Calculation of Water lnfIux for Bottomwater Drive Reservoirs D.R. Allard, SPE, Gulf Canada Resources Ltd. S.M.

Chen, ” SPE, Gulf Canada Resources Ltd.

Summary. mis paperpresents a new. water intlux model that differs from traditiO~

approaches fi fiat it ~clud~s the eff@ Of vefical flow at the reservoir/aquifer interface. The results we presented in the form of dimensionless groups, which makes the mudel readily applicable to a wide range of systems. The paper concludes with a sample calculation showing how the predictions of this new model can be significantly

dfiferent from those of conventional

radLal flow models.

[ntroductkm Petroleum reservoirs are often in contact with an aquifer that provides pressure supportthrough water intlux. Thus; the predktion of reservoir behavior USUSUYrequires sn accurate model of the aquifer. Resemoir/aquifw systems are commonly cls.vsitied on the basis of flow geumetry ss either edgewater or bottomwater drive. For edgewater drive, the most rigorous aquifer infhx model developed to date is that of van Everdmgen and Hurst, 1 which is 8ssentitiy a solution to the radial diffusivity equation. Although the assumptions made in deriving thk model are not strictly vtild for bottomwater drive systems, water influx in this case can sometimes be closely approximated by rndial flow. Therefore, bwause the rsmdts sre often quite adequate nnd for lack of a LWer model, it has been common practice to apply the vao Everdingen and Hurst method to both bottomwater and edgewater systems,

Coatsz has developeda model that takes into account vertical flow effects and has shown these effects to be fairly’ significant. The model as presented, however, has two principal liiitationx (1) the solution given applies to tic “terminal-rate” case, which ‘J30ws the usir to calculate pressure from a known influx rate rather than tbe reverse, and (2) the solmion is applicable only to intiite aquifers. This paper is essentially an extension of the work of Coats. The Wtomwater model presented here is a sohrtion to the “terminnlpressure” case md applies to bdh finite and infbite aquifers. The results ze presepted in the form of dimensionless groups that are tabulated in a manner similar to that of van Everdingen and Hurst. The paper concludes with a sample c21cu3ati0n that illustmtes the use of this new model and shows that predictions of the bottomwater model can be signiffcamly different from those of the radial flow model. The calculation of water influx is important in a number of reservoir engineering applications, such as material-bslsnce studies and the design of pressure mairrtemrice schemes. The fact that a large percentage of reservoirs have adjoining aquifers means that development of sn accurate aquifer model is criticsl to proper understanding of rer.ervoiz behavior. It is not surprising, therefore, that considerable research effort has been devoted to this subject. During the past 50 years, a kmge number of models describing water encroachment have emerged, and the majority of these have been subject to a great deal of modification. In these models, the reservoir is typicslly visualized as a right cylinder surrounded by a series of concentric cylinders representing the aquifer. Most of the models, such as tie steady-state model of Schilthuis3 or the tlniteaquifer, pseudosteady-state model of Fetkovich, 4 are applicable to only a limited range of flow conditions or reservoir/aquifer geometries. The model @at possesses the most general applicability is the unsteady-state model ,of van Everdingen and Hurst. fn fact, this model is a solution to the radisl diffusivity equation and m such is valid for all flow regimes, provided, of course, that the flow geometry is actwdly iadinl. .NOWwithZ“,ltina01]Co, Cmdght 1s88Societyof Petroleum Engineers SPE Reservok engineering, May 19S8 —.

Tbe radial flow geometry assumed by van Everdingen and Hurst is best understood by means of cm illustration. Fig. 1 shows, in cross section, a resewoir subject to edgewater drive and the idenl.jmd mdisl flow model that represents this reservoirktquifer system. The flow vectors in this case are horizontal, and water encroachment occurs acrok+ a cylindrical plsne encircling the rewrVOU. This situation can be compared to tie bottomwater drive system shown in Fig. 2. fn MS c%?, the flow vectors have a significant vertical component, and water encroachment occurs across a horizontal circular plane represendng tie oillwater contact. Thus, a rigorous bottomwater influx model must take into accotit veftical flow, and as will be shown below, the effsct of vertical flow becomes increasinsIy morepronoumed us the ratio of aquifer thickness, h, to reservoir radius, Tfl, becomes larger. The discussion below provides a detailed trsatment of the bottomwater flow model depi&d in Fig. 2. The diffusivity equation governing flow for this system is reducsd to dimensionless form by introducing dimensionless variables. The resultant eqiation is then solved witfua numerical simulator, and as in tie work of van E.erdingen and Hurst, tbe resuks are presented in the form of tables of dimensionless intlux, !7., vs. dimensionless time, tfl fncluded in the discussion is an exsmple calculation for a hypothetical h@tomwater drive reservoir. This example illustrates the use of the new model snd, by calcoladng intlux with both the radial snd bottomwater models, clsarly shows that ignotig vertical flow can result in very significant

error.

DiscussIon Basic Equations. The pam’nl-differenti$ equation governing flow of a slightly compressible fluid in a system such as that shown in Fig. 1 is the well-known radisl +ffusivity equation

azp I ap —+;%= 872

w ~;.

ap

..... .... ..... ..... .... ...

(1)

For the bottomwater flow model depicted in Fig. 2, an additions term is added to this equation5:

a~p —+; a#

azp ~4c ap 1 ap %+ Fk—=7G, azz

., ...,..............(2)

where Fk is the ratio of vertic81 to horizontal permeabflty. There are m iminite number of solutions to Eq. 2, representing 831possible reservoidaquifer cunt&Iratiom. It is possible, however, to derive a general solution that is applicable to a variety of systems by introducing the following @“ensionless varisbles r~=rlr~,

. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . (3)

zD=z/(rRFk M), . . . . . . . . . . . . . ... . . . . . . . . . . . . ...’..

. ..(4) 369

--+.

&,, ! I

[ h

I I I p--%!“ ! , i

4 e

~

~E=Rvo,R

-[

I ! ~+ ,+-

~ou,FEe

I I ! ~,

t ,

Fig. I—Idealized

WELLS ● PRODUCING Fig. 3—Numerical

radial flow model.

+

model of bottomwater

flow system.

III the derivation of an equation for cumulative water influx, it

,.4 1

I !

is convenient

to define a dmensionlcss

pressure drop as

! r.&~ fi ApD =Ap— 0.282ew#

. . ....

Eqs. 5 and 7 can be solved fort @crR2 z=tD— 6.33k

h I —

,, -.+

....

.. . ....

(7)

....

and ew, respectively,

to yield

.... ...... .... .... ..... .... ...

(8)

and r ~w.

1 I z

----------

+----

., ..,,,.,

,, ..,,..,

The equation for cumulative water inffux, difference form is given as

z/W~H\~m 1 ~“ I -r

t h

AP r#Fk % —, ApD 0.282p

We= EewAt

. . . . . . . . . ...(9)

W,, written iII fi~ie-

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..(10)

Substituting Eqs. S and 9 into this expression, Fig, 2—ldeaIized

bottomwater

we obtain

flow model. We=0.560#CrR3F#

ApE~.

. . . . . . . . . . . . . . . . . . ..(11)

and rD=K6.33MW$CrE2).

. . . . . . . . . . . . . . . . . . . . . . . . . . . ...(5)

Substitution of these new variables into Eq. 2 gives tie dimensionless form of the diffusivity equatio~

Finally, to convert this expression to a form comparable with that of van Everdmgen and Hurst, we define a water influx constant, B, and a dimemsionfess cumulative influx, W&, as B=l.119+hcrR2

azp —+~~+—= rD aTD a&

a~p a2D~

~., atD

. . ... . . . . . . . . . . . . . . ...(6)

and

@k’h

waD = —zG,

370 --

‘D

. . . . . . . . . . . . . . . . . . . . . . . . . .. .. (13)

2h

‘fkusfar, the discussion ins traced Coats’ argument quite closely, but at this point a different course wilf be followed. Rather than attempdng to formuf ate an andydc soluticm to l?q. 6 for the terminalpressure case, we will inste@ derive an eqmdon for itiux directly and use a numerical simulator to solve this equation.

. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . (12)

where his

the aquifer thickness.

This reduces Eq: ll”to

We=BAPW
--

-.

which is analogous to tie formulation of v60 Everdiogen and Hurst except that the vzlues of W.D vs. tD for the b0tt0mw6ter system will, of course, be different tlom those of the radial system. The next section will outline the method used to solve Eq. 14 for W,D witi a numerical model. At this point, however, it.shcmld be noted that W& is a function of not only t~ but also the reservoirlaquifer geometry. It is therefore necessary to introduce the following dimeosimdess constmts that describe the system geometry: rD’=rekQ

. . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . .

. . . .

(15)

and zD’=h/(rxFk%),

. . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . ..(16)

where. re is ..be aquifer radius. For fix6d values of these two parameters, W,D is a function of only zD. Solution of Equations. Before the numerical model used in this study is described, it inay be helptid to review the flow conditions @Q2ti iOthe COnS~t:tefi!Id-ra~e, W& The resemoir and aquifer are assumed to be uuhally at eqwhbnum at pressure pi. At time t =0, the .resewoir pressure is lowered by a increment, Ap, and is subsequently maintained at thk level. In response to this pressure drop, water encroaches into the resm’oir at a rate such that the cumulative influx, We, is governed by Eq. 14. For ftite aquifers, We will eventually reach a maximum value as the pressure io the aquifer approaches that of the reservoir. If a simulation model could be set up to duplicate the conditions described above, it would then be possible to evaluate Eq. 14 numerically. That is precisely the approach @ken in this paper. The numkriczl simulation model describing the idealized bottomwater flow model of Fig. 2 is shown in Fig. 3. The dimensions and properties of this model can be selected to give any desired values of rD’ and ZD’. The production routine is setup such that, at each timestep, production from reservoir gridblock is adjusted to keep the pressure at the oil/water contact constant. Determining dimensions influi is then simply a matter of ruoning the simulator and generating tables of We vs. t. These values are substituted into Eqs. 14 and 15, respectively, to produce tables of W,D vs. tD. out for various combinations Tbe above procedure was curd ofrD’andzD’, aodtheresults aregiven in Tables 1 tbrough5. Plots of W& vs. tD for selected values of TD’ and ZD’ are shown in Figs. 4 through 6. Superimposed on these plots for comparison are the dimensionless inflox curves derived from van Everdmgen and Hurst’s radial flow model. It is possible to test the validity of these results for iti]nite aquifers by comparing them directly with Coats’ analytic results. With so ffJcientlysmalld meincrements,Eq. 13c6.nbe used to convert Coats’ values of ApD to Wd values. These match quite weU the simulatorderived yalues of Table 1. Sample Calculation. To apply the results of tie c0nstan-termin2lpressure case to the general case in which pressore at the oillwater contact varies witbtime, Eq, 14ismodified byuseoftbeprincipal of superposition:

