p' = 4,437 + 2 (70) = 4,577 psi. Then, from Eq. 2.18, £=P.-Pwj-(.:1p)s p' -Pwj

k = 3.35 md. =

4,577 -3,534 -387 4 577 -3 534 , ,

Calculate s, sd' and s p; on the basis of these resulls, determine w~ether the productiviry problem results from formarlon damage or from orher causes.

= 0.629. Solution. From Eq. 2.4, -k S=I.151[Plhr-PWj -Iog(

This means that the well is producing about 621170 as much fluid with the given drawdown as an undamaged

well

in a completely

perforated

would produce. 5. £ndof Wet/bore Storage Distortion. 2.13and Examples 2.2 and 2.3, I

interval

From Eq.

Cs eO.14s = 170000 '

wbs

kh

=


]

= 1.1511 (1.940- 1,190) 50 -

-10

III.

(170,000)(0.01 1S)e(O.14)(6.37)

2)+3.23

m

(

3.35

g (0.2)(0.5)(1.5xI0-S)(0.25)2

) + 3. 23 ]

= 12.3.

(7.65)(69)/0.8

.The

=7.42 hours. This agrees closely with Ihe results of Example 2.2.

~

I.:ontribution

<:>fan in~omplelely perforated

Interval to the total skin factor IS, from Eq. 2.20, h h m ~/J= (f -1)[ln(/ V~ )-2] p

w

V

Effect of Incompletely Perforated Interval When rhe complered interval is less rhan total for~ation rhickness, the pressure d~op near Ihe well is Increased and

the

apparent

ski!)

factor

becomes

increasingly positive. In a review of technology in this area, Saidikowski6 found thai total skin factor, 5, determi.ned from ~ pressure transient re.stis related 10Irue skin factor, sd' caused by formation damage

and apparent

skin

factor,

sp'

completely perforated interval. betweenthese skin factors is hi

S=j;-Sd+Sp'

caused

The

by an in-

relationship

(2.19)

p

where hi is total interval height (ft) and hlJ is the pcrforaled interval (1'1). r

Saidikowski .also verified that Sp can be eslimated rrom the eqllatlon h I ( h r-kI ~"J= (-L -I) In -Lv ---}l- ) -2, (2.20) hp rw kv where k if is horizontal permeability (md) and k v is vertical permeability (md). Use of these equations is b~stillustrated with an example.

(

50 = 10 -I

)[In (~v50

~~

) -2 ]

.l

= 13.2. Rearranging Eq. 2.19, skin factor, sd' resulting from formation damage is h

:

l ! f

Sd= :.:2. (s-sp) hi 10

=-(12.3-13.2) 50

= -0 IS .. A~ a practical matter, the well is neither damaged nor stll1\lllaled. The obscrvcd produclivilY problem is ,-"all~cdcnl!rcly by Ihe effc,-"l~ of an in,-"omplclcly pcrloruled Inlcrval.

~I ;,i ~ f

Anulysis of Ilydruulicully "'ruclured Wells Type curves provide a general method of analyzing hydralllically fractured wells -particularly because ~

Sla'E

E.~ PWS

FR/(;~E

T

TABLE 2.4 -BUILDUP

DC».4INATB ClRVESHAPE

HYDRAULICALLY

TEST SLOPES FOR

FRACTURED

L,lr.

mm..'m'ru.

0.1

0.87

0.2

t

+6t

WELLS

0.70

0.4

0.46

0.6 1.0

0.32 0.28

logT Fig. 2.14 -Buildup curve for hydraulically bounded reservoir.

fractured

well,

finite condl,lctivity can be considered. Some convention~1 .methods are a~so o~ value for infiniteconductIvIty fractures. ThIs sectIon summarizes some of the useful conventional methods. When fractures are highly conductive (i.e., when there is little pressure drop in the fracture itself) and ~hen there is ulliform nux of nuid into the fracture, lInear now theory de~cribes well behavior at earliest t~mcsnow ill a rates buildtlp te~t. (Uniform nux mcall~ idclltical of fonnation nuid illtO thc fracturc ..w

pcr Ullit cro~s-sectlonal area at all points alollg the fracture.) From Eq. 1.46, for collstallt-rate product ion, I qB

IJ.f

Thus, L L -210g( ~ ) = -2 log ~ -210g 2r w 2 = (PI hr -Pwj) m -210g (

(~

-IOg

(~

r w)

)

IJ.C,

~ ) + 3 23

r.

.

This simplifies to log ( L j) = ~

V2

2

.

l(~

mI

~)

) I 11Lj kc, k f Forabuilduptest,forlp)..ill, +log;;;;;--2.63. (2.22) qB ill V2 .' Pws -Pwj=4.064 -(£.) .~slng Eq. 2.22, fracture I~ngth, LI' can be estimated hLj kCt if the MTR can be recognIzed, whIch allows m,PI hr' Thus, the slope ',1 of a P vs. v'A7 plot is and k to,be determined. L wYi In buIldup tests from some hydraulically fractured ) 2. (2.21) :-veils, the ~rue.middle-time line does not appear, as "'L =4.064 ~ f IlL f k4>c, Illustrated In FIg: 2.~4. (~fter~ow can cause the same From measurements of this slope, fracture length, c.urve shape.) ThIS sltua.tlon a.nse~bec~u~e, at earl~est /4f' can be estimated. This procedure requires that an tlme~, the depth of InvestIgation .'s In a region independent estimate of permeability be available -d°m.lnate~ by. the fracture; at lat.er tImes,. the.depth from a prefracture pres~ure buildup test on the well, of Investigation reaches a point dominated by for cxamr;le. boulldar~ effect~. (Se~ Fig. 2.14.) When the length L,r When linear now cannot be recognized (i.e., when of ~ vertical rracture IS greater than one-tenth of the there is f10 _early straight-line relationship between draln~ge radius r e of a7well centered in its drainage p",Sand v'A1), we can use the observation that L area, It ha~ been found that boundary effects begin = 2 r, for infinitely conductive fractures to estimat{ before the Innuence of the fracture disappears. For a fractl~'~elellgth. Rather thall calculate s directly, we given drainage radius, the greater the fracture length, can note that the greater the discrepancy between the maximum slope achieved on a buildup test and the slope of the s= 1.151 1 (PI hr -Pwj) -IOg ( -~ ) + 3 23] .true middle-time line. Table 2.4 summarizes the ratio nl JA.c,r~.' of the maximum slope attained in a buildup test to and, bccau~c the ~Iope of the true middle-time line (from the work L of RII,~~cll and Truitt') for infinitely conductive , = ~ =r e-s fracturcs. "'0 2 w , The implication is that if the test analyst simply

~

-~

Pi-PI~1=4.064-(-

1

(~

then s= -In(

does the best he can, and finds the maximum slope on a buildup test from a hydraulically fractured well and ass.ume~that this maximum slope is an,adequate approxImation to the slope of the true middle-time

L L ~ ) = -2.303 log( ~ ). 2, w 2r w

':.

~-

PRESSURE BUILDUP TESTS

35

ETR

ETR

Pws

MTR

""

t-p.

MTR

P I

L TR1/

RI

I

ws

~...

log

tp + .1t

--.11---

Fig. 2.15-Buildup test graph for infinite-acting reservoir.

log

Fig. 2.16-Buildup

tp + .1t --~t

-/

test graph for well near reservoir

limit(s).

line, then the permeability, skin factor, and fracture estimates will be in error, with the error growing as fracture length increases. Correlation of reservoir model results by Russell and Truitt 7 showed that an equation similar to Eq.

For a reservoir with one or more boundaries relatively near a tested well (and encountered by the radius of investigation during the production period), the late-time line must be extrapolated (Fig. 2.16). (This can be quite complex for multiple

2.22 can be used to estimate true fracture length even when L f > 0.1 r e. We again emphasize that all methods in this section assume highly conductive, vertical fractures with two equal-length wings. When fracture conductivity is not high, fracture length estimated by thesemethods will be too small. .For 2.9 Press~re Level In. Surrounding Formation A pressure buildup test can be used to determine average drainage-area pressure in the formation surrounding a tested well. We have seen that ideal pressure buildup theory suggests a method for estimating original reservoir pressure in an infiniteacting reservoir-that is, extrapolating the buildup t~st to infinite shut-in time (tp+dt)/dt=IJ and reading the pressure there. For wells with partial pressure depletion, extrapolation of a buildup test to infinite shut-in time provides an estimate of p., which is related to, but is not equal to, current average drainage-area pressure. In this section, we will examine methods for estimating original and current average drainage-area pressures.

boundaries near a well.) Note that our discussion is still restricted to reservoirs in which there has been negligible pressure depletion. Thus, even in the case under .consideration, the well must be relatively far from boundaries in at least one direction.

Original Reservoir Pressure For a well with an uncomplicated drainage area, original reservoir pressure, Pi' is found as suggested by ideal theory. We simply identify the middle-time line, extrapolate it to infinite shut-in time, and read the pressure, which is original reservoir pressure (Fig. 2.15). This technique is possible only for a well in a new reservoir (i.e., one in which there has been negligible pressure depletion). Strictly speaking, this is true only for tests in which the radius of investigation does not encounter any reservoir boundary during production.

Static Drainage-Area Pressure a well in a reservoir in which there has been some pressure depletion, we do not obtain an estimate of original reservoir pressure from extrapolation of a buildup curve. Our usual objective is to estimate the average pressure in the drainage area of the well; we will call this pressure static drainage-area pressure. We will examine two useful methods for making tllese estimates: (I) the Matthews-Brons-Hazebroek (Mllll)8 p. method and (2) the modified Muskat l1Il:thod.9 The p. method was developed by Matthews et al. by computing buildup curves for wells at various positions in drainage areas of various shapes and then, from the plotted buildup curves, comparing the pressure (/).) on an extrapolated middle-time line with the static drainage-area pressure (p), "r'hich is the value at which the pressure will stabilize given sufficient shut-in time. The buildup curves were l:omputed using imaging techniques and the principle of superposition. The results of the investigation are summarized in a series of plots of kh (p' -p) /70.6 q/lB vs. 0.000264 ktplt/>/lc,A. [Note that kh(p. -p) /70.6 q/lB can be written more compactly as 2.303 (p* -p) 1m. Also, the group 0.000264 ktp/t/>IJ.C,Ais a dimensionless time and is symbolized by t DA .The group kh(p*-p)/70.6 q/lB is a dimensionless pressure and is given the symbol PDMBHJ. The only new symbol in these expressions is the drainage area, A, of the tested well expressed in square feet. Figs. 2.17A through 2.17G (reproduced from the Mat-

Chapter 6

Other Well Tests

6.1lntroduction This concluding chapter surveys four well-testing techniques not yet discussed in the text: interference tests; pulse rests; drillstem lesls; and ~'ireline formation rests. These tesls and olhers covered in previous chaplers by no means exhausl the subjecl; however. the comprehensive Irealment needed by the practitioner is provided by SPE monographs 1.2 and

In an infinite-acting, homogeneous, isotropic reservoir. the simple £i-function solution to the diffusivity equalion describes Ihe pressure change al Ihe observation well as a function of lime: qBIL IP/LCr Pi -Pr = -70.6kh£i( -948 -t). ...(6.1) 1

Ihe Canadian gas well tesling manual.3 .The 6.2 Interference Testing Interference tests have two major objectives. They are used (I) to determine whether two or more wells are in pressure communication (i.e., in the same reservoir) and (2) when communication exists, to provide estimates of permeability k and porosity/compressibility product, d>£i, in the vicinity of the tested wells. An inlerference test is conducted by producing from or injecling inlo al least one well (the active well) and by observing Ihe pressure response in at least one olher ~'ell (Ihe observalion well). Fig. 6.1 indicates the typical lest program with one active ~'ell

This is simply a restatement of a familiar result. pressure drawdown at radius r (i.e., the observation ~.ell) resulting from production from the active well al rare q, slart1ng from a reservoir initially at uniform pressure Pi, is given by the £i-function solution. Eq. 6.1 assumes thaI the skin factor of the aclive well does nor affecl the drawdown al the observation ~'ell. Wellbore storage effects also are assumednegligible al bOlh Ihe aclive and observation wells "hen Eq. 6.1 is used 10 model an interference test. JargonS shows thaI bOlh Ihese assumptions can lead 10error in testanalysis in some cases. A convenienl analysis lechnique for interference lesls is Ihe use of Iype curves. Fig. 6.3 is a type curve presentedby Earlougher; I it is simply the £i function

and one observalion well. As the figure indicales. an active ~'ell starts produl.:ing from a reservoir at uniform pressure at

expressedas a function of its usual argument in now problems. 948 oJJ.'ir2/kl. Note that Eq. 6.1 can be expressed complelely in terms of dimensionless ..

Time O. Pressure in an observation well, a dislance r a\\ay, begins to respond after some lime lag (related to the lime for the radius of investigation corresponding 10 the rate change at Ihe active well to reach the observation well). The pressure in the active well begins to decline immediately, of course. The magnitude and timing of the deviation in pressure response at I~e observ~tio~ well d~~?ds on reserv?ir rol.:k and fluId propertIes In the VICInity of the actIve and observation wells. Vela and MI.:Kinley~ showed thaI Ihese properties are values from the area investigated in the test -a reclangle \\'ith sides of length 2ri and 2ri + r (seeFig. 6.2). In Fig. 6.2. ri is the radius of investigation achieved by the active well during the t.esland r is the distance bel ween active and observallon wells. The essenlial point is thaI the region investigated is much grealer than some small area bel ween wells, as intuition might suggest.

variables: I Pi -Pr = --£i qBIL 2 (141..!~) .!

I 1(-

-) 4

2 ". QILC,r'" )( ~ ).1, 0.000264 kl 'r.., ,

or I PD = --£i 2 \vhere

2 ( ~), 41D

-kh PD = (p, Pr), 141.2qBIL rD =rlr ""

, .-_(6.2)

PRESSURE

---=~lr";'

BUILDUP TESTS

~~fltff~-

_~i;""

, r -~ --1- -~ -l! ,

1---

;: 1i,

I

37

I

!

6

. I ~~

:

:

I

. .

I

!

I

-.

I

c.'m II .,a

i

~

I;

!

I

I

':

'

I

i I

: :

I

I i

i :

I

: i

i

i

/

m /1 u..o-;;;;

!

I, "

..'1"

/1..

//

/1'

.'

I

I

""

/"

-'

:

.'

0

""

, tIJ,

1 " ",-

--'

c.jCD:

-1

/1

::

--1 1

1

:

/1

i

~ I "'"

0,01--r-

01 I 01: 04

06 ~ 01

1-"'I

I

. ..1

I

I'

6 ..

0,000264 kt

~,.cA Fig,

2.178 -MBH

pressure

function

for

differenl

0.264kl pss IDA =

t;I£C,A

well

locations

in a square boundary,

general drainage-area shal'N.'Srollow eq,.ations of Ihe form

=0.1,

or. for this case (with A = 160 acres =6.97 x 106 sq

P-Pwj=

141.2~

ft)..

kh

1~In(~~ ~)2

CAr.

-

~ +s. J

4

Ipss=183 hours. "

(1.20)

The reader can verify that use or I ss in the Horner plot and in the p' method leads to t~e same results as in Examples 2.3 and 2.6. .vs. Shape Faclors ror Re~ervo.rs Brons and Millert2 observed that reservoirs with

;;

after pseudosteady state now is achieved. Values for the shape factor CA can be derived from thePDM8H I D.~ charts for the various reservoir geometries in Figs. 2.17 A through 2.17G. Note (hat (he definition of p' implies that

i f I

I

!,

II'.-.I

f

I

ii

p -p

:-'

~

,!

706QjJ.B/kh

.I:

I

" :~

---!,

:

5

--~

-':- ,.,1/ ""

4.

;\

/

;

~~..-1

~ ,,~

3 "~~--:--

.-1""

2.

-"""",:

/,

" I

""

""

~

~

:'

~ ~

/

---,

[

/'

'

~,..,.

i I

0 01

--"" 02

03 04

:' 061

QI

2

3

0.000264

4

,

I

2

3

4

,

10

.I

kt

+jJ.cA Fig,

2.17C -MBH

pressure

function

for

different

-~__~~JI_II..

well

locations

in a 2:1 reclangular

boundary

38

-WELL

~

;~~-r-]

TESTING

::rrrj--"-

706Q1/.B/kh;:i!':i

4

1---; !

'

;-!"1;--'---'

!

I

,~I

!

.2

I

1

0

I

-I

-2 06

01

2

3

4

0000264

6

1

2

3

.6

10

kt

~1/.CA Fig,

2.17D -MBH

f

p' -P":f

q 8 IL

=70.6T

pressure

function

for

different

well

locations

In(tp +~t)/~t,

~

h and lhal, allhe instant of shut-in (~t = 0),

P.q81L -p"j=70.6-

kl

f In I

-In

,~,688tPlI.£."",

I

(

k/1

kt

CA'",

-q8Jl

0.
-70.6kj;-

acting reservoirs only. Eliminating P"f between equations, p' -jJ = 70.6- q8p. In

( 10.06 A ,... _2 ) + 1.5]

) +2s. I

( -~-:-::-:! kt

These relationships result from replacing Pi with p' in Eqs. 1.11 and 2.1, which are valid for infinitei

boundary,r

in a 4:1 rectangular

In(

tPlJ.C A I

)

I :

- 70 6!!!!!:I

)

-.n

i~6si:;~~.

kh

(C

) At

DA

l '

.

0r I,

,

..!

i i

.--I -!r

!

~I

:

--' loCo

.,

,oX

a,ljQ' II ~ .' C'

'

I

.

0.1 co I

o.

,

.,

,...

,

,

"

:.

..., Fig. 2.17E -MBH

-

M pressure

I' ..., 0.000264kt function

~llcA

for

rectangles

.G of

various shapes.

r

.c ~

PRESSURE BUILDUPTESTS

I

roo

d

~

Q.m .a

0..0

4

3

2

-39

:

;

10

4IllcA

: ,

O.OCXJ264 kip cPlJ." ,A

.o'

10

= I,

GO

100

,.

,'

.'

!fit

~

p. -p PDMBIJ = 706 8 /kl =3.454. .q IJ. I Thus, In (C A (I») = 3.454 or C A = 31.6 -essentially

IDA =

determine CA. For example, consider a circular drainage ~rea with a centered well. From Fig. 2.17A, for

Fig. 2.17F-MBH pressure function in a squareand in 2:1 rectangles.

p. -p PD MBH = 70.6 q81J./kh =In(CAIDA)' This equation implies a linear relationship between

f.

~

.c I

01

0OOO264kl .,.cA

P D MBH and IDA after pseudosteady state now has been achieved. Indeed, inspection of the curves in Figs. 2.17A through 2.17G shows that, for sufficiently large IDA' a linear relation does develop. Further, any point on this straight line can be used to

,

;1'; ICI.~

-I

-3

001

0 0 po.

\D

.1 ;; Q.

:,; ; I

1Oi

"

i:;;!

.j... !;.,

III ;' t .'!!o.-""

!' :\~; 0

.0'

-':

tu',r; .-2

"

"):1' r

~ ~'

;~'

Fig, 2.17G-MBH pressurefunction on a 2:1rectangleand equilateraltriangle.

,.;:;~:;:,~:;,.

'

,.

0

: '

I

iI !',

I [

..

ii.

"C;L.L

TABLE

2.5 -APPLICA

30

4.393

40 50

4.398 4.402

72

4,407

TION OF MODIFIED

--15

(ho.;}~rS)~;i) 60

MUSKA T METHOD

4.405

~

19

29

-;,

10 6

14 10

24 20

~

17

I

1

5

15

'e::.

-i;4.408 o;;~~1f;~~=4.422 3

7

thc val'Ic

given

rclatioll...hip

in Table between

11>..t ~O.I;

ill Table

1.2, this

p~cudosteady-state the column

J .2. Note P/)~tmf

now

"Exact

al~o and

that

for '0/1

linear

begills

is the value

equation

the

IDA

cylindrical

(in

> ").

At

position now

to

of

boundaries) are

felt

at

simulate

(depth in

proximated

Fig.

which is a solution to the now well producing from a closed,

re~ervoir

constant

rate.

a buildup

and

noting buildup,

Using

rollowing

investigation

the

has

that,

once

the

boundary can

establish

effects

data

be

provide

ap-

-0.00388

k4//IJ.c,~). ~ (2.23)

F ..drainage .or a~lalysls of buildup

I

tests,

we

usually

express

t~is

as

)

QBIJ.

118.6kh

0.OO168k111 -p.cr2. , ~

ill tllc

used ..II'lt-ill

2~f) ""(',r;

tllat

this

equation

log
-PI\:f A

cq'Iatiol1 value straight p (thc

p

line

form

tested

be

that i~ too

ha~ a noticeable

in the time

a~sumed

long,

We

illustrate

now

2.lft.)

in (and

the

only

important requires

has

to

portion

of

found

to

been for

with

hydraulically

layers

only

of

different

at the wellbore.

fails. The too: (I)

reasonably

Muskat it fails

centered

drainage area in derivation

required

shut-in

times

are

frequently

this

in

method

in its

need of

Jl.C,r;)/k particularly

no

it is used

of

not this (250 im-

low-permeability

with

an example.

2.

assume

a

Solution.

a

method

value

of

well) in-

outside

of

we do

value

have

vs.

for data

not

To

well's

Ca/cli/a/e

in Drainage

Consider 2.2 and

the 2.3.

A rea

buildup

test

Estimate

drainage

area

described

the average by

using

in

presStire

the

modified

method. The

data

are those

the

of

time

not have

limit

that

the

can

in the range

IJ.C ,r;)/k.

estimates

slightly

Modified

Pressure

111 = (750

tested

7-

In this k and

range these

r~, of

complicales

(250

by

to

fortunate

to

Often,

-a

trial-and-error

data

of course,

situation

method,

this

IJ.c,r;)/k

so we can eliminate

interest. of the

the

examined

case we are

estimates

applicability

be 6./=

that

does

but one nature

lhat

of the

calculations.-Here. 750 4tIJ.C,r~ k'

~11/~

value

the

method

curvature

(750

Me/hod

Muskat

111 until

-PII'f)

(2) the

practically reservoirs.

the

assumed

well

to

of the

of log(jj

downward

of

p that

noticeable upward curvature l11e correct assumed value litlC

An

a plot

of

range

2S0 4>#(clr~ --~

An

the correct

with

sel1sitive.

producc!i

We vs.

pressure

Experience

it is quite low

applied,

it does,

drainage-area

p that

is

10g(p-PM's)

when

found.

dicatc!i

I

it

plot

form

wells

(although the as implied

and

in This

it

communicate

IJ.c,r;)/k

Problem.

constants.

how

and results;

!itatic been

6.( that

are

suggests for

has

8

for

it

correct

jJ estimates

and

area cylindrical,

Mliska/

) = /1 + 8111,

and

the

is not

the

Examples whcrc

choose (2)

two

when

when

A verage

2.24 has the

(I)

p. method disadvantages,

Exalt'p/£,

~ 111~ -k-' Eq.

has

method.

are

ral1gc

7~f) 4>/(.,r;

kNote

in developing

timc

method

and

that

Muskat

method:

In these cases, the method has serious

(2.24) Approximations

to

analysis); wells

lor modilied

properties

satisfactory

-method);

(

og(j>-PII:f)=log

valid

for

p.

reservoir

permeability

= 118.6~exp( kh

the

jJ (except

fractured B

equation

of

graph

Muskat

over

estimates

reservoir

equation

modified

advantages

stabilized

reached

2.18 -Schematic

The

super-

as

P-PII'.f

LON

the

exact

.

Eq. 1.6, for a

P

CORRECT 15

TOO

Modiried Muskat Method k h d b d I .., The d ' fi d M mo I Ie us at met 0 IS ase on a Imltlng form of equations

ASSUMED

-ASSUMED

at

at which

becomes

IINI,j

ASSUMED TOO HIGH p

~ ~

g

II;;;)

in) the

250 4tIJ.C,r~ k

is too

high

in this same of p produces

proper

time

range.

produces time range. a straight

a

---

=

(250)(1,320)2 (1.442x 107)

=30.2

hours,

and 750 C ~ IJ. , ~ = 90.6

(See Fig.

hours.

k Thu~,

we can

~

examine

~-

data

from

4/=30

hours

B

until

BUILDUP

TESTS

41

I

! TABLE

2.6 -BUILDUP

~(

DATA FOR WELL NEAR

Pws

(hours) 0

~(

Pws

BOUNDARY

~(

tOO

Pws

(psia) 3,103

(hours) 8

(psia) 4,085

1 2

3 , 488 3,673

10 12

4, 1 7 2 4,240

36 42

4 , 700 4,770

3

3,780

14

4,298

48

4,827

4

3,861

16

4,353

54

4,882

5 6

3,936 3,996

20 24

4,435 4,520

60 66

4,931 4,975

we stop We

recording make

From clearly

pressures

Table

plots the

2.5

at ~

for

of (p-Pws)' best choice;

(hours) 30

(psi a) 4,614

This

example

using

the m~thod.

5 psi

of

in such

a case,

this

method

The

trial

are

of

value

significant

the

values

of

MBH

only

from

this

section,

we

the

Muskat

Fig.

2.19 -Modified

of

built

up to withil1 of

the

to

art.

The

problems

Both

by Earlougher.

than

is

well

to a single

In Chap.

I,

superposition pressure in

l

boundary

1.50.

'

when

(such

are

P;-Pw/'=

state

based

a single

boundary

near

a well

application that from

faull)

of

is given

by

can

develop

such

a well.

an equation Note

cf>p.CI(2L)_ kip

[ -948

describing

]

k(1

--PRESSURE

can be written

Two

test

tuall~

for

k~

for

25) the

the

Ei

becomes 1 +.-11 .p .--)

I +.-11 .p '--.)]

+ In(

~

.-11

( I'P' +.-11 -.. )

qBp.

sl j

buildup slope

Eq.

of

than

the

test.

For a

for

with

For

large

approximation time

Eq.

2.1)

.this

reason,

buildup

test

time

to

ordinarily

near

be valid

and

(2)

can

be or

of

L

or

required can

available

for

~waiting

a doubl~ng

IS

necessanly

not

a

will ~q. even.2.26

values

shut-in

(2.26)

a well

curve fault,

slope to double .JI.C,L 2 /kA/
permeability,

logarithmic

on

(I)

2.26

for the 3,792

tjlp.c,L2/k.

values the

.,

slope such of as a abulldu,P s~ling

(~ompare

.-1/>1.9xIO'

longer

) ~kh

accurate

can be made:

boundary, that the double

small

cf>p.CE ) -2

that

as

observations

single shows

+~)

[ In ( ~~

large

qBp. Pws=Pi-325.2-log(lp+.-1/)/.-1/]. kh

p

q

is

qBp. ws = 70.6 -=-Iln(

the time r~qulred long -specifically,

l

sufficiently

kh.-11

that

-706~rln .kh

-70.6 (-

the equation

Eq.

,

ti.J

time

= 141.2-ln

[ ~~88cf>Jl.CA ] -2s- J P, .-= Pws

tjlp.C L 2 I --).

the

]

a buildup

792 k~

approximation

Pi -P

kip

-70.6~Ei

for

shut-in

functions,

2

We

a

This

kh

1

(2 For

the flowing a no-flow

"

...': -,"

-3 ~-

kh

to double, and distance from a

qBp. [In ( 1,688cf>p.Clr~) -2s

kh

Bp. -Ei(

( -q)

kh

we showed distance L

tjI~

on

and basedis

it becomes

-70.6-

-70.6

logarithmic

we illustrated

;I

example'

k (Ip + AI)

of

t3

as a sealing

Rearranged,

the

Eil~92

kh

boundary.

principle, a well a

to

a

for

has Much been technology developed

causes the slope of a buildup curve then develop a method for estimating tested

applied

either

techniques

presented

summarized

that

melhod

methods

pressure

with

rather

techniques data test analysis. analysis

demonstrate

hr

'

tesl.

-70.6~

briefly

on buildup-test drawdown

We

Muskal

buildup

t

70

stabilization.

deal

these

~

72 hours;

in applying final

~ ~

estimating reservoir size and distance to boundaries. These comments are introductory only, and deal only with the simplest cases. The intent is to illustrate an approach

p. 4412

I M)

2.10 ReservoirLimits Test In

~:::::~~~~~

K>

for

time

method.

when

distance

had value

p'

;~.:~~--o P' 4422

p.

the mechanics

at a shut-in is little

the

modified

only

pr~ssurc

value there

or

of

to illustrate

its static

~:=:~-Q~~~~ o

we see that p=4,412 psi is we also note the sensitivity of

application

is intended

;

I 10.-

= 72 hours.

three

this method. There is a noticeable curvature p = 4,408 psi and p = 4,422 psi (i'"ig. 2.19).

method

(/) 0.- ~

be a in a

I.

,; ~;..r:1CIO"~'~e~~oc !,.

,

, "", ""', '-

\

.' '.\';

..."'.."p".".,,. ...\"..-'. ..."--"--"-'

:. ,.

Or ~cen!!fy;~~

:1 no-now.. o '" ." :- "-~. ~'l)U!~Car\ "cr-t.'("'O,,"v SO""" "",f:t.' 2 '~-""',:.".""" '."""'-"""'\'~.".". '."-~': :'..

I'j,;"':"'r~:,~;:-".:;.~":'..':"~,.',., "' ,:?, I-\r .:,-;-

-'.;

..;

,

" --,,- .:\\\:.a~\:.es'.~'~~"':.::-:s~a!1\.".,j".. ," , ." ~.

(

",

.'.,,'

.,!.,

:,'

']

..-

.":'j

* ,'l\~

-'

P ws

-O,434£i( -3,792op.c(I_:,' \

kIp

A

~;)

~

,

""~

1""""""'-

/:

( ~~~~~\ktlt

-70 ,6~£' kJ1'

,V-rR

V

LTR.

L-

"'--~

I

,~ ~ J' ...~k,..7)

Reasons for arranging the equation in this form are as follows,

I 09

tp + ~t -~t--

1. The ~erm 162.6 (qB~/kh){log[(tp+~t)/4IJ -0:~34 £, ( -3, 79~ lPp.c,.L2 / ~t p) ] I determines the, position middle-time lane. the the Ei flilidion ofis the a constant; thus, it Note affectsthat only

FIg. 2.20- boundary, Buildup lest graph lor well near reservo'Ir

position of the MTR and has no effect on slope. 2. At earliest shut-in times in a buildup test, Ei (-3,7921Pp.c,L2/k~t) is negligible. Physically, this mcan~ that the radius of investigation has not yet, cllcountered the no-now boundary and, mathematically, that the late-time region in the buildup tc~tha~ not yet begun. Thcse observations suggest a method for analyzing

. d~llled well. ~o confirm this fault and to estimate distance from It, we run a pressure buildup test. Data from the test are given in Table 2.6. Well and reservoir data include the following.

thc buildup test (Fig. 2.20): I. Plotp"~vs.log(tJ!+~t)/~t. 2. Establi~hthemiddle-timeregion. J. Extrapolatc thc MTR into the L TR, ~. Tabulate the differences, Ap~'S' between the bu~ldup c~rve and extrapolated MTR for several p<.)lnt~(At)I':(=P":f-PMT). 5. Estimate L from the relationship implied by Eq. 2,2ft: .11':,:\ ~

70.6 fill'l

-1:.'i( --::_~~792 I/I/lc',1,2)).llic k~t

IP = Jl = = rCO ( = Aw: = Po = q = B: = J1 =

0.15, 0.6cp t7x ft10-6 psi-I ' 05 0:00545 sq ft, 54.8 Ibm/cu ft, 1221 STB/D 1:310 RB/STB, and 8 ft.

, (2.28) ,. i~ Ihc only unknown in this equation, so it can be ~ol\'cd dircctly. Remember, though, that accuracy of Illi~ cqualion requires that ~t.cft .when this condil iOiI is not satisficd, a compuf~r history match u~ing Eq. 2.25 in its complete form is required to detcrminc I~. 111i~ calculation implicd in Eq. 2.28 should be madc for several values of ~t. I f the apparent value of I. tcnds to increa~c or to decrease systematically with timc, thcrc is a strong indication that the model doc!; nol dc~cribe thc reservoir adequately (i .e., the

~'cll prodllccd only oil I1lld di~~olvcd ga~. fkforc shut-In, a total of 14,206 SloB oil had bccn produccd. Analysis of these data show that afternow distorts none of the data recorded at shut-in times of I hour or f!1°re. B?sed ?n the slope .(~50 psi/cycle) ef the earliest straight II~e, p~rm:ablhty, k, a~pe~rs to be 30 md. Depth of Investigation at a shut-In tIme of I h~ur is 1.44ft: lending confidenc~ to ~hechoic~ of the middle-time line. Pseudoproduclng time, tp' IS279.2 hours. From these data, determine whether the buildup testdata.indicate that the w~ll is beh,avingas if it were near a single fault ~nd estImate distance to an apparent fault from buildup data at several times in the LTR.~-1\'c~I..i~

not .~having a~ if it ~ere in a rcscrvoirof unllmm thlckncss and porosity, and much nearer oll,c.houndary t-,lan any others): , IIIC fulluwlng cxamplc Illustratcs this compu tatlona . I tecI1l1lque. .M~

S()luli()n.-Our attack is to plot/?~ VSi(/p+At)/~t; extrapolate the middle-time line on into the L TR; rcad pre~sures, PMT' from this extrapolated line; sublraclthosc Prcssurcs from obscrved values of /J in the LTR (Ap~ =Pws -PMT); estimate values of

kh

1

E.\'Ol11ple 2.8 -Estinloting

Distance

toaNo-Flo},'BoUlldary (Jr()I)frm. Geologists suspect a fault near a newly

L from Eq. 2.28; and assume that, if calculated va~uesof L ~re fairly constant, the well is indeed near a single sealing fault. From Fig. 2.21, we obtain the data in Table 2.7. We now estimate L from Eq. 2,28. Note that

PRESSURE

BUILDUP

TESTS

43

TABLE 3,792

4>,i.tC,

=

(3,792)(0.15)(0.6)(

1.7

k

x

10 -S)

30

~t

=1.934xI0-4.

We

first

that

estimate

the

L

At

=

approximation

10

I

hours,

=1

+

.P thts

which Al

is

assumes

adequate

in

p.

Ap;"s

= 52

=

-70.6~

(

Ei

'

792

kh

ilC L 2

'

)

kAI

-(70.6)(1,221)(1.310)(0.6) = (30)(8) 2

(

-3,792

(P..~ -PMT) = ~p:",

(t"t~tl/~t

(psi)

(psi)

6

475

3.996

3.980

(PSI) 16

8

359

4.085

4.051

34

10

289

4.172

4,120

52

243

4240

4,170

70

14

209

4:298

4,210

88

16

18.5

4.353

4.250

103

20

15.0

4,435

4,300

135

24

126

4.520

4.355

165

30

10.3

204

4,614

4,410

36

8 76

4,700

4,455

245

42 48

7.65 682

4,770 4,827

4,495 4,525

275 302 330

54

6.17

4,882

4,552

60

565

4,931

4.578

353

66

523

4,975

4,600

375

).

q"i.tC(L

'Ei

PMT'

FROM

12

case. B-3

VSIS OF DATA BOUNDARY

P..s

(hours)

at

2.7 -ANAL WELL NEAR

kAt

.

, I

-. (

-1.934x

EI

10-4

) -0.184. -

L2

10

"

Sl~ 2 L

(1.107)(10)

.1300 poi"C~"'\v/,,1

4

=

4 1.934x

.

" "" /

=5.72xIO,

.<1;

10-

C-

or

I.n

~ L=239

ft.

-.11

For larger values of I p = I p + At becomes terms

in

Eq.

satisfies

the

2.25

can

equation

boundary

be

for

for the case In has time to double, to

shut-in time, decreasingly neglected,

all

values

which the estimation

slope

is easier. time, intersect

Al x' (Fig.

distance

L

the

but of

the

no

L =240

shut-in

of the distance

of

From

find the sections

from

the approximation accurate, and

ft

time.

KX)

buildup from

buildup

tests

test well

plot,

to

the

fault

can

the

I tpt

we

at which the two straight~line 2.22). Gray 14 suggests that well

cycle

~

t

-"KtFig.

2.21

-Estimating

distance

to a no-flow

boundary.

be calculated

from

L=J~~~~~~~~!.!.

(2.29)

ct>IJ.C, In

Fig.

2.21,

shows

in

slope

did

double,

and

(/p+Alx)/A/_,,=17,

AI x = 17.45 ft,

the

that hours.

Eq.

reasonable

the

figure

from

2.29

then

shows

agreement

which

that

with

L = 225

our

previous

P ws

calculation. The

results

be used

to

compare after

of

average

of

(barrels),

AN

produced the

p

If is

between

average

then

the

sometimes

The

basic

pressure quantity

VR

Times

reservoir

production,

reservoir

reservoir,

c,.

tests

size.

a known

volumetric

pressibility,

buildup

reservoir static

production

closed,

pressure

estimate

the

constant

pressures

a material

and

PI

before balance

u

t

X

a

com-

and

and on

I

A

volume

barrels 2,

log

and from

reservoir

stock-tank I and

fluid

lp+~lx

is to

before

of

with is

can

idea

the

of

oil

P2

are

after

t log

oil

+

i1t

p i1t

,

reservoir

shows that P2

-p

I

(ANp) (Bo) V Rc «f>

'

Fig.

2.22 -Dislance

to boundary

f,om

slope

doubling.

...

or

pressed in thousands of standard cubic feet per day (~N ) (8 ) J-'R= '~'-J1'-'~o'. (p, ~P2)Ct(/)

,..,..,.,.(2.30)

(Mscf/D), and gas fC?rmation volume factor, Bg, is then expressed in reservoir barrels per thousand standard cubic feet (RD/M~cf), so that the product

Exailiple 2.9- Estimating Reservoir Size I'rcthltm. Two pressure buildup tests are run on the only well in a closed reservoir. The first test indicates . the Second I'nd,' cates an average pressure 0,f 3 000 pSI, 2,1()() psi. The well produced an average or 150 STB/lJ of oil in the year between tests. Average oil furlllationvolumcfaclor,Bo,isl,3RD/STD;total comprcssibility, ct' is 10 x 10 -6 psi -I; porosity, (/), i~ 22%; and avcragc ~and thicknc~s, h, i~ 10 fl.

q~ B~ is in reservoir barrels per day (RB/D) as in the analogous equation for slightly compressible liquids. 2, ,A}I gas prope!ties (Bg, JJ.,and ~g) are evaluated at original re~ervolr pressure, Pi. (More gener.ally, these pro:pertles shoul~ be eval~~t~d .at the uniform pressure E 2 31 In the reservoir before Initiation of flow.) In q,., 8 -178.1 Zj Tpsc(RU/M f) ~jpoT sc, I SC C.,j=C~i'\'R+CII.SII,+cf~C~j'\"~.

Eslilllalcarca,AR,ofthcrcscrvoirinacrcs. Solution. From Eq. 2.30,

3 Th f D . .e actor IS a measure 0 f non- Oarcyor turbulent pressure loss (i.e., a pressure drop in ad.

AN B --.p-~o (p I -P2)C, (/)

dilion to that predicted by Darcy's law). It cannot be calculated separately from the skin factor from a single buildup or drawdown test; thus, the concept of

(q/)B 0 (p I -P2 Xc t (/)

apparent skin factor, S'=s+Dq", is sometimes convenient since it can be determined from a single test. For many cases at pressures below 2,CXX> psi, flow

VR =

=

in an infinite acting reservoir can be modeled by =

(ISOSTB/O)(365daYS)(I.3RB/STB). (3,
2 = .2+ 1,637q"Jl.iZ;! Pwf P, kh

Thus, J'H = 3S.9x 106 bbl

( 1,688 (/)JlCti) -<::!+Dqg

./

llog

kIp

1.151

)

].

= 43,560ARh ,

(2.32)

5.615 anI.!

6

superposition Using these basic to develop drawdown equations equations, describing we can use a

,.1 = (35.9 x 10 bbl)(5.615 cu ft/bbl) H (10 ft)(43.56 x 10.1sq ft/acre)

buildup test for gas wells. Forp> 3,000 psi, q 8

=463acrcs. 2.11 Modificariol1s

P,'s=p;-162.6'fg-glr-1 I kh anI.!

for Gases

l11is scclioll prcscllts modifications of the basic ura\\uo\vn anI.! buildup equations so tha~ they can be applicl.! to analysis of gas reservoirs. These Illuuificalions are based on results obtained wilh the ga~ pscudopressure.IS although a more complete uiscll~sionoflllatsubjcctislerltoChap.S. Wattcnbarger and Ramey 16 have sho\vn that for some gascs at pressures above 3,000 psi, flow in an in finitc-act illg reservoir can be modeled accurately by tile cqualion

[ 1.og

( 1,~88q,Jl.;~!;) kIp

.k

.

~)+3.231. (/)Jl.;Ct;II' For P < 2,000 psi, q Jl.-z. T -1,637

'fgr"'1

'log

(2.34)

( 'P I +, -., ~/ )

...(2.35)

~I

and

---

2 2 ) = 1.151 (PI hr -Pwf) g

1.151

..(2.33)

k

s' =s+D(q __(S+DQg~',

~I

-Iog(

=pf

.kh

,log( I'/1'+ ~I -., )1

(p -P ) = 1,ISll \PI hr -Pwfl 111

s' =s+D(qg)

-P~I'S

-162.6QgBg;Jl.; Pll'r = P,+ .kh

-JJ..

l

mN

(2.31)

This equation has the same form as the equation for a slightly compressible liquid, but there are some important diffcrcnccs: I. Ga~ production rate, q~, is conveniently ex-

-log(

.£... -_2) + 3.23], (2.36) q,p.iCt;' w where /11N is the slope of the plot p~ Ys. log(tp+~t)/~/J,whichisl,637qgJl.;z;T/kh.

: ~

An obvious question is, what technique shollld be used to analyze gas reservoirs with pressures in the range 2,(XX)
c=(' II

S =(3.44x g/ g

10-4)(0.7)

=2.41 x 10-4 psi-I, alld

at least somewhat inconvenient, so an alternalive, approach is to use eqllalions wrillen in terms or

I ~PI hrIII-PHi)

s =s+D(qg)=1.151

either Pws or P~s and accepl the resultant inaccuracies, which, in real, heterogeneous reservoirs, may be far from the most significant oversimplification on which the te~t analy~is procedure is based. The smaller the pressure drawdown during Ihe

( ~;7?k

-log

) + 3.23 I

/ II H'

le~l, Ihe less the inaccuracy in this approach. ~(2,525 -1,801)

Example 2. 10Test Analysis

= 1.151t

Gas Well Buildup

l

?96.x 10 -4)(0.3)2 -lo' g (0. .18)(0.028)(2.41

lest. Test data include the following. l)rctltlem.Agaswclliss!luliI1roraprcssurchuildup qg Mscf/D, T = = 5,256 181°F=64I°R, h Ili Sw ~

= = = =

81

~

I

+ 3...23J = 4 84

28 ft, 0.028 cp, 0.3, 0.18,

From results of the P~vsplot, T k= I 637 q_~Jl.iZj. , mwh

Zj = 0.85, r w = 0.3 ft, Cgj = 0.344 x ~O-J psi-I,

(1,637)(5,256)(0.028)(0.85)(641) (4.8x 105)(28)

=

I

Pi = 2,906 psla, and Pwf = 1,801 psia: -=9.77md, Most of the test data fall in the intermediate pressure range, 2,(XX) psia. On a plot of Pws vs.log (tp+At)/~t, t~e MTR had a slope, m, of 81 psI/cycle; on this plot, PI hr was found to be 2,525 2 I .. Alternatively, on a plot 0f Pws psla. vs. °g (ti +~/)/~/, the MTR had a sloRe of 0.48x 10 psl1/cycleandprhr of.7.29 x 106 psi2. From these data, estimate apparent values of k and s' (I) based on characteristics of the P ws plot and (2) based on characteristics of the P~s plot.

d an

2

'=1151 s.

2

[ (Plhr-Pwf)

-IOg-~+3.23 ~JI.-c ,I.r2.H /

mw

1

6" = I 151f E.29x 10 -(1_,801)-] .t 4.8 x 105 -10

.g Solullon. From results of the Pws plot, for standard conditionsofI4.7psiaand60°F,

I

I 3..23!

9.77 --+ (0.18)(0.028)(2.41 x 10-4)(0.3)2

..

=4.27.

8 .=178.1~~ gl Pi Tsc

Neilll\.'r ~ct of r\.'slllts (k and ~") is ncc\.'~~arilymm\.' accurate than the other in the general case; as in t!lis particular case: use of an analysis procedure based on (178.1)(0.85)(64I)(J4.7) .~as.pseudopress~lre can be used 10 improv~ accura~y = If dl~agreement In results from P..'s and PHosplots IS (2,906)(520) =

0.944

unacceptably

large.

RB/Mscf,

Iii

...' ~.~~-~-~

B ..Uasi..: k= 162.6 qg-gIJl./ mh

2.12 i

Modifications

for

Multlphase

! -1

Flow

buildup and dra,,:down cqualions can be modified to model multlphase flow.17.18 For an infinite-acting reservoir, the drawdown equationI becomes

= ~162.6)(5,256)(0.944)(0.028) (8 J)(28) ,.

=9.96 md,

p

( ! ,68~q,c,r~I:

=p. + 162.6 ~ [IOg wf, )., h --,

s

.,

1

1.151

I

}., I

)

.

(2.37)

1

!

alld I Ilc buildup cljual iOll bccomc~ q Rt I +, -. AI p",~ =Pi -;- 162.6 -10g(.p Ath

AI

) .,.',.,

(2,38)

In the~e equations, tlte total nO\\' rale qRt is in rc~cr\'oir barrels per day (neglecting solulion gas liberated from produced water) R ' l/Rt =l/II/l'1 +(l/.1' -.~:~-

)8.1' +(111,1111"

E.\"a/l1p/e 2. II -;\-Ili/tip/lase Test Ana/ y sls

Blii/dlfp

P,roblem. A buildup t,estis run in a well that prod\ll:t'\ Oll~water, an~ gas simultaneously, Well, rock, an,d nuld propertIes evaluated at average reservoIr pressure during the test include the following. SO = 0.58, 5,f = 0,08,

,..,." I

.1

allutola

,. mo

-A:!I

b . I '~. I Ily,

(2.39)

"t,IS

~-".

A:8

A, -+

+

,.

., ...,

..,

(2.40)

IL" ILII' It.1' .T(ltalcomprcssibility,ct,wasdcfincdinEq.I.4. "llc~c C4lmtion~ imply that it i~ po~~iolc to uctcrmillc A, from the ~Iopc III of a buildup tc~t run 011 a \\'cll that prodllcc~ two or tllrcc pha~c~ ~imultallcoll~ly: q H, A,=162.6-. Illir ,...,...(2.41) Perrlne .17 1las s110wn t I lat It IS aIso pOSSI .bl e to c~(imate thc permeability to eoch pl,ose nowing from

..

the samc slope, ",: q B II. kn=162.6 (} () (I, ",1,

(q~-

q(}

-~

k.e = 162.6

(2.42)

R

s

) B~II.~

,f

It,} = 11."' = It" = B() = 1111'= B.1' = R~ = ~'#' -., '" ' = h =

(2.43)

1,5cp, 0.7cp, 0.03 cp, 1.3 R8/STB, 1.02 RB/STB, 1.4ROR8/M~cf, 685 scf/STB, 017

0.3ft,and 38 ft.

From plots of Bo vs. p and Rs vs. p at avcra~t' pressure in the buildup test, dR -:! =0.0776 scf/STB/psi, d an

,

",1,

5",=0.34, 36 10 -6 '-I c",=.x pSI, 35 10 -6 '-I cf = ,x pSI, C = 0.39 x 10 -3 psi -1 ,

lip

dBo -=

I 6 R 2.48 x 0 -B/STO/psl.

.

dp alld B "q", A:1I,-162.6.

1I,/tll' ...,.., (2,44) ",1, .nlc term «(I -(loR~/ I,OOO)B~, wltich appcar~ in 1:4~, 2..19 anJ 2.4.1, i~ (lte jrc'£' ga~ now ratc in tlte rt"'cr\'oir. It i~ f()llnd oy suotract ill!! thc di."s()I\.cd!!a~ ratc «(I"/?\/I,()()(») fronl tltc total ~urfacc ga~ ratc «(IE,)and con\'crting t? a reservoir-condition ba~i~. Slmullancou~ ~olutlon of Eq~, 2.37 and 2.38 re"..Its in the follo\\lng expre~sioll for tlte skin factor s. ...=1.151 /" Ihr

- ,~-log(-~)+3.23, , A

III

I

The production rates prior to the buildup tc~t \Vcrc qo = 245 STB/D, q", = 38 STO/D, and q" = 4R9 Mscf/D. A plot of P"'S vs. log (I l' + At) / ~t shows tltat tht' slope of tlte ~TR, "'~ IS 78 psI/cycle and tl~:lt 1'1 hr = 2,466,psla. FlowIng .pressure,P'vf' at thc In. sta~lt of shut-In wa~2,~28 psla. l'romtltc~cdata,cstlmatcA"ko.k""k-i,and\", Sfllution. Permeabilities to each phase can hl' determined from tlte slope ", of the MTR:

k(} =162.6

q0 8 011.0 ",II

1/>£." II'

(2.45)

(162.6)(245)(1.5)( 1.3) ==26.2md, (78)(38)

Stat ic drainage-area pressure, p, is calculated just a~ for a single-pl\a~e reservoir. In use of the MBH charts to determine p (and in the Horner plot itsel0,

kw=162.6

thc effective production time tp is best e~timated by dividing cumulative oil prOduction by the oil prodllction rate just before shut-in. An important. assumption required for accurate

=

use of these equations for multiphase now analysis is that saturations of each phase remain essentially uniform Ihroughout the drainage area of the tested \vcll.

q",8".11.'" h ",

(162.6)(38)(0.7)(1.02) 78 38 = J.49 md, ( )( )

q R (q" --fiiii k = 162.6 ' " ",h

)BgII.g

PRESSURE

BUILDUP

TESTS

I489

-47

I

(245)(6X5) (1.480)(0.03)

= 162.6

(1,000) (78)(38)

= 0 782 md ..made To calculate total mobilit A we first need lotal now rale. y, I' , q RI' = B + -(} q RS )B + qRr qo 0 qg I (XX) g q..,BI" ,

(

= (245)( 1.3) + [489 -(245)(685) 1,

2. I. In Examplc 2. I, ~'hat error arises h\.'I:"II~C:\\'\.' II~cd Eq, 2.4 to call:ulate skin'fador il1stc"d or thc more e,\"ct Eq. 2.3? What dirfercnce would it Ilavc: in the value of s had we used a shut-in timc: or 10 hours in Eq. 2.3. and lIte corre~ponJing v~lue or P\l'~? What assllmptlon have we made about dlslanl:c rrom lested well 10 reservoir boundaric:s in Exampll.' _.. ') I ? 2.2. I)rove thallhe slope of a plol of shut-in HI IP vs. log (/,,+.1/)/011 is, as asserted inlhe lext, Illc difference in pressure al two points one cycle apart, Also prove that, for .11 cC1P' w~ obtain Ihe same slope on a plot ofPM'S vs. log dr. Finally, prove that on a plot of Ph'S vs. log ~/, we obtain the same slope regardless. of Ihe units used for shut-in t!me, .~, on the plot (I.e., that ~I can be expresscd In mInutes, hours, or days \vithout affecting the slope of Ihl.'

