Water Drive Oil Reservoirs

Chapter 38

Water Drive Oil Reservoirs Daylon L. Walton, Roebuck-Walton Inc.*

2 shows that the relationship between the pressures and

Introduction Water drive reservoirs are those reservoirs in which a significant portion of volumetric withdrawals is replaced by water influx during the producing life of the reservoir. The total influx, and influx rates, will be governed by the aquifer characteristics together with the pressure-time behavior along the original reservoir/aquifer contact. Or- dinarily, few wells are drilled into the aquifer and little or no information concerning the aquifer size, geometry, or rock properties is available. However, if sufficient reservoir pressure and production history is available, the aquifer properties may be inferred from solutions of Eq. 1, the radial form of the diffusivity equation.

a% 1 ap 5h.b~ ap p+; ar=k -$ ..I.........., . (1)

where p = pressure, r = radius,

4 = porosity, p = viscosity, c = compressibility, t = time, and k = permeability.

These inferred aquifer properties then can be used to calculate the future effect of the aquifer on the reservoir performance.

Definitions Aquifer Geometry

Radial-boundaries are formed by two concentric cylinders or sectors of cylinders.

Linear-boundaries are formed by two sets of parallel planes.

Nonsymmetrical-neither radial nor linear.

‘Author of the original chapter on this topic m the 1962 edltm was Vment J Skora

Exterior Boundary Conditions Infinite-pressure disturbances do not affect the exterior

boundary of the system, during the time of inrerest. Finite closed-no flow occurs across the exterior bound-

ary. Pressure disturbances reach the exterior boundary, during the time of interest.

Finite outcropping-aquifer is finite with pressure constant at exterior boundary (i.e., aquifer outcrops into lake, gulf, or other surface water source).

Basic Conditions and Assumptions 1. The reservoir is at the equilibrium average pressure

at all times. 2. The water/oil (WOC) or water/gas contact (WCC)

is an equipotential line. 3. The hydrocarbons behind the front are immobile. 4. The effects of gravity are negligible. 5. The difference between the average reservoir pres-

sure and the pressure at the original WOC or WGC will be assumed to be zero if unknown.

Mathematical Analysis Basic Equations

Van Everdingen and Hurst ’ obtained a general solution to Eq. 1 for two cases: (1) a constant water-influx rate (constant-terminal-rate case) and (2) a constant pressure drop (constant-terminal-pressure case). By using the principle of superposition, van Everdingen and Hurst extended these solutions to include variable water-influx rates and pressure drops. Mortada’ further extended the solutions to include interference effects in homogeneous infinite radial aquifers.

Constant-Terminal-Rate Case. If time is divided into a finite number of intervals (Fig. 38. l), the average water influx in each interval can be used in Eq. 2 to calculate the pressure drop at the interior aquifer boundary. Eq.

PETROLEUM ENGINEERING HANDBOOK

ew

e wI

e w3

k

e w2

I I

I 2- INTERVAL

aewa I i I

NUMBER

Fig. 38.1 -Water influx rates-constant terminal rate case.

water-influx rates is a function of a constant m,. and a variable po. The constant m, is a function of the aquifer properties, whereas pD is a function of aquifer properties and time.

n

AP,,.,~ =mr c [c,,,~,,+,+~, -el,.,,r ,, IPD, 3 .(2)j=l

where P w,, = cumulative pressure drop to the end of

interval n, e ,,, - water-influx rate at interval n-t 1 -j, ,r,+,-,I -

PI1 m, = . 0.00,,27kha

for radial aquifers,

PM m, = . . o.ool *27kh

for infinite linear aquifers,

P WL m, = 0~00,127khb . .._................

(3)

(4)

(5)

for finite linear aquifers,

pi = dimensionless pressure term, e,. = water influx rate, RB/D, pI(, = pressure at the original WOC, psi,

k = permeability, md, h = aquifer thickness, ft, b = aquifer width, ft, L = aquifer length, ft,

FL,, = water viscosity, cp, and cx = angle subtended by reservoir, radians

-------- --

PO P

P %

p3 4-I

P_ .“OI 2 3 n-l n

INTERVAL NUMBER

Fig. 38.2-Pressure drops-constant terminal pressure case.

For calculation convenience it is recommended that time be divided into equal intervals and Eq. 6 be used.

AP..,~ =mr i e,, ,,,, +,-,,ApD, . . . j+l

=mrIelv,, 40, fe,,,,, ,, APL)-

e,, ? MD,,, ,, +e,,., APD,~ 1, (7)

where 40, ‘PO, -PO,-,

Constant-Terminal-Pressure Case. If time is divided into a finite number of intervals (Fig. 38.2), Eq. 8 can be used to calculate the cumulative water influx for a given pressure history, using average pressure drops in each time interval.

,I WC>,) =mp c Apcrr+,-,) w,D, , . . (8)

j=l

where

w,!, = cumulative water influx to end of interval,

“P = 0.17811 +c,,,har,,.’ ____._. ._. .(9) for radial aquifers,

MI] = 0.17811 $r ,,., hb 2 .(lO) for infinite linear aquifers,

AP(~~+I-~, = average pressure drop in interval n+l-j,

W PD = dimensionless water-influx term, rw = field radius, ft, and

c.,i = total aquifer compressibility, psi - ’ .

The solution of Eq. 8 requires the use of superposition, in a manner similar to that shown by the expansion of Eq. 6. A modification presented by Carter and Tracy3 permits calculations of W, that approximate the values

WATER DRIVE OIL RESERVOIRS 38-3

obtained from Eq. 8 but does not require the use of superposition. This method is advantageous when the calculations are to be made manually. since fewer terms are required.

Using Carter and Tracy’s method, Eq. I I, the cumulative water influx at time t,, is calculated directly from the previous value obtained at t,,-,

+ bpA~,,r~,, - W,,,, ,,P’D,, IVo,, -[I+,, ,, 1

PD,, -tDd”D,,

where

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

p,D =pD,, -pD,,, I, . . . . . . . . ..I..... ,> (12) ID,, -rD,,,-,,

and

Ap,,=p,-pn, . . . . . . . . . . . . . . . . . (13)

Reservoir Interference. Where two or more reservoirs2 are in a common aquifer, it is possible to calculate the change in pressure at Reservoir A, for example, caused by water influx into another reservoir, B, using Eq. 14 or 15. These are Eqs. 2 and 3 with modified subscripts.

For unequal time intervals,

A~Pnwo,, =tnr Ii [~doi‘,-,) -enB,,,JPD(A.R),~ J=I

. . . . . . . . . . . . . . . . . (14)

and for equal time intervals,

*P~(A,B),, =m, e MB j=l

,,,+,mj ,APD(A,B), > . .(I3

where PD(A,B) = dimensionless pressure term for

Reservoir B with respect to Reservoir A,

AP,~(~,J) = pressure drop at Reservoir A caused by Reservoir B, and

e,,,B = Water inflUX rate at Reservoir B.

The total pressure drop at Reservoir A at any given time is the sum of the pressure drops caused by all reservoirs in the common aquifer, or

APIA,, =AP~(A,A I,, +AP~(A.B),, +AP~(A,cJ,, +. .

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

Since dimensionless pressure differences are available only for homogeneous infinite radial aquifers, pressure- interference calculations are limited at the present time to aquifers that can be approximated by a uniform, infinite, radial system.

4\ FAULT

0 A

Fig. 38.3~Infinite aquifer bounded on one side by a fault.

Hicks et al. 4 used the past pressure and production history in an analog computer to obtain influence-function curves for each pool in a multipool aquifer. The influence function F(r) can be defined as the product of m, and PO,

F(r)=m,pD, . . . . . .(l7)

and can be substituted in Eqs. 59 and 60 to calculate the future performance.

Nonsymmetrical Aquifers. By use of the images method,2 the procedure for calculating reservoir interference can be extended to the case where one boundary of an infinite aquifer is a fault. For example, Fig. 38.3 shows Reservoir A located in this type of aquifer. To calculate the pressure performance at Reservoir A, first lo- cate the mirror-image Reservoir A’ across the fault. The water-influx history for the mirror-image Reservoir A’ will be taken to be the same as Reservoir A. Next, assume that the fault does not exist so that there are two identical reservoirs in a single infinite aquifer, with Rexr- voir A’ causing interference at Reservoir A. The pressure drop at Reservoir A now can be calculated by use of Eq. I9 (for equal time intervals).

APIA,, =mr 2 [~NzA~,,+,~, , APO, 1 J=t

Because e ,,,A =e Lr,A, ,

n APoA,, =m, c e)+,A

j=l ,,!+,-, j [APO, -APD(A.AY, 1.

.., . . . . . . . . . . . . . . . . . . (1% If other reservoirs in the aquifer also are causing reser-

voir interference at Reservoir A, each mirror image will cause reservoir interference at Reservoir A. The total pressure drop at Reservoir A, therefore, will be the sum of the pressure drops caused by each reservoir and each mirror image (see Fig. 38.4).

Nonsymmetrical aquifers will be discussed further un- der Methods of Analysis, Method 2.

38-4 PETROLEUM ENGINEERING HANDBOOK

Fig. 38.4-Dimensionless pressure drop for infinite aquifer system for constant flow rate. ,8

pn and W,~Values. Values ofpn, PD(A,B), and W,D are functions of dimensionless time rg (Eq. ZO), aquifer geometry, and aquifer size (to for radial aquifers).

Table 38.1 gives the substitution for d in Eq. 20 to calculate tD and the table, graph, or equation to obtain po, P&A-B), or W,D for various types of aquifers. The following equations are used in conjunction with Table 38.1.

0.006328kr tD = ~C~,~?ftL,d2 , . . . (20)

po=l.l284JtD, ........................ .(21)

pD=o.5(h tD+0.80!?07), ................. .(22)

pD=h ,-D, ............................. .(23)

WeD=0.5(rD’-I), . . . . . _.. . . ..(24)
‘Personal communication from Allant~c Refining Co
Ap~=-$Aro. . . . . . (25) ID

and

pD=tD+o.33333, . . . . . . . . . . . .(26)

where to = dtmensionless time, rD = dimensionless radius =T,/T,,

ru = aquifer radius, ft, rw = field radius, ft, and

d = a geometry term obtained from Table 38.1.

Methods of Analysis Reservoir Volume Known. Rigorous Methods. There are two methods for obtaining the coefficient m, and APO in Eq. 6 from the past pressures and the water- influx rates from a material balance on the reservoir. Method l* is used whenever the aquifer can be approximated by a uniform linear or radial system; therefore, published values of pD are used. If the aquifer can be approximated by a homogeneous, infinite, radial system, the method can be extended to handle reservoir interference. In Method 2,5 the product of m, and pD is replaced by Z (the resistance function).

Apwj,, = 2 e, fn+, , ,AZj, (27) j=l

where AZ, =Zi -Zj- r . Method 2 is not limited to homogeneous linear or radi-

al aquifers because the final Z is obtained by adjusting previous approximations to Z. Techniques for applying Method 2 to the case where reservoir interference exists are not available at this time, except for unusual circum- stances.

TABLE 3&l--REFERENCE TABLE FOR OBTAINING WeD AND p.

