A New Approach to Interference Test

Abstract

Summary. The line-source solution of the diffusivity equation is differentiated and rearranged so that a linear relation between the variables is obtained. A plot of log tp' vs. 1/t yields a straight line whose intercept and slope are used to estimate the transmissivity, k h/mu, and the storativity, h phi ct, respectively. The method is extended to handle two-rate tests, including buildup tests, and can he used for the analysis of the combined data of the two periods. Introduction In interference test analysis, the semilog plotting techniques are inadequate because of the invalidity of the logarithmic approximation of the exponential integral function at large times. Usually type-curve matching is used. Recently, analysis methods based on the pressure derivative, p', were introduced. Tiab and Kumar used the maximum value of p' and the time at that point to estimate the transmissivity and storativity of the reservoir. Bourdet et al. introduced type-curve matching methods that involve both pressure-drop and pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-lerivative matching. Clark and van Golf-Racht extended the pressure-derivative methods to variable-rate testing pressure-derivative methods to variable-rate testing using a superposition time function. In this work, a derivative method that yields a straight-line plot is introduced. More details can be found in Ref. 4. Method The derivative of the pressure, with respect to time at a radial distance Ar from an active well as obtained from the line-source solution, is (1) Multiplying Eq. 1 by -t and taking logarithms of both sides, we get (2) The constants A and b are related to the transmissivity, T, and storativity, S, by the following equations: (3) and (4) It is clear from Eq. 2 that a plot of tp' vs. 1/t on a semilog graph gives a straight line. The intercept and slope can be used to estimate the transmissivity and storativity of the reservoir. The intercept, A, is read directly on the logarithmic scale as the value of tp' at 1/t=0. The slope in cycles/hr would be -b/2.303. Extension to Two-Rate Tests If the rate at the active well is changed from q1 to q2 at time t, it can be shown that (5) where (6) Eq. 5 is similar to Eq. 2 and yields a straight line if Delta t(p'+C)q1/(q1 -q2) is plotted vs. 1/At on a semilog plot. In this format, the slope and intercept of the straight line are the same as those of Eq. 2. This means that data points from the first rate region (q = q1) calculated according to Eq. 2 can be combined with data points from the second rate region (q = q2) calculated according to Eq. 5 and analyzed together to obtain the values of T and S that fit the data points in the two regions. In pressure-buildup testing, q2 --0 and a semilog plot of Delta t(p'+ C) vs. 1/Delta t would give a straight line with the same slope and intercept as those of Eqs. 2 and 5. Because C is not known in advance and depends on A and b in addition to At, an iterative procedure must be used in the analysis. An initial value of C = 0 may be used, and the constants A and b are estimated either graphically or by linear regression. The values of A and b obtained are used to update C according to Eq. 6. The iteration is continued until successive values of A and b become constant within a prescribed limit. The final values of A and b are then used to estimate the transmissivity and storativity from Eqs. 3 and 4, respectively. Illustrative Example The developed method is applied to the interference test data of a gas well reported by Ramey et al. Table 1 shows the test data, calculations, and final results. The term is approximated by Deltap/Delta 1n at the geometric average time . The method of least squares was used to find the constants A and b. A graphic presentation of the data is shown in Fig. 1. Type-curve matching presentation of the data is shown in Fig. 1. Type-curve matching results reported by Ramey et al. are also shown in Table 1. Comparison with results obtained by this method indicates a difference of about 3% in the transmissivity and 0.6% in the storativity, indicating the accuracy of the proposed method. Conclusions A new approach for interference test analysis is introduced. A semilog plot of tp vs. 1/t gives a straight line from its intercept, and slope reservoir parameters can be estimated. The method can also be applied to two-rate interference tests for which an iterative procedure is used. Data points from the two regions may be analyzed procedure is used. Data points from the two regions may be analyzed separately or combined. SPEFE P. 609