~ei=B

i ~

(Apj)(W@i:j+, ): . . . . . . . . . . . . . . . . . . . . ..(17)

j=l

The use of this equation is described in several reservoir engineering texti.6,7 Consider the hypothetical reservoirb+fer system whose properties are listed in Table 6. ,~s system was simulated “numerically, and the following data were generated at selected time intemls:

PR (a=w r=e~oir pressure), P./w (average pressure at oil/w2~ and We (cumulative water influx). ‘fbe v6.lWS Ofp./w contact), werethen usedtOcalcolate cumulative influx frOm Eq. 17wi63b0th the radial and bottomwater models. As shown in T6ble 6, after a number of timesteps, tieradial model valuezof W,D give acalculated influx, We, that ismuchlager than tbe simulator values. SPE Reservoir Engineering, May 1988

TABLE I—DIMENSIONLESS INFLUX, FOR INFINITE AQUIFER

Wm,

0.1

0.3

ZD ‘ 0.5

0.1 0.2 0.3 0.4 0.5

0.05 0.700 0.793 0.936 1.051 1.158

0,677 0.786 0.926 1.041 1.155

0.508 0.696 0.634 0.952 1.059

0.249 0.547 0.692 0.812 0.918

_ 0.7 0.251 0.416 0,548 0.662 0.784

— 0.9 0.195 0.328 0.440 0.540 0.631

— 1.0 .0.176 0.295 0.396 0.466 0.569

0.6 0.7 0.8 0.9 1

1.270 1.324 1.503 1.621 1.743

1.266 1.360 1.499 1.612 1,726

1.167 1.270 1.373 1.477 1.581

1,021 1.116 1,205 1.286 1.347

0.862 0.953 1.039 1.117 1.181

0.721 0.606 0:686 0.959 1.020

0.651 0.729 0.603 0.872 0,932

2 3 4 5 6

2,402 3.031 3.629 4.217 4.784

2.393 3.016 3.615 4.201 4.766

2.268 2.685 3.477 4.04S 4.601

2.034 2.650 3.223 3.766 4.266

1.827 2.408 2.948 3.462 3.956

1.622 2.184 2.669 3.150 3.614

1.509 2.026 2.510 2.971 3.416

7 6 9 10 11

5.323 5.829 6.306 6.637 7,283

5.303 5.608 8.263 6.616 7.242

5.126 5.825 6.094 6.583 7,040

4.792 5.263 5,762 6.214 6.664

4.424 4.900 5.355 5.792 6.217

4,063 4.501 4.929 5.244 5.745

3.647 4.268 4.660 5.060 5.488

12. 13 14 15 16

7.742 8.196 8.S48 9.094 9.524

7.718 8.172 8.623 9.066 9.507

7.495 7.943 8.385 6.821 9.253

7.104 7.539 7.967 8.389 8.606

6.636 7.052 7.461 7.664 6.262

6.143 6.536 6.923 7.305 7.662

5.652 6.231 6.604 6.972 7.338

17 16 19 ?0 n

9.969 10.399 10.823 11.241 11.654

9.942 10.371 10.794 11.211 11.633

9.679 10,100 10,516 10.929 11.339

9.218. 6.656 9.626 9.046 10.029 9.432 10.430 9.815 10.626 10.194

6.056 6.426 8.793 9.156 9.516

7.699 8.057 8.411 8.763 9.111

?2 23 ?4 ?5 26

12.075 12.436 12.693 13.297 13.696

12.045 12.454 12,861 13.264 13.665

11.744 12.147 12.546 12.942 13.336

11,219 11.609 11,996 12.360 12.761

10.571 10,944 11.315 11.663 12.046

9.874 10.229 10.581 10.931 11.279

9,457 9,801 10.142 10.481 10.817

?7 28 29 30 31

14,097 14.493 14,866 15.277 15.666

14.062 14.458 14.850 15.241 15.628

13.726 14.115 14,501 14,864 15,266

13,140 12.411 13.517 12.772 ‘13.891 13.131 14.263 13.486 14.63413.648

11.625 lt.968 12.310 12.650 12.990

11,152 11.485 11.616 12.145 12.473

32 33 34 25 36

16.053 16.437 16.819 17.200 17.579

16.015 16.396 16.780 17.160 17.588

15.545 16.023 16.396 16.772 17.143

15.002 15.366 15.732 16.095 16.456

14.198 14.548 14.697 15.245 15.592

13.324 13.659 13.992 14.324 14,654

12.798 13.122 13.446 13.767 14.068

37 36 39 40 41

17.956 .16.331 .18.704 19.066 19.450

17.915 17.513 16,28917.662 18.662 18.249 19.045 16.620 19.407 16.962

16.615 17.173 17,529 17.866 18.240

15.937 14.983 14.406 16,280 “15.311 14.724 16.622 15.637 15.040 16.984 15.988 15.356 17.305 16.286 15.671

t.

42 4s 44 4S 46

19.621 19.777 20.18820.144 20.555 20.510 20.920 20.674 21.263 21,237

19.844 19.706 20.065 20.424 20.781.

16.592 17.644 18.942 17.961 19.29318,317 19.541 18,651 19.988 18,985

16.611 16.933 17.253 17.573 17.691

15.935 16.297 16.608 16.918 17.227

47 48 49 50 51

21.645 22.006 22.365 22.722 23.081

21.596 21.958 22.317 22.674 23.032

21.137 21.491 21.6cM 22.196 22.547

20,333 20,678 21.021 21.363 21.704

19.317 18.208 19.648 18.524 19.976 18.840 20.307 19.154 20.636 19.467

17.535 17.841 16.147 16.452 18.757

52 53 54 55 56

23,436 23.791 24,145 24.496 24.649

23.367 23.741 24.094 24.446 24.797

22.897 23,245 23,593 23,939 24,265

22.044 22.363 22.721 23.056 23.393

20.962 21.268 21.613 21.937 22.250

19.779 20.091 20.401 20,711 21.020

19.060 19.362 19,664 19.965 20.265

57 58 59 60 61

25.2oo 25.549 ,25.696 26.246 26.692

25.147 25.4S6 25.844 26.191 26.537

24.629 24.973 25.315 25.657 25.996

23.728 24.o62 24.395 24.728 25.059

22.5&3 22.904 23.225 23.545 23.864

21.328 21.636 21,942 22.246 22,553

20.564 20.662 21.160 21.457 21.754

371

TABLE l—DIMENSIONLESS FOR INFINITE AQUIFER ZD

IL___ 0.05 62. 26.938 63 27.283 64 27.627 65 27.970 66 28.312

0.1 26.883 27,227 27,570 27.913 28.256

TABLE I—DIMENSIONLESS FOR INFINITE AQUIFER

INFLUX, Woo, (continued)

‘

0.3

Zo

0.5 —___ 26.337 25,390 26.67625.719 27.015 26,048 27.352 26.376 27,688 26.704

0.7

0,9 —— 24.182 22.857 24.499 23.161 24.816 23.454 25.13223.766 25.447 24.068

i .0 22.04$ 22.34 22.63! 22.932 23.22!

&____ 0.05 260 86.952 265 88.351 270 89.746

INFLUX, Wgo, (continued)

‘

0.1

0.3

0.5

1.0

85.432 66,811 66.186

83.023 84,369 85.713

0.7 —_. 79.876 81,162 82,464

0.9

86.814 68.211 89.604

76.356 77.614 76.868

74.17 75.39 76.62

275 91138 90994 280 92526 92361 1.:::::::