]

.(1.480)+(38)(1.02) = 833 RB/D.

plOI), 2.3. A well produl:ing only oil and dissolved gas has produced 12,173 STD. The well has not been

Then, A = 162 6 ~ I 'mh

= ~~~~ (78)(38)

stimul"tcd, nor is there any reason to thl'rc: is a signifil:ant amount of formation

=45.7 md/cp. To calculate skin factor, S, we first need Co and C,: -~

c-0

1;~" .,'~ r " .I""": , ,

dRs dp

Bo

.I

dBo dp

B 0

-CI

= (1.480 RB/Mscf)(0.0776 scf) (1.3 RB/STB)(STB-psi)

.1

Mscf

-(2.48

1,scf

x 10-6) 1.3 (psi)

believe that damagc. A

pressure buildup test is run with lIte primary objeclive of estimating static drainage-area pressure. During buildup, there is a rising liquid level in the wellbore. Well and reservoir data are:

I

cf>= 0.14, Jl. = 0.55 CPt '-1 = 16 X 10 -6 pSI, r w = 0.5 ft, A wb = 0.0218 sq ft, r e = 1.320 ft (well centered in cylindrical drainage area), p = 54.8 Ibm/cu ft, q = 988 STB/D.

i I !

' 1

B=I.126RB/STB,and h = 7 ft.

I '

= 86.4 x 10 -6 psi -I. Data recorded during the buildup test are given in Table 2.8. Plot PI"S vs. (lp+.1/)/~1 on semilog

Then. c =5 +5 +5 + r oCo gCg I"CW Cf

10-6)+3.5x

x 10 -3)

10-6

the buildup t~st data.. 2.4. Consider the Pro?lem mation

= 860 x 10 -6 .pSI,

.-t

(

og ~.

)

AI 323 1 + .single I III

-log

78

[ (0.17)(86.0

~~I:ate

buildup the

test and

estimate

MTR

In for-

permeability.

-

original reservoir pressureestimatesIn thesecases: 45.7 x 10 -6)(0.3)2

] + 3.23 J

(I) some LTR data were obtained, but final straight line was not established; and (2) no L IR data were 0 b2.7. talne. . dConsider the buildup lest described In Problems 2.3 and 2.4. Estimate static drainage-area

= 1.50.

I

.2.6. Provethatinab.uilduptestfora.wellneara fault, the technique suggested In the text (extrapolating (he rate-time line to infinite shut-in time) is the proper method for estimating original re~e~voir pressu~e. Comment o~ the p~ssible errors in

= 1.151\ 2,466 -2,028

l

2.3.

.. described

2.5. Consider the buildup test described in Problems 2.3 and 2.4. Calculate skin factor, s; pressure drop across the altered zone. (i1p)s; now efficiency. E; and effective wellbore radius. r wu.

and m S=I.151 [ Pthr-Pwf_1

I

paper and (PI"S -PI'f) vs. .1/t' on log-log paper-yand estima!e the time at \vhich aflerno\v ceased distorting

= (0.58)(86.4 x 10 -6) + (0.08)(0.39 +(0.34)(3.6x

!

pressure for this well (I) Iising thep.

.;

method. and (2)

..

.TABLE 2.8 -PRESSURE BUILDUP TEST DATA

~t

P.,

~t

Pw.

(hours)

(pSla)

(hours)

(pSia)

709

19.7

4,198

0

197 2.95 394 492 591 788 9.86 14.8

3,169 3,508 3,672 3,772 3.813 3,963 4,026 4,133

246 296 345 39.4 -144 49.3 59,1

TABLE2.9-BUILDUPTESTDATA FORWELL NEAR FAULT

4.245 4,279 4,306 4.327 -1,343 4,356 4,375

~I (hours) 20 30 -10 50 100 200

USilig tIle mouificu Muskal melhod, 2.8, In Exalilple 2,7, explain how we could ha\'e applied the modified ~1uskat method to estimate SIalic drainage-area pressure if we had not had eslimalesofk/lt>Jl.c, orrt"

I

2 9 E . "stlmate r t f ac or rom

i

,I '

II we

f t

h

e

b pressure

.

ormation f II ' 0 owIng ' Id

UI

d

b 'l ' d k .2,11, I Ity an s In. ' I bl f aval a e rom a gas,

permea ala

I up

les

R~ = It> = r M' = II =

~t (hours) 500 ROO 1,000 1,500 2,000

" well

pressure

believed "

to

.

I

f

h

buildup

be

near

"

reservoIr,

test was ", a sealing fault EstImate

k

II

all I, given t e we ,roc, an anc.JthebuildupdatainTable2,9.

T = 199°F=659°R,

Pw~ (pSI) 2,225 2,360 2,434 2,545 2,616

748 scf/STB, 0,18, 0,3 ft, and 33 fl, A

Infinite-acting ,

Pw~ (pSI) 1.373 1,467 1,533 1,585 1,752 1,940

d

n UI'd

the

run In

on

an

,

an

011

otherwIse

dIstance

to

' propertIes

the

below

II = 34 ft, Jl.i = Sit' = ('!!; = It> = z; = rlt, = i

0.023 cp, 0.33 (water is immobile), 0.000315 psi -t , 0.22, 0.87, and 0.3 ft.

q = Jl. = It> = CI = h = P; =

Np = 84,500 51'0,

A

..

1hc wcll produccu 6,068 Mcl/O bcfore the te~l. plot ()f IIIIIJ, PIt~' v~, log (I'I+~/)/~I gavc a miuulc-time line with a slope of 66 psi/cycle. An~lysis of the buildup. curve showed. that static uralnage-~rea p~essur,e, p, was 3,171 psla. Pressure ou thc mluulc-tlme line at ~I = I hour, PI hr' wa~ 2,745 psia; nowing pressure al shut-in, Plti' wa~

940 STB/D, 50 cp, 0,2, 78xl0-6 psi-I, 195 ft, 2,945 p~i,

I

/1

= I,ll RB/STII, (I < 20 hours ,,'bs .

Bo = P; = Jl. = k = S = II = c, = It> =

~9 psi/cycle,

r It' = 0.333 ft.

Flowing

prcssllre at shill-in,

Plti'

wa~

slrniglll

lillcal~/=lhour,Ptllr.'\a~I,744psia,PlotsofB(}

anu

2.12. ' A well nowed for 10 days at 350 STB/D; it was then shut in for a pressure buildup test. Rock, " nuid and well properties include the following ' ,

2,4R6p~ia. 2,10, Estimate total mobility, X" oil, water anu ga~ pcrmeabilities, alld skin factor for a well thaI proullceu oil, waler, alld ga.~simultaneously before a press\lrc buildup lest, Production rates before the test wcre qo=276 STB/D, qlt,=68 STB/O, and ({,f =689 Mcf/O. A plot ofplt'.~ VS, log (I" +~/)/~I sho,,'cu tllal the slope m of tile middle-time line was 1,5RI p~ia; Illc prC~Sllrc 011 thc middlc-liltlc

.

) (a

1.13 RB/STB, 3,000 psi, 0,5 cp, 25 md, 0, 50ft, -6 '-I 20x 10 pSI, 0,16, and

De

' te~mlnean

d

I P,otl,

..

h

d epressure

' ISri

t

' b

'

,

utlonln

V~. P and R~ VS, P ~ho"cu thaI dR~/dp=0.263 ~cf/S1'B/psi ' and that dB(}/dp=O,248 x 10-"

Ihe reservOIr for ~h~I-!n tln~es of 0,0:1, uaY~,(~~Sllmeanlllflnll.eactlll,grese~VOI!,)

RO/51'0/psi, Rock, cludc the following,

(b) Calculate the radius ?f Investigation al ~,I, I, and 1.0days, Compare ri with the depth to which the tranSient appears to have moved on the plots prepared in Part a, 2,13, In Example 2,6, jJ was determined to be 4,411 psi, Bolh the Horner plot and the abscissa of tlJe MOH chart used tp = 13,630 hours. It can be shown that for a well centered in a square drainage

'~f) '~.l' ")'It' ('It. ,. c{

r

= = = = =

nuid, anu well properties

0.56, 0.09, 0,35, 3.5 x 10 -6 p~i -I, 3.5 x 10 -6 psi-I,

c'!! = 0.48 x 10 -3 psi -I , 1«(1 = It", = It.~ = Bf) = Bit, = IJ" =

1.lcp, 0.6 cp, 0.026 cp, 1.28 R8/STB, 1.022 RB/STB, 1,122 R B/M~cf,

-~

in-

I, and 10

area, the time required to reach semisleady state is tp.t~=(It>p.cIA/O.000264 k)(IDA)p.u and that (' /)/1) s = 0.1. Show that if tp,t\' is used instead of t p in boththe Horner plot and in the abscissa of tne MBH chart, the resulting estimate of fi is essentially unchanged, Buildup data (from the MTR only) are given in Table 2.10. Other data include:

.-

TABLE 210 -MTR DATA FROM BUILDUPTEST ~, (hours) 8 12 16 20 24

TABLE 2.11 -PRESSURE BUILDUP TEST DATA

~,

P..s (psi) 4.354 4,366 4,376 4,382 4,388

(hours) 0 0.3 0.5 1 2 3 4 5 6 8

q = 250 STB/D 8 = 1.136 RB/STO, -08 II. -.cp, h = 69 ft, ct>= 0.039, CI r

=

k

17xI0-6psi-I, I 320 fl

-7'65

d'

d

4. ~~ME(195~)I~~,91.-.I~ SIIJl.'r, 11.( ,: A SlIllplllll.'ll

an

Anilly)i~

A well producing

only oil and dissolVl.'d gas

damage believed present, the well IS shut In for a buildup test. Well and reservoir data are given below.

rw A b ;0 q B h

= = = = = =

0.17, 0.6 Cp, 18 X 10 -6 psi -I 1 "320 ft well cenle red in Squa r e draina ge area (160 acres), 0,5 ft, 0.036 sq ft, 54.8 Ibm/cu ft, I , 135 STB/D ( sta b.Ilze l. d for severa I d ays,) 1.214 bbl/STB, and 28 ft.

When the well was shut in for the buildup test, the liquid level rose in the wellbore as pressure increased. Data recorded during the buildup test are given in Table 2 II ..' DetermIne

..'. time

(a)

at

which

aftertlow

..

fl 0\\ . e ffi Iclency,

f) (

-. p using

h t

MBH e

. p

h met

0

d ' ,an

fur iI Slubilii.I.'J

~11..lllllll lIt

WI.'II,"

d

Muskat method.

References ,. Horner, D.R.: "Pressure Buildup in Wl.'lIs," PrO(.., ThirJ World Pet. Cong., The Hague(195I) Sec. II, 503-523;also Pressure Analysis Methods, Reprint Series, SPE, Dallas (1967)9, 25-43. .2. Cobb, W.M. and Smith, J.T.: "An Invesligationof PressureBuildup Tests in Bounded Reservoirs," paper SPE 5133

I'rl.'!o~url.' 1111/111111

J. P...I. Tet.h. (Sc:pt.

1971)

Conferenceand Exhibition,Las Vegas,Sepc.23-26,1979. 7. .Russell., D.G. and Truitl, N.E.:."~~ansicntPressureBc:hilvior In Vertically FracluredRe~rvolrs, J. Pet. Tn.h. (Dl:t. 1%4) -1159-1170; Trans.,AIME,23I. 8. Malthews, C.S., Bron), F., and Hazebroek,P.: "A MelhoJ for Determination of Average Pressure in a BounJ(.oJ Reservoir," Trans.,AIME (1954)201, 182-191. 9. Larson~ V.C.: "Un~erslanding t~e ~Iuskal Method .of Analyzmg PressureBullJup Curves, J. Cdn. Pet. Tech.(I'all 1963)2,136-141. 10. Matlhews, C.S. and Russell,D.G.: Pressur~ Buildllp Ulld Flclw Testsill "Ielb, MollogrilphSeries,SPE. Dalla) (1967)I. II. Pinson,A.E. Jr.: "Concerningth.: Value of Producing Tim.: Used in Average PressureDelerminations From Pressure BuildupAnalysis,"J. Pel. Tech.(Nov. 1972)1369-1370. 12. Brons, F. and Miller, W.C.: "A Simple Melhod for Correcting SPOI PressureReadings," J. Pet. Te(.h. (Allg. 1961)803-805; TruIIs.,AIME,222. 13. Earlougher, R.C. Jr.: Adl'ullce.l'

distortIon

ceased; (b) time at which boundary effects begin; (c) formation permeability; (d) radius of invesligalion al bl.'ginning and end of MTR; (I.') ~kin factor, ~P.l" and (g) p using the modified

P..

(psia) 4.272 4.280 4.287 4,297 4,303 4,308 4.313 4,317 4.320 4,322

1155-11fX>; Trulls.,AI~1E,271. 5. Agar,,'al, R.G.: "A N.:w ~1I.'thL~ To Al.'l.'ountfor PrtXlul.'ingTi/llC l:ffcl.'l) Whl.'n l)rilw"Jo\\n IYJ'l(.'(,Ilrv(.~ Arc lI!o(.."J1(1 Prc))ure Buildup anJ Olh.:r Tc)1 Data," Conf.:rcnl.'c paJ'l(.'r SPE 9289presenled atlhe SPE551h Annual Te\:hnical and Exhibition, Dallas,Sept.21-24,1980. 6. Saidikowski,R.M.: "Numerical Simulationsof the Combined Effects of presenled Wellbore Damage and Partial Penetration," pa()t:r SPE 8204 atlhe SPE-AIME 54th Annual Te\:hnicul

has produce.d 13,220 STH. To chara~lenze .Analyze t~e severe

ct>= II. = CI = r~ =

~,

(hours) 10 12 14 16 20 24 30 36 42 50

presented al the SPE-AIME 491h Annual Fall ~1"'l.'ling, I !~U!oI.~I.I, OCI.,6-9, 1974.An ~~~idgl.'dvcr!oi(~n appt."ilr!oin J. '(/. ILL". (Aug. 1975)991-()(XI,Irultl., AI~1I::,25'. 3. ~1illl.'r, C.C., Dyc:s,A.H., anJ Hutchinson, C.A. Jr.: "Eslimation of PermeabilityanJ Reservoir Pressurc From Boltom-flole Prc:ssure8uilJ-Up Chara(.1eristics," TrulIs.,

-.m, 2.14.

P..

(psla) 2.752 3.464 3,640 3,852 4,055 4.153 4,207 4.244 4.251 4.263

ill

Il'ell

Test

.. AII(/ly~is.,

Monograph St:rie),SPE,Dilllas(1977)s. 14. (ir:lY, K.E.: "Appro\i/llilling WclI-to-I';11I11 I)isl:lnl.'c I'rll/ll Prl.')sll~':1.~lIillllipI~~I),','J. Prl. I.'ch. (J!lly ..%5) 761-7.6.~.. 15.

AI-Hussamy, Flo\\. of Real

R., Raml.'), H.J. Gases Through

Jr., and Porolls

Cra\\forJ, Media "

(~1ay1966)624-636;Trulls.,AI ME, 237.

J.

P.U.. P...I.

Tilt: Tech.

'

16. Wattenbarger, R.A., Ramey,H.J. Jr.: "Gas Well Testing With Turbulence, Damage,and Wellbore Storage," J. Pel. Tech.(Aug. 1968)877-887;Trulls.,AI~1E,243. 17. Perrine, R.L.: ,. Analysisof PressureHuilJup Curves," Drill. ulld Prod. Pruc., API, Dallas(1956)482-509. 18. Murtin, J.C.: "Simplilil.'
,

'\

, \

Chapter 3 FlowTests

~ "\"\: ~~ ",

~

o

'J

-~ ,

~.\

-'

~.

\.-

i

'.' '

\

-q 3.1 Introduction This chapter discusses now tests in wells, including

estimated by qualitative comparison of a log-log plot -

constanl-rale drawdown lesls, continuously dcclining-rale drawdown Icsls, and mulliratc tcsls in infinite-acting reservoirs. T~e more g~neral (and more complex) case of multlrate tests In bounded

of (P;-Pwf) vs. t wilh lhe solulion of Fig. 1.6 or with the empirical equation based on that figure, t 2::(60+3.5s) C (1.43) D sD'

rescrvoirs is discussed in Appendix E.

or the equivalent form,

3.2 Pressure Drawdown Tests

-(200, + 12.s)Cs

I ,

I

t"'bs-

(3.2)

A pressure drawdown test is conducted by producing a well, starting ideally with uniform pressure in the reservoir. Rate and pressure are recorded as functions of time. The objeclives of a drawdown test usually include estimates of permeability, skin factor, and, on occasion, reservoir volume, These tests are particularly

kh/p. If the effective radius of the zone of altered permeability is unusually large (e.g., in a hydraulically fractured well), the duration of the ETR may depend on the time required for the radius of investigalion to exceed the fracture half-length. (More exactly, for an infinile-conductivity vertical fracture with half-

#

applicable to (I) new wells, (2) wells that have been shul in sufficiently long to allow the pressure to slabilize, and (3) wells in which loss of revenue incurred in a buildup tesl would be difficult to accept. Exploratory wells are frequent candidates for lenglhy

length Lf' shut-in time must exceed 1,260 ct>p.c(L}/k or r; must exceed 1.15Lf' The MTR begins when the ETR ends (unless boundaries or important heterogeneities are unusually near the well). In the MTR, a plot ofPwf

i .

drawdown tesls, with a common objective of dclcnllinillg minimum or lolal volume being draincd byAn the idealized well. constanl-rale drawdown tesl in an

vs. log t is a straight line wilh slope, m, give.p by qBp. 111=162.6-.kh (3.3)

; !

infinite-acling reservoir is modeled by the logarilhmic approximalion to the Ei-function solulion:

Thus, effective formalion estimaled from this slope:

Pwj=p;+162.6-log qBIL kh

l

( 1,688ct>lLc,r~)

, i

h m

J -0.869s.

permeability, k, can be

k=162.6-qBp.

kt (3.1)

Like buildup tesls, drawdown tests are more complex thanusual suggested byan simple as Eq. 3.1. The test has ETR,equations an MTR, such and an LTR. The ETR usually is dominated by wellbore unloading: the rate at which nui~ is re~oved from the wellbore exceedsthe rate at which fluid enters the wellbore until, finally, equilibrium is established. Until that time. the constant now rate at the sand face required by Eq. 3.1 is not achieved. and the straightline plot of Pwj vs. log t suggestedby Eq. 3.1 is not

achicved. Duration of wellbore unloading can be

After lhe determined.

MTR The

(3.4) is identified, skin factor, s, can be usual equation results from solving

Eq. 3.1 for s. Setling t = I hour, and letling P~ = PI hr be pressure on the MTR line at I-hour flow time, thethe result is S=I.151

[ (P;-Plhr)-IOg ( m

1

k ~~

j

I

J

'

i

)+3.23].: (3.5)

The LTR begins when the radius of investigation:

reaches a portion of the reservoir innuenced by

I

FLOWTESTS

-51

TABLE3.1 -CONSTANT.RATEDRAWDOWNTESTDATA ~(~?~r~) Pwl (psia) 0 4,412 0.12 3,812 1.94 3,699 2.79 3,653 4.01 3,636 4.82 3,616 5.78 3,607 6.94 3.600 8.32 3.593 9.99

3,586

P,-Pwl (psia) 0 600 713 759 776 796 805 812 819

t (hours) 144 17.3 20.7 24.9 298 35.8 430 51.5 61.8

826

74.2

P, -Pwl (PSIa) 839 845 851 857 863 868 875 880 886

Pwl(psia) 3,573 3,567 3,561 3,555 3,549 3,544 3,537 3.532 3.526 3,521

891

reservoir boundaries or massive heterogeneities. For a well centered in a square or circular drainage area, this occurs at a time given approximately by .1(1=

380
(3.6)

where A is the drainage area of the tested well. for

t (hours) Pwl(psial 891 3.515 107 3,509 128 3,503 154 3.497 185 3,490 222 3.481 266 3.472 319 3,460 383 3,446 460

3,429

'DA=.

983

... F?

ETR

wf

R

more general drainage-area shapes, 1(( can be calculated from the number in the column "Use Infinite System Solution With Less Than I 0/0Error for 'DA <" in Table 1.2. I The dimensionless time I DA is defined as

r

P,-Pwl (psla) 897 903 909 915 922 931 940 952 966

I0

0.CXX>264 kl
9

t

I

Fig. 3.1-Typical constant.ratedrawdowntest graph.

For this more general case. then.

I 1(1= 3.800
(3.7)

Thus. typical constant-rate draw test plot has the the shape shown in Fig. 3.1. Todown analyze the

0'

typical test. the. following steps are sug~este~. 1. Plot flowIng BHP. PwJ. vs. flowIng tIme. " on semilog paper as shown in Fig. 3.1. 2. Estimate I wbs from qualitative curve matching marks athe beginning of the (except frac(with full-size version of MTR Fig. 1.6); thisforusually

.~ .\ ~ 0.:

,'

k ;

.

tured wells). 3. Estimate the beginning of the L TR, 1((. ll~ing deviation from a match with Fig. 1.6 to confirm

)4

deviation from aninapparent semilog straight line. We mu~t be calltiou~ dr",wdowl1 tc~t aI1alysi~, though.

I

'

C

~

R-ONING TIME. hr

Even small rate changes can causea drawdown curve to bend just as boundaries do (a method of analyzing

Fig. 3.2-Semilog graph of exampleconstant-rate drawdowntest.

this possibility is presented later). 4. Determine the slope In of the most probable MTR. and estimate formation permeability from Eq.

i

3.4. -_s. Estimate the skin factorsfromEq.

3.S..~

Example 3.1- Constant-Rate Drawdown

a: ~

~~===?-.-

.:: ",

I Q.-

.u.x.--

TestAnalysis Problem. The data in Table 3.1 were recorded during a constant-~ate p:res~uredra.wdown test. The wellbore .had a failIng liquid/gas Interface throughout the drawdown test. Other pertinent data include the following.

.

. 1.l .t1' Fig. 3.3-Log.log graph of exampleconslant.rate drawdowntest.

I '

II

q = B = p. = r", = II = !/J =

250 STB/D, 1.136 bbI/STB, 0.8 cp, 0.198ft,I 69ft, 0.039, and

c, = 17 x 10 -6 psi -I .A

The tubing areas is 0.0218 sq ft; the density of the liquid in the well bore is 53 Ibm/cu ft. Determine the fonnation permeability and skin factor. Sctl"liu~. We first plot flowillg UI~P'Pwf' vs. tilllc, t, on semllog paper and (Pi -Pwf) vs. I on log-log paper. Then we determine when wellbore effects ceas.eddistorting the curve. From the shape of the semllog grapll (Fig. 3.2), Ihis appears 10 be 31 abolll 12 hours; however, we can check this assumption with Ihe log-log graph, Fig. 3.3. For several values of CD (e.g., 103 to 104), the graph shows a good fit with Fig. 1.6 for s = 5; wellbore storage distortion end~ at ~l = 5 hours, in approximate agreement with the more sellsitive semilog graph. We have no information about the location of bolllldarics; therefore, we assume that boundary effects begin when the drawdown curve begins to deviate from the established straight line on the semilog graph at a flowing time of 150 hours. This is confirmed qualitatively on the less sensitive log-log graph by noticeable deviation beginning at t.: 260 hours. The slope of the middle-time line is ", = 3,652 -J,5R2 0 / I -pSI cyc e. --7

.

At the end of the MTR (I = I SOhours), -~ 4 -ri -(1.521 X 10 )(150)l =I,510ft. substantial amount of formation has been sampled; thus, we can be more confident that the pcr-

meability of 7.65 md is representative. We next calculate the skin factor s.

I Pi -P,m

hr

.\'= 1.151

-I -.151.

-log

-log

( ;j;~k ) +3.23 ]

[ 4,412 -3,652 70

(1.442x 107) (0.198)2 + 3.23

]

= 6.37. We now can verify more closely the expected end of wellbore storage distortion from Eq. 3.2, using 25 65 A Cs.:' K'b P =0.0106 bbl/psi. -(200, (XX)+ 12,(XX)s)Cs t K'bskh/1I.

Thus, the permeability of the formation is k= 162.6~

mh

=

= (162.6)(250)( 1.136)(0.8) (70)(69) = 7.65 md.

[200, + (12,
(7.65)(69)10.8

= 4.44 hours. This closely agrees with the result from the log-log curve fit.

.. We now check the radius of invest~gation at the beginning and end of the apparent middle-time line 10 ensure that we are sampling a representative portion of the formation. At the begillning (t = 12hours), --~ -(7.65) 948 !/Jp.c,-(948)(0.039)(0.8)( I. 7 x 10-s) = 1.521 x 104,

,( ~ot )

and, from Eq. 1.23, I kl r; =...Jii4i ~

t

.Even

=..J(1-.S-21X1.04)(12) = 427 ft.

Another use of drawdown tests is to estimate reservoir pore volume, VP' This is possible when the radius of investigation reaches all boundaries during a test so that pseudosteady-stale flow is achieved. Eqs. 1.12 and 1.13 showed that, in pseudosteadyslate flow, flowing BHP, Pwf' is related linearly lo time and that the rate of change in Pwf with time is related to the reservoir pore volume. From Eq. 1.13, this relationship is -O.234qB Vp= , c where oPwj/ol is simply the slope of the straight-line Pwl vs. 1 plot on ordinary Cartesian graph paper. though Eqs. 1.12 and 1.13 were derived for a cylindrical reservoir centered at the wellbore of the tested well, the principles derived from them apply to all closed reservoir shapes. The graph of P wfvs. t is a

~.-

FLOW TESTS

53

Ihe te)1-re)ulls oblained using Ihose Icchnique~ l.:al1 lead to inlerprelations thai are seriously in error. An analysis mer hod I hal leads 10 proper inlerprelat ion is

,

available, bur il can be used only if Ihe produl.:ing rare is changing slol~'I_vand sl1/ooth~v. Abrupi rare c/1angl"swill make Ihe drawdowllle~1 uilla impos~iblc: to inlerpret using either the method discussed earlier this new method.Winestock and Colpitls2 show thai when rare is changing slowly and smoolhly, the equal ion modeling the MTR of the drawdown test becomes

.V) ~ -or ~

0

--.iCiI-

~

= 162.6~

FLONlt'IK:; TIME, hr

[IOg( ~~~t~~)

.

Fig. 3.4-Cartesian-coordinate graph of example

+ 0.869 s] + negligible terms2.

..(3.8)

conslant-rate drawdown test.

slraighl line once pseudosleady-stale now is achieved; the volume of the reservoir can be found from Eq. 1.13. It is important to remember, however, that these equations apply only to closed, or volumetric, reservoirs (i.e., they are not valid if Ihere is water innux or gas-cap expansion). Further, they are limited to reservoirs in which total compressibility,of Ct' is constant (and, specifically, independent pressure).

The analysis technique is to plot (p; -P wI) / q vs. 1 on semilog paper;-- identify the middle-time straight line; measure the slope Ill' in psi/STB/D/cycle; calculate kh from p.B kh= 162.6 -; , ", and, finally, calculates from s=).151

[( P;-PWj I W )

We will illustrate pore-volume estimates with an

q

example. -log Example3.2-Estimation

of Pore

~

-3,531-3,420 -0-500

at

-

0

222

,

0

Vp=

23 0

of Draw down

Rate

.

Test ..

B = ).) 36 bbl/STB,

p.=-0.8cp,

01

h = p = Awb = cP= ('t = rw =

(-0.234)(250)(1.136) = (1 7 )0 -5 )( 0 222) .x-.

69 ft, 53 Ib/cu ft, 0.02)8 sq ft. 0.039. ) 7 x 10 -6 psi -I .and 0.198 ft.

= 17.61 X 106 cu ft

Determine formation permeability and skin factor.

= 3.14 X 106 res bbl.

Solution. We note immediately that conventional drawdown test analysis. using an average rate, would

The method ..slluwn out lined above' d con

3.3-Analysis

T bl 2 b d. . Problem. The data In a e 3. were 0 lame In a urawdown lesl in which the rare q w.tS measllrcd as .. function of lime. Other data include the following

-.4qB

('t(~)

(3.9)

In Eq. 3.8, [(Pj-Pwj)/q]1 hr is the value of this quantity on the middle-time line or its extrapolation at a now~ng ~ime of I hour.. . We wIll Illustrate use of thIs method wIth an example.

Varying

--.psI/hr.

Thus

.,-

cpp.Ctrw

Example .JVith

t hr m

( ..L..-k _2 ) +3.23. J

Volume

P~oblem. Estimate the pore v.olume of the reservoir wIth drawdown data reported In Example 3.1. Solution. The first step is to plot Pw vs. t (Fig. 3.4). The slope of this curve is constant ~r t> 130 hours; Ihis slope, OPwj/at, is

-;-I

ted UC

of

.. a

P ermeability

applies

t a

strici

during a test -e.g..

.

, y

only . (onstan

determination

to

drawdown

t

t ra

If e.

tests t

ra

varies e

if rate declines slowly Ihroughout

.Verificalion 01 this method,s incomplete In cases with severe wetlbore slorage ellects. A nonexhaustive numerical simulation study by this author ~s thai Ille method Yluills essenllatly c(J{recl permcabillty ancl skin I~clor e."n "!IOOse c~ses '.'1... same all41ysil w"llIlUlesturaye-dum.nilleIJ

anllRamey.

lechrnque. lIala)

bul Wil.

lor a IIllIsrenl Sll!/!I"stcd eall",.

application by Glad'c"",

lan~tYllng ..,..,

J

TABLE 3.3 -DA T A FOR PLOTTING FROM VARIABLE-RATE DRAWDOWN TEST TABLE 3.2 -VARIABLE.RA TE DRAWDOWN TEST DATA '(hours) 0 0105

Pw/(psi) 'q(STB/D) '(hours) 4412 250 832 4:332 180 999

t (hours)

Pw/(psi) q(STB/D) 3927 147 3:928 145

0.105--0.151 0.217

0151

4.302

177

144

3.931

143

0313

0217 0313 0450 0648 0934 , 34 1 94

4.264 4.216 4.160 4.099 A 039 3.987 3.952

174 172 169 166 163 161 158

207 298 430 618 742 891 107

3.934 3.937 3.941 3.944 3.946 3.948 3.950

140 137 134 132 130 129 127

0.450 0648 0934 1.34 1 94 2.79 ..

279 401 5.78

3.933 3.926 3.926

155 152 150

128 154 185

3.952 3.954 3.956

(P, -Pw/ ) Iq

126 125 123

( -~

-log

3.414

3.467 3.515 3.545 3.585 3597 3.638

3.197 3.240

128 154 185

3.651 3.664 3.707

k

) (~ I h m' r

)

) + 3.23 ]

[ (0.039)(0.8)(17 7.44 x 10-6)(0.198)2 ]

-log

estimated. There is no deviation from the straight line for 1>6 hours; accordingly, we assume the MTR spans the

+ 3.23

J

= 6.02. = 0.288

.. bbl/pSI, as In Example 3.1,

Since Cs =0.0106

p.B 162.6Ill' h

I

(200 , + 12,s)Cs = M'bs-khlp.

(162.6)(0.8)(1.136)

(200,+ 12,
-(7.44)(69)/0.8

(0.288)(69)

I

= 4.5 hours.

lid

= 7.44 md,

This qualitatively storage distortion

0 ,.

t

(I) ~

confirms end.

the choice

of well bore --

q2 q

.~

q

Q.

,

Q. I

.I 0' A

I

I

f

qn-1

I I

~-I

-

207

29.8 43.0 61.8 74.2 89.1 107

3.04 = 1.151[ 0"288 .

end at approximately 6 hours; willS check this ass.umption with Eq. 3.2 when kweand have been

-

1.140

4>JJ.C1 W

technique; the first step is to tabulate (Pi -Pwf) Iq, as in Table 3.3. These data are plotted in Fig. 3.5. On the basis of curve shape, wellbore storage appears to

k=

3.299 3338 3.364

9.99 14.4

1.491 1.886 2.288 2.640 2911 3.090

s= 1.151 [( Pi -PWj q

rate decline from this time to the end of the test is only 27 STB/D (from 150 to 123 STB/D). Thus, we must use the variable-rate analysis

time range 6 hours < I < 185 hoHrs. From the plot, nl' = 3.616 -3.328 psi/ST8/D/cycie. Then,

--0.444 -8.32 0.621 0.851

4.01 5.78

be futile. Press~res fo~ now times greater than ab?ut 6 hours are Increasing even though production continues for another 179 hours and even though the

t_(h~ur_~ .!P-!-=1?:!'~!..~9-

I I

I

'

0:..-

0

r

0

-I

t

.m

FLONIt...K; TIME,

II

hr

FIg. 3.6 -Rate history for multlrate test.

Fig. 3.5 -Example variable-rate drawdown test.

~--

FLOW TESTS

55

3.3 Multirate Tests We will develop a general theory for behavior of multirate tests in infinite-acting reservoirs for ~lightly compressible liquids. In Appendix E, we extend this general theory to reservoirs in which boundary effects may become important .before the test ends. Consider a well with n rate changes during its production history, as indicaled in Fig. 3.6. Our objective is lo,delermine Ihe wellbore pressure of a well producini"- wilh this schedule. We will use sllperposilion of the logarithmic approximation to Ille Ei-function solution; 10 simplify the algebra, we will write the solution as 2 q81J.

pI -

p WJ .r= 162.6- kh

-

]

( 1,688

r

lo g

t

q

t

~

)

1J.C (r IV

kl

Fig. 3.7-: Rate history for single.rate drawdown test. .

0.869 s q8IJ.ti

= l62.6-,log

I+l.og

kh

k

2

IJ.C(r W

-3.23 + 0.869 S)

qI

,

=m' q(log 1 +sj,

t

where

q

m'=162.6~

f

kh' and

q2 =0

S=log~ -3.23+0.869s. With this lJ.c(rW nomen,clature for n rates and for I> 1n -t ' application of superposition (as in the discussion leading to Eq. 1.27) leads to Pi-pw/=m'QI(log/+sj+m'(q2-QI) .[log (/-II)+S] .[log (1-/2)+S]

qn

~

LJ

(/-I,,-I)+S].

t q

q,

(qj-qj-l)

j= 1

I I

qn

tp,--r-

.log(I-lj_I)+m's,qn~O

..(3.10)

In Eq. 3.10, we define qo =0 and 10 =0. In terms of more Pi fundamental -Pwf =m' E ~uantilies, !qj -qj-l) Eq. 3.10 becomes qn

Fig. 3.8-Rate history for buildup test following single flow rate,

..

This can be written more compactly as

-m

j= I

.

+m' (q) -q2) + ...q2

+m'(Qn-qn-I)[log

Pi-PW/ -,

t

qn

.Iog (1-lj_I)+m'

[ (

I

f1t-+

tl

t2 t

Fig. 3.9-Rate history for buildup test following two different flow rates.

log ~2 IJ.C (

-3.23+0.869S].

)

k

tP2

w

,

(3.11) "

.JV

For lhespccial caseqn =O(a pressure builduplCSl). , -, Pi-PM's=m ql(logl+S) +111 (q2-ql) .[log(I-II)+sl+ -qn-2)[log

Lel 1-12=AI. II =Ipl' 12=lpl +lp2' and I I = I p2 + AI. Then. q28p. ql I 1+1 2+AI Pi-PM's=162.6-

...+m'(qn-1 (/-ln-2)+sl-I11'qn-1

I

I since

the

well

(3,12)

producing

Appendix

E suggests a more

modeling reached.

tests in which

general

at

rate

been at rate q I for time I pi and production ju before lhe lest 10 have been al rate q2 for time Ip2' To analyze such a test, we plot

ql.

method

Pws

boundaries have been

C

0

n

t t R s an -a

t

e

P

ro

d

uc

:..(3.13) -To

p2 +

.'

Pi-Pws=162.6~1~log(lpl+lp2) kh Lq2 (I-II)]

(

)

I

I-I

I.

or

(3.14)

I

\

I -log ql

Pi-PMt='"

Ifwclclql=q,I-II=~/,andll=I"lhefamiliar serves

as

the

basis

for

Horner

..

equal

ion

~I

(

+ log. I-II 1-12

)1

ql -log

MTR

line

the

buildup

te

log (I-I.)

-r

Subtracting,

( I"l...,.-+ln2+A/ )

( -+I )

j

( I.112.+ -.41 )J ~I

ql

.

[

PM'S-Pw/=I1'tq-;log

I-II

on

1II2+AI

g

This can be written as q2

/

+ 10

-q210g (1-/2)]'

l -log ql

lhe

:

Pi -PM,~ =m .q2

I-rc!isllrc 811ildllp Tcst I-rcl'cdcd by T,,'o ~. I)irrcrcnt I-low Rates ('rom Eq. 3.12(Flg. 3.9),

Pi -P\I:t = 162.6-q2BP kl r

of

j

plot is

( I.1"I + -.. ~I )

Iql log 1+ (q2 -ql)

i !I

J

1112

plot

The

qBp. IJi-P\I,.~=162.6-log kll

( 11+12 'pi' 'pol ) +log(/,'2)+.5.

q2 the

resulls:

Pi -PI!'", =",'

,

_

qlp.B =162.6-log kJl lhal

~ ..

I p2

+I og(lp2)+S, I

kJl

ion

)]

I

. .

t .t Ion

From Eq.3.12for/l=2(Fig.3.8), p.B Pi-P\I'." = 162.6 -[q,logl-q.log

equal

( I 2 + 41 .:~=

m = 162.6~. kl, Extrapolation of the plot to AI = Q) gives Pws =) because, at AI =~, the plotting function is zer." Nole 11101 semilog polJer is "0110 be used; instead, tl,' sum or two logarithms is plotted on an ordinal' : Carlesianaxis. calculate skin faclor, s, note that at the end 4';1 h n d bef ore s h ut-m,.'~, e ow peno jUSl ,.

)

.

( I'PII'1+ 'PlI 2~+ -+Iog AI )

The slope nl of this plot is related to formatic pcrmeabililY by lhe equation

(

-3.23+0.869s. .., su e II Id T t P d d b Pr "~ r UI up es rece e y

[ -log qI q2

= 162.6 JJ.!! log I + log ~kl, p.c,r;.

q1

vs.

of

Prc!i!illrl' I)rll"'down Test (:rom Eq. 3.11, forn= I (Fig. 3.7), ~I[

(3.J

Eq. 3.15 ha~ Ihi~ applicalion: when the producil rate is changed a short time before a buildup tc begins, so that there is not sufficient time f. Horner's,approximation.to be valid,. we frequent can consider all production before time I I to ha'

(qj -qj-l)

began

+A/

AI

Eqs. 3.11 and 3.12 can be used to model several special cases of practical importance, but we must remember that they have an important limitation: the reservoir must be infinite acting for the total time

elapsed

'.Pol'-') Ip2 +AI

+log(..:2.L:-=. )J

.[log (I -In -I) + sl p.B II = 162.6 kh E j= I .log(I-I.). J I

[ -log('PI q2

kh

l

log

(Ipi + tp2)(lp2 + ~t)

]

(Ipl +lp2+~/)(lp2) (I 2)(AI) "p""-' Ip2 +41

Assume Ipl + Ip2 + 41=lpl

]+$.J

+ IPl:: and Ip2 + ~I = II

for small 13.1 (e.g" ~I = I hour). Then,

I

,

,

FLOW TESTS

57

Q.

I

t q

colltract~),

Improvement

q2

I

tI ~

F. 3 -3. Ig. .10-Rate history for two-rate flow lest. Pws-pwf=m(logl:ll+s).

(

~igllif"j~.allt

(ontheMTR

)

(

q,,-IBp.I kh.

4. Cal,-,ulatetheskinfactorsfromtheequation ( -) k [ 'Plhr-Pwfl -Iog( _2)+3.23 J S=I.ISI m tPp.C ,T w

(The derivation and assumptions implicit in this equation closely parallel those used for a buildup test preceded by two different now rates.) S. The original formation pressure Pi is the value of P W'\'on the MTR line extrapolated to X = O.

= log -~ -3.23, tPp.C,w or _ I ISI( PI hr -Pwf s- .-log m

k

)

, + 3.23 , tPp.c,rw

Two-Hale Flow Test

asbefore. We also note that duration of well bore storage distortion is calculated as in the previous analysis for buildup tests.

From Eq. 3.11, this test (Fig. 3.10) can be modeled5 by qB p. q (q -q ) Pi -Pwf= 162.6-1[ -'. log 1+ 2 I kh q2 q2

Pressure Buildup Test Preceded by (n -I) Different Flow Hales

.Iog (1- 1 ) + 10g( -~ I tPp.c ,r;.

FromEq.3.12,

-3.23+0.869S]

[~ qn-

P,.-=1626~!!~ Pws .kh

(~ )

(

)+

(

log -~

1-/

)

I-I n -3 + log I- I ~ 21

n-

...+

q n -2 q "-:.1I

2

(

1- 1n.-2 1 I

-

.. (3.17)-

If we rearrange and Introduce specIalIzed nomenbecomes

I

)

...

t log I-II

clalure, II =Ipl

.Iog

.Ifl.'

.

k

+ ~ q ,,-

ratc:~

in a,-,curacy when u~ing

2. Plot Pws vs. X on ordinary (Cartesian coordinate) graph paper. Determine the slope In of the plot and relate toI the formation permeability by the equationI /1/=162.6

If we choose i1/=1 hour,pws=Plhr line) and, for I p2 » I, -Plhr-Pw! s= m

all

this approach is questionable; fllrther, the flIndamental assumption on which Eq. 3.16 is ba~eu (that for 1=lpt +lp2 + ...+lpn-1 +i11 Ihe reservoir is infinite acting) rarely wIll be valid for large values of I. Nevertheless, when Eq. 3.16 is u~ed to model a buildup test, the following analysis procedure can be used. I. Write a computer program to calculate the plotting function, ill I I-I" -2 ---log -=+... + log -= --= )1 X. q,,-1 "I I 'n-I

I f :

t

uc:livc:rability

considered.

)1

n-

-q2 pwr-Pi-162.6-

and 1-lpl

=i1/',

(

then Eq. 3.17

!

Bp.r k ~ -3.23 kh log tPp.(ir~.

)

q I Bp. +0.869s j -162.6-

kh

1pi + i11'

/ l0g

()

-~-

i1/'

(3.16) Although we introduce no specialized nomenclature for this situation, note that 1-ln-1 =61 (time elapsed since shut-in) and that qn-1 is the production rate just before shut-in. Applications of Eq. 3.16 in which more than three terms are needed are probably rare; sometimes, though, to satisfy precise legal contracts (e.g., gas

!

+ ~ 10g(di')]. ql

,

(3.18)

This type of test can be used when estimates of permeability, skin factor, or reservoir pressure are needed but when the well cannot be shut in because loss of income cannot be tolerated. This test shares a

,.

".~o~

VI

a. .

Pws

~ t

+

At

14

log 'P .-,

.'I

20

At Fig. 3.11 -Buildup

test.

The

rate

must

constant or the test interpretation

be

kept

strictly

Fig. 3.12 -Example

Pwf

vs.

two-rate flow test.

log

)

!JJ'

q2 +

may be sub-

,

-Iog(~t

) ] .

ql

stantially in error. Eq. 3.18 is rigorously correct only when the reservoir is infinite acting for time (/pl + ~/') [just as the Horner plot is rigorous only when a reservoir is infinite acting for time (/p+~/)]. Nevertheless,

2. Determine the slope m from the plot and use it to calculate permeability, k, from the relationship B k = 162.6~. mh

application of the plotting and analysis technique suggested by Eq. 3.18 allows identification of the MTR and determination of formation permeability in a finite-acting reservoir.

3. Calculate the skin factor, s, from the equation s= I.l~[

The two-rate flow test does not reduce the duration o~wel~bor~storag~distortion-the.duration.ofthis dIstortIon IS essentIally the same as m any buIldup or drawdown test. However, the test procedure may minimize the effects of phase segregation in the wellbore, an extreme form of which is "pressure II\lmping",6 whcre high-pressure gas trapped in a \\.cllbore in poor communication with a formation may lead to pressures in the wellbore higher than formation pressures (Fig. 3.11). A buildup test with this humping is, at best, difficult to interpret; thus, a te~t procedure that can minimize these phasesegregation effects can be of value. The following method of analysis can be used for two-rate now tests. TABLE 3.4 -TWO.RATE

!

~. :.

.11' (hours)

Pwl (psi)

0 0.105 0.151 0.217 0.313 0.450 0.648

3.490 3,543 3.564 3.592 3.627 3,669 3.717

8.32 12.0 17.3 24.9 35.8 51.5 74.2

3,897 3.903 3.908 3.912 3.915 3.918 3,919

0934

3766

89

3918

1.344 1.936 2.788 4.01 5.78

3.810 3.846 3,868 3.882 3,891

-Iog

~I_~ (hours)

1

107 128 154 184.7

ql (~J!~) (ql -q2) m

(

k ~~~

) +3.23]

(3.19)

In Eq. 3.19, PI hr is the flowing pressure at ~/' = I hour on the MTR line and Pwj1 is the flowing pressure at the time the rate is changed (~t' =0). (Eq. 3.19 was derived by simultaneous solution of Eqs. 3.18 and the drawdown equation for a single rate applied at t = tpi ' at which time Pwf =Pw/l .) 4. Pi (or, more generally, p') is obTaIned by solving for Pi (p') from the drawdown equation written to model conditions at the time of the rate change. (It is implied that sand m are known at this point.)

FLOW TEST DATA

TABLE 3.5 -DATA FOR PLOTTING FROM TWO-RATE FLOW TEST

_PWI (pSi)

...,

!

Plot

q,

( tpl + ~t'

[ I.

Z8 2.1

t .3:z.log (L)t')

L)l

test with pressure humping.

second

24

tL)l'

k>g(-~)

fundamental analysis problem with the conventional drawdown

22

l

3,917 3,916 3,913 3,910

..\,' 'hours) 0 0.105 0151 0217 0313

PF -3.490 2.756 2.677 2.599 2.519

Pwl (psi) 3.543 3.564 3.592 3.627

..\,. ,hours) 8.32 12.0 17.3 24.9 35.8

PF 1.826 1.754 1.686 1.623 1.566

Pwl (psi) 3.897 3.903 3.908 3.912 3.915

0.450

2.441

3.669

51.5

1.517

3.918

0.648

2.~

3.717

74.2

1.478

3.919

0.934 1.34 1.94 2.79 4.01 5.78

2.283 2.206 2.127 2.050 1.974 1.899

3.766 3.810 3.846 3.868 3.882 3,891

1.462 1.450 1.442 1.436 1.434

3.918 3.917 3.916 3.913 3.910

89.1 107 128 154 184.7

FLOW TESTS

59

I (

) -3.23

kl,l

P, =Pw./l +m log -~

CPJl.C Ir

I

4. 1)<:ll.'r/11i/1I.'/'o.

+0.869s .

I (

IV

p' =PI':/l +m log -~kl (3.20)

) -3.23

I

+0.8695 I

Jl.({rll' = 3,490 + 70[ log[ (7.65)( 184.7)/(0.039)(0.8)

Example 3.4- Two-RateFlow Test .JrcJblem. A lwo-rale now ll.'sl was run on a wl.'ll willi properties given below. From these properties and the data in Table 3.4, determine k, 5, andp..

.(17 x 10 b)(0.1 '18)2/

ql = 250STB/D, q2 = 125STB/D, Jl. = 0.8cp, B = 1.136 RB/STB, Pi = 4,412 psi, CI = 17 x 10-6 psi -I, A wb = 0.0218 sq ft, r W = 0.198 ft, h = 69ft, P = 53lb/cu ft, q, = 0.039, and Ipl = 184.7hours.

-3.23+(0.869)(6.32)~ =4,407 psi. 5. ~e then check on wellbore-storage duration. For lhlS wel" Cs =:=25.65A lI'b/p=(25.65) (0.0218) / 53 =0.0106 bbl/psi. Then, = (200,CXX> + 12,CXX>5)Cs III'h~kh/Jl. (200,CXX> + (12,CXX»(6.32»)0.0106 =

(7.65)(69)/0.8

Solution. J. We first tabulate the plotting function (PF), PF=[log(!£!-~~)+ AI

~log(aJ')]' ql

= 4.4 hours. At this time, the plotting function is approximately 1.9;thisconfirmsourchoiceofthestartoftheMTR.

and plot Pwj vs. PF. Note that 1 1 = 184.7 hours and q2/ql ~12~/250=0.5 (Table 3.5). The data are plotted In Fig. 3.12. 2. Next, we determine permeability. Assume that the MTR spans the time range 1.5~/' >6 hours). Then the slope m=(3,9273,857)/( 1.4 -2.4) = 70 psi/cycle and k=162.6~

n-Rate Flow Test FromEq.3.11 an n-rate flow test is modeled by ' Pi -Pwj = 162 61l}! [ t (qj -qj-l) qn .kh j=1 qn

]+162.6k1;

.log(/n-lj-l)

Jl.8

mh

=

(162.6)(250)(1.136)(0.8)

.rIOg(~) cPJl.( Ir'i"

-3.23 +0.86951. ..

(70)(69) This equalion suggestsa plot of

=7.65 md.

Pi-Pili q

3. We determine skin factor, S= 1.151 r

-log.

(

qI (ql -q2) k

n

(p I hr -P w}1) m

-

[

) CPJl.C I~

') +3.23 / ,

250 (250 -125)

(qj-qj-I)lo q

J=

g (I n -I. J-I' )

n

Permeabilily is related to the slope m' of such a plot: k=162.6-;-.

Jl.8

'" h

at aJ , = I hour, PF=log(184.7+ 1)/1 +0.5 log(l) = 2.269, andPI hr = 3,869 psi (on the MTR line): s-I.151

~I vs. ""'

Irwe let funclion h' be the value thcn of (p;-Pwj)/qn plolling iszcro k '

(3,869-3,490) 70

h'=m'

when the

( d>uc.~ ) -3.23+0.869S ].

( 'Og

.')

'f'r-t

w

or -Iog

=6.32.

