Chapter 38 Water Drive Oil Reservoirs Daylon L. Walton, Roebuck-WaltonInc.* Introduction Water drive reservoirs are those reservoirs in which a sig- nificant 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 be- havior 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 cyl- inders 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 con- stant 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 prin- ciple 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. 2 shows that the relationship between the pressures and
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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 sig- nificant 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 be- havior 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 cyl- inders 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 con- stant 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 prin- ciple 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.
water-influx rates is a function of a constant m,. and a variable po. The constant m, is a function of the aqui- fer properties, whereas pD is a function of aquifer prop- erties 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
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 su- perposition. This method is advantageous when the cal- culations are to be made manually. since fewer terms are required.
Using Carter and Tracy’s method, Eq. I I, the cumula- tive water influx at time t,, is calculated directly from the previous value obtained at t,,-,
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
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, in- finite, 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 his- tory in an analog computer to obtain influence-function curves for each pool in a multipool aquifer. The influ- ence 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 inter- ference 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 cal- culate 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, as- sume 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 pres- sure 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 sys- tem 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 ge- ometry, and aquifer size (to for radial aquifers).
Table 38.1 gives the substitution for d in Eq. 20 to cal- culate tD and the table, graph, or equation to obtain po, P&A-B), or W,D for various types of aquifers. The fol- lowing 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 approxi- mated by a uniform linear or radial system; therefore, published values of pD are used. If the aquifer can be ap- proximated by a homogeneous, infinite, radial system, the method can be extended to handle reservoir interfer- ence. In Method 2,5 the product of m, and pD is re- placed 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.
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 in- terval (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 func- tion of n will be constant. Variations from a constant can result from (1) incorrect AtD, (2) production and pres- sure errors, (3) incorrect aquifer size or shape, or (4) aqui- fer 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 re- sults 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 perform- ance, 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 aqui- fer, 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
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 ex- trapolated 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.
WATER DRIVE OIL RESERVOIRS 38-7
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 in- tervals. 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 cal- culate 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 fac- tors 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 con- stant, 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 perform- ance and therefore may be used to predict the future per- formance. 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 cal- culate 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 elim- inated 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).
38-0 PETROLEUM ENGINEERING HANDBOOK
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 cor- rect, some judgment must be exercised when making these adjustments.
Approximate Methods. If the water influx rate is con- stant 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 perform- ance 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 aqui- fers, 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)
WATER DRIVE OIL RESERVOIRS 38-9
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 reser- voir 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 calcu- lated using Eqs. 7 and 42 for a given assumed aquifer permeability and various estimates. The minimum vari- ance 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 per- meability 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 pres- sure 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.
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
WATER DRIVE OIL RESERVOIRS 38-11
TABLE 38.5-DIMENSIONLESS WATER INFLUX FOR FINITE OUTCROPPING RADIAL AQUIFERS (continued)
The variance of this fit from field data can be calculat- ed 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).
36-12 PETROLEUM ENGINEERING HANDBOOK
TABLE 38.6-DIMENSIONLESS PRESSURES FOR FINITE CLOSED RADIAL AQUIFERS
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 appropri- ate 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 pres- sure, a direct solution is possible; however, if the reser- voir is below saturation pressure, a trial-and-error procedure is necessary.
WATER DRIVE OIL RESERVOIRS 38-13
TABLE 3&G-DIMENSIONLESS PRESSURES FOR FINITE CLOSED RADIAL AQUIFERS (continued)
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 appli- cation 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 satura- tions 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 por- tion 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 origi- nal 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.
38-14 PETROLEUM ENGINEERING HANDBOOK
TABLE 38.7- DIMENSIONLESS PRESSURES FOR FINITE OUTCROPPING RADIAL AQUIFERS
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 reser- voir 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 bub- blepoint 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:
The calculated water-influx rate now can be used in Eq. 58 to calculate Ap(,,, and the whole procedure is repeat- ed 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,,.)
WATER DRIVE OIL RESERVOIRS 38-15
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
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 in- terval.
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 mathe- matical 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 practi- cal 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.
WATER DRIVE OIL RESERVOIRS 38-17
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
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
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.
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.
38-20 PETROLEUM ENGINEERING HANDBOOK
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