~ ~ ~ 30 ~
3. Maier, L. F.: "Recent Developments in the Interpretation and Application of DST Data", J. Pet. Tech. (Nov., 1962) 1213-1222.
~8 ~\ '\
4. Dolan, John P., Einarsen, Charles A. and Hill, Gilman A.: "Special Applications of Drill-Stem Test Pressure Data", Trans., AIME (1957) 210, 318-324. 5. Calhoun, J. C" Jr.: Fundamentals of Reservoir Engi-
0
0 ~
~
00
neering, (1955).
0
University
of
Oklahoma
Press,
Norman,
Okla.
...
~ ~ Z5 ~ ~ $ a: zo
00 0
~
~o --2-,..a. 0
c
15 0
I Z 3 4 5 6 7 VISCOSITY OFSATURATED REKRYOIR OILAT RES. TENP.. c p Fig. 9.9 Correlation of API gravity with viscosity. (After Black.D) to gain an understanding of DST behavior and the manner in which DST pressure interpretation results should be .J d d t l. t u ge as 0 qua 1 y.
..,-, ~1"f t' )" li"",
~ "'
;"
("'~
.1,
.ii" 0/""""""
"
"
!
J ;.
.;
,;
"!'"
(:i~j!'fk
i
.t
jI:);; ~;;,,'r
--
6. Ammann, Charles B.: "Case Histories of Analyses of Characteristics of Reservoir Rock from Drill-Stem Tests", J. Pet. Tech. (May, 1960) 27-36. 7. Lebourg, M., Fields, R. Q. and Doh, C. A.: "A Method of Formation Testing on Logging Cable", Trans., AIME (1957) 210, 260-267. 8. Moran, J. H. and Finklea, E. E.: "Theoretical Analysis of Pressure Phenomena Associated with the Wireline Formation Tester", J. Pet. Tech. (Aug., 1962) 899-908. 9. Milburn, J. D. and Howell, J. C.: "Formation Evaluation with the Wireline Formation Tester-Merits and Shortcomings", J. Pet. Tech. (Oct., 1961) 987-994. 10. Odeh, A. S. and Selig, F.: "Pressure Build-Up Analysis, Variable-Rate Case", J. Pet. Tech. (July, 1963) 790794. 11. Black, W. Marshall: "A Review of Drill-Stem Testing Techniques and Analysis", J. Pet. Tech. (June, 1956) 21-30.
Chapter
10
Effect of Reservoir Heterogeneities On Pressure Behavior
The pressure analysis methods which were presented in the preceding chapters are all based on the assumption of a homogeneousformation of uniform thickness. In addition it was also assumed that the producing layer is hori;ontal and its porosity and permeability distributions are isotropic and constant. In spite of these seemingly over-simplifying restrictions, the various pressure analysis theories have proven generally to have fairly wide applicability. In this chapter we will examine reasons for this and will study caseswhere the simple theory does not apply. The engineer who has inspected subsurface cores or surface outcrops of reservoir rocks does not have to be told of the inherent heterogeneousnature of reservoir rocks. Geologic processesthemselves dictate that reservoir rocks be non-uniform. The processes of sedimentation, erosion, glaciation, etc., all act to produce reservoir rocks that are non-uniform, although the nonuniformity is, to an extent, predictable. In order to properly qualify pressure analysis results obtained in the field, one must be familiar with the pressure behavior anomalies which are engendered by the commonly encountered reservoir heterogeneities. The subject of pressure behavior in heterogeneous reservoirs has received considerable attention in the petroleum literature in recent years. With the advent of the digital computer many mathematical model stu-
The presentation of the material on reservoir heterogeneitieswill consider, first, pressure behavior for heterogeneities which occur laterally away f.rom the well. This includes faults and lateral changes In the hydraulic diffusivity such as occur at fluid contacts. 10.1 Pressure Behavior Near Faults or Other Impermeable Barriers The pressure behavior of a well' near a sealing linear fault or other flow barrier in an otherwise infinite reservoir was first presented by Horner.1 The pressure behavior in this case is derived very conve~ently by employing a technique called the "method of images". In this formulation the effect of a fault is simulated by assuming the presence of another identical well producing at a symmetrical position across the fault, and then removing the fault. The image well interacts with the actual well so that no flow occurs across the fault. The resulting pressure drop at the real well due to its own production and the "interference drop" from the image well add together to simulate correctly the pressure behavior of the real well as though it were in the proximity of the fault. Mathematically, if the well is located a distance d from the fault, then its pressure behavior during flow at a constant rate is qJl. P0! = P. +""4;;kh El -4kt c/>Jl,CT02
dies have been made of pressure behavior in heterogeneous reservoirs. These studies have ranged from investigations of behavior of wells near faults to performance of wells in naturally fractured reservoirs. Generally, one would conclude that invaluable progress ff f h ..e in understanding the e ects 0 reservoIr eterogenelties has been made. We must al~o acknQwledge th~t ~tudies of heterogeneousreservOIrSare very much limIted by our current inability to simulate them in a mathematically rigorous way. Hopefully, the detailed study
. + El -kt cpp.cd2) + 2s ] (10.)1 Note that the actual distance of the image well from the real well is 2d. Th b I ldup behavior in an ideal case can pressure u be obtained by employing, in the usual manner, the method of su e osition and Eq. 10.1 to yield p rp 2 P = p' + -.!lL [Ei -c/>,ucr0 08. 47rkh 4k (t+~t)
.
.
of the distributions of pore space parameters together with the expected larger and faster computers will bring about additiona! progre~s in the investigation of heterogeneousreservoIr behaVIor.
[ .(
)
(
.
(
-Ei -Ei
( -~ ( -~
4k~t k~t
)+ Ei ( )]
)
-c/>Jl.Cd2
k(t+ ~t)
'.
)
(10.2)
EFFECT OF RESERVOIR HETEROGENEITIES
93
For t sufficientlylarge and for all but very early shut-intimes,Eq. 10.2 canbe expressedas + c/> d2 ) PWI= Pi -~ [ln~Ei(- k(~~ 1T + Ei ( -~
)] (10.3) kdt For all but very smallvaluesof d, the last Ei-function in the abovewill be zero until dt becomeslarge.Also, the other Ei-function will be essentiallyconstantuntil dt becomeslarge. Thus, early in the buildup,
From this equationwe seethat the slopeof the second part (late time) of the buildup curve is exactlydouble that of the early part. Also, the late-time portion of the curve mustbe usedto obtainthe extrapolatedpressure. The doubling of the slope is the distinguishing
(10.4) This equationtells us that the slope of the normal pressurebuildup plot will be unchangedfor the early
featureof the pressurebehaviorof a well neara fault. A theoreticalexampleof a pressurebuildup in a well located near a fault is shownon Fig. 10.1. The characteristicchangeof slopeis clearly evident. Homer also presenteda method for calculatingthe distanceto a fault whichis basedon the shut-intime at whichthe extrapolatedearly- and late-timesectionsintersecton the basicbuildupplot. If we equatethe righthandsidesof Eqs. 10.4and 10.5,we obtainthe Homer equationfor the fault distance: . c/>,uCd2 kt ) = 2.303log~. -El -0.000264 t+dt (10.6)
part of a pressurebuildup. As dt becomeslarge,Eq. 10.3becomes
With the usualEi-functionapproximation,Eq. 10.6car alsobe written as
PWI= Pi -~
PWI-.-~ -P.
(
[ln~- dt
Ei -~
47rkh
kt
~
)J
(
( 105 )
1n dt 21Tkh
0.000264--~ kt -t+dt~ yf/>,ucd2
.
200
180
ISO
E
.-
140
0
~
; 120 Q,
100
80
60
40 7 ,.
I'
I.
6
S04
In
':
'
4
3
2
I
~ At
Fig.10.1Illustrationof thetheoretical caseof a linearbarrierfault. (AfterHomer!)
,0
94
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
These equations (in practical units) are solved by trial and error for the distance d. The value 6.lz is the shutin time value at which the straight-line sections intersect on the POD'vs log[ (1+ 6.1)/6.1] plot. The Homer method gives good results if the value of I is large. For small values of I, Gray2 has shown that the Homer method (Eq. 10.6) is inaccurate. Gray,2 in an excellent review of methods for calculating the distance to a fault from buildup tests,' has shown that the Davis and Hawkins3 formula for distance to a fault seems to give consistently acceptable results [despite a restriction that it is strictly valid for
In the case of a well located near intersecting fallits or other multiple reservoir boundaries, the pressure behavior of the well will be composed of a multiple set of transients. After the early straight-line section on a transient pressure analysis plot, a second straight-line section of slope greater than two times the early slope will usually result. Such a case is indicated schematically on Fig. 10.2. Here we show a well which is situated in proximity to three distinct sections of the reservoir boundary. A schematic pressure buildup curve for this well is also shown. Note that the second straight-line section has slope greater than twice the initial slope.
'-' pressure buildups for [( 1+ 6.lz) /6.lz] > 30. The formula
The methods which we have outlined for fault distance calculation should not be applied if multiple boundaries
is
are suspected.In such casesthe transition period from
d = ~/1.48 X 10-4( ~ .,
)
(10.7)
rpp.c
Thi s equati'.on ISaIso valid f or pressured rawdown.tests .an Another procedure for calculating fault dIstancewas presented in the paper by Gray. This method involves graphically measuring the pressure difference between the first straight-line portion and the actual buildup curve during the transition part of the curve where the image well buildup becomes significant. Gray's equation for the pressure buildup or drawdown cases is
[
-kh 70.6qp.B -El
.( -M"002~rpp.cdz )]
the first to second linear segmentsmay be of extended length because of successivearrivals of the reflections from the various boundaries. This means that the intersection point is very much a function of the distance to d h b fb d t e num er 0 oun anes.
.
In the case of multiple boundaries near a well, about the best one can hope to do is obtain an estimate of the distance to the nearest boundary. If the flow or shut-in time at which the effect of the nearest boundary is felt can be estimated (this is done by finding the time 6.1 at which a pressure buildup plot becomesnon-
-6.p (10.8)
PLANVIEWOFRESERVOIR
In this equation the 6.p value is the difference between
L..-:~G\.:~~
the first straight-line section and the actual buildup curve at shut-in time 6.1.Again, the equation must be solved by trial and error for d, and is most accurate if I is large. This formula is predicated on the existence of a single fault. di The methods outlined b . ldabove for b calculating d f hi fault stance on a pressure UI up can e use or t s purpose on other types of transient pressure tests. For example, Russe1l4has illustrated the application of the Homer technique in calculating fault distance from tworate flow tests. If the intersection method is used for finding the distance to a fault, the Davis and Hawkins formula, Eq. 10.7, is preferable because of its accuracy and ease of application. It can be applied to all types of transient tests. The principal objection to the intersecof tiontest method time ofusually findingneeded fault distance to inferis the large twQ amount straight lines correctly. Because of the time involved, it is usually best to run a pressure drawdown test instead of a pressure buildup. The "6.p method" suggested by Gray is a much faster method in terms of test time; however, to use it one must be sure that the deviation fro~ the early straight line is a result of the presenceof a sIngle fault.
~
~ WELL 0
~~N~HOUT
ATER CONTAC
pRESSURE BUILDUPCURVE
f Co ;
.
I ,
STRAIGHT LINE SECTIONS /j '"
-log
t+~t At
Fig. 10.2Pressurebuildupperformancein the caseof multiple boundaries.
EFFECT OF RESERVOIR HETEROGENEITIES linear, tance2),
as in then
95
Gray's method for estimating fault the distance to the nearest boundary
be estimated
discan
-k1 111 -~
from
-k2 112-C
~ o~ow;-~~N-.6.t d~
0.00105~.
...,
(10.9)
'
only
order-of-magnitude
tance
value
f
'
s IS an approXImate
the
buildup If
the the
minimum
II
genera
The upon
distance
pressure run, The
of additional
od infer is boundary introduced.
c
'
time
lC1
y gives
calculated
the
dis-
at which
CD +
situation
the
constant-rate best test to
a w
to be nonlinear.
a multiple-boundary to find
hi h
I
ormu
dependent
is judged
is desired
Sp --,
results.
is quite
is suspected
to the
nearest
drawdown reason
transients
and
it
t
boundary,
REGION I
test is probably for this is that a
from
the
testing
other factors ma y cause will be discussed in the
.REGION
" = -.!:!I .,", CI
meth:;;=
WELL z- .T
2
"2 = ~ .2"2 C2
i"-
or Generally fault presence speaking, uniquely it is by difficult a tran-to
test because sient P ressure ilar effects. Some of these
'
2,u22
-2C2
't',uc
Thi
'
a --0
simnext
- '""--
LINEAR BOUNOARY BETWEEN REGIONS I ANO2
section.
10.2
Effect
of Lateral
Diffusivity
on Pressure
.Discontinuities stn ution
fl d di . b Such flow
in
f
'
UI
Changes
porosity requent
in Hydraulic
,
Behavior
I
and y
permeability '
occur
wit
hi
and
in .BOUNOARY
n
REGIONS I ANO2
discontinuities, although not complete barriers (as are sealing faults) , cause discontinuous
changes
in
the
hydraulic
diffusivity,
1] =
-f-,
to REGIONI and
't',uc thereby
have
in the
hydraulic
diffusivity
occur
tween
differing
geological
depositional
changes cur
an
in porosity
at fluid
because
contacts
Bixel,
Larkin
hydraulic
difIusivity The
fluids
that
the
in the
The
formation
dent
of
sections study
sides of may
of
constant
of the
be applicable
porosity is
Some
by that and
are
constant.
are
indepen-
homogeneous,
can
they
parameters differ
on op-
schematic
cross-
situations shown
a
drawdown
compressible
These
are
for
on Fig.
which
this
The
lution
for
pressed
results the
in terms
from
pressure of
the
computer behavIor following
evaluation at
the
-if
DEPDSITIDNAL UNIT A
"'"{;""O"'f~~,'~ELL
DEPOSITIONAL UNITB
10.4.
The mathematical solution to this flow problem is presented, for the constant flow rate case, in the reference.
Larkin
in
assumed
yet
reservoir
oc-
studied
thickness.
discontinuity.
practical
and
They
formation
to be constant;
to
also
published
viscosities
and
the
by Bixel,
contacts
situation
slightly
and
reservoir studied and van Poollen,"
be-
due
discontinuities
buildup
10,3.
are
permeability
and
assumed
posite
reservoir
and
have
of linear
on Fig,
10.3 Idealized
compressibility.
reservoir
compressibilities
isotropic
units
or oil-water
Poollen5
effect
Fig.
boundary
REGION2
a
Changes
Changes
and
on pressure
pressure
at the
as gas-oil
van
is shown
the
are
such
idealized
authors
behavior.
permeability.
of the
behavior.
pressure
in viscosity and
study
on
and
of changes
theoretical
these
effect
BETWEEN
reservoIrs.
of well
parameters:
the
so-
are
ex-
I..Fig
a 10 4
reservoir Larkin
Sc
h ema
...j ti ' c
situations
cross-sec
ti. ons
for which
and van Poollen"
0 f some prac t.Ica
the theory
of Bixel,
may be applicable.
I
96
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
M = k2IJ.1/k1IJ.2 . Fig. 10.5 shows a set of theoretical pressure drawdown curves for the case of Sp= 100 and various M values. The ratio of hydraulic diffusivities, 1]2/1]1'is equal to M/Sp. A low value of M therefore corresponds to a reduced hydraulic diffusivity in Region 2, beyond the discontinuity. The greater the reduction in hydrau-
.
lic diffusivity from the region containing the well to the region beyond the discontinuity, the closer the slope change will approach a factor of two, as with faults. Gas-water contacts may not be distinguishable from a
~E ~ '::' ~.
fault in practical cases.Large increasesin hydraulic
~
diffusivity across the discontinuity will cause the pressure drop to arrest and become essentially constant.
':!.~ ~.
.:::0.::'" ,...,-,0.." uo...1
Some synthetic pressurebuildup curves from the Bixel, Larkin and van Poollen study are shown on Fig. 10.6. On these curves the hydraulic diffusivities in the two regions are assumedto be constant and the results for various mobility ratios are shown. For M = 1, the
-"
usual homogeneousreservior behavior results. For
!.:!:.M
M >1, the effect of the more mobile fluid in Region 2
At
is to cause the pressure buildup curve slope to flatten. T s e aVlor IDlg t e easty IDlsta en for the common bend-over in a buildup which occurs in a well producing from a bounded reservoir. For M < 1, the pressure buildup curve slope increases after the effect of the discontinuity is felt. Depending on the M value, the hi
b
h
..
h
b
.
1
.
k
.
b 10 6 P (After Bixel, Larkin and van Poollen.")
Fi g.
.ressure
ill
ld
up
curves
f
or.
0
001
--, <
M
<
1
000
buildup curve either steepens or flattens in the same way as does the radial pressure profile. Thus, the pressure buildup curve shape is a reflection of the reservoir
./ I. II
a..3048.. 'w'7.R-
II
,,-
II
,-" UO;I"
So'~
II II 10
!i!J1e
~ N "
~ ~
:10"
EXAMPLES
II
--
-
"
0.-
20
.i .~
01
I
~
~
..
t. =00002636 (~) + (fr.) ( Fig, 10.5 Theoreticalpressuredrawdowncurves. (After Bixel, Larkin and van Poollen.") -'c~
.
EFFECT OF RESERVOIR HETEROGENEITIES
pressure profile. This effect has been pointed out also by Weller. 56 As may be seen from these curves for large or small M values, large shut-in times may be required to extrapolate to the correct reservoir pressure. Analysis of the early part of the buildup curves (and also drawdown curves) in all cases will give correct k1h1 and skin factor values. .qtp. Bixel, Larkin and van Poollen suggestthat, to obtaIn the distance to such a discontinuity, one use an overlay technique to compare the field data with the theoretical curves. Those interested in the details of this type of analysis are referred to their paper. A large suite of
97
steady state conditions are attained. Some theoretical pressure decline curves from the subject study are shown on Fig. 10.8. During early times at which drainage boundary effects have not been felt, the pressure behavior at the well in the two-layer caseis given by Pi-Pw! -In t-lny 41T(kh)t
-1
k1h1 In ¥
theoretical pressure behavior curves is included in the paper.
+ k2h2In ¥
2
t
(kh)t (10.10)
van Poollen37 also studied the Those effect of Bixel radialand discontinuities onhave pressure behavior. Pressure Performt t d th this aspect h r of w 0 are Inferesde t In ance are re erre 0 elr pape ... ' ..p In thIS reVIew of, the effect of lateral heterogeneItIes
.. equation and w?ere (kh)t = k1h1+ ~2h2. From this FIg. 10.8, we see that pnor to boundary effects the pressure behaVIor IS that 0f a sIngle-I ayer reservoIr WIth
on pressure behavIor, we have the sImIlar appearance (qualitatively) of poInted different out situations on
is proportional to the ,kh product of that la y er. ,
pressure buildup curves. The point of the presentation thus far, and that which follows, is that unique interpretation of heterogeneoussituations from pressure behavior alone is not usually possible. However when pressure behavior is combined with geological ~nd petrophysical information, it may indeed be possible to tul t bl . d I pos a e a reasona e reservoIr mo e.
In the case sh~n In FIg, 10.8,.k:~k2 = 4, h1/h2,= 0.05 and cpl/cp2.- 2. Note that Imtially the fractio.n of tota~ flow which comes from Layer 1, ql (t) / qt, IS approXImately equal to k1h1/(kh)t = 0.166, After boundary effects. are felt and semi-steady state is reached, the fractional flow rate from ~ayer 1 becomes equal to CPlh1/(cph)t= 0.091, At semI-steadystate the fractional .. flow from Layer 1 IS proportional to the
10.3 Pressure Behavior in Layered Reservoirs Perhaps the most common type of heterogeneity we ordinarily think of is that which results from various cycles of sedimentation-a set of heterogeneouslayers. In reservoirs composed of stratified layers, the most important question is whether there is significant
pore volume of Layer 1. ~ Becauseof the rate adjustments which can occur between layers, there may be a long transition between the early transient behavior and the onset of semisteady state. For the two-layer casesstudied by Lefkovits et al., the average value of time for occurrence of
.
. .
..
.
..
kh = (kh) t. The pressure drop IS proportional to In t Ius a const ant , and thera te 0f d epIeti' on 0f each I ayer
interlayer pressure and fluid communication or lack of it. If unrestricted interlayer crossflow can occur, the reservoir behavior will be analogousto that of a singlelayer reservoir having the average properties of the layered system. If the discrete reservoir layers com-
LAYERI
municate only by means of a common wellbore, then they will perform in a much different manner, performance of bounded reservoirs composed of The stratified layers was investigated theoretically for
LAYER2
t ~~~:==~=~::===~:J'--:..1
the no-crossflow case by Lefkovits et al.6 The idealized reservoir system which they studied is shown on Fig, 10.7. Each layer was assumedto be homogeneousand isotropic but of differ~t porosity and permeability, To-
LAYERn
( A) VIEWOF RESERVOIR 1..!::~ re --I
gether with the other usual single-fluid study assumptions, a mathematical solution was found for the pressure behavior which results when the well produces at constant rate, It is important to realize that a constant producing rate from each layer is not assumed,Rather, the total rate is assumed constant. This means, then,
jJ:. IMPERMEABLE ~ BOUNDARIES ~~ ~
I I i 1: ..: r~
~I,.p! I) hi K2."2 j=th2
that..".differential depletion between the layers can cause theIr respective producIng rates to vary untIl semI-
Fi g.. 10 7 Layered reservoir . system sue t di d by L efk OVlts, .
kn,.p;) I
:I hn (B) CROSS SECTIONAL VIEWOFRESERVOIR Hazebroek,Allen and Matthews,"
~8
PRESSURE BUILDUP AND FLOW TESTS IN WELLS 60 -klhl
I-C
+ k2h2
k =
,;1~40 Co
hl+h2
V
-~lhl+~2h2
~- ~ Co
~=
hl+h2
ii
hl+h2
=10
+-
0" 20
=
00
0 I
103
104
t
-105
= Ow
106
107
108
kt
"f>JLCrw2
.15
_I
.:::
+-
00
.13
-0"
0" .11
.09 1
10
103
t
Ow Fig. 10.8 Well pressure decline and fractional
104
-105
=
kt
106
~JLCrw2
flow rate from one layer for a two-layer
Hazebroek,
108
reservoir.
(After
Lefkovits,
Allen and Matthews.")
semi-steady state was about 50 times as great as for a single-layer case with the same drainage radius. Lefkovits et al. also considered pressure buildup performance. Fig. 10.9 is a theoretical pressure buildup curve for a two-layer reservoir. As in a single-layer
After the straight-line portion, the buildup curve levels off (BC). This leveling-off corresponds in a singlelayer reservoir to the pressure's having almost reached its average value. However, in a two-layer reservoir the pressure again rises (CD), and then finally levels
reservoir,
off at the average pressure
there
is an initial
straight-line
Section
AB.
(DE).