[ -:

7.65

6

(0.039)(0.8)(17 x 10-°)(0.198)"

_

]+3.23 }

S=I.151

[~-IOg (~ )+3.23] m' 4>#lCtfw

(3.21)

Note that use of Eq. 3.21 and of the proposed

-

TABLE 3.6 -MUL TIRATE FLOW

TABLE 3.7 -DATA

FOR PLOTTING MUL TIRATE FLOW TEST

TEST DATA t (hours)

t P, -Pwf

(

PSia

)

1= 1

~,--Q,-

1

qn

I I

Pwf (psi a)

t(l)ours)

Q,,(STB/D)

P,-pwf(psia)

00.333

3.000 999

0333 0

478.5

2,001

4.18

-0.478

0.667 1.0 2.0 2333 2667 3.0

857 778.5 1.378.5 2,043 2.0675 2.094

0.667 1.0 2.0 2.333 2667 3.0

478.5 478.5 319.0 159.5 1595 159.5

2.143 2,221.5 1.621.5 957 923.5 906

4.48 4.64 5.08 6.00 5 79 5.68

-0.176 0 0.452 1.459 1 232 1.130

plotting method implies that Pi is known from inUCPClldclltmca~lIrcmcllt~. Odeh and Jones 7 discussed this analysis techniquc.

q"

RB/D_:'og(t,-t,-,)

qn =478.5 STH/D, (i = 1.0 hour,

l1tey pointed out that it can be applied to the analysis of multirate now tests commonly and oil wells. In these applications

run on gas wells of the technique,

it is essential to remember the assumption that the reservoir is infinite acting to the total elapsed time ( for all now rates combined. Further, note that the

(pi-Pwj)/qn=(3,CXX>-778.5)/478.5=4.64

~ .4J J= I

technique ignores any wellbore storage distortion c.:reatedby any discrcte rate c.:hanges.

(qj -qjq n

'

and

I) log (I n -Ii -I )

(478.5 -0)

= --og

I

( I .-0)=0. 0

-478.5

E.\-oI11ple 3. 6 -Multirote

Flow Test A 110lysis

.-r.,hlem. Odch and Joncs , prcscnt data from a 3-

Iiour drawdown tcst on an oil well; in this test, the rate during the first hour averaged 478.5 STH/D; during the second hour, 319 STB/D; and during the third hour, 159.5STB/D. Reservoirnuidviscosityis 0.6 cp; initial pressure is 3, psia; formation

1 = 3.0 hours. Here , (Pi -Pw/)/qn =(3,CXX>-2,O94)/159.5=5.68, and. n E (qj-qj-I)IOg(1 -:"1. ) i= I qn n )-1

volume factor, 8, is considered to be 1.0; and the rcscrvoir is assumed to be infinite acting for the entire test. Assume that wellbore storage distortion is minimal at all times during the test. Pressures (PHf) at various now times are given in Ta?~e 3.6: From Ihc~c data, determine the permeabliity/thlc.:kness I'roduc.:t of the tested well.

=~ (478.510g(3.0-0) + (319-478.5) 159.5 .log(3.0-1.0)+(159.5-319)

SClllIli.'II. We first prepare the data for plotting -i.e., at each tilllC, we must determine (pj-P"f)/q" and

=1.130.

~ ~ i=i.JI

t

-qi -I) -.calculations 10g(/" (i-I)' q"

at theseand other times are summarized In We Table next 3.7.I estimate the permeability/thicknessI

I

(= 0.333..'!our. Hcre, Q,,=ql

.log(3.0 -2.0)J

=478.5S'I'B/D,

product. From Fig. 3.13, (6.0 -4.2) m' = (1.459-(-0.452)J[

=0.942.I

Then,I

(" = 0.333 hour, (l'j-1111:r)/q,r=(3,()()()-999)/478.5=4.18,alld " (il'-i/'I) E ~ .'j-J'log«(,,-lj-l) j= I q" (478.5 -0) =

log(0.333 -0) = -0.478 478.5

, = 1:0 hour. Here, -I.

ILB kh= 162.6 -= m'

(162.6)(0.6)( 1.0) 0.942

=104nld-ft. Odeh and Jones do not state values for h, ~, Ct' alld , w used 10 construct these example test data. To illustrate skin-factor calculation, assume that /, = 10 ft and2lhus, that k = 10.4 md. Also assume that k/~ILCtrK.=4.81 X 10'. Then, since the graph indicated that b' =4.63 (b' is the value of (PiPll f ) Iqn when the plolting function is zero),

I

I I

FLOW TESTS

-~!~-

61

.FO~

S=I.t511l!:.--IOg(~-2)+3.231 ",'

~IJ.('" ".

I --log 4.63

=1.151

J~-~'~.;~

1

.0

(4.8xI07 ) +3.23

0.942

~i ---I

~.

c

:!it: =0.53.

~

4

I

-0:.Exercises

3.1. A constant-rate

drawdown

well with the following q = ct>= IJ. = c, = 'II' = h = Bo = Awb = P =

test was run in a

characteristics:

r

.

FI g .3.13

1(hOllrs) P.t(psil

3.8, estimate and area

formation (in acres)

rained by the well, -00218 3.2. A drawdown test in which the rate decreased

.

1 h h t . II continuous y t roug out t 1e test was run In a we with Ihe following characteristics.

ct>= 0 2 ., II. = 1.0cp, 'w h Bu A I.-b P

=

IOxI0-6psi-t, 0 25 f

=. = = = =

~

10 II

14 .

[

-xample E

IIIUItlrate I low test.

TABLE 3.8-DATA FOR EXAMPLE CONSTANT.RATE DRAWDOWN TEST

interface is in well.

From the test data in Table permeability skin factor d .' -,

02 04 ~

ll..v.(t .qJ ( .-.)/1"' q J-I '-tnJ ~ ""n.t. J-I) J=I

500 STB/D (coll~tal1t), 0.2, 08 .cp, IOXI0-6psi-t, 0.3 fl, 56 fl, 1.2 RB/ST8, 0.022sq fl, 50 Ib/cu fl, and

liquid/gas

Ct

0

t, 100 fl, 1.3 RB/STB, 0.0218 sq ft, 55lb/cuft,and II I.. d/ . t f Iqui gas In er ace I~ In wc

1(hours) p./(psil

'(hours) p.tlpsi)

0 00109 00164

3,000 2.976 2,964

0491 0.546 1.09

2,302 2,256 1,952

328 382 43.7

1.543 1,533 1.525

00273

2.953 2.942

1.64 2.18

1.828 1.768

491 546

1,517 1,511

00328 00382 00437 0.0491

2,930 2,919 2.908 2.897

2.73 328 3.82 437

1,734 1,712 1,696 1,684

655 874 1092 163.8

1.500 1.482 1,468 1,440

00546 0109 0164 0218

2,886 2,785 2.693 2.611

491 5.46 655 874

1.674 1,665 1.651 1.630

218.4 2730 3276

1.416 1,393 1.370

0273 0.328

2.536 2.469

109 16.4

1.587 1.561i

0437

2.352

27.3

1,554 ..

TABLE3.9-DATAFOREXAMPLE VARIABLE. RATE DRAWDOWN TEST

..

'(hours)

Pw/{PSI)

q(STB/OI

tiholllsl

P../IPsi)

.. qISTBID)

0 0114 0136

5.000 4,927 4,917

200 145 143

303 364 437

4.797 4797 4,7911

122 121 119

0164 0197

4.905 4,893

142 141

524 629

4798 4,798

118 117

well described in Problem 3.2. Rate was held conslanl at 600 STB/D. Afler 96.9 hours, Ihe sllrface .0340 rate was changed abruplly 10 300 STI3/0. Data lor Ihe lesls before and after Ihe rate change are givcn in

0236 0283 0401! 0490

4,881 4,868 4,856 4.844 4,833

140 138 137 136 135

754 905 109 130 156

4.799 4799 4800 41101 4.1101

116 114 113 112 110

T

0.587 0705

4,823 4815

133 132

188 225

4,1102 4.1103

109 lOll

0846 1.02 1.22 1.46 1,75

4:/j09 4,004 4.801 4.799 4,798

131 12'J 128 127 126

270 324 389 467 56.1

4.!!O3 41104 4.805 4.~ 4.~7

107 1~ 104 103 102

2.11

4.797

124

67.3

4.~7

100

4,797

123

807

4.808

99

~.9

4.809

98

From the lest data in Table 3.9, estimate formation pcrmeabililyandskinfactor. 3.3.

bl a

A constant-rate

3 e

0

F

.1.

drawdown

I t 1e

rom

d

test was run on Ihe

b ala

0

. Caine

d

. I Wit 1

600 q =

STI'/O, cst im.ltc k, .\', alld .lrc.1 of I hc rcsl'rvoir. I;'rom the data obtained in the two-rate now test arter t d 300 STO/O f. h . fk q C1ange to , con Irm t e estimates 0 and s, and calculate the current value or p.. 3

4

For

the

multirate

flow

test

described

in

..2.53

Example

3.6, (a) calculate

the value of the plotting

function at /=0.5, 1.5, and 2.5 hours, and (b) calculate the flowing bOltomhole pressure at t = 3.5 hours, assuming that there is no change in rate for 3

hours< / < 4 hours.

~

TABLE 3.10 -DATA Pwl at

Pwl at

PwI at

Pwl at

600 STBID

300 STB/D

600 STB/D

300 STB/D

~~h_O!!!:.~) 0 0.114 0.136 0.164 0.197 0.236 0.283 0.34() 0.408 0.490 0.587 0.705 0.846 1.02 1.22 1.46 1.75 2.11 2.53

~

FOR EXAMPLE TWO.RATE FLOW TEST

(~~I) 5,000 4,710 4,665 4,616 4,563 4,507 4,449 4,390 4,332 4,277 4,227 4,182 4,144 4,112 4,087 4,067 4,050 4,036 4,024

(psi) ---'-J!!_O~~~) ~,,-~IL3.833 3,978 4,000 4,025 4,051 4,079 4,108 4,138 4,167 4,194 4,219 4,242 4,261 4,276 4,289 4,299 4,307 4,314 4,319

3.03 3.64 4.37 5.24 6.29 7.54 9.05 10.9 13.0 15.6 18.8 22.5 27.0 32.4 38.9 46.7 56.1 67.3 80.7 96.9

4,012 4,002 3,992 3,982 3,972 3,963 3,953 3,944 3,935 3,926 3,918 3,909 3,900 3,891 3,883 3,874 3,865 3,855 3,845 3,833

(psi) ~,~ 4,330 4,334 4,339 4,343 4,346 4,350 4,353 4,356 4,359 4,361 4,363 4,364 4,365 4,365 4,364 4,362 4,359 4,354 4,349

He r ere.lces

r

I. [:arlougher, R.C. Jr.: Advallces ill "'ell Tesl /1l1alysis, Monograph Scrie~, SPE, Dalla~ (1977) ~. , 2. Wine~I(}\:k, A.U. and <..olpitl~, (i.P.: "Advitncc~ in [:~til11itling (,a~ Well [)cliverabilily," J, Cdll. Pel. T('('h. (July-Scpl. 1965) 111-119, Also, Gas T('('hnoloK.II, Reprint Series, SPE, Dallas (1977) 13, 122-130, 3. Uladfellcr, R.E., Will Tracy,Rcspond G.W., and L.E.: "Selecting Wcll~ Which 10 Wilsey, Production-Slimulitlion Trcalmcnl," 129.

Drill.

alld Prod. Prac., API, Dalla~ (1955) 117-

~'4. RitlllCY, H,J. Jr.: "Non-Darcy

1:10"' and Well bore Storage

Effcct~ on Pressure Buitdup and Drawdown of Gas Wells," J. Pt'l. Tech. (Feb. 1965)223-233; Trans., AIME,134. 5. R"~~II, D.(i.: "Determinalion of Formation Otaradcri~liC!i Froln Two-Rate Flow Tests," J. Pt'l. T«h. (Occ. 1963) 13471355; Trans., AIME, 228. 6. Stegemeier, G.L. and Matthews, C.S,: "A Study of Anomalous 113,44-SO. Pressure Buildup Behavior," Trans.,AIME(19-'8) .-

~ I

7. Odeh, A.S. and Jones, L,G.: "Pressure Drawdown Analysis Variable-Rate Ca~e," J. Pel. Tech, (Aug. 1965) 960-964;

:

Trails., AI~1E, 234.

J

Chapter 4

II

t Analysis

r

~-

of Well Tests

Using Type Curvesi 4.1 Introduction This chapter discusses the quantitative use of Iype curves in well test analysis. The objeclive of Ihis chapler is limited basically to illustraling how a representative sample of type curves can be used as analysis aids. Other major type curves in use todar are discussed in the SPE well testing monograph. However, type curves for specialized situations are appearing frequently in the literature, and even that monograph is not completely current. We hope Ihat the fundamentals of type-curve use presented in this chapter will allow the reader to understand and to apply newer type curves as they appear in the lilerature. Specific type curves discussed include (I) Ramey el 01.'s type curves2-4 for buildup and constanl-rate drawdown tests; (2) McKinley's type curvesS.6 for Ihe same applications; and (3) Gringarlen el 01.'s7 Iype curves for vertically fractured wells with uniform flux. 4.2 Fundamentals

of Type Curves

Many type curves commonly are used to determine formalion permeabililY and 10 characterize damagc and stimulation of the tesled well. Furlher, some are used to determine the beginning of Ihe MTR for a Horner analysis. Most of these curves were generated by simulating constant-rate pressure drawdown (or injection) tests; however, most also can be applied to buildup (or I'allorl) tesls if an equivalent ~hul-in time8 is used as the time variable on the graph. Conventional test analysis techniques (such as the Horner method for buildup tesls) share these objectives. However, type curves are advanlageous because they may allow te~t interpretalion even when wcllbore storage dislort~ mosl or all of the lesl dala; in Ihal case, conventional mclhods fail. The use of type curves for fractured wells has a further advantage. In a single analytical technique, type curves combine the linear flow that occurs at early times in many fractured reservoirs, the radial flow that may occur later after the radius of investigation has moved beyond the region influenced by Ihe fraclure and the effects of reservoir boundaries that may appear before a true MTR line is

established in a pressure Iransient lest on a fraclured well. Fundamentally, a type curve is a preplotted family of pressure drawdown curves. The most fundamental of these curves (Ramey's2) is a plot of dimensionless pressure change, PD' vs. dimensionless time change, t D. This curve, reproduced in Fig. 4.1 (identical to Fig. 1.6), has two parameters that distinguish the curves from one another: the skin factor s and a dimensionless wellbore storage constant, CsD. For an infinite-acting reservoir, specification of CsD and s uniquely determines the value of PD at a given value of t D. Proof of this follows from application of the techniques discussed in Appendix B. If we put the differential equation describing a flow test in dimensionless form (along wilh its inilial and boundary conditions), Ihen the Solulion, PD' is determined uniquely by specificalion of the independent variables (in this case, t D and rD)' of all dimensionless parameters that appear in Ihe equalion, and of inilial and boundary condition~(in this case, sand CsD)' Further, in most such ~Olillion~. we are inleresled in wellbore pressures of a te~ted well; here, dimensionless radius, r D =r/r I'" has a fixed value of unity and rhus does not appear as a parameter in the solution. Thus, type curves are generaled by oblaining solutions to the now equations (e.g., the diffusivity equation) willI specified initial alld boundary COll-i! ditions. Some of Ihese solutions are analytical; others are based on finite-differcnc.e ~pproximations generated by computer reservoIr sImulators. For~_. example, Ramey's type curves were generated from analytical ~olulions 10 Ihe diffllsivity equ3lion, wilh Ihe initial condilion thai the reservoir be al uniform pressure before the drawdown tcst, and with boundary conditions of (I) infinitely large outer drainage radius and (2) constant surface withdrawal rare combined wilh wellbore storage. which resulls in variable sandface wilhdrawal rare. A skin factor, ~'.is used to characterize wellbore damage or stimulation; as we have seen, this causes an additional pressure drop, Aps' which is proportional to the inslantaneous sandface flow rare (which changes with

10'

f

: : :::: .1 .\

-10

,

I

10

In

Q0

,

I

10', 10'

10'

10'

'0'

10'

10'

10'

'0 Fig.

time

while

wellbore

Dimensionless PD'

and

and

CsD

curve

When of

greatest

results

To

(Fig.

use a type

test,

the

now

time.

curve.

has the

values

of lI'v.

the

shape

k

thcn

in a reservoir

radial

diffusivity

is

one

or

in a sr>ccific of

a

morc can

these

-but

most plot.

have

brief

been

practice

and

is

effccts analysis

for

in

gas

well

tcsts.)

The

result

in Fig. I.

the

times

when

100%

of

of

importance

is

tests and

Ramey's

work

now

rate

slorage

for

-

is shown

is

and the of

6p)

6/

(a

Cs

6f

-( 24

-) Ap

can line

(or

in a buildup

line)

be

since

and

4.2)

from

I

Ap is since

the

is also the

determined

(Fig.

for

test),

curve

I

afternow

change

elapsed

450

on

at earliest.,

responsible

test

Ap-log61

unity

that,

is pressure

is time

log

on this

qR

that,

shut-in

follow.

solution

unloading

of 111 (~

began

curves

analytical shows

before

conslant

=

the

in a drawdown

Thus,

(6/,

of these

are based

well bore

the

a slope

('s

curves

function

test

point

of

type

equals

a linear

with

buildup

properties

Examination

the

major

for

,:'

important

which

rate

Of

4.1.

Some

the

test.

linear

\

wellbore from

the

any

,

relation

... unit.slope

in a well

with

line a liquid/gas

interface

in the

for

in

with

gas,

area

test

list

C

-V

andC

of valid

use of the

intcrprelalion.

can be removed.

bbl/

-0894

(4.1)

as \ve will

for

shapes, prior

so

or

i

( .)

M" of

Ramey's

correct

-type

different

-,

depends

the

matching

of

curves to

Cso,

type

for

on

our

find

i

of CsO to be used for

C.fv

Direct

(4 3)

curves

significantly value

val tiCS of

it is difficult

knowledge

liquid

Icf>c }rr2

analysis

for

a single-phase

.42

application t

to establish

curve

s and

with

pSI,.. C

quantitative ability

filled

"",

Successful .fD -.s

important. is not

that

bbl/psi,

a well bore

,f -CII'"

flow constant pcriod

This

assuluptions

a valid

and

wellborc or stimulatioll storage

also

,

"b P

the

now

drainage

s. is

25.65A

Cs =

inrillilc-actitlg

factor, it

the

test

that

models

durillg the puri"oses);

but

to

be used

single-phase such

tl1Cre is no assurance Icad

chapter.

can

not

drawdown

adequately

of these

this

curves

discussion

generated

compressible.

skin

in

(Ramey).

wellbore,

pr(\dllctiOlI;

limitatiol1s

later

reservoir

the

began).

These

the

princii"le

pressure

tedious,

case,

type

that

detcrmined.

at \Vellborc thc surface; damagc alld

by

assumptions

as the

will

homogeneity

rcscrvoir boulldary of interest (110 for test

ctlrvcs

be

vs.

corresponding

/)

pressure

slightly

hcrorc

characterized

(tv.

were

equation

and \\'ithdrawal concentratcd rate

drawdown

test data

and

inifinite-acting

that

Note

curves

\Vcll

type

urves

uniform

thc

actual

as this

sufficient

reservoir;

the

curve

pril,ciple Ihe

ype

with

nowing;

type

can

the

a constant-rate

liquid

(Some

and

from

type

of

ofs

C

s

~ame'y's2

of

range

Pi -P"1

paper

s, CsD'

-p"'r)]

r

'"

sItuation

Whcn

as the

as straiglltforward

3 R

values

the

change,

preplotted

is found,

del ail

4 .amey

of

same

and

~Icce~sarily ImplIes.

fixed

for

size graph

the

summarize in

for

be plotted

an actual

pressure

finds

(Pi

establishcd.

can

rate.

note

well bore,

importance,

to analyze

plots

match

sentences

the

curve

one

When

I D' drawn

practical

/, on the same

nearly

the

4.1).

allalyst

Then

differs

are

production

innuence).

at thus

time,

curves

constant

is a dominant

solutions

of elapsed

s.

lor

drawdown

by these

as a function

curves

storage

pressure

predicted

CsD

4.1 -Type

a given have

the

very

best

calculation

fit

value

of

similar without of

Cst

I Ii

I

~

i

:

I\

09 u

' V

LINE WITH SLOPE. = I CYCLE/CYCLE

:

~

p

I I

I

/

~..~ ,.=8 ..

~

MATCH

POINT

.

Fig. 4.2-Use of unit slope line to calculate wellbore storage constant,

log Po

~

.o (.\0 ~ "

.,; -'Iii:

-.D

rrD

, -f~,)fSTORA(;l W(IL~( Z.,..:..t'~ CAST(1RII(~1 -::;;--/(C~VlS'1.~TI'::Al 10 ~d9/ CUHV[ ~~ ~.u) 0)" s.o

~

,~~ .

I

,~" .

J

'-:g -;0

---t

I Fig-4.3-Use of type curves to determineend of wellbore I storage distortionr and thus CsD' from known values of A wb ~";d p or Cwband V wb does not characte~ize test conditions as well as the value of Cs determined from actual test performance as reflected in the unit-slope lines. 9

D Fig. 4.4-Horizontal and verticalshilting to Iind po:)IIIOll01 lit and matchpOllltS.

:

and Idrawdown-Albuildup' If these analogies can be used for the larger vallll:~ of At in the MTR then "'e would expect intllitively thai the approxim~tion ust.-dto develop them woliid be even better for the smalll:r values of AI in the ETR TI..e practical impli,-,ation of thi~ analysis is thi~. I"or u~e with type Cllrvt.'S,we plot actual dru\vdown tc~t data as V'i -P"1) vs. I and buildllP test data a~ (/'",\ -1'"".) vs. AI, hllt "'1: mllst rcmemhcr tll.lt AI" mllst be llsed instead of At \VII~llever~I >0.11/" 4. A log-log plot of Pv vs. IV differs from a loglog plot of (Pi-P"f) vs. I (for a druwdown .test)

~/e~.~I/(I+AI/I ),.lsused.asthetlmevarlable.An Intuitive proof of this assertion for small AI (where

only by a shift in the origin of the coordln.lte system-i.e., log I D differs from log I by a constant

.1/~~/e) fol.lows.. The equation of the MTR In a drawdown test can be expressed as

and log PD differs from log (Pi -P":f) conSlant. To show this, we note that O.CXX>264 kl

Pi-pwj=nllog/+CI'

IV=

The equation of the MTR in a buildup test is

Pv=

4>lJ.c,r;, ,

kh(Pi -Plif) 14l.2qBIJ.

.

+ AI) -Ill log AI.

I flog (I p + AI) =:log Ip(.111ad~LJllat\:.1~~lllllptit,ll for A/maxSO.l/p)' Pi-Pws=:l1Ilogll,-"/log~/ -Ill log AI-Ctt

or. (PIllS -PMf)

byaJJother

.Ind

Pi -Pws = 11110g[(lp + AI) / AI]

= (Pi -PIli/)

~,

(p i -p)lIj) drawdown-(p,,~ -P 1.1-)buildllp'

2. Well bore storage has ceased distorting the .pressure transient test data when the type curve for the value of CsD characterizing the test ~comes identical to the type curve for CsD = 0 (Fig. 4.3). (This usually occurs abollt one and a half to two cycles from the end of the unit-slope line.) Thus, thesetype curves can be used to determine how ~lllCh data (if any) can be analyze~ by. conventional methods such as the Horner .plotlor build lip te~ts... 3. The type curve~, whl,-,h wl:rc devel.opl:d 101 drawd~wn ~ests, also can. be used for b~lldlIP. test analysIs If an .equlvalent .shut-l~ time,

= nllog(/"

,-

:.

log t::.t

I

I

.'~

USE ANYffilNT (i'll 1\1') ON LINE TO CAl.CULATE Cs

I'

:

/

.'..' I

=:111log AI + C I.

Thus; the equations for MTR's in drawdown and buildup test plots have similar form if we use the .bllt ana I ogles

'llll1~, O.OOO264k log IV = log I + log ---:'.-, tI>,((,rlv

and

-,kh

10gPn=log (P;-PMf)+logI4Uqs;. The significance of this re~ult is that the plot of .in actuul.draw.down test (log I vs. log Ap) should have a shape Identical to that of a plot of log I D.vs. log P n, we have to displace both the horizontal and

~

!

~~

66

\'ertical axes (i.e.. shift the origin of the plot) to find

z, T

tlleposition?f.bestfit(Fig.4.4).. Once a fIt IS found by vertical

Pi

.Bg,=5.04~ and horizontal

shifting, we choose a match point 10 determine the relalionship bel~een actual tinle and dimensiolllcss time and between actual pressure drawdown and dimensionless pressure for the test being analyzed. AI~Yr<~intol1thcgraJ?h.papcrwill~lIfficca~a~lalch. polnl (I.e., the rcsult IS l/Jdepcndent of lhe cl'OICC01 match point). For the match point chosen, we determine the corresponding valu~s. ~f (I, ID) and (Pi-PK1),PDJ.Then, from definItion ofPD and 10' qBIJ. P k=141.2-( 0) , (4.4) h Pi-PM:! MP a/Jd -0.
k

f/J<"t -p.r~ 5

Altl .10Ug

h

tl

Ie

t

ype

curves

(4.5) were

d

eve

I

ope

d

f

rom

.10

gas no~ In terms of the p~eudopressure , 1ft(p) and c?lnpa~lson of these solutions expresse~ In lerm.s of dimensionless pseudopressure, 1fto , wIth solutions Po for slighlly compressible liquids, II show that, as a high-order-accuracy approximation for lransient n ow, lfto(ID,rD'S',CsO)

k=141.2qgIJ.,-g,( I,

average

reservoir

many

cases

for

p <

..

2 ,(xx)

psla);

be repIaced by th e defiInl t Ion 2 2

-kh(Pi PO-1422

as

-Pwf)

a

result,

E4.

4.7

(4 14)

qg IJ. I.z.T. I

'

( Pi_2_2 Po ~~~ h -Pwf )MP,

k -, - 1 422

-0.000264 k

(

(4 I 5)

I

,

)

IJ.i~

10

of

(4.16) MP

(4.6)

Determination

(4.7)

(IJ.zlp = constant or p.Z= conslanl) is valid musl be based on plots or labulations using lhe properties of lhe specific gas in lhe reservoir being tesled.

whether either approximation

Use of Ramey's Type Curves (4.8)

all d

The theoryprocedure of Ramey's type curves leadsfor lo test the following for using the curves analysis. Theliquid; procedure is through given for' slightly compressible Eqs. 4.6 4.16 show the

\.,' I' lft(p)=2dp. 'I' IIIJ.(p) Z (p)

(4.9)

changcs ncccssarywhcn a gas wcll test is analyzcd. 1 PIOl (p .- p ) vs I (drawdownlesl ) or (p .I vs. ale =all(1wf + alllp) .WS' (buildup tesl) on logPK1)

To use this result as stated would require that we log paper the same size as Ramey's type curve. (I) prepare a table or graph of values of 1ft(p) vs. P Caution: Unless a type curve undistorled in the from Eq. 4.9 based on the properties of the specific reproduction process is used, it will not have the gas in the well being tested; (2) plot [1ft(Pi) -same dimensions as commercially available graph Ift(P"1) ) vs. Ion log-log paper; and (3) find the best paper and finding a fit may be misleading or imfit just as for a slighlly compressible liquid (with the possible. The best solution is to use an undistorted values of s providing the best fit being interpreted for type curve; an acceptable alternative is to plot test the gas well as s + Dlq RI). data on tracing paper, using the grid on the distorted Sleps I and 2 can be simplified in some cases. Iype curve as a plotling aid. When ItZ is directly proportional to pressure (for 2. If the test has a uniform-slope region (458 line p>3, psia for some gases). Eq. 4.7 can be alearliestlimes).chooseanypoinl(l, (Pi-pwf)Jor replaced by the definition [61, (PM'S'-Pwf) ) on the unit-slope line and calculate -~h (Pi -Pwf) the wellbore storage constant Cs: pv, (4.10) QB( I 141.2QRIJ.iBRi

where

'1

-

50300 p scqg T , , s =s+D I qg'I

for a buildup

Thus, when p.Z= constanl, we can plot (pt -P~f) for lype-curve use. Match-point interprelalion b ecomes

kl

1ft0 = kh TSC" [1ft(Pi) -1ft (Pwf) J,

pressure

i ' I

test). In some other situations,.p.Z is constant (e.g., in

=PO (/o,ro,s,CsO) ,

..I... -_2' tPIJ.iC gir w

(4.12)

0.000264 k I u;~.(-) (4.13) IJ.i K' ID MP Nole that all gas properties are to be evaluated at originalreservoirpressureforatestinaninfinileacting reservoir (or, more generally, at lhe uniform reservoir pressure preceding the drawdown lesl or at

f/J<"ti-

10=

PD) , Pi -PK1 ~11'

tPCti=

where, for gases, 0.
(4.11)

Thus. when p.Zlp = constant, we can plot (PiPwf) for Iype-curve use juSI as for a slightly compressible liquid. Match-point interpretation is ,B '

." .can solutIons to now equallons for slightly compressIble liquids, lhey also can be used to analyze gas well tests. Transformation of the flow equations to model

" RB/Mscf.

the current

( r;;I )Mp.

~

-WELLTES

Cs=

-

24

)

Pi

-Pwf

unil-slopeline

' I

'

II

I

T ""'"

ANALYSISOFWELLTESTSUSINGTYPECURVES

67

TABLE4.1-CONSTANT.RATEDRAWDOWN TESTDATA '(hours)

Pwl(PSII

'(hourSI

Pwl

(PSI)

'(hours)

Pwl

TABLE4.2 -DRAWDOWNDATATABULATED FORPLOTTING (PSI)

P, 1(lloI.rs)

Pwl

P,

(psil

'(IIour~1

Pwl

P.

(11:'11

IIIIuur!;1

P." 111:'11

00109 0.0164

2.976 2.964

00218 164

2.611 2.693

218 273

1.734 1.768

00109

24

0164

301

218

1.232

00218 00273 00328 00382 00437 00491 00546 0109

2.953 2.942 2.930 2.919 2.908 2.697 2.686 2.765

0273 0328 0382 0437 0491 0546 109 164

2.536 2.469 2.408 2.352 2.302 2.256 1,952 1.626

328 382 4 37 491 546 655 874 109 ~64

1712 1.696 1.684 1.674 1.665 1651 1630 1614 1.587

00164 00218 00273 00328 00382 00437 00491 00546 0109

36 41 58 10 81 92 103 114 215

0218 0213 0328 0382 0437 0491 0546 109 164

389 464 531 592 648 698 744 1048 1.112

273 32U 382 431 4.91 546 655 874 109 164

1.21i6 1.2t1ti 1.:104 1.316 1.326 1.3:15 1.349 13/0 I.~ 1.413

Then calculate the dimensionless wellbore storage constant: 0.894 C C sD = -~h-T .c,

q = 4> = p. = =

500 STB/D, 0.2, 0.8 cp, lOx 10 -6 psi -I ,

, rw Note that estimates of 4>and c, are required a( this point -with implications that are discussed later. If a unit-slope line is not present, Cs and CsD must be calculated from if well bore properties, inaccuracies may result these properties do notand describe

r M' = h = Bo = Pi = .. Solution.

0.3 ft, 56 ft, 1.2 RB/STB, and 3,(xx) psia.

actual test behavior. 9 3. Using type curves with CsD as calculated in Step 2, find the curve that most nearly fits all the plotted data. This curve will be characterized by some skin factor, s; record its value. Interpolation between curves should improve the precision of the analysis, bu! may:prove difficult. Even for fixed.CsD from the unIt-slope curve, the analyst may expenence difficulty in determining that one value of s provides a better fit than another, particularly if all data are distorted by wellbore storage or if the "scatter" or "noise" that characterizes much actual field data is present. If CsD is not known with certainty, the possible ambiguity in finding the best fit is even more pronounced. 4. With the actual test data plot placed in the position fit, record values of (Pi -Pwf' of Pbest D) and (I, I D)corresponding from any convenient match point. 5. Calculate k and c, (Eqs. 4.4 and 4.5): k=141.2!!!!!--( h

ci>c,=

PD) Pi-Pwf

0.000264 k 2(-)'

, MP

I

p.rw 10 MP Eq. 4.5 does not eslablish 4>c,based on test performance unless it is possible to establish C V without assuming values for 4>C,-it simply reproduces those values assumed in Step 2. In summary, the procedure outlined in Steps 1 through 5 provides estimates of k, 5, and Cs.

Example 4.1- Drawdown Test Analysis Using Ramey's Type Curves I'rc)blem. Dctermine k, 5, and C s from the data below and in Table 4.1, which were obtained in a I1rcsslIrc drawdown tcst t}J\ ,lIt oil wcll.

We must first prepare !he ~ata for plotting (Table 4.2). The data are plotted In Fig. 4.5. From the unit-slope line (on which the data lie for I :S0.0218 hour), C -qB s -24

[

I (p. -P ). ] .. ' wf pomton unit-slope line

(500)(1.2) =

(0.0218) x

(24)

(47)

. = 0.0116 R8/psl. Then, C sD ='

0894 C 2s c,hrw

-(0.894)(0.0116) -(0.2)(1 x 10 -.5)(56)(0.3)2

,.. ..

= 1.03 x 103 =lxI03. For C sD = 103, the best-fitting type Cllrve is for 5 = 5.

,I'

A time match point is 1= 1 hour when ID=I.93XI04. A pressure match point is (p.pw/)=IOOpsi,whenpo=0.85.' .Fro~ the match, we also note that wcllbore storage dt~lortlon end~ at 1=5.0 ho\lr~ (i.e., the typt: ctlrvt: for CsO = 103 becomes identical to the type curve for CsD = 0). From the pressure match pl)int,

(

)

k= 141.2-qBIJ. PD h Pi -Pwf MP

(

0.85 = (141.2)(500)(1.2)(0.8) (56) ..00-

)

I '1

= JO.3md.

I~

--~

~

i(;"

,

;~

; " Ii

1

['C)'7W(ll~ /'~"[08_~

c.-,['r-

...

storage-dol11llKlted portloll or a test. tfle paralllctcr khAIIJlCs was much more important in determining

I

ii -IN' ~ 'Q.

-""':.':' ,'.. c"..,'n'

"'"" ,..

r

.', ..., ..,,",'y."'" \." "'llllIlitl\:U OllllUUP a/lU dra\\do\\'11 c~Jrvc,~~llo\\:d Illar. dllrillg tIle wcllborc-

!

df11qB than was the parameter kAllcPJlC,~. Accordillgly. he leI klp.ctr~,,=lOx 106 md-psi/cp-sq rt (all average value) ror all his type curves. It is important to emphasize that even when klcPJlc,r;.varies from this average value by one or two orders of magnitude. the shape of the type curves is not af-

J

fected significantly. The reason for this approximation was McKinley's judgment that the loss t

-1tJO ~ \ '- ~::~c

Fig. 4.5 -Drawdown curve.

of accuracy more than offsetthat by the theshape gain of in each sen-, sitivityin the istype curves-i.e.,

I C m FLON TIME hr .tflat test analysis with Ramey's type

curve is distinctly different at earliest times (Fig. 4.6). 4. To take into account the remaining parameters do have a significant influence on test results, McKillley plotted his type curves as AI (ordinate) vs. 5.615 C.fAplqB (abscissa), with the single parameter

I:rom thc lilnc match point,

kh15.615 CsJl. A small-scale version of McKinley's curves i~ .~hown in Fig. 4.6. 5. Note that the skin factor s docs not appear as a

O.2(>4 k I 1/>«= -~ (-) Jtr". I I) MP

parameter in the McKinley curves. Instead, McKi/llcy's curvc.'; a~~e~~damage or stimulation by nolillg that thc earlic.~t wcllbore-storage-distorted

I = (O.264)(10.3) (08)(0 3)2 (193 JO4 " .x

)

= I 96 x 10 -6 psi -I '.

t/J(',=(0.2)(1 x 10 -S) = 2 x 10

values in

s rype Curves

McKinley~ propo~ed type curves with the primary ohjective of cllara;:lcrizing damage or stimulation in a dra\\'down or buildup test in which wellbore storage dislorls most or all of the data, thus making thi~ dlaractcrization po~siblc \\'ith relatively short-term te~ts. In con~tructing hi~ typc Cllrves, McKinley observed that .toc ratio of pressure ~hange, df!, to flow rate callslllg the cflange, qB, IS a function of scvcral dimcnsionless quantities: I1IJ -=f

( k/lAI -,

kA.I -:-2:'

of thl~ quantity.

later,

after

well bore storage

formation, khlJl; this quantity also can be estimated rrom a type-curve match -but for the later data only. 6. McKinley approximated boundary effects by plotting

= values out. .,. 4.4 Mc Kiniry

data are dominated by the effective near-well transmissibility (khIJl) match of the.earliest ~ata in a wb; testthus, shoulda type-curve allow calculation distortion has diminished, the pressure/time behavior is governed by the transmissibility in ttle

Compare those with values used to determine C from C : sD s 6

..

r -.!.-,

AI

) -.and

qlJ ItC s cPJlc,rIt' r". IP Type curves with thi~ many parameters would be difficult, if not inlpossible, to u~e. Accordingly, McKinley sinlplified the problem in tIle following way. I. Hc a~~ul11cdthat thc \vell has produccd sufficicntly long (essentially to stabilization) that tIle last group, AI I II" is not imporlant. 2. He ignored bOUJldary effects except approxil1lately and, tflus, ignored r ('Ir It. in the basic logic u~ed to constrnct the type curves.

type curves for

c'!

about one-fifth log cycle beyond the end of wellbore storage distortion (where the curve has the same shape as for Cs = 0) and then making the curves vertical. This step roughly simulates drainage

the simulator-generated

",'

conditions of 40-acre spacing. Note that regions this gives the curves early-, middle-, and late-time -bul remember that the curves were designed to be used primarily to analyze earlY-lime data. When the curves are applied to drawdown tests, they"niusl be applied to early-time data only; they do not properly simulate boundary effects in drawdown tests. Use ()f McKinley's Type Curves Bcfore providing a step-by-step procedure for using McKinley's type curves, we note that he actually prepared three different curves: one for the time range

0.01 to JO minutes; Olle for 103 to J06

one for minutes.

:

I '

1 to J ,(xx) minutes; The curve for the

time range I to I,minutes is by far the most useful; accordingly, i! is the only one provided with this text. The complete set is provided with Ref. 1. The steps for using McKinley's curves follow. I. Plot A.I (minutes) as ordinate vs. Ap=P;-Pw/ (or PM~-PM'/) as abscissa on 3xS cycle log-log paper the sanle size as McKinley's type curve if uJldistorted type curves are used. Otherwise. use tracing paper for actual test data plotting. The time range on the axis should correspond exactly to one of the type curves (e.g., it should span the time range of

I

'M~

\'ILLt Il~l~

USING IYPE CURVES -69

I

0 ..,

0 ~

0

I

I f

1'

~ ~ii1

°

III

.11

.,

Ii

.

.i

11 f

-t

-1

. I

II!

I:

: ,I:

--.

I.'"

10-1

IS...

10-1

IS'"

I

10'

PRESSuRE 8UILOUP GROUP, ~ 6146 6p C. ~ q8

Fig. 4.6 -McKinley's

.R8

Iype curve.

l_minUIl'~t"O.OIIO 10minUlesor 103 to 106 J. ~,h fill' lillII.' axi~ of the lest data plot with ~ '\1..~llIlc~'~. ;\ltlVt: Ille dala along tile plOI I) tnu 1,"li(',,/ }h{jiill.!,' u/lol.'ed) untillilc 4&1, f.all illlllIg 0111.' of I hI.' typt: curves. 11«~\l111~ p..r.lml.'ll.'r \'alill.' (kh/JJ.)/5.615 C.\ "car~II)I~I.'urvl.'. -.(kh/I()j=(--) .~, ..I.IIJ 111-.1\.'11 pOInt (any ~p Irom tIle

transmissibility), shiff the data plot horizontally 10 find another type curve that better fils the later data. A shiff 10 a higher value of (kh/JJ.)/5.615 Cs indicales damage; a shift to a lower valLII.' indicale£. slimllialion. 8. Calculatt: formalionlransmis~ibilily: kh/J( x(5.615Cs)s,,:ps, 5.615 C5 SI.:p7

.'fIb I\IIlX"r illlli till.' l'()rrl:~pollding valliI.' of JitC.'IlHir,lllltIlI.'IYPl.'l:urve). IMnminl' till: \\l.'lll1orl.' ~Ioragl.' I:Onsl"11t C.\ ,.Iuc, of ~p:;: ~/IMP al1d 5. 615 ,'tlC~ hi S ~/'C °) IfIll) MI' al tile ma.ch poilll:

Nole Illal \\'1.'do I/O! find a nl.'\\' pre~sllrl.' m.ltcll point 10 redell.'rmine C5; Cs is found onct: and for all in SICp 5. In fad, if only d.lla rt:llel:ling formal ion tran~llli~,ibiliIY (afll.'r wl.'llborl.' ~Iorag\.' di~lorlion lIa~ di~appl.'arl.'d) arl.' analYlcd. errur will r\.'~ull l,~il1g IIII.'

"t. ~

.'61' -.x

\"C' /1/11).

"'"

.1I)~tt.

~It

(,1/1

McKil1ley

5.615

.Cawt" n~..r.\\1:11tran~mi~sibility. (kif/II.) K'b' ...f\AI~mctcl \O1lul.'rel:ordl-d in Step 3 and the IICW',C ,,)n~IO1ntdl.'ll.'rmined in Step 5: 4hl,.

,

(

Ik

(However.

collvemiol1al

melllod~

must be found with early, wellbore-storage-distorled data (Figs. 4.7 and 4.8). Flow efficiency also can be estimated fairly directly from the data plotted for use with MI:Kinl(y's tylX' curves.6 The definition of now efficiency, E, is

-) x5.615Cs'

...S.6IS CJ Mob .~ d.llol Irl.'l1d a~'ay from

method.

wor. kl Icre, so no pr.o bl eJu arIses. ' ) '. ' 1II.' mall: I I poInt '

the type curve

cilrli(,t fit (indicating thaI formation I)" i\ Jiff~rt:nt from effective near-well

..

E-"

-p'

P -Pwr-Aps Ap -4ps "~J -Pwf "~- -~.

The quantifies

.1p. and ~s

.

call be estimated

from

.

","'~U

~", the McKinley type curves in the (Fig. 4.9): I. ~e is the vertical asymptote in lhe McKinley plot. 2. Aps can be calculated from which the actual test data depart

~t (

)

fitting

k h/)J.

5.6/5

type

subjcctivc,

Cs

W

curve.

Picking

so no

grcat

a

following

approached by ~ Apd' the time at from the earliest-

time

accllracy

manner

of

departure

is as~lIrcd

for

is llli~

reason alone. McKinley6 states that Aps and Apd are related by

(

k wb Ap S= 1-- kf Apd'

~p

)

3. Thus, E can be calculated: Fig. 4.7-Early data fit on McKinley's type curve.

E = Ape -Ap S .

Ape

Example 4.2 -Drawdown Test Analysis Using McKinley's Type Curves Problem. &timate near-well and formation permeability and now efficiency using the data presented in Example 4.1 from a drawdown lest on an oil well.

~t (

kh/ Kn/Jj,

56/5 .5

0

)

S(tluCion. We firsl prepare lhose lie in lhe time range I minute
C f

data, most of which I , minutes for

plouing as ~ (minutes) Ys. ~ =Pi -Pwf (Table 4.3). From the dala plot (Fig. 4.10) and the match with the

best-fitling McKinley curve for the early dala,

~

( kh ) -x

P

1

= 5,.

p. wb 5.615 Cs Fig. 4.8 -later data fIt on McKinleys type curve.

We also note that the data depart from the bestfitting curve (early time) at ~/d = 100 minutes; here, Apd=I,180psi. A match point for the early fit is ApMP = 107 psi when 5.615 11pCs/qB=O.OIO. The best fit of the later dala is for (kh/lJ.)f( 1/5.615 Cs) = 10,. Thus, from the match-point data,

Cs--(5.615 i1pCs) ( ~ ) x --!!!!.- .. qB MP ~ MP 5.615 I

~t ---i)

I

td

'p*

/Y

=(0.010)

I ~ J\h

1

urd

1

~p

Fig. 4.9 -Data for flow efficiency calculation from McKinley's type curve.

( -! I )( 500X 1.2 ) 107

5.615

I

.I

=0.01 RB/psl The near-well transmissibility meability, kwL. are lhen

(~ ) p.

b W

= ( -~!!!-

and apparent per-

)

(5.615Cs) 5.615 Cs wb

= (5,
=281 md-ft/cp.

:f;I)1~

r .,.'J,;;.."t.~ ","~"o/ ri:I't3i1er: " ",

,'i'uf&ui\r::a\ly Cr.

-"c-,,;..,~, .

Then. k

b--- (281)(0.8) -..4 01 md (56)

w

~~1

Ii\1

ANALYSIS OF WELL TESTS USING TYPE CURVES

---:

---fd;

-~"I" "0000'

1~c,

C .E -TYPE-CURVE

C-=~T~E

71

Ij~

:

)

flIOM a!IGINAL MATCH

6l4 ' 100MIN 6P4'IIIOPSI

1500~

W .'

'

~ i=

TABLE 4.3- DRAWDOWN DATA FOR McKINLEY CURVE ANAL YSIS

:>

I

I~J'~,OOO

:>

~~c.

S LL '001 ~ I 0

~ 00

OX)

Pi

Fig. 4,10 -Orawdown curve.

-Pwf

' psi

test analysis

with

McKinley's

.11

P, -Pw/.11

(minutest

(pSI)

(minuiesl

P, -Pwl

(pSI!

(mlnules)

(P51!

131 164 197 229 262

47 58 70 81 92

164 197 229 262 295

464 531 592 648 698

197 229 262 295 328

1.288 1.304 1,316 1,326 1,335

295

103

328

328 654

114 215

654 984

984 131

307 389

131 ,~

..11

744

P,

P..,

393

1.349

1,048 1.172

524 654

1,370 1.3t16

1,232 1.266

984

1,413

type

The formation transmissibility and permeability are ( kh ) kh ) (khlll.x5.615 Cs)f -= x -flux II. f (khl II.x 5.615 Cs) wb II. wb

(

fractures with two equal-length wings were created. The ~urves discussed in this section assume '!"ijOffll Into the fracture (same flow rate per Unit crosssectional area of fracture from wellbore to fracture .:. tip). High fracture conductivity is required to achieve uniform flux, but this is not identical to an infinitely conductive fracture (no pressure drop from fracture tip to wellbore), as Gringarten el al. demonstrated. 7

-~~ x 281 -5 ,(XX) = 562 md-ft/cp,

08 = (562)( .-:.--) 56

The study was made for finite reservoirs (i.e., boundary effects become important at later times in the test). The reservoir is assumed to be at unirorm pressure, Pi' initiaUy. The type curve (Fig. 4.11), developed for a constant-rate drawdown test ror a slightly compressible liquid, also can be used ror buildup tests (ror .11maxsO. I lp) and for gas wells. using the modifications discussed earlier. Wellbore storage errects are ignored. .. All the dimensionless variables and parame(ers

-8 03 -.m.

considered important are taken into account ill Fig. 4.11, which is a log-log plot of Po vs. lor;; L} with

and kh kf= (-) x ~ II. f h

d

.parameter Flow efficIency becomes

xelLj- In these parameters, Lf is the fracture half-length and xe is the distance rrom the

.1p. ~ 1,500 psi (Fig. 4.10), k .1ps = (I --f) .1pt/ f

well to the side of the square drainage area in which it is assumed to be centered. Dimensionless pressure has the usual definition, P [) = kh(Pi' -Pwf) w (drawdown test),

141.2qBII. .1ps = (1- 401 ~)(1,180)

= 590 psi.

and

8.03 f l

I o~ 0,000264 kl -u- = ~..~ L2 =IOL/. (4.17) 1 IPlJCtL-f Several features of Fig. 4.11 are of interest: 1. The slope of the log-log plot is 1/2 up to 1OL ~0.16 for x!/~/> 1. This is linear flow. We ha~ shown that, In linear flow,

I 500-590 .= 0.~7. 1,500 4.5 Gringarten et al.7 Type Curves for Fractured Wells E~

Grin garten el al.7 hydraulically

fractured

developed

type curves

I

for 'ThIS slalemenl may be an orerslmplll'callOfl Some ~as wells exllltJ'l IlIlIe

wells

In

whIch

vertical

d"l"'ndellln"n.Darcyll"wlntllelracturt!.ulII'~t!llqUld~

2

~

I' I

I

..",

,L."""~U

I

0

' . .

. .

. ,

I

I

..

.

I

. ,

~

, ,

. '

.

.I0

,

f

I

I

I!, I

. ,

...

,

10... 10.1

10.'

I

Fig. 4.11 -GrillQallen et a/. Iy~ ullifo"" flux.

Pi

, ~,.

curve 'or vertically.

fractured

-PI,:r=C/I/I,

well centered

cycle type 2.

curve Select

data

plot

Altl ..10Ug

I

semllog

plot

1m.,>

is

a

1

1/210g1DL.

t

no

of

straight

for.\'('IL.r>S,

data line,

course,

when

I

a

Fig.

I

og-

4.11

signifyin.g

I l pOt

og

(p D

rad}al

vs.

log

both

Nole -PMj)

effecls

thus,

can

be

analyzed

thus conventional combines,

PII.r in vs. a single log

radial

regions

now

boundary Icngt

effects,

hs.

If

fract

1hro\tghout ncgligible allows

rather

Icn~tht

LJt

method

is

and

on

earliest

methods

and

t

poln

in

analysis

use

include Plot

the

(Pi-PMj)

Fig.

Three

fracture

I.

and

constant

storage then

of

a

and

this

permeability, the

a

type

figure

as

for

following. (drawdown

useful If

linear

or

(PM~-

[(PD)MP'

2.

~ ,

permeability

from

the

-.

from

kIMP

a

checks

]

are

half-slope

the

time

match

2

(4

data

plot, vs~

flow

..;t

somelimes

(linear replot (or

J 8)

~ata

~I

possible:

now)

);

region

from from

the the

appears region slope

on

as P wJ mL

and

theory, qB

r;;

-.:--~~, h"'L

dary test)

points

'MP).

IJ.CI (I DL f ) MP

which

test

match MP'

length

should

~I agree

with

the

result

from

the

type-

analysis. If

a

effects

radial

now

become

region important

datadeviatefromthex~/Lf=~curve),aplotofPwf

"

vertically.

~

PMOS)

curve curve

the

(Pi-PMj>MP

fracture

The

2.

of

tracing paper. the aclual test

-

test

(or

j

a 3 x 5

..

