Automatic Type, Curve Matching for Well Test Analysis

Abstract

Summary. This paper presents a general method for automatic well test interpretation. The method matches the field data with theoretical reservoir models using a constrained, nonlinear, least-squares regression technique coupled with numerical Laplace inversion of pressure-drawdown equations. Pressure gradients are computed by forward finite-difference approximations. Hence, reservoir models whose pressure gradients are difficult to obtain analytically can be readily included. Only equations for drawdown type curves of reservoir models are needed. Simulated pressure tests and actual field data are analyzed to illustrate the application of the method. The method reduces the time required to perform well test analysis and minimizes the subjectivity of interpretation. Introduction Interpretation of well test data consists of two steps: defining the well/reservoir system and computing the controlling parameter of the system. Defining the system means identifying a reservoir model that can represent the test data and yet is consistent with all available information from geology, logs. cores, etc. Usually, type-curve-matching techniques in conjunction with external information are used to determine the reservoir model. Well test data are plotted and visually compared with precomputed theoretical curves known as type curves. The reservoir model represented by the type curve that most closely matches the data is considered applicable. The controlling reservoir parameters are then calculated from a match point. Often, many adjustments in the reservoir parameters are required before an acceptable match is obtained. Therefore, interpretation of well test data after selection of a reservoir model requires a considerable amount of computation and manpower. An automatic type-curve-matching computer program offers a means to alleviate the parameter-adjustment phase of well test analysis. Such a program will shorten the analysis turnaround time and improve the quality of interpretation. Also, more complex reservoir behaviors (e.g., layered and bounded) for which the conventional graphical techniques (e.g., type curves) are inadequate or even nonexistent can be analyzed. Furthermore, tests with short durations or with variable flow rates that are usually difficult to match could be interpreted by the automated method. Different computer-aided interpretation techniques have been used in the past to analyze well test data. Most of the techniques match the pressure data with various regression schemes. History matching based on the Bayesian method also has been used. In these applications, the theoretical pressures were calculated either with numerical simulators or by simple analytical solutions, such as the exponential integral solution for a line-source well. Rosa and Horne were the first to use the general analytical solutions of homogeneous and noncommunicating layered reservoirs in their automatic type-curve matching. They computed both pressure and pressure gradients analytically in Laplace space and used the Stehfest algorithm to invert the results numerically. They considered only single-rate drawdown tests and applied their technique to single-layer homogeneous examples. Barua and Horne extended the technique to composite reservoirs. This paper presents a general technique for automatic type-curve matching of test data. The tests can include several rate changes and the test data can be matched with different reservoir models. The paper illustrates analysis of simulated and field data with different models. Finally, discussions regarding the limitations of the method, along with guidelines for its effective use, are provided. Automatic Well Test Interpretation (AWTI) Program The AWTI program is based on matching the field pressure data with results from an analytical or numerical reservoir model. The parameters of the reservoir model are systematically varied until an acceptable match is obtained. The matching process can be divided into three partsleast-squares regression or parameter estimation,pressure and pressure-gradient calculation. anddata screening. The success of the AWTI program depends on the efficiency, accuracy, generality, speed, and simplicity of each part. In the following, these three parts are described in detail. Least-Squares Regression Algorithm (Parameter Estimation). The least-squares regression minimizes an objective function defined as (1) The vector contains the reservoir parameters to be estimated. The parameters are constrained between minimum values, and maximum values, . Imposing bounds on parameters within physically meaningful regions improves the convergence of the regression algorithm. The constrained least-squares problem is converted into an unconstrained problem through the use of penalty functions. Penalty functions are terms that are added to the objective function to increase its value whenever the parameters approach their bounds. Eq. 2 is the modified unconstrained objective function after the addition of penalty functions: (2) where, and R are positive constants that control the amount of contribution from the penalty functions. Minimization of is an iterative process because the theoretical models for pressure calculations are nonlinear with respect to reservoir parameters (e.g., permeability and skin). Appendix A discusses the minimization process. Briefly, the process reduces to a system of linear equations at each iteration, which is solved by the technique of Levenberg and Marquardt. This technique ensures that the new estimates of parameters at each iteration reduce the value of . An improved version of the Levenberg-Marquardt method was developed and is presented in Appendix B. This version has a better convergence property and keeps the estimated parameter values within their specified ranges. Selection of Penalty Functions. Because the objective is to minimize, not, the value of R in Eq. 2 is decreased during each iteration to make a closer approximation to . A good choice for R for the eth iteration is (3) SPEFE P. 567^