Aquifer Type

Infinite radial Smaller t, Larger t,

Finite outcropping radial Smaller t, Larger t,

Finite closed radial Smaller lo Larger t,

Infinite linear Finite closed linear Larger to

Interference (infinite radial)

Larger to

Value of d in Eq. 20 PD WC?0

* rw Table 38.3 Table 38.3 rw Eq. 21 Eq. 21 rw Eq. 22 r, Table 38.7 Table 38.5

Table 38.7 rw rw Eq. 23

Table 38.6 Table 38.6 rw r, Table 38.3 Table 38.3 rw Eq. 25 Eq. 24 b” Lf

Eq. 21 Eq. 21+ Table 38.8

L Eq. 26 r(A.B15 Fig. 38.4 pDcA,E)

‘W) Table 38.3, Eq. 22

*r* = radus of pwl bang analyzed, f, “b = width Of aquifer. ft

+P *D = We,

§r 1 = length of aqwfei, ft

,A,Bj =distance between centers of Reservoirs A and 8. ft


TABLE 38.2-COMPARISON OF RESULTS OF METHODS 1 AND 2 FOR SAMPLE CALCULATION

QuaXer or Interval No

MZtLal Balance

(B/D) 500

1.100

663 616 599

2.476 2.550 2.615

2.449 2.646 2.828

663 672 616 630 599 614 8

9 3,100 652 2.672 3.000 652 664 IO 3,600 733 2.723 3.162 733 739

APf” Field (Psi) 55 136 318 478 581

PO AI, 210 rD =m 1.651 1.960 2.147 2.282 2.389

Z” fi

(psi/B/D) 1.000 1.414 2.732 2.000 2.236

Method 1 Mzi%d 2 4PW”

(Psi) (psi) 55 55 136 135 318 317 478 477 581 584

11 3,500 761 2.770 3.317 761 761 12 3,600 803 2.812 3.464 803 607 13 3,800 858 2.851 3.606 858 860 14 4,100 928 2.887 3.742 928 934 15 3,900 949 2.921 3.873 949 946

The procedure for both methods can be illustrated best by an application to a single-pool aquifer. Assume that a reservoir has produced for 15 quarters and that Cols. 2 and 3 in Table 38.2 are, respectively, the pressures at the end of each quarter and the average water-influx rates obtained by material balance for each quarter.

Example Problem 1. Method 1. From the following assumed best set of aquifer properties, check Table 38.1 for the substitution of d in Eq. 20.

c,,, = 5.5X10-’ psi-‘, /.i,,, = 0.6 cp,

h = 50 ft , 01 = 27~ radians, k = 76 md, q5 = 0.16,

r,, = 3,270 ft,

and the aquifer geometry is infinite radial. Calculate a convenient value (to minimize interpolation)

of dimensionless time interval (AZ,) for the quarterly interval (Ar=91.25 days) by varying the permeability (if necessary) in Eq. 20. In this case, AID = 10, correspond- ing to k=91 md, was selected. A check of Table 38.1 shows that pi is to be obtained from Table 38.3 (also tabulated in Table 38.2, Col. 4).

m APS,, ?I= ), . (28)

where Ape is the known field pressure drop at original woe.

Calculate ApD as a function of interval number. Then calculate m, as a function of interval number using Eq. 28 and plot m, as a function of n (Curve 1, Fig. 38.5). Fig. 38.6 shows an example of the calculation procedure for n=5 using equal time intervals.

If the AZD selected is the correct value, m, as a function of n will be constant. Variations from a constant can result from (1) incorrect AtD, (2) production and pressure errors, (3) incorrect aquifer size or shape, or (4) aquifer inhomogeneities. An examination of the m, plot will aid in the analysis of the cause.

r~=3(A’i,Af~)“.30’ . . . .(30)

for NirAtD 63.4, where N;, is the time interval number where m, vs. n increases from a constant value.

In this example, m,. increased with n (Fig. 38.5. AtD = 10). Therefore, AtD was decreased from 10 to 1 (large changes are recommended) and m, for At, = 1 was calculated (Curve 2). Now m, is constant until about In- terval 9 and then increases, indicating the possibility of a finite-closed aquifer. Using Ni, =9 and AtD = I in Eq. 29 gives a first approximation of 7 (rounded from 7.2) for rD. The m,. calculated for AtD = 1 and rD =7 is rem duced after Interval 9 (Curve 3) but is still too high and therefore indicates that the aquifer is still too large. An rg of 6 is taken for the next approximation, and this results in a constant value of m, (Curve 4). This shows that the past field behavior (Col. 3, Table 38.2) can be dupli- cated by assuming a finite-closed aquifer where AtD = 1 and rD=6 (Col. 6, Table 38.2). Because these aquifer properties gave the best match to the past field performance, they should be taken as the best set for predicting the future performance.

Value of m, Possible Remedy

increase with II decrease with At, decrease with n increase AtD constant, then increasing finite-closed aquifer constant, then decreasing finite-outcropping aquifer

For a finite-closed aquifer or finite-outcropping aquifer, Eq. 29 or 30 is used to find rD.

rD=2.3(NilAtD)0.518 . . . . . . (29)

for N;,At, ~3.4, and

38-6 PETROLEUM ENGINEERINGHANDBOOK

TABLE 38.3-DIMENSIONLESS WATER INFLUX AND DIMENSIONLESS PRESSURES FOR INFINITE RADIAL AQUIFERS w eD

0.112 0.278 0.404 0.520 0.606

2.5x 10 -' 3.0x10-' 4.0x10-' 50x10- 6.0x 10 -'

7.0x10-' 8.0x10-' 9.0x10-' 1.0 1.5

0.689 0.758 0.898 1.020 1.140

1.251 1.359 1.469 1.570 2.032

2.0 2.5 3.0 4.0 5.0

2.442 2.838 3.209 3.897 4.541

6.0 5.148 7.0 5.749 8.0 6314 9.0 6.661 1.0x10' 7417

1.5x10' 2.0x10' 2.5x10' 3.0x IO' 4.0x10'

9.965 1.229x10' 1.960 40~10~ 1.455x10' 2.067 5.0~10~ 1.681~10' 2.147 6.0~10~ 9.113x104 60x10' 5.368~10' 2.088~10' 2.282 7.0~10" 10.51 x105 7.0~10~ 6.220~10'

5.0x10' 6.0x IO' 7.0x10' 8.0x10' 9.0x10'

1.0x10* 1.5x10* 2.0x 102 2.5x102 3.0x10'

4.0x10* 5.0x10* 6.0x 10' 7.0x 102 80x102 9.0x10'

2.482~10' 2.388 8.0~10~ 11.89 x105 8.0~10' 2.860x10' 2.476 9.0x10" 13.26 x105 9.0x10" 3.228~10' 2.550 1.0~10" 14.62 x105 1.0~10'~ 3599x10' 2.615 1.5~10" 2.126~10~ 1.5~10'~ 3.942x 10' 2.672 2.0 x lo6 2.781x lo5 2.0~10'~

4.301x10' 2.723 2.5 x IO6 3.427x lo5 2.5x 10" 5.980x10' 2.921 3.0 x106 4.064x lo5 3.0x IO" 7.586~10' 3.064 4.0 x lo6 5.313x105 4.0x10'" 9.120x10' 3.173 5.0x lo6 6.544~10~ 5.0~10'" 10.58 x10' 3.263 6.0 x IO" 7.761 x IO5 6.0~10'"

13.48 x10' 3.406 7.0 x106 8.965x10' 7.0~10" 16.24 x10' 3.516 8.0~10" 10.16 x106 8.0~10'" 18.97 x10' 3.608 9.0x106 11.34 x106 9.Ox1O'o 21.60 x.10' 3.684 1.0x10' 12.52 x106 1.0~10" 24.23 x10' 3.750 26.77 x10' 3.809

PO t, ~___ 0.112 1.5x103 0.229 2.0 x 103 0.315 2.5x IO3 0.376 3.0 x 103 0.424 4.0 x IO3

0.469 5.0 x103 0.503 6.0 x IO3 0.564 7.0 x lo3 0.616 8.0 x103 0.659 9.0 x lo3

0.702 1.0 x lo4 0.735 1.5 x lo4 0.772 2.0 x lo4 0.802 2.5 x lo4 0.927 3.0 x IO4

1.020 4.0x10" 1.101 5.0x104 1.169 6.0 x lo4 1.275 7.0~10~ 1.362 8.0x lo4

1.436 9.0 x IO4 1.500 l.OxlO~ 1.556 1.5~10~ 1.604 2.0~10~ 1.651 25~10~

1.829 3.0x105

W c?D t D 4.136x10' 1.5~10~ 5.315x10" 2.0x107 6.466x IO2 2.5x10' 7.590x10' 3.0x107 9.757x10' 4.0x107

W eD

1.828~10~ 2.398x106 2.961~10~ 3.517x106 4.610~10"

to 1.5x 10" 2.0x IO" 2.5x 10" 3.0x 10" 4.0x 10"

W

1.17xs100'" 1.55x 1o'O 1.92x10'" 229x1o'o 3.02~10"

11.88 x103 5.0x107 5689x10' 5.0x10" 3.75xlO'O 13.95 x103 6.0~10~ 6.758~10~ 6.0x IO" 4.47x 10" 15.99 x103 7.0x107 7.816~10~ 7.0x IO" 5.19x IO'O 18.00 x103 8.0~10~ 8.866x10e 8.0x IO" 5.89x 10'0 19.99 x 103 9.0x107 9.911xlO~ 9.0x IO" 6.58~10'~

21.96 x102 1.0~10~ 3.146~10~ 1.5~10' 4.679x103 2.0~10~ 4.991 x103 2.5~10' 5.891 x IO3 3.0x 10'

10.95 x 106 1.0~10'~ 7.28x IO" 1.604x 10' 1.5x10" 1.08x10" 2.108x 10' 2.0~10'~ 1.42~10" 2.607~10' 3.100x10'

7.634~10~ 4.0x10* 4.071x10' 9.342x103 50x108 5.032~10~

11.03 x104 6.0~10" 5.984x10' 12.69 x104 7.0~10' 6.928x10' 14.33 x104 8.0~10' 7.865~10'

15.95 x104 9.0x10* 17.56 x104 1.0~10~ 2.538~10~ 1.5~10' 3.308x104 2.0x10" 4.066x IO4 2.5~10"

8.797x10' 9.725x10' 1.429x10n

4.817~10~ 3.0~10" 2.771~10' 6.267~10~ 40x10' 3.645~10' 7.699x IO4 5.0~10~ 4.510x108

l.OxlOJ 29.31 x10' 3.860

7.066~10' 7.909x 108 8.747x10B 1.288~10" 1.697x10"

2.103~10~ 2.505~10~ 3.299x10" 4.087~10" 4.868~10~

5.643~10" 6.414~10~ 7.183~10~ 7.948x10'

If an infinite aquifer had been indicated, it may be de- sirable in some cases to predict the future performance assuming first an infinite aquifer and then a finite-closed aquifer having a calculated rg based on the best estimate of AtD and setting N;, equal to the last interval number in Eq. 20 or 30.

Note that, in general. the plot of m,. will not be a smooth plot because of errors in basic data. The first few values are particularly sensitive to errors and generally may be ignored.

If it is possible to obtain a relatively constant value of v?,., check the production and pressure data for errors. If the production and pressure data are correct, try Method

2. If it appears that the production and/or pressure data may be in error, refer to the following discussion of Errors in Basic Data.

Example Problem 2. Method 2. This method is based on the following principles: (I) the slope of Z (m, times J>I)) as a function of time is always positive and never increases; (2) a constant slope of Z vs. time indicates a finite aquifer (see Eqs. 25 and 26) and therefore the extrapolated slope is constant; and (3) a constant slope of Z vs. log time indicates an infinite radial aquifer (Eq. 22). Extrapolation of this constant slope continues to simulate an infinite aquifer.


0.18

0.1 6

0.14

E 3.12

0.10

0.08

0.06 3 5 7 9 II 13 I5

TIME INTERVAL YUMBER

Fig. 38.5-Estimation of m,, N,, and roP for data in Table 38.2 (Method 1).

As in the first procedure, time is divided into equal intervals. The first approximation to 2 can be obtained as in Method 1 or by arbitrarily using the square root of the interval number (Col. 5, Table 38.2, and Trial 1, Fig. 38.7). A fitting factor m is calculated as a function of time for Trial 1 in exactly the same manner used to calculate M r in Method 1.