The- rise in the por-
tion CD is due to the repressuring
of the more depleted,
more E
permeable
layer
by the less depleted,
less per-
meable layer. The Section BC may have a slope only slightly less than that of Section AB, and thus, the two sections may be indistinguishable in some practical situations as shown in Fig. 10.10. See also Fig. 10.14, noting that one might draw a single straight line through the early
Pws
sections in a practical case. The slope of this straightline section is used to calculate (kh ) t in the usual manner. Obviously, value for p* ervoir. plot -3 10
10
-2
-I 10
AL t + tJ. t Fig. 10.9 Theoretical pressure buildup curve for two-layer reservoir. (After Lefkovits, Hazebroek, Allen and Matthews.") ~-
I
of
conventional extrapolation to obtain a cannot be carried out in this type of res-
Lefkovits log
et al. showed
(p -PW8)
vs ~t
that the trial-and-error which
Section 3.9 can be used to determine sure in these cases.
was
discussed
in
the average pres-
In the case where no barriers to vertical flow of fluids between the layers are present, the pressure behavior of the well ~ll be .considerably ~ifferent fro~ that which we have Just discussed. Dunng the penod 1960 to 1962, seven papers (Refs. 7 through 13) were pub-
EFFECT
OF RESERVOIR
HETEROGENEITIES
99
I I
lished on the theoretical behavior of reservoir systems composed of inter-communicating layers. Russell and Prats14summarized the practical aspects of the findings of these papers in a later paper. A schematic cross-sectionof the reservoir situation of interest is shown on Fig. 10.11. In the case of constant producing rate, it has been found that the flowing bottom-hole pressureperformance of a two-layer crossflow
lent homogeneous system of identical radial dimensions, and with (kh) t and «I>h)t substituted for kh and h,respectively, in the homogeneous-caseformulas. Thus, the transient bottom-hole pressure performance of a well in a reservoir with crossflow is given by PtO!= Pi -162.6q,p.B (kh) t
[ log
_J~2~ -3.23 «I>h)tp.crto
system can be represented almost exactly by an equiva-
}
.
(10.11)
""
..-
,-
300 CI
a.
.STRAIGHT-LINE w
SECTION
D::
=>
~ 2900 lLJ
D::
c.. lLJ
,;. .I;
0-.J
"'-/" ;., ~1
:J:
f
I
~ 0
.' :'
"?,.
2800
,
';'1"'\
i-'
1-.
,;71
'0
:!"j'q
CD
, ,
PRESSURE
2700
I
100 SHUT-IN TIME, minutes
1000
10,000
Fig.10.10Buildup in a two-layer reservoir. (AfterMatthews.")
-
---
---
CAPROCK
DIRECTION OFOIL FLOW-
i
i
RCK
---
I
--
---
~
---
-
--
Fig.10.11Schematic cross-section of a portionof a two-layer :::~~~
interlayer crossflow. (AfterRU-:~d
Prats.H)
100
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
In this equation,
t is expressed
in hours.
For large times (semi-steady h d b d b aVlor IS escn e y
..
state),
which
the pressure
be-
.
[ 0.000528
-.-141.2qp.B Pw! -P.
(kh)t t ( c/>h) t p.cr e2
(kh ) t
is producing
where
t is again expressed
The time
at which
..(10.12)
c.an WIth
properties
For
semi-steady
state flow,
flowing bottom-hole given by
-
1
=
0.07455
I
s ope -.2
783
pressure
the slope of the plot of
flows,
and is
re c
'I'
h)
t
buildup
be-
from
this type of reservoir of
reservoir
should
possess the
appearance
and should
(kh) t.
one
should
be able
to
detect
crossflow
either
justed to produce at constant rate, the linear coordinate plot of flowing bottom-hole pressure vs time should be ... linear after the semi-steady state IS reached. The the-
B .l.
that pressure
from pressure drawdown or from pressure buildup tests. In the case of a drawdown test with the well ad-
qB . /d ( h) pSI ay, r e C c/> t (
case pe~ormance reservoir theory
From the differences between pressure behavior with and without crossflow, it is sometimes possible to infer the presence or absence of crossflow. If the well
.(10.13)
vs time is constant
2 q
indicates
classical homogeneous yield an estimate
( h) 2 47.35 J?_(if>£!.!.-(dayS). t
measure-
The fact that the constant-rate be represented by homogeneous
buildup state begins is given
by
t ~
Field
havior in a layered reservoir with crossflow is similar to that in a homogeneous reservoir. Thus, a pressure
in hours.
semi-steady
rate.
type
equivalent + 1n!.!!- -0.75], rw
at constant
ments of flowing pressure. in reservoirs ~f this should P ossess the properties shown on this plot.
psi/hour.
oretical slope and time of onset of linearity. have been discussed previously. The drawdown curve at semi-
(10.14)
.
From these expressions it is apparent that so-called reservoir limit tests in reservoirs with crossflow should
steady state will have the same slope whether crossflow occurs or not; however, the time to the onset of linearity without crossflow will be of the order of 50
yield
times
accurate
measurements
of
the
total
productive
pore volume. Fig. 10.12 pressure plot
that required
expected shows an idealized flowing bottom-hole for a two-layer reservoir with crossflow
time
with
crossflow.
of linearity
from
By computing
the
equation
the
on Fig.
10.13 and comparing with the observed time, the presence or absence of crossflow can be judged. The the-
~
It: VI ~ ::)
TRANSIENT
162 6 Pwf = PI -'-Tk~~~
PERFORMANCE:
[
B
(kh)t log ~h~r:
t -3.23
]
Q.
~
SEMI STEADY-STATE
-J
~ 1 2
P
.p.wf
I
PERFORMANCE:
141.2nuB -.,r-lkhlt
[
0.000528 (~hl 't't
(k h)+ t .+
r. --0.75 rw
In
IAocr I. e
]
0 lI-
g l!) z ~
TIME AT WHICH SEMI STEADY IS REACHED: '4>hI JJ.c r 2 t~ 1136 8f""'trVOe hr
0
TATE q B
(khlf"
SLOPE
=
1.783
Ii
re
=
~ c
0.07455 rl
0
Dsi
2 (l#Ih)
t
B q~ c (l#Ihlt
DAY
. ~ hr
TIME
." 10.12 IdealIzed constant-rate flowmg bottom-hole pressure performance in two-layer reservoir' FIg. with crossflow. (After Russell and Prats.")
i~!;;, i
EFFECT OF RESERVOIR HETEROGENEITIES
101
oretical behavior in these cases is depicted on Fig. 10.13.
DECLINE
The pressure buildup characteristics of a well in a d .. h fl .. 1 thosem.(ljIh)t/Lcre layere reservoir WIt cross ow are S1In1arto h . A f h a omogeneous reserv01r. companson 0 t e pressure buildup behavior in a two-layer reservoir with and without crossflow is presented on Fig. 10.14. The case without crossflow has an initial straight-line section, then a slight flattening and next a rising portion. By qualitative comparison of observed curves with the curves in Fig. 10.14 and with other h thexamplesfl in Figs. 10.9 and 10.10, 1 d d one can ec1 ewe er cross ow 1Soccumng m a ay-
.
2
.
CROSSFLOW
TIME, DAYS 2 t~4735 (ljIh),/Lcre(DAYS) I ( k h) t
'i a.
out crossflow is often a near-exact representation for the behavior of wells in which production from two or more separate reservoirs is being comingled within the ., .'lg. wellbore. Future study of additional representative reserv01r mo~els WIll undoubtedly contnbute to further understanding.
~
a.
..
.
qB
e C{ljIh)t DAY
in
reservoir and well behavior. For1 example, the theoretid th ca1 mo d e1 emp1oye d m stud ying ayere reservOirS WI -
.
.
1783
.
I
(kh)t
ered reservoir. The studies which have been summarized in this section constitute the bulk of our knowledge of layered
reservoir behavior. Although the studies are highly theoretic a1, t h ey have been 0f va1ue m und erstandmg .r2
-
(.h)t~re(ln reo-0.75)DAY
t~4735(DAYS) Q
;. .; co -NO
...
.
O,O1267( kh)t 2., ---,
EXPONENT:.,.
SSFLOW . ~8~~~u~0 TIMESTHATWITHCROSSFLOW NO CROSSFLOW I TIME, DAYS
I,. 1o.13 Pressure perf ormance at constant pro d uction . rate with and without crossflow in bounded layered reservoir. (From Russell and Prats.")
5500
550
BUILDUP WITHCROSSFLOW
551
.:: .c
5515
oM
; ~ ~ .;.
STRAIGHT-LINE SECTIONS
-IQ a. . ~ 5520
8UILDUP
WITH
NO CROSSFLOW ~
[AFTERLEFKOVITS ET AL (REF6)] ~
0.000264Ikh)t t
8 5.5X109
\4-hltjJcrJ
552
k, -8 kl h
4'
" -80.4 .1
-1- 82.5,7-
553
hi
5535 -6 10
-5 10
10 -4
10-3
r W
.2000
10-2
6t
t+6i Fig, 10.14 Comparison of pressure buildup behavior in two-layer reservoir with and without crossflow. (After Russell and Prats!')
K)-I
102
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
10.4 Pressure Behavior in Naturally Fractured Formations ...two Most reservoIr engIneenng theory has been founded on the assumptionof a homogeneousand isotropic porous medium. However, many prolific reservoirs produce from naturally fractured and jointed formations. The occurrence of a high-permeability, secondary porosity flow system of this type generally requires analysis by something other than the normal homogeneousreservoir theory. Early contri~utors to f.ractured reservoir flow and pressure analysIs theory Include Pollard,I5 Freeman and Natanson,16and Samara.17 Perhaps the most extensive theoretical work on behavior of naturally fractured reservoirs is that of Warren andofRoot.IS These authors formulated theoretical model a single-fluid fractured reservoirasystem and solved mathematically for the pressure behavior. In the text which follows we shall outline their model and discuss some of their results. The
...a Idealization
of a heterogeneous
porous
medium
which was used by Warren and Root is shown schematically on Fig. 10.15. The primary porosity systemis homogeneous and isotropic, and is contained within an array of identical parallelepipeds. All of the secondary porosity is contained with an orthogonal systemof continuous, uniform fractures of uniform (anisotropic) permeability. Flow can occur between the secondary and primary porosities, but flow through the gross medium to the well can occur in the fracture systemonly. It is also assumedthat semi-steadystate flow occurs on a local basis between the primary and secondary sys-
tems; i.e., flow between the two systems at any point is proportional to the pressure difference between the systems at that point. The mathematical solution presented by Warren and Root for the case of pressure behavior at constant flow rate will not be repeated here. Rather, we shall present some results from numerical evaluation of the solution. All the results shown are for the infinite reservoir case and are described by two basic parameters: /I)= !J>2C2/ (!J>lCl + !J>2C2) and ,\ = ak1 rw2/k2 , where C1= total compressibility, primary system, ... C2= total compressIbility, secondarysystem, k1 = matrix permeability, k -ff . bili' 2 -e ective permea ty, fractures, and = shape factor systems.
controlling
flow
between
Fig. 10.16 shows a set of theoretical pressure buildup curves from this study. The pressure buildup for early time is the ~ame a~.in ~ homogeneous.reservoir. Then, as more rapId stabIlIzation of pressure In the more permeable secondary porosity system occurs, the pressure buildup lags. As inflow from the primary porosity system progresses,the pressure buildup assumesa trend parallel to that for early time. A field example of a buildup curve from a fractured reservoir displaying the parallel sections is shown on
,
Ii .
VUGS
i
I !
MATRIX
ACTUAL
RESERVOIR
FRACTURE
MATRIX
MODEL
FRACTURES
RESERVOIR
i
;
two
Fig. 10.15 .Idealization of a naturally fractured heterogeneous porous medium. (After Warren and Root.]")
EFFECT
OF RESERVOIR
HETEROGENEITIES
103
~ig.. 10.17. Odeh19 has studied a theoretical m~del simIlar to that of Warren and Root and has emphasized that fractured. reservoirs ~requently behave as homogeneous res~rvolrs. ~deh cites several field examples to support his conclusions. Thus, we conclude that naturall~ fractured reservoirs mayor may not display clear evidence of their pore space distribution on transient pressuretests. In the casewhere a "parallel section" type of pressure buildup does result, valuable reservoir information can probably be obtained with the Warren and Root theory. Pollard
l5 h
. d t t . as carne ou an ex enslve
stu
d
y 0
f
pres-
buildup is typified by the occurrence of a "tail" on the buildup. When the well is closed in for a pressure buildup, the pressure builds up first in the fracture system, giving the straight-line section as shown. Then the less permeable matrix, which is at higher average pressure, begins to feed fluid into the fractures. This causesthe rise above the straight-line section. Finally, the pressure will build up completely, as shown on the end of the buildup curve. The occurrence of the type of buildup displayed ~y Pollard
is quite
common
in fissured ...
carbonate
reservoirs
sure buildup behavior in fissured limestone reservoirs in conjunction with evaluation of acid treatments. A
in Venezuela and the United States. ~his ~pe, In fact, seems to be much more common t an t e type sug-
typical field example of a buildup from a fissured lime-
gestedby the study of Warren and Root.
stone reservoir is shown on Fig. 10.18. This type of
Pollard also presented a method for analyzing pressure buildups in fissured limestone reservoirs which has proved to be quite useful. The Pollard method of analysis is based on the assumption of semi-steadystate flow during the pressure buildup period. One plots log (p-P",.) vs ~t as in the extended Muskat method (Chapter 3). From this basic plot it is possible to obtain estimates of fracture system volume and the extent of any damage zone. An example of the Pollard method is shown on Fig. 10.19.
4000
3
CI 3
It should be noted that pressure behavior in naturally fractured reservoirs is similar to that obtained in
W. 0
~ ~ ~ ..; 3850 ~ ~ ~ Q. 3
layered reservoirs with no crossflow. In fact, in any reservoir system with two predominant rock types, the pressure buildup behavior is similar to that of Fig. 10.18. The geometry of the fractured system, the permeabilities involved, and the pore volumes of each rock type combine to yield systems which are far too complex for precise analysis with presently known techniques. There may be a future for probabilistic
~ ~
reservoir models in aiding description and analysis of these complicated systems.
.J
.J
~ 31
10.5 Pressure Behavior in Hydraulically Fractured Wells The discussion of reservoir heterogeneities thus far has considered pressure behavior for naturally occuring heterogeneousreservoir situations, e.g., faults, layers, natural fractures, etc. A very important and manmade type of reservoir heterogeneityis the hydraulically induced formation fracture. A large percentage of
3 INFINITE RESERVOIR 1.5.10-1FOR ALLCASES q. 115 STB/D t .21 DAYS
3
10 A Fig.
:
10.16 Theoretical Warren
+
1<5" 10t
buildup and
present-day well completions employ the hydraulic fracturing technique. It is important that we have a good understandingof the effects of hydraulic fractures on the pressure behavior of wells. I"
The orientation of hydraulic fractures is dependent on the stress distribution in the formation and the steps which may be taken to alter the stress distribution locally in the region near the well. If the leastl
curves.
Root.'")
(After
principal
stress
a vertical
fracture
in
the plane
formation usually
is results.
horizontal, If
then
it is vertical,
1
104
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
(\IE 450
WELL
A
"
Co)
~CI
~440
w a: => cn cn w 430 a: a..
, ""
/ ,,/
'
-l
~ w
"
~ 410
,, 0
.I,,'
,,
"
,,
~ 420 0 m
/
",
'
"
"
, ',
""
'
,"
,
"
,
"
"
I i
:
400
108
107
~
106
105 t + ~t
104
103
102
10'
~t
Fig. 10.17 Field buildup curve. (After Warren and Root.'")
2000
1900 (/)
Co
~ 1800 ::>
cn U>
~ 1700 a.. w
~ 1600
LI NEAR ~RTION
~ ~ 1500 ~ 0
ro
140 1300 I
10
CLOSED-IN TI ME, '- i:l!J..! ""'"
~
Fig. 10.18 Buildup in a fissuredlimestone reservoir. (After Pollard.'")
0
EFFECTOF RESERVOIR 'HETEROGENEITIES
_1~
1000
-
1
In
,Co
-
In
-
~
Co
I ICo
1
Will
A
-Skin resistance = 264 psi at 261 m3/D 3 Coarse nssure resistance = 50 psi at 261 m ~D Fair prasped far acidizatian ta remave ,kin (law FPI preclude, a large praductian increa,.) I
50
100
150
SHUT-IN
200
TIME IN HOURS
,
Fig. 10.19Curveshowingboth skin and coarsefissureresistance, WellA. (After Pollard.lI) then a horizontal (bedding plane) fracture usually occurs. Theoretical studies of pressure behavior for both vertical and horizontal fractures have appeared in the literature. First we shall consider pressure behavior for wells with vertical fractures. Prats et al.2O first discussed the performance of vertically fractured reservoirs for the caseof a compressible fluid. These authors considered large-time (semisteady state) constant-production-rate behavior for vertic ally fractured wells; however, transient pressure behavior at constant rate was not investigated. McGuire and Sikora21and Dyes, Kemp and Caudle22 employed an electrical analog to investigate the influence of artificial vertical fractures on well productivity and pressure buildup. They found that fractures which extend beyond 15 percent of the drainage radius away from the well alter the position and slope of the straightline portion of the buildup curve. They concluded that these effects must be considered both in the determina-
tion of the effective permeability of the formation and in any calculations of final buildup pressure. In a more recent paper, Scott23reported the results of an investigation of the effect of vertical fractures on pressurebehavior. This study was conducted with a heat flow model. Russell and Truitt24 obtained comprehensive information on transient pressure behavior in vertically fractured reservoirs through study of a mathematical model of a vertically fractured system. A schematic view of the reservoir situation which they studied is depicted on Fig. 10,20. A horizontal reservoir which is homogeneous, isotropic and completely filled with a fluid of small and constant compressibility and constant viscosDRAINAGE ~BOUNDARIES , J
#1
4
'1-~" ""/,,
h ! ~
ELEMENT
SYMMETRY
,'-, ''
J--
RE
I
~
FRACTURE
f -Xe
-
' /'~IL-BEARING
STRATUM BOUNDING SURFACES OF DRAINAGE VOLUME
Fig. 10.20 Schematic view of fractured well and accompanying reservoir drainage volume. (After Russell and Truitt."')
Fig. 10.21 Plan view of fractured reservoir showing position of symmetry element. (After Russell and Truitt.2')
106
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
ity is assumed. The reservoir is initially at uniform pressure Pi. Gravity effects are neglected. The drainage area of the well is assumed to take the form of a square. Thus, the situation is analogous to one well in a pattern of wells, as would be found normally in an oil reservoir. A plane, vertical fracture is assumed which extends over the entire vertical extent of the formation, is parallel to a drainage boundary, and is located symmetrically within the square drainage area. The effects of pressure drop within the fracture and production into the wellbore other than from the fracture are neglected. Because th e frac ture ext end s from to p to bottom of the f ti d orma on, an graVI t y e ffec ts are, n eglected the P robI b t d t di n . al dom "' n as
.
em
h
s o~.
Ii I
.
can
e
.F. ~
represen
Ig.
e
. m
wo-
me
Sion
...,
10 21
four fracture penetrations. From it we can obtain a better idea of the early-time behavior of the pressuredrop function. The pressure-drop function is the pressure drawdown expressed in dimensionless form, (PiPwf)/(q/l./4kh). Th.e very early-ti~e beha~or of ~e pr.ess~re-drop ~ction !usually at times of little practica! sIgnificance) I~ essentially as ~OUgh the flow were linear. ~t la~e tImes, after semi-steady state ~ress~re decline IS ~eached, t.he rate of pressure decline IS constant and IS ~roportional to the hy~ocarbon-fi1led pore volume, as IS the case for pure radial flow. Thus, we are assured h li mIt tests sh 0uld give valid resuIts m t at reservoir hi h fiCIa 11y fracture. d Th e mterme wells w care arti diate-
.
. . .
.
.
.
...
time from
behaVIor of the pressure function can Figs. 10.22 and 10.23 to depend greatly
be seen on frac-
UtiliZIng the a!i.sumptions and geometry ou1?ned above, a .mathe~atical model for the case of a smgle compressIble flwd was formulated and solved for pressure behavior.-at the well fo~ consta~t flow rate. F~r the mathematical and numencal details, the reader IS referred to the paper. A coordinate plot of the pressure-drop function vs
ture penetration. The deeper the fracture penetration, the nearer the performance approaches that for linear flow. The curve for linear flow is not shown but would lie slightly below the curve for Xf/Xe = 0.7. For small fracture penetration the pressure-drop performance is nearer that for radial flow. By superposing these pressure-drop functions, one
dimensionless time for various fracture "penetrations" is shown on Fig. 10.22. Fracture penetration is defined as the ratio of fracture length to length of side of the square drainage area pattern, and is equal to Xf / Xe. A coordinate plot of this type best shows the intermediate to long-time behavior of the pressure-drop function. Fig. 10.23 is a plot of the pressure-drop function vs the logarithm of dimensionless time for the same
can obtain illustrative pressure buildup curves. A set of pressure buildup curves for various fracture penetrations is shown on Fig. 10.24. This figure illustrates the effects of fracture penetration on the slope of the buildup curve. For fractures of small penetration, the slope of the buildup curve is only slightly less than that for the unfractured (radial flow) case. However, for large fracture penetrations, the slope of the buildup
2.
,
kt : D '~C"2.
FOR tD
> 0.7,
APD IS GIVENBY
2.
,
I
t +l4127
2.
~
~
I 0
D
51
I.
.d
Xe
xf
.D+Q8036
I. D+0.5371
I.
D+0.3980
I. O. °,-.
0 0 0
.00
.,
0.4
..7
tD
Fig. 10.22 Vertically fractured reservoir, dimensionless pressure drop vs dimensionless time. (After Russell and Truitt.")
EFFECT OF RESERVOIR HETEROGENEITIES
107
2.2 2.
t
I.
kt : D .ucxeG X f-!e.t
I.
[;]
I.
0
-
.: I. I.
1-4 xf
o. o. o. o. 0 10-6
10-5
10-3
10-
10-
1.0
tc
Fig. 10.23 Vertically fractured reservoir,dimensionlesspressuredrop vs logarithm dimensionless time. (After Russelland Truitt.") ~.9 4.0 4.1 4.2 2
4.3
c 1.0 (LINEAR
xe : 0.7
4.4 4. '" ' c.~ 0" ~ ::l .c
l
I
4.6
v
a-
4.7
"0
c. ~
4. ~x
4.
5.
'Q
L_~J
5.
5. 5. 105
104
10
10
10
t + ~t ~It
Fig. 10.24 Vertically fractured reservoir,calculatedpressurebuildup curves,tD= kt/~,u.cx.' = 4. (After Russelland Truitt. ")
I
108
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
curve becomes progressively smaller. This means that in field interpretation of buildups, the effects which are introduced by the fractured reservoir flow geometry can lead to erroneous calculation of kh, skin factor and
pressure buildup curve, the following equation is suggested:
average pressure values. The deeper the fracture penetration, the greater the over-estimation of the kh product from conventional pressure buildup analysis. This effect at least partially accounts for the appearance of "new pay" after some fracture treatments. This effect has been observed quite frequently in restimulation of older wells in tight reservoirs. Fig. 10.25 is a plot of fracture penetration vs the ratio of true to apparent kh value obtained by conventional analysis of fractured reservoir synthetic pressure buildups. This figure shows, for instance, that for a fracture penetration of 0.5, the apparent kh product as calculated from pressure buildup would be about 2.5 times the true kh product. Russell and Truitt24 also calculated apparent skin factors from analysis of fractured reservoir synthetic pressure buildup curves. Then they calculated apparent fracture length by means of the relationship
(10.16) ... After obtaImn~ (Xt).,I. from .Eq. 10.16, one can obtain (Xt)tru. from FIg. 10.26 by trial and error. For determination of average pressure, Russell and Truitt found that the Muskat method of plotting log (p -PW8) vs .o.tgives good results when the straight-line sectionof the buildup curve is difficult to identify. When the straight-line section is identifiable, the usual method of extrapolation to p* and correction to p may be used. The behavior of horizontally fractured wells has been investigated by Hartsock and Warren25 for the case of semi-steady state flow. An idealized cross-section through their model is shown on Fig. 10.27. They assumed the reservoir to be homogeneous, of constant thickness h, of anisotropic permeability kr and kz, and
(Xt)..I. = 2rwe-8.
true length of a hydraulically created fracture from a
-ACTUAL
~~
~
~
@
-0.8 fJ
~
f
5
::>-w
RELATIONSHIP
!:?!::
-)(
)(
0.6
5J x
~
~
~~
~
,"
0.9
---APPROXIMATION
~~
"'Q;~ ~ ~ 0.5 ...0
1,
~ ~ 0.7
0.9
0.6
4k -3.23 + 10g""i;;"C
u
1.0
0.7
I.
