Abstract
This paper describes the application of regression analysis for obtaining a two - dimensional areal description of heterogeneous reservoirs from short-term pressure-time data such as that obtained in interference tests. The method replaces the time-consuming trial-and-error procedure commonly used to match field data on an electric analyzer or digital computer with a systematic search which is programmed for a computer.
The computer program adjusts the properties of a reservoir model automatically until a least-squares fit is obtained between observed and calculated pressure data. The reservoir is simulated by a single-phase, compressible, two-dimensional model. It is divided into a number of homogeneous blocks whose transmissibility (kh/f) and storage (fch) values are varied to obtain the least-squares fit. The reliability of these values is determined from their standard deviations and correlation coefficients.
Although the method is rigorously applicable to single-phase flow only, multiphase flow can be handled provided saturation changes are small during the test. Possibly the method can also be used to obtain a reservoir description from pressure-production history, but this application is outside the scope of this work.
The paper includes, in addition to a description of the numerical procedure, a discussion of some of the problems associated with the method. Rules are given to help in selecting the number of homogeneous blocks and deciding upon their arrangement. The uniqueness of a reservoir description is considered. Finally, the use of the method is illustrated by the interpretation of field data from two interference tests.
INTRODUCTION
Pressure data from short-term transient tests, such as single-well and interference tests, are widely used to obtain reservoir properties. These tests are usually analyzed by assuming a simple reservoir model; very often, a homogeneous one is used. As a result, analysis of the transient data from each well frequently gives different values for reservoir properties. The problem then arises to combine all these differing results into a more detailed picture of the reservoir.
One technique is to simulate the reservoir with a digital computer or with an electrical analyzer and to adjust the reservoir parameters by trial and error until the simulated pressure data are in reasonable agreement with the observed pressure data for all wells. Although this method has been used for both transient tests and pressure-history data, it is time-consuming and subjective.
A second technique uses regression analysis to replace the trial-and-error procedure with a systematic search that can be programmed for a digital computer. Use of regression analysis in reservoir description was proposed recently by Jacquard and Jain.1 They divided the reservoir into a number of homogeneous blocks whose properties are varied until a least-squares fit is obtained between observed and calculated pressures. However, they did not consider their technique to be operational, mainly because of "...the lack of experience in using the method. . . notably for the improvement of convergence; andlimitations imposed by the insufficiency of available computers".1
While the analysis presented in this paper applies the same general principle used by Jacquard and Jain, the specific method is significantly different. Some differences arethe regression problem is solved in a different way which requires less computer time in most cases;a stepwise solution, in which the detail in the reservoir description is increased from step to step, is used to improve convergence; andthe reliability of the estimated reservoir properties, as measured by their standard deviation and correlation coefficient, is estimated.
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