APf,, mn= n

“.“““““.‘.

(31)

c e,,,,+,m,,AZ, j=l

However, instead of m being plotted, m is used to calculate the next approximation of Z by use of Eq. 32.

New Z, =m,(old Z,,). . . .(32)

The new values of Z are plotted as a function of n (Tri- al 2, Fig. 38.7), and a smooth curve is drawn through the points, making certain the slope is positive and never increases (Principle 1). This procedure is repeated with values of 2 from this smoothed curve until the fitting factors are relatively constant and equal to 1 (Trial 3, Fig. 38.7).

The final 2 curve then is extrapolated to calculate the future performance as follows.

1. If the final slope of Z as a function of time is constant, extrapolate Z at a constant slope (Principle 2).

2. If the final slope is not constant as a function of time but is constant as a function of log time, first assume that the aquifer is an infinite radial system and will continue to behave as such (Principle 3) and extrapolate Z as a straight line as a function of log time; then assume that the aquifer is immediately bounded and extrapolate Z as a straight line on a linear plot of time using the last known slope (Principle 2).

3. If the final slope is not constant for either time or log time, extrapolate Z as a straight line using half the last known slope.

l-l e “15

e t %+I-,

e, *p, 5

ew AP 4 D2

i

e -3 ApD

e *2

AP D4

e II

*P %I

1 Apo I

=6 108.7

= 1050.6

= 467.5

= 148.5

= 53.5

n=5 581

m =--0074 r5 7828.8 .

u .087 I= 7828.8

Fig. 38.6-Sample pressure-drop calculation

Fig. 38.7 shows that three trials were needed to obtain a constant value of 1 for m. Col. 7, Table 38.2, shows that the final Z’s will duplicate the past pressure performance and therefore may be used to predict the future performance. Because Z becomes a straight line as a function of n, a finite-closed aquifer is indicated (Principle 2). Therefore, Z can be extrapolated as a straight line to calculate the future performance.

Errors in Basic Data. Good results were obtained for both methods, since accurate water influx and pressure data were used. In many cases a solution for m, and Ape in Method 1 or Z in Method 2 is impossible because of errors in basic data. In these cases the errors may be eliminated by smoothing the basic data or may be adjusted somewhat by using Eqs. 33 and 34.5

m, -m 6Apf,, = -0. l- Apf,, . (33)

m,

“0 2 4 6 8 IO 12 14 ” n

Fig. 38.7-Estimation of Z for data in Table 38.2 (Method 2).


0.1

0.06

0.04

EL 0.02

0.0 I

0.006

TIME ( QUARTERS 1

Fig. 38.8-Estimation of mF and F function for approximate water drive analysis of data in Table 38.2.

and

--!---&e AZ I j=2

n,i,i+,-, , AZ,, . . . .(34)

where

@f” = correction to Apf,, ,

6e% = correction to eM? , and n ti = average value of m.

In applying Eqs. 33 and 34 to Method 1, replace m by m, and AZ by ApD. Note that, since Eqs. 33 and 34 im- ply that the last values of Z (or APO) are reasonably correct, some judgment must be exercised when making these adjustments.

Approximate Methods. If the water influx rate is constant for a sufficiently long period of time, the following equations can be used to estimate water drive behavior roughly.

A P w, ,, =mFervr,,F . . . (35)

and

W 1

e,,,m,l,=- s ‘2 4M.r

- . . . . . . . . . . . . . . .

mF, I F ’ (36)

where F is an approximation to pD and a function of the type of aquifer and m,G is a proportionality factor. See Table 38.4 for function and aquifer type.

TABLE 38.4-WATER DRIVE BEHAVIOR EQUATIONS

Type Aquifer Basis

Infinite radial lo t ;

Eq. 22 Infinite hear Li Eq. 21 Finite outcropping L Eq. 23 Finite closed t Eq. 25 or 26

The equations for the infinite-radial and finite- outcropping aquifers are commonly referred to in the liter- ature as the “simplified Hurst” and “Schilthuis”6 water drive equations.

The procedure consists of calculating mF for the past history using Eq. 35 or 36, plotting mF as a function of time, and extrapolating m,V to predict the future water drive performance. Since the method assumes a constant water influx rate, the use of these equations should be limited to short-term rough approximations of future water drive behavior. Large errors may be obtained if the method is used to predict the behavior for large changes in reservoir withdrawal rates.

Fig. 38.8 shows a comparison of mF as a function of time for various values of F and the data in Table 38.2. These curves seem indicative of either an infinite linear or radial aquifer (the curves for these assumptions more nearly approach a constant value), whereas the more rigorous analyses indicated a finite aquifer. The selection of the best curve to use in predicting the future performance is difficult because of the fluctuations in the curves caused by variations in water influx rates. Note that this difficulty would be compounded if there were errors in the production and pressure data.

Fetkovitch’ presented a simplified approach that is based on the concept of a “stabilized” or pseudosteady- state aquifer productivity index and an aquifer material balance relating average aquifer pressure to cumulative water influx. This method is best suited for smaller aquifers, which may approach a pseudosteady condition quick- ly and in which the aquifer geometry and physical properties are known.

In a manner similar to single-well performance, the rate of water influx is expressed by Eq. 37.

ew,=Ja(Pa -p,), . . . . . . . . . . (37)

where e wp = water influx rate, B/D, J, = aquifer productivity index, B/D-psi, p, = average aquifer pressure, psi, and P W’ = pressure at the original WOC, psi.

Combining Eq. 37 with a material-balance equation for the aquifer, the increment of influx over a time interval t,, -t,- 1 is given by Eq. 38.

Aw = wet[Pa(n-j) -p wn [l -,(-J,*‘,)‘((,,V,,)] e

Pd

. . . . . . . . ..~......_...._.___ (38)