89556 90926

87053 88391

63782 35073

80119 81367

7784 7906

92.292 93.654 95.014 96.370 97.724

89.725 91.056 92.385 93.710 95.033

86.371 87.660 88.948 90.232 91.514

62.612 83.655 85.095 86.323 87.568

80.27 81.48 62.70 83.90 85,11

67 68 69 70 71

28.653 2&596 28,994 28.936 29,334.29.275 29.673 29.614 30.011 29.951

28,024 28,359 26.693 29.026 29.359

27.030 27.356 27.681 28.006 28.329

25,762 26.075 26.389 26,701 27,013

24.369 24.669 24.969 25.268 25.566

23.51[ 23.81 [ 24.101 24.391 24.681

235 290 295 300 305

72 73 74 75 76

30.349 30.836 31,022 31.357 31.692

30.268 30.625 30.960 31,295 31,629

29.691 30.022 30,353 30.682 31.012

28.652 28,974 29,296 29.617 29.937

27.324 27.834 27.944 28.254 28,562

25,864 26.161 26.456 26.754 27.049

24.971 25.26C 25.54? 25,636 28.124

310100.79 315102.16 320103.52 325 104.8S 330106.24

100.54 102,00 103.36 104.72 106.08

99.07 100.42 101.77 103.11 104.45

96.35 97,67 98,99 100.30 101.61

92,79 94.07 95.34 96.62 97.69

88.80 90.03 91.26 92.49 93.71

86,32 87.52 66.72 89.92 91.11

77 78 79 80 ’61

32.026 32.359 32.692 33.024 33.355

31.963 32.296 32.626 32,959 33,290

31.340 31.668 31.995 32.322 32.647

30.257 30.576 30.895 31,212 31.580

26.870 29.178 29.485 29.791 30,097

27344 27.829 27.933 28.226 28.519

26.41C 26.697 26.98S 27,28E 27.552

335107.60 340108.95 345110.30 350111.65 355113.00

107.42 108.79 110.13 111.48 112.62

105,79 107,12 108,45 109.78 111.11

102.91 104,22 105,52 106,62 108.12

99.i5 100.42 101.68 102.94 104..20

94,93 96,15 97.37 96.58 99.80

92,30 93,49 94.68 95.67 97.06

82 63 64 85 86

38.686 24.016 24.345 34,674 35.003

33,621 33.950 34.279 34.608 34,933

32,973 33,297 33.622 33,945 34.268

31.646 32.183 32.478 32.793 33.107

30,402 30,707 31.011 31.315 31,618

28.812 29.104 29.395 29.686 29.976

27.337 28,121 28.404 28.687 26.970

360114.34 365115.66 370117.02 375116.36 380119.69

114.17 115,51 116,84 118.18. 119.51

112.42 113.76 115.06 116,40 117.71

109.41 110.71 112.00 113.29 114.57

105.45 “106,71 107,96 109.21 i 10.46

101.01 102,22 103,42 104.82 105.83

98.24 99.42 100.60 ,101.78 102.95

87 88 89 90 91

35,330 35,657 35,964 36.310 36.636

35.268 35.589 35.915 36.241 36.566

34.590 34.912 35.233 35.554 35,874

33.421 33.735 24.043 34.360 24.672

31,921 32,223 32,525 32.826 33.127

30,266 30,556 30,845 31.134 31,422

29.252 29.584 29.815 30.096 30.376

385121.02 390122.35 395123.68 400125,00 405126,33

120.84 122.17 123,49 124.82 126.14

119.02 120.34 121.65 122.94 124.26

115.86 117.14 118.42 119.70 120.97

111.70 112.95 114.19 115.42 116.67

107.04 108.24 109.48 110.63 1 i 1.32

104.i3 105,30 106,47 107,64 108.60

92 93 94 95 96

36,960 37.265 37,609 37.932 36,255

36.890 37.214 37.538 37,861 38.163

36.194 36.513 36.832 37.150 37.487

34.983 35.294 35,604 35.914 36.223

33.427 33.727 24.026 34.325 34.623

31.710 31.997 32,284 32,570 32.857

30,656 30,935 31,215 31.493 31,772

97 98 99 100 105

38.577 38.699 39.220 39.541 41.136

38.505 38.826 39.147 39.467 41.062

37.735 38.101 38,417 38.733 40.305

36.532 36.841 37.149 37,456 38,9S7

24.921 35.219 35.516 35,813 37,290

33.142 33.427 33.712 33.997 35.414

110 115 120 125 130

42.724 44.299 45.664 47.420 43.966

42,845 44.216 45,781 47.834 48,879

41.865 48.415 44,956 48,467 48,009

40.508 42.018 43.520 45.012 46.497

36.756 40,216 41.666 43.107 44.541

135 140 145 150 155

50.504 52,033 53,555 55.070 56.577

50.414 49.523 51.942 51.029 53.482 52.526 54.974. 54,019 56.479.55,503

47.973 49.441 50.903 52.357 53.605

160 165 170 175 180

56,077 59,570 61.056 62.539 64.014

57.977 59,469 60,954 62,433 63.906

56.981 56.452 59,916 61.375 62.629

!85 :90 ’95 !00 !05

65.484 66.948 68.406 69,660 71,309

65,374 66,836 66.293 69.744 71.191

!1O !15 !20 !25 !30

72.752 74.191 75.626 77,056 73.462

!35 !40 !45 !50 !55

79.803 ,61.321 82.734 64.144 85.550

372

93.911 95.293 96.672 98.048 99,420

83.764 95.144 96.521 97,895 99,266

1::::::””: 41012765 41512897 420130.28 425131.60 430132.91

12746 12678 130.09 131.40 132.72

12556 12686 128.16 129.46 130.75

12225 12352 124.79 126.06 127.33

11790 11914 1:20.37 121.60 122.63

11302 11421 115.40 116.59 117.77

10997 11113 112.30 113.46 114.62

32.050 32.327 32.605 32,861 34.260

435134.22 440135.53. 445136.34 450136.15 455139.45

124.03 135.33 136.64 137.94 139.25

132,05 133..34 134.63 135,92 137,20

128.59 129.86 131.12 132.38 133,84

124.06 125.29 126,51 127.73 128.96

118.86 120.14 121.32 122,50 123,68

115.77 118.93 118,08 119,24 120,39

36.821 38.221 39.612 40.895 42.372

35.630 36.993 36.347 39.694 41.035

460140.75 465142.05 470143.35 475144.65 480145,94

140.55 141.85 148.14 144.44 145,73

138.49 139.77 141.05 142.33 143.61

134.90 136,15 137.40 138.66 139.91

130.13 131.39 132.61 133,62 135.04

:124.86 126.04 127.21 126.38 129.55

121.54 122.69 123.64 124.96 126.13

45.967 47.366 46,796 50.204 51.603

48.741 45.104 48,460 47.610 49.155

42.368 43.696 45.017 46.333 47.642

485147.24 490148.53 495149.82 500151.11 510153.66

147.02 143.31 149.60 150.89 153.46

144,89 146,16 147.43 148.71 151.24

141.15 142.40 143.85 144.89 147.38

136.25 137,46 138.67 139.86 142.29

130,72 131,89 133.06 134.23 136.56

127.27 126.41 129.56 130.70 132.97

55,246 56,661 56.110 59.534 60.952

52.996 54.384 55.766 57.143 58.51.4

50.494 51,827 53,156 54.479 55.798

48.947 50.247 51.542 52.632 54.116

520156.25 530156.81 540161.36 550163.91 560156.45

156.02 i56.56 161.13 163.68 166,22

153.78 156.30 158.82 161.34 163.85

149.85 152.33 154.79 157.25 159.71

144.70 147.10 149.49 151:88 154,27

136.68 141.20 143.51 145.62 148.12

135.24 137.51 139,77 142,03 144,26

84.276 65.718 67,156 66,568 70.015

62,365 63.773 65.175 66.573 67.967

59,881 61,242 62.600 63.952 65.301

57.112.55.399 58.422 56,876 59.727 57.949 61 .02S 59.217 62.326 60.482

570166.99 580171.52 590174.05 600176.57 610179.09

168.75 171.28 173.60 176.32 176.83

166.35 168.85 171.34 173.83 176.32

162.16.156.85 164.61 159.02 167,05 161.39 169.46 163.76 171.92 166.12

72.633 74.070 75.503 76.931 76,355

71.437 72.855 74,269 75.678 77.083

69.355 66.645 70.740.67.965 72.120 69.321 73,496 70.653 74.868 71.981

63;619 64,903 66.194 67.476 68,755

61.744 83.001 &L255 65.506 66.753

620161.60 630164.10 640166.60 650189.10 660191.59

181:34 i 83.85 186.35 188.84 191.33

176.80 181,27 183.74 166.20 168.66

174.24 176.76 179.18 181,60 164.00

166.48 161.85 170.83.164.13 173.18 166.40 175.52 168.66 177.66 170.92

157.71 159,93 162.15 164.37. 166.58

79.774 61.190 82.602 84.010 65.414

78.434 79.681 81.275 82.664 84.050

76.236 77.601 73.962 60.319 61.672

70.030 71.302 72.570 73.736 75.098

67.997 69.238 70,476 71.711 72.942

670194.08 660196.57 690199.04 700201.52 710203.99

193.61 196,29 198.77 201.24 203.71

191.12 193.57 196,02 196,46 200,90

186.41 168.61 191.21 193.60 195.99

160.20 182.53 184,66 187.19 189.51

168.79 170.99 173.20 175.39 177.59

73.306 74.627 75,945 77,259 78,570

150.42 146,53 152.72 148.77 155.01 .151 .01 157.29 153.25 159.56 155.46

173.16 175.44 177,69 179,94 182,18

3PE Reservoix Engineering, MaY 1988

TABLE I—DIMENSIONLESS FOR INFINITE AQUIFER

TABLE I—DIMENSIONLESS FOR INFINITE AQUIFER

INFLUX, W’eD, (continued)

INFLUX, W,., (continued)

fD

0.05

0,1

0.3

Zn ‘ 0.5

_ 0.7

—0.9

1,420 1,430 1,440 1,450 1,460

371.64 373,82 376.10 378..36 380.65

371.08 373.35 375.63 377.90 380.17

366.40 368.65 370.90 373.15 375.39

358.19 360,40 362.61 364.81 367.02

347.32 349.47 351.62 353.76 355.91

334.94 327.15 337.03 329.19 339.11 331.23 241.19,333.27 343.27 335.31

362.44377.64 384.71 379.88 386.96 382.13 389,25 384.37 394.90 389.96

369.22 371,42 373.62 375.82 381.31

358.06 360,20 362.34 364.46 369.82

245.35 247.42 249.50 351.58 356.78

337.35 339.38 341.42 343.45 348.52

— 1.0

770 780 790 800 810

218.73 221.17 223.61 226.05 228.48

218.42 220.87 223,31 225.74 226.17

215,45 217.86 220.27 222.68 225.08

210,24 212.60 214.96 217.32 219.67

203.36 205.66 207.95 210.24 212.53

195,57 197,80 200.01 202.23 204.44

190,69 192.67 195.04 197.20 199.37

1,470 1,480 1,490 1,500 1,525

382.92 385.19 387.46 389.73 395.39

820 830 640 850 860

230.91 230,60 233.33 233.02 235.76 235.44 238.18 237.86 240.59240.27

227.48 229.87 232.26 234.65 237.04

222.02 224.36 226.71 229.05 231.38

214.81 217,09 219.37 221$4 223.92

206.65 208.86 211.06 213,26 215.46

201.53 203.69 205.85 208.00 210.15

1,550 1,575 1,600 1,625 1,650

401.04400.55 406.66 406.18 412.32 411.81 417.94417.42 423.55 423.03

395.55 401.12 406.69 412.24 417.79

386.78 375.16 392,25 380.49 397.71 385.80 403.16391.11 408.60 396.41

361.93 367.09 372.24 377.39 382.53

353.59 356.65 363.70 368.74 373.77

670 680 690 900 910

243.00 245.41 247,82 250,22 252.62

242.68 239.42 245.08,241.80 247.49.244.17 249,89 246.55 252.28 248.92

233.72 226.19 236.05.228.45 238.37 230.72 240.70 232.96 243,02 235.23

217.65 219.85 222.04 224.22 226.41

212.30 214.44 216.59 218.73 220.87

1,675 1,700 1,725 1,750 1,775

429.15 434.75 440,33 445,91 451.46

428.63 434.22 439.79 445.37 450.93

423.33 428.35 434.37 429.89 445,39

414.04 419.46 424.87 430.28 436,66

401.70 406.99 412.26 417.53 422.79

387.66 392.76 387.89 403.00 408,10

376.60 383.82 386.63 393.64 398.84

920 930 940 950 960

255.01 257.41 259.80 262.19 264.57

254.68 257.07 259.46 261.84 264.22

251.26 245.24 253.65.247,66 256.01 249,97 258.36 252.28 260.72 254,59

237.49 239.74 241,99 244,24 246,48

228.59 230.77 232.95 235.12 237.29

223.00 225.14 227.27 229.39 231.52

1,860 1,825 1,850 1,875 1,900

457,04, 462,59 468.13 473.67 479.19

456.48 462.03 467.56 473.09 476.61

450.86 456.37 461,85 467.32 472.76

441.07 446.46 451.83 457.20 462,56

428.04 433.29 438.53 443.76 448.98

413.20 418,28 423.36 428,43 433.50

403.83 408.82 413.60 418.77 423.73

970 980 990 1,000 1,010

266.95 269.33 271.71 274.08 276,35

266.60 268.98 271.35 273.72 275.99

263.07 265.42 267.77 270.11 272.35

256,89 259,19 261,49 263.79 285.99

248,72 250.96 253,20 255.44 257,58

239.46 241.63 243.80 245.96 246.04

233.65” 235.77 237.69 240.00 242.04

1,925 1,950 1,975 2,000 2,025

484.71 490.22 495.73 601.22 506.71

464.13476.24467,92 439.63 463.69 473.26 495.13 469.13 478.60 500.62 494.56 483.93 506.11 499.99 489.26