0.000264

LJIt=

nontype-curve

the

DLj)

formation oint:

4 064

fracture k.

of

on

version

use sliding

and

[(I

1 (drawdown

abscissa

undistorled

Otherwise, malch by

:

hydraulically of

to

the has

an

vs.

the

(PD)MP.

f-

neither),

estimation

4. I I

4.1 and J

wilh

data,

superior

L

various

analysis

in Chap. of

Fig. flow

the

of

wellbore

formation

discussed

Stcps

if

by

plots). linear

is high

specifically,

frequently

value

Eslimate

Fig. 4.11 can. flow region

region

effect

conductivity

complete

well-

a

the

test

effect a

fractured

J.

ure

the MP)

h

permeabililY

I or graph, Horner the

(and and

the

for

if

k=141.2~

Important,S.

wllh radial

ordinate

P

[ (and,

no wellbore storage.

on

horizontally

P

flow.when,

b~com~

test data data in lhe

the test)

paper

4. Eslimate ressure match

a

on

is availablc. the besl

Thestralght-llnet.ermlnates,

boundary actual of

(Pi

?n

In

1lJ/~,=2.

but a match of sho\v the amount

t

appar~n

lhe

square.

(buildup

log-log

3.

,

test)

~/t'

"z.

t

2

or

I

+

in closed

(buildup

test)

logp/J=logc'

10"

.,.C,L,.

PwJ)

p/)=C'I/)/., 111cn

G

I

or

of

10 000026370'

appears -that

(before is,

before

bounthe

'i '

ANALYSIS OF WELL TESTS USING TYPE CURVES

73

,

of;-

';;~ 1i;

""oil

U)

a. j

a. I

. ~

CTABLE 4.4 -FRACTURED WELL BUILDUP

I

TEST DATA

~ l

~~Lh,!u!S) 0

Pws -Pwl --(psi) 0

~t (hours) 0.75

0.0833 0.167 0.250 0.330 0.417

31 43 54 66 66

0833 0.917 1.00 1.25 2.00

100 100 100 114 136

0.500 0.583 0.667

72 78 83

2.50 4.00 4.75 6.00

159 181 206 218

Example fora

4.3 -Buildup

Vertically

Problem.

of xe to check the quality

10

Gringarten

01

MAltHfQHT A -01 P"-P:f'150~ :

of the

~Vell

q = 2.750 STO/D. p. = 0.23 Cpo 8 ~ 1.76 RB/STB. h = 230 ft, t/> = 0.3, and c( = 30x10-6 psi-I. S I r F 4 12 I f AI -..~ 0"100. tg. .lsapoto ...,}-PII'~-PII'1'V!i.ul. adequate fit is characterized by the maid. point!i (1=0.062 hour, IDL=O.OI) and (AjJ= 15.2 p!ii, Po =0.1). From the p/essurematch point,

.

.

.

k=141.2(i!~~~.~~!!!~t'-'-

el ul.13 presented build IIp tC!it

II

=

Producing time, 1p' was significantly maximum shut-in tIme, so that ~1=~le.

=4.5 md.

..

(~p) MP

(141.2)(2.750)( 1.76)(0.23)(0.1) (230)(15.2)

greater than

TABLE 4.5 -CONSTANT. RATE DRAWDOWN TEST DATA Pwl (psi) 2,976 2,964 2,953 2,942 2,930 2,919 2,908 2,897 2,886 2,785 2,693

t (hours) 0.218 0.273 0.328 0.382 0.437 0.491 0.546 1.09 1.64 2.18 2.73

II)

Fig. 4.12 -Buildup test analysis for vertically.fractured well with Gringarten type curve.

data for a well believed to be fractured vertil:ally. From these data, presented below and in Table 4.4, estimate fracture length and formation permeability.

t (hours) 0.0109 0.0164 0.0218 0.0273 0.0328 0.0382 0.0437 0.0491 0.0546 0.109 0.164

--

~ t , hr

Test Analysis

Fractured

TIM~O~;~~lroINT 6l"O062 Iv

Fff::~

vs. log 1 (pw!" vs. log ~I or log (lp+~I)/~I) should show that k = 162.6 q8p.lmh. in agreement with typecurve analysis. 3. If a well proves to be in a finite-acting reservoir. it may be possible to estimate xe from a matching parameter, xelLf' to compare with the known (or assumed) value match .An

2::.::

Pws -Pwl (psi) 89

Pwl (psi) 2,611 2,536 2,469 2,408 2,352 2,302 2,256 1,952 1,828 1,768 1,734

t (hours) 3.28 3.82 4.37 4.91 5.46 6.55 8.74 10.9 16.4 21.8 27.3

Pwl (psi) 1,712 1,696 1,684 1,674 1,665 1,651 1,630 1,614 1,587 1,568 1,554

t (hours) 32.8 38.2 43.7 49.1 54.6 65.5 87.4 109.2 163.8 218.4 273.0 327.6

Pwl (psi) 1,543 1,533 1,525 1,517 1,511 1,500 1,482 1,468 1,440 1,416 1,393 1,370

. :

c ,:;t;

TABLE 4.6 -BUILDUP

~I (flours)

Pws (psi)

~, (hours)

Pws (psi)

0.0 0.109 0.164 0.218 0273 0.328 0.382 0.437 0.491

1,370 1,586 1.677 1.760 1.834 1.901 1,963 2,018 2.068

0.546 109 164 218 2.73 328 3.82 4.37 4.91

2,114 2.418 2.542 2.602 2.635 2.657 2.673 2.685 2.695

Froln the time match point, 0.264kAt Lf=( MP

[

(0000264)(4

)

5)(0062)

.i

'12

]

(0.3)(0.23)(3 x 10- )(0.01) -59 7 f -.t. Exercises E .MBH xam~les 4.1 and 4.2 were based on a portIon of the followIng ~ala for a drawdown test followed by a J1rcs~llrc btliidup tcsl.

r

q = 5(x) STOll) (con~tanl), ct>= 0.2, Jl = 0.8 cp, ", = lOx 10-6 psi-I, .Pi = 3, psi, , '" = 0.3 fl, h = 56 ft, 80 = 1.2 RU/Sl.U, A 11"11 = 0.022 sq ft, p = 50 Ibm/cu ft, single-phase liquid, liquid/gas interface in wellbore, and well centered in a cylindrical drainage area with , t' = I ,ft. The drawdown (esl da(a are presented in Table 4.5; buildup lest dala (I = 327.6 hours) are given in Tablc 4.6. p 4.1 Using conventional analysis techniqucs (P"i v~.. log t p'IOl in ETR and MTR, PMi vs. I plot in I.IR). c~tlmatc k, S, E, tIl"II.~' Vl"'t (assumc cylindrical reservoir), and 'i allhe beginning and end

"

~I (ll0urs)

546 655 8.74 109 16.4 21.8 27.3 32.8 38.2

Pws (pSi)

2,703 2,717 2,737 2.752 2.777 2,793 2.805 2.814 2,822

.}I (flours)

437 49.1 54.6 655 87.4 109.2 163.8 218.4

2.828 2,833 2,837 2.844 2,853 2,858 2.863 2,864

the L TR be analyzed with these curves? Why? 4.3 Analyze the drawdown test data using McKinley's type curves. Estimate k, k wb' and E. Can the data in the L TR be analyzed with these curves? Why? 4.4 Analyze the buildup test using the Horner plotting technique. Estimate (I) k, s, E, 'wbs' and'i at the beginning and end of the MTR, (2) jJ from the and modified Muskat techniques, and (3) reservoir pore volume (using jJ before and after the drawdown lest). 4.5 Analyze thc buildtlp lest as completely as pos~iblc using Ramcy's typc curvcs. Is there a shut-in time, Almax' beyond which the type-curve technique should not be used? Why? 4.6 Analyze the buildup test using McKinley's type curves. Estimate k M'b' k, and E. Is there a shutin time beyond which the type-curve technique should not be uscd? Why? 4.7 In the buildup lest analyzed in Example 4.3, does a linear flow region appear? If so, analyze the data using the conventional equations for linear flow in a reservoir. Does a. radial flo~ region appear? If so, analyze the data usIng conventlon~1 methods: 4.8 A d~awdown test was ru~ In. a vertIcally fractured oIl well; the results are gIven In Table 4.7. Using the Gringarten et 01. type curve, estimate f.racture length and ~ormation pe!meability. I~ntify linear flow and r?dlal. now reglo?s and vertf~ the type-cur~e analysIs with conyentlo.nal analysl~ of these regions. As part of the conventIonal analysIs of the radial now region, estimate 'i at the beginning and end of the MTR and estimate fracture length from skill-factorcalculalion. Thclesl dala were as follows.

TABLE 4.7 -FRACTURED WELL DRAWDOWN TEST DATA t (hours)

'pWI (P~~

of the MTR for the drawdown test. 4.2 Analyze the drawdown test data as completely as possible using Ramey's type curves. Can data in

V2

ct>JlC I (t OL/) MP

='

TEST DATA

Pwl (psi)

, (hours)

Pwl (psi)

t (hours)

Pwl (psi)

0

4,000

1.5

3,932

20

3,823

0.15

3.982

2.0

3.922

30

3.803

0.2 0.3

3,978 3,975

3.0 4.0

3,907 3,896

40 50

3,789 3,778

0.4

3,969

5.0

3,886

60

3,768

0.5 0.6 0.8

3,965 3,960 3.957

6.0 8.0 10

3,879 3,866 3,856

80 100

3,755 3,744

1.0

3,950

15

3,837

-'"

I

ANALYSIS OF WELL TESTS USING TYPE CURVES

-r---"

q = 200 STB/D (constant). B = I 288 RB/STB h = 12f' t, 41 = 0.1. II- = O.S Cpo C = 20xI0-6psi-l.and I llb

~e

I ore

un

oa

d . Ing

d .. Istortlon

6. ~t..'J.;lnlc.':-,R.~I. "Esllmalinf fir-" Effl':lcn..'~ from Aflerno,,-Disloned Pressure Buildur Dala," J. Pt'l. T«h. (June 1974)696-697. 7. Grln~arten, A.C., Ramey, H.J. Jr., and Raghavan, R.: "Unsleady.Slate Pre~sure Distributions Cruled by a We]! With a Sinile Infinile-Conducti\ily Vertical FraCture," Soc. Pel.Eng.J.(AugI974)347-360;Trans,AI~IE,2S7. neg

I .. bl II 19l e at a

tImes. References I. Earlougher, R.C Jr.: Advances in H.ell Tesl Analysis, Monograph Series,SPE. DaJlas(1977)5. 2. Ramey, H.J. Jr.: "Shon- Time Well Tesl Data Interpretation in the Presenceof Skin Effecl and Well boreStorage," J. Pel. Tech.(Jan. 1970)97-104; Trans., AIME, 249. 3. Agarwa], R.G., AJ-Hussainy, R., and Ramey, H.J. Jr.: "An Investigation of Well bore Storage and Skin Effect in Unsteady Liquid flo~': I. AnaJ)1icaJ Treatment," Soc. Pel. Eng. J. (Sept. 1970)279-290; Trans.. AIME, 249. 4. Wattenbarger, R.A. and Ramey, H.J. Jr.: "An Investigation of Well bore Storage and Skin Effect in UnsteadyLiquid flow: II. Finite-Difference Treatment," Soc. Pel. Eng. J. (Sept. 1970)291.297; Trans.. AIME, 249. 5. McKinley, R.M.: "WeUbore Transmissibility from Afternow-Dominated Pressure Buildup Data," J. Pel. T«h. (July 1971)863-872; Trans., AIME, 251.

l

-75

8.

Agar"al, R.G.: "A Ne" Method Time Effects When Dra~do~-n

To Accounl for Producing Type Curves are Used to

Ana]yzc Pressure Buildup and Other Test Data," paper SPE 9289 presented at the SPE 55th Annual TechnicalConference and Exhibition,-held in Dallas, Sept. 21-24, ]980. 9. Ramey, H.J. Jr.: "PracticaJ Use of Modem Well Test Analysis," paper SPE 5878 presented at the SPE-AIME 51st Annual TechnicaJ Conference and Exhibition, New Orleans. Oct. 3-6, 1976. 10. AJ-Hussainy, R., Ramey, H.J. Jr., and Cra~.ford, P.B.: "The Flq~ of Real Gases Through Porous Media," J. Pel. Tech. (May 1966)624-636; Trans.. AIME, 2.31. II. Wattenbarger, R.A. and Ramey, H.J. Jr.: "Gas Well Testing With Turbulence, Damage, and WeUbore Storage," Trans., .AIME,(I960) 243,877.887. 12. Holditch, S.A. and Morse. R.A.: "The Effecu of Non-Darcy flo~' on the Behavior of HydraulicaJly Fractured GasWells," J. Pel. Tech. (Oct. 1976)1169-1178. 13. Gringanen. A.C.. Ramey, H.J. Jr., and Raghavan, R.: "Pressure AnaJysis for Fractured "'ells." paper SPE 4051 presented at the SPE.AIME 47th AnnuaJ fall Meeting, San Antonio, Oct. 8-11,1972.

Chapter 5

Gas Well Testing

S.l Introduction This c.hapte~di~cussesdeliv:rability tests of g.aswells. The dISCussionIncludes basic theory of transient and

I,(.() = 1,(.(-) -50 300 & P"'f P , T

!!.&:!. kh n r

rLl ( ~'" )

pseudosteady-state flow of gases, expressed in terms of the pseudopressure I,(.(p) and of approximations to the pseudopressure approach that are valid at high and low pressures. This is followed by an examination of flow-after-flow, isochronal, and modified isochronal deliverability tests. The chapter concludes with an introduction to the application of pseudopressurein gas ~ell test analysis.

Eqs. 5.1 and 5.3 provide Ihe basis for analysis of gas well tests. As noted in Sec. 2.11, for P > 3,CXX> psi, these equations assume a simpler form (in terms of pressure, p); for p < 2,000 fsi, they assume another

S.2 BasIc.'. Theory of Gas Flow In Reservoirs Investigations 1.2 have shown thaI gas flow in in-

procedures testswe with equations simple formfor(inanalyzing terms of gas p ).~'ell Thus, can develop intermsOf~(pj,p,andp2.1nmostofthisl.'haPter,finile-ac[ing

reservoirs can be expressed by an equation similar to that for flow of slightly compressible liquids if pseudopressure I,(.(p) is used of pressure:

our equations will be written in terms of p2 -nol because p2 is more generally applicable or more accurate (the e~uatio;}s in I,(.best fit this role), butnstead because the p equations illustrate the general

p

~(P"1)

q T[

=1,(.(Pj) +50,300-!f..:!:K.:.

Tsc kh

sc

]

-0.75 +S+D/qg /.

method and permit easier comparison with older methods of gas well test analysis that still are used

1.151

widely. Before

I .Iog(~

688

~#l..c.r I ~

2 )-(S+D/qg/)

] ,

kt

I,(.(p)=21

J

P -dp.

JJ.Z

Eq. 5.1 uniform

developing to drainage-area model a

the

equations,

drawdownpressure

test

let

us

starting that
generalize from may

any ~

much lower than initial pressure (p,) after years of

(5.1) ~Ierethe pseudopressure is defined by the integral ,P

(5.3)

production: ~(P"'f) = ~(jJ) + 50,300~

!ff sc

lr1.151

(5.2)

PJ

e term Diqgl reflects a non-Darcy flow pressure 5-i.e., it takes into account the fact that, at high near the producing well (characteristic of g~ gas production rat~s), ~arcy's law does not 'dJct correctly the relationshIp between flow rate pressure drop. As a first approximation, this litional pressure drop can be added to the Darcy's pressure drop, just as pressure drop across the zone is, and D can be considered constant. ~ absolute value of q" Iq I, is used so that the n D!q,! is positive for eitter production or inion. 'or stabilized flow3 (r, ~ r ~\

'Iog

(~8 (j)#l.DCtD~ kt )-(S+D/qg/)

].

(5.4) where p=jJ for ail r at tp=0. For p<2,1XX> psia, pZg ~constant =.upZpg for most gases; in this case,I 2 2 2 ~(p) = -(~ -~ ). #l.pZP8 2 2.:red Subslituting into Eq. 5.4, q #l.-Z- T Pwf2 =jJ2 + 1,637~X"'~~2I..:. r log( '1,688~#l.-c ~'---,~"'D.t~

-

)

! j '

GAS WELL TESTING

~IJ

_ ( s+Dlq.rI

1.151

)]

(5.5)

For stabilized now,

~4

I

[ ln(!L) rw

p",j2=p2_I,422q.r~pZP.rT kh -0.75+S+Dlq,l}

(5.6)

qg

~I

Eq. 5.6 is a complete deliverability equation. Given a value of flowing BHP, Pwj' corresponding to a given pipeline or backpressure, we can estimate the rate q, at which themust well be willdetermined deliver gas. However, certain parameters before the equation

I t,

p~-pwj2=aqg+bqg2,

~ -Z -T a=I,422"'P..pg-ln kh

[

(5.7)

(

r -!-rw

)-

-0.75+s,

J

(5.8)

t2

T

! i'

~

I

f?vf Wf ,4 tl

~-z-

l4

p

and

b=I,422"'P~pgkh D.

t3

t

can be used in this way: I. The well flowed at rate q until rj ~r e (stabilized flow). In this case, note ttat Eq. 5.6 has the form

where

77

t

(5.9)

The constants a and b can be determined from flow tests for at least two rates in which q and the corresponding value of P wj are measure~; p also must be known. 2. The well flowed for times such that rj ~r e (transient flow). In this case, we will need to estimate kh, S, and D from transient tests (drawdown or buildup) modeled by Eq. 5.5 (or some adaptation of it using superposition); these parameters then can be combined with known (or assumed) values of p and rein Eq. 5.6 to provide deliverability estimates. The gas flow rate qg' used in Eqs. 5.1 through 5.7, should include all substances that are flowing in the vapor phase in the reservoir, with their volumes ~.xpressedat standard conditions. These substances Include the gas produced as such at the surface, and condensate and liquid \\'ater produced at the surface that existed in the vapor phase in the reservoir. ~alculatio~ of the vap,or equivalent of condensate is discussed In Appendix A of Ref. 4. Craft and Hawkins5 summarize the calculation of the vapor equivalent of produced fresh (non formation) water. Most of the remainder of this chapter provides detailed information on testing procedures that lead to estimates of the parameters required to provide deliverability estimates- This discussion is based on recommendations in the ERCB gas well testing 5.3 manual.4 Flow-Arter-Flow Tests

In this testing method, a well flows at a selected constant rate until pressure stabilizes-i.e., pseudosteady state is reached. The stabilized rate and pressure are recorded; rate is then changed and the

l2

t3

4

.

Fig. 5.1 -Rates and pressures in flow-after-flowtest.

+ : i p2 -P

2 wf

~~:~!_~~.! I I ,.'i,! iSlOPE ;; STAB" E '-'I"", : -!~L'".E~A~llITY I : :uRvE i :; ;/ : ..-,_AesCL~E OPEN.. ,.~ FLOW JTE~TIAL rAQF)

a -g Fig. 5.2-Empirical del!verability plot for flow.after-flow test,

-

~ 78

WELL TESTING

TABLE 5.2 -STABILIZED FLOW TEST ANAL YSIS TABLE5.1-STABILIZEDFLOW TEST DATA ~ 1 2 3 4

P., (PSIa) 403.1 394.0 378.5 362.6

P., ~ 4082 4031 394.0 378.5 362.6 14.7

qg (MMscf/DI 4.288 9.265 15552 20.177

well flows until the pressure stabilizes again at the new rate. The process is repeated for a total of three or four rates. Rates and pressures in a typical test follow the pattern indicated in Fig. 5.1. Two fundamentally different techniques can be used to analyze these test data. Empirical Method An empirical observation -with a rather tenuous theoretical basis -is that a plot of ~2 = p2 -Pwl vs. q (Fig. 5.2) on log-log paper is approximately a straig~t line for. many wells in w~ich the pseudosteady IS reached at each rate In aline flowafter-flow teststate sequence. The equation of the in this plot is q g = C( yn1- p w12 ) n

(5 10)

This plot is an empirical correlation of field data.

Example

, !

!

Unfortunately,

such an extrapolation

5.1-

Stabilized

stabilized (r/~rt') during the testing period used to construct the plot. If this is not the case, stabilized deliverability estimates from the curve can be highly misleading.

Flow

(p2_p2.,)/qp (psia2/MMScf/D) 9649 1,229 1.502 1.742 -

Test Anal

I. Empirical Method. From a plot of (jJ2 -Pwf) vs. qg on 10g-lo~ paper, and extrapolation of this plot to p2_Pwj =166,411 (where Pw1 =0 Psig or 14.7psia),AOF~60MMscf/D. The slope of the curve, l/n, is log IV~ (p2 \P -P -Pwj 2)12 -log -IV& (jJ2 \P -P -Pwf 2)I(

I/n=

is

frequently required. To estimate the absolute open flow potential (AOF) -the theoretical rate at which the ~'ell would produce if the flowing pressure P"'j were atmospheric -it may be necessarv to extrapolate the curve far beyond the range ortest data. An AOF determined from such a lengthy extrapolation may be incorrect. The constants C and n in Eq. 5.10 are not constants at all. They depend on fluid properties that are pr~ssure (and, th~s, tim.eJdependen.t. According,ly, ,if thIs type of dellverablllty curve IS used, perIodic retesting of the ~ell will sho~' changes in C and perhaps in n. We must emphasize that deliverability estimates based on this plot assume that pressures were

p2_p.,2 (pSia2) -4.138 11.391 23.365 35.148 166.411

y:sis Problem. The data in Tabl.e 5.1 w~re reported for a flow-after-flow (or four-point) test In Ref. 6. At each rate, pseudosteady ~tate was- reached. Initial (i.e., before the test) shut-In BHP, p, was determined to be 408.2 psia. Estimate the AOF of the tested well using (I) the empirical plot and (2) the theoretical flow equation, In addition, plot deliverabilities estimated using the theoretical equation on the empirical curve plot. Solution. We prepare a table of data (Table 5.2) to be plotted for .,both empirical and theoretical analyses.

As in any other empirical correlation, there js substantial of error extrapolating a large distance risk beyond the in region in which the dataplot were obtained.

qp (MMscf/D) 0 4288 9265 15.552 20.177 AOF

log qg.2 -log qg.1 I og = log

(

105

103

)

( ~1.77 )

= 1.449.

Thus, n = 0.690. Then, q C= -2"Og 2 n (p -Pwf )

= Th

us,

..

42.5 ~ 0690 = 0.01 508. (10-) , th

., ld I. b.r .. e empmca e Ivera I Ity equatIon IS

01 ~08{,;2 -2 Pwf )°.690. qg = 0 .-V'

Theoretical ~ethod

These data are plotted in Fig. 5.3.

Eq. 5.7 suggests that we plot (p2 _p",j2)/qg vs. qg; the result (for pseudosteady-state flow) should- be a straight line with slope b and intercept a. Becausethis line has a sounder theoretical basis than the log Ap2 -log qg plot, it should be possible to extrapolate it t.o det~r!Dine ~OF with less erro~ and to correct dellverabll!ty estImates for changes In JAp' 'Pi' etc., more readIly.

2. Theoretical Method. The theoretical deliverability equation is (p2 -Phf2)/qg

=a+bqg'

Fig. 5.4 is a plot of(jJ2 -Pwl)/qg vs. qg for the test data. T~'o points on the best straight line through the data are (2.7; 900) and (23.9; 1,900). Thus,

' i ;

..

~

GAS WELL TESTING

79I

I, I

0

'

!

L1::

U

I

U')

..:::

"

"""

II

THEORETICAL ACF

-"..

PwI.14 7 DO'.

N

N

I

-!

I

"+~

;

I

j

I

It

I

0-

I

-I 0"

r

N

i

I

~

I~

IC-

!

!

!

I

!

I ;

I,

I

.i

NI

'

VI

~

.i-I I

IJ

tt

~ :::== ~

I I

2 I~

8.

j

~ 00::::

! i

!

I~

I: (\J

10-

'-'"

I .

I

! "EMPIRICAL ACF I /

"

()

qg

100

,MMSCF/D

Fig, 5.3-Stabilized gas well deliverability test,

900 = a + 2.7 b, 1,900 = a + 23.9 b. Solving for a and b, we find that a = 773 and b = 47.17. Thus, the theoretical deliverability equation is 47.17 q g 2 + 773 q g = (p2 -P wf2).

0

4

1

qg

16

20

24

,MMSCF/D

Fig. 5.4-Stabilized deliverabilitytest, theoretical flow equation. constantdetermination,

lower-permeability reservoirs. where it frequently is impractical to achieve 'j =, l' during the test. An isochronal test is conducted by flowing a well at a fiXed rate, then shutting it in until the pressure builds up to an unchanging (or almost unchanging) value, jJ. The well then is flowed at a second rate for the same length of time, followed by another shut-in, etc. If possible, the final flow period should be long enough to achie..e stabilized flo~'. If this is im-

5.4 Isochronal Tests The objective of isochronal testing 7 is to obtain data

possible or impractical, it is still possible to predict the stabilized deliverability characteristics (with increased potentiaJ for error, of course). In obtaining data in this testing program, it is" essential to record flowing BHP, P \of' as a function of time at each flow rate. Fig. 5.5 illustrates rate and pressures in an isochronal testing sequence.This figure illustrates the following important points about the isochronal testing sequence. 1. Flow periods, excepting the final one, are of equallengt~ [i.e.,.fl = (f3 -f~) = (f 5.- f~)~ (/7 -16.»)' 2. Shut-In penods have the objectIve of letting P ~ jJ rather than the objective of equal length. Thus, in general, (f2 -fl) ~ (f4 -f3) ~ (/6 -f5)' 3. A final flow period in ~'hich the well stabilizes (i.e., 'j reaches, t at time I,) is desirable but not essential.

to establish a stabilized deliverability curve for a gas well without flowing the well for sufficiently long to achieve stabilized conditions (r j 2::,l') at each (or, in some cases, any) rate. This procedure is needed for

The most generaJ theory of isochronal tests is based on equations using pseudopressure. However, we will once again present the theory in terms of the low-pressure approx.imations (0 these equations ~

We can solve this quadratic equation for the AOF: 47.17qg2+773qg-166,411=O. The solution is qg = AOF = 51.8 MMscf/D. We also can determine points on the d~liverability curve as calculated from the theoretical equation: p2 -P\of2 = 47.17 qg 2 + 773 qg' See Table 5.3. These results are plotted In Fig. 5.3. The plot is almost linear, but there is sufficient curvature to cause a 15.8070error in calculated AOF.

-".-

P

80

WEll TESTING "

I

q

q q qg

Q.

l, TABLE 5.3 -THEORETICAL DEUVERABIUTIES

qg

(MMsC/D)

11,210

(

to. t2

l3

23,430 34,800

30

65,640

40

106,400

49.8 = AOF

166,600

t6

t7

t.

ls

l6

t7

W

4,182

9265

t

~ f

(psia2)

15552 20 177

l.

t

p2 -Pw/2

4.288

l2 t,

t Fig. 5.5 -Rates

~

and pressures

In isochronal

test.

equations) because (I) they are somewhat simpler and less abstract than equations in pseudopressure and (2) they allow direct comparison with more conventional analysis methods 7 based on plots of

points (q , p2 -PWf2) obtained at that time at several 'different rates, and a truly stabilized deliverability curve can be drawn when r j ~ r ~. These assertions can be made more quantitative if

(jJ2 -Pwf2) vs. q on log-log paper. Eqs. 5.5 and !.6 provide the basic method for interpreting isochronal For transient flow (rj tests. < r ~),

we note that for flowing time t, at each rate, there corresponds a drainage radius, rd=crj, that is independent of rate. Admittedly anticipating a log con-r venient result, we let r d = 1.585 rj (but the p vs.

2 -2 p.-Z -plot P wf = P + 1,637 ~ Tq g

constructed in the solution to Problem 1.2 shows that such ~ r d approxi~ates qu.ite.clos.ely the .point beyond whIch no apprecIable fluId ISbeIng drained). At time tt,

l

)_(~..:.!!lg-zJ )J . lOg ( 1,688
2- -2 ~ Pwf -P +1,422 kh

[

(5.5) For stabilized flow (rj ~r ~),

.T

-.-Tq '..g Pwf-., =jJ2 -1,422 p. '-P"PE

kh

I ,'

[In( -!-)r

; ~ I

Tqg

" I

I 688'"

In (.,

-(S+Dlqgl)

-c

-r

2

I

"p.p~/p.w ) kll

A

I

J.

rw 1

-O.75+(s+Dlqgl).

Because

I

(5.6),

J

"9,

rd2=(1.585)-r;2

In addition to the flow equations, an important theoretical consideralion in understanding isochronal tests is the radius-of-investigation concept. We observed previously that the radius of investigation achieved at a given time in a flow test is independent of flow rate and, thus, should be the same at a given time for each flow rate in an isochronal test. Further, the radius of investigation at a given time can be considered to be proportional to a d~ainage radius at that time, because it is near (but slightly less than) the point beyond which there has been no appreciable drawdown in reservoir pressure a?d thus no fluid drainage. Accordingly, at a given tIme, the same portion of the reservoir is being drained at each rate and, as a good approximation, stabilized flow conditions exist to a point just beyond r=rj. Thus., a deliverability curve can be drawn at each fixed tIme (hence, the name isochronal) through

".,1"-

..I

<2 = (1.58~) kt I

! !

948 (j>p.pCtp -kt

-,

j i

I 377 (j>ppctp

we may write Ihe transient flow equation as 2 -2 "PpZpg Tq, r 'rdl Pwf = P -1,4,,2 kh lln(rw

)

. -O.75+(s+Dlq

,

OJ.:

Compare with the stabilized deliverability equation: 7j Pwf2=p2_I,422PpZPb.q~ [ ln(.!.!kh rw )

I

1:'.11

GAS WELL TESTING

[

81

-2

]

-2

P

TABLE

5.4-ISOCHRONAl

TEST

Pw/ DuratIon

---I

Test

I

SL

Initial

I I

'+-

~

=

I

POINT

I

I

N

t2:

10-

I

t I

shut-In

First First

l3~ I

~

Qg a)

(MMscf/D)

48

1.952

-

12 15

1.761 1.952

2.6 3.3

flow shut.in

Second

flow

12

1.694

Second

Shut.in

17

1.952

-

12

1.510

5.0

Third

flow

Third

shut-In

18

1.952

-

flow

12

1.320

6.3

1,151

6.0

1.952

-

Fourth

~

Extended

I

Final

flow

(stabilized)

72

shut-In

100

: I

-:

~g

=

AOF

TABLE

5.5

-ISOCHRONAL

qg

9

506

-Empirical

deliverability

-0.75

( s+D

+

plot

are

isochronal

stabilized

identical

in

form

because

we

have

-2

(

)

kt

709.000

273.000

941.000

285.000

Y2

5.0

1,530.000

306.000

6.3

2.070,000

328.200

~d

transient

stabtltzed

-2

flow

Pwf

written

in

the

form

flow.

.

-+b

P

)/qg

(psia2/MMscf/D)

2.6

For

definedatime-dependentdrainageradius,rd'as

YSIS

(~2 -Pw/2

(psia2)

flow

.below.

ANAL

3.3

test.

I)]

I

q,

equations

for

TEST

~2 -Pw/2

(MMscf/D)

q

10 9

The

Pws

(pSi

I

0

Fig.

or

(hours)

.-

I-STABILIZED

N

DATA

-(14.7)

2

-oq,

(5

7)

;

qg'

"

where

.

, rd=

.. 3774»#J.pctp

Thus,

we

analysis

of

should

that,

different

be

except, tb ' l o s a I Ize

b

'

stabilized

as

the o

Ivera

l

I

data

each

fixed

used

just

course, e

at

rates

possible

of ddl

#J..Zo=1422"'P~pg.ln--O.75+s

0

conclude

it

in

is

Th

are

available

test

yield

o.

a .

IS

IS

or

an

poSSI

if

bl

e

they

nl

[

(

r,

)

]

5.8

r w

,(

:, )

.'

d -an

*

truly 0

T

.kh

test

stabilized

not

curve.

t I'

isochronal

a

will

o

Ity

an

for

data

tIme

#J.pZpg '

y

I

can

be

f

b=I,422

.. (5.9)

kh

.

F

estimated.

TD

or

fl

transIent

ow,

," p2_Pwf2=Otqg+bqg2, Analysis One

of Rate

Test

7

results

can

using

the

I.

time

be

<.02

paper,

just

each 3.

qg)

empirical

b

and

rates

as

used

data

fixed

in

are

when

through

a

the

value

the

should the

and

isochronal

be

drawn

slope

tin

on a

for

with

the

the

Once

the

single

be

Iin

stabilized

given

flow

by

..

2

I( n

)

kt 1,688

+

4>#J.pcrprw2

s.

]

Thus.

values

of

established

p2_PWf2=0/qg+bqg2,rj
and from

stabilized established

the

the

IS

(qg'

p2

drawn

curve usual

way,

P

curve. is as

An

determined,

with

indicated

in

method based

on

for the

theoretical

analyzing

._~ =oqg+bqg

isochronal equations

<.02 2.

for

Pw!~]'

analysis the

I. of

is

22 -Pwf

,rj:2:r,.

(5.7)

-Pw!)'

deliverability

deliverability in

then

point,

stabilized

(5.11)

for

determined

curves

stabilized the

theoretical data

for is

(5.12)

5.6. The

[~

.kh

-2-

establishes

test

time,

log-log

curve. slope

fIXed-tIme

This

is

as

of

422!.£!:iK!.

0/-

...0

Fig.

meaning

of

stabilized

several

should

deliverability

line

nonstablllzed,

AOF

same function

curve.

t,

through

a

isochronal

plotted

analyzing

the or'

method

drawn

at

has

where

-I is

obtained

The

Lines

A

satisfactory

the

line

different

deliverability 2.

with

straight

the programo

where

reasonably

procedure.

-Pwf2,

with

Stabilization that

obtained

best

testing

time

to shows

following

The

points

Data:

Continued

E.xperience

(5.11)

method

of

theoretical

For

a j"rxed

-Pwf2)/qg Using

value

of

vs. the

determine

isochronal

equations

consistent

t,

determine

b

from

a

plot

q-K'

stabilIzed 0

tests

follows.

data

point

{qgs'

<.02-

from

.

.~

I.

GAS WELL

TESTING

83

/

t S'

l

q 9

...... ~S

I= S

__~~__r~4 EXTEND q, q iW PERIOD

ql P" P~L

Pws

0

!?,s3

~S4

Pwf2

~~~

Pw

qg

PWf3 ~~S(STA8LE)

~

TIME Fig.

5.9 -Skin

factor determination,

Fig.

where 1,422

"'p~pg'

T[

In(-!-)-O.75+s,

kh

r

]

following (5.8)

rw

and -1l-Z -TD b= 1,422-'""P~pg' (5.9) kh We also noted that the transient equation has the form p2

and pressures

in modified

isochronal

dra",'do\\n le~ts run at different rate~ or buildup lests Il'Z'

a=

5.10 -Rates test

.

-pwf2=a{Qg+bQg2,

(5.11)

where

a {,= I 422!:J!..~ kh

then

dra",'do",'n plot

s'

tests vs.

QI!;

kt 1,68841p.-c{-r

2

)+ ] S

'

The objective of determining Ihe stabilized flow equation I:an be a~hieved if the constants a and b can be delermined. We note that Ihe ~onstant b ~an be determined from the iso~hronal test data as illuslrated [b~' plolling (p2 -p..,{2 ) /QI! vs. QR for fi,xed value of ( and determining the slope. b. of the resulting beSI straighl line]. Constant a, ho\\ever, is more troublesome: The onl~' salis factory means of determining a is through knowledge of each term in Ihe defining equation for Ihis quantilY. Thus, we need estimate~ of kh. s. and r 1" (Other quantities in Eq. 5,8 are usually available.) Since an isochronat test-consisis of a series of drawao\\n and buildup lests, kh and s usually can be determined from them. Delermination of kh is straightfor""ard in prin~iple; determination of s is less straighlforward. Recall that a single test provides only an estimate of s' =S+DQRi. Accordingly, 10 determine s, we must analyze at least two tests: either

q'!

We =0

~an

pro\'ides

Isochronal

10 be plolted

.

h t

e

~ometlme~

I

hh engt

~an

be

Tests

The objective of modified iso~hronal oblainlhesamedataasinanisochronaltestwithout u~lng

(5.8)

rates. 10

QI! also is desired. data points determined from theequalion.

ppw ,

different

an eslimate of true skin fa~tor. s (~ee Fig. 5.9). The drainage radius r e mu~t be estimated from expected ""ell spa~ing (or kno",'led~e of re~ervoir geometry in a small or Irregular reservoir). The ~onstants a and b determined in thi~ way then ~an be substiluled inlo the stabiliz;d deli~erability equalion. Eq. 5.7. If a plol of log (p- -P'I'f-) vs.log

5.5 Modified

[ ~2 In(

al

extrapolation

y

' ~

ut-in

' perlo

tests d

is to .

s

requlr

ed

for pre~~ureto slabilize complelel~' before ea~h no",' lest is run, -. In the modified isochronal test (Fig, 5.10), shut-in periods of the same duration as the now periods are used, and the final shut-in BHP (P..'s) before the beginning of a new now period is used as an approximation to p in the test analysis procedure. For exam~le. for the first now period, use (i>2Pwf-' )=(PwS,12 -tWf,,2); for the. second flo:-" perIod, use (P~,2 -Pwf.22). OtherwIse, the analysIs procedure is the same as for the "true" isochronal test. Note that the modified isochronal procedure uses approximations. Isochronal tests are modeled exactly by rigorous theory (if reservoir and fluid properties cooperate); modified isochronal tests are not. However, modified isochronal tests are used widely because they conserve time and money and because they have proved to be excellent approximations to true isochronal tests.

84

WELL TESTING

TABLE 5.6 -MODIFIED ISOCHRONAL TEST DATA

P.,

Duration (hours)

Test Pretest shut.in

First flow First shut.in Second flow Second shut-in Third flow Third shut-In Fourth flow E~tended flow (stabilized) Final shut-In

or P.. (psia)

20

1.948

12 12 12 12 12 12 12 81 120

1.784 1.927 1.680 1.911 1.546 1.887 1.355 1.233 1.948

qg (MMscf/D) -

4.50 -TABLE 560 6.85 -(MMscf/D) 8.25 8.00 -6:85

5.8 -THEORETICAL STABILIZED DELIVERABILITIES

qg

p..

J.P~

(psial

(Psla2)

4.50 560 68S

1.948 1.927 1.911

uBi 1.680 1546

612.048 890.929 1,261.805

136.011 159.094 184205

i:L:: U (/)

8.25 8.00

1.887

1.355

1.724.744

209.060

~ ~

.HOle (stabilized) thaI I' test

P., Pw.2_p.,2

the1.948. true

current1.233 reservOIr

2.274.415 pressure.

(Pws2-Pw,2)/qg

IS used

lor

the

p2 -P ,2 (Psla~)

~.~

1 ~.~ 1:794:000 2.274.000 3.660.000

8.0 10.8

TABLE 5.7 -MODIFIED ISOCHRONAL TEST ANAL YSIS

~~~}

qg

0

stabIlIZed

N

analysIs


V5

.I

Q.. C'

I

-~

I

N

"--_!!'_:'!.~_7.._~!..~ :

N

~I

I


'

: :

U)

i

:

Q.. N

T)£OfIET

I

Iv-

! :

"

:

.',I

: TRANSIENT: DELIVERASILlTY :

:

0

2

.6

i

:

:

i

10'1

., I :.. ; ACF. 108MMSCF/D

~VM~r-,~g qg'

g

.

.

-

MMSCF/D

I

I

STASlL'ZED: i OEUVERAel-rrY

N ." ~ ",

~ QS.

.q

,

~ I

I

:

-..a'

-'

NIJ)

Fig.

5.12 -Modified isochronal test analysis. theoretical flow eq uation, constant determination.

this value

,

For

i

con~tant

~

MMSCF/D

Pwf b= using

Fig.5.11-Modifiedisochronaltestanalysis.

a=..

IS A

the

OF

= 1 0 .sc. 8 MM

theoretical

b

from.

the.

slope

)/qgvs.qg;mthlscase, 243,(XX} -48,(XX) 10 data from ( -2 IP -Pwf --J~

flD

method,

we

of

a

establish

plot

of

the


= 19.500,

..

Fig. 5.12. Then, 2 ) b 2 Js-vq.s -~.

qgs -2,274,415

Example

5.3 -Modified

Test Analysis Problem. Estimate 5.6 obtai!1~d both emplTlcal

the

Isochronal

AOF

in a modifi~ and theoretical

from

Solution. We first prepare (Table 5.7). Fig. 5.11 shows

same point.

the

slope

the data the data

of the curve,

and

in Table

test,4

empirical method. This is a plot of
-8.0

the data

isochronal methods.

for riot

using

plotting for the

-Pwf2) vs. are used to

a line

with

the

slope is drawn through the single stabilized The AOF is the value of qg when Pws 2 -p,":~2

= p2 -Pw!

=

1,9482

-14.72

=

-(19,500)(8.0)2

3,790,(XX)

psia;

= 128,300. Thus. curve

is

the

equation

p2 -PWf2 Solving PWf2

this

stabilized

the

stabili

= 128,300

qg + 19,500

equation

for

= 3. 790.(xx}),

MMscf/D. It is also

of

of

the

we find interest

deliverability

to curve

ze

qg2.

AOF

that

d d r b1" e Ivera I Ity

(q

when

it is lqual

calculate

points

and to plot

them

p2 -

to on

11.0 the

on the

GAS WELL TESTING

'...'1111-

85

TABLE 5.9 -GAS PROPERTIES FOR EXAMPLE 5.. P

1'9

pllAZ

(psia) (Cp) z --;-so 0:01238 ~ 300 450 600 750 900 1,050 1,200 1.350 1,500 1,650 1.800 1,950 2.100 2.250 2.400 2.550 2.700 2,850 3.000 3,150

0.01254 0.01274 0.01303 0.01329 0.01360 0.01387 0.01428 0.01451 0.01485 0.01520 0.01554 0.01589 0.01630 0.01676 0.01721 0.01767 0.01813 0.01862 0.01911 0.01961

TABLE 5.10 -PSEUDOPRESSURE FOR EXAMPLE 5..

0.9717 09582 0.9453 0.9332 0.9218 0.9112 0.9016 0.8931 0.8857 0.8795 0.8745 0.8708 0.8684 0.8671 0.8671 0.8683 0.8705 0.8738 0.8780 0.8830

~(P}

(psia/cp) 12.290

P (pSia) 150

24.620 36.860 48.710 60..70 71.790 83.080 93.205 104.200 114,000 123.400 132.500 140,900 148.400 154,800 160.800 166.200 171.100 175.200 178,800 181,900

300 450 600 750 900 1.050 1,200 1,350 1,500 1.650 1,800 1,950 2,100 2.250 2.400 2.550 2.700 2.850 3.000 3,150

For p = 150psia '

5.6 Use of Pseudopressure

1J(150)= 2JP

[

E. + E. =2 i~~£.~~~~J 2

Calculation of Pseudopressure Gas pseudopressure, y, (p) , is defined by the integral .p pj

p -dp,

2 - 1 844 -.x

1J(300)= 1.844 x 106

IlZ

low base pressure. To

+2

evaluate .",{p) at some value of p, we can evaluate the integral in Eq. 5.2 numerically, using values for p. and z for the specific gas under consideration, evaluated at reservoir temperature. An example will illustrate this calculation. 5.4 -Calculation

of Gas

bl

C

I

h

d

( 12,290 +2 24,620 )(300-150)

6 .2 = 7.381 x 10 psla /cp. .., . ProceedIng m a similar way, v.:ec~n construct Table 5.10. These results are plotted m Fig. 5.13.

Transient 1

..

Drawdo~n Test Anai)'sis Using Pseudopressure

Pseudopressure P

106 pSI.2 a / cpo

For p = 300 psia,

{5.2)

where PJ IS some arbitrary

Example

(150-0)

= 2[0 + 12,290] (150)

cludmg systematic development of working equations and application to drawdown, buildup, and deliverability tests, is provided in Ref. 4.

y,(p)=2~

E. dp

P~ ilZ

Accuracy of gas well test analysis can be improved in some cases if the pseudopressure y,(p) is used instead of approximations writt~n in t~rrns of p~essure or pressure squared. In thIS sectIon, we discuss the calculation of pseudopressure and provide an introduction to direct use of this quantity in gas-well dra~down test analysis. Detailed discussion, in-

.I. {

) f

t

ro em. a cu ate t e gas pseu opressure 'Y p or a reservoir containing 0.7 gravity gas at 200"F as a_. function of pressure in the range ISO to 3, ISO psia. Gas properties as functions of pressure are given in

,

Table 5.9.

:

7.381 X 106 1.660 X 107 2.944 X 107 4.582 X 107 6566 X 107 8.888 X 107 1.154x1OS 1..51x1OS 1.779 X 108 2.135x108 2.518x 1OS 2.929 X 108 3.363 X 108 3.817 X 108 ..291 X 1OS 4.781 X 108 5.287 x 108 5.807 X 108 6.338 X 1OS 6.879 X 108

empirical plot.plotted The values are given in Table 5.8. Thesedata data are in Fig. 5.11.

in Gas Well Test Analysis

(pSia2/cp) 1.~ X106

Solution. We will select Pp =0 and use the fact that, as Pp -0, p/ilZ-O. We will use the trapezoidal rule

acting

flow

at a constant

gas reservoir

y,(Pwj) -y,(p/)

is modeled

rate

from

an

infinite-

by Eq. 5.1.

+ 50 300~ , T

~ kh sc

r

'll.15110g

[ 1,688~Il;CI;rW

2

kt

for our numerical integration.

I

--'"

."~

I

. 86

WELL TESTING

TABLE 5.11-

DRAW DOWN TEST DATA

'(hours)

P./(psla)

0024 0.096 0244 0.686 2.015 6.00 17.96 53.82 161. 281 401 521 641 761 881

2.964 2.920 2,890 2.866 2.848 2.833 2.817 2.802 2.786 2.777 2.771 2.766 2.763 2.760 2.757

0

3.000

-(S+Dlqg l !

TABLE 5.12 -DRAWDOWN TEST DATA FOR CURVE MATCHING

~(P., )

(Psia2/cP)

6.338x 108

, (hours)

6.210 x 108 6.055 x 108 5.947 x 1oB 5864 x 108 5.801 x 108 5.747 x 108 5.693x1oB 5.640 x 108 5.585 x 108 5.553x1oB 5532x108 5517x108 5505x108 5.494 x 108 5.485 x 108

0.024 0.096 0.244 0.686 2.015 6.00 17.96 53.82 161 281 401 521 641 761 881

I)]}

(5.1)

This equation describes the MTR in a gas well test, just as Eq. 3.1 describes flow of a slightly compressible liquid. In general, of course, drawdown tests in gas wells also have ETR's (usually dominated

Exomptet 55.-no A

IYSlS ..i" G rlI OJ as,.,. eII

Swi :

0:21 i,

I

Vw = h = T = rw = IJ.i =

286 cu ft, 10 ft, 200.F, 0.365 ft, 0.01911 cp,

i

qg :

I ,(xx) Mscf/D,

, I

"tg -0.7, Cti=0.235xIO-3psi-t,

:

Cf' Q X

:

Q.. U

6-

1

/

I ! i:

.., I :

I I

I I

drainage area = 640 acres (square), and well centered in drainage area. This gas is the same as that analyzed in Example 5.4;

ilh: ~".

~ U")

,.1;1! :'" ; II

41-

I

-0:

.

~'

I

'

,

0

!I

~"r Q.'

I

u

i-!

:

.. ..

j

N

~

I

p = 3 (XX)psia ~ -0'19 '

variable is illustrated in Example 5.5.

3-

(psla2/cP)

0.128 x 108 0.283 x 108 0.391 x loa 0.474 x 1oB 0.537 x 108 0.591 X 108 0.645x1oB 0.698 x 1oB 0.753 x 108 0.785 x 108 0 806 x 108 0821x108 0.833 x 108 0.844 x 1oB 0.853 x 108

Drowdown Test Using Pseudopressures Problem. A constant-rate drawdown test was run on a gas well with the properties given below. results are shown in Table 5.11. '

by wellbore storage distortion) and L TR' s (in which boundary effects become important). Analysis of a gas well test using pseudopressure as the dependent

Q..

~(P,I-(P.,)

I ~ I """"

~, I =3.

2

-! 0

'l_~

0

!
2000

"~,."" -.""' '

~

--~ 00

'0

::::--~ 0

Q)

~

3(XX)

FLOWING TIME, hr

PRESSURE,

PSI A

! Fig. 5.14 -Drawdown

Fig.

5.13 -Pseudopressure

ys. pressure.

pressures.

test analysis using pseudo-

!! !

I

I

' I

I

-

GASWELLTESTING

I

87

TABLE5.13-STABILIZED DELIVERABILITY TESTDATA

Rate

p.,

Test (MMsCf/D) (psia) initial buildup -3.127 1 3.710 3.087 2 5980 3.059 3

8191

4

14.290

3.035

2.942

TABLE5.15-GAS WELL BUilDUP TESTDATA ~t (days)--p.s (PsiaL

TABLE5.14-ISOCHRONAL DELIVERABILITY TESTDATA q (Mscf/D) 983 977 970 965 2.631 2,588 2.533 2.500 3,654 3,565 3,453 3.390 4.782 4.625 4.438 4,318

P (psia) 352.4

352.3

351.0

349.5

P., (psia) 344.7 342.4 339.5 3376 329.5 322.9 315.4 310.5 318.7 309.5 298.6 291.9 305.5 293.6 279.6 270.5

t (hours) 0.5 1.0 2.0 3.0 0.5 1.0 2.0 3.0 0.5 1.0 2.0 3.0 0.5 1.0 2.0 3.0

accordingly, we can determine 1t-(pw/) foreachpw/' These values are included in Table 5.11. From these data, determine formation permeability and apparent skin factor. Solution. The first step in the analysis procedure is to plot 1t-(pw/) vs. log t. This plot is shown in Fig. 5.14. The curve shape suggests wellbore storage distortion up to t ~ I hour, and boundary effects starting at about 200 hours. A log-log plot of ~~ = [1t-(Pi) -k ~(Pw/)J is useful to confirm this suspicion. Thus, we tabulate (Table 5.12) and plot ~1t-vs. t. Qualitative curve matching of log ~~ vs. log I with Ramey's type curv.e (no~ shown) indicates an end ~o wellbore storage distortion at about I hour for skin factors in the range 0 to 5 regardless of CsD. confirming the indication on the 1t-(p w/) vs. log I plot. The absolute value of the slope of the MTR line is

0001883 0 0.003392 0.005738 0.009959 0.01903 0.04287 0.1144 03289 0.9724 2.903 7.903 12.90 17.90 22.90 27.90 32.90 37.90 .-(I

,637)(1 ,(XX»(660) = (11.18 x 106)(Im = 9.66 md.