where WC,, = ~C..,P,,, total aquifer expansion capacity,

bbl, IJ’,~,; = initial water volume in the aquifer, bbl, PO1 = initial aquifer pressure, psi, and c ,I’, = total aquifer compressibility, psi -1 .

~~~~,~,,=p~j[l-~], .t..., . (39)

7.08x 10 -’ kh Jo = . . ~,,,(ln rD-0,75) (40)

for a closed radial system, and

Jo = 3(1.127x IO-‘)kbh

(41) P J

for a closed linear system.

Original Oil in Place (OOIP) Occasionally. it may be necessary to estimate the OOIP and to make a water drive analysis simultaneously. In general. the methods available are very sensitive to errors in basic data so that it is necessary to have a large amount of accurate data. Also, since the expansion of the reservoir above the bubblepoint is relatively small, generally only the data obtained after the reservoir has passed through the bubblepoint will be significant in defining the OOIP. In the three methods to be discussed, the aquifer will be assumed to be infinite and radial.

Brownscombe-Collins Method. This method’ assumes that the OOIP and the aquifer permeability are unknown and that the reservoir and aquifer properties other than permeability are known.

The pressure performance and the variance are calculated using Eqs. 7 and 42 for a given assumed aquifer permeability and various estimates. The minimum variance from a plot of variance vs. OOIP (Fig. 38.9) will be the best estimate of OOIP for the selected permeability.

c2=i -$ (AP.~, -a~,,.). (42) /

This procedure is repeated for various estimates of permeability until it is possible to obtain a minimum of the minimums. The permeability and the OOIP associated with this minimum should be the best estimates for the assumptions made.

It is possible to calculate the best estimate of OOIP for each selected permeability by the following procedure. Using the best available estimate of OOIP. calculate the reservoir voidage and expansion rates as a function of time. Select an aquifer permeability and use these rates in place of the water influx rates in Eq. 6 to calculate pressure drops Ap, ,, and APE,, The estimated OOIP mul-

RESERVES IN)

Fig. 38.9-Estimation of reservoir volume and water drive (Brownscombe-Collins method).

tiplied by the factor X calculated by Eq. 43 gives the best estimate of OOIP for the selected permeability. Eq. 44 gives the minimum variance for this permeability.

-*of, WPE,

x=“- n . . (43)

c (APE,)~ j=l

and

.d i W~+P~,-XA~~,)~, . . . . n j=1

where A~,z = total pressure drop at original WOC (field

data), psi, Ap, = total pressure drop at WOC (calculated

using reservoir voidage rates), psi, and ApE = total pressure drop at WOC (calculated

using reservoir expansion rates). psi.

van Everdingen, Timmerman, and McMahon Method. This method9 assumes that the OOIP, aquifer conduc- tivity k/m/p, and diffusivity kI(@pc) are unknown. Com- bination of the material-balance equation and Eq. 8 and solving for the OOIP yields Eq. 4.5.

N=A +m/,F(t), . . . . . (45)

where

1 A= U’,JvB, +N,,(R,, -R,)& + w,,l.

V’V- 1P,;

. . . . . . . . . . . . . . . . . . . . . . .._... (46)

1 II F(t) =

[ C *PC,,+ 1-j) Wa/, , (47)

CFVmllBoi j=I 1 F”=Ph-P -+I, . . . . . . . . . . . . . . . . . . . . . . . . ..(48)

PY

38-l 0 PETROLEUM ENGINEERING HANDBOOK

TABLE 36.5-DIMENSIONLESS WATER INFLUX FOR FINITE OUTCROPPING RADIAL AQUIFERS

To =I.5 70 =2.0 rD =2.5 fD =3.0 rD =3.5 rD =4.0 rD =4.5

t, weD to W eD t, W eD t, weD tD weD tD weD tD w,D --~~ 5.0x 10 -? 0.276 5.0~10~' 0.278 1.0x10-' 0.408 3.0x10m 6.0x10-* 0.304 7.5x10-" 0.345 1.5x10-' 0.509 4.0x10- 7.0x10-2 0.330 1.0x10-' 0.404 2.0x10-' 0.599 5.0x10 - 8.0x10-' 0.354 1.25x10-' 0.458 2.5x10-' 0.681 6.0x10 - 9.0x10m2 0.375 1.50x10-' 0507 3.0x10-' 0.758 7.0x10 -

1.0x10-' 0.395 1.75x10-' 0.553 3.5x10-' 0.829 8.0~10~

0.755 1.00 1.571 2.00 2.442 2.5 0.895 1.20 1.761 2.20 2.598 3.0 1.023 1.40 1.940 2.40 2.748 3.5 1.143 1.60 2.111 2.60 2.893 4.0 1.256 1.60 2.273 2.80 3.034 4.5

1.363 2.00 2.427 3.00 3.170 5.0 0-l 0.897 9.0x10 -' 1.465 2.20 2.574 3.25 3.334 5.5 11x10~' 0.414 2.00x10-' 0597 4.0x

1.2x10-' 0.431 2.25x10-l 0.638 4.5x 1.3x10-' 0.446 2.50~10 -' 0.678 5.0x 1.4x10-' 0.461 2.75x10-l 0.715 5.5x

2.835 3.196 3.537 3.859 4.165

1.563 2.40 2.715 3.50 3.493 6.0 1.791 2.60 2 649 3.75 3.645 6.5 1.997 2.80 2.976 4.00 3.792 7.0

4.454 4.727 4.986 5.231 5.464

3.00 3.098 3.25 3.242 3.50 3.379 3.75 3.507 4.00 3.628

4.25 3.932 7.5 4.50 4.068 8.0 4.75 4.198 8.5 5.00 4.323 9.0 5.50 4.560 9.5

5.684 5.892 6.089 6.276 6.453

4.25 3.742 6.00 4.779 10 6.621 4.50 3.850 6.50 4.982 11 6.930 4.75 3.951 7.00 5.169 12 7.200 5.00 4.047 7.50 5.343 13 7.457 5.50 4.222 8.00 5.504 14 7.680

3.317 6.00 4.378 8.50 5.653 15 3.381 6.50 4.516 9.00 5.790 16 3.439 7.00 4.639 9.50 5.917 18 3.491 7.50 4.749 10 3.581 8.00 4.846 11

3.656 8.50 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

6.035 20 6.246 22

7.880 8.060 8.365 8.611 8.809

12 5.321 17 13 5.385 18 14 5.435 20 15 5.476 22 16 5.506 24 17 5531 26 18 5.551 30 20 5579 34 25 5.611 38 30 5621 42 35 5.624 46 40 5.625 50

6.425 24 6.580 26 6.712 28 6.825 30 6.922 34 7.004 38 7.076 42 7.189 46 7.272 50 7.332 60 7.377 70 7.434 80 7.464 90 7.481 100 7.490 7.494 7.497

8.968 9.097 9.200 9.283 9.404 9.481 9.532 9.565 9.586 9.612 9.621 9.623 9.624 9.625

10-l 0.962 1.00 0-l 1.024 1.25 0-l 1.083 1.50

0-l 1.140 1.75 0-l 1.195 2.00 0-l 1.248 2.25 0-l 1.229 2.50

1.5x10m' 0.474 3.00x 10 -' 0.751 6.0x 1.6x10-' 0.486 3.25x10-l 0.785 6.5x

2.184 2.353 2.507 2.646

1.7x10m1 0.497 3.50x10-' 0.817 7.0x 1.8~10~' 0.507 3.75x10-1 0.848 7.5x

2.772 1.9x10-' 0.517 4.00x10 -'

2.0x 10 -' 0.525 4.25 x 10 -' 2.1x10-' 0.533 4.50 x IO -' 2.2x10-l 0.541 4.75 x IO -' 2.3~10~' 0.548 5.00 x 10 -' 2.4x10-l 0.554 5.50x10-'

2.5~10.' 0.559 6.00x10 -' 2.6x10 -' 0.565 6.50x IO-' 2.8x 10 -' 0.574 7.00x10m' 3.0x 10 -' 0.582 7.50x10-' 3.2x 10 -' 0.588 8.00x10 -'

3.4x10-' 0.594 9.00x 10-l 3.6~10~' 0.599 1.00 3.8x10-' 0.603 1.1 4.0x10m' 0.606 1.2 4.5x10-' 0.613 1.3 5.0x10m' 0.617 1.4 6.0x10-' 0.621 1.6 7.0x10 -' 0.623 1.7 8.0x10-' 0.624 1.8

2.0 2.5 3.0 4.0 5.0

0.677 8.0x10 -' 1.348 2.75

1.395 3.00 1.440 3.25

0.905 0.932 0.958 0.982

8.5x10-' 9.0x10 -'

2.886 2.990 3.084 3.170

1.484 3.50 1.526 3.75

1.028

9.5x10m' 1.0 1.1 1.605 4.00 3.247

1.070 1.2 1.679 4.25 1.108 1.3 1.747 4.50 1.143 1.4 1.811 4.75 1.174 1.5 1.870 5.00 1.203 1.6 1.924 5.50

1.253 1.7 1.975 6.00 1.295 1.8 2.022 6.50 1.330 2.0 2.106 7.00 1.358 2.2 2.178 7.50 1.382 2.4 2.241 8.00 1.402 2.6 2.294 9.00 1.432 2.8 2.340 10.00 1.444 3.0 2.380 11.00 1.453 3.4 2.444 12.00 1.468 3.8 2.491 14.00 1.487 4.2 2.525 16.00 1.495 4.6 2.551 18.00

3.894 3.928 3.951 3.967 3.985 3.993 3.997 3.999 3.999 4.000

1499 5.0 1.500 6.0

7.0 8.0 9.0 10.0

2.570 20.00 2.599 22.00 2.613 24.00 2.619 2.622 2.624

w

4a

and

y= ph-p P(FV-,). ~.~..............,._.._,,,,

FV = ratio of volume of oil and its dissolved original gas at a given pressure to its volume at initial pressure,

N = OOIP. STB, N,, = cumulative oil produced, STB, W,] = cumulative water produced. bbl, R,, = cumulative produced GOR, scf/STB. B,, = oil FVF, bbl/STB, B,q = gas FVF. bbhscf, and p/1 = bubblepoint pressure. psia.

Generally, Y is calculated with laboratory-determined values of FV - 1. Because Y vs. p is generally a straight line, smoothed values of Ycan be calculated with Eq. 50:

Y=b+m, . . . (50)

here h= intercept and m =slope. The equations for obtaining the least-squares tit to Eqs.

6 and 47 for a given dimensionless time interval, At,. nd n data points are

II

nN= c A,-m, i F(t), .(51) j=l J=I


TABLE 38.5-DIMENSIONLESS WATER INFLUX FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)

r, = 5.0 rD = 6.0 rD = 7.0 rD =8.0 rD =9.0 r, =lO.O

to W ell tD W eD tD weD tD weD rD weD rD weD ___~ _-_

3.0 3.5 4.0 4.5 5.0

5.5 6.0 6.5 7.0 7.5

8.0 a.5 9.0 9.5 10

3.195 6.0 3.542 6.5 3.875 7.0 4.193 7.5 4.499 8.0

4.792 8.5 5.074 9.0 5.345 9.5 5.605 10.0 5.854 10.5

6.094 11 6.325 12 6.547 13 6.760 14 6.965 15

5.148 9.00 6.861 9 6.861 10 7.417 15 9.965 5.440 9.50 7.127 10 7.398 15 9.945 20 12.32 5.724 IO 7.389 11 7.920 20 12.26 22 13.22 6.002 11 7.902 12 a.431 22 13.13 24 14.09 6.273 12 6.397 13 8.930 24 13.98 26 14.95

6.537 13 a.876 14 9.418 26 14.79 28 15.78 6.795 14 9.341 15 9.895 28 15.59 30 16.59 7.047 15 9.791 16 10.361 30 16.35 32 17.38 7.293 16 10.23 17 10.82 32 17.10 34 18.16 7.533 17 10.65 18 11.26 34 17.82 36 18.91

7.767 18 11.06 19 11.70 36 18.52 38 19.65 8.220 19 11.46 20 12.13 38 19.19 40 20.37 8.651 20 11.85 22 12.95 40 19.85 42 21.07 9.063 22 12.58 24 13.74 42 20.48 44 21.76 9.456 24 13.27 26 14.50 44 21.09 46 22.42

11 7.350 16 9.829 26 13.92 28 15.23 46 21.69 48 23.07 12 7.706 17 10.19 28 14.53 30 15.92 48 22.26 50 23.71 13 8.035 18 10.53 30 15.11 34 17.22 50 22.82 52 24.33 14 8.339 19 10.85 35 16.39 38 18.41 52 23.36 54 24.94 15 8.620 20 11.16 40 1749 40 18.97 54 23.89 56 25.53