454.20 459.41 464.61 469.61 475.00

436.56 443.61 448.66 453.70 458.73

428.69 433.64 438.59 443.53 448.47

1,020 278,72 1,030 261,08 t ,040 283,44 1,050 285,81 1,080 288,16

278.35 280.72 283.08 285.43 287.79

274.69 268,29 277.03 270.57 279.36.272.86 281.69 275.15 284.02 277.43

259.81 262.04 264,26 266.49 268.71

250.19 252.35 254.50 256.66 258.81

244.15 246.26 248.37 250.43 252.58

2,050 2,075 2,100 2,125 2,150

512.20 517.67 523.14 528.60 524.05

511.58 517.05 522.52 527.97 533.42

505.41 510.82 516.22 521.62 527.02

494.58 489..29 505.19 510.49 515.78

480.18 485.36 490.53 495.69 500.85

463.76 466.76 473.80 476,81 463.81

453.40 458.32 463.24 468.15 473.06

1,070 1,080 1,090 1,100 1,110

290,52 292.87 2S5.22 297.57 299.91

290.14 292.49 294.84 297.18 299.53

286.35 288.67 290.99 293.31 295I33

279,71 281.99 284.26 286.54 288,81

270,92 273.14 275.35 277.57 279.76

260.95 263.10 265.24 267.38 269.52

254.69 256.79 258.89 260.98 263.08

2,175 539.50 2,2o0 544.94 2,225.550.38 2,250 555.81 2,275 561.23

538.86 544.30 549.73 555.15 560.56

532.40 537.78 543.15 548.52 553.88

521.07 526.35 531.62 536.88 542.15

506.01 511.15 516.29 521.42 526.56

486.81 493.81 498.79 503.78 506.75

477.96 482.85 467.74 492.63 497.51

1,120 1,130 1,140 i ,150 1,160

302.28 301.87 304.60 304.20 306.93 306.54 309.27.308.87 311.60 311.20

297,94 300.25 302,56 304,87 307,16

291.07 293.34 295.61 297.87 300.13

281.98 284.19 286.39 238.59 290.79

271.86 273.80 275.93 278.06 260.18

265.17 267.26 269.35 271.44 273,52

2,300 2,325 2,350 2,375 2,400

566.64565.97 572.05 571.38 577.46 576.78 582.85 582.17 588.24 567.55

559.23 547.41 564.56.552.66 569.92 557.90 575.26 583.14 580.59 568.37

531.68513.72 536.80 518.69 541.91 523.65 547.02 528.61 552.12 533.56

502.36 507.25 512.12 516.98 521.63

1,170 1,160 1,190 1,20.0 1,210

313.94 316.26 318.59 320.92 323.24

313.53 315.86 316.18 320.51 322.63

308,46 311.78 314.08 316,38 318.67

302.38 304.64 306,89 309,15 311,38

292.99 295.19 297.38 299.57 301.76

282.32 284.44 286.57 288.69 290.81

275.61 277.69 279.77 261.85 283.92

2,425 2,450 2,475 2,500 2,550

593.63 599.01 604.38 609.75 620.47

585.91 591.23 596.55 601 .S5 612.45

1,220 1,230 1,240 1,250 1,26o

325.56 327.66 330.19 332.51 324.82

325.14 327.46 329.77 332.08 334.39

320,96 323.25 325.54 327.83 330.11

313,64 315,69 318,13 320.37 322,61

303.95 306.13 306.32 310.50 312.68

292.93 295.05 297.16 299.27 301.38

266.00” 268.07 290.14 292.21 294.28

2,600 2,650 2,700 2,750 2,600

1,270 1,260 1,290 1,300 1,310

337.13 339.44 341’:74 344.05 346.36

336.70.332.39 339.01 334.67 341.31 336.95 343,61 339.23 345,91 341.50

324.85 327.06 329.32 331.55 333.76

314.85 317.03 319.21 321.38 323.55

303.49 305.60 307.71 308.81 311.92

296.35 286.41 300.47 302.54 304.60

1,320 1,330 1,340 1,350 1,360

348.65 350.95 353.24 355.54 357.63

346,21 350,50 352,80 355,09 357.38

343.77 346.04 348.31 350.58 352.64

336.01 338.23 340.46 342.66 344.90

325.72 327.89 330.05 332.21 334.38

314,02 316,12 318.22 320.31 322.41

1,370 1,380 1,390 1,400 1,410

360.12 362.41 384.69 366.98 369.26

359.67 361.95 364,24 366.52 368,80

355.11 357,37 359.63 361.88 364.14

347.12 349.34 351.56 353.77 355.96

336.54 336.70 240.85 243.01 345,16

324.50 326.59 326.66 330.77 332,86

8PE Reservoir Engineering, MaY 1988

592.93 596.31 603.68 609.04 619.75

573.60 557.22 578.62 562.31 584.o4 567.39 589.25 572.47 599.65,582.62

538.50 543.45 546.38 ‘553.31 563.16

526.66 531.53 536.37 541.20 550.66

631.17 630.43 623.03 641.64 641.10 633.59 652.50 651.74644.12 663.13.662.37 654.64 873.75 672.97 665.14

610.04 620.40 630.75 641.07 651.36

592.75 602.66 612.95 623.02 633.07

572.99 562.80 592.60 602.37 612.13

560.50 570.13 579.73 589.32 598.90

2,650 2,800 2,850 3,000 3,050

6S4.34 694.92 705.48 716.02 726.54

683.56 694.12 704.67 715.20 725.71

675.61 686.07 696.51 706.94 717.34

661.67 671.94 682.19 692.43 702.65

643.11 653,12 663.13 673.11 663.08

621.68 631.60 641.32 651.01 660.69

608.45 617.99 627.52 637.03 646.53

306.65 308.71 310.77 312,62 314,87

3,100 3,150 3,200 3,250 3,300

737.04 747.53 756.00 768.45 778.69

736.20 746.68 757.14 767.58 778.01

727.73 738.10 748.45 758.79 769,11

712.85 723.04 733.21 743.36 753.50

693.03 702,97 71,2.89 722,80 732,69

670.36 680.01 689.64 699.27 708.87

656.01 665.48 674.93 684.37 693.80

316,92 318.97 321.02 323,06 325,11

3,350 3,400 3,450 3,500 3,550

789.31 786.42 799.71 798.81 810.10 809,19 820.48.81 9S5 830.63 629,90

779.42 769.71 799.99 810,25 820,49

763.62 773.73 783.62 793.90 803.97

742.57 752,42 762,28 772.12 761.94

718.47 728,05 737.62 747.17 756.72

703.21 712.62 722.00 731.38 740.74

; I

373

TABLE I—DIMENSIONLESS

INFLUX,

WO,

FOR INFINITE-AQUIFER

(continued)

ZD’

374.

830.73 640.94 651.15 881.24 671.51

0.5 _—— 614.02 824.06 834.08 644.09 854.09

791.75 801.55 511.33 821.10 830.86

.786.24 775.76 765.27 794,76 804.24

— 1.0 750.08 759,.43 768.76 778.08 767.38

891.70 901.85 912.19 922.41 932.62

881.68 891.83 901.96 912.09 922.20

864.06 674.05 884.01 693.96 903.89

840.81 850.34 860.06 869.77 879.47

613.71 623.17 632,62 642.06 851,46

796,68 605.96 815.23 824,49 633.74.