(10.15)
From this work, they obtained the curve of fracture penetration vs the ratio of calculated to true fracture length, which is shown on Fig. 10.26. For a fracture ~en~tr~tion of 0.1 or less, the calculated fracture length IS wIthin 10 percent of the true value. To estimate the
0.8
p -p log (Xl).,I. = 0.5 [--~~
~
0.5
1-1 xf
~
0.4
a.
~ a. ~ 2. ~ 0.4 II
0
0.1
0.2
0.3
0.4
0.5
x FRACTURE PENETRATION,..;-1xe Fig. 10.26Effect of fracture penetrationon fracture length
a:
as derived from skin factor formula, pressure
buildup. (After Russelland Truitt.") 0.3
I
h/Z+ I \
l'
j 0 j
.I
h.
I I
Lh..
THROUGH ALL PROOUCTION FRACTURe
RAOIAL FOR r.
0.2
! FLOW" r.
:I
~
0
0.2 0.4 0.6 0.8 FRACTURE PENETRATION, XfIxe
1.0
Fig. 10.25 Vertically fractured reservoir,pressurebuildup interpretation. (After Russelland Truitt.")
-h/ZOr.
r,
r-
r.
rt
Fig. 10.27Idealized cross-sectionof horizontally fractured reservoir. (After Hartsock and Warren.")
EFFECT OF RESERVOIR HETEROGENEITIES completely
penetrated
horizontal
symmetrical
by a well of radius fracture
109 rw. A single,
of radius rf and flow ca-
of horizontally
fractured
wells
ren26 in his discussion
was extracted
of Coats'
by War-
mathematical
model
pacity (kh) J is located at the mid-point of the reservoir. They assume there is no flow across the drainage ra-
for water movement about bottom water-drive reservoirs.27 Warren observed that a segment of Coats' re-
dius
suIts could
re and
is purely tion for
that
beyond
some
critical
radius
rc, flow
radial. A numerical solution of the flow equathese conditions was carried out. From the
numerical
results,
for various
apparent
combinations
A set of numerical
skin factors
were calculated
of parameters.
infinite nary
results for the case of rf/rw
= 200
fracture transient
reduction
in productivity.
conductivity. radial
than the fracture
radius,
dependent that the apparent of drainage skin fracture Further
treatment
the skin factor
factor radius. curves They
M.
A (M/2)
pressure
expression
-In!L
that the ordi-
with
-0.4045 r '0
t
s = ~ 2 M.
A (M)
-In.!.L
'
skin
behavior
effect
[ fracture
in
[
on
center
-0.4045, r,o
fracture
~
]
] ~
~op or ottom
(10.17) can be used to calculate these expressions,
of the transient
to tran-
is in-
can also beconcluded used in
design.
knowledge
He showed
flow
From the results shown and others, the authors concluded that for a radius of drainage at least four times greater
as being applicable
given by
is shown on Fig. 10.28. As pointed out by Hartsock and Warren, these curves show that a poorly designed fracture treatment can yield an increase in apparent skin effect and a consequent
be interpreted
sient flow in a horizontally fractured situation similar to that considered by Hartsock and Warren and with
geometrical
M
transient
flow
= h yk;Jk-;/rJ'
constant
obtainable
behavior.
and
In
A(M)
from Table
is a
2 of Coats'
paper.
The papers which have been noted representthe 6
"state in
5
of the art"
as far as transient
horizontally
fractured
have been additional for different fracture
4 R .!L. f
wells
steady-state orientations,
It is apparent that additional of transient pressure behavior
200
r.
models
could
3
knowledge.
2
10.6
contribute
Pressure
1
This
topic
reservoir 10
havior
is
investigations mathematical
to
the
industry's
in Non-
perhaps
Areas
incorrectly However,
in non-symmetrical
that for more
drainage
ideal situations,
classified since
areas departs
non-symmetrical
age areas can probably be thought of as being more naturally caused reservoir heterogeneities.
-2
was studied
-3
broek.28 By employing late reservoir pressure
-4
the pressure drop for any reservoir very early times, is given by
different
-5
in non-symmetrical
in detail
reservoir
Pi-Prof
-6~
behavior
by Matthews,
(kh)
l
/~
Fig. .10.28 Apparent slonal fracture
akin to areas
and Haze-
[1n~
kt
established
that
shape, and all but
+ 41T~
1n~+
kt
0.809
~ .."
from
the method of images to calcubehavior for a large number of
-F(~)+ ).=
a
be-
drain-
drainage Brons
shapes, these authors
q,u = 4-;kji""
as
pressure
-1
Pressure
There
model investigations fracture shapes, etc.
theoretical in realistic
Drainage
heterogeneity.
behavior
concerned.
significantly
Behavior
Symmetrical
pressure
is
+ 2S], (10.18)
",",
skin factor as a function flow capacity. (After Warren.2') ~~--
of limen-
Hartsock
where
A is the area of drainage
and
and shape-dependent
time function
given by
F (~)
is a
EFFECT OF RESERVOIR HETEROGENEITIES completely
penetrated
horizontal
symmetrical
by a well of radius fracture
109 rw. A single,
of radius rf and flow ca-
of horizontally
fractured
wells
ren26 in his discussion
was extracted
of Coats'
by War-
mathematical
model
pacity (kh) J is located at the mid-point of the reservoir. They assume there is no flow across the drainage ra-
for water movement about bottom water-drive reservoirs.27 Warren observed that a segment of Coats' re-
dius
suIts could
re and
is purely tion for
that
beyond
some
critical
radius
rc, flow
radial. A numerical solution of the flow equathese conditions was carried out. From the
numerical
results,
for various
apparent
combinations
A set of numerical
skin factors
were calculated
of parameters.
infinite nary
results for the case of rf/rw
= 200
fracture transient
reduction
in productivity.
conductivity. radial
than the fracture
radius,
dependent that the apparent of drainage skin fracture Further
treatment
the skin factor
factor radius. curves They
M.
A (M/2)
pressure
expression
-In!L
that the ordi-
with
-0.4045 r '0
t
s = ~ 2 M.
A (M)
-In.!.L
'
skin
behavior
effect
[ fracture
in
[
on
center
-0.4045, r,o
fracture
~
]
] ~
~op or ottom
(10.17) can be used to calculate these expressions,
of the transient
to tran-
is in-
can also beconcluded used in
design.
knowledge
He showed
flow
From the results shown and others, the authors concluded that for a radius of drainage at least four times greater
as being applicable
given by
is shown on Fig. 10.28. As pointed out by Hartsock and Warren, these curves show that a poorly designed fracture treatment can yield an increase in apparent skin effect and a consequent
be interpreted
sient flow in a horizontally fractured situation similar to that considered by Hartsock and Warren and with
geometrical
M
transient
flow
= h yk;Jk-;/rJ'
constant
obtainable
behavior.
and
In
A(M)
from Table
is a
2 of Coats'
paper.
The papers which have been noted representthe 6
"state in
5
of the art"
as far as transient
horizontally
fractured
have been additional for different fracture
4 R .!L. f
wells
steady-state orientations,
It is apparent that additional of transient pressure behavior
200
r.
models
could
3
knowledge.
2
10.6
contribute
Pressure
1
This
topic
reservoir 10
havior
is
investigations mathematical
to
the
industry's
in Non-
perhaps
Areas
incorrectly However,
in non-symmetrical
that for more
drainage
ideal situations,
classified since
areas departs
non-symmetrical
age areas can probably be thought of as being more naturally caused reservoir heterogeneities.
-2
was studied
-3
broek.28 By employing late reservoir pressure
-4
the pressure drop for any reservoir very early times, is given by
different
-5
in non-symmetrical
in detail
reservoir
Pi-Prof
-6~
behavior
by Matthews,
(kh)
l
/~
Fig. .10.28 Apparent slonal fracture
akin to areas
and Haze-
[1n~
kt
established
that
shape, and all but
+ 41T~
1n~+
kt
0.809
~ .."
from
the method of images to calcubehavior for a large number of
-F(~)+ ).=
a
be-
drain-
drainage Brons
shapes, these authors
q,u = 4-;kji""
as
pressure
-1
Pressure
There
model investigations fracture shapes, etc.
theoretical in realistic
Drainage
heterogeneity.
behavior
concerned.
significantly
Behavior
Symmetrical
pressure
is
+ 2S], (10.18)
",",
skin factor as a function flow capacity. (After Warren.2') ~~--
of limen-
Hartsock
where
A is the area of drainage
and
and shape-dependent
time function
given by
F (~)
is a
110
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
( kt ) p* -p F T~ = qp.c ~
vestigations are usually portrayed in terms of rock properties as a function of effective stress on the rock. The effective stress is simply the difference between
,
which IS the familiar pres~ure.correction function of Matthews et al. presented In FIgS. 4.3 through 4.9. Brons and Miller 29and also DietzSOhave shown that for semi-steadystate conditions
(
F ~
kt
)
= 1n~
C kt
'
..(10.19)
where C A is a shape-dependentconstant whose value has been tabulated. On Fi .10.29 is shown a tabula-
.
f dr . h g nd assocI ' ated C values 0 ., amage area s apes a A
tion from
Co~bInation ~f Eqs. 10.18 .and 10.19 YIelds the folloWIng expreSSIonfor the semI-steadystate case. = ~ qp. [ 47r~ kt
+ 0.809 +28]
-InCA
+ 1n~A
(10.20)
'
then Eq. 10.20 becomes P-PIO! = ~ [ ln~ + 0.809 + 2S] 41Tkh CArlO (10.21) At semi-steady state. therefore, the difference between the average pressure p and the flowing pressure PlOtdepends directly on the shape of the drainage boundary as evidenced by the shape factor CA, These shape factors given in Fig. 10.29 may be used to calculate p from a buildup curve as shown by Dietz. The implications of reservoir shape in pressure calculations were discussed in Chapter 4 and also in the references which have been cited. In addition, a comprehensive review of these considerations has been prepared by Ramey.81 10.7
and permeability. ... The effective s~ess which correspond~ to th~ maXl-
mum depth of bunal the rock has expenenced In geologIC time IS qwte Important. If one can esta bli s h th at
.
.. not expenence stressesd unng d epIeti on any rock will greater than those it has previously experienced, then it is unlikely that rock skeleton failure will be a prob-
lem. However, which today at their maximum depth inof sediments burial, depletion of fluidare from their
pores will subject the rocks to progressively greater stressesthan they have yet withstood. This may cause rock failure. The variation of porosity and permeability with ef-
If we note that ---qt Pi -P -~~
up to a high enough level, failure of the rock skeleton may occur ~d cause drastic reductions in porosity
.
DIetz ...a paper.
Pi-PIC!
the conmrnwng or overburden pressure and the pressure within the pores of the rock. As reservoir pressure (pore pressure) declines, effective stress increases forcing skeletal changes in the rock and consequentreductions in permeability and porosity. If effective stress builds
Effect of Pressure-Dependent Rock Properties All of the pressure analysis theories we have presented have been predicated on the assumptionof pressure-independent porosity and permeability. For the most part this is not a greatly restrictive assumption. However, we do know from laboratory studies and from observed pressure behavior in some wells that both porosity and permeability decrease as reservoir pressure declines. For reservoir rocks which are "normally" compacted, these effects are usually less than for those which have unusually high pore pressure, i.e., geopressuredreservoirs. Vander Knapp82and others 8. have studied the effect of pressure on rock properties. The results of these in-
fective stress generally displays hysteresiseffects. That is, restoration of pressure in a rock back to its original level generally will not bring permeability and porosity back to their original levels. Sandstonesand other clastic rocks tend to be more elastic in their behavior than carbonate rocks. Limestones often are somewhat plastic in their behavior. What does all this have to do with pressure analysis techniques? In general we should expect to observe a decline in calculated permeability from successivetransient pressure tests run throughout the life of a well in depletio~ reservoirs. In. ~any cases we do. It i~ not unusual In low-permeabIlity, geopressuredreservoIrs to observe declines in kh values from transient tests o~ the order of 30 to 50 percent over the first 2,000 .pSI of pressure ~op. !n norm~ll~ pressured reservoIrs, however, nothing this dramatic ISobserved. In fact, declines of 10 percent or so. 1!laybe observed, but because of variations of other kinds (two-phase flow effects, etc.) quantitative evaluation becomesdifficult. The differential equation for flow of a single, slightly compressiblefluid with pressure-dependentrock properties is identical in mathematical form with that for flow of a non-ideal gas with constant rock properties. Predictions of pressure behavior incorporating laboratory determined curves of porosity and permeability vs pressure can be carried out in a manner analogous to the non-ideal gas studies of Russell et al.ss 10.8 Concluding Comments The intent of this chapter on reservoir heterogeneity has been to display to the reader the "state of the art" as far as understanding and predicting pressure behavior is concerned. We believe it is essential for en-
EFFECT OF RESERVOIR HETEROGENEITIES
~
III
-5!-
Stabilized conditions kt for->
cpp.cA
~
In bounded reservoirs
0
EE3
3.45 31.6 0.1
S
Stabilized conditions kt for->
cpp.cA
I
2.38
10.8
0.3
I
1.58
4.86
1.0
0.73
2.07
0.8
1.00
2.72
0.8
2 I
EB
2 D
3.43
30.9
0.1
~I
2
()
3.45
8
3.32
31.6
27.6
0.1
1_I
+ I 4
I, I
I
I.
I' -1.46
0.232 2.5
-2.16
0.115 3.0
0.2
4 Eo.
I:i7
3.30
3.09
27.1
0.2
21.9
0.4
W ~
I
\:==-1 I
-I
I
.I'
.~I
4
600
-.t(~
I
1.22
3.39
0.6
1.14
3.13
0.3
2 EHE 3.12
22.6
0.2
I
-0.50
0.607 1.0
2
~
2 I
.II
1.68
5.38
0.7
EEIiE'
4 IL
.II ~~
2.36
0.7
EE
[ A )4
3 .t=~
In water-drive
2.56
12.9
0.6
[~EJ
[IJ
-2.20
0.111 1.2
-2.32
0.098 0.9
2 0.86
reservoirs
(~":"\ U
2.95
19.1
0.1
25
0.1
In reservoirs of unknown production charocter
1.52
4.57
0.5
(:\ '---=..J
Fig. 10,29 Table of drainage area shapes. (After Dietz. SO)
:
3.22
112
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
gineers who are responsible for analyzing and interpreting pressure behavior to develop an intuitive feel for the effects which the various reservoir heterogeneities can cause. Th t d b h h e s u les on pressure e avlor m eterogeneous reservoirs, although highly idealized, do represent reservoi~ models which are p.hysicallymore rea~stic th.an the sImple model upon which pressure analysIstheones are based. What has been the value of these theoretical studies?We believe them to be valuable for severalreasons. First, they have shown how various reservoir heterogeneities affect pressure behavior and how analyses based on a homogeneous interpretation theory are affected. Secondly, they have shown that in some cases augmentation of pressure data by geological and h I f .'. ..Based petrop yslca m ormation may aId us m forrntng Ideas .e of reservoIr geometry and pore spacedistribution. Also, the studies have served to temper the thinking of those who mi~t ten~ to regard pressure analysis results as always bemg uruque and exact. We hope that additional theoretical studies may be of further aid in understanding and better defining the utility of pressure analysis techniques. I h ld b d db " t s ou e un erstoo y all that a uruque mterpretation as to the cause of pressure behavior of a certain
.
..
. .
type
...0 IS very
difficult
to
make
from
pressure
data
alone.
In fact, there are so many heterogeneitieswhich manifest themselves similarly that an attempt at unique interpretation without adequate supporting data from other sources is foolhardy. What the engineer should strive to develop is an objective interpretation of reservoir and well behavior which integrates all available pressure, petrophysical and geological information. R f
e erences
.
Pressure
Build-Up
Analysis",
Trans.,
AIME
(1959)
216, 38-43. 16. Freeman,H. A. and Natanson,S. G.: "RecoveryProblemsin a Fracture-PoreSystem-Kirkuk Field", Proc.. Fifth World Pet. Congo(1959) II, 297. 17. Samara,H.: "Estimation of Reservesfrom Pressure Changesin Fractured Reservoirs",Second Arab Pet. Cong., Beirut (Oct., 1960). 18. Warren,J. E. and Root, P. J.: "The Behaviorof Naturally Fractured Reservoirs",Soc. Pet. Eng. J. (Sept.,
1963)245-255.
1. Horner, D. R.: "Pressure Build-Up in Wells", Proc., Third World Pet. Cong., E. J. Brill, Leiden (1951)II, 503. W II t F It D t 2 G K E "A ' . .ray, ..: pproxlmatlng e -0- au ISance from Pressure Build-Up Tests", J. Pet. Tech. (July, 1965)761-767. G d J dH k M F J ." L near 3.aVIS, D . E .ra y, r., an aw inS, .., r.. I Fluid-Barrier Detection by Well Pressure Measurements", J. Pet. Tech. (Oct., 1963) 1077-1079. 4. Russell,D. G.: "Determination of Formation Characteristics From Two-Rate Flow Tests", J. Pet. Tech. (Dec., 1963) 1347-1355. 5. Bixel, H. C., Larkin, B. K. and van Poollen, H. K.: "Effect of Linear Discontinuities on PressureBuildUp and Drawdown Behavior", J. Pet. Tech. (Aug., 1963)885-895. 6. Lefkovits, H. C., Hazebroek, P., Allen, E. E. and Matthews,C. S.: "A Study of the Behaviorof Bounded Reservoirs Composed of Stratified Layers", Soc. Pet. Eng. J. (March, 1961)43-58. 7. Jacquard, P.: "Etude Mathematique du Drainage d' un ReservoirHeterogene",RevueI.FP (1960)XV, No. 10.
.
.
8. Katz, M. L. and Tek, M. R.: "A Theoretical Study of Pressure Distribution and Fluid Flux in Bounded Stratified Porous Systemswith Crossflow", Soc. Pet. Eng. J. (March, 1962)68-82. 9. Russell,D. G. and Prats, M.: "Performance of Layered Reservoirs with Crossflow-Single-Compressible-Fluid Case",Soc.Pet. Eng. J. (March, 1962)53-67. 10. Vacher,J. P. and Cazabat,V.: "Ecoulementdes Fluides dans les Milieux Poreux Stratifies. ResultatsObtenus sur Ie Modele du Bicouche Avec Communication", RevueI.FP (1961)XVI, No. 14. 11. Pelissier,F. and Seguier,P.: "Analyse Numerique des Equations des Bicouches", Revue I.FP (1961) XVI, No. 10. 12. Maksimov, V. A.: "The Influence of Nonhomogeneities on the Determination of ReservoirParameters on Data on Unsteady-StateFluid Influx Into W IIs. A C ase 0f a Two-L ayer Format." lon, I.zvest. Akad. Nauk SSSR,Otdel. Tekh. Nauk, Mech. (1960) No.3. 13. Pendergrass,J. D. and Berry, V. J., Jr.: "Pressure Transient Performance of a Multilayered Reservoir With Crossflow",Soc. Pet. Eng. J. (Dec., 1962) 347354. 14. Russell,D. G. and Prats, M.: "The Practical Aspects of Interlayer Crossflow", J. Pet. Tech. (June, 1962) 589-594. 15 P IIar,d P.: "Ev alua tIon 0 f A Cl.d T rea tIn en t s f rom
.
19. Odeh, A. S.: "Unsteady-StateBehavior of Naturally Fractured Reservoirs",Soc.Pet. Eng. J. (March, 1965) 60-66. (Includesdiscussionof the paper by J. E. Warren and P. J. Root.) 20. Prats, M., Hazebroek,P. and Strickler,W. R.: "Effect of Vertical Fractures on ReservoirBehavior-Compressible-FluidCase", Soc. Pet. Eng. J. (June, 1962) 8794 -. 21. McGuire,W. J. and Sikora,V. J.: "The Effect of Vertical Fractures on Well Productivity", Trans.,AIME (1960)219, 401-403. 22. Dyes,A. B., Kemp, C. E. and Caudle,B. H.: "Effect of Fractures on Sweep-OutPattern", Trans., AIME (1958)213, 245-249. 23. Scott, J. 0.: "The Effect of Vertical Fractures on Transient PressureBehavior of Wells", J. Pet. Tech. (Dec., 1963)1365-1369. 24. Russell,D. G. and Truitt, N. E.: "Transient Pressure Behavior in Vertically Fractured Reservoirs",J. Pet. Tech. (Oct., 1964) 1159-1170. 25. Hartsock,J. H. and Warren,J. E.: "The Effect of Horizontal Hydraulic Fracturing on Well Performance", J. Pet. Tech.(Oct., 1961)1050-1056.
J
EFFECT OF RESERVOIR HETEROGENEITIES
113
26. Warren, J. E.: "Discussionon a Mathematical Model for Water Movement about Bottom-Water-DriveReservoirs", Soc. Pet. Eng. J. (Dec., 1962) 367-368. 27. Coats, Keith H.: "A Mathematical Model for Water MovementaboutBottom-Water-DriveReservoirs",Soc. Pet. Eng. J. (March, 1962)44-52. 28. Matthews, C. S., Brons, F. and Hazebroek,P.: "A Method for Determination of Average Pressurein a Bounded Reservoir", Trans.,AIME (1954) 201, 182191. 29. Brons, F. and Miller, W. C.: "A Simple Method for Correcting Spot Pressure Readings",J. Pet. Tech. (Aug., 1961)803-805. 30. Dietz, D. N.: "Determination of Average Reservoir PressureFrom Build-Up Surveys",J. Pet. Tech. (Aug., 1965)955-959. 31. Ramey, H. J., Jr.: "Application of the Line Source Solution to Flow in Porous Media-A Review", paper SPE 1361 presentedat joint SPE-AIChE Symposium
..
held during 58th Annual AIChE Meeting, Dallas,Tex. (Feb. 7-10, 1966). 32. van der Knapp, W.: "Nonlinear Behavior of Elastic Porous Media", Trans.,AIME (1959) 216, 179-187. 33. Russell,D. G., Goodrich, J. H., Perry, G. E. and Bruskotter,J. F.: "Methods for Predicting Gas Well Performance",J. Pet. Tech. (Jan., 1966)99-108. 34. McLatchie,A. S., Hemstock,R. A. and Young, J. W.: "The Effective Compressibilityof ReservoirRock and Its Effects on Permeability", Trans., AIME (1958) 213, 386-388. 35. Matthews,C. S.: "Analysis of PressureBuild-Up and Flow Test Data", J. Pet. Tech. (Sept., 1961)862-870. 36. Weller, W. T.: "Reservoir PerformanceDuring TwoPhaseFlow", J. Pet. Tech. (Feb., 1966)240-247. 37. Bixel, H. C. and van PQollen,H. K.: "PressureDrawdown and Buildup in the Presenceof Radial Discontinuities", paper SPE 1516 presentedat 41st Annual SPE Fall Meeting, Dallas,Tex. (Oct. 2-5, 1966).
