Improved Estimation Algorithms for Automated Type-Curve Analysis of Well Tests

Abstract

Summary. The analysis of well-test data by automated type-curve match by use of computers is a subject of current interest. Although the automated type-curve-matching technique is often an improvement over conventional methods, there are certain practical problems that occur in its application. Except in the case of a homogeneous reservoir, many reservoir models require the estimation of several parameters. Unless the initial guess is very good, the estimation procedure for many parameters may fail to converge. Even in cases that have only a few parameters. contours of the objective function show that some parameters are inherently ill-defined. To date, most applications have used the Gauss method or its modifications, such as the Levenberg-Marquardt algorithm. The Newton method should be better suited to the estimation of ill-defined parameters because it uses a search direction that is aligned in the direction of these parameters. This study investigates the application of the Newton method (and an important modification. the Newton-Greenstadt method) to automated well-test analysis. It shows that the added expense of the Newton-Greenstadt method is often justified by the improvement in performance achieved. In many cases, the Newton-Greenstadt procedure performance achieved. In many cases, the Newton-Greenstadt procedure converges almost as fast as the Gauss-Marquardt method, yet is more reliable in cases where one (or more important, more than one) parameter is ill-defined. Introduction Although the use of computer-assisted well-test interpretation is not new (for a brief overview of earlier work see Refs. 1 through 6), it has become more popular now with the increasing availability of computing power. Automated well-test interpretations are ideal applications for desktop microcomputers, and their use is likely to become widespread. The advantages of automated well-test analysis over either type-curve analysis or straight-line analysis techniques have been clearly demonstrated by Barua and Horne, Barua et al., and Guillot and Horne, among others. The advantages include a much higher resolution than type-curve analysis, a reduction of the danger of choosing incorrect straight lines. and ease in handling of multiple-flow-rate history. Last but not least, automated match frees the analyst from numerical or procedural errors. This is not an insignificant advantage given the growing complexity of models available and their attendant interpretation procedures. In the past, one of the difficulties with automated well-test interpretation was the evaluation of the reservoir response functions in closed form. Rosa and Horne showed that by numerical inversion from Laplace space of both function values and gradients, it is possible to fit well-test data to the most complicated of mathematical models by the least-squares method, and generally do a much better job in parameter estimation than is possible with manual methods. parameter estimation than is possible with manual methods. A further impetus to this method has been the reliability of the Stehfest numerical inversion scheme for the inversion of the Laplace transforms. Despite the advantages of automated match and the ability to match very complex models, there remains a need for improved estimation algorithms. Barua and Horne showed that as the number of parameters becomes large, it is often difficult to get good matches. Also, some parameters are inherently ill-defined and prove to be difficult to determine even if the number of parameters is not large. A typical case of such a difficulty is where a particular parameter is a function of only a specific range of the data (e.g., wellbore storage in the early data and reservoir limit in the late data). If the required range of data is missing in the well-test measurements (perhaps the test was stopped too early or started too late), then the parameter estimation algorithm may "crash," even though the other unknown parameters could easily be determined. This is a major deficiency that is easily avoided by performing a visual type-curve or semilog analysis. If automated well-test procedures are to become reliable general-purpose reservoir engineering tools, algorithms that are sensitive to these difficulties must be developed. Many of the applications to date have used the Gauss method modified by Marquardt's algorithm (also called the Levenberg and Marquardt algorithm), which has proved both popular and easy to use in least-squares estimation. In view of the difficulties sometimes experienced with it, a need exists for better estimation algorithms. This study was undertaken to try different algorithms in an effort to solve these problems. In the first instance, second-order methods based on Newton's method were examined to investigate their performance compared with the first-order Gauss method and its close derivatives. This choice was prompted by the observation that second-derivative methods, being second-order, should be better than the first-derivative methods currently used. In theory, some of the second-order methods are less prone to the difficulties of parameter insensitivity and should be useful in well-test applications. Specifically. the Greenstadt modification to Newton's method should avoid parameter insensitivity by adjusting the eigenvalues of the solution matrix in such a way that the insensitive parameters are not included in the search for the optimum values of the unknown reservoir parameters being estimated. At the same time, the functions in use in well-test analysis are peculiar to our discipline and it is not clear whether the desired improvements in performance of the algorithms will be offset by increases in computation required. Because of the generally complex nature of functions involved in parameter estimation, it is generally impossible to prove analytically the performance of a method on a given problem. Instead. conclusions as to the relative merits of the different estimation algorithms are reached by trying them on the chosen function(s). SPEFE P. 186