948,66 954,06 964.25 974.42 984.58

942,82 953.01 963.19 973,35 983,50

932.30 942.39 952.47 962.53 972.58

913.62 923.73 933.83 948.52 953.40

889.16 898.84 908.50 916.16 927.80

660.90 870.30 67S.89 689.08 698.45

842.99 852.22 861.44 670,65 879.85

4,350 4,400 4,450 4,500 4,550

994.73 1,004.9 1,015.0 1,025.1 1,035.2

993,64 1,003,8 1,013,9 1,024,0 1,084.1

982.62 992.7 1,002.7 1,012.7 1,022.7

963.27 973.1 963.0 992.8 1,002.6

937.43 947.1 956.7 966.3 975.9

907.81 917.2 926.5 935.9 945.2

669,04 698.2 907.4 916.6 925.7

4,600 4,650 4,700 4,750 4,600

1,045.3 1,055.4 1,065.5 1,075.5 1,085.6

1,044.2 1,054.2 1,064.3 1,074.4 1,084.4

1,032,7 1,042.6 1,052,6 1,062.6 1,072,5

1,012.4 1,022.2 1,032,0 1,041.6 1,051.6

985.6 995.0 1,004.6 1,014.1 1,023.7

954.5 963,8 973,1 982.4 991.7

934.9 944.0 si53.1 962.2 971.4

4,650 4,900, 4,950 5;000 5,100

1,095.6 1,105.6 1,115.7 1,125.7 1,145.7

1,084.4 1,104.5 1,114.5 1,124.5 1,144.4

1,082.4 1,092.4 1,102.3 1,112.2 1,132.0

1,061.4 1,071.1 1,060.9 1,090.6 1,110.0

1,033.2 1,042.8 1,052.3 1,061.8 1,080.8

1,000.9 1,010.2 1,019.4 1,028.7 1,047.2

960.5 969.5 998.8 1,007.7 1,025.8

5,200 5,300 5,400 5,500 5,600

1,165.6 1,185.5 1,205.4 1,225.3 1,245.1

1,164.4 1,184.3 1,204.1 .1,224.0 1,243.7

1,151.7 1,171.4 1,191.1 1,210.7 1,230.3

1,129.4 1,148.6 1,168.2 1,187.5 1,206.7

1,099.7 1,118.8 1,137.5 1,158.4 1,175.2

1,065.6 1,064.0 1,102.4 1,120.7 1,139.0

1,043.9 1,062.0 1,060.0 1,098,0 1,118.0

5,700 5,800 5,900 6,000 6,100

1,264.9 1,284.6 1,304.3 1,324.0 1,348,6

1,263.5 1,283.2 1,302.9 1,322.6 1,342.2

1,249.9 1,269.4 1,288.9 1,306.4 1,327.9

1,226.0 1,245.2 1,264.4 1,283.5 1,302,6

1,194.0 1,212.8 1,231.5 1,250.2 1,268.9

1,157.3 1,175.5 1,193.6 1,211.9 1,230.1

1,134.0 1,161,9 1,189.8 1,187.7 1,205.5

6,200 6,300 6,400 8,500 6,600

1,363.2 1,362.8 1,402.4 1,421.9 1,/441 .4

1,361.8 1,381.4 1,400,9 1,420.4 1,439.9

1,+47.3 1,366.7 1,386.0 1,405.3 1,424.6

1,321.7 1,340.8 1,359.8 1,378.6 1,397,8

1,287.5 1,306.2 i ,324.7 1,343.3 1,361.9

1,246.3 1,266.4 1,284.5 1,302.5 1,320.6

1,223.3 1,241.1 1,258.9 1,276.6 1,294.3

6,700. 6,800 6,900 7,000. 7,100

1,460.9 1,480,3 1,499.7 1,519.1 1,538,5

1,459,4 1,476.8 1,498,2 1,517,5 1,536.9

1,443.9 i,463.1 1,42.4 1,501.5 1,520:7

1,416.7 1,485.6 1,454.5 1,473.4 1,492,3

1,380.4 1,398.9 i ,417.3 1,A35,6 1,A54,2

1,338.8 1,356.6 1,374.5 1,392.5 1,410.4

1,312.0 i ,329,7 J ,347.4 1,365.0 1,362.6

7,200 7,300 7,400 7,500 7,600

1,557,8 1,577,1 1,596,4 1,615,7 1,634,9

1,556.2 1,575.5 1,594.8 1,614.0 1,633.2

1,539.6 1,559.0 1,576.1 1,597.1 t ,616.2

1,511.1 1,529.9 1,546.6 1,567,4 1,586,1

1,472.6 i ,491,0 1,509.3 1,527.6 1,545.9

1,428.3 1,A46.2 1,464.1 1,181.9 1,499.7

1,400,2 1,417,8 1,485,3 1,452,6 1,470.3,

7,700 7,800 7,900 8,000 8,100

1,654,1 1,673.3 1,692,5 1,711,6 1,730.8

1,652.4 1,671.6 1,690.7 1,709.9 1,729.0

1,335.2 1, S54.2 1,673.1 1,682.1 1,711.0

1,604.6 1,623.5 1,642.2 1,660.8 1,678,4

1,564.2 1,562.5 1,600.7 1,619.0 1,637.2

1,517.5 1,535.3 1,553.0 1,570.8 1,568.5

1,467.8 1,505.3 1,522.7 1,540.1 1,557,6

8,200 8,300 8,400 8,506 8,600

1,749,9 1,768.9 1,786,0 1,807.0 1,826.0

1,748.1 1,767.1 1,766.2 1,805.2 1,824.2

1,729.9 1,748.8 1,767.7 1,786.5 1,605.4

1,693.0 1,716.6 1,735,2 1,753.7 1,772.2

1,655.3 1,673.5 1,691.6, 1,709.6 1,727.9

1,606.2 ‘1,623.9 1,341.5 1,=9.2 1,676.6

1,574.9 1,592.3 i ,609.7 1,627.0 1,644,3

6,700 8,800 8,900 9,000 9,100

1,M5.O 1,364.0 1,863.0 1,901.9 1,620.8

1,143.2 1,S62.1 1,861.1 i ,900.0 1,918.9

1,624.2 1,642.9 1,861.7 1,880.5 1,899.2

i ,790.7 i ,809,2 1,827.7 1,846.1 1,664,5

1,748,0 1,764.0 1,762,1 1,800,1 1,818.1

1,694.4 i ,712.0 1,729.6 1,747.1 1,764.7

1,661.6 1,676.9 1,696,2 1,713.4 i ,730.7

~_

0.05

0.1

3,600 3,650 3,700 3,750 3,800

841.16 651,51 861.83 872.13 882,41

840.24 650.56 860.86 871.15 661.42

3,850 3,800 3,950 4,000 4,050

692,69 902.95 913.20 923.42 933.65

4,100 4,150 4,200 4,250 4,300

0,3

0.9

0.7

,

‘

3PE R&ervOii tigine.tig,

MaY. 1988

TABLE 1–DIMENSIONLESS

INFLUX,

Woo, FOR” INFINITE

AQUIFER

(continued)

tn

0.05

0.1

0.3

Zn ‘ 0.5

0.7

0.9

1.0

9,200 9,300 9,400 9,500 9,600

1,939.7 1,958,6 1,977.4 1,996,3 2,015.1

1,937.4 1,956.6 1,975.4 1,994.3 2,013.1

1,917,9 1,936.6 1,955,2 1,973.9 1,992.5

1,882.9 1,901.3 1,919.7 1,936.0 1,956.4

1,836.1 1,854.1 1,872.0 1,890.0 1,907.9

1,782,2 1,799.7 1,817.2 1,634,7 1,852.1

1,747.9 1,765.1 1,782.3 1,799.4 1,816.6

9,700 9,800 9,900 1.00XI04 I.25x104

2,033.9 2,052.7 2,071.5 2.090 X 10S 2.553 X 103

2,031.9 2,050.6 2,069.4 2.088x 109 2.551 X 103

2,011.1 2,029,7 2,048.3 2.067 X 103 2.526 X 103

1,974.7 1,993.0 2,011.3 2.029 X 103 2.4s1 XI03

1,925.6 1,943.7 1,961.6 1.979x 103 2,421 X 103

I,50XI04 1.75XI04 2, OOX1O4 2,50x 104 3,00 XI04

3.009 x 103 3.457XI03 3.900 x 103 4.773XI03 5.630 X 103

3.006 3.454 3.897 4.766 5.625

2.977 3.421 3.860 4.724 5.574

2.925 X 3.362x 3.794 x 4.646x 5.483 X

103 103 103 103 103

2,855 X 103 3.264x 103 3.707 x 103 4.541 x 103 5.361 X 103

3.5 OX1O4 4.00 XI04 4,50x 104 5.00 XI04 6,00 x104

6.476 X 103 7.312 x103 6.139 x103 8.959 x 103 1.057XI04

6.470 X 103 7.305 x 103 8.132x IOS 8,951 X 103 1.057XI04

6.412x IOS 7.240 X 103 6.060 x 103 6,872 X 103 I,047XI04

6.309 x 103 7.125x IOS 7.933XIOS 8.734xiOa 1.031 XI04

6.170 x103 6.970 X 103 7.762 X 103 6.546x103 I.oloxlo”

6.009 X 6.790x 7.564 X 8.331 X 9.848x

103 IOS 103 103 IOS

5.906 x 10S 6.675x103 7.437 x 103 6.193 x103 9.664 X 103

7,00 XI04 8,00 x104 9,00 XI04 1.00 XI05 1.25x 105

1.217x104 1.375XI04 1.532x104 1.687x 104 2.071 X 104

I,217x104 I.375X104 1,531 x lo~ 1.686XI04 2,069 x lo~

I,206x104 1.363 x104 I.518x104 1.672 XI04 2.052 X 104

1.i88x104 I.342X 104 1.496x 104 1.547XI04 2.023x 104

1.163 x104 1.315 x-io4 1.465 x104 1.614x 104 1.982x 104

1.134XI04 1.283x 104 1.430 XI04 1.576x 104 1.936x 104

1.116 x104 1.262x 104 1.407XI04 1.551 xlo~ 1.906%104

1.50 XI05 2.00 XI05 2.50 X105 3.OOX1O5 4.00 XI05

2.443x 104 3.190XI04 3.918x 104 4.636 X 104 6.048 X 104

2,446x 104 3,188x 104 3,916x104 4.633 X 104 6,044x 104

2.427x 104 3.163x 104 3.865 X 104 4.598 X 104 5.999 x lo~

2.392x 104 3.li9xlo4 3:632x 104 4,536 X 104 5.920 x 104

2.245XI04 3.059 x 104 3.760x 104 4.452x 104 5.612 x104

2.291 X 104 2.989 X 104 3.676 X 104 4.353x 10? 5.687x la4

2,256 2,945 3.622 4.290 5,606

5.00 XI05 6.00 x105 7.00 XI05 8.00 x105 9. OOX1O5

7.436x 104 6.805 X 104 1.016x 105 1.150 XI05 1.283x 105

7,431 x 104 8.796 X 104 1.015 XI05 1.149 XI05 1.262x 105

7.376 X 104 S.735XI04 1.008x 105 1.141 X105 1.273x 105

7,260 X 104 8.623 X 104 9,951 x 104 1.127x 105 1.257x 105

7.150XI04 S.471 x 1o~ 9.777XI04 1. I07XI05 1.235 xI05

6.998 X 10g 8.293 X 104 9.573 x 104 I,064x105 I.2IOX1O5

6.900 X 104 8,178 x104 9.442x 104 1.070XI05 1.194 XI05

I.ooxloo 1.5OX1O6 2.00 XI08 2.50 x106 3.OOX1OO

1.415XI05 2.059 X 105 2.695 X 106 3.320 Xl 05 3.937 x 105

.I.412X105 2.060 X 105 2.695 X’105 3.3 I9X105 3.936 X 105

1.404XI05 2.041 X 105 2.676 X 105 3.286 x10S 3.909 x 105

1.367x 105 2,016xlb5 2.644x IOS 3.254x 105 3,664x 105

1.383XI05 I.962x105 2.601 X 106 3.202 X 105 3.803 x 10’