Chapter
11
:
"
i\O'/.(, ';"-'
:J :Tr;IjfJJA
1,~~'~~i;:I,fl:;} ~:,oJ
'..
'tic",,~t.~
Practical Aspects of Pressure Analysis
,;:~;:\"i'~::-": !1rf1'lfiM ,-"
""'"h~~.1:1 -
~c:
If,': Q~
11.1 Choice of Tests in Flowing Wells The standard transient pressure test in flowing wells will probably always be the pressure buildup test. This is the most direct method of obtaining an averagepressure for reservior analysis. The test is operationally simple, and the theory is well developed. In certain cases it will be desirable to carry out a two-rate flow test rather than a buildup test. This will be the case if phase redistribution masks the true buildup behavior and causes "humps". The two-rate test will usually eliminate these. In addition, if the well is in a non-prorated area, the two-rate test will occasion less income loss than a closed-in test. The two-rate test usually will require somewhat more supervision since one needs to make sure that two constant rates (or near constant) are actually obtained.
omalous behavior, it is often helpful to record casingand tubing-head pressuresas a function of time in addition to bottom-hole pressure.Some applications of such data were discussed in Section 3.6. These data are also helpful for checking on the progress of long buildups or flow tests. In the course of collecting data on flowing (and pumping) wells, it is advisable to check for extended periods of shut-in. Monthly production data alone may suggestthat production was continuous, when in actuality a shut-in period was sandwiched between two months. It may be necessaryto account for this by superposition (Sections 2.8 and 3.8) if the shut-in occurred near the time of buildup or flow test.
Long flow tests are very useful for finding reservoir limits. High precision in pressure measurementis required for such usage because the rate of change of pressure with time is small at the extended test times required to delineate reservoir boundaries. A surfacerecording bottom-hole pressuregaugeI is very helpful in obtaining and recording accurate data during such tests. An accuracy and precision better than 1 psi for a 10,000-psi reading have been obtained by careful calibration and operation of this gauge. Interference tests serve a different purpose than pres-
In injection wells, one of the most useful tests is the two-rate injection test. This test does not usually require running a bomb and thus allows use of a very accurate surface gauge or dead-weight tester. An important practical consideration in a two-rate test is that injection wells are normally tied together through an injection header. In running two-rate tests, one should make certain that rates of other wells are not changed during the test period. If rates in other wells are changed,the observed surface pressure behavior will be difficult to interpret.
sure buildup tests. They are designedto indicate degree of reservoir connectivity and directional trends in permeability. They also enable determination of effective reservoir porosity. The latter determination is especially useful in fracture porosity reservoirs. where core analysis is of little aid in determining the amount of oil in place. When testing wells which have previously given an-
In carrying out a pressurefall-off test, it is important to close in the well for a sufficient length of time. Generally speaking, longer closed-in times are required for water injection wells than for new oil wells in equivalent formations. The reason for this is that the relative permeability to water (at residual oil saturation near the well) generally will be only 1/10 to 1/3 that of oil at its original saturation. Other factors being equal, it will take a pressure response 3 to 10 times as long to
*Interference methods plied quantitatively to
be
using the fractured
Ei-function reservoirs
can be apwhere the
fractures are numerous and well connected. For reservoirs
with only a few large fractures and some matrix porosity, only. interference tests should be used in a qualitative sense
11.2 Choice of Tests in Injection Wells
felt
in
the
water
injection ...
case
as
in
the
oil-filled
case, and thus the requIred longer closed-m time. Early in an injection m the testslifef orofti mes varyIng project f rom 1, one t 0 4 should days t0run detclosedermtne .
..
.
PRACTICAL ASPECTS OF PRESSURE ANALYS1S
115
how long a time will be required. Why not close in for times greater than 4 days?First, the authors have observed few injection caseswhich required more time than this. Secondly, the pressure changes after 4 days are usually small and, when .-the used m plots considerable times
of the log
(PW8-p)
scatter. Thus, pressures
often
are
not
of
much
value
vs ~t type,
lead to
taken at such large unless
one
of
In many pumping wells the rate of influx is low and, conseq~en~y, the rate of buildup in pr.essureis low. Cl?sed-m time~ of the. order ?f 4 days WIll often be requlred to obtain suffic~entpo~ntsto draw the. slope accurately. Often there ~sc?nslderab~euncertamty as to average gas saturation m the drainage area of pumpThi alue is best obtained by a rough ma-
.II ten
m~
al we b a S l ance .S
f or v
th
e
reservoir.
.
the
highly accurate subsurface gaugesis used. If the mobility of oil is considerably less than that of water the authors recommend use of theofnon-unit mobility , ratio interpretation method. Use the single-
11.4 Required Closed-In Times Wellbore Storage Effects A well should be closed in for pressure buildup long
fluid case may indicate that a wellbore clean-up to eliminate "skin" is not needed when use of the correct method would show that stimulation is desirable.
enough to allow the str~ght-li~e section.to be clearly delineated. The longest time WIll be required for deep, low-productivity pumping wells not equipped with pack-
11.3
ers since in these a long period of "afterproduction" wili be needed to fill the wellbore with liquid and com-
Tests in Pumping Wells
Buildup tests in pumping wells may be made in two general ways: (1) by measuring pressuresin the annuIus, or (2) by pulling pump and rods and running a bomb in the tubing. Annulus measurements may be made by running a bomb if there is sufficient clearance, by using echo-sounding devices, or by using permanently installed devices which are surface-indicating, such as the Maihak or Ball Bros. gauges. Each system of measurementshas its advantagesas well as its problems. Annulus bomb runs are probably the best. However, these cannot be made in deeppumping wells or in wells where the annulus clearance is small. Sonic measurementshave been useful in some cases, but the accuracy is not so great as is desirable. Permanently placed instruments are usually satisfactory; however, the higher fixed cost must be borne by a few measurements. Tubing measurements using a conventional pressure bomb are quite accurate, but they are costly unless run at the time of a routine rod-pulling job. Further, pulling the pump dumps the liquid contents of the tubing into the well and introduces a new transient. This should be taken into account for accurate results. In addition, when the bomb must be run in through the tubing after a pulling ~ob, no pressure measurement can be made at the time the well is closed in. Thus, the skin effect cannot be calculated. In summary, the best system is some sort of annulus measurementwhen conditions allow this. Morse and Ott2 have given examples of a novel method of testing pumping wells in water injection projects. They first shut in the producer for 2 days for pressure stabilization, and then inject water down the annulus for several hours until the surface injection pressure stabilizes. At such time they ceaseinjection and observe the surface pressure fall-ofl in the well. Interpretation of this test allows calculation of kh and skin. Although the averagepressure obtained from such a test may not be very accurate, this is often of secondaryimportance to the diagnosis of causes for poor productivity which can be made from knowledge of kh and skin. -~
pressedgas. Only after the influx into the wellbore becomes small can the simple theory of pressure buildup be applied. A simple method of determining when the rate of influx has become small is to determine casing- and tubing-head pressures (CHP and THP), as well as bottom-hole pressure (BHP), during buildup. The difference between BHP and THP or CHP is directly proportional to the mass of fluid between these two levels. A curve showing (BHP-THP) and (BHPCHP) can be plotted vs time. When the slope of each difference curve falls to, say, only 10 percent of its initial value, the influx is probably small enough that the simple theory can be applied. At this time, the influx rate into the wellbore is only 10 percent of the production rate at time of closing-in. RameyShas presented a useful equation for estimating the time at which wellbore storage effects become negligible during buildup or drawdown. The equation for "time of afterflow" is: taft = 2XI05~,
hours. ..(11.1) kh where V w is the volume (bbl) and c is the compressibility (psi-I) of fluid in the wellbore. If there is a large volume of gas in the wellbore, it will usually be satisfactory to take c = cu. Also, when two or more phases are flowing into the wellbore, the quantity k/ p.should representthe total mobility, (k/ p.)t. Eq. 11.1 applies both to gasesand liquids and their mixtures. It assumesthat there is no skin. To illustrate the use of this equation, we choose the data from Appendix B, Example 2: (k/ p.)t = 2,159 md/cp, h = 20 ft, C = Cg= 0.000854 psi-I, and V w = 36 bbl (9,150 ft of 2-in. tubing). Then, t
aft
= 2X 105(36) 0.000854 2,159 (20) , = 014 hours or 8 minutes. .
Eq. 11.1 is useful in two ways. First, it enables esti-
11
116
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
mation of the minimum length of time required for a pressure buildup or flow test. Then, having made the test, and having obtained kh/ JL from its analysis, one can .recalculate taft and decid.e whether ~e correct portion of t~e curve was used m ~e analysIs. For example, for FIg. 3.7, the afterflow time calculated above was 0.14 hours. As may be seen from Fig. 3.7, the afterflow appears to die out at about 0.6 hours. This agreementis about as good as we should expect from Eq. 11.1. In general, Eq. 11.1 should be used in a qualitative manner, as in this illustration. Another useful test for choosing the proper straightline portion of the buildup curve may be obtained from the flow efficiency, as calculated in Examples 1, 2 and 3, Appendix B. A c~lc~lated flow e~ciency much greater than two often Indicates that the Incorrect portion ...question of the buIldup curve was used as the straIght-line portion. The early portion of the ~uIldup .curve for a da~aged well usua~y has a section which appeals ...presence straIght, but actually IS much steeperthan the true slope (see FIg. 3.3). Use of this portion of the curve leads ...suc to a I?W cal:ulated. effective permea~ility and a ~orrespondingl.yhig~ estimate of flow efficIency. Thus, If the flow efficIency IS around two and the calculated effech I h d ..m bl tive permea Iity IS muc o",!er t an exp~cte , It IS d that too early a portion of the buIldup curve probable was use.
for our analysis. As an example, consider the buildup curve in Fig. 3.3 (treated as Example 1, Appendix B). The maximum dimensionlesstime of shut-in, ~tDe' is: 0.000264 (7.65) 92 = 0050 ~tDe= 0.039 (0.80) 17 X 10-6 (2,640)2 .. ... Since most of the pOInts used m the analys~sare within the interval 0.005 to 0.1, the correct portion of the pressure buildup curve was analyzed. A calculation such as the above should be made before conducting a buildup or flow test on a new well to determine the shut-in or floyvtime required to obtain the correct portion of the curve for analysis. 11.5 Radius of Investigation "How far into the reservoir have I investigated with my test.?', A precIse answer c annot be m o'ven to this .. SInce any pressure d ISturbance I' S felt to a small extent throughout the reservoir. Further, the existence of layers of different permeability and the 0f 0th er het erogenelti es make any answer to h ti. dt de of magm aques on goo oanor r tude only. An estimate can be based on the fact that equations for pressure behavior in an infinite reservoir are applicable . .til a finit e dramage area un a dimensionlesstime of about 0.1 (see Ref. 9). After this time, the pressure drop is greater in the finite reservoir than in the infinite case. Fur ther, setnl-steady state WI11 st art m a bounded ,
Boundary and Interference Effects Curve A, Fig. 3.13, as developed by Miller, Dyes and Hutchinson,4 shows that boundary and interference
cylindrical reservoir at a time IDeof about 0.3.* We now choose (after van PoollewO) a time IDeof 0.25, int:rmed~ate.to the$e two times, and calculate a radius of Investigation, Tiny,from
.
. ..
effects limit the usable straight-line portion of a pressure buildup curve taken in a bounded reservoir to a time, ~tDe = 0.000264 k ~t/ct>fl.CTe2, less than about 0.075. Curve B for constant pressure at the outer boundary shows deviation from the straight line at about ~t = 0.4. Since most drainage areas are at least pa~~lly bounded (see Fig. 4.10), use of ~tDe = 0.1 gives an approximate upper limit to required shut-in times. Most of the points which will be used in pressure buildup analysis should fall in the range of dimensionlesstime ~tDefrom 0.005 to 0.1. Interference tests will normally ~equire that the observation well be closed in for a time ~tDe = 0.2-0.3. Odeh and Nabor6 show that the straight-line section of a drawdown test persists for a time IDeof about 0.3 in a circular, homogeneous, bounded drainage area. This is about four times as long as for a buildup curve in the same drainage area. Strictly speaking, both drawdown and buildup tests deviate from unsteady-state theory at about the same time. However, the rate of deviation is considerably smaller for drawdowns than buildups. Because of this, the apparent straight-line sectionis much longer for drawdowns than for buildups The cntena we have discussed may be used to test whether we have used the correct portion of a curve
.
.
..
.
Tiny="
.
.
.
~10.000264 kt _I kt 0.25 ~IJ.C = ,,0.00105~,
ft. (11.2)
.., In this equation, t is the time of floWIng for a floWIng test or the time of shut-in for a buildup test. As an e~ample, choose the d~ta for the pressure fall-off test m Example 2, Appendix F, where k = 51! /45 = 11.4 md, ct>= 0:3 and fl. = 0.9 cpo Fo~ the oil and water banks we estimate c = 6 X 10-6 PSI-I. At a shut-in time of 20 hours, TinT-,
I
11.4 (20) 0.00105 0.3 (0.9) 6 X 10-6 -386
ft.
This is well past the estimated outer radius of the oil bank (82 ft). This indicates that the fall-off test was carried out long enough to reflect the pressure Pe at this outer radius. Note that past this outer radius there will be a gas saturation and thus a much higher compressibility than used above. If one used this higher value for compressibility in Eq. 11.2, one would obtain a much smaller radius of investigation than the 386 ft calculated above. Thus the true radius of investigation *Jones'suggests0.38 for this time. No precisevalue can be given for either the end of "infinite-reservoir" behavior or the start of "sem~-stea~y state"..See 9deh and Nabor8 for referencesand diScussIonon thIS subJect.
-
PRACTICAL ASPECTS OF PRESSURE ANALYSIS
117
is much less than the 386 ft calculated above. Because of such factors as change in saturation and change in reservoir properties with position, Eq. 11.2 should generally be used in an order-of-magnitude manner, as in this example.
indicates that oil will tend to flow from that high-pressure region to a lower-pressure region at that same datum level. This is not true of uncorrected level pressures. Thus, degree of reservoir connectivity can be judged by comparison of corrected datum-level pres-
11.6
sures..
Notes on Fractured and Other H eterogeneousR eservolrs .The ..approxImately
reason
In the case of wells which have been heaVIly fractured by hydraulic means, the slope of the apparent straight-line portion of a buil~up curve will be affected. The greater the fracture radius, the greater the overesti~ation of kh from the slope. This eff:~t at lea~~ partially accounts for the appearance of new p~y 5 after some fracture treatments. ~ussell and Truitt have.pres~nted a method of correcting the apparent kh obt8;ined m such well.sto the ~rue k~, for the case of a vertical fracture, as discussedm Section 10.5. A note of caution on closed-in times should be added for heterogeneousreservoirs, particularly for fractured reservoirs with a tight matrix, and for layered, stratified reservoirs. As discussedin Chapter 10, a long time will be required in these cases both to reach semi-steady state during flow and to approach averagepressureduring buildup. The criteria given in Section 11.4 (AtDe= 0.005-0.1) are still applicable in such cases for defining the "straight-line" portion of the buildup curve. Much longer times (10 to 50 times as long) are required to observe the later rising and then flattening portions (see Fig. 10.10). As this latter portion defines p, one should plan long buildups in such reservoirs. .density 11.7 Correction of Pressure to a Datum It is common practice to correct all pressures obtained in a given reservoir to a common datum level in that reservoir. Although this is not strictly necessary when pressures are to be used to obtain values for kh and s, it is necessarywhen these pressures are to be used to obtain jj or p*. As this sometimes causesconfusion, the practice of correcting all pressures has arisen. Ideally, the datum level is chosen so that about half the oil volume of the reservoir is above and half below, as shown in Fig. 11.1. Comparison of pressures corrected to this common datum indicates oil flow tendency directly. Thus, a high pressure at a given datum level WELL A
WELL B DATUM -LEVEL
CHOOSE DATUM LEVEL
SO THAT APPROXIMATELY HALFOF THE OIL-ZDNE VOLUME IS ABOVE AND HALF IS BELOW
Fig, 11.1Choiceof a datumlevel.
for choosIng the datum level IS soabove that half the 011reservoIr volume and half below is to make the resultant averagepressure at this datum level reflect the volumetric average oilzone pressure. This is especially important when the pressures are to be used in material-balance calculations. When the pressuresare to be used for comparative purposes only, the choice of datum level is not so important since the differenceswould be the same.However, the absolute value Qf the pressure is important in material-balance calculations and, thus, the greater importance of choice of datum level. In practice, changes in datum level of 50 to 100 ft will normally have small effect on results of a material balance. In water influx calculations, pressure differences are used rather than absolute values of pressure. It is convenient to use the common field datum level for these pressures also, since the datum level is immaterial as long as the same level is used for all wells. The datum level chosenas discussedwill tend to give an average oil-zone pressure. If there is a gas cap as shown in Fig. 11.1, the average pressure at the midpoint of the gas volume can be obtained from the oil datum-level pressure by correcting from that level using the oil density to the gas-oil contact and the gas above. Pressuresare ordinarily corrected from the measurement level to the datum level by using a so-called "tubing gradient". To measurethe tubing gradient, pressures are measured at selected levels in the tubing (usually at one level 100 ft above the lowest measurementlevel and then every 1,000 to 2,000 ft) by holding the bomb at these levels for an identifiable time period (5 to 15 minutes, depending on the clock) while coming out of the hole 'after a buildup. Pressuresmeasured at the different levels are used to calculate the tubing gradient. This gradient is defined as the pressure drop over 100 ft of tubing. Values obtained for the gradient range from 2 to 50 psi/100 ft and reflect the average density of the fluid betweenthe two levels. Gradient surveys often indicate that the tubing contains water at the bottom, oil next and gas at the top, as shown in Fig. 11.2 (water and oil only). Water is often found in the bottom
of the hole even though the well produces no water. Two of these tubing gradients are used in correcting to datum-the gradient just above the bomb, and the oil gradient. The reasonfor determining the pressure at. several levels in the tubing is to make sure that the 1 gradi ent .IS measure. d The 011 gtadi ent IS 01 .e 10und b y noting which of the gr adients corresponds to the expected oil density. After obtaining the oil gradient and the bottom-hole
.
L
118
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
gradient (the two may be the same if there is no water in the hole), one can correct pressuresto datum. The first step is to correct the measured bottom-hole pressure to the pressure at a point in the producing interval. This is done by use of the gradient obtained just above the bomb. When the datum level is in the producing interval, as with Well A, Fig. 11.1, the correction is made from measurementlevel to datum level in this one step. If the datum level is not in the producing interval, as with Well B, Fig. 11.2, a second step is required. The second step is to correct from the point in the producing interval (the top of the open interval is a convenient point as shown in Fig. 11.2) to the datum level. This final correction is made by use of the oil gradient, as noted in Fig. 11.2. Although the oil density in the tubing will not be exactly the same as that in the formation, it will usually be close enough for practical purposes if datum-level corrections are a few hundred feet or less. A more accurate density for second-step datum-level corrections can be obtained from PVT data for the oil in question. The true oil density is used only in the second step of datum-level correction. The density of fluids in the tubing is used in the first. In the case of gas wells, the same discussion applies
NOTE BOMB
WELL
B (SEE
except that the .'gas gradient" is used instead of the "oil gradient". As should be evident from the discussion, the pressure bomb should be run as close to the formation interval as possible to minimize correction errors. Note that, even if the bomb is run exactly to datum level in the tubing, a correction may still be required if the tubing contains water, as for Well B, Fig. 11.2. If the tubing contains only oil between datum level and the open interval (or gas in the case of a gas well), no correction will be required for a bomb run at datum. As the tubing may be partially filled with water below the bomb level, the bomb should be run as close as possible to the open interval to minimize possible error. A gas well may contain condensate and/or water below the bomb level, and therefore in this case also, the bomb should be run as close as possible to the open interval. 11.8 Well Stabilization When conducting a pressure buildup test, it is important that the production rate be stabilized for approximately a week before the test. The stabilized rate should be approximately the same as the normal production rate. This allows the simplified method of calculating production time (Eq. 3.6) to be used. It may be desirable in some casesto "stabilize" a well at
THAT WATER AFTER WELL
STANDS ABOVE IS CLOSED IN
CORRECT FROM BOMB LEVEL TOP OF FORMATION USING
FIG.II.I)
DOWN WATER
TO
GRADIENT (OBTAINED JUST ABOVE BOMB). THEN CORRECT uP TO DATUM LEVEL USING OIL GRADIENT
DATUM LEVEL FIRST CORRECTION (WATER GRADIENT)
SECOND CORRECTION (OIL
GRADIENT) TOP
OF
BOTTOM
FORMATION
OF
Fig. 11.2Correctingpressuresto a datum level. --
FORMATION
PRACTICAL ASPECTS OF PRESSURE ANALYSIS
119
a much higher than normal rate to obtain a measurable buildup slope. If this is done, the method of superposition (see Section 3.8) should be used in plotting the buildup curve. Odeh and Nabor6 have presented an excellent discussion of the errors brought about by changing rates prior to a buildup test. In some reservoirs the permeability is so high and/or the allowable production rate so low that the pressure buildup curve will have no measurable slope. This can also happen in tight reservoirs wheref .there is a low th th nil .I
0
Gauges
e Ine
be no particular problems of measurement or inter-
.
.
.
d
an
)
.
Th
h ave a state d
di
ms tru ments
ameter
di
m
-In.
ame er
I
m
full
Smker bars Will some-
times be required to lower the bomb mto a floWing well. I,n some cases mfo~ation on p, kh and ~kin Will be ~eslred from a well ~~ch has been closed m for some time. In such cases it lS usually better .first to measure the bottom-hole pressure over a penod of, say, 24 hours. to make sure the pressure is constant or is changing only very slowly. Then, a pressure drawdown test should be conducted and interpreted as in Chapter 5. This procedure Will give the required information b fasterbuild much than Will a stablhzed flow penod and a su sequent up. For wells producing by continuous gas lift, the production rates will usually be steady and there will
.. In.
-74
The most widely used bottom-hole pressure measuring instrument is the Amerada RPG-3 gauge* shown in Fig. 11.3. This gaugeis lowered into the well through tubing or annulus on a piano-wire line. Fluid is admitted at the lower end of the instrument where it actuates the bellows, which, in turn, introduces fluid into the helical Bourdon tube. Motion of the Bourdon tube in responseto a pressure change is recorded by a stylus on a clock-driven chart. The chart is made of brass and is coated on one side with a special darkgray paint. A steel, sapphire or diamond stylus burnishes a fine bright line on this chart. The clock and the pressure element can be replaced by other devices of different ranges. Clock ranges vary fr~m 3 to 180 hours. Press~reelemen~s?ave ranges va~~g from 500 to 25,000 pSI. A magmfymg chart reader l~ n~essary to enable the charts to be r~ad to their mherent acc~rac~. The ch~rts have a 2-m. pressure scale and a 5-m. time scale. During a bottom-hole pressure run, the maximum temperature is recorded by a maximum recording thermometer placed in the pressure section. This is since the calibration of the pressure element is a slight function of temperature, especially at temperatures greater than 200F. Calibration of the pressure elements should be carried out at frequent intervals. A dead-weight tester is needed for this operation. Procedures are given by the manufacturer. . B th th Am d RPG 3 ( I , /" RPG 4 ( 1 di t ) ' f
m
us,
ng.
rea
e
a,4 000-pSteIemen,t the accuracylS 8 pSI. The precision ..' -sca
.0
bl
..
li
d
d
and possible hang-up.