From Eq. 5.1, we see that the apparent skin factor, $' =$+ D!qg I. is , 5 $ = 1.1 I -m 1t-(Pi)-1t-(PI ---=-I hr)

[

-IOg(;;:..~ -2)+3.23 ~Jl.ictir w

[

T P sc !!-I.:(1.151). T sc kh

'

x 10 -4)(0.365)2

]

The well isofapparently slightly stimulated. Radius investigation at the beginning and end of the MTR is found from Eq. 1.25: r.= t

k V2 ~ (948 t ) ~Jl.iCti

Thus. for Psc = 14.7 psia and T sc = 520oR, k=I,637~

q T

]

+3.23 = -0.21

= 11.18x 106 psia2/cp-cYcie.

m=50,300-

..I

-10

108J/4

Eq. 5. I shows that the proper interpretation of this slope is

].

Thus, , 6.338 X 108 $ = 1.151 -11 18-5.833 x 106- X 108 .

g[ (0.19)(0.01911)(2.35 ~;:;"ft ft.-. _._9~~ m=[(5.944-5.497)X

2542 2.430 2.600 2.650 2.692 2.726 2.756 2.785 2.814 2.843 2,872 2.896 2.907 2,913 2.917 2.920 2.921 2,922

[

= ("948)(0.19)(0.0191 1)(2.35 x ~ (9.66)(1)

] Yz

--

BB

;,'

= 109 ft at start of MTR (t = I hour), and 200 Y2 r 1= (109)( -) I = 1,550 ft at end of MTR. The distance xe from the well to the edge of the 640-acre square in which it is centered is 2,640 ft; thus, the time at which the observed deviation from the MTR occurs agrees qualitatively with the time at which boundary effects should begin to appear.

Ex

.written

erclses

_

I"

0.--

~.,

-

WELLTESTING

In the 214-hour test, the rate was 1,156 Mscf/D, the shut-in pressure was 441.6 psia, and the flowing BHP was 401.4 psia. Using the data in Table 5.14, (a) determine the AOF with both empirical andi theoretical methods, and (b) establish plots (on the same graph paper) of the empirical and theoretical stabilized deliverability curves. 5.3. Confirm o/I(p) results stated in Example 5.4 for pressures in the range 450 :5P ~ 3,150 psia. 5.4. The well discussed in Example 5.5 was produced at 2,000 Mscf/D for 90 days and then shut in for a pressure buildup test. Data obtained in the buildup test are given in Table 5.15. Determine formation permeability and apparent skin factor using an analysis procedure based on equations in terms of pseudo pressure, o/I(p).

5.1. The data in Table 5.13 (from Ref. 6) were References obtained on a well believed to be stabilized at each I. AI-Hussainy, R., Ramey,H.J. Jr., andCrawford,P.B.: "The rate. Using equations in p2 (strictly speaking, not Flowof RealGasesThroughPorousMedia," J. hI. Tech. applicable in this pressure range), estimate the AOF (May1966) 624-636;Trans.,A/ME,237. . using (a) the empirical method and (b) the theoretical 2. W~ttenbarger, R.A. and Ramey,H.J. Jr.: "Gas Wc;I!Testing h d With Turbulence,Damage,and WellboreStorage, J. P~t. met 0 '. .Tech. (Aug. 1968)877-887; Trans.,AIME,143. Also, do the following: (c) plot the theoretical 3. Dake,L.P.: Fundamentalsof ReservoirEngineering, Elsevier deliverability curve on the same graph paper as the ScientificPublishing Co., Amsterdam (1978). empirical curve; curat at th O

(d) since p2 equations are not acI I d I and outll .ne a

e .IS pressure eve, eve '?P .Calgary, theoretical method based on equations In p; and (e) apply equations in p to these data; in particular, calculate the AOF. 5.2. Cullender7 presented data from an ..Isochronal test and from an earlier, longer test th at led to approximate stabilization in 214 hours test time.

4. Theoryand Practicr ofth~ T~ting ofGas W~IIs,Ihi.rdedition, Pub. ECRB-7.5-34,Energy Resources and ConservationBoard,

Alta. (197.5). .5. Craft, B.C. and Hawkins,M.F. Jr.: Applied htro/~um Reservoir Enginnring, Prentice-Hall Book Co., Inc., Englewood Oiffs, NJ(19.59). 6. Back Pressur~T~~tfor Natural Gas W~IIs,Revisededition, RaIlroadCommIssion of Texas(19.51). 7. CullenderM.H.: "The IsochronalPerfonnanceMethodof Delermini~gthe FlowCharacteristics of GasWells," Trans., AIME(19.5.5) 204,137-142.

. 90

WELL TESTING

a3SERVATION ~LL

and

BOTTCf.4HCX..E ~SSURE \ TIME LAG "..

10 =

"-"-"___fACTIVE WELL

--

RATE AT ACTIVEWEU.

--Fig.

6.3 can be used in the following way to analyze interference tests. I. Plot pressure drawdown in an observation well,

q

TIM

0.000264 kl .,. 4Il/.c,r,:,

~=Pi-Pr' vs. elapsed time I on the same size loglog paper as the full-scale, type-curve version of Fig. 6.3 using an undistor.ted curve (the reader can prepare such a curve easily). Slide the plotted test data over the type curve .1 d (H ' I d . h . f unt. a matc IS oun. onzonta an vertlca I sliding both are required.) 3. Record pressure and time match points. ., (Po) MP' ~~P and [(lolrfJ)MP,/MP]' 4. Calculate permeability k in the test region from the pressure match point:

E.

..2. Fig. 6.1 -Pressure ,esponse In Interferencetest.

k=141.2:!!!!!-

(Po)~p.

h

(~)~p

5. Calculate oct from the time match point:

- ( 0.000264k )1 ..,. IMP /PC,-2 2

IJ.T

rj

I.

(lolrfJ)MP

r

E.\"ample6.1-1nterference Test 2r. + r

in WaterSand

I

Problem. An interference test was run in a shallowwater sand. The active well, Well 13, produced 466 STB/D water. Pressure response in shut-in Well 14, which was 99 ft from Well 13, was measured as a function of time elapsed since the drawdown in Well 13 began. Estimated rock and fluid properties include 1/ = 1.0 cp, B... = 1.0 RB/STB. h = 9 ft, r... = 3 in., and 0=0.3. Total compressibility is unkno\\'n. Pressure readings in Well 14 ~.ere as given in Table 6.1. Estimate formation permeability and total compressibility. Solution. We assume that the aquifer is

Fig. 6.2 -Region investigated in interference test.

r

rOlo"

=:::::::::::==i'ot

c,o

107

'0'

0'

homogeneous,

I I

to be plotted are presented in Table 6.2. The data fit the Ei-funclion type curve ~'ell. A patr of march points are (..l/=128 minutes, lolrfJ=10) and (~= 5.1 psi,po = 1.0). (See Fig. 6.4.) Thus,

.the

:

isotropic,

Ei-function

,

0

a:

type curves

k= 141.2~

KI"

I

KI

t 0 /r

"'

, ,.

M)Z

KI'

co

=

to estimate

(141.2)(466)(1.0)(1.0)

..

we use

k and c,.

Dala

(1.0)

(5.1)

(9.0)

02

infinite-acting;

(PO)~1P (~) ~IP

h 10"

and

= 1,433 md,

Fig. 6.3 -Exponential integral solution.

.

and c, =

0.000264 k

.1._2 -,.

IPr

!,~~1'~ti;"~

~~--,.,

c.:c,"

I/.

(/Mp/60)

,_2,

(lolro)MP

I i

I

THEAWELLTESTS

=

91

(0.~264)( 1 433)( 128/60) 11\.",nn';/1 1\~/1i\' (0.3)(99) (1.0)(10)

TABLE6.1 -PRESSURE/TIMEDATA FROMINTERFERENCE TEST

.)t

= 2.74 x 10 -S psi -I .(minutes)

D 5 25 40 1~ 200 300 400 580

~:~:~~ 139.72 13870 137.99 13712

I 6.3 Pulse Testing ~ulse tests6 have the same obj.ectives as conventio~al Interference tests -to determIne whether well pairs are in pressure communication and to estimate k and tpc, in the area of the tested wells. The tests are conducted by sending a coded signal or pulse sequencefrom an active well (producer or injector) to a shut-in observation well. The pulse sequence is created by producing from (or injecting into) the active well, then shutting it in, and repeating that

TABLE6.2-INTERFERENCETESTDATA FORLOG.LOGPLOT

s~quenc~ i~ a regular pattern. An example is indlcated In Fig. 6.5. The reason for the sequence of pressure pulses is that we readily can determine the effect of an active well on an observation well amid the established

.)t (minutes! 0 5 25

trend in reservoir pressure and random perturbations (noise)to that trend. Highly sensitive pressure gauges usually are required to detect these small coded pulses, which may have magnitudes of less than 0.1 psi. Pulse testing has several advantages over con-

~ 100 200 300 400 580

ventional interference tests: I. It disrupts normal operations much less than interference testing does. It hours lasts to a minimum which may range from a few a few days.time,

, !

r

0 u; a.

daries be taken into account. Analysis techniques for pulse tests usually are

.. '+-

position to model the rate changes in the pulsing sequence. From the simulations of pulse tests, charts relating key characteristics of the tests to reservoir properties have been developed.7 Before we discuss Ihese charts (Figs. 6.7 through 6.14) and their application. it will be useful to introduce nomenclature used in pulse test analysis, using the system of Earlougher I (and his schematic pulse-test rate and pressure-response history). Fig. 6.6 illustrates the time lag I L which is the time elapsed between the end of a pulse and the pressure peak caused by the pulse. The radius-of -investigation concept prepares us to expect a time lag. A finite period of time is required for a pulse caused by producing an subsequent active welltransient to movecreated to a responding well. and the by a shut.in period also requires a finite time period to affect pressure response. The amplitude L\p of a pulse can be represented

I

'-

~=P, -P., (psla) --0 0 4.01 ;.~ 74; 9.20 10.22 1093 11.80

I

2. There are fewer interpretation problems caused by random noise and by trends in reservoir pressure asthey affect pressure response at observation wells. 3. Pulse test analysis usually can be based on simple solutions to the flow equations -specifically, superposition of Ei-function solutions, which assume infinite-acting, homogeneous reservoirs. In many' cases, longer interference tests require that boun-

based on simulating the pressure response in an observation well with the familiar Ei-function solution to the diffusivity equation, using super-

P..

(pSla) 148.92 148.92 144.91 14372

-

10

4

..,,-o":~;;--or~:~~

~

A~ .'51 psi to 0 I

I 6--~. II G-

POINT ..

At...128min 10


0I .K)

100 ~

t

' m in .

Fig. 6.4-Interference test data from waterreservoir.

~

OTHERWELLTESTS

=

91

(0.(xx)264)( I 433)(128/60) 11\"/nn';/1 1\\/ln\ (0.3)(99) (1.0)(10)

TABLE6.1 -PRESSUREITIMEDATA FROMINTERFERENCE TEST .It 0 5 25

P.. (pSla) 148.92 14892 144.91

1~ 200 300 400 580

~:~.~~ 139:72 13870 13799 137.12

= 2.74 x 10 -S psi -I .(minutes)

6.3 Pulse Testing

40

Pulse tests6 have the same objectives as conventional interference tests -to determine whether well pairs are in pressure communication and to estimate k and tbc( in the area of the tested wells. The tests are conducted by sending a coded signal or pulse sequence from an active well (producer or injector) to a shut-in observation well. The pulse sequence is created by producing from (or injecting into) the active well, then shutting it in, and repeating that

14372

TABLE6.2-INTERFERENCETESTDATA FORLOG.LOGPLOT

s~quenc~ in. a regular pattern. An example is indlcated In Fig. 6.5. The reason for the sequence of pressure pulses is that we readily can determine the effect of an active well on an observation well amid the established

.It (minutes) 0 5 25

~=P, -P., (psia) --0 0 4.01

trend in reservoir pressure and random perturbations (noise) to that trend. Highly sensitive pressure gauges usually are required to detect these small coded pulses, which may have magnitudes of less than 0.1

;g 100 200 300

~.~ 745 920 10.22

psi.

400

1093 1180

.580

Pulse testing has several advantages over conventional interference tests: I. It disrupts normal operations much less than interference testing does. It lasts a minimum time, which may range from a few hours to a few days.

I

.

2. There are fewer interpretation problems caused by random noise and by trends in reservoir pressure as they affect pressure response at observation wells. 3. Pulse test analysis usually can be based on

I

I,

simple solutions to the flow solutions, equations -specifically, superposition of Ei-function which assume

0

cases, longer interference testsreservoirs. require that infinite-acting, homogeneous In bounmany.

0in

daries be taken into account. Analysis techniques for pulse tests usually are

~ '+-

based on simulating the pressure response in an observation well with the familiar Ei-function solution to the diffusivity equation, using superposition to model the rate changes in the pulsing sequence. From the simulations of pulse tests, charts relating key characteristics of the tests to reservoir properties have been developed.7 Before we discuss these charts (Figs. 6.7 through 6.14) and their application, it will be useful to introduce nomenclature used in pulse test analysis, using the system of Earlougher 1 (and his schematic pulse-test rate and

' 10

A --,,"",c-::!~-:

'II ~:~~

~

~INT

A~ .51 psi lo .10

~ 0- -~ II 0-


.r

..

At ./28min ! 10

pressure-response history).

Fig. 6.6 illustrates the time lag t L which is the time elapsed between thepulse. end of a pulse and the pressure peak caused by the The radius-of-investigati?n concept prepares us to expect a time lag. A finIte period of time is required for a pulse caused by producing an subsequent active welltransient to movecreated to a responding well, and the by a shut.in period also requires a finite time period to affect pressure response. The amplitude Ap of a pulse can be represented

0.1

.

0

100 ~

t

KXX>

' m in .

Fig. 6.4-Interference test data from waterreservoir. ~i,

~\.

..LI'

""~~:"~ilil ,. TESTING WELL

92

; I

I

RATE IN ACTIVE

q

q

q

q

RATE

q

IN

I

3

0

~~t.-J

TIME 6.5 -Typical

rate schedule

TIME-

in pulse

test

Fig. 6.6 -Pressure

conveniently as the vertical distance between two adjacent peaks (or valleys) and a line parallel to this through the valley (or peak), as illustrated in Fig. 6.6. The length of the pulse period and total cycle length (including both shut-in and flow periods) are represented by Alp and Arc' respectively. Analysis of simulated pulse tests shows that Pulse I (the first odd pulse) and Pulse 2 (the first even pulse) have characteristics that differ from all ... I responses, aII sub sequent pulses; beyond these mltla odd similar I pulses aJ hhave .. 1 hcharacteristics and all even pu ses so ave slml ar c aractenstlcs. W~ used

no:"", m

Figs.

define 6.7

dimensionless through

~ariables 6.14,

whIch

tha.t were

and [(tL )DlrbJ, estimates of k and

which
are

k=]41.2~D(tL~¥,

in pulse test.

.

.

S I r d 6 6 013 U.yze thF.e f our 6 th 13pu fi Ise, weduse FIgs. . U Ion. T 014 an aJ A."D (an ,., t I ".. )2. S dlngh kg.. Irhst to etermme L "'" c ' an t us , we note t at

'

are

then

response

was 26 ft; and porosity, ~, was 0.08. In the test following rate stabilization, the active well was shut in for 2 hours, then produced for 2 hours, shut in for 2 hours, etc. Production rate, q, was 425 STB/D and formation volume factor, B, was 1.26 RB/STB. The amplitude Ap of the fourth pulse (Fig. 6.15) was 0.629 psi, and the time lag was 0.4 hour. From these data, estimate k and It>c,.

F'

=

Ar

desIgned

~or quamitative analysis of pulse tests: dimensi~nless ume lag, (tL)D=0.000264 ktL/~.IlC,r;; dlmensionless distance between active and observation wells, r D = r I r IV; and dimensionless pressureresponse amplitude, ApD =khAp/141.2 qBp. (with sign convention that Aplq is always positive). Figs. 6.7 through 6.14 provide the correlations to be used in pulse test analysis. Figs. 6.7 and 6.8 are to be used for the first odd (i.e., first) pulse; Figs. 6.9 and 6.10 for the first even (i.e., second) pulse; Figs. 6.11 and 6.12 for all other odd-numbered pulses (third, fifth, etc.); and Figs. 6.13 and 6.14 for all other even-numbe~ed pulses (fourth, sixth, etc.). The figures use the rauos (I) F' = pulse length (~t p) to cycle length (At c) and (2) time lag (t L) to cycle length (~t c). The figures appropriate for a given pulse number are used to obtain values of tlpD(tLI.ltc)2 used to provide

! 5l~6

ACTIVE WELL

WELL

Fig.

4

,

I

At

=

2/(2

+

2)

=

0

5

pc.

,

I Ar = 04/4 = 0 I IL c. .. Then, from Fig. 6.13, 2 ApD (t L l.:lIc) = 0.00221, and qB Ap (t I AI ) 2 k=141.2 -f"/~ I~.jhAp(ILIAtc) (J41.2)(425)(1.26)(0.8)(0.00221) 2: (26)(0.629)(0.1) -817 d -m. =

F

F. rom

, /

..

614 .,

Ig.

(tL)Dlrb=O.09I.

hAp(tLIArc)

Thus, 0.000264 kt L 11>c,= --.,

0.000264kl L

-2-.

~cl=

p.r-[ (I L ) DI r DJ

-,

-2

p.r-{ (fL )DlrDJSxample

6.2 illustrates how these figures are applied. (0.000264)(817)(0.4) In Q\/Q~~\2,/\ "",' (0.8)(933) (0.091)Jroblem. = 1.36x 10-6 =

:xample6.2-Pulse

Test

Analysis

A pulse test was run in a reservoir in which distance between wells, r, was 933 ft. Formation viscosity, p.,was 0.8 cp; formation thickness, h,

Then,

I

Iluid he ;

!

--1 1

1.11

I~

OTHER WELL TESTS

0003 N

r-a

"

~ 0.00 oJ

.=., ~ 0.0025

~

1&1

a

::> 0.0020

~

J

~

~

0.9.

ct 0.001

.

1&1

II!

Z

a

~

0.0010

1&1

It

1&1

II! 0.0005 .J ::>

~

0

171t

S17.t 10-1

I

(TIME LAGI/(CYCLE LENGTH). tL/6tc Fig.

6.7 -Relation

between

time lag and response

amplitude

for first odd pulse.'

0.2

N

0

0.17

,

~

0

--

oJ

0.15

13 ct

.J

0.125

1&1

~

~

II!

0.100

II! I&J

.J

Z Q II! Z

0.075

I&J

~

is o.o~o 0.02~

7 ..I

to.'

(TIME LAG)/(CYCLE LENGTH). tL/6tc Fig.

6.8 -Relation

between

time lag and cycle

length for first odd pulse. 1

WELL'

0.004~ N

Y

0.0040

~

~

oJ

0

0.003

Q.

~

!oJ 0 ::>

0.00

~

oJ Q.

~

0.002

4

!oJ VI

Z ~

0.0020

VI !oJ

~

!oJ VI

0.001~

oJ ::>

Q.

'8'

0.0010

45.'8' 10-1

(TIME

Fig.

6.9

-Relation

between

I

LAG)/(CYCLE

time

lag

and

LENGTH).

response

tL/OtC

amplitudes

for

first

even

pulse.

0.200 N

0.17 0

'-~

oJ

O.I~

..

-~

~

.

4 oJ

.

0.12'

!oJ

~ ~ VI

0.10

VI !oJ oJ

Z

~ z

0.07'

:.

!oJ

~ 0

;

0.0'0

0.025

'8'

'8' 10-1

(TIME

Fig.

6.10

-Relation

between

1

LAG)/(CYCLE

lime

lag

.:

LENGTH).

and

cycle

length

tL/OtC

for

first

even

pulse.'

1

v. OTHER WELL TESTS

95

0.003 N ~ U

~

"-

0.003

.J

~

Q

Q. <3

0002

I&J

0 -;

0.0020

J

a-

~ 4

0.00"

\oj I/)

~ a-

I/) I&J

0.0010

a:

\oj I/)

-l

0.0005

! 0

4517..

10-1 (TIME LAGI/(CYCLE

Fig.

6.11-

Relation

between

Z

1

LENGTH I. tL/c.tC

time lag and response

amplitude

for all odd pulses

after the first.'

0.20

N

'"..-

0.17

Q

Q

-

-:, 0 4 -l

I&J

0.1'

0.12

..

~

~ I/)

0.10

I/) I&J -l

Z Q

0.07'

I/)

Z

\oj

~ 25

0.0'0

5 I 7..

0.02'

I

(TIME LAGI/(CYCLE Fig.

6.12 -Relation the first.'

between

LENGTH). tL/c.tC

time lag and cycle length for all odd pulses after

~_IIIIIIIII'I

r"".'

WEllTESTING

0.00.' N

r-I U

..0.00

~ .J

...=, Q

Q.

0003

q

I&J

0

:> ~

~ ~ 4

0.003

0.002

I&J (/)

Z

~ 00020 (/)

I&J

~

I&J ':'J

O.COIS

!

1 I . 10-1

00010

(TIME LAG)/(CYCLE Fig.

6.13 -Re!ation pulses

between

I

LENGTH).

time lag and response

tL/OtC

amplitude

for all even

alter the flrst.'

0.200

N

0.17S

..Q

"Q

..

-.,

O.ISO

Ii4 .J

.. 0.12S

I&J ~

-.

~

(/)

0.100

(/)

I&J .J

Z

~

0.07'

Z

I&J

~ 0

O.OSO

0.02Ss

I'

~"-"5'.'" 10-1

Fig.

6.14 -Relation

the first_'

I

between (TIME time LAG)/(CYCLE lag and cycleLENGTH). length for tL/OtC 'all even pulses after

OTHER WELL TESTS

97

i

I

f

~

~

,..)J;;i

,F

U)

,;--!i.,\ IJ H

W g::

A

DE

K

t Fig. 6.15-Schematic of pressure response in pulse test.

c{ =

1.36 X 10 -6

= 17 x 10 -6 psi -I.

0.08 .slightly 6.4 Dnllstem Tests A drillstem test (DST)8.9 provides a means of estimating formation and fluid properties before completion of a well. Basically, a DST is a temporary completion of a well. The DST tool is an arrangement of packers and valves placed on the end of the drillpipe. This arrangement can be used to isolate a zone of interest and to let it produce into the drillpipe or drillstem. A fluid sample is obtained in the test; thus, the test can tell us the types of fluids the well will produce if it is completed in the tested formation. With the surface-actuated valves on a DST device, it is possible to have a sequence of flow periods follo~'ed by shut-in periods. A pressure recorder on the DST device can record pressures during the flow and shut-in periods. The pressures recorded during the shut-in periods can be particularly valuable for estimating formation characteristics such as permeability Ithickness product and skin factor. These data also can be used to determine possible pressure depletion during the test. To illustrate how a typical DST is performed, we will examine a schematic chart (Fig. 6.16) of pressure vs. time from a test with two flow periods and two shut-in periods. At Point A, the tool is lowered into the hole. Between Points A and B, the ever-increasing mudcolumn pressure is recorded; at Point B, the tool is on bottom. When the packers are set, the mud column is compressed and a still higher pressure is recorded at Point C. The tool is opened for an initial flow period, and the pressure drops to Point D as sho"'n. As fluid accumulates in the drillstem above the pressure gauge, the pressure rises. Finally, at Point E, the well is shut in for an initial pressure buildup test. After a suitable shut-in period, the well is reopened for a second final flow period, from Point G to Point H. This final flow period is followed by a final shut-in period (from Point H to Point I). The packers are then released, and the hydrostatic pressure of the mud column is again

TIMEFig. 6.16-Schematic 01arilistem test pressurechart.

imposed on the pressure gauge. The testing device is then removed from the hole (Point J to Point K). The initial flow period is usually brief (5 to 10 minutes); its purpose is to draw down the pressure near the well bore (perhaps letting any mudfiltrate-invaded zone bleed back to or below static reservoir pressure). The initial shut-in period, often 30 to 60 minutes, is designed to let the pressure build back to true static formation pressure. This initial shut-in pressure on a DST may be the best measurement made of static reservoir pressure. The second flow period is designed to capture a large sample of formation fluid and to draw down the pressure in the formation to the maximum distance and extent possible within the time that is possible to allow for the DST -frequently 30 minutes to several hours. The second shut-in period is designed to obtain good pressure buildup data so that formation properties can be estimated. In addition, comparison of the final (or extrapolated) pressure from the second shut-in period to the initial shut-in pressure can indicate that pressure depletion has occurred during the DST and that the well thus has been tested in a small, noncommercial reservoir. The desired length of the second shut-in period varies from equal to the second flow period (for highpermeability formations) to twice the length of the .. second flow period (for low-permeability formations). Theory much like that used for an ordinary pressure buildup test following production at constant rate is used for analyzing the shut-in periods on a DST. This is true even though the flow rate preceding a shut-in period in a DST usually decreases continuously. Usually, the average production rate can be used as a good approximation in buildup test analyses; this average rate of production is determined by dividing the fluid recovery by the length of the flow period. To analyze the buildup test, we plot Pws vs. log (tp+A/)/.lt, wherelp isnowtheactua!.n°wi~gtime at the average rate q. The permeabilIty/thIckness product is found from the relationship kh = 162.6 qBJJ./m. Usually, a fluid sample will not yet have been analyzed in the laboratory; accordingly, correlations (Appendix D) relating JJ.and B to produced fluid properties must be used.

, j j

J

~

, I I

t

98

WELLTESTING

TABLE6.3-DRILLSTEMTESTDATA ~t

c

pressure is recorded. Follo~'ing sample collection, shut-in pressures are recorded as they build up with ' tIme. Flow into the sample chamber is probably approximately spherical (i.e.. into a point rather than spread uniformly across an entire productive interval). For this reason, the shut-in test cannot be analyzed as in a DST, although theory based on steady-state spherical flo~' may explain ~;reline test buildup pressure satisfactorily in some cases. t I The

p~s

(minutes) --0-5 10 15 ~~ 30 35

IPSI) 3S-O965 1.215 1.405 1,590 ~.~~ 1:740

40 45

1.753 1.765

Static reservoir pressure is found by extrapolating the buildup tests to infinite shut-in time. In a dual shut-in test. we have the opportunity to extrapolate both buildup tests to infinite shut-in time and to compare the estimated static reservoir pressure. If the static reservoir pressure from the final shut-in period is significantly lower than that from the initial shut-in period, it is possible that the reservoir was partially pressure depleted even during the relatively short DST, implying that the formation tested is probably noncommercial. Skin factor is calculated from the conventional skin-factor equation: (p -p ) s= 1.ISI[ \PI hr -Pwfl m

device is useful for obtaining samples of formation fluid and estimating initial formation pressure; extensive use of the device for this latter application has beenreported in the literature. II . Exercises 6.1 Determine the duration of an interference test required to achieve a pressure drawdown of 25 psig at the observation well for the reservoir described in Example 6.1 if the active ~'ell produces 500 STB/D throughout the test. What will be the radius of investigation at this time? If the skin factor in the active well is 2.0, what will be the drawdo~'n in the active well? 6.2 For the pulse test described in Example 6.2. given the results of the test analysis of the fourth pulse, determine the time lag and pressure response for (a) the first odd pulse; (b) the first even pulse; (c) the third pulse; (d) the fifth pulse; and (e) the sixth pulse.

6.3 JohnstQn-Schlumberger9 reports data below ,k, -log (~

) + 3.23j.

(2.4)

#J.Ctw

There is a further complication in DST analysis. At the time of the test, reservoir rock and fluid

from a DST: initial flow period = S minutes, Inltla Ihs ut-in = minutes, ' ' peno . d30 ... final flo.w per~od = 60 m~nutes,and final shut-In perIod = 4S minutes.

properties that appear in Eq. 2.4 may not yet be known accurately. This is particularly true of porosity, ct>, and total compressibility, Ct. Accordingly, one may be forced to use the best available estimates and to recognize that skin-factor and radius-of-investigation calculations, which also depe.nd on these p.roperties, may be s.ubject to consIderable uncertainty. Use of Eq. 2.3 Instead of Eq. 2.4 (Eq. 2.3 includes the sometimes important ~ermlog (tp+;~t)/tp]toc~lculatesm.aybejustified In. some cases If other data In the equatIon are kno~'n

Data obtained in the final shut-in period were as given in Table 6.3. In the initial shut-in period, the pressure reached and remained stable at 1,910 psi. Total fluid recovered in both initial and fmal flow periods filled 300 ft of 2 Vl-in.-lD drill collars (0.0061 bbl/ft) and 300 ft of 4'il-in. drillpipe (0.0142 bbl/ft). The produced fluid '''as 35°API oil with a measured .. gas rate of 47 Mscf/D al Ihe surface (assumed solution 2as). Formalion temperature was measured at l20°F~ PorosilY is estimaled to be 10;'-0; total compressibility is 8.~ x 10 -6 psi -I; wellbore radius

with unusual accuracy. 6 5 W. ..Estimale .Irellne Formation Tests In many areas. hole conditions prohibit use of DST's

is 4.5 in; and formation thickness is 10 ft. formation permeability, skin factor, flow efficienc\", and radius of invesligation achieved in the lest. .

as temporary wellbore completions. In these areas, and in others where the costs of Ihe required numbers of DST's for complete evaluation are prohibitive, ~'ireline formation tests 10.11frequently are used in formation evaluation work. A wireline formation tester is, in effect. a sample chamber of up to several gallons capacity combined

6.4 Smolen and Litsey II propose that "boreholecorrected," stead\"-state, spherical flow into the wireline formation"tester can be modeled by

wit~ pressure gauges. The.test chal.nbers are forced against the borehole wall In a sealing pad, and the formation is perforated by firing a shaped charge. The signal t.o fi~e the charge is Ir~nsmitted o.nlogging cable. fluid IS collected durIng sampling, and

k = 3,300 qlJ./~.

, 1

where k = permeabililY, md, q = flow rate, cm3Is (reservoir conditions). Jl. = fluid viscosity (usually mud filtrate), cp, and .I ~ = drawdown from formation pressure. psI.~

I

I

OTHER WELL TESTS

99

A test showed that formation pressure was 3,850 psi. Pressure was dra~.n down in the sample chamber to an apRroximatelv constant 1 850 psi by withdrawing 3 ., ., 10 cm of filtrate (JJ.= 0.5 cp) from the formatIon In 16 seconds. Estimate formation permeability from the test data. ...Properties Derive an equatIon for steady-state sphencal flow and show that it has the form k = constant x qBIJ./(r (Pi -Pw/)]. State the assumptions required for the equation to model a wireline formation tester flow

4. Vela, S. and \1cKinley, R.\1.: "How Areal Heterogeneities Affect Pulse.Test Results," Soc. P~t Eng.J. (June 1970)181191;Trans..AI~E. 249. 5. Jargon, J.R.: Effect of ~ellbore Slorage and Wellbore Damageat Ihe Active Well on ImerferenceTest Analysis," J. P~t. T~ch.(Aug. 1976)851.858. 6. Johnson,C.R., Greenkorn, R.A., and .~oods, E.G.: "PulseTesting:. A New Method for Descnblng Reservoir Flow BetweenWells," J. Pet. Tech. (Dec. 1966) 1599I~; Trans.,AI ME, 237. 7. KamaJ,\1. and Brigham, W.E.: "Pulse-Testing Responsefor UnequalPulse and Shut-In Periods," Soc. Pet. Eng. J. (Oct. 1975)399-410; Trans.,~1\1E. 259.

test

8. Edwards, A.G. and Wlnn, R.H.: "A Summary of \1odern and Techniques Used in Drillstem Testing," Pub. T-

.Tools

References 1. Earlougher, R.C. Jr.: Adl'anc~s in ".~/I T~st Analysis. Monograph Series,SPE, DaJlas(1977)5. 2. \1atthews, C.S. and Russell, D.G.: Pr~ssure BuIldup and Flow T~stsin W.ells,\1onograph Series,SPE, Dallas (1967)t. 3. Theoryand Pract;~ o/the TestIngo/Gas Wells,third edition, Pub. ERCB-75-34. Energ~. Resourcesand Conser\'ation Board, Calgary, Alia. (1975).

4069. Halliburton Co., Duncan,OK (Sept. 1973). 9. .'Revie-. of Basic Formation Evaluation," Form J.328, Johnston-Schlumberger,Houston (1976). 10. Schultz, A.L., Bell, W.T., and Urbanosky, H.J.: "Advancemems in Uncased-Hole, Wireline Formation-Tester Techniques,"J. P~t. T~ch.(Nov. 1975)1331-1336. 11. Smolm, J.J. and Litsey, L.R.: "Formal ion Evaluation Using ~.ireline Formation Tester PressureData," J. Pet. Tech. (Jan. 1979)25-32.

m ~."." !

.., i

..,.

, I

Appendix A

Development of Differential Equations for Flow in Porous Media I

I

t

Introduction

I

In this appendix, we develop some of the basic differential equations that describe the no~' of fluids in porous media. Results presented include equations

=pu-,.~YI1Z+PUyAK.lz+pu;:.lx.1y-

+ ~(pU.f) ]~YI1Z -[PUy + ~(pu.y) ]AKAz

for three-dimensional flow of slightly compressible liquids, and for radial flo\\! of slightly compressible

-[PU: + ~(pU;:) ]AK~Y

liquids, gases, and simultaneous flow of oil, water, and gas. In developing these equations, we start with continuity equations (mass balances); then we introdu~e flow l~ws (such as Darcy's. law) ~nd appropnate equatIons of state for the fluId considered. I, Continuity Equation for ! Three-Dimensional Flo,,' To develop continuity equations, we use a mass balance on athesmall element of porous material. The balance has following form.

f

(rate of mass flow into element) -(rate o~ mass flow out of element~ = ~rate of accumulation of mass wIthin element). Our element is shown in Fig. A-I. It has dimensions .:lx,f ~.v, and .1.-:, . in theh x, y, and d .' ~ coordinate system; or convenIence, t e coor mate system IS oriented such that gravitational forces are in the ( -)~ direction. We denote the components of the \'olumetric rate of flow per unit cross-sectional area (cu~ic feet .per ~our-s9uare feel or feet per hour are typlcalenglneenngumts)byux,u.,andu;:. The rate at which mass enters ihe element in the .\' direction is pu ~v~ (Ibm/cu ftxft/hrxsq ft= ."C

-I

lbm/hr);. the rate. at ,!"hi~h the mass leaves the element m the x dIrectIon IS [pux+~(pux)].ly.1:.. Similar expressions describe rates of mass entering and leaving in the y and ~ directions. The result of adding these expressions is (rate of mass flow into element) -(rate of mass flow out of element)

[pux

= -.1 (pu x) ~Y~ -~(pu

I ; I

\I) A.\',lz -~ (pu.) ."

'AK~y. To determine the rate at which fluid accumulates within the element, we first note that the mass within the element (of porosity 0) at a given time is p(!J~.ly,lz (Ibm/cu ft x cu ft = Ibm). Thus, the rate at which this mass changes over a time interval ~/ is (pO 1+,),1 ~r -pti> I) Ax.1_~',lz,

j i !

..

where time, r, is in hours. Accordingly, the mass balance becomes

.i ! !

-~(pu.\.).1.v.:lz-~(pUy)~.l.:-~(pu;:)AK~y p~ .I.;1+J.r - p tl>1 = ~'~.~',lz. ~/ If we divide each term by A.\".l.~',lzand take the limit as .1r,:l.\',.l)', and 11Z-0, the result is ~

~ a .+a lX'

~ ~

+

= -~ a~

ar

( ,pq,). (A.I)

I i

: !

Continuitv

Equation for Radial Flow ... From a mass balance sl.mll.arto that ~sed to develop E.q. ~.l, the. comlnulty equation for onedImensional, radIal flow can be sho\\"nto be ~~(rpur)=-~(tI>p), r ar at

(A.2)

! ijj

DEVELOPMENT

OF DIFFERENTIAL

t +-.d(f

tu

EOUATIONS

FLOW

/ tUx

t

+-6(tu>Uy Y

6z

)1

t Ux+-A('Ux)

/'

~.-

.~~1:?j;;~,;i{~J;" MEDIA

IN POROUS

'"

101

introduce additional assumptions. First, we restrict our analysis to slightly compressible liquids -those with constant compressibility. c. where c is defined by the equation

-~Uy-

~

FOR

tlUt

x

-I dV_1 c=v"dj;-pdP'

dp

(A.6)

For constant compressibility A.6 gives

"1

c, integration of Eq.

p=poeC(P-P,,).

(A.7)

.where Fig. A.1 -Element of porous medium used for mass balance.

Po is the value of p at some reference pressure p.0 Eq. A.7 describes most single-phase liquids adequately. If \\'e now introduce Eq. A- 7 into Eq. A.4 and

where u, is the volumetric flow rate per unit crosssection area in the radial direction. Eq. A.2 is less general than Eq. A.I; in particular. radial flow onl.v

assu~e. that (I) k.f = k.".= k;: = constant, (2) gravitatIonal forces are negligIble, (3) ct>= constant, and (4).Il = constant, Eq. A,~ becomes

is assumed [i.e., there is no flow in the Z or 8 "direction" in a cylindrical (r,z.8) coordinate

0 ~

l

oP

eC(P-p,,) ~

I+

0 l oP a ec(p-p,,) a

system],

Y

Flow La,,'s

+-e0

Liquid flo\\ usually is described by Darcy's law. This law. when applied in the coordinate system orientation "e have ~hosen to .describe three=dimensional flow, becomes, In field Units. k op U.f = -0.001127 -:! -, -/l ox

o.\"-

~ /l OZ u.=-0.001127~(-+0.OO694p

I '.'

), d.

(A.3) d

Th k h b' e i are t e permea I Itles In Irection i an p denotes pressure (psi). In Eq. A.3, we ha\e assumed that the .\' and y directions are horizontal, so that gravity acts on!y i~ the.: direction.. .C After equation, substituting theseisequations Into the continuity the result

( ~ ~ ) 0 ( k "p ~ 0\' a\" + -;- --;k.p

/l

oP ) IiY

~

-Ilj~'~

+c

-

I(

ox

)

( op

+ -

--:; + ;-::2 + ;::r =

oY

)

2

1

2/

OZ) ' "

o.v

OZ

,I

(A.S)

0.264 k 01

loop ct>/lC op ~ a;: (r a;:) = O.264 kat.

0

of

( op

+ -

For radial flow, the corresponding equation is

(A.4)

(A.S)

Slightly

Compressible Fluids To solve Eqs. A.4 and A.S analytically, we must I

~

If we further assume that 0 ~ 0 2 0 2 I( .:E I + ( -E) + ( -E) o.\" oy OZ\\'ith J is negligible compared other terms in the ..

OX-

For radial flow, the result is I I 0 rpk ap I 0 I -a(--!-a)=-(Pct». r r /l r O.264 ot Flow ..constant

.

OZ op 2

-I. ~ 'I'/lC ~ 0.000264 k ot '

=

op

= 0.000264 at (ct>p).

Single-Phase

02p

oy-

equation (w~ich requires either small C or small pressure gradients, or both), then .., o-p o-p 02p OjlC op

+ a:: 1-;- ( az + O.00694p)J I

- P,,) op /

Sim plifvin ., g

oY op

dY

I

OZ

02p -;:-;+-,.+;::r

k.

Y

-ct>/l ~ -0.000264 k ot [e c(p-P..) J.

~,

./l

a

C (p

OZ

-02p

u" = -0.00 1127 ~

..Il'

I

I !I !'

Eqs.

A.S

and

A.9

are

""."'"

(A.9)

diffusiyity

equations.

Analytical solutions to Eq. A.9 are known for several simple boundary conditions; these solutions are used for most \vell test analysis. We must remember that Eqs, A.Sand A,9arenotgeneral; instead, they are based on several important assumptions, including (I)

the

single-ph~e .l}quid flo~ing has small and compressibility; (2) k IS constant and the

same in all directions (isotropic); (3) ct>is constant; and (4) pressure gradients are small. .,

! IAj

~

102

WELLTESTING

Single-Phase Gas Flow For gas fl ow characterlze . db v D arcv ' s I aw andf or a gas described by the equation'of stat~, .\1 p

P=--, RT

ctI

~(.

l?-).

(A.IO)

O.264 k at Zg ..pressure-dependent In this section only, we use Zg to denote the real gas law compressibility factor so that we can ~ontinue to use the symbol Z for one of the space variables. For gases, p.and Zg are functions of pressure and cannot be assumed constant ~xcept. i~ special cases. To reduce Eq. A.IO to a form sImIlar to Eq. A.8, we define a pseudopressure, I ~ (p), as follows: .p

d(l?-) -Zg Z.~) -dp

(A.II) Now,

.Zg since

ap at

tIme-dependent saturatIon (Soo S and SoS); formation volume factor (80, B ,,'0and B ); and pressure-dependent viscosity (p.oo p , and p.g 1. When gravitational forces and capillary pressures are negligible. the differential equation describing this type of flow is I a ap 264 Al at

-~ ~ ~ AI- p.o + p.g + p

co

,

(A.IS)

(A.16)

-I

dBo+ ~ dRs dp

Bo

dp

(A.I/)

--

Note in Eq. A.17 that effective oil compressibility depends not only on the usual change in liquid volume ~ith pressure but also on the change with pressure of the dissolved GOR. RS' A similar expression can be written for water

I d( l?- ) C = -~ = ~ Zg .~ P dp p dp Al so note th at a I a.I ~ -;-~- -:;- ¥,up -a = --,2p ap

compressibilitv: . -I dB... ~ dRSI" C"' + B... dp B ...dp

(A.18)

IJoZg at

and

Exercises

a.o -2p

ap

A.I Derive

a; -~ ~ .required S'.I ' .A.2 Iml ar expressIons apply for a~/ay and a~/az. Thus, Eq. A.IO becomes ~

water. and gas, and that each phase has saturation~ependent effecti\e perr:neability (koo k",o and kg);

and

at'

t

equation describing radial, simultaneous flo~ of oil, gas, and water. More complete discussions and more general equations are given by Matthews and ~.and Martin. 3 We assume that a porous medium contains oil,

cI=Soco+S...c..+SgC~+cf'

( l?-

cpap = ~ -Bo

a

and Gas

~'here

p

~(P)=2) -;-dp, P-Zg ~'herePJ isalowbasepressure. Pi,

up

Flow of Oil.

In this section. we outline a detailed derivation of an

=

(It

Simulraneous \\'ater,

Z

Eq. A.4 becomes, for constant ctIand k and negligible gravitational forces, ~ ~ ~ ~ ~ ~ Q p dp Q P dp Q ( P dp -( --+ ---+ ---Russell a.\" 'lJoZg a.\") ay (. IJoZgay) az IJoZgaz)

~ al

to Eqs. A.8 and A.9 prove to be accurate approximatesolutionsofEqs.A.12andA.13.

( a~ )

a

( a1f, ,

a

a~.)

a;+a;a;)+az(a:;:: -ti>,uc I{ a~ -0.264k at.

(A.12)

For radial flo\\, the equivalent of Eq. A.12 is I a (. a~ ) ctllJ.Cg a.,; ; a; r ~ = 0.000264 kat.

in.this Derive required in this A.3 Derive

Eq.

A.2;

derivation. Eq. A.S; derivation. Eq. A.9;

state the

assumptions.

. state the assumptIons state the assumptions

required in this derivation. A..4o.erive. Eq: ~.13; state the assumpt!ons requIred In thIs derivatIon. Compare the assumptIons required to derive Eq. A.9 with those required to deriveEq.A.13. References

(A.13)

Eqs. A.12 and A.13 are similar in form to Eqs. A.8 and A. 9, but there is one important di fference. The

." coe ffi IClent ctI,uc/0.000264 k IS constant In Eqs. A.8 and A.9; in Eqs. A.12 and A.13, it is a function of

I. AI-Hus':liny.R..R:lmey.H.J.Jr..andCra~(ord.P.B.:"The Flo~.of Re:l~Gasc:ThroughPoro~sMroi:l." J. PI'I. Tn.h. (May1966)6~4-636. Truns..AI\IE. 237. 2. Malthe\,s.. C.S. andRussell.D.-G.:Prl'ssurl'Buildl/!!andRull. Tl'sls.lIIlll'lls. Mono~~:lph Series. .SPE.Dalla~(196,)I. .

3. Marlin. J.C.: Equallons of FloworIn Multipha.~ Gas Dri\e Reser\oirs and"Slmplltiro (he Theorelil:al Foundalion Pr~surc: Buildup ~al\'si~." Trails.. AI ME (1959) 216. m-

Ii i

I

Appendix B Dimensionless

Variables

I

Introduction It is convenient and customary to present graphical or tabulated solut~ons t~ flow equ~tions, such.as Eq. A.9,intermsofdlmenslonlessvanables.lnthlsway, ... bl I I t. f a It IS paSSI e to present compact y so u Ions or wid.e range of parameters cp, /l, C, and k, and variables r, p, and t. In this appendix, we show how many of the dimensionless variables that appear in the. welltesting literature arise logically and directly In the' differential eq~~tions (and in. their i".itial and bou~dary condItIons) that descrIbe flow In parous medIa. Radial

or ap -I ar

qB/l =-. (8.4) 0.00708 khr w 'w This boundary condition arises from Darcy's law in the form similar to that used in Eq. A.3: B u = -0.001127 ~ ~ = ~ . /l ar 2rhr Our objective in this analysis is to restate the differential equation, and initial and boundary conditions in dimensionless form so we can deter-

Flow of a Slightly -mine ..charactenze Compressible Fluid In this section, we identi~y the dimensi.onless variables and parameters requIred to charactenze the solutions to the equations describing radial flow of a slightly compressible liquid in a reservoir. We assume that Eq. A.9 adequately models this flow. Specifically, we analyze the situation in which (I)

the ~ime~sionless .vari~bles and parameters that thIs flow situatIon and that can be used to characterize solutions. These dimensionless parameters and variables are not unique (i.e., more than one choice can be made for each). Further, we want to emphasize that these dimensionless variables' are defined rather than derived quantities. These ideas will become clearer as we proceed. We define a dimensionless radius, rD=r/rw (any

pressure throughout the reservoir is uniform before production; (2) fluid is produced at a constant rate from a single well of radius r w centered in the reservoir; and (3) there is no flow across the outer boundary (with radius r e) of the reservoir. Stated mathematically, the differential equation, and initial and boundary conditions are I a a~ CPIi.C a~ --r= -, r ar ar 0.000264k at

other convenient reference length, such as r e' could have been used). From the form of the differential equation, we also note that a convenient definition of dimensionless time is tD = 0.000264 kt I o/lcr!. The initial and boundary conditions suggest that a convenient definition of dimensionless pressure is

( )

(8.1)

PD = 0.OO708kh(Pi-P) qB/l \JII.ith this definition, B.4) becomes

" " ~. ,

..

.

.. the boundary condItion (Eq.

-qBIJ apD' 0.OO708khr...~i,

att=O,p=pjforallr,

!

-qBIJ = 0.00708khr".

,

(1.1

atr=re,q=Ofort>O,

(8.2)

or ap -I =0, ar ,

,.. (B.3)

~

or simply a a;: I

=1.

'D=} ... Expressed In terms of dimensionless variables, the differential equation and its initial and boundary conditions become

atr=rw,q=

..

:

-0.001127(2rrwh)

k ap

B;a;,.fort>O

I

I

a

~~(rD~

aPD

)=ar;' apD

(8.5)

j

104

",,-

po=Oforallro

-:-

atlo=O.

ap 0, ~I

(8.6)

WELL TESTING

andl:ompressibilityhasthe

I

fundamental units

Compressibility:.[lt~/m). =Oforlo>O.

(8.7)

Then.

'n"..'o"IJ.' 10=

-IapO aro'

.. [l-)[t)

0.264kl

=lforlo>O.

[I) I --m It

~~cr;

(8.8)

lllt2

1(l " )

[1).

m

'n. I

The implication is that any solution, Po, of Eq. 8.5 is a unique function of r 0 and 10 for fixed r £¥; no other dimensionless variablesequation appear inoreither dimensionless differential in the the dimensionless

initial

or

boundary

conditions

describing this particular problem. Thus, if we wish to present solutions to Eq. 8.5, we could do so compactly either by tabulating or by plotting Po at ro=1 (r=r...) as a function of the variable 10 with the parameter r[H. In fact. such tables and graphs have been prepared and presented; this problem is precisely the ~'.ell known van Everdingen a~d Hurst constant terminal-rate problem. for which they present solutions as functions of 10 and r£¥. We have implied that the groups of parameters and varia.ble~ ~~at arise from the diff.e~ential equ:ation and tts InItIal and boundary condItIons are dlmen~ionless. .We ~'ill now provide the proof for the case just considered. Dimen~ionless variables are ro =rlr ..., 0 264 k tD =

q ~ m [l ~](l)1 ~ I -It

I l3 ! , t

-(I]. m (I) lt 1

!

Thus. Po also is dimensionless. Again. ~'e stress the rea~on for introduction of some~hat unnatural dimensionless quantities. They allo~' solutions for wide ranges of k. h, c. I, rt' r ..., q, ~. and B to be presented I:ompactly (tables or graphs) as functions of a minimum number of variables and parameters. Such tabulations and graphs are in widespread use in well testanal~"Si~.

dimensionless

op.Cr..,:

is

Radial flow With Constant BHP To illustrate further ho~' the choice of appropri.ate

.t "

,

0 00708 kh ( .P,

variables

depends

on

the

specIfic

differential equation, and initial and boundary conditions, we now determine appropriate dimen-

and Po =

Thus, 10 has units of unity, or. more plainly. dimensionless. Similarly, Po = 0.00708 kh B (p I -p)

,-

) P.

qBIJ. Obviously, r 0 is dimensionless. To sho~' that 10 and Po are dimensionless. we introduce the symbol (), which denotc~ "has units of." let m denote mass. l. length. and t. time. The quantities that appear in 10 and Po have the following basic units: k -(l-),

sionless variables for radial now of a slightly compressible liquid through a ~'ellbore of radius r...

from a reservoir of radius r t. There is no now across the outer boundary (at r= r t); initial.pressure,.p" is uniform before ~roduction; and nowlng 8HP In ~he wellbore. P../. IS he.ld constant .once production begins. The mathematIcal problem ISthen I a ap 0IJ.C ap (r-)= -, (8.9) r ar ar O.264 k at ..

1 -(tJ.

0 -(IJ(i.e..dimensionless). IJ. -(miLt). c -(lt~/mJ. r.., h P q B

-(LJ, -(lJ, ..~I -(m/Lt-J, -(l3/t), and -(I).

.parameters A co~~e.nt on t~e Units of I?ress~r_eiP. and compressIbIlity. c ~whlc~ ha~ the units pSI) .may be helpful: Pressure IS defIned as force .per unIt ~rea. From Ne~'Ion's second law, force IS mass times f I

acceleration; thus, force has the fundamental units Force:. [ml/t 2J. Thus, pressure has the fundamental units Pressure:. (ml/t2l2J or (m/lt2), ~--

'- Append'. C

p=p,at/=Oforallr.