16 8.879 22 il.74 45 18.43 45 20.26 56 24.39 58 26.11 18 9.338 24 12.16 50 19.24 50 21.42 58 24.88 60 26.67 20 9.731 25 12.50 60 20.51 55 22.46 60 25.36 65 28.02 22 10.07 31 13.74 70 21 45 60 23.40 65 26.48 70 29.29 24 10.35 35 14.40 80 22.13 70 24.98 70 27.52 75 30.49

26 10.59 39 28 10.80 51 30 10.89 60 34 11.26 70 38 il.46 80

11.61 90 11.71 100 11.79 110 11.91 120 11.96 130

il.98 140 11.99 150 12.00 160 12.0 180

200

14.93 90 16.05 100 16.56 120 16.91 140 17.14 160

17.27 180 17.36 200 17.41 500 17.45 17.46

22.63 80 26.26 75 28.48 80 31.61 23.00 90 27.28 80 29.36 85 32.67 2347 100 28.11 a5 30.18 90 33.66 23.71 120 29.31 90 30.93 95 34.60 23.85 140 30.08 95 31.63 100 35.48

42 46 50 60 70

23.92 23.96 24.00

160 30.58 100 32.27 120 38.51 180 30.91 120 34.39 140 40.89 200 31.12 140 35.92 160 42.75 240 31.34 160 37.04 la0 44.21 280 31.43 180 37.85 200 45.36

80 90 100 120

17.48 17.49 17.49 17.50 17.50

320 31.47 360 31.49 400 31.50 500 31.50

200 38.44 240 46.95 240 39.17 280 47.94 280 39.56 320 48.54 320 39.77 360 48.91 360 39.88 400 49.14

220 17.50 400 39.94 440 39.97 480 39.98

440 49.28 480 49.36

and

J=i j=l J=f

The variance of this fit from field data can be calculated by Eq. 53.

02=1 i {A,,-N+m,[F(r)],}? n /=I

(53)

The minimum in a plot of variance vs. various assumed values of At, will be the best estimate of At, and can be used in Eqs. 51 and 52 to solve for the best estimate of N and m,, (see Fig. 38. IO).

I u I Id

BEST ESTIMATE

I OF At,

Ato Fig. 38.10-Estimation of reservoirvolume and waterdrive(van

Everdingen-Timmerman-McMahon method).


TABLE 38.6-DIMENSIONLESS PRESSURES FOR FINITE CLOSED RADIAL AQUIFERS

ID = 1.5 rD =2.0 r,=25 rD = 3.0 rD = 3.5

to PO tLl PO tD PO tD PO tD PO __- -~- 6.0x10-' 0.251 2.2x10-' 0.443 4.0x 10-l 0.565 5.2x10 0.627 1.0 0.802 8.0x10-' 0.288 2.4x10-l 0.459 4.2x10-l 0.576 5.4x 10 0.636 1.1 0.830 1.0x10-' 0.322 2.6x10-l 0.476 4.4x 10-l 0.587 5.6x10 0.645 1.2 0.857 1.2x10-' 0.355 2.8x10-l 0.492 4.6x 10-l 0.598 6.0x10 0.662 1.3 0.882 1.4x10-l 0.387 3.0x10-' 0.507 4.8% lo-' 0.608 6.5x10 0.683 1.4 0.906

1.6x10-' 0.420 3.2x10-l 0.522 5.0x lo-' 0.618 7.0x10 0.703 1.5 0.929 1.8x10-' 0.452 3.4x10-l 0.536 5.2x 10-l 0.682 75x10 0.721 1.6 0.951 2.0x10-' 0.484 3.6x10-l 0.551 5.4x 10 -' 0.638 8.0x 10 0.740 1.7 0.973 2.2x10-l 0.516 3.8x10-l 0.565 56x10-' 0.647 8.5x10 0.758 1.8 0.994 2.4x10 -' 0.548 4.0x10 -' 0.579 5.9x10-' 0.657 9.0x IO 0.776 1.9 1.014

2.6x10-l 0.580 4.2x10-l 0.593 6.0x 10-l 0.666 9.5x10 0.791 2.0 1.034 2.8x10 -' 0.612 4.4x10-' 0.607 6.5x 10-l 0.688 1.0 0.806 2.25 1.083 3.0x10 -' 0.644 4.6x10-l 0.621 7.0x 10-l 0.710 1.2 0.865 2.50 1.130 3.5x10 -' 0.724 4.8x IO-' 0.634 7.5x10-' 0.731 1.4 0.920 2.75 1.176 4.0x 10 -' 0.804 5.0x10-' 0.648 8.0x IO-' 0.752 1.6 0.973 3.0 1.221

4.5x10m' 0.884 6.0x IO -' 0.715 8.5x10-' 0.772 2.0 1.076 4.0 1.401 5.0x 10 -' 0.964 7.0x 10-l 0.782 9.0x10-' 0.792 3.0 1.328 5.0 1.579 5.5x10m' 1.044 8.0x10-' 0.849 9.5x 10-l 0.812 4.0 1.578 6.0 1.757 6.0x10-' 1.124 9.0x10-' 0.915 1.0 0.832 5.0 1.828

r. =4.0 rn =4.5

tD PO --

1.5 0.927 1.6 0.948 1.7 0.968 1.8 0.988 1.9 1.007

t, PD

2.0 1.023 2.1 1.040 2.2 1.056 2.3 1.072 2.4 1.087

2.0 1.025 2.5 1.102 2.2 1.059 2.6 1.116 2.4 1.092 2.7 1.130 2.6 1.123 2.8 1.144 2.8 1.154 2.9 1.158

3.0 1.184 3.0 1.171 3.5 1.255 3.2 1.197 4.0 1.324 3.1 1.222 4.5 1.392 3.6 1.246 5.0 1.460 3.8 1.269

5.5 1.527 4.0 1.292 6.0 1.594 4.5 1.349 6.5 1.660 5.0 1.403 7.0 1.727 5.5 1.457

1 .o 0.982 2.0 1.215

2.0 1.649 3.0 1.596 3.0 2.316 4.0 1.977 5.0 3.649 5.0 2.358

8.0 1.861 6.0 1.510

9.0 1.994 7.0 1.615 10.0 2.127 8.0 1.719

9.0 1.823 10.0 1.927 11.0 2.031

12.0 2.135 13.0 2.239 14.0 2.343 15.0 2.447

Havlena-Odeh Method. In this method, lo the material- balance equation is written as tire equation of a straight line containing two unknown constants, N and m,, Com- bination of the material-balance equation and Eq. 8 yields Eq. 54. (See Fig. 38.10.)

vR,, Nfm, c *PW I -;) WA,

j=i . . . . (54)

EN,, EN,,

where

E,tr Bf, I/ =B,-B, +p

’ I-S,,. (cf+Sw~w)(P; -P,,)

VR,, = cumulative voidage at the end of interval II, RB.

EN = cumulative expansion per stock-tank barrel OOIP. RB,

B, = two-phase FVF, bbl/STB. W,, = cumulative water produced, STB, Wi = cumulative water injected. STB. G, = cumulative gas injected. scf. B,, = water FVF, bbl/STB,

cf = formation compressibility, psi t , Cl, = formation water compressibility, psi t , s,,. = formation water saturation, fraction, and

m = fitting factor.

Eq. 54 is the equation of a straight line with a slope of mP and a y intercept of N.

Estimates of TD and Are are made and the appropriate values of W,D are obtained from Table 38.3 or 38.5, according to system geometry. The summation terms in Eq. 54 then may be calculated and a graph plotted, as shown in Fig. 38.11. If a straight line results, the values of mp and N are obtained from the slope and intercept of the resulting graph. An increasing slope indicates that the summation terms are too small, while a decreasing slope indicates that the summation terms are too large. The procedure is repeated, using different estimates of TD and/or Ato until a straight-line plot is obtained. It should be noted that more than one combination of i-o and AND may yield a reasonable straight line-i.e., a straight-line result does not necessarily determine a unique solution for N and mp.

Future Performance The future field performance must be obtained from a si- multaneous solution of the material-balance and water drive equations. If the reservoir is above saturation pressure, a direct solution is possible; however, if the reservoir is below saturation pressure, a trial-and-error procedure is necessary.


TABLE 3&G-DIMENSIONLESS PRESSURES FOR FINITE CLOSED RADIAL AQUIFERS (continued)

rD = 5.0 rD =6.0 rD =7.0 rD =8.0 rD = 9.0 rD = 10.0

t, PO ‘0 PO t, PO t, PO tD PD t, PO __~ 3.0 1.167 4.0 1.275 6.0 1.436 8.0 1.556 10.0 1.651 12.0 1.732 3.1 1.180 4.5 1.322 6.5 1.470 8.5 1.582 10.5 1.673 12.5 1.750 3.2 1.192 5.0 1.364 7.0 1.501 9.0 1.607 11.0 1.693 13.0 1.768 3.3 1.204 5.5 1.404 7.5 1.531 9.5 1.631 11.5 1.713 13.5 1.784 3.4 1.215 6.0 1.441 8.0 1.559 10.0 1.653 12.0 1.732 14.0 1.801

3.5 1.227 6.5 1.477 8.5 1.586 10.5 1.675 12.5 1.750 14.5 1.817 3.6 1.238 7.0 1.511 9.0 1.613 11.0 1.697 13.0 1.768 15.0 1.832 3.7 1.249 7.5 1.544 9.5 1.638 11.5 1.717 13.5 1.786 15.5 1.847 3.8 1.259 8.0 1.576 10.0 1.663 12.0 1.737 14.0 1.803 16.0 1.862 3.9 1.270 8.5 1.607 11.0 1.711 12.5 1.757 14.5 1.819 17.0 1.890

4.0 1.281 9.0 1.638 12.0 1.757 13.0 1.776 15.0 1.835 18.0 1.917 4.2 1.301 9.5 1.668 13.0 1.801 13.5 1.795 15.5 1.851 19.0 1.943 4.4 1.321 10.0 1.698 14.0 1.845 14.0 1.813 16.0 1.867 20.0 1.968 4.6 1.340 11.0 1.757 15.0 1.888 14.5 1.831 17.0 1.897 22.0 2.017 4.8 1.380 12.0 1.815 16.0 1.931 15.0 1.849 18.0 1.926 24.0 2.063

5.0 1.378 13.0 1.873 170 1.974 17.0 1.919 19.0 1.955 26.0 2.108 5.5 1.424 14.0 1.931 18.0 2.016 19.0 1.986 20.0 1.983 28.0 2.151 6.0 1.469 15.0 1.988 19 0 2.058 21.0 2051 22.0 2.037 30.0 2.194 6.5 1.513 16.0 2.045 20.0 2.100 23.0 2.116 24.0 2.090 32.0 2.236 7.0 1.556 17.0 2.103 22.0 2.184 25.0 2.180 26.0 2.142 34.0 2.278

7.5 1.598 18.0 2.160 24.0 2.267 30.0 2.340 28.0 2.193 36.0 2.319 8.0 1.641 19.0 2.217 26.0 2.351 35.0 2.499 30.0 2.244 38.0 2.360 9.0 1.725 20.0 2.274 28.0 2.434 40.0 2.658 34.0 2.345 40.0 2.401

10.0 1.808 25.0 2.560 30.0 2.517 45.0 2.817 38.0 2.446 50.0 2.604 11.0 1.892 30.0 2.846 40.0 2.496 60.0 2.806

12.0 1.975 13.0 2.059 14.0 2.142 15.0 2.225

45.0 2.621 70.0 3.008 50.0 2.746

There are several methods of solution because there are several possible combinations of the various material- balance and water drive equations. However, only one combination will be used to illustrate the general application to (1) a reservoir above the bubblepoint pressure, and (2) a reservoir below the bubblepoint pressure. In either case, it will be necessary to know (1) the saturations behind the front from laboratory core data or other sources, (2) the water production as a function of frontal advance, and (3) the pressure gradient in the flooded portion of the reservoir.

Pressure Gradient Between New and Original Front Positions. Eq. 55 shows that the difference between the average reservoir pressure and the pressure at the original WOC is a function of water-influx rate, aquifer fluid and formation properties, and aquifer geometry.

where FG is the reservoir geometry factor. The linear frontal advance is given by

FG= L.f 0.001127hb

and the radial frontal

FG= 27r In@, irf)

0.00708ha :

.,_...,.....,..........I (56)

advance is given by

.____.____............ (-57)

,’

00 0

1 AP%, e EN

Fig. 38.11-Estimation of OOIP and mp.


TABLE 38.7- DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS

r,=1.5 rD =2.0 r. =2.5 rD =3.0 rD =3.5 rD = 4.0

to PD to PD t, PD ‘D PO tD PO tD PO

5.0x10-' 0.230 2.0~10~' 0.424 3.0x10-' 0.502 5.0~10~' 0.617 5.0x 10 -' 0.620 1.0 0.802 5.5x10-2 0.240 2.2x10-l 0.441 3.5x10-' 0.535 5.5~10~' 0.640 6.0x10-' 0.665 1.2 0.857 6.0x10-' 0.249 2.4~10~' 0.457 4.0~10~' 0.564 6.0~10~' 0.662 7.0x10-' 0.705 1.4 0.905 7.0x10 -2 0.266 2.6x10-l 0.472 4.5~10~' 0.591 7.0x10m' 0.702 8.0x10 -' 0.741 1.6 0.947 8.0x10-' 0.282 2.8x10-' 0.485 5.0x10-' 0.616 8.0x10-' 0.738 9.0x10-' 0.774 1.8 0.986

9.0x10-' 0.292 3.0~10~' 0.498 5.5x10-l 0.638 9.0x10m' 0.770 1.0 0.804 2.0 1.020 1.0x 10-l 0307 3.5~10~' 0.527 6.0~10~' 0.659 1.0 0.799 1.2 0.858 2.2 1.052 1.2x10-' 1.4x10-' 1.6x10-'

1.8x10-' 2.0x10m' 2.2x10-l 2.4x10-' 2.6~10~'

2.8~10~' 3.0x10-' 3.5x10m' 4.0x10-' 4.5x10-'

5.0x10 -' 6.0x IO-' 7.0x lo- 8.0x10-'

0.328 4.0x IO-' 0.552 7.0x 10-l 0.344 4.5x10-l 0.573 8.0x10-' 0.356 5.0~10~' 0.591 9.0x10-'

0367 5.5x10-l 0.606 1.0 0.375 6.0x10-' 0.619 1.2 0381 6.5~10~' 0.630 1.4 0.386 7.0~10~' 0.639 1.6 0390 7.5x10-' 0.647 1.8

0.393 8.0x10-' 0.654 2.0 0.396 8.5x 10-l 0.660 2.2 0.400 9.0x IO-' 0.665 2.4 0.402 9.5x10-' 0.669 2.6 0.404 1.0 0.673 2.6

0.405 1.2 0.682 3.0 0.405 1.4 0.688 3.5 0.405 1.6 0.690 4.0 0.405 1.8 0.692 4.5

2.0 0.692 5.0

2.5 0.693 5.5 3.0 0.693 6.0

0.696 1.2 0.850 1.4 0.904 2.4 1.080 7.5 1.516 0.728 1.4 0.892 1.6 0.945 2.6 1.106 8.0 1.539 0.755 1.6 0.927 1.8 0.981 2.8 1.130 8.5 1.561

0778 1.8 0.955 2.0 1.013 3.0 1.152 9.0 1580 0.815 2.0 0.980 2.2 1.041 3.4 1.190 10.0 1.615 0.842 2.2 1.000 2.4 1.065 3.8 1.222 12.0 1.667 0.861 2.4 1.016 2.6 1.087 4.5 1.266 14.0 1.704 0.876 2.6 1.030 2.8 1.106 5.0 1.290 16.0 1730

0.887 2.6 1.042 3.0 1.123 5.5 1.309 18.0 1.749 0.895 3.0 1.051 3.5 1.158 6.0 1.325 20.0 1.762 0.900 3.5 1.069 4.0 1.183 7.0 1.347 22.0 1.771 0.905 4.0 1.080 5.0 1.215 8.0 1.361 24.0 1.777 0.908 4.5 1.087 6.0 1.232 9.0 1.370 26.0 1.781

0.910 5.0 1.091 7.0 1.242 10.0 1.376 28.0 1.784 0.913 5.5 1.094 8.0 1.247 12.0 1.382 30.0 1.787 0.915 6.0 1.096 9.0 1.240 14.0 1.385 35.0 1.789 0.916 6.5 1.097 10.0 1.251 16.0 1.386 40.0 1.791 0.916 7.0 1.097 12.0 1.252 18.0 1.386 50.0 1.792

0.916 8.0 1.098 14.0 1.253 0.916 10.0 1.099 16.0 1.253

rD =6.0

tD PO ~___ 4.0 1.275 4.5 1.320 5.0 1.361 5.5 1.398 6.0 1.432

6.5 1.462 7.0 1.490

where Lf = linear penetration of water front into

reservoir, ft, rf = radius to water front after penetration. ft,

and (Y = angle subtended by reservoir, radians.

Note that FG is a function of distance traveled by the front so that, if the pressure gradients between the reservoir and the original reservoir boundary are known for the past history, F, may be evaluated as a function of frontal advance. Future values of FG then can be obtained by extrapolating FG as a function of frontal advance on some convenient plot (linear, semilog, etc.)

Reservoir Above Bubblepoint Pressure. Above the bubblepoint pressure the total compressibility can be assumed to be constant; so the material-balance equation

APO,, = (qr,, -e,,,8 W

+Apo,,,- ,/, . . vl7co,

(58)

where *P,,,, = total reservoir pressure drop from initial

pressure at end of interval n,

q,,, = total production rate, RB/D, V,, = total reservoir PV, bbl, and c 0, = total reservoir compressibility, psi - ’ ,

can be combined with Eqs. 6 and 5.5 and solved for the water-influx rate:

e w,, =

*P (,,,, ,) +(*tqr,r/V,+-,,,)-mr 2 oil ,,,,, ,,*PD, ., = 2

%*PD, +(*tlv,,~,,,)+(ll.,,.F,B/~I, 1

. . . . . . . . . . . . . . . . . (59)

The calculated water-influx rate now can be used in Eq. 58 to calculate Ap(,,, and the whole procedure is repeated for the next time interval. If Eq. 27 is used instead of Eq. 6, mr= 1 and ApD is replaced by AZ in Eq. 59.

Reservoir Below Bubblepoint Pressure. To simplify the calculation procedure, it was assumed that (1) uniform saturations exist ahead of and behind the front, (2) the saturations do not change as any portion of the reservoir is bypassed, and (3) the changes in pressure are selected small enough that the changes in oil FVF’s are very small. Fig. 38.12 shows the saturation changes as the front ad- vances into the unflooded reservoir volume I/,- 1 during time interval n.

The following equations will be used in this method. Water influx rate:

II

.I -

” (60)

m,ApD, -(p,,,.FGlk,,.)


TABLE 38.7- DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)

ID =8.0 r,=lO ,,=I5

to PO tD PO tD PO 7.0 1.499 10.0 1.651 20.0 1.960 7.5 1.527 12.0 1.730 22.0 2.003 8.0 1.554 14.0 1.798 24.0 2.043 8.5 1.580 16.0 1.856 26.0 2.080 9.0 1.604 16.0 1.907 28.0 2.114

9.5 1.627 20.0 1.952 30.0 2.146 10.0 1.648 25.0 2.043 35.0 2.218 12.0 1.724 30.0 2.1 I1 40.0 2.279 14.0 1.786 35.0 2.160 45.0 2.332 16.0 1.837 40.0 2.197 50.0 2.379

18.0 1.879 45.0 2.224 60.0 2.455 20.0 1.914 50.0 2.245 700 2.513 22.0 1.943 55.0 2.260 800 2.558 24.0 1.967 60.0 2.271 90.0 2.592 26.0 1.986 65.0 2.279 10.0x10 2.619

28.0 2.002 70.0 2.285 12.0~10 2.655 30.0 2.016 75.0 2.290 14.0x10 2.677 35.0 2.040 80.0 2.293 160x10 2.689 40.0 2.055 90.0 2.297 18.0~10 2.697 45.0 2.064 10.0~10 2.300 200x10 2.701

50.0 2.070 11.0x10 2.301 22.0x10 2.704 60.0 2 076 12.0x 10 2.302 24.0x10 2.706 70.0 2.078 13.0x10 2.302 26.0~10 2.707 80.0 2 079 14.0x10 2.302 28.0x10 2.707

16.0x 10 2.303 30.0x10 2.708

r,=20 r,=25 r,=30 r,=40

tD PO tD PO to PO to PO 300 2.148 50.0 2.389 70.0 2.551 12.0x IO 2.813 35.0 2.219 55.0 2.434 80.0 2.615 14.0~10 2.888 40.0 2.282 60.0 3.476 90.0 2.672 16.0~10 2.953 45.0 2.338 65.0 2.514 10.0x10 2.723 18.0~10 3.011 50.0 2.388 70.0 2.550 12.0x 10 2.812 20.0x10 3.063

60.0 2.475 75.0 2.583 14.0x10 2.886 22.0x 10 3.109 70.0 2.547 80.0 2.614 16.0x 10 2.950 24.0x 10 3.152 80.0 2.609 85.0 2.643 16.5x 10 2.965 26.0x 10 3.191 90.0 2.658 90.0 2.671 17.0x 10 2.979 28.0x 10 3.226 10.0x10 2.707 95.0 2.697 17.5x10 2.992 30.0x 10 3.259

10.5x10 2.728 10.0x10 2.721 18.0x10 3.006 35.0x 10 3.331 11.0x10 2.747 12.0x10 2.807 20.0x10 3.054 40.0x 10 3.391 11.5x10 2.764 14.0~10 2.878 25.0~10 3.150 45.0x10 3.440 12.0~10 2.781 16.0x10 2.936 30.0x10 3.219 50.0x10 3.482 12.5x10 2.796 18.0~10 2.984 35.0x10 3.269 55.0x10 3.516

13.0x10 2.810 20.0x10 3.024 40.0x10 3.306 60.0x 10 3.545 13.5x10 2.823 22.0x10 3.057 45.0~10 3.332 65.0x 10 3.568 14.0~10 2.835 24.0~10 3.085 50.0x10 3.351 70.0x10 3.588 14.5x10 2.846 26.0x10 3.107 60.0x10 3.375 80.0x 10 3.619 15.0~10 2.857 28.0~10 3.126 70.0x10 3.387 90.0x10 3.640

16.0~10 2.876 30.0x10 3.142 80.0~10 3.394 10.0x10' 3.655 180x10 2.906 35.0~10 3.171 90.0x10 3.397 12.0x10' 3.672 200x10 2.929 40.0x10 3.189 10.0x10* 3.399 14.0x10~ 3.681 240x10 2.958 45.0~10 3.200 12.0~10' 3.401 16.0x10* 3.685 28.0x10 2.975 50.0x10 3.207 14.0~10' 3.401 18.0x10* 3.687

30 0x10 2.980 60.0x 10 3.214 20.0 x 10' 3.688 40.0~10 2.992 70.0x10 3.217 25.0x 10' 3.689 50.0x10 2.995 80.0x 10 3.218

90.0 x10 3.219

T

L

Flooded and unflooded volumes:

Al’,, = (e I\.,, - 4 it ,, W,,

f~(I-sj,,.-sor-s~,) ,,-, “.‘..“’

and

V,,=V,,-, -AL’,. . .

Oil saturation in V,:

+ ~RAV, [So,,vm,, -S,,,, I

B -q,,,At, . .

C’,,

Gas production:

aGPft = vrz[s,,,t ,, -s,,? 1 B h’w,,

+ fRAv&,,, I, -‘<v,> 1 +q At jj B

II ,, !I .’ 8, K I,

(61)

(63)

(64)

S On-l

% n-l Siw

s On-l Siw

S gn-I

'

S S On Orn

r s4'" %

:

S wn Sii

(b)

Fig. 38.12-Saturation change with frontal advance.


TABLE 38.7-DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)

r,=50 rD =60 r,=70

t D PD t, PO to PO 20.0x 10 3.064 3.0 x 10' 3.257 5.0x10" 3.512 22.0x 10 3.111 4.0x10' 3.401 6.0 x 10’ 3.603 24.0 x10 3.154 5.0x IO2 3.512 7.0x10' 3.680 26.0x10 3.193 6.0 x IO* 3.602 8.0~10' 3.746 28.0 x10 3.229 7.0 x10* 3.676 9.0x 10' 3.803

30.0 x10 3.263 8.0 x lo* 3.739 10.0~10' 3.854 35.0x10 3.339 9.0 x 102 3.792 12.0x 102 3.937 40.0x10 3.405 10.0xlo2 3.832 14.0x 10' 4.003 45.0 xl0 3.461 12.0~10~ 3.908 16.0x 10' 4.054 50.0x 10 3.512 14.0~10~ 3.959 18.0~10~ 4.095

55.0 x10 3.556 16.0x IO2 3.996 20.0~10~ 4.127

r,=80 r,=90 r,=lOO

tD PO

6.0x IO* 3.603 7.0x 10" 3.680 8.0x10' 3.747 9.0x10' 3.805 10.0x10' 3.857

12.0x IO" 3.946 14.0x 102 4.019 15.0x lo2 4.051 16.0x 10' 4.080 18.0x IO' 4.130

20.0x 10' 4.171 60.0x 10 3.595 18.0~10~ 4.023 25.0~10' 4.181 25.0x 10' 4.248 65.0x 10 3.630 20.0x10* 4,043 30.0~10~ 4.211 30.0~10~ 4.297 700x10 3.661 25.0x IO2 4.071 35.0~10' 4.228 35.0x 10' 4.328 75.0 x 10 3.668 30.0 x IO2 4.084 40.0~10' 4.237 40.0~10~ 4.347

t, PO 8.0 x10* 3.747 9.0x10' 3.806 1.0~10~ 3.858 1.2x 103 3.949 1.3 x IO3 3.988

1.4~10~ 4.025 1.5 x IO3 4.058 18~10~ 4.144 2.0 x103 4.192 2.5 x103 4.285

3.0 x 103 4.349 3.5 x102 4.394 4.0 x lo3 4.426 4.5 x103 4.446 5.0 x103 4.464

t, PD 1.0x 10" 3.859 1.2x 103 3.949 1.4x lo3 4.026 1.6x IO* 4.092 1.8x IO3 4.150

2.0x IO3 4.200 2.5x IO3 4.303 3.0x IO3 4.379 3.5x 103 4.434 4.0x lo3 4.478

4.5x 103 4.510 5.0x IO3 4.534 5.5x IO3 4.552 6.0x IO3 4.565 6.5x lo3 4.579