..accuracy
.,
..
b
b
th
thickness, q/h, is small. In such casesone cannot calculate kh and skin from the test; only an over-all productivity index can be obtained. Good results can be obtained on wells which produce only part-time during a month by proper scheduling of flow and buildup tests. When there are numerous short flow and shut-in periods, the method of superposition should be used in pressure analysis. Nisle7 has shown that a short shut-in period (such as to change a valve before a buildup test) is not detrimental to a subsequent buildup provided the short shut-in is followed by a flow period 10 times as long. Wells which intermit or "head" are poor candidates for pressure buildup or flow tests. The taking of accurate oil, gas, and water production records is indispensable, of course, both before and during the test. During two-rate flow tests, as well as during drawdown tests, production rate determinations should be made every few hours so that the true rate behavior will be known. Metering separators are handy for this purpose. An average production rate determination for severalhours preceding a buildup test is usually sufficient for this test if the well has been stabilized. ...necessary 11.9 Other Considerations In Well Tests It is necessary to obtain the bottom-hole flowing pressure ~rior to. buildup in order to calculate th~ skin effect. .This requ~resthat the pressure.bomb be mtroduced mto a floWing well. In general, this can be accomplished without difficulty. However, if a well is subject to paraffin deposition, the paraffin should be cut before e om lS mtro uce to e mmate pOSSle difficulty m msertion
'
O
urnt
t
per
0
ux
a
1
era
0
2
e rate
.percen
at
e
so
0
pressure
Measuring Instruments Wr
f
.
11.10
0
reservoir
pretation. These wells often will be producing at high water cut, and it will be necessaryto include water in calculation of total compressibility and total mobility. Wells on intermittent gas lift do not give a steady flowing pressure. An average value is usually satisfactory for calculating the skin effect, however. Buildup pressures are usually smooth and satisfactory on these wells.
of the measurements can be considerably better than this, depending partly on the care taken in using the instrument and in reading the charts. It is mainly precision (not accuracy) which is involved in determining slopes of pressure curves, and thereby determining the value of kh. Th K t KPG ** . I I e us er gauge lS comp ete y mterchangeable With the Amerada RPG-gauge. 3 Both 0f t hese I 74-m. , /" .m diamet er gauges use a helicaI N i-span C
.
.
.
Ok;~anufactured by GeophysicalResearchCorp., Tulsa,
.
h Cal1f . ** M anuf actured by Kust er C0., L ong B eac,
120
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
WIRELINE
OUTER
SOCKET
CLOCK
HOUSING
INNER
HOUSING
INNER HOUSING COVER
CHART
ASSEMBLED
HOLDER
GAUGE
CHART LOADING SADDLE
STYLUS
ELEMENT
HOUSING
MANDREL
ARM
PRESSURE
a
ASSEMBLY
ELEMENT
. ., j
,
BELLOWS 01 L TRAP L
THERMOMETER THERMOMETER
WELL
Fig. 11.3RPG-3 pressuregaugemanufacturedby GeophysicalResearchCorp. -~
-~
,,"c
PRACTICAL
ASPECTS
OF PRESSURE
ANALYSIS
121
Bourdon tube as pressure element. Other Kuster gauges are available in diameters of % and 1 in. These two gaugesare somewhat shorter than the Amerada gauges (42 in. vs 75 in., approximately). Their stated accuracy of 0.25 percent of full scale is comparable to that (0.2 percent) of the Amerada gauges and of the Kuster KPG gauge. The Leutert-Hugel wireline gauge* has now been made available in the U. S. This gauge is a spring-andpiston type device in which the piston is rotated continuously during the test by a clockwork mechanism. This enables an accuracy of 0.025 percent of full scale, or about 10 times that of the more commonly used subsurfaceBourdon gauges.This higher accuracyshould be very helpful in many cases, such as in tests in highpermeability wells where the slopes of the pressure curves. are small,. and in int~rf~rence tests.wher~ the ~ota~dIsturbance IS small. ThIS. mstrument IS avaIlable m,di~meters of 1.26 ~nd 1.42 m., an.d?as a length of 139 ~n. Costs of .run.mngshould be sImIlar to those of runnIng other wirehne gauges.
(Boulder, Colo.) and by the H. Maihak Co. (Hamburg, Germany). The Ball Bros. gauge is basically a helically wound Bourdon tube whose disp!acementis telemetered to the surface via a single conductor cable. The Maihak gauge utilizes the pulsed-vibration frequency of a diaphragm-actuated taut wire for telemetering. In the Ball Bros. gauge,the deflection of the Bourdon tube is measured by the down-hole rotation of a code wheel. Data are telemetered to surface in a code form. Conversion tables are used to change the telemetered code readings to equivalent pressures. The code readings are a series of zeros and ones, such as 0110100. This reading, for example, represents 135 psi on a 500-psi MK- 7 gauge. There are two basic Ball Bros. instruments -the MK- 7 which has the seven-digit code as above, and the MK-9 which has a nine-digit code. With the sevendigit code, an accuracy of about 1 percent of full scale is obtained, while with the nine-digit code 0.2 percent of full scale is realized. The latter is required for most pressure buildup and flow tests.
Permanently Installed Surface-Recording Instruments There are two types of permanently installed bottomhole pressure instruments which are usually satisfactory for pressure buildup and drawdown tests. These are the gaugesmanufactured by Ball Bros. ResearchCorp.
Due to the construction of the down-hole code reading device, the pressure information received at the surface may have a slight "stair-step" appearance.This is not usually a drawback if care is taken in drawing a smooth curve. An example of data from a Ball Bros. gauge is given in Fig. 11.4. The Maihak surface-indicating bottom-hole pressure
*Manufactured by Lueneburg, Germany; Long Beach, Calif.
F~ied.rich dIstrIbuted
L.eutert, 3141 Erbstorfm' U. S. by Kuster Co.,
gauge
.consists
of
a permanently
installed
bottom-hole
t ranSmitt er and a po rtable surface receIver. Th e two .8
1200
I
100
'
.~ In
,
~a. 80
:
a: :::> U> U>
W
~
60
'"
",'
-J 0 Z
I
400
2 0
PRODUCING
BOTTOM-HOLE
PRESSURE
tt-
o
m
200
0 0
20
40
60
80 HOURS SHUT
100 IN
120
140
Fig. 11.4Builduptest of a producingwell, Ball Bros. gauge.
160
180
122
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
components are connected by a single conductor cable strapped to the production tubing. Transmitters have ranges varying from 1,100 to 5,900 psi. The down-hole transmitter consists of a sealed cylindrical steel body closed on one end by a steel diaphragm
Other surface-recording instruments which can be run on conductor cable have been made available by both Amerada and Kuster. These instruments employ the standard helical Bourdon tube for down-hole pressure measurement, telemetering to surface via a single
which is deformed by externalpressure.A taut steel
conductorcable. Their accuracyis stated to be 0.2
wire is attached to the diaphragm and the body. Changes in pressurechange the tension of the wire and, thus, its vibrational frequency. The wire is vibrated by a coil and magnet, which also act as pickup to transmit the vibration to the surface. The surface receiver consists of a similar magnetic pickup and a vibrating wire whose tension can be varied. Frequency matching is noted on a cathode-ray tube in the portable field receiver. An automatic re1 bl h' h ' '. d 1 cor er IS a so aval a e w IC gives output on a stnp h 1 cart.
percent of full scale. A small-diameter (42-mm) Maihak and a small-diameter (11/4-in.) Ball Bros. gauge also belong to this category of cable-hung devices. The surface-reading devices are very useful for special purposes such ~s reservoir limit tests. They have found only limited general application becauseof the increased operating costs of running the conductor cable and sealing it at the wellhead. 11 11 Q l t t I t t t ua I a lye n erpre a Ion 0f B UIld up Curves
.
K
Ib l
.
h
0
f as
accuracy
0 full
f
cent 0
d oun
.
th
t a ,
.
f th M h k e 1
sca e.
al A
a
. th WI
ms
t
proper
ca
t '
rumen '
IS a
b
l 'b ti. I ra on,
ou
O
th
As
.perb 'ld
f
a
e
t or
full
care
mos l 'b
b
'ld UI up t d
d
fl
an di
t owes d
t
'f I
s b
K
th e
gauge
Ideal
made
curves,
aid,
by
well effect,
.
wellhead
achieved
with
lubncator
this
c
t e Instrument
11.6,
ideal
Of
course,
actual actual
with
these
curves
curves
may
combIne
Interference.
the
reader
For should
a
further
consult
the
, . discussion .., section
of
each
Indicated.
3. Ramey, H. J., !r.: "Non-Darc.y Flow and Wellbore Storage Effects m Pressure Build-Up and Drawdown of Gas Wells", J. Pet. Tech. (Feb., 1965) 223-233.
IS used
tohi seal h hat the ...1 wellhead around the conductor cable on w
Fig.
1. Kolb, R, H.: "Two Bottom-Hole PressureInstruments Providing Automatic Surface Recording", Trans., AI ME (1960)219, 346-349. 2. Morse, J. V. and Ott, F.: "Field Application of UnsteadyStatePressureAnalysesin ReservoirDiagnosis", pape~SPE 1514, presentedat 41st Annual SPE Fall Meeting, Dallas, Tex. (Oct. 2-5, 1966).
The precision instrument designed by Kolbl for surface recording of subsurface pressuresuses a helical Bourdon tube, a single conductor cable and a telemetering systemto measure very accurately the angular d I f h B d b A f ISpacement 0 t e our on tu e. n accuracy 0 been
in
References
Conductor Cable
has
show
companng
Surface-Recording Instruments Run on
0.01 percent of full scale .., Instrument. A grease-type
we
several of the features shown, such as wellbore fillup
.and IS
Ib
y cal ra e , as scusse yo.
be
..
o
f
practical
'.
may
companson 0 a pressure UI up
h b th A d d M . h k d .. h k sown t .taIn FenIg.' Wit 11..e50 Th meraM al.haa kanpressuresal a are evlcesquI .teIS accep -,
bl
.
a
1 b examp e Ulld up curves. Sorne qua1ItatIve Interpretation
e
t 0 25
.
o
o
4
IS run.
M
ll er, C ..,C
o
"The
Estimation
D yes, A ..an B of
d H utc h.mson,
Permeability
and
C ..,A
Reservoir
Jr.: Pres-
(/)
0W
a:
60
.MAIHAK
PRESSURE
POINTS
~ ~
oAMERADA
PRESSURE
POINTS
~50 w
~ ~
0' ~
AMERADA BOMB
400
PRESSURES
2
~
m
300
'-MAIHAK DEVICE
~ I I-
::I
x (/)
PRESSURES
200
I
2
4
6
B 10
20
40 60 80 100 TIME, MINUTES
Fig. 11.5 Pressurebuildupcomparison.
200
400 600
1000
2000
PRACTICAL ASPECTS OF PRESSURE ANALYSIS
123
""," Pws
/
/
/
10Q[(1.41)/41J
Pws
~
Pws
10Q[(I.41)/41J
IDEAL- Sec.3.1
o
o;:~~
"""
1
10Q[(I.41)/41J
SKIN AND/OR WELL FILLUP- Sec.3.2,3.6
1
DEEP PENETRATING HYDRAULIC FRACTURESec. 10.5
"",,"'" P ws
P ws
10Q[(I,(oI)/41J
1
,/"""""""
Pws
loq [(I.41)/41J
BOUNDARY (one well in a bounded reservoir) -wells Sec. 3.3
~
Pws
FAULT OR NEARBY
BOUNDARY-Sec. 10.1
~
~
" p ws
10Q[(I.41)/41J STRATIFIED
1
PHASE SEPARATION IN TUBING -Sec. 3.6
p ws
10Q[(I.41)/41J
10Q[(1.41)/41J
INTERFERENCE (mulliple in a bounded reservoir)- Sec.7.2
.-t:
/
~
LAYERS
OR FRACTURES WITH TIGHT MATRIXSec. 10.3, 10.4
""
/""..,/-"".000-
10Q[(I.41)/6.IJ LATERAL
1
INCREASE
IN MOBILlTYSec. 10.2
Fig. 11.6Example buildupcurves. sure from Bottom Hole Pressure istics", Trans., AIME (1950) 189, Build-up 91-104. Character5. Russell,D. G. and Truitt, N. E.: "Transient Pressure Behavior in Vertically Fractured Reservoirs",J. Pet. Tech. (Oct., 1964) 1159-1170. 6. Odeh, A. S. and Nabor, G. W.: "The Effect of Production History on Determinationof Formation Characteristics From Flow Tests", J. Pet. Tech. (Oct., 1966) 1343-1350.
7. Nisle, R. G.: "The Effect of a Short Term Shut-In on a
Subsequent Pressure Test on an Oil Well" ' Trans.,AI ME (1956)Build-up 207, 320-321. 8. Lozano,G. and Harthorn, W. A.: "Field TestConfirms Accuracyof New Bottom-HolePressureGauge",J. Pet. Tech. (Feb., 1959)26-29. 9. Jones,L. G.: "ReservoirReserveTests",J. Pet. Tech. (March, 1963)333-337. 10. van Poollen, H. K.: "Radius of Drainage and Stabilization Time Equations", Oil and Gas J. (Sept. 14,
1964) 133.
Chapter 12
. Conclusion
-
12.1 The State of the Art What is the "State of the Art" as far as our knowledge and understanding of pressure behavior is concerned? Do the theories and analysis methods which we now possessenable us to do a competentjob, or are they too highly idealized to generate realistic results? To answer these basic questions we must first of all define what we as engineersare seekingto accomplish. Essley,l in his paper entitled "What is Reservoir Engineering?", has stated that the goal of engineering is optimization. He points out the fact that maximization of the economic recovery from a petroleum reservoir requires early and accurate identification of the reservoir system. Wyllie2 has made an eloquent case for what he calls "the holistic approach" to reservoir mechanics which relies on physical reservoir system measurementsmade in situ. We believe that, fundamentally, we are seeking through our knowledge of pressure behavior an accurate definition of the reservoir system. Analysis of pressure measurementsmade in wells is an in-situ technique for accomplishingthis task. We may ask ourselves, then, how well the present theories and analysis techniques permit us to define reservoir systems. As we have seen, we believe that under favorable circumstancesgood estimatesof formation permeability, well damage and averagepressure in the drainage volume of the well can be obtained by transient pressure test analysis. Of equal importance is the fact that such estimates are unique in value. But what of our ability to interpret pressure behavior to infer the nature and distribution of the pore space heterogeneities within a reservoir? The answer here is that from pressure behavior alone we cannot obtain unique interpretations. So far as we can see, it is not possible to infer heterogeneity type and distribution solely from pressure observations. Except for this question of uniqueness in the interpretive portion of pressure analysis techniques, we belieye that the theory has been developed to a high level and that the analysis methods do a reliable job if used with discretion.
The conclusion that the theory and practice of transient pressure testing techniques is in good shape, with the exception of the uniquenessproblem associated with heterogeneousreservoirs, may be viewed by some as an inconsistent statement. One could argue that all reservoirs are heterogeneousto a degree and, therefore, all transient pressure test results are non-unique. In reply, we must first acknowledgethe fact that to varying degreesall suchresults are non-unique. It is not possible for even the most experienced reservoir engineer to analyze a transient pressure test and, in the absolute absence of other geological and petrophysical data, give a unique interpretation. However, this is not the way that we should use the information. Transient pressure test data must be used with geological and petrophysical data in an integrated approach to reservoir characterization. Then, it is possible in many cases to obtain unique interpretations. In Qther cases even the integrated approach will not supply a unique interpretation. However, the engineercan evaluate the economic consequencesof the various reservoir models which fit the data and recommend a course of action which should be followed in each instance. This type of analysis is not new to most engineers.We believe that when transient pressure analyses are used in conjunction with all other available data, our conclusion stated at the beginning of the paragraph makes sense. With such an approach we do not believe the uniqueness problem to be overwhelming. Another way of portraying the "State of the Art" is to say that it is good if viewed as a part of the holistic approach to reservoir mechanics, i.e., one of a suite of tools which the engineer must apply objectively and intelligently in his job of reservoir systemdefinition. Throughout this Monograph we have emphasized that engineersmust use all available petrophysical; geological and pressure-production behavior information in an integrated approach in order to define the reservoir systemand predict its behavior. Since the early work on pressure behavior in the
CONCLUSION
125
1930's, we have made substantial and satisfactory progress in developing and applying our understanding of reservoir pressure behavior. We are not, however, in the euphoric state of having all our problems solved. To the contrary, there areattack important and challenging problems which we must successfully in order to maintain our progress. In the section which follows we shall discuss a few of the areas in which additional work is needed. 12.2
Current Problems and Areas for Further Investigation
Reservoir Heterogeneities It should be clearly evident to all who have read Chapter 10 that the heterogeneousreservoir situations for which pressure behavior has been studied are highly idealized. Thus far, we have been limited in our ability to describe reservoir heterogeneitiesin a rigorous manner. Hopefully, through geologic studies of various depositional units and the developmentof faster computers with larger memories, we may be able to study more realistic situations. For example, how will the pressure behave in a well which is completed in a shaly, lagoonaI-type sand traversed by a stream-channel deposit? Studies of pressure behavior based on more realistic geological models are a must. Also, the influence of multiphase flow is important. We need further studies aimed at improving our ability to detect fluid contacts in more realistic geometries. Also, more rigorous treatment of hydraulically
that we make in analyzing these wells and some suggestionsas to how to improve our techniques. Rigorous Treatment of Borehole Effects From the early recognition of the importance of afterproduction in pressure buildup behavior, through the work of Stegemeier and Matthews,3 and to the present, our understandingof the nature and magnitude of borehole effects on pressure behavior has steadily increased. At present we can do a reasonablygood job on pressure buildup analysis taking into account afterproduction, phase redistribution, etc. This is not so, however, in the case of flowing transient pressure tests. The borehole flow and reservoir flow need to be combined to produce better interpretation theory. Reservoir m~chanics and vertical lift perfo~mance are complexly I?terrel.ated and together constitute the over-all system m which we seek to operate. We possessthe ability and understanding from each viewpoint to integrate our knowledge and develop better theory. Perhaps it is too much to hope that eventually a suite of testing tec?niques for producing wells could be developed WhICh are as successful as our present bottom-hole pressure-basedmethods and which employ only surface measurements of pressure. However, a start in this direction has been made by Kern and Nicholson.4 One can point to pressure fall-off analysis methods to contend t~at succe~sful combination of wellbore and reservoIr flow ~ght le~d to surface-based measurement and analysIs theones.
fractured wells should be encouraged. The effects of pressure drop within the fracture and the heterogeneous ..mce nature of the fracture itself should be considered. The
Computer-Based Analysis Methods the ad vent 0f t he d.. IgIta1 computer as a comS t 1 .. 1 d t 1 y use 00 m pe ro eum engIneenng most com-
.
studIes of naturally fractured formations should be .mon extended
by
removal
of
some
of
the
more
.'
..parnes
mathematical assumptions. Authors should be encour-
aged
to
k
follow
Th
through
h
k
more
t
f
completely
It
b
.
in
d
.
restnctive.
f
future
h
wor s. e c ec ou 0 resu s 0 tame rom mat eh ld b f companson .pu WIth field behavIor matlcal Investigations by s ou ecome more 0 a routine matter. P .W II umptng e s We can, at present, do a reasonably good job of pressure measurement and analysis on wells which produce by artificial lift if pressures are obtained by means of permanently installed, surface-recordingpressure gauges. In the more common cases in which such equipment is not available, the value of our present techniques is significantly reduced. For instance, if we run a pressure buildup in a pumping well, then we usually must pull the rods before we can begin pressure measurements. In doing this we miss the important early-time portion of the buildup and we cannot dete~ne t~e ski? effect .Mth much precision. Mathematlc~l simulati?n st~dies of pr~ssure behavior in pumpIng wells mIght YIeld for us estimates of the errors
have
endeavored
.
.
to
reduce
.
the
routIne
. engIneer-
appli cation 0f t his new too1 .mg wor k 1oad by JUdiCIOUS As
far
as
pressure
analyses
are
concerned,
.
there
have
been, .m most quarters, efforts to reduce these to comtenzed techniques. We are a11 "10r t his; we urge, . however, that such approaches be camed out in a rational manner lest both the engineerand his management become dubious of the value of this approach. What we are saying is that there are instances in which computerized pressure analyses can be applied because of the routine nature of the job. There are also cases in which the interpretive aspects of the analysis techniques so dominate the analysis that a great danger exists of erroneous results from a straightforward computer-based analysis. In reservoirs in which more-or-less ideal pressure buildup behavior is obtained, one might feel confident in using a computer-based analysis as a routine tool. However, in less ideal cases the chief value of the computer may be to organize and display the data graphically as required by the analysis technique being applied so that the engineer is freed from the more
.
-
-'",'" 126
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
burdensome tasks. We believe that this in itself is a worthwhile use of computers which should be pursued more fully. Of course, in any analysis method which is based on trial-and-error procedures, the computer can save even greater amounts of time for the user. Thermal Recovery Applications A field which is, as yet, virtually untouched as far as applications of pressure analysis methods are concerned is thermal recovery. Undoubtedly, pressure behavior analyses can be used to aid i~ ~he .control ~f thermal recovery processes and maXImIZation of 011 recovery. There is a danger involved in straightforward application of present theories to thermal recovery operations. This is due to the fact that all of our presentpressure analysis techniques are based on isothermal reservoir conditions. Before we can apply our present analysis methods or designnew, more rigorous methods, theoretical studies of pressu~e~eh~vior which take i~to account the temperature distnbutlon must be camed out. Examples of studies of this ~Yrehave been~resented by v~n ~~0Ilen5 and KazemI, who h~ve dIscussed the applIcabIlIty of pressure fall-off techmques for 10cating a burning front in in-situ combustion operations. O~e can visu~lize in steam soak ~perations t~at t?rough suIta.blymo~fied pressure analysIs methods It mIght b~ possIble to Infer the shape and extent of the heated 011 zone at the end of a period of steaminjection. For those interested in extension of pressure analysis techniques, the development of suitable theories for application in thermal recovery operations should present a challenging and rewarding goal. 12.3 Value of Pressure Analysis Methods To the Petroleum Industry " .12.4 In the preceding section we have outlined a few of the important problems in pressureanalysis which presently exist. Is it really worthwhile to encourage work on these problems or should we simply leave well enough alone? Will these problems be so hard to solve that the cost of the necessaryresearch is prohibitive? To answer these questions is difficult, to say the least. However, by attempting to assessthe value of present pressure analysis methods we may be able to make an educated guess at whether further work is worthwhile. The cost of a transient pressure test may range from $200 to $300 for a 48-hour pressure buildup to as much as $5,000 to $10,000 for extended reservoir limit tests. These costs are simply average costs and can vary appreciably depending on operating conditions. Do we get our money's worth from such tests? Even considering the fact that there will be an occasional test which fails to meet its objectives, we believe the answer is yes. For purposes of discussion, consider the case of a well which has beencompleted but the productivity ~--"" ~"
is not as high as was anticipated. Should we spend, say, $10,000 for a stimulation treatment, or is the formation permeability so low that this is the controlling factor and a treatment to remove a "skin" would be of no value? This is clearly a case where a transient pressure test and analysis can provide the answer, and most operators would be willing to spend several hundred dollars to obtain the neededinformation. As an example of value in such cases,Stalnaker7notes that the success ratio for fracturing operations in the West Burkburnett field improved by a factor of 4 to 5 after pressure buildup analysesbegan to be used as a basis for selection. But what of the more expensive tests such as those costing severalthousand dollars? In the case of a reservoir limit test, the reservoir size and/or the reservoir recovery mechanism might be established. The monetary value of a reservoir limit test can far exceed its cost when drilling and development decisions must be made.s Another way of attempting to assessthe value of pressure analyses to the industry is to ask whether or not alternatives exist for characterizing a reservoir. As we see it, pressure analysis techniques are an indispensible part of the packageof tools which the engineer must use to describe and characterize the reservoir system. Without an efficient set of pressure analysis techniques, we do not believe it is possible to achieve the goal of optimization of the economic recovery of hydrocarbons from a reservoir. Herein lies the value of pressure analysis methods to the industry. Based on the foregoing we must conclude that research efforts on pressure behavior analysis at least as ambitious as those carried out in the past should be maintained. We believe that through further work on the more pressing problems the value of pressure analysis methods can become even greater. Where Do We Go From Here? The "State of the Art" as far as pressure analysis methods are concerned is good. We have a variety of analysis techniques and testing procedures which do a good job, provided they are used with a reasonable amount of professional judgment. It is not a dead art, however. Many fundamental problems remain to be solved. At present we are on a sort of plateau as far as our theories and methods are concerned. With the theoretical foundation upon which our present theories are based, we have built quite a framework. It appears to us that movement forward from this plateau will come only as the result of well-conceived fundamental research on pressure behavior for various types of reservoir systems. The burden is not entirely on the researcher. Operations engineers must devote some extra effort to evaluation of present techniques in welldefined field cases so as to provide a guide for research efforts. -~'"
CONCLUSION
127
Finally, we hope that from the objective treatment
f
.
