SPE 12565-PA Analytical Representation of the van Everdingen Hurst Aquifer Influence Functions for Reservoir Simulation ---Fanchi J.R.

Analytical Representation of the van
Everdingen-Hurst Aquifer Influence
Functions for Reservoir Simulation
J .R. Fanchi, * SPE, Keplinger and Assoc. Inc.
Abstract
Linear regression analysis has been used to develop some
simple analytical expressions for the van EverdingenHurst aquifer influx influence functions. Regression
results are presented for a variety of aquifer radius/reservoir radius ratios. The regression equations are designed
for use in reservoir engineering applications, especially
reservoir simulation.
information about dimensionless pressure P tD and its first
derivative P~D as functions of dimensionless time t D.
Usually, the relationship between t D and P tD is available
in the reservoir simulator in tabular form for the infiniteacting constant terminal rate case only. The program
determines PtD and P~D for a given tD by using a
numerical interpolation scheme. An alternative approach
that requires less computer work while providing
equivalent or greater accuracy than the table look-up
method is presented here.
Introduction
A reservoir-aquifer system can be modeled by using a
reservoir simulator in which small gridblocks define the
reservoir and increasingly larger gridblocks define the
aquifer. This approach has the disadvantage of increased
computer storage and computing time requirements
because additional gridblocks are needed to model the
aquifer. A widely used and more cost-effective means of
representing an aquifer is to compute aquifer influx with
an analytical model. Among the more popular analytical
aquifer models in use today is the Carter-Tracy
modification 1 of the van Everdingen-Hurst 2 unsteadystate aquifer influx calculation.
, The Carter-Tracy aquifer influx rate calculation requires
Description of Method
A linear regression analysis 3 has been used to develop
analytical representations of the Carter-Tracy influence
functions. The regression equations, the regression coefficients, correlation range limits, and measures of the
linear regression validity are presented in Table 1 for a
number of commonly encountered r e/r w cases. Plots of
these expressions are shown in Figs. la and lb. Differences between the van Everdingen-Hurst tabular values
and the calculated values based on linear regression
analysis are insignificant within the correlation range
TABLE 1-CARTER·TRACY INFLUENCE FUNCTION REGRESSION COEFFICIENTS FOR THE CONSTANT TERMINAL RATE CASE
Regression Equation: PtO
=80 +8 1 t O +8 2
In to
+8 3 (ln
t o)2
Regression Coefficients
Case
r.lr w
1.5
2.0
3.0
4.0
5.0
6.0
8.0
10.0
8 1
8 2
83
1.66657
0.68178
0.29317
0.16101
0.10414
0.06940
0,04104
0.02649
-3.68x10- 4
-0.04579
-0.01599
0.01534
0.15812
0.30953
0.41750
0.69592
0.89646
0.28908
-0.01023
-0.01356
-0.06732
-0.09104
-0.11258
-0.11137
-0.14350
-0.15502
0.02882
_8_0_
0.10371
0.30210
0.51243
0.63656
0.65106
0.63367
0.40132
0.14386
0.82092
00
Correlation
Range of
to
0.06 to 0.6
0.22 to 5.0
0.52 to 5.0
1.5 to 10.0
3.0 to 15.0
4.0 to 30.0
8.0 to 45.0
12,0 to 70.0
0.01 to 1,000,0
Objective
Function
Multiple
Correlation
Coefficient
Number
of
Input
Values
S'
R2'
N'
1.7x10- 6
3.9x10- 5
1.4x10- 5
6.7x 10- 6
4.5x10- 6
1.5x10- 5
4.8x10- 6
4.6x10- 6
8.2x10- 3
0.99999
0.99999
0.99999
0.99999
0.99999
0.99999
0.99999
0.99999
0.99978
19
23
19
22
29
25
24
26
30
Standard
Error of
Estimate'
3.2x 10- 4
1.4x10- 3
9.1 x10- 4
5.8x10- 4
4.1x10- 4
8.1 x10- 4
4.7x10- 4
4.3x 10- 4
1.71 x10- 2
Average
Deviation
From
Actual
~
0.06
0.16
0.09
0.03
0.02
0.04
0.02
0.01
1.50
-NOTE: These symbols are defined in Ref. 3 as follows. The linear regreSSion objective function S is derived by
N
s. ~ (Y,-Y,)'.