80.0x10 3.713 35.0x 102 4.090 45.0~10' 4.242 45.0x 10' 4.360 6.0 x lo3 4.482 7.0x lo3 4.583 85.0 x10 3.735 40.0x 10" 4.092 50.0~10~ 4.245 50.0x IO2 4.368 7.0 x103 4.491 7.5x IO3 4.588 90.0x10 3.754 450x10 4.093 55.0~10' 4.247 60.0~10~ 4.376 8.0~10~ 4.496 8.0x IO3 4.593 95.0x10 3.771 50.0x102 4.094 60.0~10' 4.247 70.0~10" 4.380 9.0 x lo3 4.498 9.0x IO3 4.598 10.0x 102 3.787 55.0~10' 4.094 65.0~10~ 4.248 80.0~10~ 4.381 10.0~10~ 4.499 10.0~10~ 4.601

12.0x10' 14.0x 102 16.0~10~ 18.0~10~ 20.0 x102

22.0x 10' 24.0~10' 26.0~10~ 28.0~10'

3.833 70.0x102 4.248 90.0x102 4.382 11.0x103 4.499 12.5~10~ 4.604 3.662 75.0x102 4.248 10.0~10~ 4.382 12.0~10~ 4.500 15.0x IO3 4.605 3.881 80.0~10' 4.248 11.0~10~ 4.382 14.0~10~ 4.500 3.892 3.900

3.904 3.907 3.909 3.910

GOR (relative permeability):

(65)

GOR (production):

AGn R,=---- qo,, At, . . . . (66)

For these equations, fR = fraction of reservoir swept, S, = oil saturation, fraction, S, = gas saturation, fraction, S,,. = water saturation, fraction, and

Sj,,. = interstitial water saturation, fraction.

One method for solutions using equal time intervals is as follows.

1. Estimate the pressure drop during the next time interval.

2. Calculate the water-influx rate with Eq. 60. 3. Calculate AL’, and V, with Eqs. 61 and 62. 4. Calculate the oil saturation in V, for the predicted

oil production during Interval n with Eq. 63. 5. Calculate gas production with Eq. 64.

6. Calculate the GOR with Eq. 65. 7. Calculate the GOR with Eq. 66 for average values

of pressure and saturation. 8. Compare the GOR’s obtained in Steps 6 and 7 and,

if they agree, proceed to the next interval. If they do not agree, estimate a new pressure drop and repeat Steps 2 through 8.

If the water drive equation for unequal time intervals is used, the need for re-evaluating the pressure functions for each trial in a given interval can be eliminated. This procedure calls for selecting a given pressure drop and estimating the length of the next time interval in Steps 1 and 8 and this program. The remaining steps are un- changed.

Reservoir Simulation Models. The capability of mathematical simulation models to calculate pressure and fluid flow in nonhomogeneous and nonsymmetrical reservoir/ aquifer systems has been thoroughly described in the liter- ature since the early 1960’s. Widespread availability of computers and models throughout the industry has helped to remove many of the idealizations and restrictions re- garding geometry and/or homogeneity that are a practical requirement for analysis by traditional methods. These models have the capability to analyze performance for vir- tually any desired description of the physical system, in- cluding multipool aquifers. See Chap. 48 for more information.


TABLE 38.7-DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUlFERS(contlnued)

rD =200 fD =300 rD =400

to PO t, PO t, PO 1.5~10~ 4.061 6.0 x lo3 4.754 1.5x104 5.212 2.0x103 4.205 8.0~10~ 4.896 2.0~10~ 5.356 2.5x lo3 4.317 10.0~10~ 5.010 3.0~10~ 5.556 3.0x 103 4.408 12.0~10~ 5.101 4.0x104 5.689 3.5x 103 4.485 14.0~10~ 5.177 5.0~10~ 5.781

4.0x 103 4.552 16.0~10~ 5.242 6.0~10" 5.845 5.0x10" 4.663 18.0~10~ 5.299 7.0~10~ 5.889 6.0~10~ 4.754 20.0~10~ 5.348 8.0~10~ 5.920 7.0x103 4.829 24.0~10" 5.429 9.0x104 5.942 8.0~10~ 4.894 28.0~10" 5.491 10.0~10~ 5.957

fD = 500 r,=600 rD = 700

to PO t, PO t, PO 2.0x104 5.356 4.0~10~ 5.703 5.0~10~ 5.814 2.5~10~ 5.468 4.5~10~ 5.762 6.0~10~ 5.905 3.0 x lo4 5.559 5.0~10~ 5.814 7.0~10~ 5.982 3.5x104 5.636 6.0~10~ 5.904 8.0~10~ 6.048 4.0 x lo4 5.702 7.0~10~ 5.979 9.0~10~ 6.105

4.5x IO4 5.759 8.0x10" 6.041 10.0~10~ 6.156 5.0x104 5.810 9.0x104 6.094 12.0~10~ 6.239 6.0~10~ 5.894 10.0~10~ 6.139 14.0~10~ 6.305 7.0x104 5.960 12.0~10~ 6.210 16.0~10~ 6.357 8.0x10" 6.013 14.0~10~ 6.262 18.0~10~ 6.398

25.0~10~

9.0~10~ 4.949

5.264

30.0~10~

10.0~10~

5.517

5.702 200x10"

11.0~10~ 5.967

5.991 30.0x103 5.282 12.0~10~ 5.703 24.0~10~ 5.991 35.0x 103

10.0x103

5.290

4.996 40.0~10"

140~10~

5.606

5.704 26.0~10~

12.0~10~ 5.975

5.991

40.0x 103 5.294 15.0x10" 5.704

12.0x103 5.072 50.0~10~ 5.652 12.5~10~ 5.977 14.0x103 5.129 60.0~10~ 5.676 13.0~10~ 5.980 16.0~10" 5.171 70.0~10" 5.690 14.0~10~ 5.983

18.0~10~ 5.203 80.0~10~ 5.696 16.0~10~ 5.988 20.0x 103 5.227 90.0x103 5.700 18.0~10~ 5.990

25.0~10~ 6.211 50.0~10~ 6.397 60.0~10~ 6.550 30.0x104

9.0x104 6.055

6.213 60.0~10~

16.0~10~ 6.299

6.397

20.0~10~

70.0~10~

6.430

6.551 35.0~10~ 6.214 80.0~10~ 6.551

40.0~10" 6.214

10.0x10' 6.088 18.0~10~ 6.326 25.0~10~ 6.484 12.0~10" 6.135 20.0~10~ 6.345 30.0~10~ 6.514 14.0~10~ 6.164 25.0~10~ 6.374 35.0~10~ 6.530 16.0x10" 6.183 30.0~10~ 6.387 40.0~10~ 6.540

18.0~10~ 6.195 35.0~10~ 6.392 45.0~10~ 6.545 20.0~10~ 6.202 40.0~10~ 6.395 50.0~10~ 6.548

Nomenclature

A = constant described by Eq. 46 b = intercept

B, = gas FVF, bbl/STB B, = oil FVF, bbl/STB B, = two-phase FVF, bbl/STB

B,,. = water FVF, bbl/STB cf = formation compressibility, psi -I C (,, = total reservoir compressibility, psi-’ c,~ = formation water compressibility, psi -I

cwt = total aquifer compressibility, psi - ’ d = geometry term obtained from Table

38.1 e,,. = water influx rate, B/D

e WB = water influx rate at Reservoir B, B/D e I, ,111, ,I = water-influx rate at interval n+ 1 -j,

BID c 1v1 ,, = total water influx rate at interval

n, B/D E,li = cumulative expansion per stock-tank

barrel OOIP, bbl f~ = fraction of reservoir swept

F = approximation to po and a function of type of aquifer

FG = reservoir geometry factor F(r) = influence function FV = ratio of volume of oil and its dissolved

original gas at a given pressure to its volume at initial pressure

G, = cumulative gas injected, scf !I = aquifer thickness, ft j = summation of time period 1 fo,,

J, = aquifer productivity index, B/D-psi k = permeability, md L = aquifer length, ft

Lf = linear penetration of water front into reservoir, ft

m = fitting factor (see Page 38-7); ratio of initial reservoir free-gas volume to initial reservoir oil volume; slope

mF = proportionality factor

mrJ = influx constant, bbl/psi (see Eqs. 9 and IO)

m,. = rate constant, psiibbl-D (see Eqs. 3 through 5)

n = interval N = OOIP, STB

N,, = time interval number y,, = cumulative oil produced, STB P ‘I = average aquifer pressure, psi

PN, = initial aquifer pressure, psi

ph = bubblepoint pressure, psi pi = dimensionless pressure term

PD(A,B) = dimensionless pressure term for Reservoir B with respect to Reservoir A

P II’ = pressure at original WOC, psi P II’,, = cumulative pressure drop at the end of

interval n, psi Ape = known dimensionless field pressure

drop at original WOC APO, = dimensionless pressure drop to time

period i


TABLE 38.7- DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)

rD = 800 rD = 900 rD =I,000 rD =I,200 fD =1.400 rD =1,600

to PO tLJ PO tL7 PO t, PO t, PO t D PO

7.0x10" 8.0~10~ 9.0x lo4 100x10~ 12.0x104

140x104 16.0~10~ 180x104 20.0x104 250x10"

30.0x104 35.0x104 40.0x lo4 45.0 x lo4 50.0x10"

550x104 60.0x lo4 70.0x10" 80.0 x lo4

100.0x10"

5.983 8.0x 10' 6.049 1.0x IO5 6.161 2.0 x105 6.507 6.049 9.0 x104 6.106 1.2~10~ 6.252 3.0x 105 6.704 6.108 10.0x lo4 6.161 1.4~10~ 6.329 4.0 x lo5 6.833 6160 120~10~ 6251 1.6~10" 6.395 5.0 x 105 6.918 6.249 14.0x lo4 6.327 1.8~10~ 6.452 6.0~10~ 6.975

6322 6.382 6432 6.474 6551

6.599 6.630 6.650 6.663 6.671

6.676 6.679 6.682 6.684 6.684

160~10~ 18.0x lo4 20.0 x lo4 25.0 x lo4 300x10"

6.392 6.447 6.494 6.587 6652 4.0~10" 6.781

40.0 x104 6.729 4.5x lo5 6.813 45.0x10" 6.751 5.0~10~ 6.837 50.0x10" 6.766 5.5~10~ 6.854 55.0x10" 6.777 6.0~10~ 6.868 60.0~10" 6.785 7.0~10~ 6.885

70.0 x104 5.794 80.0x IO4 6.798 90.0 x IO4 6.800 10.0 x IO5 6.801

8.0~10~ 6.895 9.0x lo5 6.901 10.0~10~ 6.904 12.0~10~ 6.907 14.0~10~ 6.907 16.0~10~ 6.908

2.0~10~ 6.503 2.5~10~ 6.605 3.0x105 6.681 3.5~10~ 6.738

2.0x lo5 6.507 2.5~10~ 6.619 2.5~10~ 6.619 3.0x 105 6.710 3.0x IO5 6.709 3.5x105 6.787 3.5x 105 6.785 4.0x105 6.853 4.0x105 6.849 5.0x lo5 6.962

7.0x10" 7.013 5.0x105 6.950 6.0~10~ 7.046 8.0x10" 7.038 6.0~10~ 7.026 7.0x 105 7.114 9ox105 7.056 7.0x IO5 7.082 8.0~10~ 7.167 10.0x10~ 7.067 8.0x lo5 7.123 9.0 x 105 7.210 120x105 7.080 9.0x105 7.154 10.0x lo5 7.244

14.0x105 7.085 10.0x 105 7.177 15.OxlO~ 7.334 16.0 x lo5 7.088 15.0x IO5 7.229 20.0x IO5 7.364 18.0~10" 7.089 20.0~10~ 7.241 25.0~10~ 7.373 19.0x105 7.089 25.0~10~ 7.243 30.0~10~ 7.376 20.0 x 105 7.090 30.0~10~ 7.244 35.0~10~ 7.377

21.0x105 7.090 31.0~10~ 7.244 40.0~10~ 7.378 22.0x105 7.090 32.0~10~ 7.244 42.0~10~ 7.378 23.0 x10' 7.090 33.0x 10' 7.24 44.0x IO5 7.378 24.0 x lo5 7.090

APO, =

APL =

Apy =

*PO,+I-.;) = AP,,,A,B) =

APIA,, =

A,-.],. =

Yo,, =

r,,, =

J/, = R = .’ 3,

St, = fD =

AIn = VP = VR =

dimensionless pressure drop to time