1
0
pressure
ana
th
YSIS
me
d 0
s
w
h . ch 1
we
have
en
3. Stegemeier.G. L. and .Matthews.C: ~;: "A Study of
deavored
Anomalous (1958)
213
to give in this Monograph, the usage of the methods WI
.
11 b
e
s
ti
mu
1
a
t
e.
d
Al
so,
w
e
ho
p e
that
in
some
measure
the development of more rigorous methods will result. We cau~on. that pres~ure a~alysi~ tec~niques mu.stbe u:;ed objectively and In conjunction WIth all avaIlable reservoir information. Our goal is optimization of recovery through characterization of the reservoir system. References 1. Essley,
P.
L.,
Jr.:
"What
Is
Reservoir
Engineering?",
J. Pet. Tech. (Jan., 1965) 19-25.
Pressure 44-50.
BuIld-Up
BehavIor,
Trans.,
'. 4.
Ke~n, catIon
C. of
AIME
.".
P. and NIcholson? F. Calculated Mulnphase
R.,
.
Jr.. .PractIcal ApphFlowmg Bottom-Hole
Pressures",J. Pet. Tech. (Dec., 1965) 1373-1378. 5. van Poollen,H. K.: "TransientTestsFind Fire Front in an In Situ CombustionProject", Oil and GasJ. (Feb. I, 1965) 78. 6. Kazemi,Hossein:"Locating a Burning Front by Pressure Transient Measurements",J. Pet. Tech. (Feb., 1966) 227-232. 7. Stalnaker, Successful
D. B.: Shallow
"West Project
Burkburnett Waterflood-A in North Texas", J. Pet. Tech.
(Aug., 1966)919-923.
2. Wyllie M. R. J.: "Reservoir Mechanics-Stylized Myth or Potential Science?",J. Pet. Tech. (June, 1962) 583588.
8. Jones,P. and McGhee,E.: "Gulf CoastWildcat Verifies ReservoirLimit Test", Oil and Gas J. (June 18, 1956) 184.
~
,
i
.I (
':
..
~, -1, :,
"'..
(1\
-';c;,
:"1)
c,
..(:
~':"" , ..!
,,
~( ;,f
'"J i;F:i:..:r/1~ ;!fJ(,i
".
..''I.' ","" "' i';,Jf" hr! ""~ ",,'J f
l' r(,!iJ1'$11,,~,;''" :j'.}(i[.1'(.;'!t!;b' i;:r,"'II.;:;i'i"':'i'..,r"OfT,t..!i:,':
1j] "i;
""J!'\(,;...'f.1..1';':, ?f..~ii; 10 irl~i:J"j
.
,.. "" "","Tq ,,"'c' ' , J' l",..(,:;..' ..,., b.'r' !t ,'..
'i~'['
",'
","m ,...,f;:" ',"."
c'" ~ J
.,),
\!10~t ~}.',;-,
,
7'
;'{Ii !t
c,
~i[;,
i.
;.I',';!.!' -
"""~, "'fi:;(lii"'i.J..'.,',,
Nomenclature
"" 'i, .
Only practical oilfield units are given after symbols; for Darcy units, see Section 2.10. al = distance between observation well and production well 1, ft; similarly for a2, a3' etc. A = drainage area of well, sq ft .pressure b = Intercept at .6.t=0 of plot of log .6.pvs .6.t, .Piw pSI .Pwf b1 = Intercept at.6.t = 0 of plot of log(Pw8-Pe) vs .6.tfor injection wells, psi B = formation volume factor c = compressibility psi-l , Cf = effective formation (rock) compressibility, pSi-l Ct = total compressibility, psi-l d diameero t f t b..pressure t= u 109, In. D = non-Darcy flow constant, (B/D)-l fu ti' 1 tt d . F ' 8 9 8 10 nd F = thencOnpOelnlgs.." 8.11 a g = accelerationdue to gravity h = formation thickness, ft i = injection rate, BID at surface conditions I d BID ' = In]ectlVlty In ex, -pSI / = productivity index, BID-psi k
=
f
orma
ti.
on permea
b ' lit d 1 y, m
m = absolute value of slope of linear portion of pressure buildup or flow test curve, psil 10glocycle M = mobility ratio, (kl,u)1/(kl,u)2 M = molecular weight of a gas Pe = external boundary pressure at radius re, psi
--~-
Pi = initial reservoir pressure, psi P~c= pressure at standard conditions, psi Pt = tubing-head injection pressure at time of closing in, psi Pw = bottom-hole pressure, psi; in two-rate flow tests and in all injection tests, Pw is the at time of change in rate = b0ttom- h01e In]ecti on we11 pressure, pSI
.. .
.
= bottom-hole flowing (or pumping) pressure psi ' Pl br = pressure read from linear portion of pressure buildup curve at I-hour closed-in time, psi; also refers to pressure read from linear portion of drawdown test curve two-rate flow test curve or Pres, , sure falloff curve, at I-hour test time p* = pressure obtained when linear portion of buildup curve, PW8vs log[ (.6.t+ .6.t)l.6.t], .IS extrapolated to (t + .6.t)I .6.t= 1; correspondsto pressureobtained after infinite closed-in time in an infinite rese~volr, pSI; see Seetion 3.3 for a disCUSSIon on p* -. P = averagepressure, pSI A P = Pw at semi-steadystate (Eq. 5.5), psi P = Laplace transform of .6.PD(see Appendix A) .6.PD= pressure drop, dimensionless, (Pi -Pw) / (q,u/21Tkh);
since
Miller-Dyes-Hutchin-
son define .6.PD as (p -PW8) I (q,ul 21Tkh), we have used this definition in presenting their method in Section 3.9 and Figs. 3.13 and 3.14; seealso Section 10.5 for the definition of .6.PDfor hydraulically fractured wells ... .6.P8kln = pressuredrop In "skin" region next to wellbore, psi
\
NOMENCLATURE
129
q = production rate of well, B/D conditions
at surface
z = gas deviation factor (compressibility factor, z = pV /nRT)
rD = dimensionlessradius, r/Tw 1 f d f 1 roD = dimens10ness ratio 0 1nner ra IUS 0 01 bank to outer radius (see Fig. 8.4) b d di ft re = external oun ary ra us,
,8 = absolute value of slope of log Ap vs At curve, hours-1
.
.
/ reD
=
re
r,o,
d .. ImenS10n
1
.
.
.
,8L = absolute value of slope of linear plot of well pressure vs time, hours-1
S
,81
=
[absolute
es
value
of
slope ...
of
log
(pw.
-Pe)
vs At curve for mjection wells] .2.303, hours-1
rw = wellbore radius, ft R = universal gas constant
"Y= ratio of total compressibility in oil bank to
R. = gas solubility in oil, bbl/bbl
total compressibility in water bank
R.w = gas solubility in water, bbl/bbl
"Y= Eulers constant, y = 1.78; In"y = 0.5772
s = skin factor, dimensionless
p.= viscosity, cp
oS"= apparent skin factor, dimensionless (s' = s + Dq) S = saturation, fraction of pore space
p = density, gm/cc (in injection well analysis)
t = time of flowing, hours tDe= dimensionlesstime of flowing based on re, kt/c/>p.cre2; see note at end of Nomenclature tDw= dimensionlesstime of flowing based on rw, kt/c/>p.crw2; seenote at end of Nomenclature At = closed-in time, hours
ft h .os, At' = flowmg (or mjection) time a er c ange m rate, hours AtDe= dimensionless closed-in time based on re,. kAt/c/>p.cre2; see note at end of Nomenclature
c/>= porosity, fraction 00
.
-EI(-X)
=
x Subscripts i = initial o,W,g = oil, water, gas; W also refers to well when used with p and r ws,gs = oil, water, gas at standard conditions or, gr, = oil and gas at residual conditions sc = standard conditions t = total; refers to tubing when used with d or p
T = absolute temperature, oR .Note: T,c = absolute temperature at standard condiR .O
tions, u = volumetric rate of flow per unit cross-sectional area V = 01 .1 vo1ume bbl 0, V" = pore volume, bbl V /V = ratio of volume of oil bank to volume of ° w water bank (see Fig. 8.4)
Wi = cumulative water injection, bbl
f sds e-'
.. Dimensionless ..
Quantities
.
D ImenS10n 1ess time m the D arcy syst em 0f urnts (darcy, sec, cp, cm, atm) .1StD = kt/c/>p.cr2. The quantity r may be chosen either at r e or rw. In one case we obtain IDeand in the other tDto.For the practical oilfield units used in this Monograph (md, hr, cp, ft, psi), IDebecomes 0.000264 kt/c/>p.cre2. The constant 0.00105 in Eq. 7.1 is 4(0.000264). In the Darcy system of units the flow rate is usually written as qp./47rkh, which has the units of atmospheres.
In practical oilfield units in this report, this quantity is 70.6qp.B/ kh, which has units of psi. Also,
Y(t) = function giving effect on well pressure of reservoir boundary
70.6qp.B -162.6qp.B kh In t -kh log10t.
..,
\
-~
Appendix A
Solutions for Radial Flow of Fluids of Small and Constant Compressibility
Constant Rate, Infinite Reservoir Case The initial value problem represented by this case is presented in Chapter 2. In summary, the mathematical problem which we must solve is:
~or2+ ~r ~=~ or k ~ 01'
."
Then Eq. A.3 becomes y!!E:.+(I+y)p'=O
for I>
O.
Separation of the variables and integration yield p'=
As mentioned in the text, several slightly different approac~es to t~e solution of this problem have appeared In the lIterature. We have chosen to present the approach of Polubarinova-Kochinal because it is quite straightforward. We are indebted to H. J. Ramey, Jr., for calling this approach to our attention. To develop the solution, we first replace the second boundary condition by the condition
This imation
r
boundary to the
op -q,u a:;;- -~,
for
condition original
is the boundary
1 >
"line-source"
shown to yield identical results (from a practical stand-
Fundamental to the solution is the use of the Boltzmann Transformation, -cp,ucr2
"
(A.2)
Substitution of Eq. A.2 into the differential Eq. A.l and accompanying boundary conditions gives y~+~(I+y)=O, dy2 dy with
.(A.3)
+ c
of
this
expression
Jim 2yliYdp = y--+-O
q,u 2-;kh
with
=
Eq.
Jim y--+-O
(A.S)
A.S
shows
that
2C1 e -II.
been
point) with those obtained from solution of the problem with the original, less-tractable condition (see Mueller and Witherspoon5).
y -4F
y -y
P' = ~= ~e-ll. dy y C and C1 are constants of integration. .. From boundary condition (2) above, Jim 2y ~ = -!!L. y--+-O dy 21Tkh
approxIt has
-In
or
Comparison
O.
condition.
(A.4)
dy
P --+-Pi as r--+-oo for all 1 .In
Jim r --+- 0
=~dP.
To solve Eq. A.3Ietp'
p=Piat/=Oforallr.
(2) ( r ~ )rw = &(3)
p--+-Pi as y--+- 00, Jim dp q,u (2) y--+-O 2y-ay-= ~ .
(A .y1)
Boundary and initial conditions: (1)
(1)
Thus, C = ~ 1 41Tkh'
(A 6) '.
Eq. A .now S becomes dp -q~
e-ll
"tiY-4;lChy'
which can be integrated to yield p = -&
I
T
dy + C2. "
(A.7)
The lower limit of the integral in Eq. A.7 can be assignedarbitrarily. We choose y = 00 and obtain
APPENDIX A
131
1le-li
J -dy
P = ~
~+~~=~
+ C2,
41Tkh y
OrD2
.
00
or
(1) qp.
JydY e-li
00
P = -4;kji"
(2)
+ C2 .rD
11
This last equation can be rewritten as P = -&Ei
rD
OrD
OtDW'
With
(-y)
(A.8)
(a-OAPD) ---1,
for tDw > O.
].
( -:ar;;aAPD )r
--0, for all tDw. eD The Laplace transform of ApD is given by (3)
+ C2.
ApD = 0, at tDw = 0 for all rD'
00
If we fi d apply boundary we n
condition
(1)
J
p ( rD , S) -ApD -
[ -Ei ( -~
, t) = ~41Tkh
)]
4kt'
., (All) where S IS the Laplace transform van able (not to be (A.9)
.
confused with skin factor). Application of the transform to the differential equation and boundary con-
hi h E 2 31 f th t t w c IS q. .0 e ex. This solution can also be obtained by physical and mathematical arguments using the instantaneous point-
ditions yields
source solution as a basis. For an interesting presentation of this approach, the reader is referred to the
with
book by Collins.2 Constant Rate, Bounded
(3)
rw
0 (J!..) or re
= .-!!1!:-for 21Tkh
.(A.13)
= O. (2) ( ~ ) drD reD
( )
differential equation. Eq. A.12 is simply a form of Bessel's equation and
P = Pi at t = 0 for all r.
or
(A.12)
t
-- 1 S
dr~].
,..
The condition at tDw = 0 has been accounted for in the transformation of the time derivative in the partial
with initial and boundary conditions:
(r ~ )
(-=dP )
Reservoir Case
02p 1 op cf>p.C op ¥ + r ar = kat'
(2)
d2P 1 dP d;:;f+"t:;dr; = SP
(1) Circular
The ~nitial value problem for this case (from Chapter 2) IS as follows:
(1)
( rD, tDw) e -stn.. d tDw,
0
Finally C2=Pi' we obtain , PI -p(r
to the above,
possesses a general solution
t > O.
P = A 10 (rDYs) + B Ko (rD~)' .(A.14) -where 10 (rDYs) and Ko (rDYs) are modified Bessel functions of the first and second kind, respectively, of order zero. A and B are arbitrary constants whose
= 0 for all t.
As mentioned in the text, this problem has been solved by several different authors. We shall present here the solution of van Everdingen and Hurst3 which employs the Laplace transform. To facilitate solution of the problem, we introduce the following dimensionless variables:
values must be determined through application of the boundary conditions (Eq. A.13). Differentiation of Eq. A.14 with respect to rD and substitution into the conditions (Eq. A.13) give Aysl].
(ys)
-BysK].
(ys)
1
= --
S Pi -P
A ysl].
(reDYS) -BysK].
(reDYS) = 0 .
ApD = ~ 21T
(A.I0)
rD = r/rwkt tDw
=
A. 'j'p.crw
If we solve these equations for A and B and substitute the values into Eq. A.14, we obtain P = K1 (reDYS) I~ (rDYs) ~ 11 (reDys)_Ko
2
S3/2
[I].
(reDYS)
K].
(ys)
-K].(reDys)I].(ys)]
(rDYs)
'
(A.lS) Substitution of these new variables into the differential equation and boundary conditions yields
Eq. A.lS is the Laplace transform of the general solution for the pressure behavior in a circular, bound-
APPENDIX
A
133
Constant Rate, Constant Pressure Outer Boundary Case The mathematical statementof this problem is as fol-
case we find that the only singularities of Eq. A.25 in the complex plane are simple poles at the origin and along the negative real axis. At the origin
lows. -1
o2p 1 op -tPp.c op a;:2"+rar-k~'
.h
p = -(In reD -In rD) S and, therefore, for large time
(A.26)
WIt
(1) P = Pi = Pe at t = 0, for all r. (2) (3) If
( pI
~PD = In reD -In
op ) qp. r -:a-:;:r.. = 2;kh , for t > O.
interested in the details are referred to Carslaw and Jaeger.4
.
.
.
t th bl t f th d m nless we recas e pro em m erms 0 e 1 ensio .pressure vana bles 0f Eq..,A 10 we 0btam
.
.!!:!!:!!-Eo + -1--~= orD2 rD orD
Finally, we obtain the following expression for the . behaVIor at the well.
~ otDw'
Pw! = Pi -~
with
I
d2P 1 dP -analysis -a;:-2+ r ~ = sP,. D D D
(2) PI
= 0
..
b
.
.
Pw, = Pi + h1 e-Pl~t.
)
.(A.24)
Since only the first term in the series expansion has been retained, this expression is valid for large values of shut-in time only.
-Bys
K1 (ys)
= -+,
A 10 (reD yS) + B Ko (reDyS) = O. S I f A d B d b ti t ti mo. t 0 VIng or an an su sung yields
.
. ..
Eq A. 14
References 1. Polubarinova-Kochina,P. Ya.: Theory of Ground Water Movement,Translated from the Russian by J. M. fi9~2e~~9~' Princeton University Press,Princeton,N.J. 2. Collins, R. E.: Flow of Fluids Through Porous Materials, Reinhold Publishing Corp., New York (1961). 3. van Everdingen,A. F. and Hurst, W.: "The Applica-
'
tion of the Laplace Transformation to Flow Problems in Reservoirs", Trans., AIME (1949) 186, 305-324.
KO(reDYS) + K1(ys) 10(reDYS)] (A.25)
4. Carslaw,H. S. and Jaeger,J. C.: Conduction of Heat in Solids,Oxford at the ClarendonPress (1959) 89.
p = 10(reDysL KO(rDYs~
S8/2[/1 (ys)
.
E A 28 d I q. .IS 1 entica WIth Eq.. 2 38 0f t he text. In Ref. 4 of Chapter 8 a slightly different form of Eq. A.28 is employed to provide a basis for pressure fall-off
ore radIus (rw~O) assumption. The pressure falloff equation which is obtained is of the form
Again, Eq. A.14 is a general solution of the differential equation and the conditions (Eq. A.24) must be used to evaluate the constants A and B. In this case
I
.(A.28)
prior to reservoir fillup in the unit mobility ratio case. This form is based on the vanishingly small
.(A.23)
r.D
A ys 11 (ys)
; n=l
where ,8nis a root of ]1(,8n) Yo(,8nreD)-Y 1(,8n) ]o(,8nreD)= O.
.Application of the Laplace transform to the above gives
~ = _1-~ !rD S
[ In reD-2
]
(2) ~o~p = -1, for tDw> O. rD 1 (3)" ~PD = 0 at rD = reD,for all tDw'
(1)
27rkh
e-P.2tD..]02 (,8"reD) ,8n2[]12 (,8n) -]02 (,8nreD)] ,
(1) ~PD = 0, at tDw= 0 for all rD.
.well WIth
...(A.27)
To obtain the complete solution we again need to find the singularities along the negative real axis. Those
--Pi, f or all t.
r.
rD.
-KO(reD~)
10(rDY~
which is the transformed solution to our problem. " ..sure Proceeding as WIth the bounded cIrcular reserVOIr
I
5. Mueller, Thomas D. and Witherspoon,Paul A.: "PresInterference Effects Within Reservoirsand Aquifers", J. Pet. Tech. (April, 1965) 471-474.
I i I
Appendix B
Example Calculations for Pressure Buildup Analysis I
This appendix contains three example pressure buildup analyses:
:,
Example I-Reservoir
Above Bubble Point,
Ii
--
f ,I
Examples 2A and 2-Reservoir Below Bubble Point, Examples 3A and 3-Gas Reservoir.
\
il
fi :i
Results for each case are presented on a form sheet, based on equations given in Chapter 3, and designed especially for routine pressure buildup analyses.The curve analyzed in each case is noted on the form sheet. Calculations were usually made on a slide rule, and thus the reader should expect only this level of accuracy in the results shown.
II
1
At,
hours
10
100
4600
440 CURvE
1hour
co .In
42
>0
Q.IDZ
W.-to "~ ,
'
Z
, 'U)~I-
::) U)
0,.,0 -w~-
U)"
4000
,
-'
::)-tu
W
~
~ !l.
-t~::)
"00
u
-t
0
'
Q: Zw!l.
O~Q: -ow I-IDI-
Q:-l"°-l-t I-wo
38
~~ 0
+
, , I
360
,
" "
3400 100.000
"i
'
"
10,000
1000
(t + At) fAt Fig. 3.3 Pressurebuildupshowingeffectof wellboredamageand afterproduction. --
100
APPENDIX 8
135
Example Calculation 1: Reservoir Above Bubble Point (Based on Fig. 3.3) Test Data: Test Date January 4, 1951 Producing Formation H I S' ( h ) - 4 "/CDolomite ),1.,".,:...::: ..., rr ,0 0 e Ize mc es -/4":; ! Cum. Prod. Np (bbl) 142,01-0,;:..,:,
.
Stabilized Daily Prod. q (bbl) 250 Effective Prod. Life t (hr) =24 N,,/q
Company Shell Lease Lend Well No.1 Field Center State .Texas
,~-
n
13,630
I. Calculation of kh (md-ft) and k (md): kh = .!~~~~ h q
\
; k=~.
69.0 250
ft B/D
p. B m
.
kh = 162.6X(250) X (0.80) X(I.136)
= 5277 d-ft ~ m,
(70)
k =
0.80 1.136 70
(527.7)= (69)
cp psi/cycle
~ 765 md.
II. Calculation of Skin Effect, s; and PressureLoss Due to Skin, ~P8kln(psi): s = 1.151[~~Og(*)
+ 3.23].
~P8kln= (m) X 0.87 (s). k c/> p. c
~
7.65 0.039 0.80 17 X 10-6
s -1.151
~ md
'to PI hrPtO! m
cp psi-l
[ (4,295) (701-log -(3,534)
2.375/12 4,295 3,534 70
]
+ 3.23 -~
(7.65) (144) (0.039)(0.80)(0.000017)(5.64)
ft psig psig psi/cycle .
~P8kln= (70) X 0.87 (6.37) = ~psi. III. Calculation of Productivity Index (B/D-psi) and Flow Efficiency:
1
1(actual)-q - p* -(Ideal) Pto! ~P8kln 388 q 250
-(250)
I(actual) -(4,585) -(250) 1(ldeal) -(1,051) ,
Flow
Efficiency
-(3,534)-
-.
p* Pto!