(8.10)

p=p...ratr=r...forallt>O.

(8.11)

=Oforallt>O. (8.12) ar ' '.. As before. our approach ~'ill be to eliminate all and variables \\'ith dimensions from the differential equation, and initial and boundary conditions. The appropriate definition of r 0 is again r = rlr .; for t ,it is 0 ..0 10 =.

O.264 kt olAL'r!

The~e definitions will eliminate all parameters from ~q. ~.9..The appropriate definition of Po for this situation I~

I

1

-DIMENSIONLESS

PD -Pi

VARIABLES

'.1:~' -:,;;;eo

( 0.00)264,

-P

Pi-Pwf With these definitions,

(

~

.!:.- a'D .!-.- 'Da'D 'D

)),1

qdl.

4>JJ.c,,; 0

the mathematical statement

of the problem becomes.

k

105

Therefore,

)-~

-aiD'

tbCh,: (Pi -Pw Qp=I.119~"fw\i-pWflQPD'

(8.13)

PD=OatID=Oforall,

,

(8.14)

The implication production, Q

>0,

(8.IS)

mined QpD'

D PD=lat'D=lforalll D

a -1!.Q ! -it a'D I -0 for all ID >0. (8.16) ro.,..".=,o.. Thus, dimensionless pressure PD is a function of dimensionless time, ID' and dimensionless radius'D for a given value of the dimensionless parameter, D since no other variables or parameters appear in the differential equation, or initial or boundary conditions. However, the appropriate definition of dimensionless pressure is different in this constantpressure case from the constant-rate case-with the appropriate definition dictated by the boundary conditions. 8ecause (his cons(ant-pressure case is of considerable prac(ical importance, \\'e proceed further with our analysis. It is of in(eres( (0 de(ermine ins(an(aneousvolume production andIns(an(aneous cumula(ive produced for thisrarecase. production rare q is 0.001127 (2~, wh) k ap q = -B -a I ' JJ. , r.

(8.17)

fr~m "'hlch

)

(8.20)

of our analysis is that cumulative (stock-tank barrels), can be deter-

~mensionless IS based on

cum~lative solutIons

to

production, Eqs. 8.13

through 8.16. Since QpD is based on these solutions, is a function of I D and, [N only (for, D = I); thus, it appears possible that Q D could be tabulated (or plotted) as a function of~D for selected values of 'IN. In fact, this has been done by van Everdingen and Hurst; this problem is their well known constantpressure case. The QpD values also can be interpreted as dimensionless cumulative water influx from a radial aquifer into a reservoir if , w is interpreted as the reservoir radius and, e as the aquifer radius.

.

E

xercise 8.1 Consider (he radial flow of a gas expressed in terms of (he pseudopressure, ;(t, by Eq. A.13. Define dimensionless pseudopressure as -;(tD=S0300kh T sc

T(;(ti-;(t),

-,qgPsc dimensionless time as -0.00)264 kl ID2' oJJ.iC Ii' h.

and cumula(ive production Qp is

and

I .1

QP=- 24 ~ qdl.

.~ ...T I( IS con~'eruent to define a dimensIonless productIon rate, qD. qBII. q D = 0.00708 kh (p, -P -). Th I WJ en, qD = ~ I a'D ro- I

(8.19)

, \\e also defIne a dImensIonless cumulatIve produced volume, QpD. ,ID B QpD = \ q D d'D = I .11. 1 Q 0.OO708Ah (Pi-Ph/)

..,

---~I

!

'D='/'".,

(8.18)

T sc PIc qg lI.i

= = = = =

reservoir (emperature, .R, s(andard-condi(ion (empera(ure, .R, standard-condition pressure, psia, 2a -, s tlO\\. rate Ms cf/ D , gas viscosity evaluated at original reservoir pressure, cp, and Cgl = gas compressibility evalua(ed at original reservoir pressure, psi -1

The cylindrical reservoir is initially at uniform pressure. Pi; there is no flow across the ou(er boundary of radius 'e: and flow rate into the well bore of radius, h' is constant (expressed at standard I.'onditions). Write the differen(ial equation, and initial and bou~dary c~ndi.tions. in dimensional form; (hen re\\Tltethemmdlmenslonlessform.

I

~:.. ~; """': I ..

'AP"~\,.,t(" C " In'

.~

S'~;'. " .-tMl1

as

.."

Appendix C

Van Everdingen and Hurst Solutions to Diffusivity Equations Introduction In Appendix B, we sho"'ed that solutions to differential equations describing flow in a petroleum reservoir for given initial and boundary conditions can be expressed compactly using dimensionless variables and parameters. In this appendix, we examine four of these solutions that are important in reservoir engineering applications. C:°nstant Rate at Inner Boundary, No Flow Across Outer Boundary This solution of the diffusi\ity equation models radial flow of a slightly compressible liquid in a homogeneous reservoir of uniform thickness; reservoir at uniform pressure Pi before production; no production flow across at constant the outerrate boundary q from (at the r single = r e); well and (centered in the reservoir) "'ith wellbore radius 'MI. The solution-pressure as a function of time and radius for fixed values of re' '.." and rock and fluid pr°J:'erties.- is expressed most conveniently in terms of dImensionless variables and parameters: PD=/(fD,rD"eD)'

'..'.."...'.".'

.(C.I)

h were 0.OOi08 kh(Pi -p) q8J1.'

ID <0.25 r;D. 3. For 100
Table C.I can be (C.3)

(This equation is identical to Eq. 1.10 for 5=0. It begins to become slightly less accurate at ID=0.0625

loss in accuracy is not of sufficient magnitude to cause problems in most practical applications.) 4. Table C.2 presents PD as a function of 1D for 1.5<'eD<10. (a) For values of 1D smaller than the value listed

-'" r D -r/r

for a given reO' the reservoir is infinite acting, and Table C.l should be used to determinePD.

...,

(b) For values of 1D larger than the largest value listed for a given reD (or, more correctly, for 25< 1D

and -and 'eD-rt'/r.." Eq. C.l states that PD is a function of the variables 1D and r D for a flXed value of the parameter, eD. The most important solution is that for pressure at the wellbore radius (,=, MIor'D = I): D.I

=f(/D"~D)'

When expressed in terms of dimensionless pressure

~

PD =2.fiD/r. (C.2) 2. Table C.l IS valid for fimte reservoIrs WIth

-0.IJ.C r~ '

PDI,

':

r;D' but there is no simple approximation between this value and 1D =0.25 ~D. Fortunately, this slight ..

PD =

,

evaluated at 'D = 1, Eq. 1.6 shows the functional form of /(1 D" eD) -an infinite series of exponentials and Besselfunctions. This series has been evaluated I for several values of, eD over a wide range of values of ID. Chatas2 tabulated these solutions; a modification of Chatas' tabulation is presented in Tables C.l and C.2. Some important characteristics of this tabulation include the following. I. Table C.l presents values of PD in the range 1D < I,(XX) for an infinite-acting reservoir. For ID <0.01, PD can be approximated by the relation

0.25 '~D < 1D)' PD can be calculated from2 PD =

2(1D + 0.25) 2 1 ' eD -

(3':D-4(~n

4r:D In' eD-~D -I) -1 )2' (C.4) 4(eD-I) (c) A special case of Eq. C.4 arises when r;D.I; then,

i I

~I

, VAN EVERDINGEN

AND HURST SOLUTIONS TO DIFFUSIVITY EQUATIONS

107

TABLEC.1 -Po VI. to -INFINITE RADIALSYSTEM, CONSTANTRATEATINNERBOUNDARY

Estimate the pressure on the inner boundary of the sand pack at times of 0.00 I, 0.0 I, and 0.1 hour.

-.!.S!0 0.0005 0.001 0.002 0.003 0.004 0.005 0.006 0.007

-P..P- ..!.sL -P..P- ~ -p..p0 0.15 0.3750 60.0 2.4758 0.0250 0.2 0.4241 70.0 2.5501 0.0352 0.3 0.5024 80.0 2.6147 0.0495 0.4 0.5645 90.0 2.6718 0.0603 0.5 0.6167 100.0 2.7233 0.0094 0.6 0.6622 150.0 2.9212 0.0774 0.7 0.7024 200.0 3.0036 0.0845 0.8 0.7387 250.0 3.1726 0.0911 0.9 0.7716 300.0 3.2630

Solution. We first calculate 10 and r ~D:

0.008

0.0971

1.0

1.2

0.8672

0.8019

350.0

3.3394

0.01

0.1081

1.4

0.9160

450.0

3.4641

0.009

0.1028

400.0 3.4057

0.015 0.02 0.025 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1

0.1312 2.0 1.0195 500.0 3.5164 0.1503 3.0 1.1665 550.03.5643 0.1669 4.0 1.2750 600.0 3.6076 0.1818 5.0 1.3625 650.0 3.6476 0.2077 6.0 1.4362 700.0 3.6842 0.2301 1.4997 800.0 750.0 37505 3.7184 0.2500 7.0 8.0 1.5557 0.2680 9.0 1.6057 850.0 3.7805 0.2845 10.0 1.6509 900.0 3.8088 0.2999 15.0 1.8294 950.0 3.8355 0.3144 20.0 1.9601 1,000.0 3.8584 30.0 2.1470 40.0 2.2824 50.0 2.3884 NotesFor'O<0.01.PO.2"o" For100<10<0.25 '~D-PO.0511n '0+ 0809071

r~D=IO/I=IO, 0.264 kl 1D = ~ --2 ~~ I' MI (0 264)(1
Then, the folloWIng data result. I

1 (hour)

,"

0.001

ID -4

Po 1.275

0.01 0.1

40 400

2.401 9.6751

PD=~+lnr~-¥4. (C.5) r~D 5. The PD solutions in Tables C.! and C.2 also apply to a reservoir of radius, MIsurrounded by an aquifer of radius e when there is athe constant rateFor of water influx from rthe aquifer into reservoir. this case, values of, eD in the range 1.5 to 10.0, as in Table C.2, are of practical importance. For most well problems, r ~D is larger than 10.0, and the aQproximations given by Eq. C.3 for lOO
E.\"ampleC.l- Use of PD Solutions for No-Flow Boundary Problem. In a large laboratory flow experiment, fluid was produced into a I-ft-radius perforated cylinder from. a sand-packed model with a rad~us of 10 ft. No fluId flo~ed across the external radIus of the. m.odel. PropertIes o;f the sandpack and produced fluId mclud~ the followIng. k = h = p. = ~ = q = B= cP = c1 =

I darcy, 0.5 ft, 15 psia, 2 cp, 1.0 STB/D, I.ORB/STB, 0.3, and 0.11 x 10 -3 psi -I.

TableC.I (reservoir infinite acting) Tabl e C .(r 2 ~D = 10) Eq. C.4

- (025 )( 10)2 -.: = 25; thus, Eq. C.41s usedto calculate PD. 2(1D + 0.25) PD = 2 ~D-I

Note th at f or 1 D =. 400'D1 >0 .~D 25 ~

(3r:D -4r:D 21

i 1

SourceofpD

In r ~D -U;D

-4(r;D

-I)

-1)2

=9.6751.

.

k, J.

-

,.~ I: 1

A rearrangement of the definition of PD results in qB P =P 1-141.2-.!:-PD kh (141.2)(1.0)(1.0)(2) = 15-(1
P (psia)

0.001

14.28

~.~I ..

I~.~

Constant Rate at Inner Boundary, Constant Pressure at Outer Boundary This solution of the diffusivity equation models radial flow of a slightly compressible liquid in a homogeneous reservoir of uniform thickness; reservoir at uniform pressure Pi before production; unchanging pressure, also Pi' at the outer boundary (at r=, ~); and production at constant rate q from the single well (centered in the reservoir) with wellbore radius ~MI'The solutions, PD (evaluated a.t' D = I), as a function of 1D for fiXed values of r ~ In the range I.S<'~D<3,, are given in Table C.3. The dimensionless variables PD' 'D' 'D' and r ~ have the same definitions as in the previous section. I

I

~

-2.0

'.0 = 2.0 -

'.0 = 2.5

'.0 = 3.0

'.0 = 3.5

5.0

3.649

5.0

2.398

1.5 0.927 1.6 0.948 1.7 0.968 1.8 0.988 1.9 1.007 2.01.025 2.2 1.059 2.4 1.092 2.61.123 2.81.154 3.0 1.184 35 1.255 4.01.324 4.51.392 5.0 1.460 5.5 1.527 6.0 1.594 6.5 1.660 7.01.727 8.0 1.861 9.0 1.994 10.0 2.127

'.0 = 4.0

TABLE C.2 -Po Ys. to -FINITE RADIAL SYSTEM WITH CLOSED EXTERIOR BOUNDARY, CONSTANT RATE AT INNER BOUNDARY '.0 = 1.5

1.444

!.P---EJL-!.P---ER-!.P---ER-!.P---ER-!.P---ER-!.P---ER0.06 0.251 0.22 0.443 0.40 0.565 0.52 0.627 1.0 0.802 0.08 0.288 0.24 0.459 0.42 0.576 0.54 0.636 1.1 0.830 0.10 0.322 026 0.476 0.44 0.587 0.56 0.645 12 0857 0.12 0.355 0.28 0.492 0.46 0.598 0.60 0.662 13 0.882 0.14 0.387 0.30 0507 0.48 0.608 0.65 0683 1.4 0.906 0.160.420 0.320.522050 0.618 0.70 0.703 1.5 0929 0.18 0.452 034 0536 0.52 0.628 0.75 0.721 1.6 0.951 0.20 0.484 0.36 0.551 0.54 0.638 0.80 0.740 1.7 0.973 0.220.516 0.38 0.565 0.56 0.6470.850.758 1.8 0.994 0.240.548 0.400.5790.58 0.6570.90 0.776 1.9 1.014 0.26 0.580 0.42 0.593 0.60 0.666 0.95 0.791 2.0 1.034 0.28 0.612 0.44 0.607 0.65 0.688 10 0.806 225 1.083 0.300.644 0.460.621 0.700.710 1.2 0.8652.50 1.130 0.350.724 0.48 0.6340.750.731 1.4 0.9202.751.176 040 0.804 0.50 0.648 0.80 0.752 1.6 0.973 3.0 1.221 0.45 0.884 0.60 0.715 0.85 0.772 2.0 1.076 4.0 1.401 0.50 0.964 0.70 0.782 0.90 0.792 3.0 1.328 5.0 1.579 0.55 1.044 0.80 0.849 0.95 0.812 4.0 1.578 6.0 1.757 0.60 1.124 0.90 0.915 1.00 0.8325.0 1.828 0.65 1.204 1.0 0.982 2.0 1.215 0.70 1.284 2.0 1.649 3.0 1.506 0.75 1.364 3.0 2.316 4.0 1.977 0.80

...

C'-!~:~

1.651 12.0 1.732 1.673 12.5 1.750 1.693 13.0 1.768 1.713 13.5 1.784 1.732 14.0 1.801 1.750 14.5 1.817 1.768 15.01.832 1.786 15.5 1.847 1.803 16.01.862 1.819 17.0 1.890 1.835 18.0 1.917 1.851 19.0 1.943 1.867 20.0 1.968 1.897 22.0 2.017 1.926 24.0 2.063 1.955 26.0 2.108 1.983 28.0 2.151 2.037 30.0 2.194 2.096 32.0 2.236 2.142 34.0 2.278 2.193 36.0 2.319 2.244 38.0 2.360 2.345 40.0 2.401 2.446 50.0 2604 2.496 60.0 2.806 2.621 70.0 3.008 2.746 80.0 3.210 2.996 90.0 3.412 3.246 100.0 3.614

'.0=4.5 '.0=5.0 '.0=6.0 '.0=7.0 '.0=8.0 '.0=9.0 '.0=10.0 !.P---E-L!.P---ER-!JL-ER-!.P---EJL!.P---EJL!.P---ER-~~ 1.023 3.0 1.167 4.0 1.275 6.0 1.436 8.0 1.556 10.0 1.040 3.1 1.180 4.5 1.322 6.5 1.470 8.5 1.582 10.5 1.056 3.2 1.192 5.0 1.364 7.0 1.501 9.0 1.607 11.0 1.702 3.3 1.204 5.5 1.404 7.5 1.531 9.5 1.631 11.5 1.087 3.4 1.215 6.0 1.441 8.0 1.559 10.0 1.653 12.0 1.102 3.5 1.227 6.5 1.477 8.5 1.586 10.5 1.675 12.5 1.116 3.6 1.238 7.0 1.511 9.0 1.61311.0 1.69713.0 1.130 3.7 1249 7.5 1.544 9.5 1.638 11.5 1.717 13.5 1.144 3.8 1.259 8.0 1.57610.0 1.66312.0 1.73714.0 1.158 3.9 1.270 8.5 1.607 11.0 1.711 12.5 1.757 14.5 1.171 4.0 1.281 9.0 1.638 12.0 1.757 13.0 1.776 15.0 1.197 4.2 1.301 9.5 1668 13.0 1.810 13.5 1.795 15.5 1.222 4.4 1321 100 1.698 14.0 1.845 14.0 1.813 16.0 1.246 4.6 1.340 11.0 1.757 15.0 1.888 14.5 1.831 17.0 1.269 4.8 1.360 12.0 1.815 16.0 1.931 15.0 1.849 18.0 1.292 5.0 1.378 130 1.873 17.0 1.974 17.0 1.919 190 1.349 5.5 1 424 14.0 1.931 18.0 2.016 19.0 1.986 20.0 1.403 6.0 1.469 15.0 1.988 19.0 2.058 21.0 2.051 22.0 1.457 6.5 1.513 16.0 2.045 20.0 2.100 23.0 2.116 240 1.510 7.0 1 556 17.0 2.103 22.0 2.184 25.0 2.180 260 1.615 7.5 1598 180 2.160 24.0 2267 30.0 2.340 28.0 1.719 8.0 1.641 19.0 2.217 26.0 2.351 35.0 2.499 30.0 1.823 9.0 1.725 20.0 2.274 28.0 2.434 40.0 2.658 34.0 1.927 10.0 1.808 25.0 2.560 30.0 2.517 45.0 2.817 380 2.031 11.0 1.892 30.0 2.846 40.0 2.135 12.0 1.975 45.0 2.239 13.0 2.059 50.0 2.343 14.0 2.142 60.0 2.447 15.0 2.225 70.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.2 3.4 3.6 3.8 4.0 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 11.0 12.0 13.0 14.0 15.0

NotesFor10smatlerIhanva'lJ~s listed,nthIstablefora gIven reO reservO" IS,nl.nlleactIng ~lndPO,nTableC1 For2S<10 and'0 largerthan.aluesonlable .IriO -11 - 3f~O-.f~oln'eO-2f~O-1 ~fio -1)2 (';'+2/01 PO-' ForWellsonbOunded r.servoofs WIth 2 .1. feD Po,2/0+lnf.O-" f~O

..

.

..

.

I I

I i

-

VAN EVERDINGEN AND HURST SOLUTIONS TO DIFFUSIVITY EQUATIONS

TABLE

C.3 -Po

YI. to -FINITE

AT EXTERIOR

'_0 = 1.5 to

Po

'_0 = 2.0 to

Po

RADIAL

BOUNDARY,

SYSTEM

CONSTANT

WITH RATE

109

FIXED

'-0 = 2.5

'_0 = 3.0

'_0 = 3.5

to

to

to

Po

Po

CONSTANT

AT INNER

PRESSURE

BOUNDARY

'_0 = 4.0

Po

to

Po

O:OSOo:2"3O"f;iii"~~o:5O"2-O:-SO00-O:-SO0:620~D:8()24:0~ 0.055 0.240 0.22 0.441 0.35 0.535 0.55 0.640 0.60 0.665 1.2 0.857 0.060 0.070 0.080 0.090 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 0.35 0.40 0.45 O.SO 0.60 0.70 0.80

0.249 0.24 0.266 0.26 0.282 0.28 0.292 0.30 0.307 0.35 0.328 0.40 0.344 0.45 0.356 O.SO 0.3670.55 0.3750.60 0.381 0.65 0.386 0.70 0.390 0.75 0.393 0.80 0.396 0.85 0.400 0.90 0.402 0.95 0.404 1.0 0.405 1.2 0.405 1.4 0.405 1.6 0.405 2.0 1.8 2.5 3.0

0.457 0.40 0.472 0.45 0485 0.50 0.498 0.55 0.527 0.60 0.552 0.70 0.573 0.80 0591 090 0606 1.0 0.6191.2 0.630 1.4 0.639 1.6 0.647 1.8 0654 2.0 0.660 2.2 0665 2.4 0.669 2.6 0.673 2.8 0.682 3.0 0.688 3.5 0.690 4.0 0.692 0.692 4.5 5.0

0.564 0.591 0.616 0.638 0.659 0.696 0.728 0.755 0.778 0.815 0.842 0.861 0.876 0.887 0.895 0.900 0.905 0.908 0.910 0.913 0.915 0.916 0.916

0.693 5.5 0.693 6.0

0.916 8.0 0.916 10.0

'_0=8.0

'_0=10.0

'_0=15.0

to

to

to

Po

Po

Po

7-:0- 1:""499 10:0 1-:651"""20:0 ~ 7.5 8.0 8.5 9.0 9.5 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 30.0 35.0 40.0 45.0 SO.O 60.0 70.0 80.0

Noles i

~

.

1.527 1.554 1580

12.0 1.730 14.0 1.798 16.0 1.856

22.0 2.003 24.0 2.043 26.0 2.080

0.60 0.70 0.80 0.90 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0

0.662 0.70 0.702 0.80 0.738 0.90 0.770 1.0 0.799 1.2 0.850 1.4 0.892 1.6 0.927 1.8 0.955 2.0 0.980 2.2 1.000 2.4 1016 2.6 1.030 2.8 1.042 3.0 1.051 3.5 1.069 4.0 1.080 5.0 1.087 6.0 1.091 7.0 1.094 8.0 1096 9.0 1.097 1.097 10.0 12.0

0.705 0.741 0.774 0.804 0.858 0.904 0.945 0.981 1.013 1.041 1.065 1.087 1.106 1.123 1.153 1.183 1.225 1.232 1.242 1.247 1.2SO 1.251 1.252

1.098 14.0 1.099 16.0

1.253 1.253

1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.4 3.8 4.5 5.0 5.5 6.0 7.0 8.0 9.0 10.0 12.0 140 16.0 18.0

0.905 0.947 0986 1.020 1.052 1.080 1.106 1.130 1.152 1.190 1232 1.266 1.290 1309 1.325 1.347 1.361 1.370 1.376 1.382 1.385 1386 1.386

'_0=20.0

'_0=25.0

'_0=30.0

to

to

to

~

Po

35.0 2.219 40.0 2.282 45.0 2.338

1.604 18.0 1.907 28.0 2.114 SO.O 1.627 20.0 1.952 30.0 2.146 60.0 1.648 25.0 2.043 35.0 2.218 70.0 1.724 30.0 2.111 40.0 2.279 80.0 1.786 35.0 2.160 45.0 2.332 90.0 1.837 40.0 2.197 50.0 2.379 100.0 1.879 45.0 2.224 60.0 2.455 105.0 1.914 50.0 2.245 70.0 2.513 110.0 1.943 55.0 2.260 80.0 2.558 115.0 1.967 60.0 2.271 90.0 2.592 120.0 1.986 65.0 2.279 100.0 2.619 125.0 2.002 70.0 2.285 120.0 2.655 130.0 2.016 75.0 2.290 140.0 2.677 135.0 2.040 80.0 2.293 160.0 2.689 140.0 2.055 90.0 2.297 180.0 2.697 145.0 2064 100.0 2.300 200.0 2.701 1SO.0 2.070 110.0 2.301 220.0 2.704 160.0 2.076 120.0 2.302 240.0 2.706 180.0 2.078 130.0 2.302 2600 2.707 200.0 2.079 140.0 2.302 280.0 2.707 240.0 160.0 2.303 300.0 2.708 280.0 300.0 400.0 500.0

Po

Po

i148 -so-:o2:"389-ro:-o ~ 2.388 2475 2.547 2.609 2.658 2.707 2.728 2747 2.764 2.781 2.796 2.810 2.823 2.835 2.846 2.857 2.876 2.906 2.929 2.958 2.975 2.980 2.992 2.995

55.0 2.434 60.0 2.476 65.0 2.514 70.0 75.0 80.0 85.0 900 95.0 100.0 120.0 140.0 1600 180.0 200.0 220.0 240.0 260.0 280.0 3000 350.0 400.0 450.0 500.0 600.0 700.0 800.0 900.0

80.0 2.615 90.0 2.672 100.0 2.723

2.550 120.0 2.583 140.0 2.614 160.0 2.643 165.0 2.671 170.0 2.697 1750 2.721 1800 2.807 2000 2.878 250.0 2.936 300.0 2.984 3500 3.024 400.0 3.057 450.0 3.085 500.0 3.107 600.0 3.126 700.0 3.142 8000 3171 900.0 3.189 1.000 3.200 1.200 3.207 1.400 3.214 3.217 3.218 3.219

2.812 2.886 2.950 2.965 2.979 2.992 3.006 3.054 3.150 3.219 3.269 3.306 3.332 3.351 3.375 3.387 3.394 3.397 3.399 3.401 3.401

'_0 = 6.0 to

Po

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 10.0 12.0 14.0 16.0 18.0 20.0 22.0 24.0 26.0 28.0 ~.O 35.0 40.0 SO.O

1.320 1.361 1.398 1.432 1.462 1.490 1.516 1.539 1.561 1.580 1.615 1.667 1.704 1.730 1.749 1.762 1.771 1.777 1.781 1.784 1.787 1.789 1.791 1.792

-

'_0=40.0 to

'

Po

120 2:B13 140 2.888 160 2.953 180 3.011 200 220 240 260 280 300 350 400 450 500 550 600 650 700 800 900 1,000 1.200 1,400 1.600 1,800 2.000 2.500

3.063 3.109 3.152 3.191 3.226 3.259 3.331 3.391 3.440 3.482 3.516 3.545 3.568 3.588 3.619 3.640 3.655 3.672 3.681 3.685 3.687 3.688 3.689

.-

:

..

For to sm.ller th.h values IISled in thIs table '°' a 9,ven '_0 'eservo.' IS InfInIte aCllno Find Po In Table C 1 For to '.'ger

Ihan values IISled in Ihis lable. Po i In '_0

IIJ

~TABLE C.3 -CONTINUED

'.0=50.0 '.0=60.0 '.0=70.0 '0 Po to Po to Po 200 ~ ~ i2S7 ~ 3.512 220 3.111 400 3.401 600 3603 240 3.154 500 3.512 700 3680 260 3.193 600 3602 800 3.746 280 3.229 700 3.676 900 3803 300 3.263 800 3.739 1.000 3.854 350 3.339 900 3.792 1.200 3937 400 3.405 1.000 3.832 1.400 4.003 450 3.461 1.200 3.908 1.600 4.054 500 3.512 1.400 3.959 1.800 4.095 550 3.556 1,600 3.996 2.000 4127 600 3.595 1.800 4.023 2.500 4.181 650 3.630 2.000 4.043 3,OCKJ4.211 700 3.661 2,500 4.071 3.500 4.228 750 3.688 3,000 4.084 4.OCKJ4.237 800 3.713 3,500 4.090 4.500 4.242 850 3.735 4.000 4.092 5.000 4.245 900 3.754 4.500 4093 5.500 4.247 950 3.771 5.000 4094 6.OCKJ 4.247 1.000 3.787 5.500 4094 6.500 4.248 1.200 3.833 7,000 4.248 1.400 3.862 7,500 4.248 1.600 3.881 8.OCKJ 4.248 1,800 3.892 2,000 3.900 2,200

'.0=80.0 to Po 600 3.603 700 3.680 800 3.747 900 3.805 1.000 3.857 1.200 3.946 1.400 4.019 1.500 4.051 1.600 4.080 1.800 4130 2.000 4.171 2.500 4.248 3.000 4.297 3.500 4.328 4.000 4347 4,500 4.360 5.000 4368 6.000 4.376 7,000 4.380 8,000 4.381 9.000 4.382 10.000 4.382 11.000 4.382

'.0=90.0 to Po 800 3.747 900 3.806 1.000 3.858 1,200 3.949 1,300 3.988 1.400 4.025 1.500 4058 1.800 4.144 2.000 4.192 2.500 4285 3.000 4.349 3.500 4.394 4.000 4.426 4.500 4.448 5,000 4.464 6.000 4.482 7.000 4.491 8.000 4.496 9.000 4.498 10.000 4.499 11.000 4.499 12.000 4.500 14.000 4.500

'.0=100.0 to PD 1.000 3859 1.200 3 949 1.400 4026 1.600 4.092 1.800 4.150 2.000 4200 2.500 4303 3,000 4379 3,500 4.434 4.OCKJ 4478 4.500 4.510 5,000 4.534 5.500 4.552 6,000 4.565 6.500 4579 7.000 4583 7,500 4.588 8.000 4.593 9.000 4.598 10.000 4601 12,500 4604 15.000 4.605

'.0=200.0 to Po 1.500 4.061 2.000 4.205 2.500 4.317 3.OCKJ4.498 3.500 4.485 4.OCKJ4.552 5.OCKJ4.663 6.OCKJ4.754 7,OCKJ4.829 8.OCKJ 4.834 9.OCKJ 4.949 10.OCKJ4.996 12.OCKJ5.072 14.000 5.129 16.OCKJ5.171 18.OCKJ5.203 2O,OCKJ5.227 25.OCKJ5.264 3O.OCKJ5.282 35.OCKJ5.290 40.OCKJ5.294

-

3.904

2.400 3.907 2,600 3.909 2.800 3910

'.0 = 300.0

'.0 = 400.0

'.0 = 500.0

-.!.J?- -E.!L -.!.J?- -.PJL ~ 6.000 8.000 10.000 12.000 14.000 16.000 18.000 20.000 24,000 28.000 30.000 40.000 50.000 60,000 70.000 80.000 90.000 100.000 120.000 140,000

4.754 4.898 5.010 5.101 5177 5242 5.299 5.348 5.429 5.491 5.517 5606 5652 5.676 5.690 5.696 5.700 5.702 5.703 5.704

150,000 5.704

'" ;

it

15.000 5.212 20.000 5.356 30.000 5.556 40.000 5.689 50.0005.781 60.0005.845 7000C 5.889 80.000 5.920 90.000 5.942 100.000 5.957 110.000 5.967 120.000 5.975 125.000 5.977 130,000 5.980 140.000 5.983 160.000 5.988 180.000 5.990 200.000 5.991 240.000 5.991 260,000 5.991

'.0 = 600.0

-.PJL ~

20.000 5.365 25.000 5468 30.000 5.559 35.000 5.636 40.000 5.702 45.0005.759 50000 5.810 60.000 5.894 70.000 5.960 80.000 6.013 90.000 6.055 100.000 6088 120.000 6.135 140.000 6.164 160.000 6.183 180.000 6.195 200.000 6.202 250.000 6.211 300000 6.213 350.000 6.214 400.000 6.214

'.0 = 700.0

-.PJL -.!.J?- -E.!L

40.000 5.703 45.000 5.762 50.000 5.814 60.000 5.904 70.0005979 80.000 6.041 90.000 6.094 100.000 6.139 120.000 6.210 140.000 6.262 160.000 6.299 180.000 6.326 200.000 6.345 250.000 6.374 3OO.OCKJ 6.387 350.000 6.392 400.000 6.395 500.000 6.397 600.000 6397

50.000 5.814 60,000 5.905 70,000 5.982 80.000 6.048 90.0006.105 100.0006156 120.000 6.239 140.000 6305 160.000 6.357 180.000 6.398 200.000 6.430 250.000 6.484 300,000 6.514 350,000 6.530 400.000 6.540 450.000 6.545 500.000 6.548 600.000 6.550 700.000 6.551 800.000 6.551

'.0 = BOO.O

to-.PJL 70.OCKJ5.983 SO.OCKJ 6.049 9O.OCKJ6.108 100.OCKJ6.160 120.OCKJ6.249 140.OCKJ6.322 160.OCKJ6.382 1SO.OCKJ 6.432 2OO,OCKJ 6.474 250.OCKJ6.551 300.000 6.599 350.OCKJ6.630 400.000 6.650 450,OCKJ6.663 5OO.OCKJ 6.671 550,OCKJ6.676 600.000 6.679 700.000 6.682 800.000 6.684 1.000.000 6.684

..

r

VANEVERDINGEN ANDHURST SOLUTIONS TODIFFUSIVITY EQUATIONS

111

TABLEC.3-CONTINUED

I

'.0 =900.0

, i

to a~ 9.0(104) 1.0(105) 1.2(105) 14(105) 1.6(105) 1.8(105) 2.0(105) 2.5(105) 3.0(105) 4.0(105) 45(105) 50(105) 55(105) 60(105) 70(105) 8.0(105) 9.0(105) 10(106)

'.0 = 1.000.0 '.0 = 1.200.0 '.0 =1.400.0 '.0 = 1.600.0 '.0 = 1.800.0

Po to 6-:-o-:i91~ 6.108 1.2(105) 6.161 1.4(105) 6251 1.6(105) 6.327 1.8(105) 6.392 2.0(105) 6447 2.5(105) 6.494 3.0(105) 6.587 3.5(105) 6.652 4.0(105) 6.729 4.5(105) 6.751 5.0(105) 6.766 5.5(105) 6.777 6.0(105) 6.785 7.0(105) 6.794 8.0(105) 6.798 9.0(105) 6.800 1.0(106) 6.801 1.2(106)

Po i161 6.252 6.329 6.395 6.452 6.503 6.605 6.681 6738 6.781 6.813 6.837 6.854 6.868 6.885 6895 6.901 6.904 6.907

to Po to ~ 6:""5():; ~ 3.0(105) 6.704 2.5(105) 4.0(105) 6.833 3.~105) 5.0(105) 6918 3.5(105) 60(105) 6975 4.~105) 7.0(105) 7013 5.~105) 8.0(105) 7038 6.~105) 9.0(105) 7056 7.~105) 1.0(106) 7067 8.~105) 1.2(106) 7.080 9.~105) 1.4(106) 7.085 1.~106) 16(106) 7088 15(106) 1.8(106) 7089 2.~106) 1.9(106) 7089 2.5(106) 2.0(106) 7.090 3.~106) 2.1(106) 7.090 3.1(106) 2.2(106) 7.090 3.2(106) 2.3(106) 7090 3.3(106) 2.4(106) 7090

Po to Po 6:""5():; i"5(iO5) i619 6.619 3.0(105) 6710 6.709 3.5(105) 6.787 6.785 4.0(105) 6853 6849 5.0(105) 6.962 6.950 6.0(105) 7.046 7026 7.0(105) 7.114 7.082 8.0(105) 7.167 7123 9.0(105) 7.210 7.154 1.0(106) 7244 7.177 1.5(106) 7.334 7229 2.0(106) 7364 7.241 2.5(106) 7.373 7.243 3.0(106) 7376 7.244 35(106) 7.377 7244 4.0(106) 7378 7.244 4.2(106) 7.378 7244 4.4(106) 7378

to ~ 4.0(105) 5.0(105) 6.0(105) 7.0(105) 8.0(105) 9.0(105) 1.0(106) 1.5(106) 2.0(106) 3.0(106) 4.0(106) 5.0(106) 5.1(106) 5.2(106) 5.3(106) 5.4(106) 5.6(106)

Po 6-:7"10 6854 6.965 7.054 7.120 7.183 7238 7.280 7.407 7.459 7489 7495 7495 7.495 7.495 7.495 7.495 7.495

'.0 = 2.000.0 '.0 = 2.200.0 '.0 =2.400.0 '.0 = 2.600.0 '.0 =2.800.0 '.0 = 3.000.0 to 4~ 5.0(105) 6.0(105) 7.0(105) 8.0(105) 9.0(105) 1.0(106) 1.2(106) 1.4(106) 1.6(106) 1.8(106) 2.0(106) 2.5(106) 3.0(106) 3.5(106) 4.0(106) 5.0(106) 6.0(106) 6.4(106)

Some solutions

Po 6:8"54 6.966 7.056 7.132 7.196 7.251 7.298 7.374 7.431 7.474 7.506 7.530 7.566 7.584 7.593 7.597 7.600 7.601 7.601

to ~ 5.5(105) 6.0(105) 6.5(105) 7.0(105) 7.5(105) 8.0(105) 8.5(105) 9.0(105) 1.0(106) 1.2(106) 1.6(106) 2.0(106) 2.5(106) 3.0(106) 3.5(106) 4.0(106) 5.0(106) 6.0(106) 7.0(106)

Po to 6:""966 6~ 7.013 7.0(105) 7.057 8.0(105) 7.097 9.0(105) 7.1331.0(106) 7.167 1.2(106) 7.199 1.6(106) 7.229 2.0(106) 7.256 2.4(106) 7.307 2.8(106) 7.390 3.0(106) 7.507 3.5(106) 7.579 4.0(106) 7.631 5.0(106) 7.661 6.0(106) 7.677 7.0(106) 7.686 8.0(106) 7.693 9.0(106) 7.695 9.5(106) 7.696

8.0(106)

7.696

important properties include the following:

of

these

Po to ~ ~ 7. 134 8.~105) 7.200 9.~105) 7.259 1.~106) 7.3101.2(106) 7.398 1.4(106) 7.526 1.6(106) 7.611 1.8(106) 7668 2.0(106) 7.706 2.4(106) 7.720 2.8(106) 7.745 3.~106) 7.760 3.5(106) 7.775 4.~106) 7.780 5.~106) 7.782 6.~106) 7.783 7.~106) 7.783 8.~106) 7.783 9.~106) 1.0(107)

tabulated

I. For values of tD smaller than the smallest value listed for a given reD' the reservoir is infinite acting, and Table C.I should be used to determine PD' 2. For values of 1 D larger than the largest value listed for a given r (or for 1 >~ ) eD D eD' PD~lnreD'

(C.6)

Po to 7:1"34" 8~ 7.201 9.0(105) 7.259 1.0(106) 7.312 1.2(106) 7.4011.6(106) 7.475 2.0(106) 7.536 2.4(106) 7.588 2.8(106) 7631 3.0(106) 7.699 3.5(106) 7.746 4.0(106) 7.765 5.0(106) 7.799 6.0(106) 7.821 7.0(106) 7.845 8.0(106) 7.856 9.0(106) 7.860 1.0(107) 7.862 1.2(107) 7863 1.3(107) 7863

Po to "i2O1 1~ 7260 1.2(106) 7.312 1.4(106) 7.403 1.6(106) 7.5421.8(106) 7.644 2.0(106) 7.719 2.4(106) 7.775 2.8(106) 7.797 3.0(106) 7.840 3.5(106) 7.870 4.0(106) 7.905 4.5(106) 7.922 5.0(106) 7.930 6.0(106) 7.934 7.0(106) 7.936 8.0(106) 7.937 9.0(106) 7.937 1.0(107) 7.937 1.2(107) 1.5(107)

c, = 20 x 10 -6 psi -1 , r w = 0.45 ft,

..

/" = 0.8 cp, h = 12 ft, and B = 1.25 RB/STB. .. Calculate pressure In the wellbore for production afterO.I, 1.0, and lOO.Odavs. SolUtIon. Smce ID- 0 .VVV264 k/1 ct>/"c,rw2 (I .In

.

.-

~

hours), then Example for

C.2 -Use

Constant-Pressure

of PD Solutions

ID =

Boundary

following. ct>= 0.21, k = 80 md,

(0.00)264)(80)(24)_1

days -

(0.21 )(0.8)(20 x 10 -6)(0.45)2

Problem. An oil well is producing 300 STB/D from a water-drive reservoir that maintains pressure at the original oil/water contact at a constant 3,00) psia. Distance from the well to the oil/water contact is 900 ft. Well and reservoir properties include the

Po 7:3-;-2 7.403 7.480 7.545 7.602 7.651 7.732 7.794 7.820 7.871 7.908 7.935 7.955 7.979 7.992 7.999 8.002 8.004 8.006 8.006

=7.45 x 105t. Also

'

reD-

-r

e -900 r", 0.45

2,vvv. /VVI

Thus, we obtain the following.~

1 12

WELL TESTING

TABLE C.4 -OpO

Ys. to -INFINITE

RADIAL SYSTEM. CONSTANT

Dimension. less Time

DimenSIon. less CumulatIve ProductIon

DImenSIon. less Time

D,menSion. less CumulatIve Production

-'0

__~P-e-

_!O_-

-~p'p-

--'-P--

-9-Po-

'0-

42.433 42781 43129 44858 46574 48277 49968 51648 53317 54976 56625 58265 59895 61517 63 131 64737 66336 67928 69512 71090 72661 74226 75785 77338 78886 80428 81965 83.497 85023 86.545 88062 89575 91084 92589 94090 95568 97081 98.571 100057 101540 103019 104495 105968 107437 108904 110.367 111.827 113284 114738 116189 117638 119083 120526 121966 123403 124838 126270 127699 129126 130550 131972 133391 134808 136223 137635 139.045 140453 141859 143.262 144.664 146.064 147461 148856 150.249 151460 153029

740 750 760 770 775 780 790 800 810 820 825 830 840 850 860 870 875 880 890 900 910 920 925 930 940 950 960 970 975 980 990 1.000 1.010 1.020 1.025 1.030 1.040 1.050 1.060 1.070 1.075 1.080 1.090 1.100 1.110 1.120 1.125 1.130 1.140 1.150 1160 1.170 1.175 1.180 1.190 1.200 1.210 1.220 1.225 1.230 1.240 1.250 1.260 1.270 1.275 1.280 1.290 1.300 1.310 1.320 1.325 1.330 1.340 1.350 1.360 1.370

226904 229514 232.120 234.721 236020 237318 239912 242501 245086 247668 248957 250245 252819 255388 257953 260515 261795 263073 265629 268 181 270729 273.274 274545 275.815 278353 280888 283.420 285948 287211 288473 290995 293.514 296.030 298543 299799 301053 303.560 306.065 308.567 311~ 312.314 313.562 316055 318.5015 321.032 323517 324760 326000 328.480 330.958 333433 335.906 337142 338376 340.843 343.308 345.770 348.230 349460 350688 353.144 355597 358048 360496 361720 362942 365386 367828 370267 372.704 373.922 375139 377.572 380003 382.432 384859

1.975 2.000 2.025 2.050 2.075 2.100 2. 125 2.150 2.175 2.200 2.225 2.250 2.275 2.300 2.325 2.350 2.375 2.400 2.425 2.450 2.475 2.500 2.550 2.600 2.650 2.700 2.750 2.800 2.850 2.900 2.950 3.000 3.050 3.100 3.150 3.200 3.250 3.300 3.350 3.400 3.450 3.500 3.550 3.600 3.650 3.700 3.750 3800 3.850 3.900 3.950 4.000 4.050 4.100 4.150 4.200 4.250 4.300 4.350 4.400 4.450 4.500 4.550 4.600 4.650 4.700 4.750 4.800 4.850 4.900 4.950 5.000 5.100 5.200 5.300 5.400

0.00 001 005 0 10 015 020 025 0.30 040 050 060 070 080 0.90 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 Noles

For/O'

0.000 0112 0278 0404 0520 0606 0689 0758 0898 1020 1 140 1251 1359 1469 1569 2447 3.202 3.893 4539 5153 5743 6314 6.869 7.411 7940 8457 8964 9461 9949 10434 10913 11.386 11855 12319 12778 13233 13684 14131 14.573 15.013 15450 15883 16313 16.742 17 167 17590 18.011 18429 18845 19259 19671 20080 20488 20894 21298 21701 22101 22500 22.897 23291 23684 24076 24466 24855 25.244 25.633 26020 26.406 26791 27.174 27555 27935 28314 28691 29068 29.443

98 99 100 105 110 115 120 125 130 135 140 145 150 155 160 165 170 175 180 185 190 195 200 205 210 215 220 225 230 235 240 245 250 255 260 265 270 275 280 285 290 295 300 305 310 315 320 325 330 335 340 345 350 355 360 365 370 375 380 385 390 395 400 405 410 415 420 425 430 435 440 445 450 455 460 465

0010pO=2'/01'

For /0 . 200 0pO'

-.29881 + 202566' 0 Into

DimenSion. less TIme

DImenSion. less Cumulative PrOductIon

PRESSURE AT INNER BOUNDARY

DimenSIOn. less Time

D,menSion. less Cumulative PrOduction

DimenSIon. less Time

_°2.0-

-.!JJ--

528337 534145 539945 545 737 551522 557299 563~ 568830 574585 580 332 586072 591 086 597532 603252 608965 614672 620372 626~ 631755 637437 643113 648781 660093 671379 682640 693877 705
8.900 9.000 9100 9.200 9.300 9.400 9.5OC 9.600 9.700 9.800 9.900 10.000 12.500 15.000 17.500 20.000 25.000 30.000 35.000 40.000 50.000 60.000 70.000 75.000 80.000 90.000 100.000 125.000 1.5(105) 2.~105) 2.5(105) 3.~105) 4.~105) 5.~105) 6.~105) 7.~105) 8.~105) 9.~105) 1.~106) 1.5(106) 2.~106) 2.5(106) 3.~106) 4.~106) 5.~106) 6.~106) 7.~106) 8.~1061 9.~106) 1.~107) 1.5(101 2.~101 2.5(107) 3~107) 4.~107) 5.~1071 6.~101 7.~101 8.~107) 9.~101 1.~108) 1.5(108) 2.~108) 2.5(108) 3.~108) 4~108) 5.~108) 6.~108) 7~108) 8.~108) 9.~108) 1.~109) 1.5(109, 2.~109) 2.5(109) 3.~1091

DimenSIon. less Cumulative Production

CpO 1.986796 2.~.828 2.026438 2.046227 2.065996 2.0857« 2.105473 2.125.184 2.144878 2.164555 2.184.216 2.203861 2.688967 3.164.780 3.633368 4.095800 5.005.726 5.899508 6.780247 7.650096 9.363.099 11.047.299 12.708358 13.531.457 14.350.121 15.975389 17.586.284 21.560.732 2.538(104) 3.308(104) 4006(104) 4.817(104) 6.2671104) 7699(104) 91131104) 1.0511105) 1.1891105) 1326<105) 1462t105) 2.126(105) 27811105) 3427(105) 4.064!105) 5.3131105) 6.~105) 7.7611105) 8.965(105) 1016(106) 1.134{106) 1252<106) 1828(106) 2398(106) 2961(106) 3517(1061... 461~106) 5.689(106) 6758{106) 7816(106) 8866(106) 9.911(106) 1095(107) 1.6O4{107) 2. 108{107) 2.6071107) 3. 1~107) 4071(107) 503 (107) 5.98 1107) 69281107) 7865(107) 87971107) 9725(107) 1429(108) 1~108) 2.3281108) 2.771(108)

VAN EVERDINGEN AND HURST SOLUTIONS

DImenSIon. less TIme

-.!.~

D,menSIon. less CumulatIve ProductIon

D,menSion. less TIme

TO DIFFUSIVITY EQUATIONS

D,menSIon. less CumulatIve Production

-_
'0

CpO

63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96

29818 JO192 JO565 JO937 31308 31679 32048 32'17 32785 33 151 33517 33883 34247 34611 34974 35336 35697 36058 364'8 36777 37.136 37494 37851 38207 38563 38919 39272 39626 39.979 40331 40684 41034 41385 41735

470 475 480 485 490 495 500 510 520 525 530 540 550 560 570 580 590 600 610 620 625 630 640 650 660 670 675 680 690 700 710 720 725 730

1~.416 155801 157184 158565 159945 161322 162698 165444 168 183 169549 170914 173639 176357 179069 181774 184473 187166 189852 192533 195208 196544 197878 200542 203201 205854 208502 209.825 211145 213784 216.417 219046 221670 222980 224289

97

42084

(

TABLE C.. -CONTINUED D,mens,on. D,menSIon. less less Cumulative T,me ProductIon

_'0 1.375 1.380 1.390 1.400 1.410 1.420 1.425 1.430 1.440 1.'50 1.460 1470 1475 1.480 1.490 1.500 1.525 1.550 1.575 1.600 1.625 1.650 1.675 1.700 1.725 1.750 1.775 1.800 1.825 1.850 1.875 1.900 1.925 1950

Pwf

-.113

D,menSIon. less Time

CpO

'0

386.070 387283 389705 392125 394 543 396959 398167 399373 401786 404 197 406606 409013 410214 411418 413820 416220 422.214 428196 434168 440128 446077 452016 457945 463863 469771 475669 481558 487437 493.JO7 499167 505019 510861 516695 522.520

5.500 5600 5.700 5.800 5900 600) 6.100 6200 6.300 6400 6500 6600 6700 6800 6.900 700) 7.100 7200 7300 7400 7500 7.600 7.700 7800 7900 8.00) 8.100 8.200 8.300 8400 8500 8.600 8700 8800

D,mensIon. less CumulatIve Production

_f?pp

D,menSIon. less TIme

---~Q-

1.296893 1.317709 1338 486 1359225 1.379927 1400593 1421224 1441820 1462383 1.482912 1503408 1523872 1.544305 1.564706 1585077 1.605418 1.625729 1.646011 1666265 1.686490 1.706688 1.726859 1.747002 1.767120 1.787212 1807278 1.827319 1.847336 1.867329 1.887298 1 .907243 1.927166 1.947065 1966942

DimenSion less CumulatIve Product,on

-C~o

4~109) 5.~109) 6~109, 7~109) 8~10Y) 9~10~ 1~10' ) 1.5(10'°) 2.~10'0) 2.5(10 'O) 3~10'0) 4~10'0) 5~10'0) 6~10'0) 7~10'0) 8~10'0) 9.~10'0) 1.~10") 15/10'" 2.~10") 2.5/10'1) 3.~10") 4.~10'" 5.~10") 6.~10"1 7~10") 8.~10") 9.~10") 1.~10'2) 15/10'2) 2~10 12)

3645(106) '51~108) 53681108, 62~108) 7 ~10S) 79O9110S, 8747(108) 128&109) 1697(109) 21031109) 2505(109) 3299(109) '0871109) 4868(109) 5643110~) 6.414(109) 7183110~) 7948110~) 117/10'°) 155110'0) 192(10'°) 2.29110'°) 3.02(10'°) 375110'0) 447'10'°) 519110'°) 5.89110:O) 658110°) 728110'°, 108110'" 142110")

i

l

and

I

constant BHP Pwf at the single producing well (centered .in the reservoir) with wel.lbore ra~ius r "'. The solution -pressure as a function of time a~d radius ~or f~ed values of r~, r w' and .rock 3:"d fluid pro~rtles.-ls expre~sed most conveniently In terms

I. ., I II

~D Note that for r~D=2,OOO, the reservoir is infinite

of dimensionless variables and parameters: PD=!«(D,rD,reD)'

I ",..~

acting at t D = 7.45 X 104. This means that Table C.I rather than Table C.3 is to be used to determine P D; however. f?r this (D;' Eq. C.3, which "extends:'

where

(days) ~

(D 7.45 X 104

PD ~

1.0 100.0

7.45 x 10S 7.45 x 107

7.161 7.601

Source (psia) 0.5 (In t -i735 + 0.8~) Table C.3 2,684 In r 2,665

TableC.I,lsappropnate.For(D=7.45xIOS,PDIS found in Table C.3; for (D=7.45XI07, calculated from Eq. C.6. Note also that

PD is

II.