I,335XI05 I:943XI05 2.551 X 105 3,141 X 105 3,731 x 105

1.317XI05 I.9I8X105 2.516 x105 3.101 X108 3.624x10S

4.00 X106 6.OOX1O6 6,00 x106 7.00 XI06 8.00 XI06

5. I54X105 6.352x 105 7.536x 105 6.709 X 105 9.972 X 105

5.152x 105 6.249 x105 7.533 x 105 8.705 X 105 9.667 X 106

5.116x105 6.308 X 105 7.465XI05 8.650 X 105 9.806 X 105

5.060 6.236 7.402 8.556 9.699

x 105 X 105 X105 X 10$ X 105

4.981 X 105 6.142x10S 7.290 X 105 6.427 X 105 9.555 x105

4,688 x 105 6.029 X 105 7,157XI05 8.276 X 105. 9.364 x”105

4.828 X 105 5.956 X 105 7.072 X 105 6.177x 105 9,273x 105

9.00 XI06 1.00 XI07 I,50XI07 2.00x I07 2,50 X107

1. I03XI05 1.217x106 1.782x 108 2.337x 108 2.884x 10@

I.102x106 1.217x 106 1.761 X108 2.336 X 10e 2.882 X 106

1.095 X’106 1.209x1o6 1.771 Xlos 2.322 X 106 2.666 X 106

1.084XI06 1.196x IOE I.752x108 2.298 x 106 2.837 x 106

1.067x 108 1.179XI06 1.727x 106 2.266 x 106 2.787x10e

1.049 XI06 l,156x10e I.697x106 2,227x 106 2,750 X 106

1.036 x106 1.144XI06 1.678x 106 2.202 x 106 2.720 X 106

3.00 XI07 4.00XI07 5.00 XI07 6,00x 107 7.00XI07

3.425 X 4.493 X 5.547x 6,590 X 7,624x

108 106 106 106 106

3.423 X106 4.491 x 106 5544x 106 6.587 X 106 7,620 X 106

3.404X 108 4.466 X 10 E 5.514XI06 6.551 X 10s 7.579XI08

3.369 4.422 5.460 6.466 7.507

3.323 4.361 5.386 6.401 7.407

3.268 4.290 5.299 6.299 7.290

X 106 X 108 x 106 X 108 X 106

3.232x 106 4.244x10e 5.243x 106 6.232 X 106 7.213x 106

8,00x 107 9, OOX1O7 1. OOX1O8 1.5OX1O8 2.00XI08

6,651 X 108 9,671x106 1.069 x107 1.567 x107 2.059 X 107

6.647x106 9,666 x106 1.067x 107 1.567x 107 2.059 X 107

8.600 x i Oe 9.615x 108 1.062x 107 1.555%107 2.048 X 107

6.5~9 X 10e 9.524 X 106 1.052x 107 1.541 xio7 2.029 X 107

8.407x IOE 9.400 x 108 1.039,XI07 1.522x 107 2.004x 107

6.274x 106 9.252 X 106 1.023 x107 1.499 XI07 1.974XI07

8.136x 106 9.156x10e I. OI2X1O7 I.483 x107 1.954XI07

2.50x I08 3. OOX1O8 4.00 XI08 5.00 x 10S. 6.00x 108

2.546x 107 3.027 X 107 3.979 x 107 4.920 X 107 5.852 X 107

2.545x 107 3.026 X 107 3.978 X 107 4.916x107 5.850 X 107

2.531 X 107 3.010XI07 3.956 X 107 4.694x 107 5,821 X 107

2,:07 X 107 2,984x 107 3,923 X 107 4.651 X107 5.771 XI07

2.476x 2.947x 3.875 X 4.793 x 5.702x

2.429 2.904 3.819 4,724 5,621

x107 X 107 x107 X 107 X 107

2.415x107 2.675 X 107 3.782 X 107 4.678 x107 5.568x 107

7.00% 108 8.00x10S 9.00XI08 1.00XI09 1.5 OX1O9

6.777 X 107 7.700 x 107 8.609 X 107 9,5i6x107 1.401 x 108

6.774x107 7.693 X 107 3.606 X 107 9.515XI07 1.4OOX1O8

6,741 X 107 7.655 X 107 6.564x 107 9.469 X 107 1.394XI08

6.834x 107 7.590 x 107 6.482 x107 9.390%107 1.382x108

6.605 X 107 7.501 x 107 8.393 X 107 9.261 X107 1.367x 108

6,511 X107 7.396 X 107 6.275 x107 9.151 XI07 1.348X108

6.450 X 107 7.327 X 107 8.198x 107 9.066 X 107 1.336x 108

X x X X X

103 103 10S 103 103

X 103 X 103 X 103 X 103 x loo

x 106 X 10 s X 10s X 108 x 106

X 106 X 106 X 106 X106 x 10 S

107 107 107 107 107

1,669.6 1,887.0 1,904.4 1.922XI03 2.352XI03 2.775 x103, 3.193XI03 3.605 X 103 4.419XI03 ‘5.219x 103

“

1,833.7 1,650.9 1,868.0 1.885x IOS 2,308 X 103 2.724x 103 3,135XI03 3,541 XI03 4.341 x 103 5.129xi03

X 104 X 104 x104 X 104 X 104

375

3PE Resemoir Engimering, May 1938

.. . . . .

TABLE I–DIMENSIONLESS

INFLUX,

w.,,

FOR INFINITE ZD

t. 2.00 x 1’09

0.1

0.05

,2.50x10° 3.00 XI09 4. OOX1O9 5. OOX1O9

1&43x108 2.281 X 108 2.714x 10a 3.573 X 108 4.422 x108

8.00x 109 7.00 XI09 8.00 XI09 9.00 XI09 I. OOXIO1O

5,285 X 108 6,101 x108 6.932x108 7.760 X 108 8.583 X 108

1.843x 2.280 X 2.713x 3.572x 4.421 X 5.262 6.098 6.930 7.756 8.574

108 106 IOB IOS 108

X 108 X 108 X 108 X 108 X 108

AQulFER(continued)

‘

0.3 _ 1.634XI08 2.269 X 108 2.701 X 108 3.556 x 108 4.401 x 10S

— 0.5 1.819 x108 2.251 X 108 2.660 X 108 3.528 X 108 4.367x 108

0.7

0.9

1.0

1.799XI08 2.228x 108 2.650X f08 3.489x 108 4.320 X 108

I.774X108 2,196x108 2.615 x108 3.443 x108 4,263 X 108

1.758 x10a 2,177x 10a 2,592 X 108 3.413XI08 4,227x 108

5.240 6.072 6,900 7.723 8.543

5.199x 6.025 X 6.847x 7.664 X 8.478 x

108 108 108 108 10s

5.14+x 108 5.961 X 108 6.775 x108 7,584 X 108 8,389 X 108

5.077 X 108 5.885 X 108 6.686 x 10S 7,487 x 108 6.283x108

5.033 x 108 5.835 X 108 6.632x 108 7.424 X 106 8.214x10B

X 108 X 108 X 108 X 108 X 108

1.263 x109 1.666XI09 2.085XI09 2.458x 109 3.240 X 109

1.264x 109 1.666x10g 2.063 X 10s 2.456 X 109 3,239 X 109

1.257x 109 1.659 x109 2.055 X 109 2.447 x10g’ 3.226 X 109

1.247 x109 1.646 x10g 2.038 X i 09 2A30XI09 3,203 X 109

1.235 x109 1.630 x10g 2.016x 109 2,405 X 109 3. I7IX109

1.219 x109 1.610 x10g 1.993XI09 2.376x 109 3.133X 109

1.209x 109 1.596 xiOg 1.977 XI09 2.357 X 109 3.106x 109

4.014 XI09 4.782 X 109 5S46X10g 6,305 X 109 7.060 X 109

4.013XI09 4.781 X 109 5.544x 109 8.303 X 109 7.058 x 109

3.997 4.762 5,522 6,278 7.030

3.968 4.728 5.483 6.234 6,982

3,929 x 109 4.682x 109 5.430XI09 6.174x 109 6.914x 109

3.883 X 10s 4.627 x 109 5.366 X 109 6.102x10g 6.634 X 109

3.832 4.591 5,325 6.055 6,782

l.ooxloj~ 1.50x ioi1 z,ooxlol~ 2.50x 10ii 3.00 XI011

7.613 x.f Og 1.154x1040 1,522 x1010 l,866xlof0 2,246 X 1010

7.810x 109 1.153 XIO:0 1.521 x1040 1.885x1040 2.247 X 10 ‘0

7.780 X 109 1.149x1010 1.515x1010 1.878 x10’0 2.239 x10i0

7.726 x.1 09 1.141 x10[0 1.505x1010 1.666 XIO$Q 2.224 x10f0

7.652x 109 1.130x1010 7.49 IX1O’O 1.649x10%0 2.204 X 1010

7.584X109 1.118 x1010 1.474 x10’a 1.623 x10’0 2,179x1010

7.506 X 109 1.109 XIO1Q I.463x101Q 1,614 x1010 2.163x1010

4.00 XI011 5.00 XI011 8.00x I011 7.00x lo~~ 8.00x 101’

2.965x1010 3.677x1010 4,383x 1010 5.085x1010 5.763x1010

2.964 X 10 ‘0 3.675 X 10 ‘0 4.361 xlOiO 5.082x1010 5.781 XIOIO

2,953 x 1010 3.662x1010 4,365x 10(0 5,064 x10[o 5.760 x101U

2.934x1010 3.636x1010 4.337x lo~o 5.032 x101D 5.723x1010

2.907 3.605 4.298 4.987 5.673

2.S76x1010 3,566x1010 4,252x1010 4.933x1010 5.612x1010

2.855 x10i0 3.540 xlo~Q 4,221 X 1010 4,898x1010 5,572x 1010

9.ooxlol~ 1.00 XI012 1.50 X1012 2,00 XIO!2

6.478 7.171 1.060 1.400

6.476x1010 7,163 x101Q 1.060x101’ 1.399 xlot~

6A53x101D 7.143%1010 I. 056X1 O’* 1.394 xlo~;

6.412 xiOt0 7.096x 1010 I. O5OX1O” 1.366 x1011

6.355 X1010 7.035 x 1ON 1.041 Xlo” I,374x1011

6.266x1010 6.961 X 1010 I,o3ox1O” 1.359 xlot~

6.243 x10i0 6.912 x10
1.5oxlo~Q 2.00 x IO”lQ 2.50x1010 3. OOXIO1O 4.00 XIOID ‘S.ooxiojo 6.00x iOi0 7.00 x 1010 8.00 X 10<0 9.00 x 1010

X 1010 xiO<0 x10i1 x10i1

x 109 X 109 x109 X 109 x 10$

X 109 X 10s X 109 X 109 X 10s

X1010 X 10>0 X 1010 X1010 X 1,010

x109 x 109 X 109 X 109 X 109

,,

. 2,

m

0. 0,1

!

,0

100

t. Fia. 4—Plots

376

of

W.. vs. t. for infinite aauifer.

Fig. 5–PlOts

of W..

vs. t.

for finite aquifer

(r.’ +S).