(p *
- Pto!)
-A
...akin p
4,585 3,534
psig .psig
~B/D-pSI.
-. -(388)-~B/D-pSI. =
I(actual)=~= 1(ldeal)
Note:
psi B/D
-q -
0.631. 0.377
-
1. Compressibility is obtained from C";" ~ Ct = SoCo+ StOCto +C! = 0.85(11 X 10-6)+ 0.15(3 X 10-6) + 7.2 X 10-6 = 17 X 10-6.
i
The value of Cois obtained from PVT analysis, cf from Fig. G.5; CWis an averagevalue for water. 2. p* is obtained by extrapolating two cycles to the right on Fig. 3.3. p* = 4,445 + 2m = 4,445 + 2(70) = 4,585 psig.
136
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
Example 2A: For Reservoir Below Bubble Point I. Obtaining Total Mobility and Total Compressibility for Fig. 3.7:
[~ ]
,u t
=~ + ~ + ~ IJ.o ,uu.uw
,
." (f
] = [~ ,u t
Values of (dR./dp) and (dBo/dp) are obtained as the slopes of laboratory-determined curves of R. and Bo vs p; the slope is drawn at the estimated averagepressure (1,300 psig on Fig. 3.7) or 1,315 psia. Since we do not have a gas analysis, we estimate T c = 464°R and Pc = 655 psia from the gas gravity of
~ [Boqo + Bu (qut-qoR.) + B",q",]y mh 162.6 = (135) (20) [1.227 (924) + 12.9 X 10-3
0.93 (Fig. G.6). Then Tr.= 720/464 = 1.55, Pr = 1,315/655 = 2.01. From FIg. G.7A, Cr = 0.56. Then Cu= cr/Pc = 0.56/655 = 0.000854 psia-l. We estimatec'" = 3 X 10-6 psi-I, and from Fig. G.5,
(2.740 X 106 -924 = 2,159. B (1P R 1 dB
Cf = 4 X 10-6 pSi-I at cf>= 0.15. Then Ct = Soco+ SgCu+ S",c", + Cf, and for So = 0.546, Su = 0.204 and S", = 0.25, Ct = 0.546 (0.0003622) + 0.204
Co =
-1fT-BY 0
X 53.1) + 0] '
I
'-s 0
(0.000854) 0.000376.
p
= 12.9 X 10-3 (00455) 1.227 .II. -~(O 0001425) 1.227 .f = 0.0003622 psi-I.
+
0.25
(3
X
10-6)
+
4
X
10-6
=
Obtaining p* for Fig. 3.7: p* is obtained by extrapolating two cycles from the pressureread at [(I + ~t)/~t] = 100. Thus, p*=1320 + 2m = 1320 +2 (135) = 1,590 psig.
.~. i
bot. hours 1
10
1
12
II
10
I
9
8
1 100,000
10,000
1000
1
(t+At)/4t
Fig. 3.7 Pressurebuildupin a reservoirwhen both oil and gasare flowing.
10
APPENDIX B
137
Example Calculation 2: Reservoir Below Bubble Point (Based on Fig. 3.7)
TestData: Company
Shell
Test Date April 1, 1956 Lease Weller Producing Formation Sandstone Well No.4 Hole Size (inches) 12 Field Edd Cum. Prod. Np (bbl) 33,300 State California Stabilized Daily Prod. q (bbl) 924 oil, 15.38 MMcf gas (2.740 MM bbl gas) Effective Prod. Life t (hr) = 24Nplq 865 I. Calculation of kh (md-ft) and k (md) : ! kh = ~~~~~-; k =~.
h q
20 924
Bg 12.9 X 10-3 Rs 298 ft3/bbl or 53.1 11.0 0.675cp Bo 1.227 m 135
ft BID
kh = 162.6 X (924) X (0.675) X (1.227) = 922 (135) _m,
bbl/bbl bbl/bbl
psilcycle
d-ft . k = ~(20) = ~ 46 1 md.
II. Calculation of Skin Effect, s; and PressureLoss Due to Skin, ~PBkln(psi): s = 1.151 [.p..~~~..!!.!!!!--IOg(~)+
3.23].
~PBkln= (m) X 0.87 (s). kll1.
2,159
c/>
0.15
c
0.000376
.
md/cp
rOD
6/12
PI hr
psi-I
[
-(240) s = 1.151 (1,195) (135)
1,195
POD! m -log
(144) (36) (0.15)(2,159) (0.000376)
240 135
ft psig
psig psilcycle
]
+ 3.23 -~
~PBkln= (135) X 0.87 (2.43) = ~psi. III. Calculation of Productivity Index (BID-psi) and Flow Efficiency:
1(actual)=
q P*. -POD!
~P8kln q
285 924
I(actual) = 1(ldeal)
=
1(Ideal)--q psi BID
p' POD!
(924) (1,590) -(240)= (1,350) (924)-(285)
Fl ow Effi clency ' -I(actual) _ I
(Ideal)
.
~B/D-pSI.
= ~
BID-psI.
0 788 -- 0 868 -.. -0.684
.-
.
(p *
-
.
POD! ) -" L.lp skin
1,590 240
psig psig
138
PRESSURE
Example 3A: For Gas Well P,C p* + P~f 2
T
= 520oR IC. P.c = 14.65 psia
Pseudocritical temperature T c = 420R P d " I 663 ' seu ocntica pressure Pc = psia .Ct ReservoIr temperature T = 200 + 460 = 660R p* + P~f _(2,895)
IN WELLS
from Ref. 12, Chapter 3; cf from Fig. G.5 at I/>= 0.16. Then, forSw = 0.29 andSg=0.71, = SgCg + SwCw + Cf,
+ (2,422)
= 0.71(0.000347) +0.29(13X10-6) +4X10-6,
-2 = 2,658 psig = 2,673 psia. T 660 Tr = -== 1.57 T c 420 + p~f)/2-
FLOW TESTS
660 14.65 Bg = 0.809 X 520 X 2673 = 0.00563 . ' Using T,. and P,., and a gas gravity of 0.8, /Lgis obtained from Figs. G.3A and G.3B as /Lg= 1.7XO.01185 = 0.0201 cpo Also, Cgis obtained from Fig. G.7A as 0.23/663 = 0.000347 psi-I. The compressibility of gassaturated water at pressure (p* + Pwf)/2 is estimated
2
-(p* pr -Pc-
AND
z = 0.809 (from Fig. G.6) .
I. Obtaining Bg, /Lg,and Cgfor Fig. 3.8: B = z -.:!g T,c
BUILDUP
= 0.000254 psi-I. II. Obtaining p* for Fig. 3.8: p* is obtained by extrapolating three cycles from
2,673 -the 663 -4.03
pressure read at [(t+~t)/~t] = 103. Thus,.p* = 2,844 + 3 m = 2,844 + 3 (17) = 2,895 pSlg.
6 t, hours 1.0
3000
2900
2~
27
2600
2500
2400 10'
10
105
(t+6t)
/6t
Fig. 3.8 Pressurebuildup in a gaswell.
10
10
APPENDIX 8
139
Example Calculation 3: Gas Well (Based on Fig. 3.8) Test Data:
Company Test Date November 16, 1956 Producing Formation Sandstone Hole Size (inches) 7 Cum. Prod. N,,(bbl) 1.138 X 109 (6,390 MMcf) Stabilized Daily Prod. q (bbl) 536,900 (3.01 MMcf/D) Effective Prod. Life t (hr) = 24N,,/q 50.8 X 108
Lease ~ellNo. Field State
Shell
Orr 3 Left Texas
I. Calculation of kh (md-ft) and k (md): (
kh = 162.6 q~.
' k =!!!.h .
m
h q
84 536,900B/D
ft
IL, B, m
l/'
kh =162.6 X (536,900) Tli~.0201)
X ~Q.00563)=~md-ft;k
0.0201 0.00563 17
=_i~~J-
cp cu ft/cu ft psi/cycle
= !!;!!:!:md.
II. Calculation of Skin Effect, 8; and PressureLoss Due to Skin, ~P.k!n (psi):
[
8 = 1.151 PI hr -Pro! -log m
(~
) + 3.23 ].
cJ>/lcrro2
~P.kln = (m) X 0.87 (8). k cJ> /l C
6.92 0.16 0.0201 0.000254
md
rtD Pt hr Pro! m
cp psi-t
[
-(2,422) s = 1.151 (2,815) (17)
-log
3.5/12 2,815 2,422 17
ft psig psig psi/cycle
(6.92)(0.000254) (144) (0.16) (0.0201) (12.25) + 3.23J = ~
~Pskln= (17) X 0.87 (21.12) = ~psi. III.
,~
Calculation of Productivity Index (BID-psi) and Flow Efficiency: ](actual) -q- p* -
](Ideal) -q-
P ro!
~P.kln 312 q 536,900
psi B/D
(.";"
(P * -
p* POD!
Pro!
)
-"
~
p skin
2,895 2,422
psig psig 1('01-
(536,900) l(actual) = (2,895) -(2,422) ] (Ideal)= Flow
(536,900) (473) -(312)=
.J EfficIency
=
=~ 3 335 2--
(actual) = 1135 --!-](Ideal) 3,335
B/D-psi..
BID-pSI. .
I~
= 0.340. -
I
Note: At high rates of gas flow, one will obtain the apparent skin factor 8', instead'of 8, from the above calculations. Consult Section 3.5 for a method of obtaining 8 in such cases.
~
.,.
~~
!C
d
0-."
-0'
;, -" a, '~",\;3' -~ .r: "",.,it ""'0 : """,,"'/.t.;;o/,~~ .'u"
.
Appendix C
Example Calculation for Average Pressure
Matthews-Brons-Hazebroek Method
Prom the information in Example 1, Appendix B, we
.
As an example for consld er PIg. 3 .,3 To obtain p* we must extrapolate two cycles to the right to [(I + ~t)/~t] = 1. Thus, p* = 4,445 + 2m '. -' determInIng p,
have, taking A = 160 acres = (2,640)2 sq ft , 0.000264 kt 0.000264 (7.65) 13,630 I ~p.cA_- = l[O391(O-:-SO)17 X 10-6 (2,~
= 4,445 + 2(70) = 4,585 psig.
1
= 7.45.
At,hours
10
100
460
440
,
r
42
I
4000;,
T~
38 -I
,
36
I
3400
100,000
10,000
100
(t + At) tAt Fig. 3.3 Pressurebuildupshowingeffectof wellboredamageand afterproduction.
APPENDIX C
141
Then from Fig. 4.3, for the square we read
line sectionof the buildup curve; obtain re2=J~~2.:.;
(p*-p)/(70.6 q,uB/kh) = 5.45. The slope of the buildup curve m is 70 psi/cycle. Now since "J'; i;, , /2 303 (70.6qp.B/kh) = m. , = 70/2.303, -4 -30.
~lQ~ !:~2. ~3.3. PIQ' at ~O hours l~ ~.362 PSl_~.!!2m Fig. 3.13, Curve A, no flow over drainage radius, ~PD
,
then
= 0.90. Then p* -p
= 5.45 (30.4) = 166,
:9_~_.Q.Q1L!~-:=_M_l~f,.:~s
and
with the value of p obtained above. '\ If we had used Curve B, Fig. 3.13, for constant pressure at the drainage radius, we would have obtained ~PD = 1.62. Then, P = 4,362 + 1.62 (70)/1.15 = 4,461 psig. This value lies in betweenthe value of p* and that of p for no flow over the drainage radius.
Miller.Dyes.Hutchinson Method Choose ~t = 10 hours as a point on the straight-
Calculations in this appendix, as in the others, reflect slide-rule accuracy.
i
, ~"" i
,. I \
~.-t
!-
!;
t
, i: , [.
l
t
,
t
i
,
t.
~2". agrees closely
p = 4,585 -166 = 4,419 psig. Note that the pressure buildup curve in Fig. 3.3 appears to be approaching, after 90 hours' shut-in, an average pressure of about 4,430 psig, some 11 psi greater than this calculated average.This difference reflects the accuracy which should be expected from use of methods for calculating p.
!
i
then ~ i " 0.000264 k~t = 0.000264 (7.65) 10 (1T) cpp.cre2 0.039 (0.80) 17 X 10-6 (2,640)2 = 0.0172. ; ~ ~-
".
',,3 ~a~,."" ..,,"" !.eo
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NI 3Wll
0001
001
01009
008
.." r
0
0001
~
z Q
~ 0
~
0021
~ 0 E I :I: 0
r
,., OOt'1 ~
,., (/) (/)
c: ~
,.,
0091
:0-
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.c
0081
0002
'SIAOnOJ S~ al~ suop~In:>I~;)
'aI:>A:>/!Sd ZIZ
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palnst!aw
10Id
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s~"J uo
S~IA a}~l
1A0g aq.L
'a:>!Aap alnssald
'8!d
lsa"J aq} }noqgnolq"J
01.1'}J 8 = l{ ',1'0
lua!sueJl
.A:>t!ln:>:>t! aInl
,
UMOqS }sa"J S~} 10J suopt!In:>It!;) aIqt!"Js a}!nb
aloq-Wo}}oq
8U!
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= cp 'd:> 0'1
= 'Ii 'Q/H.LS
:SMOnOJ St! alt! }sa"J aq"J 01 }uauplad
008 = b
t!"Jt!Q 'l!OAlaSal
U!st!H laAuaQ t! U! naM t! U! }sa"J UMOpMt!lp lnoq-os t! wol} pau!t!"Jqo alaM t!}t!p alnssald aq.L 'pa1uasald aJt! S la"Jdt!q;) Jo spoq}aw
aq} 8u!sn t!}t!p 1sa"JUMOpMt!lp
a1!nS t! Jo S!SAlt!ut! aq} Jo sI!t!"Jap aq"J x!puaddt!
]0
s!q} uI
S!S~leUV UMOpMeJOl aJnSSaJd JOJ SUO!~elnJle3 81dwex3
a x!puaddv
APPENDIX
D
From Eq. 5.3, kh = 162.6qp.» ,
s = 1.15 [ Pi -Plhr-log~+
"
m
~~
m
k
=
767
=
96
3.23
],
q,}.tcrw
[
-1,690 s = 1.15 1,895212
162.6 X 800 X 1.0 X 1.25 kh = 212' kh
143
i' ,.'
md-ft,
-log
0.14
X
1.0
X
96
17.7
X
10~6
X
0.11
+
3.23
md.
From Eq. 5.,4
s = -5.0.
1000
b=320 /.",
/~=1300 0
000
""0"',.. """"-0 '\.
0, 0'0
100
'"'"
---8;
""0'
""
'0\'\
"""
'\ ~ -0 cn
_D',~
---0
: = 1400
\ \ \
a..
ea.-
\
~ =slope.
\
,
-\ a.~
I
fi
= 0.138
)l:{
!"
,
/,(r
Af
"Jr,.
\ \
,,.
\\
A
\
10
p=1460
\
\
\
\
0
\ A p=1490 ,
I 0
2 flowing
time
-hours
Fig. 5.6 Late transientanalysisplot, extendedpressuredrawdown test, Denver Basin Muddy SandstoneweU. --
J
,
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= 1J'I
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r
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gOl
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P u~
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= "A
3~nSS3~d
V171
': APPENDIX D
145
Since the theory for transient analysis assumesradial flow, the kh value derived from a transient analysis will b I I hi h A fl d h di I e anoma ous y g. s ow time procee s, t e ra a flow in the region away from the fracture becomesdoininant and 1ate-transi~nt anal--sis which is also based on ra al flow theory more nearly represe e values of the reservoIr parameters. Thus, in the case of a fractured well we believe the late transient results are probably more representative.
.
Semi-Steady State Analysis (Reservoir Limit Test) The linear plot of P,vfvs t is found on Fig. 5.5. This plot appears to be linear for times greater than 15 hours. From the slope of the plot of Fig. 5.5 and Eq. 5.14, we find -qB V" -0.0418 fj";c' -of V" = 0.0418 X 800 X 1.25 -6' 15.8 X 17.7 X 10 V" = 0.149 X 106 reservo~rbbl.
The equivalent drainage area is approxim;t;iyi7~cres. D..
ISCUSSlon
Th. e suIte of ~ata analyzed here is typical of the ty~e .that t?e .engIneermust analyze whenever a reserVOIr IS of lImIted extent. Actually, it is seldom that all the sep~ra.tefacets of the t~ansient, late transient, and reservoIr lImIt analysesare In perfect agreement. In the case we have seen here, there was disagreement between the transient and late transient analyses which was readily explained by fracturing. The belief that the late transient results are more representative is further supported by the almost-exact agreement in reservoir size calculated from the late transient and reservoir limit analyses. As is discussed in the text, these conclusions were substantiated by subsequent production performance. There are instances in which unique interpretation pressure drawdown data is not possible. More than one set of conclusions may be feasible. In such cases, the economic consequencesof each answer must be considered.
,
VI;
.-.. /
f;.' ow
-I
~.~ '"~"
'f .,~;~
I
,\::uV7,."::~
,1,
}11:1
All. LL --
I.
'
\ .
I
Appendix E
Example Calculations for Multiple-Rate Flow Test Analysis
In this appendix the details of the calculations required in the analysis of two different types of multiplerate flow tests are presented. The first of these is a two-rate flow test. The second is a gas-well open-flow potential test. Slide-rule accuracy is maintained in the results.
kh = 174 md-ft, k = 3.0 md. T~e ~ext step in ~he analysis proce~ure is the deterffilnation of ~e skin factor s. For this purpose, Eq. 6.10 of the text IS used:
Two-Rate Flow Test The well is a flowing producer from a typical lowpermeability limestone reservoir in the Permian Basin region of West Texas. In this particular reservoir, pressure buildup data usually are poor in quality because of long, low-rate afterproduction periods which occur when the wells are closed in. Pressure buildups of 72 hours' duration, or in some cases even longer, are required to obtain interpretable data. I . f n preparation or the two-rate flow test, the well was stabilized at a rate of 107 STB/D on a 12/64-in. choke. The rate was reduced to 46 STB/D by a reduction in choke size to 7/64-in. The well is equipped with d .324 a pro uction packer. Wellhead pressureswere not recorded during the test; however, producing rates were measured by means of a continuous-recording metering oil.26400 and gas separator. Flowing bottom-hole pressureswere measuredin the
.-3220
well for a period extending from 3 hours prior to the h 2 rate c ange until 2 hours after the rate change.The resuIting flow test plot and data pertinent to the flow test analysis are shown on Fig, 6.5. As is suggestedby the appearance of the flow test plot, the producing rate stabilized at 46 STB/D very soon after the rate change.
'ti ~ 3210 ~ ~ ~ 3200 ~ ~ 3190
This was confirmed by the metering separatormeasure-
::
.
ts men.
,[.
s = 1.151 [( ~ ) (fu ~ ql -qz m
From the basic flow test plot of Pw!vs {log [(t+~t')/ ~t'] + (QZ/ql) log ~t'}, the value of m is 90 psig/cycle. Thus, from Eq. 6.9 of the text, kh = 162.6 Ql,uB m ' kh = (162.6) (107) (0.6) (1.5) 90'
-log~
+ 323 c/>,ucrw2"
[(
s = 1.151
-log (0 --.)06 s = -3.6.
107 107 -46
)
]
)( 3,169 -3,118 90 3
(0.6)
(9.32X
) +
10-5)
3
(0.04)
23 .,
]
3250 ql '107STB/D Pw'3//BpliV Q2'46STB/D h'59fl .-1 c,'. 9 32xl0 -5 P"' rw'. 02fl /J.'0.6Cp B'I.5 -#I'0.06 Np'26,400STB 1.-=tar .24 =5922hr BASIC DATA-WELL A
323 c-
~
SLOPE' 90pII
g 31BO 317
, RESULTS
316
k =3.omd .=-3.6 . p*.354Bpslg
3150 3.0
..3.3
3.4 !..tA! '
3.5
3.6
.39
.9.i. I
log 61' + Ql
og 61'
Fig. 6.5 Two-rate flow test plot, Permian Basin well.
148
PRESSURE BUILDUPAND FLOWTESTSIN WELLS
Having found values for k and s, we may now proceed to determine p*. By Eq. 6.11 of the text, we have
[
p* = Pw + m log '" kt 2 -3.23
0.017
]
+ 0.87s ,
0.01
't'p.cr'/J
0.01
p* = 3,118 + (90)
[log (0.06)
-3.23
0.014 (3) (5,922) (0.6) (9.32 X 10-5) (0.04)
+ (0.87) (-3.6)
],
p* = 3,548 psig. The pressure drops across the skin at rates ql and q2,respectively, are: dP(skin) = 0.87 (m) (s),
,0.013 ~
0.01
~ 0.011 -i ~ ~ 0.010 6:-
psig,
dp(skin) = 0.87 (q2/ql) (m) (s), = 0.87 (0.43) (90) (-3.6), = -121
psig.
RESULTS kgh =140md-ft k g =35md .
000 .s
= 0.87 (90) (-3.6), = -282
m'=0.02904
= -4.7
0.00 0.007
I b =0.00625
/ 1 0.006 0
0.1
The minus sign indicates that, because of an enlarged well radius, the pressure drop near the well is less than normal.
Multi-Point Open-Flow Potential Test In this casethe well is a gas producer in the MorrowChester sandstone in the Anadarko basin of Oklahoma. The data were obtained on a four-point OFPT run
~ i-I .
(~~ ) qn
0.3 I
(t -t.
0.4 )
og n J-I
Fig. 6.12 Calculation of k.h and s from OFPT data, Anadarko Basin well. kgh =
upon completion of the well. General data pertinent to the test are as follows: rw = 0.23 ft,
0.2
(~8,958) (0.017) (8.28X 10-8), 0.02904
kgh = 140 md-ft, kg = 3.5 md.
From Eq. 6.,14
4>=0.16, S - 0 20 to-.,
]
k g 2+ 3.23 , s = 1.151 [-;hi -log", m 't'p.gCVW
h = 40 ft, s
p.g
=
0.017
=
1.151-
[
cp,
Ct = 6.89 X 10-4 psi-l,
0.02904
3.5 log (0.16) (0.017) (6.89 X 10-4) (0.052) + 3.23 ] ,
Bg = 8.28 X 10-3 cu ft/cu ft, .s gas gravIty = 0.7. T he total vanation m pressure dunng the test was 82 psi. Accordingly, the IJ.gBgproduct was evaluated at 1,650 psia, the mean pressure. The pressure and production rate data are given in Table E-l, and the I ul ti th th d necessary ca c a ons to carry out e me 0 proposed in Section 6.5 are given in Tables E-l and E-2. From the plot (~eeFig. 6.12) we have m'=0.02904, hi = 0.00625. Thus, from Eq. 6.13.
.
kgh = 28,958IJ.gBg m'
0.00625
= -4.7
.
TABLE E-1~PRESSURE AND PRODUCTION RATE DATA FOR GASWELLANDCALCULATION OFORDINATE ' IN FIG. 6.12
Time ~ -0 t, t, t, t.
t Lengthof Elapsed Flowat Flow Assoc. q Prod. Time Rate Rate p.. (hours) (hours) (Mcf/D) ~~~ -0 1691 1.25 1.25 1048 1682 2.25 1.0 2101 1667 3.25 1.0 4167 1637 4.25 1.0 5116 1609
PI -p../. q. 0.00859 0.01142 0.01295 0.01603
149 TABLE E-2-CALCULATION
OF ABSCISSA FOR FIG. 6.12
t. -t,..-
log(t. -tl-,)
n=1
n=2
n=3
n=4
n=l
n=2
n=3
n=4
1=1
1.25
2.25
3.25
i=2
-1.0
2.0
4.25
0.09691
0.35218
0.51188
0.62839
3.0
-0
0.30103
0.47712
1=3
--1.0
1=4
---1.0
2.0
--0
0.30103
---0
.t.=O
q.