-0.00708 khPD

=3.CXX>-44.14PD' Pressure

at Inner

Boundary

, No Flo'" Across Outer Boundary This solution of the diffusivity equation models radial flow of a slightly compressible liquid in a homogeneous reservoir of uniform thickness; reservoir at uniform pressure Pi before production; I

~

,

.. -.

Pi-P"'f 0.264 kt -,__2' ~#J.Clr", rD=r/r"" 'D=

(300)(1.25)(0.8) = 3.CXX>-(0.00708)(80)(i2)PD

Constant

Pi-P

PD='-

B

-q P"'f""Pi

no flow across the outer boundary (at r=r~);

i

r~D=r ~/r ",. For this problem, instantaneous rate q and cumulative production Qp are of more practical importance than P D' and these quantities can be derived from the fundamental solutions, PD' In Appendix B, we showed that a dimensionless production rate, q .and dimensionless cumu!ative production, QpD' c~n be defined as qB qD = II. . 0.00708 kh (Pi -Pwf)

I I'

,'~)"I

..114

,'"

WELL TESTING

and

,

I

For reD = ~ and t D ~ 100. q D can be approximated

as3 QpD = I ID q D dt D 0

,

1

-( -4,29881 + 2.02566 t D) QpDlntD .

B = 111"..L_2 /_\Qp' 1.119 (j>cI hr w (Pi -P wf) Dlmenslon!ess cumulative production. QQ' IS pres~nted I.n Tables C:.4 and C.5. Table. C.41s ~or Infimte-actlng reserVOirs, and Table C.5 IS for fimte reservoirs with 1.5 ~ r eD~ I x 1~6. For r ~D ~ 20, values for both qD and Q D are given. In Table C.5, f~r values of t D smalle! t~a~ the. small~st value for a given r~D' the reservoir IS Infimte-actlng and Table C.4 should used.forFora values of t D oflarger those in thebetable given value r~D' than the

Since both qD and QpD are based on solutions (PD) that, for rD=1 (r=rw)' depend only on tD and r~D' q D and Q~D also should depend only on t D and reD' Table C.5 confirms this expectation. For given reD and tD' QpD is determined uniquely. Although the QI!D solutions can be used to model individual well problems. they more often are used to model water influx from an aquifer of radius r ~ into a petroleum reservoir of radius r w for a fixed reservoir pressure. Pwf' Superposition is used to model a variable pressure history, as illustrated in Example C.4.

I

reservoir has reached steady state, and

Example C.3 -Use of QpD Solutions

.

QpD = (~D -1)/2.

I : I I : !.

Problem. An oil well is produced with a constant

'.

TABLE C.5 -Cpo Ys.to -FINITE RADIAL SYSTEM WITH BOUNDARY CLOSEDEXTERIORBOUNDARY. CONSTANT PRESSURE AT INNER

'.0=1.5

'.0=2.0

'.0=2.5 ~~

'.0=3.5

!.fl.-~-.!LE.eJL-.!LE.eJL-.!L~-.!L!}..pJ; 0.05 0.276 0.06 O.~ 0.07 0.330 0.080.354

I

'.0=4.0 !L.9.PJl..

0.05 0.075 0.10 0.125

0.278 0.345 0.404 0.458

0.10 0.408 0.15 0.509 0.20 0.599 0.250.681

0.30 0.755 1.00 0.40 0.895 1.20 0.50 1.023 1.40 0.60 1.143 ~1.60

1.571 1.761 1.940 2.111

2.00 2.20 2.40 2.60

0.09 0.375 0.150 0.10 0.395 0.175 0.11 0.414 0.200 0.12 0.431 0.225 0.130.446 0.250 0.14 0.461 0.275 0.150.4740.300 0.16 0.486 0.325

0.507 0.553 0.597 0.638 0.678 0.715 0.751 0.785

0.30 0.758 0.35 0.829 0.40 0.897 0.45 0.962 0.50 1024 0.55 0.60 1.083 1.140 0.65 1.195

0.70 1.256 0.80 1.363 0.90 1.465 1.00 1.563 1.251.791 1.50 1.997 1.752.184 2.00 2.353

1.80 2.00 2.20 2.40 2.60 2.80 3.00 3.25

2.273 2.427 2.574 2.715 2.849 2.976 3.098 3.242

2.80 3.034 3.00 3.170 3.25 3.334 3.50 3.493 3.753.645 4.00 3.792 4.253.932 4.50 4.068

I .

0.17 0.497 0.350 0.817 0.18 0.507 0.375 0.848 0.19 0.517 0.400 0.877

0.70 0.75 0.80

1.248 1.299 1.348

2.25 2.50 2.75

3.50 3.379 3.75 3.507 4.00 3.628

4.75 4.198 5.00 4.323 5.50 4.560

I

0.20 0.525 0.21 0.533 0.22 0.541 0.23 0.548

425 4.50 4.75 5.00

600 6.50 7.00 750

4.779 4.982 5.169 5.343

I : .~ ; !

3.2J7 550 4.222 800 3.317 6.00 4.378 8.50 3.381 650 4.516 9.00 3.439 7.00 4.639 950 3.491 7.50 4.74910 3.581 800 4.846 11 3.656 850 4.932 12 3.717 9.00 5.009 13 3.767 9.50 5.078 14 3.809 10.00 5.138 15 3.843 11 5.241 16 3.89J 12 5.321 17

5.504 5.653 5.790 5.917 6.035 6.246 6.425 6.580 6.712 6.825 6.922 7.004

3.928 13 3.95114 3.967 15 3.985 16 3.993 17 3.997 18 3.999 20 3.999 25 4.000 30 35 40

7.076 7189 7.272 7.332 7.377 7.434 7.464 7.481 7.490 7.494 7.497

2.507 2.646 2.772

0.425 0.450 0.475 0.500

0.905 0.932 0958 0.983

0.85 0.90 0.95 1.0

1.395 1.440 1.484 1526

3.00 2.886 3.25 2.990 3.50 3.08J 3.75 3.170

0.24 0554 0550 0.25 0559 0.600 0.26 0.565 0.650 0.28 0.574 0.700 0.300.5820.750 0.32 0.588 0.800 0.34 0594 0.900 0.36 0599 1.000 0.38 0.603 1.1 0.400.606 1.2 0.45 0.613 1.3 0.50 0.617 1.4

1.028 1.070 1.108 1.143 1.174 1.203 1.253 1295 1.330 1.358 1.382 1.402

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 2.0 2.2 2.4 2.6

1605 1.679 1.747 1.811 1.870 1.924 1.975 2.022 2.106 2.178 2.241 2.294

400 4.25 4.50 4.75 5.00 5.50 6.00 6.50 700 7.50 8.00 9.00

0.60 0.621 1.6 0.7006231.7 0.80 0.624 1.8 2.0 2.5 3.0 4.0 5.0

1.432 1.444 1.453 1.468 1.487 1.495 1.499 1.500

2.8 3.0 3.4 3.8 4.2 4.6 5.0 6.0 7.0 8.0 9.0 10.0

2.340 2.380 2.444 2.491 2.525 2.551 2.570 2.599 2.613 2.619 2.622 2.624

10.00 11.00 12.00 14.00 16.00 18.00 20.00 22.00 24.00

3.742 3.850 3.951 4.047

5.385 18 5.43520 5.476 22 5.506 24 5.531 26 5.551 30 5.579 34 5.611 38 5.621 42 5.624 46 5.625 50

2.442 2.598 2.748 2.893

. I j

.

I :,

..'

1

I

'I ; I i .

I

II i

I

i j !

'

i,.I

I

I

EVERDINGEN AND HURST SOLU~10NS TODIFFUSIVITY EOUATIONS .BHP a for hour from a reservoir initially of at2,
r:.

115

I,
and well properties include the following. B = 1.2 RB/STB,

There is no entry in Tablr; C.5 at this (D. Thus, the reservoir is infinite acting, and from Table C.4,

II. = I cp,

QpD =44.3.

'wk = = 0.294md, O.Sft,

Then,

Qp = 1.1 19 (j)cth'w2 (p/-Pw/)

h = IS ft, ~ = 0.15,

QpD/B

=(I,119)(0.IS)(20XI0-6)(IS)(0.S)2

c t = 20 x 10 -6 psi -I , and 't = 1,
production,

.(2,SOO-2,
in barrels.

= 0.233 STB.

Solution. will calculate t D and from eitherWe Table C.4 or Table C.S. reD and read Q pD t D = 9-:~~~ ~IJ.Crr w =

.-1_-

(0.
Example

C.4 -Analysis

Pressure

History

With

of Variable QpD Solution

Problem. A well is completr;d in a reservoir with an initial pressure of 6,000 psi. The well can be considered centered in the cylindrical reservoir; there is no flow across the outer boundary, Reservoir, fluid, and well properties include the following.

= 103

TABLE C.5 -CONTINUED

'.0=4.5

'.0=5.0

'.0=6.0

'.0=7.0

~E-ESL~.-9-PJL-.!Q.-~l~lI2-~~~!JL~ 2.5 2.835 3.0 3.195 6.0 5.148 9.00 6.861 3.0 3.196 3.5 3.537 4.0 3.859 4.5 4.165 5.0 4.454 5.5 4.727 6.0 4.986 6.5 5.231 70 5.464 75 5.684 8.05.892 8.5 6.089 9.0 6.276 9.5 6.453 10 6.621 11 6930 12 7.208 13 7.457 14 7.680 15 7.880 16 8.060 18 8.365 20 8.611 22 8.809 24 8.968 26 9.097 28 9.200 30 9.283 34 9.404 38 9.481 42 9.532 46 9.565 50 9.586 60 9.612 70 9.621 80 9.623 90 9.624 100 9625

'-

3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10 11 12 13 14 15 16 18 20 22 24 26 28 30 34 38 42 46 50 60 70 80 90 100 120

3.542 3.875 4.193 4.499 4.792 5.074 5.345 5.605 5.854 6.094 6.325 6.547 6.760 6.965 7.350 7.706 8.035 8.339 8.620 8.879 9.338 9.731 10.07 10.35 10.59 10.80 10.98 11.26 11.46 11.61 11.71 11.79 11.91 11.96 11.98 11.99 12.00 12.00

6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5 11 12 13 14 15 16 17 18 19 20 22 24 25 31 35 39 51 60 70 80 90 100 110 120 130 140 150 160 180 200 220

5.440 5.724 6.002 6.273 6.537 6.795 7.047 7.293 7.533 7.767 8.220 8.651 9063 9.456 9.829 10.19 10.53 10.85 11.16 11.74 12.26 12.50 13.74 14.40 1493 1605 16.56 16.91 17.41 17.27 17.36 17.41 17.45 17.46 17.48 17.49 17.49 17.50 17.50 17.50

9.50 10 11 12 13 14 15 16 17 18 19 20 22 24 26 28 30 35 40 45 50 60 70 80 90 100 120 140 160 180 200 500

'.0=8.0

9 6.861 7.127 10 7398 7.389 11 7.920 7.902 12 8.431 8.397 13 8.930 8.876 14 9.418 9.341 15 9.895 9.791 16 10.361 10.23 17 10.82 10.65 18 11.26 11.06 19 11.70 11.46 20 12.13 11.85 22 12.95 12.58 24 13.74 13.27 26 14.50 13.92 28 '15.23 14.53 30 15.92 15.11 34 17.22 16.39 38 18.41 17.49 40 18.97 18.43 4520.26 19.24 50 21.A2 20.51 55 22.46 21.45 60 23.40 22.13 70 24.98 22.63 80 26.26 23.00 90 27.28 23.47 100 28.11 23.71 120 29.31 23.85 140 30.08 23.92 160 30.58 23.96 180 30.91 24.00 200 31.12 24031.34 280 31.43 320 31.47 360 31.49 400 31.50 500 31.50

'.0=9.0 '.0=10.0 10 7.C17 15 9.945 20 12-26 22 13.13 24 13.98 26 1C.79 28 15.59 30 16.35 32 1710 34 1782 36 1852 3819.19 40 19.85 42 20.48 44 21.09 46 2169 48 22.26 50 22.82 52 2336 54 2359 562A39 58 2488 60 2536 65 26..18 70 2752 75 2848 80 29.36 85 30.18 90 30.93 95 31.63 100 32.27 120 34.39 140 3592 1603704 180 37.85 200 38.« 240 39.17 280 39.56 320 39.77 360 39.88 400 39.94 440 3997 480 39.98

15 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 65 70 75 80 85 90 95 100 120 140 160 180 200 240 280 320 360 400 440 480

9.965 12.32 13.22 14.09 14.95 15.78 16.59 17.38 18.16 18.91 19.65 20.37 21.07 21.76 22.42 23.07 23.71 24.33 2494 25.53 26.11 26.67 28.02 29.29 30.49 31.61 32.67 33.66 34.60 35.48 38.51 40.89 42.75 44.21 45.36 46.95 47.94 48.54 48.91 49.14 49.28 49.36

..

i~\/t

,

116

WELL TESTING

k = <> = rw = r~ = 80 = h = 1J.0 = c( =

31.6 md. 0.21. 0.33 ft. 3.300 ft. 1.25 RB/STB. 20 ft. 0.8cp.and 20xl0-6psi-l.

from three wells. each beginning to produce when P wj is changed and each producing with pressure drawdown equal to the difference in pressures before and after the change: Weill producesfor(18-0)=18months with (Pi-Pwjl)=6.-5.500=500psi.

The well produ.ced for 6 months with. flowing BHP Pwj of 5.500 pSI. for 6 more mon~hs WIth Pwf = 4.5~ PSI. and for 6 more months WIth Pwj=5.(XX) pSI. Calculate cumulative production after 18 months of production from this well. Solution. problem. production

.For Superposition is required to solve this We can calculate the cumulative by adding the cumulative

Well 2 produces for (18-6) = 12months with (Pwj I -Pwj2) = 5.500 -4.500= Wel~3producesfor(18-12)=6months with (Pwj2 -Pwj3)=4.500-5,

1. psi. = -500psl.

.

. well. reO = 3.300/0.33 = 10.CXX>. this rD =' 0 CXX>264 ., k t

production

<>IJ.C {r ~

TABLEC.5-(CONTINUED) qo and CpoY5.to -FINITE RADIALSYSTEMWITH CLOSEDEXTERIORBOUNDARYCONSTANT

PRESSURE AT INNER BOUNDARY

'.0=20

'.0=50

-.!P-

-9.P-

-EoPJL

100 130 160 200 240 300 400 500 600 700 800 1.000 1.300 1.600 2.000 3.000

0.3394 0.3174 0.2975 0.2728 0.2502 0.2197 01770 0.1426 0.1148 0.0925 0.0745 0.0483 0.0483 0.0132 00056 0.(XX)6

42.91 52.76 61.98 73.38 83.83 97.91 117.7 133.6 146.4 156.7 165.1 177.1 187.8 193.4 1969 199.2

to

~

600 800 1,000 1.300 1.600 2.000 2.400 3,000 4.000 5,000 6.000 8.000 1x10. 1.3 x 10. 1.6x10. 2 x 10. 2.4 x 10. 3 X 10.

'~0=200 to 1x10. 1.3x10"

0.1943 0.1860

2.19x103 277x103

1.6x10. 2x10. 2.4x10~ 3x10. 4x104 5 x 10. 6x10' 8x10. 1x105 1.3x105 1.6x105 2x105 2.4 x 105 3x105 4x105

01820 3.33x103 0.1742 4.04 x 103 0.1668 4.72x103 0.1562 5.69 x 103 0.1401 7.17x103 0.1236 8.50 x 103 0.11269.68x103 0.0905 1.17x10. 0.07281.33x10. 0.0524 1.52x10. 0.0378 1.65x10. 0.0244 1.78x10' 0.0138 1.86 x 10' 000821.92x10. 0.0028 1.97x10" 1.99 x 10'

I; 1. For,O smaller than.alues I,SledIn thISlablefora g'ven',0' re_' 2. For,O larger than'aluesllSled In th,slable OpO; V.o,2 -1/2 3 For,O larger I~n .alues "Sled ,n IhlSlableqo' 00

~.

0.2652 0.2915 0.2393 0.2220 0.2060 0.1865 0.1682 0.1543 0.1133 0.0833 0.0682 0.0418 0.0254 0.0120 0.0056 00021 0.0006 0.0002

189.0 241 290 359 473 502 573 667 795 895 974 1,082 1.148 1.201 1.227 1.241 1.246 1.249

-9.P-

2.000 3,000 4,000 5,000 6,000 8,000 1x10. 1.3x10. 1.6x10. 2 x 10. 2.4 x 10. 3 x 10. 4x10. 5 x 10. 6x10. 8 x 10. 1 x 105 1.1 x 105

0.2304 0.2179 0.2070 01967 0.1869 0.1686 0.1536 0.1304 0.1118 0.0910 0.0741 0.0645 0.0326 0.0195 0.0117 0.0042 0.0015 0.0009

to

~

Opo 532 757 969 1.171 1.363 1.718 2.000 2.461<103 2.82~103 3.23 x 103 3.56 x 103 3.94 x 103 4.37x103 4.62 x 103 4.77x103 4.92 x 103 4.97 x 103 4.98 x 103

'.0=1.000

~-

Opo

0.0009

~-

to

'.0=500

-9.P-

5x105

'.0=100 -EoPJL

~-

Opo

to

1.75x10. 2.21x104

3x10. 4x10.

0.1773 01729

~

Opo

1x105 1.3x105

0.1566 0.1498

1.6x105 2x105 2.4x105 3x105 4x105 5 X 105 6x105 8x105 1x106 1.3x106 16x106 2x106 2.4 X 106 3x10° 4x1OS

0.1435 2.65 x 104 0.1354 3.21x104 0.1277 3.73x10. 0.1170 4.47x10. 0.1012 5.56 x 104 0.0875 6.50 x 104 0.07567.31x104 00565 8.62 x 10. 0.04229.60x10. 0.0273 1.06x105 00176 1.13x105 0.0098 1.18x105 0.0055 1.21 x 105 0.0023 1.23x105 0.0005 1.25x105

5x104 1x105 2x105 3x105 4x105 5 X 105 6x105 7x105 8x105 9x105 1x106 1.4x106 2 X 106 24x106 3x106

0.1697 9.35x103 0.1604 1.76x10. 0.1518 3.32 x 10. 0.1464 480x10. 0.1416 6.24 x 10. 0.1371 7.64 x 10. 0.13278.98x10. 0.1285 1.03x105 0.12441.16x105 0.1204 1.23x105 0.1166 1.40x105 0.1024 1.83x105 00844 2.39x 105 0.0741 2.71x105 0.0610 3.11x105

0.00011.25x105

4x106 5 x 106 7 X 106 8.4 X 106 1 X 107 1.4 X 107 2 X 107 3 X 107

0.04423.63x105 0.0320 4.01 x 0.0167 4.48 x 0.0106 4.67 x 0.0063 4.80 x 0.0017 4.95 x 0.0004 4.99 x 0.0000 5.00 x

5x106

IS,nl,n'le acllng F,ndOpOin TableC.

5.89x103 764x103

105 105 105 105 105 105 105

.

-"

VAN EVERDtNGENAND HURST SOLUTIONSTO DIFFUSIVITYEOUATIONS

=

(0.00)264)(31.6)(730 hr Imonth)(t months) (0.21)(0.8)(2xI0-5)(0.33)~

117

Q D = 1.06 x 107, ~p=(8.19Xl0-6)(-500)(I.06XI07)

= 1.664 X 107 (t months).

= -0.434x

105.

Then ForWelll,tD=(I.664xI07)(18)=3.0xl08. From Table C.5,

Qp=Qpl +Qp2+Qp3 = 1.04 x 105 + 1.55 x 105 -0.434 x 105

QpD=2.54XI0',

Q

=2.16xl05STB.

p =I.119ct>c{hr;

(Pi-Pwf)Q

E

pDIB

.xercises

=(I.119)(0.21)(2xl0-5)(20)(0.33)~ .(p

-P I

.) Q ..1

C.l

11.25

dary

= 8.19 x 10 -6 (Pi -Pwf) Qpo =(8.19x 10-6)(500)(2.54x 10')

I07)(I2)=2.0x

ft)

with

the

no. fluid.

reservoir.

~rossing

Initial

th:

outer

reservoir

boun-

pressure

is

following: It> = 0.21, k = 80md, CI = 20xlO-6psi-l,

108.

QpD=1.89xIO',

rw=0.45ft, IJ. = 0.8 cp, h = 12 ft, and B = 1.2 RB/STB.

107)

, 10 )(6)= 1.Ox 108.

For Well 3, CD=(I.664x

of

3,00) psia. Pressure in the well bore is maintained at 2,00) ~sia. Well and reservoir properties include the

= 1.04x 105STB.

Qp =(8.19x 10-6)(l,oo)(1.89x = 1.55 x 105 STB.

An oil well is producing from a reservoir

(r e = 900

pD

ForWeIl2,cO=(I.664x

.

Calculate instantaneous rate and cumulative production after 1.0 day of production.

TABLE C.5 -(CONTINUED) -'.0 to 1 X 105 2 X 105 3x105 4x105 5x105 6x105 7x105 8x105 9x105 1x106 1.3x106 1.6x106 2x1OS 2.4x106 3x106 4x106 5 x 106 7 x 106 1x107 1.3x107 1.7 x 107 2x107 2.4x107

:2. --.9£-Opo 0.1604 0.1520 0.1475 0.1445 6.27 x 10. 0.1422 7.70x10. 0.1404 9.11x10. 0.1389 1.05 x 105 0.1375 1.19x105 0.1363 1.33x105 0.1352 1.46x105 0.1320 1.86x105 0.1291 2.25 x 105 0.1254 2.76x1OS 0.1216 3.26 x 105 0.1166 3.97 x 105 0.1084 5.10x105 0.1008 6.14 x 105 0.0872 8.02 x 105 0.07011.04x106 0.0563 1.23x106 0.0421 1.42 x 106 0.0339 1.53x106 0.0253 1.65x106

to 9 x 105 1 x 106 1.3x106 1.6x106 2x106 2.4x106 3x106 4x106 5x106 6x106 8x106 1x107 1.2x107 1.4x107 1.6x107 18x107 2 X 107 2.3 X 107 3x107 4x107 5 X 107 7x107 8x107

'.0 :4. -..:9.POpD 0.1366 0.1356 0.1333 0.1315 2.26 x 105 0.1296 2.78x105 0.1280 3.30 x 105 0.1262 4.06 x 105 0.1237 5.31x105 0.1215 6.54 x 105 0.1194 7.74x105 0.11551.01x106 0.1118 1.24x106 0.1081 1.46x106 0.1046 1.67x106 0.1012 1.87x106 0.0979 2.07 x 106 0.0948 2.27 x 106 0.0902 2.54 x 106 0.08033.14x106 0.0681 3.88 x 106 0.0577 4.51 x 106 0.0415 5.49 x 106 0.0352 5.87 x 106

.1%/t'? 1lt'/N 1;7-"%/1/'".9%/pl tltJ8

6,lPXJp6 24x1p8 o.~

4"KI't1'( G1'tt'5 /4YK~ (KrvrII' tf'tfS//6#

5X107 7X10s 1x10 1.3x108

0.0038 0.0009 0.0001 O~OO~

1.95X10: 1.99x10 2.00 x 106 2.00 x 106

-4

1.2x10s 1.4x10s 1.7x10s 2x10s 2.3x10s 2.6 x 10s 3x10s 5 X 10s

0.0181 6.90 x 106 0.0130 7.21x106 0.0079 7.52 x 106 0.00487.71x106 0.0029 7.82 x 106 0.0018 7.89 x 106 0.0009 7.94 x 106 0.0002 8.00 7.99 x 106 0.0000

1 FFOf'Osm~lIerth~nv~lu 2 I 8 Sli Sled Inlh'SI~ble'or~glvenrfO.'eservolrls,n"nlle~ctlngFindO .Of'O ~rge' than values lIsted In thIs labIa. °pO- (1_0,2_112

'.0: 1.0(10.) to -9-9Opo 3 x 106 0.1263 4.06x 105 4 x 106 0.1240 5.31 x 105 5x106 0.1222 6.76x1OS 6x106 0.1210 7.76 x 105 8x106 0.1188 1.02x106 1x107 0.11741.25x106 1.2x107 0.1162 1.49x106 1.4x107 0.1152 1.72x106 1.6x107 0.1143 1.95x106 1.8x107 0.1135 2.17x106 2x107 0.11282.40x106 2.4x107 0.1115 2.85 x 106 3x107 0.1098 3.51x106 4x107 0.10714.60x10& 5x107 0.1050 5.66 x 106 7x10;" 0.0998 7.70x106 8 X 107 0.0975 8.69 x 10& 9 X 107 0.0952 9.65 x 106 1x10s 0.0930 1.06x107 1.2x10s 0.0887 1.24x107 1.4 X 10s 0.0846 1.41 x 107 1.7x10s 0.0788 1.66x107 2x10s 0.0734 1.89x107

k//

4x10s 5x10s 6x10s 7x10s 8x108 1 X 109 1.4x109 32 X 109

/~

..

211xl07

dr'

0.0458 3.06 x 107 00362347x107 0'0286 3'79x107 0:02264:04x107 0.0178 4.24x107-0.0111 4.53 x 107 0.0043 4.82 x 107 0.0011 5.00 4.96x 107 107 0.0001

.'I.S

pO inT~bleC.

3 For,O larg-r thanvalueslisted in thislable. "0. 00

-

I

I

.--~

1 18

WELL

C.2 A single oil well is producing in the center of a circular. full water-drive reservoir. The pressure at the.oil/~ater contact is c.onstant at 3.340 p:sia and the radial distance to the oil/water contact IS 1.500 ft. The well produced for the first 15 days at a rate of 500 STB/D. the next 14 days at 300 STB/D. and ~he last day at 200 STB/D. Calculate the cumulative production and well bore pressure at the end of 30 days. 41 = k = CI = r w = !l = h = B =

fl

radius) radius)'.

References at a pressure. Pi' of

2.734 psia. If the boundary pressure is suddenly lo".ered 10 2.724 psia and held there. calculate the ..

0.2 83 r'nd 8 x 10~6 psi-I. 3(xx) ft (reservoir 30.(xx) ft (aquifer 0.62 cpo and 40 ft.

C.4 If. in Exercise C.3. the reservoir boundary pressure suddenly dropped to 2.704 psia at the end of 100 days. calculate the total water influx at 400 days total elapsed time.

0.20. 75 rod. 17.5xI0-6psi-l. 0,5 ft. 0.75 cp, 15 ft.RB/STB. and 1.2

C.3 An oil reservoir is initially

I

d> = k = C( = r = r: = !l = h =

TESTING

..

f

100

cumu atl\e water In ux Into the rese!volr a ter .' 200. 400, and 800 days. ReservOIr and aquifer properties include the following.

I. van E'erdingen. A.F.and Hurst. W.: "The Application of the Laplace TransformatIon to Flow Problems in Reservoirs,. Trans.,AIME (1949)186,305-324. . 2. Chatas. A.T.: "A Practical Treatment of Nonsteady-State Flow Problemsin ReservoirSystems," Pel. Eng (Aug. 1953) B-44through B-56.

3. Edwardson, M.J. el al.: '.Calculation of Formation Ternperature DisturbancesCaused by Mud Circulation," J. Pel. Tech.(April 1962)416-426; Trans.,AIME. 225.

TABLE C.5 -(CONTINUED)

'eO=2.5(10.)

'0

--.9..P.-- ODO

3 X 101 0.1103 4 x 101 0.1086 6 x 101 0.1064 6.77 x 106 7 x 10:' 0.1054 7.83 x 106 8 x 107 0.1047 8.86 x 106 9 x 107 0.1041 9.93 x 107 1x10B 0.10351.10x107 1.4x10G 0.1016 1.51X107 2x108 0.09932.11x107 2.6x10G 00973 2.70x107 3 x 108 0.0960 3.09 x 107 3.3 X 10G 00950 3.37 x 107 3.6x10. 0.0940 3.66 x 107 4 x 100 0.0927 4.03 x 107 4.4 x 100 0.0915 440 x 107 5x10G 0.0896 4.94x107 5.4x10c 0.08045.30x107 6 x 100 0.0866 5.82 x 107 64x10. 00855 6.17x107 7x10. 0.08376.67x107 74 x 100 0.0826 701 x 107 8x10. 0.08097.50x107 84 x 10" 00798 782 x 107 9x100 0.0782 8.29x107 1x103 0.0756 9.06x107 1.3x103 00683 1.12x108 1.6x10' 0.0616 1.32x108 2 x 10' 0.0538 1.55 x 108 2.4x109 0.0469 1.75x10B 3 x 109 0.0382 2.00 x 108 4 X 109 0.0272 2.33 x 108 5 X 109 0.0193 2.56 x 108 6 X 109 0.0138 2.72 x 108 8x109 0.0070 2.92 x 108 1 x 10'0 00035 302 x 108 1.4x10'c 0.0009 3.10x108 2x10'O 0.0001 3.12x108 3 x 10'0

'_0=1.0(105)

'0 1.4 x 108 2 x 108 2.4 X 108 3 X 108 3.5 X 108 4 X108 5x108 6x108 7x108 8x108 8.4 X 108 9 x 1oB 1x109 1.4 X 109 2 X 109 3x109 4x109 5 X109 6x109 7x109 8 X 109 9x109 1 X 10'0 1.3x10'O 1.6x101O 2x101O 2.4x10'0 3 X10'0 4x10'O 5 X 10'0 6 X 10'0 7 X 10'0 8 X 10'0 9x10'0 1 X 10" 1.3x10" 1.6x10"

0.0000 3.12 x 108 2.4x10" 2 X 10" 3 x 10"

-.9JL

'eO=2.5(105)

ODO

0.1017 0.100) 0.0990 0.0980 3.10 x 107 0.0971 3.59 x 107 0.0966 4.07 x 107 0.0956 5.03x107 0.0948 5.98x107 0.0941 6.93x107 0.0935 7.87x107 0.0933 8.24 x 107 0.0930 880 x 107 0.0925 9.73x107 0.0911 1.34 X 108 0.0896 1.80 x 108 0.0877 2.77x108 0.0861 3.64x108 0.0845 4.49 x 108 00829 5.33x108 0.08146.15x108 0.0799 6.95 x 108 0.07847.75x108 0.0770 8.52 x 108 0.0728 1.08x109 0.0689 1.29x109 0.0639 1.56x109 0.0594 1.80x109 0.0531 2.14 x 109 0.04412.62x109 0.0366 3.03 x 109 0.0304 3.36 x 109 0.0253 3.64 x 109 0.0210 3.87 x 109 0.0174 4.06x109 0.0145 4.22 x 109 0.0083 4.55x109 0.00484.74x109

'0

-.9JL

ODO

2 x 109 0.0897 3 x 109 0.0881 4 x 109 0.0870 5 X 109 0.0861 4.51 x 108 6 X 109 00854 5.37 x 108 7 X 109 0.0849 6.22 x 108 8x109 0.0844 7.07x108 1X10'0 0.0836 8.75x108 1.4x10'0 0.0824 1.21x109 2x10'O 0.0809 1.70x109 3 x 10'0 0.0787 2.49 x 109 4 X 10'0 0.0766 3.27 x 109 4.4x10'0 0.0757 3.58x109 4.7 X 1010 0.0751 3.80 x 109 5 x 10'0 0.0745 4.03 x 109 5.4X10'0 0.0737 432x109 6x101O 0.0725 4.76x109 7 x 10'0 0.0705 5.48 x 109 7.4x10'0 0.0698 5.76x109 8x10'0 0.0686 6.17x109 8.4 x 10'0 00679 6.42 x 109 9x101O 0.0668 6.85x109 1 X 10" 0.0658 7.51 x 109 1.1x10" 0.0632 8.15x109 1.3x101' 0.0593 9.38x109 1.6x10" 0.0551 1.11x101O 2x10" 0.0494 1.32x10'0 2.4 X 10" 0.0443 1.51 x 10'0 3x10" 0.03761.75x10'O 4 X 10" 0.0286 2.08 x 10'0 5 X 10" 0.0217 2.33 x 10'0 7 X 10" 0.0126 2.67 x 10'0 1 X 10'2 0.0055 2.92 x 10'0 1.3x10'2 0.0024 3.04x10'O 1.6 X10'2 00011 3.09 x 10'0 2X10'2 0.00043.11X10'O

0.0023 x 109 0.0011 4.88 4.94x109 0.0004 4.98 x 109

'_0=1.0(10&)

'0 2 x 10'0 3 x 10'0 4 x 10'0 6 X 10'0 8 X 10'0 1 X 10" 1.3x1011 1.6x10" 2x10" 2.4x10'1 3 X 10" 4 X 1011 5x10" 6 X 10" 7 X 10" 8x10'1 1x10,2 1.2 X 1012 1.4x10,2 1.5x10'2

-.9JL

ODO

0.0813 0.0800 0.0791 0.0778 0.07700.0763 7.95 x 10'0 0.0756 1.02x10'O 0.0750 1.25x10'O 0.07431.55x10'O 0.0737 1.84x10'O 0.0730 2.28 x 10'0 0.0719 3.21 x 10'0 0.0709 3.72x10'O 0.0697 4.42 x 10'0 0.0686 5.11 x 10'0 0.06766.32x10'0 0.0656 7.13x10'O 0.0636 8.42 x 10'0 0.0617 9.67x10'0 0.0607 1.03x10,2

..

>1.5x10,2no1determined.

i . 0

3.4 x 10" 0.0002 4.99 x 109 4 X 10" 0.0001 5.00 x 109 Noles 1 For,Osmalle'IlIln .alueslisled.nIlloslablelorlo.ven'_0. reservoIr oS.nfinlleaClono F.ndOpO.nTlbleC. 2 F"'.Ola,oerlllan,"lueSIISledlnlll'slable.OpO'

(/eOI2_,

3 Fc':Ola,oe,Illan,"IueslisledInIII.s'able.QO'00

12

i I I I~ I ~

I,J,,6

rr(ji:"~'rt,, t~tirt.at. "rv,"..' ,.".1f,;~- " ~

~"11Y. ana ,I' \c1$e i)f 5Ia

",. {~\'~\'edc. '-f"" .f '-'-1;:, ..;,' !i.~':) ::11( ~

~ ~...' ~ut!Ofl... GO 1\.. .".,..-~ ~~-

.-,';,,!Wf.il'.

.1 -EsrimQ/i

Appendix D

Rock and Fluid Property Correlations Introduction

Pressure transient test analysis requires knowledge of reservoir fluid properties such as viscosities, com-

Tec' and pressure. p~, of an undersaturated crude oIl with gravity of 30 API (specific gravity = 0.876 at

pressibilities, and formation volume factors. In I . d f addition, f formation compressibility is I a. rock F property requent y requIre or test ana yslS. or most of these properties, laboratory analysis

60°F). Sol U Ion. -285 Ppc -psla.

provides the most accurate answer; however, in many cases, laboratory results are not available. and the test analyst must use empirical correlations of experimental data. ~ This appendix provides a summary of correlations that have proved useful for test analysis. These correlations are selected from those presented by Earlougher; 1 his collection of correlations is probably the best and the most complete in print at the time of this writing. We assumethat the reader of this text has completed a studv of the fundamentals of reservo.r fl ' d t .. s h d.. d Ih UI proper les. ucd stu I.les bprovl d .1 .' . .l. ef etal on t e meaning, origin. an app Ica I Ity 0 these correlations. Readers not familiar with this basis are referred to texts bv Amvx et al.2 and McCain.3 this appendix.collection 'we si~plY figures fromIn Earlougher's andreproduce illustrate

r

I. ,

...

1300 E I-Go 1200

!:.IITT

: .!:

1 ': ,

I- 700 :! 100 i: 500'

~ specific gravity oof the undersaturated reservoir liquid :; corrected to 60 F. Example 0.1 illustrates use of the correlation.

~ i 400 i a.. U III 300

: ,

8~

I

D.l-

..

I

Estimation T

of

dP

:s-eu ocrltlca I emperature an ressure for Undersaturated Crude Oil Problem: Estimate the pseudocriticaJ temperature.

L.,

200

T i

.,I I .

dersaturated crude oil. as illustrated by Example 0.5. To use Trube's correlation, one must know the

d

d

~Sl'ClDoc*,rlcAL Tl'.~f"*ArCl*l' ,, ' ,

are required to estimate compressibility of an un-

P

T.or --an1160o R

..

~ i 1100 0~ g~ 1000 III Ifj III a: '°0 Go ~ Go 100

Example

0 - 1,

~

Pressure

Pseudocritical temperature (T pc) and pressure (ppc) of undersaturated crude oils can be estimated from presented an approximate in Fig. correlation 0-1. Thesedeveloped pseudocritical by Trube4 properties and

F.Ig.

Pressure of Crude Oil .. The test anaJ~stmay need to estimate the. sa.turatlon o.r bubble-.polnt pressure of a cr.ude oil In some clrcum~ta~ces -e.g., to determine whether ,a reservOir IS satu.rat~d or und~rsa~u~ated at a certaJ.n pr~ssure. ~tandlng s corre~atlon IS. useful for thIs estimate; Fig. O-~ sh,owsthis c~rrelatlon. To. use Standing s co~r~latlon, one m,ust k".ow solution G
i °. Uu

Hydrocarbons

.".

Bubble-Point

the use of each with an example. Pseudocritical Temperature and of Liquid

From

, ~~f"ClDoc*'rICAL ,~*l'ssu*l" , .I ! ., '

: : I

:J III'(/)U) 0.12 0.11 0.70 Q" 0.71 Qat 0.11 : ~ SP£CIFIC GRAVITY OF~DERSATURATED RESERVOIR GoLIOUIOATRESERVOIR PRESSURE CORRECTED TO60°F Fig. D.1 -Approximate correlationof liquid pseudocritical pressureand temperaturewith specific gravity.'

1.

120

WELL TESTING

Example D.2 -Estimation of Bubble-Point of Crude Oil P bl. , ro e~. Estimate the, bubble-I?olnt pressure of a crude 011from a reservoIr producIng at a GOR of 350 scf/STB (believed to be solution gas only) with gas

To determine solution GOR, one must know bubble-point pressure, reservoir temperature, stocktank oil gravity, and gas gravity. Example D.3 illustrates use of Standing's5 correlation for t" at solut"on GOR es 1m mg I .

gravity 0.75 and oil gravity 30. API from a reservoir with temperature 200.F. Example Solution. On Fig. D-2, we start on the left by extending a horizontal line from the assumed GOR of

Problem.

350

from 0

f/STB

h

sc

to

t

e

10 Ine

f

"

or

a gas

gravIty

0

f 0 75 .;

a

D.3 -Estimation 0 0 Estlm,ate ~he solution reservoir

wIth

.F

200

of Solution ~R

bubble-poInt

GOR . of a crude 011

.

pressure

1,930

01 . 30 API ,an d gas ,01 gravIty

f h" 0 d . al I.me th at terrom t IS pOInt, we raw a vertic minates at the line for a stock-tank oil gravity of 30.

pSla, temperature 0 075 gravIty 0 .

API; we then draw a horizontal line from that point that terminates at the intersection with the 200.F reservoir temperature line. Finally, we draw a vertical line from this point that intersects the bubblepoint pressure scale at 1,930 psia. Thus, the estimated bubble-point or saturation pressure of the crude oil is 1,930 psia.

Solution. In Fig. D-2, we start at the lower right by extending a vertical line from the bubble-point pressure to the 200.F reservoir-temperature line. We next draw a horizontal line to the left that terminates at the intersection with the line for tank-oil gravity of 30. API. We then draw a vertical line from this point to intersect the 0.75 gas-gravity line. Finally, we draw a horizontal line from that point to the GOR scale on the left and read the GOR to be 350 scf/STB.

S I .

GOR

0 otlon

Frequently. the saturation pressure of a reservoir oil is known, but solution GOR at saturation pressure. (required for some test analyses) is unknown. Fig. D2 also can be used for this estimate.

. 011 Formation Volome Factor Fig. D-3 can be used to estimate the formation

£'--1.£ ~£Q"'.£D ,. -' ~

." -~ Ao .110 c... .,.. .-" *' .--"

.' ~-..

, , ,.,.. .' 'r ., a", oN ., ~'AP,

...,ccou-c

s ,." .' .._f, ~ -"N'r

-.,

, ,..

"'. .I.5D

-_., u. ..~. ---,oak? -"-', .f,.0" ...»"API Ny '~. -*' ""~'r KoN .,.. .w...

-~. -,...

,

'.

'-'0 P$JA

Copyriv"t

/952

C"e.ron Researc" Company Reprinted

Fig. 0-2 -Bubble-point pressureor dissolvedGORof oilo"

br PermissIon

ROCK AND FLUID PROPERTY CORRELATIONS

121

EXA~Pr.(

~£OtJ'~ED

~,..,.." ,. -", "..,.~ ,.I".. -.."'. .' 100." i' '.'" ." JSOCI"8..~, "'.~" .,. 0-7J, ~ .1.,...", ." JO'AP' PROCEDURE

S,.".'" -"w

.f "" I." ,.-. .' "" ,AG... """."'."" ..", ...JSO cr..

' "".""

0.7S ".-- .."

P_..~ -' ...,-'-'."" """."" .". ""'. ...,-"k.' JO .API

""~'" M. 100'" r". ,-.,.. -"." .."'.. .."",..~ /0.. , ..,'., -, ..',." of"..". ..,

Copyr;9'"

19S2

C"evron Re.earc" Company Reprinted ~y Perml..ion

I"ORMATION OO,UME.f 1U8., E POINT "Ou'DFig. D.3-Oil fOfmationvolumefactor.' 10-1 .estimate, .'\. u~

"\." "

>-" g

...I

'$E'IIDO~E'DUCE'D -'

"

~

rl'.'E'~ATII~E'. T,r

'"\

~

'l

'~'

~"'"-'

volume factor 80 of saturated crude oil, For this one must kno~. solution GOR, gas gravity, stock-tank oil gravity, and reservoir temperature. Example 0.4 illustrates use of this correlation, which also was developed by Standing.

:.

;',

I

,

:;

~ ~,'

E.\"amp/eD.4-Estimation

:

Formarion.Vo/umeFacto:

~ 10-1 ~ .salurated 0

u

"~

..gravity

0 III

u :>

0 ~

..,

, 1 2-r-./-'!

g

"

~ 10-S

~

'

!,

'"

~

I

...j. I

'\.1

1./0

'" I".j r '\.."" ' "-J

~'" 0'0

i ~1 ~ ~ I

"".""

;"

-.

00

I

--~.:.+~ --~ -0.10 .o~~ T ~ -:-- '. ..II '. 10

--.,

and reservoir temperature of

F

scf/STB ;80

\\.e

.0..70 .I .'

OO'

of 30' API,

Solution. In Fig. 0-3, ~'e start at the upper left and e.xtenda horizontal line from a solution GOR of 350

~~,

,..l",""

i

Problem. EstImate the formatIon volume factor of a crude oil from a reservoir wilh solution .. GOR of 350 scf/STB, 2as 2ravitv of 0.75, tank oil

I

'\.~,

"...

I

ofOi/

.04.. 102

then

to intersect extend

the line

a vertical

for line

gas gravity

of 0,75.

from

point

this

to

intersect the 30' API tank-oil-gravity line. From this point of intersection, we dra~ a horizontal line to inlersect the line for a reservoir temperature of ~OO'F. Finallv, we dra~' a vertical line from that point to the formalion-volume-factor scale at the lower right and read 80 = 1.22 RB/STB.

PSEUOOREOUCEOPRESSURE,Ppr

Fig. D.4-Correlation of pseudoreduced compressibility for an undersaturated 011.'

Com

pressibility of UndersaturatedOil

The correlation in Fig. 0-4, developed by Trube,4 can be used to estimate the compressibility, co' of an

L

,

-.-"

~

r

122

WEll TESTING 10

, ,

I

, 1

'='

,

4. .

In

~ ... ~

I

~ U

In 10"

-,

.-' ...6 C~ ~ ---2:

.