8PE Rem’vou Engineering, May 1988

TABLE 3-DIMENSIONLESS

INFLUX,

W.D, FOR r~’ = 6

ZD r

0.05

0.1

6 7 6 9 10

4.780 5.309 5.799 6.252 6.750

4.782 5.289 5.776 6.229 6.729

TABLE 2—DIMENSIONLESS

INFLUX,

WK,, FOR r~’ = 4

ZD ‘

11 12 13 14 15

7.137 7.569 7.967 6.357 8.734

0.05

0.1

0.3

0.5

0.7

0.9

I .0

2 3 4 5 6

2.398 3.006 3.552 4.053 4.490

2.389 2.993 3.528 4.017 4.452

2.284 2.874 3.404 3.693 4.332

2,031 2.629 3.158 3.627 4.047

1.824 2.390 2.693 3.341 3.744

1.620 2.149 2.620 3.045 3.430

1.507 2.012 2.466 2.876 3.248

7 8 9 10 11

4.867 5.191 5.464 5.767 5,964

4.829 5.157 5.424 5.739 5.935

4.715 5.043 5.322 5.59S 5.829

4.420 4.757 5.060 5.319 5.561

4.407 4.437 4.735 5.000 5.240

3.77S 4.096 4.385 4.647 4.834

3.587 3.898 4.184 4.443 4.6S1

12 13 14 15 16

6,188 6,380 6S59 6.725 6.876

6.158 6,350 6,529 6.694 6.844

6.044 6.240 6.421 6.589 6.743

5.7S0 5,9Si 6.171 6.245 6.506

5.463 5.670 5.863 6.044 6.213

5.107 5.316 5.611 5.695 5.S67

4.903 5,113 5.309 5.495 5.671

17 18 19 20 22

7.014 7.140 7.261 7.376 7.518

6.983 7.113 7.240 7.344 7.507

6.S85 7.019 7.140 7,261 7,451

6.656 6.792 6.913 7.026 7,227

6.371 6.523 6.663 6,785 6,982

6.030 6.187 6.334 6,479 6,691

5.836 5.999 6.153 6.302 6.524

24 26 2S 30 34

7.618 7.697 7.752 7.608 7.664

7.607 7.665 7.752 7.797 7.664

7,518 7.607 7.674 7.741 7.819

7,361 7.473 7.563 7.641 7..741

7,149 7.283 7.395 7.484 7.61S

6.870 7.026 7.160 7.283 7.451

6.714 6.681 7.026 7.160 7.350

38 42 46 50 60

7.909 7.931 7.942 7.954 7,968

7.909 7.931 7.942 7.954 7.988

7.675 7.908 7.920 7.942 7.965

7.608 7.864 7,69S 7.920 7.954

7.719 7.797 7.842 7.875 7.931

7.585 7.685 7.752 7.806 7.898

7.486 7.618 7.697 7.764 7.S64

70 80 90 100

7.976 7.982 7.987 7.987

7.976 7.962 7.967 7.987

7.976 7,987 7,987 7.987

7.968 7,976 7.924 7.987

7.965 7.976 7.983 7.987

7,942 7.865 7.976 7.983

7.920 7.954 7.965 7.976

120

7.987

7.987

7,987

7.887

7.987

7.887

7.967

tD

0,9 ——

1.0

0.5

0.7

4.597 5.114 5.595 6.041 6.498

4.265 4.779 5.256 5.712 6.135

3.953 4.422 4.875 5.310 5.719

3.611 4.053 4,476 4.888 5.278

3.414 3.837 4.247 4.642 5.019

7.116 7.545 7.916 8.334 8.709

6.916 7.325 7.718 6.099 6.467

6.546 6.945 7.329 7,699 S.057

6.110 6.491 6.858 7.214 7.557

5.646 6.009 6.359 6.697 7.024

5.376 5.72S 6.067 6.395 6.713

16 9.093 17 9.442 18 9.775 19 10.09 20 10.40

9.067 9.416 9.748 10.06 10.37

6.819 9.160 9.485 9,794 10.10

s.398 8.730 9.047 9.443 9.646

7.834 6.204 S.510 6.S02 9.067

7.336 7.641 7.934 8.214 8.487

7.017 7.315 7.601 7.874 8.142

22 24 26 26 30

10.99 11.53 12.06 12.52 12.95

10.96 11.50 12.03 12.49 12.92

10.67 11.20 11.72 12.17 12.59

10.21 10.73 11.23 11 .6S 12.09

9.631 10.13 10.62 11.06 11.46

9.009 9.493 9.964 10.39 10.76

8.653 9.130 9.594 10.01 10.40

35 13.96 40 14.69 45 15.27 50 15.74 60 16.40

13.93 14.66 15.24 15.71 16.38

13,57 14.33 14.94 15.44 16.15

13.06 13.84 14.48 15.01 15.S1

12.41 13.23 13.90 14.47 15.34

11.70 12.53 13.23 13.84 14.78

11.32 12.15 12.87 13.49 14.47

~.

0.3 __—

70 60 90 100 110

16.87 17.20 17.43 17.56 17.71

16.85 17.18 17.42 17.56 17.69

16.67 17.04 17.30 17.46 17.63

16,38 16.80 17.10 17.34 17.50

15.99 16.48 16.65 17.12 17.34

15.50 16.06 16.50 16.83 17.09

15.24 15.63 16.29 16.66 16.93

120 130 140 150 175

17.78 17.S4 17.88 17.92 17.95

17.78 17.84 17.86 17.91 17.95

17.73 17.79 17.85 17,88 17.94

17.63 17.73 17.79 17.84 17.92

17.4S 17.62 17.71 17.77 17.87

17.29 17.45 17.57 .17.66 17.61

17.17 17,34 17.48 17.58 17.76

200 225 250 300 350

17.97 17.97 17.99 17.98 17.96

17.97 17.97 17.98 17.9S 17.98

17.96 17.97 17.98 17.98 17.98

17.95 “17.96 17.97 17.98 17.98

17.93 17.85 17.96 17.98 17.98

17.86 17,93 17,95 17.97 17.98

17.66 17.91 17.95 17.97 17.96

400 450 500

17.98 17.98 17.96

17.98 17.9S 17.96

17.93. 17.96 17.96

17.98 17.93 17.98

17.98 17.98 17.98

17.98 17.98 17.96

17.98 17.98 17.88

It is interesting to observe what effect this error has on a materialbakmce calculation. The material-balance equation for an undersaturated reservoir is

60 50

NBoiceAPR=NPBo–W,.

. . . . . . . . . . . . . . . . . . . . . . . . .. (18)

Ag& cumulative inthx values from both models were entered into Eq. 18, which wassolved fortbeorigind oiltipla%N. It is appment in Table 6 that &r a few timesteps the ~ulative water influx predicted by the radial mcdel is in fact larger than the cumuIative oiLproduction (lxXhexp ressedin= semoirvolumes), and consequently tie cdcolated value of Nbecomes negative. The dial flow model isclearly nonapplicable totili system, The bottomwater model, ontheotherha.nd, gives avalueof Nthat uickly .$ converges tothevohunetricv sdueof25M MSTB[4.OXIO stock* m3].

,0 %D 30

,0

10 0 m

,00 b

Fig. 6—Plots “of W.D

8PE Reservoir En@ering, —.

—.

vs. t~ for various aquifer sizes.

May 1988

,,00

Conclusions 1. A rigorous water influx model for bottomwater systems must into account vertical flow in the vicini~ of the reservoir. 2. Such a model, based on a modified form of the unsteady-stats radial dfisivity equation, is Ny developed and presented in this paper. 3.’ For bottomwater drive reservoirs, the calculation of water ioflux with a mcdel that ignores verdcsd flow effects IMY lead to considerable error.

*

317

TABLE 5-DIMENSIONLESS

INFLUX, ‘D

W.D, FOR r~’=S

‘

0.05

0.1

0,3

0.7

0.9

1.0

6.301 6,628 7,250 7.725 8.173

6.27S 6.807 7.229 7,700 8,149

6,066 6.574 7,026 7,477 7.919

5.756 6.205 6.650 7.0S6 7,515

5.350 5.763 6.204 6.621 7.029

4.924 5.336 5.732 6.126 6.514

4.675 5.072 5.456 5.S36 6.210

.9.619 8.594 9,056 9,032 16 9,485 9,45S 17 9.907 9.879 18 10.32 ,10.29

8.355 8.723 9.202 9.613 10,01

7,937 6.351 8.755 9,153 9.537

7.432 7.828 8.213 8.594 6.961

6.895 7,270 7.634 7.997 6.343

6.578 6.940 7.293 7.642 7.979

19 10,72 20 11.12 22 11.69 24 12.63 28 13,36

10.69 11.08 11.86 12,60 13.32

10.41, 10,80 11,55 12.27 12.97

9.920 10.30 11.02. 11.72 12,40

9,326 9.687 10.38 11.05 11.70

8,691 9.031 9.686 10.32 10.94

8.315 8.645 9.280 9.896 10.49

23 30 34 38 40

14.06 14,73 16,01 17.21 17,80

14.02 14,69 15,97 17.17 17.75

13.65 14.30 15.54 16,70 17,26

13.06 13.68 14.S6 15.99 1&52

12.33 12.93 14.07 15.13 15.64

1i.53 12.10 13,18 14.18 14,66

11.07 11.62 12.67 13.85 “ 14.12

45 50 55 60 70

19,15 20.42 21.46 22.40 23.97

19.10 20.3’6 21.39 22,34 23,92

18.56 19,76 20,80 21.75 23,36

17.76 16.91 19.96 22.55

16,83 17.93 18.97 19.93 21.58

15.77 16.80 17.83 18.78 20.44

15.21 16.24 17.24 16.79 19.86

60 90 100 120 140 160 180 200 240 280

25.29 26.39 27.30 26,61 29,55 30.23 30.73 31.07 31.50 31,72

25,23 26.33 27.25 28,57 26.51 30.21 30.71 31.04 31.49 31.71

24.71 25.86 26.81 28.19 29.21 29.96 30,51 30,90 31,39 31.66

23,94 25.12 26,13 27.63 2S.74 29.57 30.i8 30.63 31.22 31.56

23.01 24.24 25.29 26.90 28.12 29.04 29.75 30.26 30.98 31.39

21.91 23.18 24.29 26.01 27.32 26.37 29.18 29.79 30.65 31.17

21,32 22.61 23.74 25.51 26.90 27.99 2S.84 29.51 30.45 31.03

320 360 400 450 500

31.65’ 31.90 31.94 31.96 31.97

31.34 31.90 31.94 31,96 31,97

31.80 31.88 31,93 31,95 31,96.