period j total pressure drop at WOC (calculated

using reservoir expansion rates). psi

total pressure drop at original WOC (field data), psi

average pressure drop in interval, psi pressure drop at Reservoir A caused

by Reservoir B, psi total pressure drop at Reservoir A at

end of interval H. psi total pressure drop at WOC (calculated

using reservoir voidage rates), psi total oil production rate at end of

interval n. BID total production rate. B/D aquifer radius, ft dimensionless radius=r,,/r,,. radius to water front after

penetration, ft field radius, ft cumulative produced GOR, scf/STB average solution GOR at end of

interval n, scf/STB gas saturation, fraction interstitial water saturation, fraction oil saturation, fraction residual oil saturation at end of interval

n. fraction formation water saturation, fraction dimensionless time dimensionless time interval total reservoir PV. bbl cumulative voidage, bbl

v = M, w =

W rD =

we,, =

w,, =

w; =

w,, = Y= z=

z,, = CY=

6e ,,,,, =

@?f,, =

Pl!, = 02 =

dJ=

initial water volume in the aquifer, bbl aquifer width, ft dimensionless water-influx term cumulative water influx at end of

interval n, bbl W,.,,,p,i, total aquifer expansion

capacity, bbl cumulative water injected, bbl cumulative water produced, bbl constant described by Eqs. 49 and 50 resistance function new values of Z angle subtended by reservoir, radians correction to e,,.,, correction to A pi,, water viscosity, cp variance porosity, fraction

TABLE 38.8-DIMENSIONLESS PRESSURES FOR FINITE-CLOSED LINEAR AQUIFERS

to PO -!k- PO o.005 0.07979 0.18 0.47900 0.01 0.11296 0.20 0.50516 0.02 0.15958 0.22 0.53021 0.03 0.19544 0.24 0.55436 0.04 0.22567 0.26 0.57776

0.05 0.25231 0.28 0.60055 0.06 0.27639 0.30 0.62284 0.07 0.29854 0.4 0.72942 0.08 0.31915 0.5 0.83187 0.09 0.33851 0.6 0.93279

0.10 0.35682 0.7 1.03313 0.12 0.39088 0.8 1.13326 0.14 0.42224 0.9 1.23330 0.16 0.45147 1.0 1.33332


TABLE 38.7-DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)

r,=1,800

tD PO

3.0~10~ 6.710 4.0~10~ 6.854 5.0x IO5 6.965 6.0~10~ 7.054 7.0x 105 7.120

8.0~10~ 7.188 9.0x IO5 7.238 10.0x lo5 7.280 15.0x 105 7.407 20.0x 105 7.459

30.0 x lo5 7.489 40.0x105 7.495 50.0x lo5 7.495 51.0x105 7.495 52.0x i05 7.495

53.0x 105 7.495 54.0x lo5 7.495 56.0x IO5 7.495

rD =2,000 rD =2,200 rD =2,400 rD = 2,600

to PO t, PO tD PO tD PO

4.0x105 6.854 5.0~10~ 6.966 6.0~10~ 7.057 7.0~10~ 7.134 5.0x105 6.966 5.5~10~ 7.013 7.0~10" 7.134 8.0~10~ 7.201 6.0x105 7.056 6.0~10~ 7.057 8.0~10~ 7.200 9.0x105 7.259 7.0x 105 7.132 6.5~10~ 7.097 9.0x105 7.259 10.0~10~ 7.312 8.0~10~ 7.196 7.0~10~ 7.133 10.0~10~ 7.310 12.0~10~ 7.401

9.0 x lo5 7.251 7.5x105 7.167 12.0x lo5 7.398 14.0~10~ 7.475 lO.Ox10~ 7.298 8.0~10" 7.199 16.0~10~ 7.526 16.0~10~ 7.536 12.0x105 7.374 8.5~10~ 7.229 20.0~10~ 7.611 18.0~10~ 7.588 14.0x105 7.431 9.0x105 7.256 24.0x IO5 7.668 20.0x lo5 7.631 16.0~10" 7.474 10.0~10~ 7.307 28.0~10~ 7.706 24.0~10~ 7.699

18.0~10~ 7.506 12.0~10~ 7.390 30.0~10" 7.720 28.0~10~ 7.746 20.0 x lo5 7.530 16.0~10~ 7.507 35.0~10' 7.745 30.0~10~ 7.765 25.0~10" 7.566 20.0~10~ 7.579 40.0~10" 7.760 35.0~10~ 7.799 30.0x10" 7.584 25.0~10~ 7.631 50.0~10" 7.775 40.0~10~ 7.621 35.0x105 7.593 30.0~10~ 7.661 60.0~10" 7.780 50.0~10~ 7.845

40.0x10" 7.597 35.0~10" 7.677 70.0~10~ 7.782 60.0~10~ 8.656 50.0x10" 7.600 40.0~10" 7.686 80.0~10" 7.783 70.0~10~ 7.860 60.0~10" 7.601 50.0x IO5 7.693 90.0x10" 7.783 80.0~10~ 7.862 64.0x IO5 7.601 60.0~10" 7.695 95.0x10" 7.783 90.0x105 7.863

70.0 x105 7.696 1O.OXlO~ 7.863 80.0~10" 7.696

rD =2,800

t, PD

8.0x lo5 7.201 9.0x lo5 7.260 10.0x IO5 7.312 12.0x105 7.403 16.0~10~ 7.542

20.0x lo5 7.644 24.0~10~ 7.719 28.0x105 7.775 30.0x 105 7.797 35.0x lo5 7.840

40.0x lo5 7.870 50.0x 105 7.905 60.0x lo5 7.922 70.0x IO5 7.930 80.0x i05 7.934

90.0x lo5 7.936 10.0x 10" 7.937 12.0x 10" 7.937 13.0x IO6 7.937

rD = 3,000

tD PO

1.0~10~ 7.312 1.2x106 7.403 1.4~10~ 7.480 1.6~10" 7.545 1.8~10~ 7.602

2.0 x 10" 7.651 2.4 x IO6 7.732 2.8 x 106 7.794 3.0 x106 7.820 3.5~10~ 7.871

4.0 x IO6 7.908 4.5x106 7.935 5.0x106 7.955 6.0x lo6 7.979 7.0x106 7.992

S.OXlO~ 7.999 9.0x106 8.002 10.0~10~ 8.004 12.0 x 106 8.006 150x10~ 8.006

a

w

Key Equations With SI Units The equations in this chapter may be used directly with practical SI units without conversion factors, except for certain equations containing numerical constants. These equations are repeated here with appropriate constants for SI units.

P II

112 r =

8.527~10-~ kha’ .“““’

P ,I’ mr= 8,527x10-” kh’ ...“.“’

!J ,J t?lr=

8.527x10-” khb’ “‘....’

m,,=(l)& ,,‘, bar,,?,

m,,=(1)r#x,,.,hb2, .

8.527 x 10 -s kt tD =

(#)(‘b,,,p ,,p ’ .

5.36x 1O-1 kh Jo = p,,,,(ln rD -0.75) -

. (3)

.

(4)

(5)

(9)

(10)

(20)

(40)

J,= 3(8.527 x 10 -5)kbh

, . tLM.L .(41)

FG= Lf 8,527x,o-5 hb, . . . . .

nd

FG= 2a In(r,/rf)

5,36x1o-4 ha, . . . . . . . . . . . . . . . . . . .

here k is in md, h is in m, b is in m, L is in m,

rD is dimensionless, r,,. is in m. p,,. is in mPa*s, c,,., is in kPa - ’ , J, is in mj/d*kPa, ~1,. is in kPa/m3 *d, tnp is in m3/kPa, FG is in m-‘, and

01 is in radians.


References 1. Van Everdmgen. A.F. and Hut-Q. W.: “The Appltcatton of the

Laplace Transformation to Flow Problems in Reservoirs.” Twns., AIME (1949) 186. 305-24.

2. Mottada, M.: “A Practical Method for Treating Oillield Interference in Water-Drive Reservoirs,” J. Per. Twh. (Dec. 1955) 217-26; Trurts.. AIME. 204.

3. Carter, R.D. and Tracy, F.W.: “An Improved Method for Calculatmg Water Influx,” J. Pet. Tech. (Dec. 1960) 58-60; Trms., AIME. 219.

4. Hicks. A.L. ( Weber, A.G., and Ledbetter, R.L.: “Computing Tech- mques for Water-Drive Reservoirs,” J. PH. Twh. (June 1959) 65-67; Trum.. AIME. 216.

5. Hutchwon. T.S. and Sikora. V.J.: “A Generaltzed Water-Drive Analysis.“J. Prt. T&r. (July 1959) 169-78; Trclns.. AIME, 216.

6. Schilthuis. R.J.: “Active Oil and Reservoir Energy.” 7rctn.s.. AIME 11036) 118. 33-52.

7. Fetkovich. M.J.: “A Simplified Approach to Water lntlux Calculations-Finite Aquifer Systems.” J. Pc~t. T&I. (July 1971) 814m28.

8. Brownscombc. E.R. and Collins. F.A.: “Estimation of Reserves and Water Drive from Pressure and Production Hratory,” Trtrnv., AIME (194Y) 186, 92-99.

9. Van Everdingen. A.F.. Timmerman. E.H., and McMahon, J.J.: “Application of the Material Balance Equation to a Partial Water- Drive Reservoir.” J. Prr. Tech. (Feb. 1953) 51-60; Trm\., AIME. 198.

IO. Havlena. D. and Odrh. A.S.. “The Material Balance as an Equation of a Straight Line.” J. &f. Twh. (Aug. 1963) 896-900: Trwrc.. AIME. 228.

General References Chatas, A.T.: “A Practical Treatment of Nonstcady-State Flow Prob-

lems in Rew-voir System-I.” Per. Enx. (May 1953) B42-

Chatas, A.T.: “A Practical Treatment of Nonsteady-State Flow Prob lems in Reservoir System-II,” PH. Enq. (June 1953) B3X-

Chatas. A.T.: “A Practical Treatment of Nonsteady-State Flow Prob- lems in Reservoir Systems-III.” Per. Eng. (Aug. 1953) B46-

Closman. P.J.: “An Aquifer Model for Fissured Reservoirs,” Sue. Pet. Eng. J. (Oct. 1975) 385-98.

Henaon. W.L., Beardon, P.L., and Rtce, J.D.: “A Numertcal Solutton to the Unsteady~State PartiallWater-Drive Reservoir Performance Prob- lem,” .Soc. Per. Eng. J. (Sept. 1961) 184-94; Trans., AIME. 222.

Howard, D.S. Jr. andRachford, H.H. Jr.: “Comparison of Pressure Dis- tributions During Depletion of Tilted and Horizontal Aquifers,” J. Per. Tech. (April 1956) 92-98; Trans., AIME. 207.

Hurst, W.: “Water Influx Into a Reservoir and Its Application to the Equation of Volumetric Balance.” Trans., AIME (1943) 151, 57-72.

Hutchinson. T.S. and Kemp, C.E.: “An Extended Analysis of Bottom- Water-Drive Reservoir Performance,” J. Pet. Tech. (Nov. 1956) 256-61; Trum., AIME, 207.

Lowe. R.M.: “Performance Predictions of the Marg Tex Oil Reservoir Using Unsteady-State Calculations,” J. Per. Tech. (May 1967) 595-600.

Mortada, M.: “Oiltield Interference in Aquifers of Non-Uniform Prop- c&s.” J. Pej. Tech. (Dec. 1960) 55-57: Trms AIME, 219.

Mueller, T.D. and Witherspoon, P.A : “Pressure Interference Effects Within Reservoirs and Aquifers.” J. Per. Tech. (April 1956)471-74; Trum., AIME, 234.

Nabor. G.W. and Barham, R.H.: “Linear Aquifer Behawor.” J. Per. Tdr. (May 1964) 561-63: Truns., AIME. 231.

Odeh. A.S.: “Reservoir Simulation-What Is It’?” J. Prr. Twh. (Nov. 1969) 13X3-88.

Stewart, F.M.. Callaway. F.H., and Gladfelter. R.E.: “Comparisons ot Methods for Analyzing a Water Drive Field. Torchlight Tensleep Reservoir. Wyommg.” J. Per. Tech. (Sept. 1954) 105-10; Trms.. AIME, 201.

Wooddy, L.D. Jr. and Moore, W.D.: “Performance Calculations for Reservoirs with Natural or Artificial Water Drtves.” J. PH. Twh. (Aug. 1957) 245-5 I; Trans., AIME, 210

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Water Drive Oil Reservoirs