..

l&-

~~~

I

.(J

10-

5

.J m

~ I~.(/) .., ~ ~ ..m

.m

(aRs_'

I

\dP)T:

R,

4

(O.83p + 21.751 3

10') 10

1.0 I

., 102 ., 10J GAS IN SOLUTION, Rs. SCF/STB

o'

1.2 1.4 1.6 1.8 2.0 OIL FORMATION VOLUME FACTOR, Bo. RES BBL/STB

Fig. D.S-Change of gas in solution in oil with pressurevs. gas in solution.'

Fig. D.6-Change. of oil formation volume factor wi,thgas In solution vs. 011formationvolumefactor.

undersaturated crude oil. The basic information required to use this figure is specific gravity of the undersaturated crude oil,

By definition, Co = Cpr/Ppc , so C = 0.001/285 = 3.51 x 10 -6 psi -1. 0

these reservoir values, temperature, we can and estimate reservoir pseudocritical pressure. Given tem-

Compressibility

perature (T pc) and pressure (P~), pseudoreduced temperature (T pr = T/ T pc:) and pressure
The apparent compressibility of saturated crude oil is significantly higher than that of undersaturated ?il. The reason is that a pressure drop results!n evol~t~on of gas from the oil; the total volume of oil remaJrnng actually decreases with pressure decline (although the of the remainin~ liquid oil actually decrease.s slightly). The net result IS that the total volume of oil

Example D.5 illustrates this sequence of calculations.

and evolved gas becomes greater as pressure drops, leading to an apparent compressibility of the system that is appreciably higher than that of liquid oil .. alone. In equation form,

Example

D.5 -Estimarion

of

Undersaturated Oil Compressibility Problem. Estimate the compressibility, co' of an

of Saturated

C = -..!.- ~ +!!L ~ 0 Bo dp Bo dp

Crude 011

",

The first term accounts for the volume change in the liquid caused by (I) vaporization of some. of the liquid and (2) increase in density of the remaJnder of the liquid. The. de~ivative .dBo /dp is a positive number, so vaporization domm~tes. The second term accounts for the volume o~cupled by gas evolved.as pressure decreases ~or dlssolve~ as .p~essure I~creases). term is positive. The derivative Further, dR sldp its numerical lS positIve,value so this is

and

greater than that of the term -(I/Bo) (dBoldp). Ramey6 proposed correlations that lead to an estimate of Co for a saturated oil; these correlations are given in Figs. D-5 and D-6. To use these correlations, we must note that Eq. D.I can be written as

cpr = 0.001.

""", ~;,

.'.

(D.l)

undersaturated crude oil with 30° API gravity (0.876 specific gravity) at a re~ervoir temperature of 200°F and pressure of 5,000 psla. Solution. In Example D.I, we found that the pseudocritical temperature of a 30. API oil was T pt. = 1180°R and that the pseudocritical pressure was p = 275 psia. Thus, Tpr=TITpc=(200+460)/I,I60=0.569, pc

Ppr =Plppc = 5,000/285 ='7.5. From Fig. D-4, then,

.j!

'I i

.IL

ROCK AND FLUID PROPERTY CORRELA TIONS

123

c = -~ ~ + ~ ~ 0 Bo dp Bo dp

= ~ ~(Bg-~). Bo dp

'0.; :

(D.2} dRs

,

Fig. 0-5 provides an estimate of dRsldp; Fig. 0-6 estimates dBoldRs' To estimate co' one needs values of reservoir pressure, p, solution GOR, R s (which, in turn, can be estimated from Fig. 0-2), tank-oil specific gravity ("Y0) and gas gravity ('Yg >, oil formation volume factor, Bo (which can be estimated from Fig. 0-3), and gas formation volume factor. Bg. (Bg can be calculated from reservoir temperature, pressure, and gas gravity, which leads to the real-gas-law deviation factor z: Bg = 0.00504 Tzlp RB/scf.) Example 0.6 illustrates use of these correlations.

~ i = '0 ~ ! ~ ~ ~ ~ '0

I

."

~

ExampleD.6 -Estimation of Saturated Oil Compressibility Problem. Pressure in an oil reservoir has dropped below the initial bubble point to 2,500 psia. Reservoir fluid characteristics include the following: "Yo = 0.825 (400 API), "Yg = 0.7,

,

T = 200°F = 66OoR,and z = 0.851. Estimate current saturation the apparent pressure compressibility of 2,500 psia. of this oil at its Solut~on. To evaluate co' we must determine each term In Eq. 0.2.

C0 = B~ ~d p ( B g -~

000. _h.

-aT 00".-a_~

--

Fig. 0.7 -Dead oil viscosity at reservoir temperatureand atmosphericpressure.'

).

dR a s From knowledge of p, 'Yg' 'Yo' and T, we can estimate Rs from Fig. 0-2, as in Example 0.3; the result is R s = 640 scf/STB. From knowledge of Rs' 'Yg' 'Yo' and T, we can estimate Bo from Fig. 0-3, as in Example 0.4; the result is Bo = 1.36 RB/STB. We can calculate Bg since T, p, and z are known: Bg =0.00504 Tzlp = (0.00504)(660)(0.851)/2,500

dB t:;:~..J..:!.!L X 104 = 5.6. dRs 'Yg Thus, dB

I

--2- =5.6x 10-4,\-2.:L dRs io

=0.001132RB/scf.

=(5.6xI0-4>..j-

rOT 0.825

From the inset in Fig. 0-5, dRs dP = (0.83pRs + 21.75) \

-

= 0.516 x 10 -3 RB/scf. We no\\' can calculate co: C0

640

---I -

I dRs!

B

dp

IB e' --I

dBo'

-dR

--a

J s

-[(0.83)(2,500) + 21.75] = 0.3052 scf/STB-psi. (This result also could be read from the curves plotted in Fig. 0-5.) From Fig. 0-6,

= (~

I

)<0.3052)(0.001132-0.0005 16)

=0.138 x 10-3 psi-I

~

.~

,

124

WELL TESTING

. .

100

.E.AM~LE:

,

PWoeLEM:

FINO

A C-uOE

.Cu

FT

.AT

OIL /

BeL

THE

THE

GAS.SATURATEO

HAVING AND

SAME

A

SOLUTI~

DEAD

OIL

VISCOSITY

OF

_ATIO

OF

GAS/OIL VISCOSITy

OF

,~O

600

C~,

All

TEM~ERATURE.

~ROCEOURE:

lOCATE

I.~O

C~

ON

THE

DEAD

OIL

VISCOSITY

I SCALE

(ABSCISSA!

GAS/OIL ;:;

.READ

~ III III OJ

~ATIO THE

OIL

AND

GO

liNE.

THEN

ANS.E~,

VISCOSITY

UP

0.~8

SCALE

VE_TICAllY

GO CP,

lEFT

ON

TO

THE

500

HO~IZONTAllY

THE

TO

GAS-SATURATED

10~0INATEI

~

a. a. V

10

j~ 0'4 ..

0~

'

..4

~ III . ..Z

.

1 III" 4 ~ 0 ;)

I

0

~ 4

.. ..4 0 ~

.

..

,.. a. ~ 2

.."

0"

..-

v~ >

> . 0

10

... ~

III .., ~ ..4

J:.

. 01

...,..

...,..

OS

10 VISCOSITY

(AT

Fig.

Oil

D-S -Viscosity viscosity

~ES:~VOIR

TEMPERATURE

for plus

crude oil can be estimated dataS';' (Fig, (Fig. D- 7) D-8). combined Viscosity with

undersaturated an additional

gas-saturated p!essu.re;. and

oil

require

correlation

oil Fig.

of

[he Beal's,

point and reservoir illustrates this estimation

pressures. procedure.

7 -Estimation

Problem. Estimate oil at a reservoir saturation

ATMOSPHERIC

PRESSVRE)

of Oil

of 1,930 psia.

is 30. API,

and

reservoir

IS 2.15

VIS~OSlt~

s~turated

cp'.

From

FIg.

D-8,

.011 (#l.ob) at th~

the

bubble

pol.nt

of

IS. 1.0

gascpo

Finally, using da:a from Fig. D-9, the VISCOSIty?: the undersaturated 011 (#1.0) at a pressure of 5,000 pSI IS ,5000-1 #1.0= 1+ (0.067)('

930 ,)

= 1.21 cpo

1,000 of

GaR, bubble-

Example

D.7

VIscosity

Solution

oil gravity

Dead oil

Fig.

the viscosity of an undersaturated pressure of 5,000 psia and at the

pressure

and pressure.

same

pressure. knowledge

reservoir temperature, oil gravity, solution and, in the case of an undersaturated oil,

oil's

AND

,~ CP

o~1 VIS~OSlty, #l.od'

viscosity, #l.ob' at saturation D-9 provide.s an estimate of

VISCOSIty Increase above bubble-point These viscosity estimates require

D.

OIL,

temperature is 200.F. ~olutlon. From FIg. J?-7, the dead

D-9. Fig. D- 7 provides an estimate of dead (gas-free) loil viscosity, #l.od; Fig. D-8 provides an estimate of

Example

DEAD

is 350 scf/STB,

Viscosity of saturated Chew Beal's from and Connally's correlation

figures

OF

of gas.saturated crude oil at reservoir temperature from laboratory data or from Fig. 0-7.1

Viscosity

estimates

.14"'"

14

GaR

... Solubility

of Gas

Solubility of natural from correlations

In Water gas in water of Dodson

D-IO and D-II.

can be estimated and Standing, 9

presented

in Figs.

solubility Fig. D-II

of natural gas in pure (nonsaline) water; provides a means of correcting solubility in

pure water for brine salinity. To estimate solubility of gas know

reservoir

temperature

and

Fig.

D-IO gives the

in water,

one

must

pressure

and

total

--~-

'" ~OCK ANDFLUID PROPERTYCORRELATIONS

--

!if, ::

125

.J

m m

'- 20 ~

10.

:)

u !

.a:

iii

Q:

.

2

:) V! V!

Q ~ ~

102

'&:' 0.

a: 12

.a:

~

~

Z

0

~

1

0.

III

'" ~

-I

Z 0 ~

:)0. II

-I

J: O

2

0-

a:'

0

111

II. 0. U

I

c

.TEMPERA~E.

III I/)

~ )0

...

ii 0 >u V!

0

.260

III

,

-F

2

Fig. 0.10-SOlubility of naturalgas in purewater.1

F4~~~~~~~

Ja:

10-1

.III .ZIIIIO 1

CORRECTIONFOil BRINE SALINITY

Q a: m -l- ~

10 10-1 2

..I

1

..10

1

..101

1 ~

\lISCOSITY OF GAS-SATURATED CRUDE AT BuBBLE POINT PRESSURE, CP

2-.-ccr ~

~Z Z ~-a: J J

0.9

a: ;

08

~ 100'¥" 0

10 20 TOTAL SOLIDS IN BRINE.

30 ppm .IO-~

40

Fig. 0-9 -Rate of increase in oil viscosity above bubblepoint pressure-1

Fig. 0-11-Correction of natural gas solubility for dissolvedsolids,'

!

solids content of the water. Example 0.8 illustrates this estimation procedure.

0-11, to This estimatethe requires the same information used estimate solubility of gas in water (reservoir temperature and pressure and total solids

': .

E

co~tent. of the water), Example 0,9 illustrates the estimatIon procedure.

I D 8_£ xamp e ,'.

' ..r .stlmatlon

Gas Solubility In Water

OJ

Problem. Estimate the solubility of natural gas in a formation temperature "-ater with 20,fXXJ ppm dissolved solids, reservoir of 200°F, and reservoir

..1 E.x-ampleD.9-Estlmatlon

pressure of 5, psia. Solution. From \Fig. 0-10, the solubilitv (R ,) of natural gas in pure water at 200°F and 5,OOOs~iais 20.2 scf/STB. From Fig. 0-11, the correction factor

Formation Volume FaClor Problem. Estimate the formation volume factor of brine with 20,~ ppm dissolved solids. at a temperature of 200 F and pressure of 5,fXXJpsla.

(Rs.._IRs..p) for 20,OOO-ppmsalinity at 200°F is 0.92. Thus,

Solution. From Fig. 0-12, the formation volume factor of pure water is 1.021 RB/STB; for gas-

R s..- = (R s..p ) (R sWI R sw ) -., p -(.0.2)(0.92)

~

"

= 18.6 scf/STB.

Water Formation Volume Factor The formation volume factor Bw can be estimated by using Fig. 0-12, an additional correlation developed by Dodson and Standing,9 along with Figs. 0-10 and

I .II

r

4

.:) 0

~

8

~

10

IIUj. m

I

16

-\1

J ~'"\

I

'ii,; Ii", .i

of Water

saturated pure. water, it is.1 .030 RB:STB. Br!ne with 20,fXXJ ppm dIssolved solids contains less dIssolved gas than pure water; from Fig. 0-11, the ratio of

this solubility value in to brine interpolate to that between in pure water volume is 0.92. factors Using for pure and gas-saturated water, then, Bw = 1.021 + (1.030 -1.021)(0.92) = 1.029 RB/STB.

'

126

..WELL TESTIN . ~;;" [;

-:.. 40 C/I Q.

.Q

.3

loe

J u

~

>...

~'"

C/I

-~

" 1.05

= 3.2

~

~

g:. .-NRC .~

~

---WATER

1.04

" ~~ :'+ '

WATCR PLUS NATURAL""

..-,~

Vi

C/I

~

Q. 28

~ 0

.I.)

~ "I.)

g: '"

4

~

I.; 1.03 .~

~

-~ ~-

>

1.02

-,--

---,._-,--,.

0

--T

-'-

-1$0

0 g: 1.01 Ii.~:';:

-,

~

g:.~

++J

~ ~

--I-

~ 1.00

~ --1 ~ 1000

;.;

C/I'"

2000 PRESSURE.

~

"~"., ""

Q.a:

g: ~~ u 0

~ E Hf:!:J~

~ '" ~i

;zc "O,

I.) 1.1

-"

~

I

Vi 12

Qa:::J 4Q.

~

lOoO-

,

260

>- 1.3 ...>-

..::J

iiij

,...

220

8F

m..J -M

~

~ ~'-

::;-:~

180

..J-

;~ i+1-j

-,.-

140

~

~

"

100

FIg. D.13-Compressibility of gas-freewater.' L.o..'

,.

'"

.0

60

.TEMPER.ATURE.

.~

~

~~

0" --'-~

24

!.J.;

..J O Z

~

~

,.c!'

~ a: ~

:;';,:'

!,"', ~i.:. 1.00

p, PSIA

5

10

GAS-wATER

15

RATIO,

20

\c'

CU FT leeL

. .,-

Fig. 0-12-Formation volume factor for pure water, gasfree andgas-saturated.'

Fig. D.14-Effect of dissolved gas on water compressibility.'

Compressibility of Water in Undersaturated Reservoirs

S,(XX) psia. Oil in the reservoir has a saturation pressure of 1,930 psia.

The formation water that occurs in an undersaturated oil reservoir will not release gas as pre~sure is d~creased; this wo.uld l.eadto formati.on of or Increase In a gas saturation In the reservoir. In

Solution. We found in Example D.8 that solubility of gas in water at the stated conditions is 18.6 scf/STB. From Fig. D-13. the compressibility (cwp) of pure (gas-free) water is 2.96 x 10 -6 psi -1 .At a gas/water

such a case. we ~'ill assume that the formation water is saturated with gas at reservoir pressure. One implication of this is thai, as gas is released from solution in the water, it is assumed 10 be redissolved

ratio of 18.6 scf/STB. the correction factor for gas in solution is 1.16(Fig. 0-14). Thus.

t

in the undersaturated oil. For. this assumed system behavior. Dodson and Standlng's9 correlations (given in Figs. D-13 and D14) can be used to estimate the compressibility of water in an undersaturated oil reservoir. These water-compressibility estimates require knowledge of reservoir temperature and pressure and formation--:vater salinity. E.xample 0.10 illustrates the calculatIon procedure. Example D. la-Estimation of Water CompressibllilY i a Und I I d R .n n ersa ura e

c w = C"'p (c ,,/c "p)

"

=(2.96xIO-6)(1.16) -4 -6.-1 -3. 3 x 10 pSI.

Compressibility of Water in a Saturated Reservoir In a saturated reservoir, gas released from solution in the formation water either ~'ill begin to form or will increase a gas saturation as reservoir pressure is ~owered; as Ramey6 pointed out, thi~ ~~amaticall.y Increases the apparent ~a~e.rc?mpresslblllty. In thIs case. the water compressIbilIty IScalculated from

eSerVOlr

the compressibility of a formaPro blem. Estimate .I tion water containing 20,(XX}ppm dissolved solids in a reservoir with temperature 200.F and pressure

1 L

..

~'~ .Ii

c w = ---+

dB".

Bw dp The term -(I/B".)

--aB. dRSIII Bill dp (dBw/dp)

(D.3) is still determined

.,I

:1

"

i

"' I

ROCKANDFLUIDPROPERTY CORRELATIONS

using Fig. 0-13 (for gas-free water) and Fig. 0-14 (to correct for the effect of gas in solution). Ramey's correlation,6 presented in Fig. 0-15, is used to estimate dRsw/dp for fresh water, and Fig. D-II is f

127

o. -= r Q

used to correct for the effect of salinity on dRsw/dp. This compressibility estimate requires knowledge of formation-water salinity, reservoir temperature and pressure, and formation volume factor of the gas dissolved in the water. Example 0.11 illustrates this estimation procedure.

~ ! ~

QOO4

-:t

n

~ ~Q. QOO2

~ Example D.llEstimation of Water Compressibility in a Saturated Reservoir

00

1000 ~

Problem. Estimate the apparent compressibility of

~

~

~

PRESSuRE,PSIA

for.ma~ion water containing. 30,(XX) ppm dissolved solIds In an undersaturated 011reservoIr at 200.F and

Fig. D.15-Change of naturalgas In solution in formation water with pressurevs. pressure.'

21

20

r~'I..'ro ~ I

Go

roooo

'r.'

,.-

I

."..110.

'...

~...

'IO..I".!... 1'1..000. 10...

I.

!... !...

U .'"

:

17

w ~

'

., ., ~

I

J

W

-

1

Go

U

14

W i:

.,

13

i ~

IZ

0Z

1.1

i

I

'or~~..orcooorc'lo.'.c'oo III 'oo..'ro .S '."' .ors...ro ."",c r '0 .a,.rs .",

C ~W ~J

.

.0' co."o.ro rl.tO,.t.'."., 10

..'cos". ." r..r..'to "t,s"ar "P.' .,., f..'

~ 0' w Go

2 w ~

-0&

~ g

~

07

/

If)

IaI

~ ~ c

>-

oa

w,sco'", I,..) .' I.,.

'.rs",.r .t..o. I't".

., s."'O."O. ..rs'...( 0' ..'r. ..o.r lit"

0$

~ Vi

0

04

(,) If)

>:

03

02

r

01 00

.0

ac

ac

tOO

120

140

lac

lac

200

220

240

TE~PERATuAE, -,.

fig. D-16-Water viscosity atvarioussalinities and temperatures.'

"I 128

WELL TESTING

!

700

Example

oJ~

c

.w'Sl'CLLANCO(l.

..:so

.C.t'$A'l"'1;

Problem. Estimate the viscosity of water containing 20.00) ppm (2070)dissolved solids at 200"F and 5,

It'tVID,f

u ..

of

W'ater Viscosity

G'ASCS

~ ~ 6~0 ~ Q.

D. 12-Estimation

0111 0«600 ~~ ~~ ~ ~50

psla. Solution. From Fig. 0-16. the viscosity IJ.r" of water with 211io NaCI at 200" F and atmospheric pressure is 0.32 cpo The correction factor f for 5,-psia pressure is 1.016 (inset at upper right of figure). Thus.

~oo «

/l... =/lr"f=(0.32)(1.016)

=0.33 cp,

0

oJu.

C

4~0

~ ~Q.

Pseudocritical

i ~ g ~ 400

Pseudocriticaltemperature, Tpc:' and pressure, Ppc. are useful quantities that allow application of

g: ~!' 3~0 Go ~ ~ 300

generalized correlations of gas properties needed in applications. The most accurate values of these quantities are calculated from compositions of gas mixtures.2.3 Approximate values can be determined from a correlation developed by Brown et 01.II This correlation, presented in Fig. 0-17, is based on gas gravity, 'Yg; values of critical properties

~.

-natural 0.6

0.1 OJ 0.9 1.0 1.1 GASGRAVITY,Yv' (AIR 81)

1.2

Fig. D.17-Correlation of pseudocritical properties of condensate well fluids and miscellaneous naturalgaseswith fluIdgravity.'

Properties

of Gas

depend on whether the gas is from a gas condensate reservoir (the "condensate well tlwds" curve) or from a relatively dry gas reservoir (the "miscellaneous gases" curve). Example 0.13 illustrates use of this correlation.

2,5~ psia.. Formation volume factor of the 0.7. gravltygaslsO.001132RB/scf.

P

Solution.

Problem. Estimate Tpc and Ppc for a dry gas of

-:seu

From

Fig.

0-15,.

for

dRs...ldp = 0,0033. The correction effect of salinity (Fig. 0-11) is 0.875;

fresh

water,

factor for the thus, for brine,

Example

d

D. J3 -Estimation ,. IG P

ocntlca

gravitv(f .g

)07

as

.,

dRs",ldp = (0.0033)(0.875) =0.0029 scf/STB-psi. From Fig. 0-12, using procedures outlined in Example 0.9, B", = 1.033RB/STB. From Figs. 0-13

Solution. From p pc = 665 psia.

andO-14,usingproceduresofExampleO.l0,

G

-~

~ =(3.13x 10-6)(1.11) B... dp =3.47xl0-6psi-l.

Th us, c

'"

= -~

~ + ~ ~ B... dp B... dp

= 3 47 x 10 -6 + (0.001132)(0.0029) .(1.033) = 6.65 x 10 -6 psi -I.

Water0-16, Viscosity Fig. first presented in the Ilteraiur-eby~Mii:-~thews and Russell,lO allo\\s estimates of water viscosity as a function of reservoir temperature, salinity, and reservoir pressure. Example 0.12 illustrates use of this correlation.

of

t. roper les

Fig.

0-17,

Tpc=390oR

and

..

L D t as- aw evla Ion Fac t or ( 1.-Factor) and Gas Formation Volume Factor Application of the real gas law, pV=znRT, (0.4) to relate pressure, volume, and temperature for gases .. requires values of the deviation factor z. One application of particular importance is in calculating gas formation volume factor, Bg, from Bg = 0.00504 TZlp (RB/scf). (0.5) Fig. 0-18, developed by Standing and Katz, 12 can be used to estimate z. One must know pseudoreduced pressure, Ppr' and pseudoreduced temperature, Tpr' to determine z. By definition, Ppr=plppc and. T -TIT

(0.6) ~,~

(0 7)

pr -PC" In these definitions, pressures must be expressed in psia (psig + 14.7) and temperatures in degrees Rankine (8F + 460). To use this correlation, one must know reservoir

~ --~J

: i I I I

,

ROCK AND FLUID PROPERTY CORRELATIONS

1.1

0

I

&

S

129

4

~£IIDO-~£D(JC"D

r£MP£~A~£, r,

5

7

1./

.

".0

6.. 6"

10

.0

i4 6.6

0..

0.. ,

~

.,J

NO.7

,..

~

U ~ .1:

I."

0..

,.-

Z 0

~ 4

~ 0 1/1

I.' 0.5

1.4

o.

l.a

4

~ oJ 4 .., ~

.

,./

.. OS

1.2

I I

1.1

41.0

{'II

1.0

I. I

1.06

0.'

0...10

II

1&

13

14

I.

PSEUDO-REDUCED PRESSURE, Ppr Fig.

0-18 -Gas-law

deviation

factor

for natural gases as a function

of pseudoreduced

pressure and temperature.'

I ! I

I I.

---~

I

s

?

...k

130

Q.

u >-" -1&1

~

~ !

~

..7 Q

U >-

m

.J

... gj

~

U

1&1

:) 0

1&1

0

~ 0

of pseudoreduced gases.1

10

compressibility

.0

Q.

'

I

10" 3

...1

00

.,

.1

80

.7

Q

§~ C..

..

~II

. !.; "-

10

u> ..

00

,

80

~

..

0..

15

WELL TESTING

10

I

-

CO.

a..

compressibility

PSEUOOREOUCEO PRESSURE. Ppr

...,»

-

Fig. 0-198 -Correlation of pseudoreduced for natural gases. 1

1.0

.0

00

GASGMVITY. Y,. IAIRa'.OOOI-

.S"

.""co,

,..

'

~ . Q. ~

0

0

U

f 0

U

0 :)

D.19A -Correlation for natural

o.

so

cn

1&1

0

Z

0

10

of natural gases at 1 atm.'

MOLECULAR WEIGHT. M

0-20 -Viscosity

.0

PSEUOOREOUCEO PRESSURE. Ppr

I&J

U :)

0

1&1

I&J

I

~ 5

I

~ I ~ Q.

0 :) I&J

10-1

Fig.

00'. OOt. ' 00'

a. u 0.005

..." &

:to QO'

OJ

~ "'ax . 0011 ~ .. C 00'0

101

..000. C .. ..

U

-000. .. 0 ..> 000"

QOO

0

Fig.

I:":

IfIit' ~£

,~

,.-

"

,;'

.i

.II

.1

"

~ ROCKANDFLUIDPROPERTYCORRELATIONS

131

temperature and pressure and pseudocritical temperature and pressure (from either composition or gas gravity). Example D.14 illustrates use of this correlation.

'0

"

ExampleD. 14-Estimation of Gas-Law Deviation Factor Volume Factor

and Gas Formation

"

Problem. Estimate.: for O.7-gravity gas in a reservoir with temperature 200°F and pressure 2.500 psia. Use this value of z to determine the gas formation volume. factor Bg. Solution. In Example D.13, we T pc = 390"R andppc =665 psia. Then.

found

that

Tpr=TITpc=(200+460)/390=1.69. and Ppr =plppc =2.500/665=3.76.

~ ~" g4

I'

~ ~ "0

j

>

From Fig. 0-18, ,=0.851.

Then,

Bg =0.00504 T,/p

00

= (0.00504)(660)(0.851 )/2,500 = 0.001132 RB/scf. 00

Gas Compressibility Figs. D-19A and O-I9B (developed by Trubel3) lead to estimates of gas compressibility, Cg' In these figures (which cover different ranges of the independent variables), pseudoreduced compressibility, cr. is plotted as a function of pseudoreduced pressure, Ppr' with the parameter pseudo reduced temperature, Tpr' Pseudo reduced compress!b!l!ty.is defined as cpr =C"Ppc; thus, gas compressIbIlity ISfound from the relatIon -viscosity_' Cg-Cprlppc'

10

'0,

'0 ~S~UOO"~DUC~D T~..PE"ATU"~. T..

Fig. D.21A-Effect of temperature and pressure on gas (D.8)

Use of Figs. D-19A and D-19B requires kno\\'ledge of reservoir temperature and pressure and pseudocritical temperature and pressure of the gas (from either composition or gravity). Example 0.15 illustrates use of this correlation. ..for ExampleD.15-Estlmatlon of Gas Compressibility Problem. Estimate the compressibility of a 0.7gravity gas at a reservoir temperature of 200"F and pressure of 2,500 psia. S I I E I D 14 f d h 0 utlon. n xamp e ., \\'e oun t at T pr = 1,69, Ppr = 3.76, and Ppc = 665 psia for these conditions. From Fig. D-19A, cpr =0.26. Thus,

.

Cg =cprlppc =0.26/665 = 0 00039 .-1 .pSI. Gas Viscosity Figs. D-20, D-21A, and 0-21 B (from the work of Carr (Or01.14)can be used to estimate gas viscosity at

reservoir conditions. From knowledge of reservoir temperature and gas gravity (or its equivalent, molecular \\-eight). we can estimate the viscosity of a hydrocarbon gas, /lgi' at atmospheric pressure... Insets in Fi2. 0-20 allo\\' corrections to this viscosity nonhyd;ocarbon components of the gas. Figs. D21A and D-218 (two different \\'ays of plotting the same data) permit calculation of gas viscosity at reservoir temperature and pressure, given viscosity at atmospheric pressure. pseudoreduced temperature, Tpr' and pseudoreduced pressure. Ppr' Example 0.16 illustrates application of these figures. Example

D.16-Estimation

of Gas

Viscosit.v Probiem. Estimate the viscosity of a -o.7..gravity hydrocarbongas(noHzS,N2,orCOz)at200°Fand 2,500 psia. Solution. From Fig. D-20, the viscosity It 1 of 0.7gravity gas [molecular weight = 0.7 x 28.~ = 20.3 Ibm/(lbm-mole») at 200°F and atmospheric pressure

i iI

-~

, 132

WELL TESTING

e.

'.0

0

~

~ "~ ~

4.0

0

.i.-

~ ~ ~

).. I-

--.---

U) 0 (.)

II)

>

3.0

2.0

, i

1.0

I

J

4

'8'1'

I

1.0

10

14

,1

"po"

!

PSEUOOREOUCEO Fig. 0.21 B -Effect

of pressure and temperature

Ppr

on gas viscOSity.'

is 0.01225 cpo In Example D.14, we found that. at these conditions. T pr = 1.69 and Ppr = 3.76. Thus,

Formation compressibility is a complex function of rock type, porosity. pore pressure, overburden

from Fig. D-2IA or 0-218, J.lg/J.lgl= 1.45. At 200°F and 2,500 psia, gas viscosity is -( / ILg-J.lg J.lgi)J.lgl = (0.01225)( 1.45)

pressure, and, in general, the stresses in different directions in the formation. No reliable correlation of this quantity with the controlling variables has been presented in the literature; indeed, laboratory determinations of cf are difficult, and many reported

= 0.0178 cp

values .because

... " ,!

of this quantity are doubtless erroneous conditions in the field were not duplicated in

.by Formallon Compressibility

the laboratory. A much-used correlation, developed Hall. IS is presented in Fi2. 0-22. This correlation relates Cf to a single variabl~-porosity. As reported

-~~

Formation compressibility, Cf' is defined as I 0V C f = --( .:..:..2.) , V p op T

by Earlougher, I this correlation is known to be incorrect by an order of magnitude or more in specific situa(ions. Thus, \\'hile (he correlation is easy to use, the result may be seriously in error for IftJ~

.

where V p = pore volume of porous medium.

~.-

PRESSURE,

-,

(D.9)

"--

given applica(ion.

I"j

ROCK AND FLU::::~:RELATIO:..~I.IUse of [his correlation is illustrated 0.17. The result may be of no greater simply assuming Cj:4x 10-6 of the many variables affecting

in Example accuracy than

psi-l,sinceonlyone C f has been taken

into

account.

ExampleD,17-Estimationoj Formation

Compressibility

Problem, Estimate for a reservoir with Solution,

From

Fig,

Exercises. Results of pressure are

combined

calculate

.T ., vI

compressibility

D-22,cf=3,6XIO-6

Cf

psi-[.

T

I \.~\l.~

-'=i

I

-i

The

transient

with

.! J

~i

1

rock

test analysis and

fluid

sometimes

+cwSw

properties

+CgSg

exercises

,

qo

=

100 STB/D,

=

20 STB/D,

qg

= qoRs

(reservoIr

1.. ater sa tnlty = ko = 20 md, k... k

calculation

of

=

lOOSTB/D,

=

5 STB/D

..3.

produces

for

an underproperties.

dIssolved

gas

qg

psia,

25 rv,,-,

,vvv ppm

.I

I

0

'--1 10 II

II

I.

.

D-22 -Formation

Water

salinity

Sw

= 0.25,

S S~

= 005 = 0:70:

compressibility.

= 27,500

15

ppm,

and

[.

Earlougher,

R:C.

Jr.:

Adl.'ances

... In

"'~I

Test

AnalYSIS,

= 3S' APi, = O.S,

= 1100 md' 0 .181, k... = 3.3 md, kR = 7.25 md,

Petroleum Publishing Co., Tulsa (1973).

of Petroleum

. Fluids,

Oil and It~, Associated Gases at Oil-Field Temperatures and Pressur~s, Trans., AI~IE (1946) 16~: 94-~IS. ..

sa[uration),

= 250 Mscf/D,

Book Co., Inc., Nev.. York City (1960). McCain, W.D. Jr.: 77Ie Propernes

341-3~.. . 5. Standing, M.B.: Volumetric and Ph~e BehaVIor of Oil Field Hydrocarbon Systems, Reinhold Publishing Corp., New York City (1952). 6. Ramey, H.J. Jr.: "Rapid ~Iethods for Estimating Reservoir Compressibilities," J. Pet. Tech. (April 1964) 447-454; Trans., AIME. 231. 7. Seal, C.: "The Viscosity of Air, Water, ~alural Gas, Crude ..

(2 5 U7 N Cl) ..0 a ,

Reservoirpressure Reservo.r temperature = 2,OO)psia, = 200.F k

1

.'-"-

4. Trube, A.S.: "Compressibility of Undersaturated Hydrocarbon Reservoir Fluids," Trans., AIME (1957) 210,

= 220.F,

".,

Oil gravity Gas gravity

.

q Rt'

8. Chew, J. and Connall)., C.A. for Gas-Saturaled Crude Oils,"

So = 0.65,

q

li--

!,i.i

It> = O.IS.

IS

q 0

~

'O-Ollff-'tIU.f

+cf'

S..' = 0.35, and S = O. g 0.2. Calculate q Rt' At, and Ct for reservoir with the following properties:

I

o_..;-~

=1

References

= 4,00)

= 0.93 md, = 0 (no free-gas

g -0 4> -.,

I

,1

..~.::

Fig.

require

Reservoir [em perature R = 400 scf/STB s 0 7 ' i' g = ., i'n = 0.S5,

W

~...::..;:

,i

Monograph Senes, SPE, Dallas ([977) 5. 2. Arnyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petrolt'Um Reservoir Engineering: Ph_vsical Proper/ies, McGraw-Hili

only), pressure

Reservoir

i

to

Ct for two cases.

q w

e ~-!-

quantities:

0.1. Calculate qRt' At, and Ct saturated""bil reservoir with the following

I

'1:;

;'~;

:

00

mobIlity,

following and

.:I

I

.f ~,

At=kol#l.o+kwl#l.w+kgl#l.g, Total compressibility,

~t'

""".

~.-Iu

the following

Ct =coSo

~

I.-i .:

I~

., ~.

Total reservoir flow rate, qRt =qo.l!o +qwB... + (qg -Rsqo/l,(xx»Bg. Total

.~~~l -; .i ~

f

lit

the formation 20070 porosity.

-:-

Jr.: A VIscosity Correlation Trans., AIME (1959) 216, 23-

25.

a saturated

oil

9. Dodson, C.R. and Standing, M.B.: "Pressure-VolumeTemperature and Solubility Relations for Natural-Gas-Water Mixtures," Drill. and Prod. Prac., API (1944) 173-179. 10. Matthews, .C.S., and Russell, D.G..: Pressure Buildup and Flow Te~ts In Jfells, Monograph Senes, SPE, DaJlas (1967) 1, Appendix G. II. Brown. G.G., Karz, D.L., Oberfell, G.G., and Alden, R.C.: .Vatural

Gasoline

Gasoline

Assn. of America,

and

the

Volalile

Hydrocarbons.

NaturaJ

Tulsa (1948).

12. Standing. M.B. and Katz, D.L.: "Density of Natural Gases." Trans., AI ME (1942) 146,140-149. 13. Trube, A.S.: "Compressibility of Natural Gases," 14. AIME(1957)210,355~357. Carr, N.L., Kobayashi, R., and Burrows, D.B.:

Trans.,

ViSCOSityof

Hydrocarbon Gases Under Pressure," Trans., AIME (1954) 264-272. IS. Hall, H.N.: "Compressibility of Reservoir Rocks," Trans., AIME(1953) 191, ~.311.

:J ;} j;!:$tr\ioii i~

I.;~"{, ..'" t"" ""~ l;..'co

~"~I,"",, ""t.."" """

Appendix E

A General Theory of Well Testing The purpose of this appendix is to summarize an a.pproach to well test analysis that is not limited eIther to wells in infinite-acting reservoirs or to wells centered in cylindrical reservoirs. This theory is stated more clearly in terms of dimensionless variables, which we have largely avoided in the text because they are abstract and thus are less easily understood than other approaches. For some applications. though, use of dimensionless variables is essential-and We have seen general that for theory I,S 948 is such q,JJ.C an (r e2 area. / k (infinite-

or 0 00708 kh (p -p ) , ~~~.. \Yi ~f'wf~ = q8jJ. 21 ( :-4 + In r De -0.75) + s rDe -+ I >0 25 2 -PD s'D .rDe' .

acting reservoir) and for constant-rate flow for a well centered in a cylindrical reservoir.

(I II) and

for

I>

948

Pi -Pwf=

q,p.c

Even more generally. we can wnte 0.00708 kh (Pi -Pw~ --

2 ) -2s ].Thus. [ln ( !.688q,JJ.C(r~ kl

P-I -P wf= -70 .kh 6~

I r e2 I k

(pseudosteady

8 -PD(ID)+S. (E.3)givenq ajJ.value ID' there exists a rule for. determining PD for a well centered in a cylindrical reservoir-either expression general

state),

q8jJ. 0.527 kl 141.2-[ .; -2 -The kh q,p.c(r e

3 -n (1.12 modified)

We can generalize by writing these equations in terms of dimensionless variables: 0.00708 kh (p -P _-'::--=-=-:':T' ~ q jJ.

)

Eq.

I 0.264 kl = -(In .; -2 2 q,jJ.c(rw

I =-(lnID+0.809)+s

changes in rate in the producing

(

-~_o-

history

of a

in terms of dimensionless variables. is found by superposition. as we showed in Chap. 3 for the..

. ,

(I D») I:

2'

q8jJ. (Pi-~

0.000264 kl .~:;:..-2q,JJ.C(rW

this and

~-'~~~"~i-f'Wf~=(ql-O)[PD(ID-O) jJ.

I

=PD+S,ID,S0.25rDe2. and~708kh

Thus, flow

well(Fig.E-I). total pressure drawdown at the well. expressed

0.00708kh[Pi-Pw

q8jJ.

Eq. transientE.2.

general.

or )

or

special caseof transient flow. Here. note that rate ql acts for time. I; (q2 -ql) for time (I-II)' ...; and (qn-qn-l) for time (I-In-I)' Thus. we write, in

+ 0.809) + s.

0.00708kh(p,-p ""iYwf'

E.I includes

pseudosteady-state flow as special cases. value of this generalization becomes clearer if now we reconsider a subject we introduced in Chap.

, r +In(-!..-)-0.75+s]. r w -The

(E 2)

(E.I)

+In

( ---+s, re ) r...

-ql)[PD(ID-IDI)+S)+

+ (qn )(PD (ID -ID.n-1 or.morecompact~. ~00708 kh -qn-1 [Pi ~Pwf(~Q!.! = n

= [ (r -Zelr w) 2

)

+s)+(q2

[ j=1L'

8jJ.

3 4

]

(

PD ID-IDj

)] +qns,

'" )+sJ,

~j

j'. (E.4)

.

:I'!

~

A GENERAL THEORY OF WELL TESTING

135

where Aqj =q; -qj-l

(and qo -0),

tLX)-0. Eq. E.4 is general- i.e., it applies to a reservoir in which, for some values of t D -t Dj' P D can be the pseudosteady-state solution, and, for other values of t D -t Dj' P D can be th~ transient solutio~.. As an example, consider a pressure buildup test In a cylindrical reservoir. tDI =tpD+AlD;andtm-tDI 0.00708 kh (Pi -Pws) BII.

Let q I = q; =AlD.Then,

Q.n,.t

I Q.

-

~

3

4

q2 = 0;

n-1 n

t

Fig. E.1- n rate changes in well's producinghistory.

=qPD (tpD +AlD)

-qPD(.1tD)'

I -2

If and only if flow is transient tpD+AlD,then 0.00708 kh (Pi -Pws) -~ BII. -2q

[In (tD +AlD)

+0.809] +PD(tD+AlD)'

for total time

[1

, Now, for In(tD+~tD)

n (tpD+AlD)

some small =In(tD)

,

(E.6)

values of 61/D:S61/Ds' and PD(tD+~tD) =

PD(tD)' For olt D:S AIDs' then,

+ 0.8(1)] -~ q(ln I1tD + 0.809) 2

" 0.00708kh

or

qBII.

p;-pws=70.6-ln

.~

-qBII. -162.6k;;-log

-In I

=

qBIL

( .pU' t D+~D -u

kh

(Pi-Pws)

2

)

( t+~ -+PD(tD) )

I1

AI

I

AID

-2

(~AI ) .2

= ~ In

In fact, the arguments leading to Eq. E.4 understate its generality. For constant-rate flow in cylindrical reservoirs, PD can be calculated for all t D from simple equations -but the method is not restricted to cylindrical reservoirs. It applies to any drainage configuration for which P D can be calculated as a function of t D (using finite-difference simulation or any other convenient means). We now examine a useful method 1-3 for determining P D as a function of t D for more general reservoir shapes; this method uses the MatthewsBrons-Hazebroek functions4 developed for use in determining average reservoir pressure. We start by a pressure buildup test in a reservoir no thlasthfor 0 f ting genera ape,

[In (2.246 tD)]

(~AI ) +constant..

(E.7)

The implication of Eq. E.7 is simply that for sufficiently small AI, a plot of P ws vs. In [( t + ~) / ~J or log [( t + ~) / ~J will be linear and will have a slope, m, related to permeability. This linear relationship exists, of course, only for sufficiently small values of 11t. Once we have established the linear relationship, tho~gh, we can extrapolate it to larger times. In particular, we ca4ne~traP.olatePws to (t+~) / ~ = I. Matthews et al. did this and chose to call the extrapolated pressure p'. Eq. E.7 shows that 0.00708 kh I B (Pi -p') = PD (t D) --In (2.246 tD)' ..i q JJ. 2 ...,

,

,

,

. i

"

I

(E. 8)

A material balance shows that ( .) 000708 kh .P, Pws B=PD(tD+~D)q IL P D (~D

c, V p (Pi -jJ) =5.615 qBt/24 --c,Ah(p;-p),

),

, (E.5)

where A is the area drained

(E.9)

by a well (square

(Eq. E.9 is valid regardless of drainage-area

For ~ID sufficiently small (e.g., fJD :s0.25 r[)e2 in a cylindrical reservoir), flow will be transient regard-

.No.\\" define tDA as 0.~264 stltutlng for I from Eq. E.9 gives

less = 1/2(lnAlD+O,809).lfwenowaddandsubtract of drainage-area configuration, and PD ~AlD)

tDA= 0000264 '= kt

1/2 In (t D +.1l D) on the right side of Eq. E.5 written at these sufficiently small values of ~ D' the result is 0.00708 kh (Pi -Pws) = ! In (~)

1LC,A

qBIL

2

~

feet),

I j

shape.)

kt/ILC,A. Sub-

(0 .(p;-p) 000264) (kh) (24) (5.6 I 5)(qBIL)

-0.00708 kh (Pi -p)

-2rqBIL

' I

I

I

:r!l~

.. ' :.

," ",;;r:'{; 136

~~

:~;'

or

WELL TESTING

been used to determine the values in Table 1.1 in the -column

"Use Infinite

2 r t DA _0.00708kh(pj-p) -qBp.

(EIO)

of the relationshipPD(tD)=1/2 reservoirs of general drainage-area

Then, using Eqs. E.8 and E.ID,

-I 0.00708 kh (p -p')

+ 0.00708 kh B (p.I -p)

qBp.

=

q II.

pointed out that, at pseudosteady state, PDMBH (tDA) = I n (C AIDA )

.6

q

=lnCA+lntDA' = 47ft

DA

+

In

(2.246

t D)

-2

P D (t

D)

p.

where the

=PDMBH(tDA)'

(E.II)

Note that the left side of Eq. E.II is the ordinate of the MBH plots (thus, we give it the name PDMBH)' Note also that tDA =0.000264 ktltPlJ.C(A is the abscissa of the MBH plots. Eq. E.II allows us to use the MBH charts to determine P D (t D) for the drainage-area configurations considered in constructing the charts. To understand why this is so, note that Eq. E.II can be put in the form PD(ID)

I = 2 'l't DA + -In 2

I --PDMBH(IDA)' 2

ForID
(E 12)

ID
then,

P D (I D) -I -21n

I -2PDMBH

a

sh~pe draInage

factor

(E.14)

whose area

value

configuration.

~epends

o,n ThIs

-1/2In(C

.

-

A tDA)

=2rIDA +1/2In(~!.Q) C A IDA or

2.246 A PD (tD) =2rIDA + 1/21n (-;:;:-::-r). CArw

(E. IS)

Thus,beweestablished have established a rule drainage by which shape PD (I D) can for general in

(2.246 I D)

=2 'KtDA+2 ~ I n (2.-ID "~6

specific

is

equation implies a linear relationship between PDMBH and In tDA' with intercept In CA varying from case to case. The \-1BH charts (Fig. 2.12) show that this linear relationship does exist and that the intercept does depend on the specific drainage-area configuration. Further, the time at which PDMBH becomes a linear function of In t DA establishes the beginning of pseudosteady-state flow. The task at hand is to show that Eq. E. J4leads to a method for determining PD(ID) for general drainage-area shape for pseudosteady-state flow. If we substitute the value of P~BH (IDA) from Eq.

For time, ID' sufficiently small that no boundary effects have appeared (transient flow), I

C A

E.14 into Eq. E.12, the result is _ P D (I D) -rl2 DA + 1/21n.(2 246 t D )

(2.246ID)

PD (I D) = 2 In (2.246ID)'

in

we turn ourP D attention to developing a method forNow determining (t D) after pseudosteady:state

Simplifying, (p' -p)B

(In tD+O.809) configuration.

conditions have been established. Brons and Millers

I -PD(tD)+-ln(2.246tD)+2rtDA' 2

70

System Solution With Less

thanlOJoErrorfortDA<,"thusprovidinguswitha means of determining the upper limit of applicability

)

(IDA)'

pseudosteady-state flow, Table 1.1 gives the times .. ("Exact for IDA <" and "Less Than.JOio Error for tDA<")atwhichEq.E.15canbeapplled.. Finally, there is the problem of how to establish PD (I D) for general drainage-area co~fi.8uration when there is a gap between the upper limIt of applicability of the transient solution and the lower limit of Eq. applicability the pseudosteady-state solution. E.12, whichofapplies at all limes, can be

or P DMBH (IDA) = 4rt DA

used to fill this gap: PD (t D) =2'Kt DA + 1/21n (2.2461 D)

(p' -p) =,lD
-1/2PDMBH ( IDA' ) d h . h . To avoid both tDA an ID on t e rig t Sldeo f the same working equation, we can rewrite this result as 2 PD (I D) =2rlDA + 1/21n (2.246 IDAA/r. ) -1/2p (I)... (£.16) -~, DMBH DA . Thus, values of PDMBH could be read from their charts at a desired value of IDA' and PD (/p> could

.!..A:

,

"

I'

r

AGENERALTHEORYOF WELLTESTING

be calculated

137

for use in subsequent reservoir analysis.

~lougherprovides.vaJuesofPD(tD) ~orgeneraJ draInage-area shapes In Ref. 6 AppendIx C. For applications. the reader should find the desired PD values in that reference. In summary. this appendix has shown that well test

aJ

L

.

h

.

,..

d

II

ed

.an yS!S t~ rnques ar~ not Iml~e t? we ~ center In cylIndrIcal reservoIrs or to InfinIte-actIng reservoirs. One can derive or find in the literature dimensionless-pressure values for general drainagearea shapes; one can combine these values (using k . b .5. .. s~perposltlOn) to ta e I.nto account any ar Itr~y rate hIstory before and durIng the test. Thus. the title of the appendix: A GeneraJ Theory of Well Testing.

References I. Ramey. H.J. Jr. and Cobb. W.M.: "A General Pressure Buildup Theorv for a Well in I Oosed DrainageArea," J. Prt. Tech. (Dec. 1971)1493-1505;Trans.. AIME, 251. 2. Cobb. W.M. and Dowdle. W.L.: "A Simple Method for Determining Well Pressurein Oosed RectangularReservoirs." J. Prt. Tech.(Nov. 1973)1305-1~. 3. Dake, L.P.: Fundamentals af ResenlOirEngineering. Elsevier Scientific Publishing Co., Amsterdam(1978). 4. Matthews, C.S., Brons, F., andHazebroek.P.: "A Method for Determination of Average Pressurein a BoundedReservoir," Trans.. AIME (1?5.J)201,187:191: . Brons, F. and MIller, W.C.: A Simple Method for CorrectlDl Spot PressureReadings," Trans.,AIME (1961)222,803-805. 6. Earlougher. R.C. Jr.: Adl.oonm in Well Test Analysis, Monograph Series,SPE, Dallas(1977)s.

Appendix F

Use of Sf Units in Well-Testing Equations

This Appendix summarizes the changes required to solve the equations stated in the text by using International 5ystem (51) metric units. To show the necessary changes and to allo~' the interested reader to apply the 51 unit system to typical well-testing problems. this Appendix has four major parts: (I) conversion factors from "oilfield units" to 51 units are tabulated in Table F-I for the units used in the text; (2) a summary of preferred 51 units for major variables is given in Table F-2; (3) major equations in the text are restated in Table F-3. with constants given in both <:,ilfield and 51 unit.s: an~ (4) ans~ers to all examples In the text are given In 51 units. In addition. in the Nomenclature. preferred 51 units are given (in parentheses) for each quantity used in the text. TABLEF-I

To Convert ~~ acre!i bbl cp cp cu fl ft md psi oR 5Qfl

-CONVERSION

~ m~ m' mPa.s "Pa.!i m' m m.m~ kPa K m~

~ulliJ1lyBY 4.04; E + 03 1.590 E-OI 1.0 1.0 E+03 2.832 E -02 3.048 E-OI 9.869 E-OI 6.895 5.555 E-OI 9.24X>E-02

FACTORS

tnver!il:_2.471 E -04 6.21XJ 1.0 1.0 E-03 3.532 E +01 3.281 1.013 I..SO E-OI 1.80 1.076 E+OI

A more complete table of conversion factors. emphasizing application to well-testing problems. is given by Earlougher.1 Table F-2 summarizes oilfield (customary) and preferred 51 units (practical) for single variables and groups of variables of major importance in well test analysis.

TABLE F-2 -CUSTOMARY

A~D \IETRIC UNITS

.,

fOR ~IAJOR VARIABLES IN EQUATIONS

Compre~sibiliIY.(i

CuSI(\mary Practical tJnil Metric Unit ~i ; kPa I

DensilY.p Gas no~ rare.q. Gasvi!ico~iIY.,.. Liquid no~ raft. q.. and q.. Liquidvisco,iIY." Permeability. k Pre!i\ure.p Pseudopres~ure;~(p) Radius.r Slope. III Temperalure. T Thickness.h Time. ( Volume. V Wellboresforageconslanl. C,

Ibm,cuft Mscf D ,,-p BID cp md ~i ~i~;...p fl ~i/,,~...le oR ft hr bbl bbl/psi

kg/m' m'!d "Pa.' m'/d mPa.s md kPa MPa~/Pa.s m kPa/cyde K m b m' m'!kPa

-

. ~~

USEOFSIUNITSINWELL-TESTINGEaUA::~~

TABLE F-3 -~AJOR

EQUATIONS

WITH

CONSTANT

VALUES

139-

IN CUSTOMARY

AND

Numerical Value of Constants in Order of Appearance

Equalion

Number

("I' "2' "3 .-.)

in TexI

Equalion .,

1.1

1.6

a. P I ap -,+--=--

or.

Customary

~IJ.C' ap

r ar

"Ik

at

qBIL [ 2to

Pwf=Pi-"I-

3

-:-y+lnrrO-rrO

kh

4

, ., ~ e-ao'oJ.(a r > + 2 '""' I n rO i.J22 2 n=1 an fJI (anrrO> -JI(an>])

1.7

P=P+"

1.11

I 12 .WJ

kh

~s="I~(~-I)ln(2.) kh ks Pi -P..f=

-"I

( ""OIL",r;, ..-ls )

kh

kt

~=_C-J!!!! at CI VP

1.16

P-Pwf=c.-ln qBIL / kh

1.19

( ---+s rr )

3

r..,

4

P-P '"

qB1L c2 kt f =c l -.,+In kh ~ r:

Js

q

=

1.21

J=

II BIL -In( 2

1.26

9.33x 102

948

7.036 X 104 1.866xI03

706..X

933

1.688

1.253x 105

0.527

I(

)

3

kJh

102

I I

7.1 X 10-6 4.168xI0-2

141.2

1.866x 103

141.2

1.866xI0

0.527

141.2

)

3

7.lxIO-6

1.866x 103

31 4

( 10.06 A ,... _2)C.4 rw

".kh 10.06 A 3 -, ) --+

CAr...

rr

r---+s 4 w

.. 3 -+5 4

I

141.2 0.00708

4

CS="I~!£

1.866x 103 5.356xI0-4

I

s

25.65 p

1.29

70.6

3

1.866x 103

I (

"I BIL In --rr

qB1L1 1-In kh 2

P-P..rj=".-

1.866xI0

1

r .4

r... 1.20

r

( ---141.2 rr ) 3

1J.C"t'

P-P"f

141.2

-0.234

1

1.17

3.557xI0-6

141.2

-In 1 qB1L

1.13

0.264

r ".

c2kt_2 +In qB1L1 .Lkh ~"" r r- It'..

P ",=P,-clI

51

' I \

., { -"2~1L"lr ) kt

I -£1 qBIL

I

1.9

SI ~ETRIC

101.98

g

Po=c,kh(Pi-PW>

0.00708

5.356x!0-4

qjBp.

1.46

pj-Pwf=cl kt

1.47

r -i:.-.

qB -(-) hLf '/j

p.t

'/: 4.064

6.236

kq,c,

4

r;=()

948 c I ~IJ.C',

if!1- -".u

:::;],. -~;.,." -,..

.

7.036x10 ,'~-~ 1:

.-

'!'"'l_::~

:i~'F~

140

WEll

TABLE F-3 -MAJOR

EQUATIONS

WITH

CONSTANT

VALUES

IN CUSTOMARY

TESTING

AND SI METRIC(Conld.)

NumericalValue of Constants in Orderof Appearance

Equation Number

(CJ'C2'C3 ...)

in Text

Equation

Customary

51-

1.48 Is=CltPlJi:,r;/k

948

7.036x 104

1.52

lp =cINp/qlasl

24

24

2.1

qBIJ. P"'S=Pi-C/~log

162.6

2.149x103

2.2

m=c.- qBIJ.

162.6

2.149x103

+3.23

5.10

0.00708

5.356x 10-4

0.000264

3.557x 10-6

0.894

0.159

[ (lp+;1/)/;1/ ]

kh

2

.4

2.7

s=I.151

I (Plhr-P"f)

(

In

-Iog.~.

k

)+cll

PD= clkh(Pws-Pwj) qBIJ.

2.8

ID= ~ tPlJ.'.,rIt.

2.9

CsD= ~ « , h It'

2.10

.1/t'=~/(1

2.11

C

s

+A//lp)

--

= qB --24 AI

24

ci ~

2.121D=50CsDeo.14S

--

CseO.14S

2.13

I '"b s =c J (kh/lJ.)

2.14

r",o=r",e-s

--

2.16 2.15

Lj=2r",o (~)s=0.869m(s)

--

2.17 E= ~

170,000

= P-P"f-

Jideal

(~)s

2.247x 106

--

P-P",j

.

--

s= hI-sd+sp

--

2.18 'E=P.-P",j-(~)s P -Pwl

2.19

hp

2.20

sp= ( !!!. -1)[ In( !!i-Jiji) -21 hp rw kv

2.21

mL =C, -(-)

qB

hLf

2.22

IJ.

--

1/2

4.064

6.236

kcPc,

10g(LI) = ~ 1( PWj-Plhr ) +IOg.-!!.-CI 2 m cPlJi: I

-;c-.:

I

2.63

4.497

-

USE OF SI UNITS IN WELL. TESTING EOUA TIONS

TABLE F.3 -MAJOR

141

EQUATIONS WITH CONSTANT VALUES IN CUSTOMARY AND SI ,\fETRIC(Contd,) Numerical Value of Constants in Order of Appearance (CI'C2'C3.")

Equation Number

in Text

-Equation

--~

~~iomar;

.

2.24

10g(jJ-pws)

c2k~

kh 2.25

QBIJ. [ -In kh

Pi -Pws = -ci

-c,

kh

kh 2.29

L=J~l~§~
2.30

VR=

-Ei

kh

0.00168

-0.8385

70.6

9.33 x 102

1.688

1.253 x 105

3.792

2.814x

10S

)

k~1

-Ei(~~
{

227.37

1

1

( -C3
BIJ.

(-q)

~"ws=-CI~

( c2
In --~~

k(lp +~)

118.6

}

Eil ~3
kh -c,

4>IJ.C fr*e

I r~ / -2.s c2
BIJ. /

(-q)

-ci ~

2.28

/

s;t-

--

(cl --., QBIJ. )

=Iog

-J

70.6

9.33xI02

3.792 0.(XX)148

~4Np)(Bo~

2.814x 105 1.944 x 10-6

--

(P2-P,)Cf