S1 .74 31,85 31.90 31.94 31.96

31.64 31.76 31.86 31.91 31.95

31.49 31.68 31.79 31.88 31.93

31.39 31.61 31.75 31.85 31.90

550 600 700 600

31.97 31.97 31.97 31.97

31,97 31,97 31,97 31.97

31.97 31.97 31.97 31.97

31,96 31,97 31.97 31.97

31,96 31,97 31,97 31.97

31.95 31.96 31.97 31.97

31.94 31,95 31.97 31.97

‘D

9 10 11 12 13 14 15

0.5 _—_—

20.91

Nomenclature

0.3

0.7

0.9

1.0

22 12.07 24 12.88 26 13.65 28 14.42 SO 15.17

12,04 12,33 13.62 14.39 15,13

11.74 12.62 13.29 14.04 14.77

11.21 11.97 12.72 13.44 14.15

10.56 11.29 12.01 12,70 13.36

9.865 10.55 11,24 11.90 12.55

9.449 10.12 10.78 11.42 12.05

32 34 36 38 40

15.91 16.63 17.33 16.03 18.72

15.87 16.59 17.29 17.99 18.6S

15.49 16.20 16,89 17.57 18.24

14.85 15.54 16.21 16.S6 17.51

14.05 14.71 15.35 15.98 16.60

13.16 13.81 14.42 15.02 15.61

12.67 13.26 13.67 .14.45 16.02

42 44 46 48 50

19.38 20.03 20.67 21.30 21.92

19.32 19.99 20.62 21,25 21,87

16.89 19.5S 20.15 20,76 21.S6

18.14 1S.76 19.36 19.95 20.5S

17.21 17.60 16,3S 16,95 19.51

16.19 16.75 17.30 17.84 1S.36

15.58 16,14 16.67 17.20 17.72

52 54 56 58 60

22.52 23.11 2S.70 24.26 24.62

22.47 2S.06 23.64 24.21. 24.77

21.95 22.53 23.09 23.65 24.19

21.10 21.66 22.20 22.74 23.26

20.05 20.59 21,11 21.63 22.13

16.69 19.40 19.S9 20.39 20.87

18.22 1S.72 19.21 19.68 20.15

65 70 75 60 S5

26.18 27.47 28.71 29.89 31.02

26.12 27.41 28.55 29.82 S0,95

25.50 26.75 27.94 29.0S 30.17

24,53 25.73 26,66 27,97 29.01

23.34 24.50 25.60 26.65 27,65

22.02 2S.12 24.17 25.16 26.10

21 .2S 22.36 23.39 24.36 25.31

90 95 100 110 120

32.10 33.04 3S,94 35.55 S6.97

32.03 32.96 33.85 35.46 36.90

31,20 32.14 33.03 34.65 36,11

30.00 30.95 31.85 33.49 S4.98

28.60 29.54 30.44 S2.06 3S,56

27.03 27,9S 26.82 S0.47 31 .9S

26.25 27.10 27.96 29.62 31.14

130 ?40 150 170 190

38.28 39.44 40.49 42,21 43.62

36.19 39.37 40.42 42.15 43.55

s7.44 38.64 39.71 41.51 42.98

36.23 37.56 38.67 40,54 42.10

.34.96 36.23 37.S8 39.33 40:97

33.3S 34.67 35.S8 37.89 39.62

32.55 33.65 35.04 S7.11 38.9o

210 230 250 270 290

44.77 45.71 46.48 47.11 47.61

44.72 45.67 46,44 47.06 47.58

44.19 45.20 46.01 46.70 47.25

43.40 44.48 45.36 46.13 48.75

42.36 43.54 44.53 45.36 46.07

41.11 42.S6 43.47. 44.40 45.19

40.42 41.74 42.87 43.34 44.66

310 3S0 35o 400 45o

48.03 48.38 46.66 49.15 42:46

46.00 48.S5 46.64 49.14 49.45

47.72 48.10 4S.42 43.99 49.35

47.26 47.71 46.08 46.74 49.17

46.66 47.16 47.59 4S,36 43.91.

45.67 46.45 48.95 47.39 48.55

45.41 46.03 46.57 47.60 48.31

500 600 700 800 900

49.65 49.64 49.91 49.94 49.86

49.64 49.84 49.91 49.94 49,96

49.58 49.81 49.90 49.93 49.94

49.45 49.74 49.67 49.92 49.84

49.26 49.65 49.82 49.90 49,93

48.98 49.50 49.74 49.65 49.91

46.82 49.41 49.69 49.S3 49.90

1,000

49.96

49.96

49.96

48.96

49,84

49.93

49.93

i ,200

49.96

49.96

49.96

49.96

49.96

49.96

bbl/STB

Bw = water FVF, c = effective

[m3/stock-tank m3] [m3 /stock-tank

bbl/STB

[m3 /stock-tank

aquifer compressibility, reservoir

co = 01 compressibility,

compressibility, psi-l

PdW = average PIeSSUm at OWwater c0nt2ct, psia tkpal pR = average reservoir pressure, psia &Pa]

m3.]

m3]

psi-1

[kPa -

psi -1

11

[kPa -11

@-1]

ew = rate of water infhx, B/D [m3 /d] Fk = ratio of verticti to horizontal pcrmeabili~, dimensionless k = horizontal permezbili~,

dzrcies

m31 NP = cumulativeoil production, STB [stock-tankm3] N = initial oil in place,

Ap = pressure increment,

psia NPa]

APD = dmensiodess

pressure increment ApR = reservoir pressure increment, psia NPa] r = radius, R [m] rD = dimensionless

radius

rD = ~ensionless mdius comtmt r, = aquifer rzdius, * [m]

h = aquifer thickness, ft [m]

373

46.96

p = pressom, psia &Pa]

Boi = initial 01 FVF, bbl/STB

Ce = effective

0.05

pb = ~a~ation preSSU1., pSia Ha] pi = nutd presure, psia l’kpa]

B = water influx constant, bbl/psi [m3 /kPa] II. = oil FVF,

ZD’ 0.5

We~, FOR r.o‘ = 10

0.1

tD

TABLE 4—DIMENSIONLESS

INFLUX,

STB [stock-@k

rR = reservoir rdus, t = time, days tD = dmensiordess

ft [m] time

TABLE 6-SAMPLE

CALCULATION

Sy2tem Par2meter2 2,000

r., ft

.

:ft k, dxcies F,

200 0.050 0.04 0.10 0.395 6X I0-6 .15x10-6 22 XIII-6 3,000 1.000

d p, Cp c,

psi’1

co,

psi-l

co,

psi-t

p,, psia pb, psia

1.0

bbl/STS Bw, Bd, bblLSTS N, MMBTB

1.432 25

C0n8tants

Oimenslonlezs

rO’=ZD’ = 0.5 tD = o.25t S=716 STS/psi Water Influx Calculation SOttOmwater Radial Model (d:ys)

30 60 90 120 150 160 210 240

mol.

Q

Pd. (psia)

(psla)

Wm

7.5 15.0 22.5 30.0 37.5 45.0 52.5 60.0

2,956 2.917 2;877 2,644 2,811 2,791 2,773 2,755

22,0 41.5 39.5 36.5 33.0 26.5 19.0 16.0

6.028 9.949 13.439 16.472 i 9.676 22.897 25.627 28.691

Simulator

(M%B)

(M%) 3% 678 1,103 1,594 2,126 2.676 3:250

Material-Salance

24.726

110 314 606 964 1,362 1.629 2:295 2.785

2:782

Calculation Bottomwater Model

(;&) 2,950 2,908 2,661 2,625 2,768 2,766 2,749 2,729

60 90 120 150 160 210 240

At = time increment, AtD = dimensionless We = cumulative

ZD

= tiemionkss

~ = fmation

1$ 175 212 232 251 271

1,082 1,403 1,730 2,090

days bbl [m3]

curmdative water influx

2 = verf-ic8J distaoce coordinate, ZD, = dimemiouless

(MM{TS)

7

time inorement

water infiux,

W,D = dimensionless

$%)

(d;ys) 7

ft [m]

vertical distance ~oor~~ thi&h5.2s cOnsMt Q Pa.

water viscosity,

S1

+ = formation porosity, fraction

1.4256 1.4371 1.4875 1.4272

- 5.9 -15.0 -22.5 -26.7

AJME 3. SCbilibuis, R.J.: “Active Oil and Rtsenoir Energy,,, Tm., (1936) 118,37-52. 4. Fe!kovich, M.J.: “A 3implified Apprcach to Water Inilux CakukatioII— Finite Aquifer Sysmns,,, JPT (July 1971) 814-28. 5. Muskal, M.: The Flowof Homo~eneomFluidsZhmugk PomuJMedia, J.W. 2dwards fllC., Ann Arbor (1946). 6. Craft, B.C. and Hawkins, M.F. Jr.: AppliedPetroleumReservoirEngineeting, Premice.Hafi fro., Englewcd Cliffs, NJ (1959) 205-24, 7. D&e, L. P.: Fmdmrenrah ofRe.w,-voirEn@zeetig, Elsevier Sciendlic FubtiSbiD8Co., New York City (1978) 315-24,

S1 Metric Conversion Factors

Acknowledgments

bbl X 1.589873 Cp x 1.0*

We thank Joy A. Ostmnder for typing tie manuscript and GuJfCanada Resources

Ltd. for ~rmission

46.3 37.0 33.5 31.6 30.2 27.4 27.1 26.1

to publish this paper.

ft X 3.(248* psi X 6.894757 psi-l x 1.450377

References Tmn.?1. vanEverdigen, A.F. and Hurst, W.: “Application of the Laplace fo~on to Flow Problems in Resavoirs,” Tram., AfMS (1949) 186, 305-24. 2. tiati. K. H.: “A MatbematicaJ Mcdel for Water Movement About Reservoirs,” SPEJ (March 1962)44-5% Trans., Bg;-Z-Drive ,.

E–OI E–03 E–01 E+OO E–01

.Convmbn

facm

= = = = =

m3 Pas m kPa kFa-l SPERJ3

Is exact,

Oligl”dSPE m.

nmn”s?iPt ,ocdved b, IWOW S,F,L 16,1984 P.w.er accqtti for PubliwMarch 3!, 1987. Revised man!mrlp mmived Aug. 11,1927. Paper (SPE131 70) firs!

Pmen:ed at the 1934 SPE Annual Technlm sem. 16-19.

Con ferenc@ and Exhlbiilon held In Houston, . .

8PE Reservoir En.gineeriIIE,May 1988

379