i-1
where Y, is the van Everdingen-Hurst value of Pta. 'Yi is the regression equation value of P2'D' and N is the total nu~ber of van Everdingen-Hurst pairs of (P IO and tD)' A positive value of S near
zero indicates a good fit with S - 0 being a perfect fit. The "multiple correlation coefficient" R of Ref. 3 is defined by R = sum of squares using regresston equation values divided by sum of squares
using van Everdingen-Hurst values. or
N
~ (1',_1')'
i_1
R2 - - N - - - - - '
L;
where
Y is the average van
Everdingen-Hurst value=.2..
N
~
(N
iz1
Yi
)
•
(y,_y)2
;_1
The range of R 2 is between 0 and' with R 2 = 1 being a perfect fit. The standard error of estimate is defined by
'Now with Marathon Oil Co.
JUNE 1985
.JS/(N - 2). where S is the objective function.
Copyright 1.985 Society of Petroleum Engineers
405
0.00.1l
.10
.0
.0
at • f., . . Is- 'It
100.0
I
!
g
)
A
Dimensionless Time tD
Fig. 1-Aegression equation fit of the van Everdingen-Hurst influence functions. The smooth curve
is computed from the regression equation.
limits tabulated by van Everdingen and Hurst for all cases
except the infinite ratio case.
The average deviation of regression values from van
Everdingen-Hurst tabular values for the infinite ratio case
is 1.5% (Table 1). Most of this error appears at early times
(0.01 < t D < O. 1) when aquifer influx is often small
relative to influx for the total life of the reservoir. A correlation for the infinite ratio case with an average deviation of 0.2 % and a standard error of the estimate of
1.1 x 10 - 3 has been presented by Edwardson et al. 4
Their additional accuracy requires approximately twice
the computational labor as the correlation presented in
Table 1. They do not present correlations for other r efr w
values.
It is interesting to note that the form of the regression
equation in Table 1 is the same for each r efr w case. This
simplifies the coding needed to incorporate these expressions into a reservoir simulator such as that described in
Ref. 5. Besides simplifying the programming effort, the
computer work is lessened because it is no longer
necessary to perform a table look-up. Furthermore, the
derivative P~D is obtained directly from differentiation of
the regression equation for P tD by t D. This avoids the
necessity of performing a numerical differentiation and
ensures that a mathematically smooth function is always
used.
Nomenclature
P tD = dimensionless pressure
P~D
re
rw
tD
= derivative of PtD with respect to
= external aquifer radius, ft [m]
= external reservoir radius, ft [m]
= dimensionless time
tD
References
1. Carter, R.D. and Tracy, G.W.: "An Improved Method for
Calculating Water Influx," J. Pet. Tech. (Dec. 1960) 58-60; Trans.,
AIME,219.
2. van Everdingen, A.F. and Hurst, W.: "The Application of the
Laplace Transform to Row Problems in Reservoirs," Trans., AIME
186 (1949) 305-24.
3. Kuester, J.L. and Mize, J.H.: Optimization Techniques With Fortran, McGraw-Hill Book Co. Inc., New York City (1973) 205.
4. Edwardson, M.J. et al.: "Calculation of Formation Temperature
Disturbances Caused by Mud Circulation," J. Pet. Tech. (April
1962) 416-26.
5. Fanchi, J.R., Harpole, K.J., and Bujnowski, S.W.: "BOAST: A
Three Dimensional, Three-Phase Black Oil Applied Simulation
Tool," Vols. I and II, U.S. DOE Report DOE/BC-lOO33-3,
Bartlesville Energy Technology Center, Bartlesville, OK (1982).
SPEJ
Original manuscript (SPE 12565) received in the Society of Petroleum Engineers of·
fice Sept. 15. 1983. Paper accepted for publication June 4.1984. Revised manuscript
received July 9. 1984.
SOCIETY OF PETROLEUM ENGINEERS JOURNAL