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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