~cf/D) 1048 0
ql-q,..
(Mcf/D)
(q,-q,-,)/q.-!!.=.L
1048
1.0
2101
1053
-0.50123
4167
2066
--0.49580
5116
949
,, ~
;=1
(q
)
1- q,.. log(t.-t,-,) q.
n=2
n=3
n=4
0.49885
0.25150
0.20485
0.25270
0.20582
0.17568
0.40383
0.20481
---0.18550
0.09691
0.34849
.q. = 0
The four numbersin the last columnof Table E-2 are the summationsfor n= 1, n=2, n=3 and n=4, respectively.For example,the summationfor n=3 is obtained from the values in the columns for n=3 as follows. Summation=0.25150(0.51188)+0.25270 (0.30103)
~
+ 0.49580 (0) = 020481 .. In Fig. 6.12 the four numbersin the last columnof Table E-2 are plotted vs the four numbersin the last columnof Table E-l.
Appendix F
Example Calculations for Injection Well Analysis Example lA: Pressure Fall-Off Analysis, Liquid-Filled
Since 70.6 ip./kh = m/2.303,
Case,Unit Mobility Ratio
Determining p for Fig. 8.3-The
-
first step is to
then p -p*
= 7.91 (m/2.303),
obtain koutfrom the Islope of the fall-off curve.1.This is carried in Part of Appendix F, Example Using
= 7.91 (130/2.303) ,
this k and other data given in this example, we calculate dimensionlessflowing time for a 40-acre pattern flood (injection area A of 20 acres):
= 447 psi; ..* and, obtaining p
0.000264 kt = 0.000264 (21.8~ 40,100 = 393. cpp.cA 0.16 (0.6) 7 X 10- (871,200) From Fig. 8.2, (p-p*)/(70.6 300
P = -322 + 447 = 125 psig. This resu1t and 0ther resu1ts In A ppendix F are liInlted to slide-rule accuracy.
6t, hr 10
.
.
ip./kh) = 7.91.
I
. from FIg. 8.3, we find
100
0 0 0
0
20
"'" ~~
CALCULATED AVERAGE PRESSURE ;;::""
10
"
co --"
,
a. on
w
, "
It:
(/) :J (/)
"-
" "
w It: Q. -10
, " ~-+-"-9" 4..0'0,
~4;:" .'11' /0'
,
-20
"
, -30
"
",
p*=-322
-40 10
10
10
t+6t
I
6t
Fig. 8.3 Examplepressurefall-off curve.
:
j'.
" psig
1
APPENDIX F
151
Example 1: Pressure Fall-Off Analysis, Liquid-Filled Case, Unit Mobility Ratio (Based on Fig. 8.3) Test Data: Test Date October 30, 1964 Producing Formation Sandstone Hole Size (inches) 8.5 Cum. Inj., Wi (bbl) 2,380,000 Stabilized Daily Inj.,i (bbl) 1,426 Effective Prod. Life t (hr) = 24 W i/i
Company Shell Lease Zipper Well No.4 Field Bent State Illinois 40,100
I. Calculation of kh (md-it) and k (md); k is permeability to water, kw: kh = J~~-~h i
;
49 1,426
k = ~
.
it BID
kh = 162.6 X (1,426) X (0.6) X (1.0) = 1 070 (130) -!.--m
d-f ' t,
po B
0.6 1.0
cp (Fig. G.4)
m
130
psi/cycle
k = (1,070) = 21 8 (49) ~m
d = kw.
II. Calculation of Skin Effect, s; and PressureLoss Due to Skin, ~Pskln (psi): s = 1.151 [~~~-IOg(k)
+ 3.23J.
~Pskln= m X 0.87 s. k cf> po c
21.8 0.16 0.6 7.0 X 10-6
md cp psi-I
[ (525)(130) -(273)
s = 1.151
'w PI hr Pw m
III.
it psig psig psi/cycle
]
(21.8) (0.16) (0.6) (7.0 (144) X 10-6) (18.1) + 3.23 = =~
-log
~Pskln= (130) X 0.87 (-3.73)
4.25/12 273 525 130
= -~psi
(well had been fractured).
Calculation of Injectivity Index (BID-psi) and Flow Efficiency: I(actual)= ~P.kln i
1
(actual)
i -1(ldeal) Pw -P -421 psi 1,426 B/D
-(1,426) -(525)
-(125)
1(Ideal)= (400) (1,426) -(-421)
- i (Pw -p) -~P.kln 'iJ 125 Pw 525 =
3
56B/D
---=--
. -pSI.
.
1 73 BID-pSI. = -:.-
Flow Efficiency = I(actual) = ~= 1(ldeal) 1.73
2.06. -
Note: Assuming So = 0.20, Sg = 0 in the swept zone, we have C = Ct = SoCo+ SwCw+ Ct, = 0.20 (3 X 10-6) + 0.80 (3 X 10-6) + 4.0 X 10-6, = 7.0 X 10-6psi-I.
.
psig psig
152
PRESSURE
Example
2: Pressure Fall-Off Analysis Prior to Reservoir Fillup, Unit Mobility Ratio
BUilDUP
AND
kh = !!!:. .(1 -C1 -C2) b1 (1 -C3)2
Fig. 8.8 shows an example pressure fall-off curve. The value of Pewas known for this well from a previous
FLOW TESTS
.f(fJ) = 1,020(0.9). 347
(0.961) (0.937)2 .176.5 = 511 md-ft,
lengthy shut-in period. This value (32 psi) was used in plotting Fig. 8.8. The wellhead injection pressure was
d an
zero, so we will apply the case where the wellhead pressurefalls to zero shortly after closing-in. We have the following data on this well: i = 1,020 B/D, h = 45 ft, dt = 6.366 in. (casing diameter, since no tubing in this well), p", = 598 psi, Pt = 0 (zero wellhead injection pressure), Wi = 6,097 bbl, p. = 0.9 cp (from Fig. G.4 at 88F and 8 percent NaCl), p = I gm,/cc, ~'o = 1 ft .(estimated on basis of sand removed dunng swabbmg), = 0.3, S", = 0.32, S9 = 0.12, Stir = 0, So = 0.56, and Sor = 0.2. From Fig. 8.8, we find b1 = 347 psi and,Bl = 2.303' (log 347 -log 122)/20 = 0.0522 hours-I. We have for the case where the pressure drops to zero shortly f I a ter c osmg m, f rom Eq. 8 .,5b
$ + In ~ = 0.007~8(P",-Pe) r,o lp./kh -0.00708(566) -1,020(0.9)/511 = 2.23 . According to Eq. 8.4,
..
C1 = 0.0538 ~t2 ,Blb1 pi -0.0538 -1
(6.366)2(0.0522) (1,020)
(347)
,
= 0.0386.
.
C3 = C1. ¥
-66 1
re = _I Wi(_5.615), 17r(S9-Stlr)h £f 6,097(5.615) = 1 ..(0.3) (0.12) (45) = 82 ft. T herefore, $ = 2.23 -2.303 log (82/1) = -2.18. Example 3: Pressure Fall-Off Analysis, Non-Unit Mobility Ratio From the information given for the preceding example, for /10= 12 and for k",/ko = 0.3, we have ~= V,o
C 2 = O, sInce Pt = 0 .
= 0.0386 (~) 47
= 0.0628.
Then,
So -Sor= S9 -S9r
3 and M =~= ko p.",
The ratio k",/ko is obtained by measuring core permeability to water at the saturation in the water bank and core permeability to oil at the saturation in the
0.0386(0.937) 2(0.961)=0.0188.
According to Eq. 8.9, 1 rD 0 ==~/~+ 1 1 V",
From Fig. 8.6, f(fJ) = 176.5. Therefore, ,!
1 = 05
-..
y4
600
. Further,
~ ~
0.3(12) = 4. 0.9
oil bank. See Fig. 8.5. C1(I-C3) fJ=2(I-C1-C2)=
, ,
IN WEllS
50
M
=
4
and
y
=
1
smce
f Co
~
c,o
h . or
t
IS
d
ea
d
oil. Therefore, using Eq. 8.7 and reading F from Fig. 8.9, we obtain
L
~
40
.
k",h = ~
.2F . bl
'. 30 :: a.-
= 1,020(0.9).2(220) 347
.. ~ ,
'
= 1,170 md-ft. 20
This value of k,ch is 2.28 times as large as the value obtained for the single-fluid case. The skin effect is found from Eq. 8.11.
$= 0.00708(p", . -Pe) k",h
I 100
0
l",p.",
CLOSED-IN TIME,HOURS Fig. 8.8 Pressurefall-off curve.
M -1 -In 2
(-+1 V 0 V,o
) -In-,
re
r",-~..
APPENDIX F
153
= 0.00708(566) 1,170_!!:..-=!ln4 1,020(0.9) 2
-ln~
1 '
= 5.10 -2.08 -4.41 , = -1.39. This value of s is less negative (indicating a smaller effective wellbore radius) than the value obtained in the single-fluid case. Thus, use of the single-fluid case has given too large a value for effective wellbore radius and, as noted above, too small a value for kh. This is the result one finds when the water mobility is greater than the oil mobility (M> 1). By obtaining too large an effective wellbore radius from use or the single-fluid case, the engineer may incorrectly decide that there is little possibility of injectivity improvement by well stimulation. Use of the pro.per mobility ratio would lead to a proper recommendation. Example 4: Two-Rate I.nj.ect~onTes~ ..b An example two-rate mjection test IS shown m FIg. 8.12 for a 10,000-ft well. In preparation for the test the well was stabilized at an injection rate of 2,563
B/D. To obtain the transient pressure data the rate was reduced to 742 B/D. By trial and error as shown on the plot, the average pressure in the region around the well was found to be 3,600 psig. The average pressure is chosen as the highest value for which the plot in Fig. 8.12 is linear at large time. Note that the curves bend down at large time for greater assumedvalues of p. Further note that an injection time of 48 hours after the rate change was required to obtain this value of static pressure. At 32 hours injection time, the plot is linear for values of p < 4,200 psig, as shown. Thus one might have estimated the average pressure as some 600-psi higher at the shorter time. Data pertinent to the analysis of this test are shown in Fig. 8.12. From the solid line on this figure we find = 740 psig (intercept), and fJ (absolute value of slope) as fJ = (log 740-log 460)/50=0.00413
hours-!.
10,000
DATA
.~
il=2563B/D
~
, r
i2=742 LL =037 rW .
~
Ct=7.0XI0-6psi-1 . Pw .6777 psig 4> =0.244 h =31 11 rw = 0.3 It
10-
1 ~
~ N -a .';'1.+ 1000 00.
'---'I
:!:.
Co
00 a a a a a a
00 a 00 00
BID Cp
000
a
00000
a
A 00
000
0 0000 OOOOOOOG-E)..e-&e-o-e-ee-e-o-
P=3400psig
e 000000
a
P=3600psig
000 00
a 00
OOOOOOO~ 00000
00
0000
00 a Oc-e-e
-e-&~-e-E)
&\9-o-E)-e-g.
00 00
0
p=3800psig
--e
-9--.
9-G-e:
a 000
G-'O~"G
--
P=4000psig ---9-
6-e"$oE)'"9'9 So&..1000
-e--
--
--
P=4200psig
4
50 A t: HOURS
Fig. 8.12Two-rate injectiontest.
-~
154
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
To determine the formation permeability, we use Eq. 8.13 of the text. kh-
1812 ( .. -b
.'l-'zp.
) '
= 181.2 (2,5~~~742) 0.37, = 165md-ft. k = 5.32 md. We use Eq. 8.17 to find the skin factor.
of-1.283
(~ b ) . (~4 )
-1.1511og
(
0.000664 k 2' fJcf>p.crw
= 1.283 6,777-3,600 740.
) ( 2,563-742 2,563 ) -1.151
log
0.000664 (5.32) 4.13XI0-s (0.244) 0.37 (7.0XI0-6) 0.32' of= -4.36. These values agree quite well with those determined by other methods for wells in this reservoir.
Appendix G
Charts and Correlationsfor Use in Pressure Buildup and Flow Test Analysis 10POO 8000 6000 4000
'"
2000
~ 1000 ~ 800
;: 600 z ~
-'
0
i
~
400 200
10
U
8
...6
:
4
,;,
"
20
.. ~
RESERVOIR TEMPERATURE
10 8
)-
~
6
8
4
'"
>' ...2
.~
I
'" "
08 n6
m
0.4 02
0.10
IS
20
CRUOE
OIL
25 GRAVITY
30 'API
35 AT
40 60
~
45 ANO
30
ATMOSPHERIC
55
60
65
PRESSURE
Fig. G.l Viscosityof gas-freecrude oil at oilfield temperatures.From Beal, Trans., AIME
(1946) 165, 94.
100 80
~
60
0
40
ICY
In In In
"""
~ ~ 00.
n.
v
20
0. o::~ z-
~ ~
-", .~
O~
10 8 6
oln
~O "Z
"0-'"
4
~4 4
~
In"
I ~
IL
2
BL
In~
4", tOo. ...2
0'" >o-~ iii ~ o~ ~ In
.-
"",
>'"
I 08 06 04
~
~
-02
. 01
0.4
os 0.8 I
(AT
RESERVOIR
2 VISCOSITY
20
40
60 eo 100
OF DEAD OIL, CENTIPOISES
TEMPERATURE
AND
ATMOSPHERIC
PRESSURE)
Fig. G.2 Viscosityof gas-saturated crude oils. From Chewand Connally,
Trans., AIME (1959) 216, 23.
..-
~
156
PRESSURE
6.0
II Note: obtain ~l from Figure G.J8
--1-..'" ---/
5.5
--'5
I 1/
-Q.
f
1/
II 1 /
1 I 1/
I I
/ 1/
-~
'5
-QO
~~I g 1 .:) 1 11.1 K!
I
/
/
1
/
/ I
""'I
~ ~
-
1 1
I !/
1
I
/ II
I
/
11)/ / ~I
~
0 .:: ~
"I,
/
-~
4.0
I
kI
-11.1
I
/
0
/ ,/
3.5
1/
cv ...:/
/ /
I
/
>-
~/ -'
/ 1
1
I-
u:; 3.0 0
/
'
.///"
I
/
I
1/
/
/
V /
/
"7 r1;~ /
/
L
~~
V
-'"
~ ///v"",""
/
'\I;~I
/
/
/
~91
7 ~I ~I
-""'"
-~ .2.3.4
I ('II
I
/
:r0'0
I
/
I,
I / II
/
1(0
/
/
I
15 .1/
--
.I
I I
"
!/
/
I""
/
I
1 1 1
I
I
I
I
1.0.1
!J
-..:/ II
I
2.5
~
I
~
/
/
0 ~I
//
>
I
I
I
0
2.0
IN WELLS
I
1 1
-
FLOW TESTS
I 1
/
4.5
AND
/
--: 50 .~
BUILDUP
-'"
~
~ .5.6.7.8.9
1.0
2
3
~
~
-6 7 89100
20.0
PSEUDOREDUCED PRESSURE,Pr Fig. G.3A Viscosityratio. From Carr, Kobayashi and Burrows, Trans.,AIME (1954) 201, 264.
.
APPENDIX
G
157
~ o.
~
W)
01
~--
0 2(.)
~ 0
01
I
.
W)i -I
r
!!
-
-
.J
(.)
0 Iff
0 0
*
-~
.~
::f
"-
0
'
-0
8.d')-',)5Ih'O.l 8:
0
~ ,,-.,
01
~
§
0300~
~ '-'
NOI.l03YYOO
J
.trJ
~
-I
<
,,). ~
~
W) ,"I.
~
0
--'I ~ -~
:-
m
§ ~
-' 01
I
..-2z
rj
W) Z
;;
--I g -J--.J -O!, ,
01
,
-1 0 0 0 ...I
~ =- -'
. ~!
I ,
,
';'I:';J-'
: : i
W) 8
"
0300Y
.::PI
q
:I
,
;
-,
,
t,
-1-'
'I' -~Ji
,I'
NOI.10iYYOO
:
~
f 'r
Q ~
,
..J
.~ ,(.)
'. I I ___, I' ~ ,I I -I !,-
"'_"'~ ' I , '
(I)
: it,'
,I .I 'J l'
r
,
,
!/:-,
I'
',1
LLIT-,
1:
1-.'( I'
"
" "
(,
~
I :/
1',
: '
.I,
.I'
\
"
,
.L
i-
L // T7-,
"L , y'" , :
, I
"
V)
..If)
q
q
.dO':"
OiOOY
I' '"
-0
>-
0
l -I~
I' I!
,
~~ 0
--o-!.'.'
.: .J --~ _0 2 -.;
'I
1"1
-0
---u ,:
' I
-1-
I
~-
i
-,
~
~ 0 0 -05Ih'
O.l
NOI.lOiYYOO
Ji-t-T1-1-1-ritTitTt;tI I I ...CD
0
m
q 3SI0dl.1.N3~c q O§;;\~ W1~ -~ I 1'0"8 A.lISOaSI~ 8 8
g.
." =
-~ =
~
=ro:~
.~
-~ 1 V)
> '"
-,
::::::
_r~.jI_lf-l-_I_I_IIIII_IIII_--=1-
no
~
---~ae-
~ 0 0
:
I
+
g
t\I
,.: '
II-,
! H
d 0 .0
-C
-~ L
~: O:/'"'
-+
~ 0 0
"I
-i--
-, 1 CD
,,'
,-'-
I I !
"
-r
-' '
t
I
" ""
I
q
I'
"
-.,y,
I
-I
."
~
W)
~ '
.'.I
O.
-;-
f
i
-.I.
'I ,-i:.-
--, t---V! 'A %a
,J -1-'.1' -r--:/--;-
Q-
~ I
~
I
1.1
.I,
r-
r
13 ~ -
... oj
I
I
,I
~ ~
~ co
I
, Ii/...
U e
~
I
,
ft
'~
0
.'/
,
, ,
0
: , ,I
1,-
r: I:
,
W)
.In
'..
I,{
, ,
,
I"
I
0 -'
0
!I'11 ,
,
..J
,
.-d
-I 'I
~ 0
I
r-
~ :a rI) (U '1: 0 ~
;
'I
I
-+-
,
~ ~
; -~
~- -II I'
> C ~ ~
0
, '-
I
, 'I
--I-
'::
-:
, ~L
:
--I'
e ~ c~
'... ':r.
I -f-
-~ ,~
01
1I I
>-
/!'
oo! 051h
!,
'I
0 0 q d 0-
~ --f=i ,
\
at, W)2'
, -.-0 ~-
~
158
PRESSURE BUILDUP AND FLOW TESTS IN WELLS
2.1
2
1.14 ESTI~ATEO
I
TE~P
ERROR
IJ.*
40°-120° 120°-212° 212°-400°
I.
~
f
1%
~%
~% 10%
~% ~%
1.12 1.10
I. f I
I.
1.4
1.02
1.3
1.00 0
'"
T,o,
~
0 A. ;: ~
u
1.2
PRESSURE FOR WATER
1.1
PRESU~EO
.NOT
CORRECTION VS.
'ACTOR
APPLICABLE
CONFIR~EO
(f )
T,oF TO
BRINES
BUT
E~PERI~ENTALLY
*;t. ..1.0
VISCOSITY
0in
IJ.p,T
AT ELEVATEO = IJ.
*T
PRESSURE
.fp,T
0
u on 0.9 > O.B
0.7 0.6
VISCOSITY (IJ.*) AT SATURATION
/
AT I AT~ PRESSURE BELOW 212° I PRESSURE OF WATER ABOVE 212°.
O.~
0.4
0.3
0.2
0.1
0 40
'" 60
BO
100
120
140
160
IBO
200
220
240
260
2BO
300
320
340
360
3BO
400
T, of
Fig. G.4 Water viscositiesfor various salinitiesand temperatures.From Chesnut,unpublished,Shell DevelopmentCo.
~
:,-
APPENDIX G
C.f
.159
---
10
9 >-
~ .J
~ w
8
-~
m
~
7
w
a: a.
~7 0 0
0
.z
6
--&&. In S
a.
W
C --
I
4
Z
~
w -~-~ > 3 -~o&&.-
oc~-
0
w8- c- ~ .
&&._0...
I-
W LL
-
~
-c cOm
a: 0
LL
c
9 ~
...Q;
~ ow
0
i:
-
C
~~
oJ-lAIN
2
u
.J-
U
W
I
0
LIMESTONE
.SANDSTONE
"
00
2
4
6
8
10
12
14
16
18
20
22
24
26
POROSITY, percent ;
Fig. G.5 Effective formation (rock) compressibility.From Hall, Trans.,AIME (1953)198. 309. -.1-1
-
160
PRESSURE
Pseudoreduced 0
1.1
I
2
3
BUILDUP
AND
FLOW TESTS
IN WELLS
Pressure 4
S
8
7
8U
T P RA R
I.
s 8~
0.9 N
0.8 L
N
-
'1.7
..
I.
..L
0
0
~
-I-J
-I-J 0.7 U
1.8 U ro
ro
LJ..
",
,.
LJ..
~
0.6
I.S
C
0
C 0
.-
.-,. -I-J o.
~
ro
-I-J 1.4 ro
.-
.-> >
Q)
0
Q) 0.
-.30
. 0.3
.2
0.2 .1
I
0
.0
.7
~
10
Pseudoreduced
II
12
13
.9
Pressure
Fig. G.6 Deviation factors for natural gases.From Standingand Katz, Trans.,AIME (1942) 146, 140.
~
APPENDIX G
0
s
.. ~
+'
C
-L~
/
161
J -(.Pc..
I'O9
0.8 0.7
.-PSEUDOREDUCED r-f 0.6
T E M PER AT U R E
...a
.-0.5 (/) (/)
a.>
L
0.4
c.. E 0
(..)
0.3
"'C a.> (,)
~ "'C a.> 0.2 L 0
"'C
~
a.> (/)
Q.-
0.1 I
2
Pseudoreduced
3
4
5
6
Pressure,
7
p
8
-/
r
Fig. G.7A Reducedcompressibilityfor gases.From Trube, Trans.,AIME (1957)210,355. Cr= c.P..
l~~;
.
9
10
162
PRESSURE
BUILDUP
AND
FLOW TESTS
IN WELLS
0.10 0.09 0.08 L0
0.07
..
~
0.06
M
Q05
+' ...c cn
0.04
cn Q)
L0-
E
0.03
0 (..)
""C Q)
0 ::)
Q02
""C Q)
L0 ""C ::) Q)
cn
a-
QOI '3
4
Pseudoreduced
5
6
7
8
Pressure,
9
10
15
p
1
r Fig. G.7B Reducedcompressibilityfor gases(low range). From Trube, Trans.,AI ME (1957) 210, 355. Cr = c.P..
APPENDIX
G
-~
163
IQ..
ci
0;
0;
0;
N
-00
0;
0;
~
IQ..
0;
0;
N
0
0;
-
0
0
0;
0;
., 1
N
. ci
.. ci .. ci ,++
-I-
,
H
0
d
)(
0 OJ:: U d
='
'+;'
,
~0
~ ~ .
t-'
~ ~
. ci
0;
.. 0; .. Q.
~ 1 -,
-il ' 0
N
-~
0 W I
ci
0
N
-:"
0
q
0
