Automatic History Matching With Variable-Metric Methods

Abstract

Summary. Variable-metric minimization methods are introduced for use with optimal-control theory for automatic reservoir history matching. Variable-metric methods are more efficient and robust than the previously used minimization methods, such as steepest-descent and conjugate-gradient methods. In addition, variable-metric methods can be used effectively with parameter-inequality constraints. Two variable-metric methods-the Broyden/Fletcher/Goldfarb/Shanno (BFGS) method and a self-scaling variable-metric (SSVM) method-are tested with hypothetical two-phase reservoir history-matching problems. Estimated rock/fluid properties include permeability, porosity, and relative permeabilities. The SSVM method is more efficient than the BFGS method when the number of unknown parameters is large. Both methods perform better than the steepest-descent and Nazareth's conjugate-gradient methods except when the performance index is nearly quadratic, where the conjugate-gradient methods may be more efficient than the BFGS method. A constrained BFGS algorithm is tested to he successful in problems where the unconstrained algorithms have failed. Introduction Reservoir simulations are routinely used today to predict reservoir performance under different operating scenarios. Probably the primary factor that affects the reliability of predictions of reservoir behavior by numerical simulation is the accuracy of estimates of reservoir rock properties. Because of the relative inaccessibility of petroleum reservoirs to sampling, well pressure and production data are important sources of information for estimating the reservoir properties. The process of estimating the properties by adjusting parameters in the numerical simulator so that simulated pressure and production "match" the field data is known as history matching. History matching can be carried out either manually or automatically on computers. Manual history matching, usually a trial-and-error process, is difficult and often painstaking because reservoir behavior is complex and the parameters to be estimated may be highly interacting. To the best of our knowledge, general guidelines for manual history matching do not exist; therefore, manual history matching requires a great deal of experience and depends heavily on personal judgment. The need for the involvement of human experience is not eliminated with automatic history matching; engineering skills are still needed in many important aspects of history matching. Automatic history matching, however, does have the potential to save significant amounts of manpower and to provide more accurate estimates than manual history matching. In automatic history matching, estimates are usually chosen as those parameter values that minimize a performance index. With a suitable choice for the performance index, the estimates so obtained have certain desirable statistical qualities. A least-squares performance index is normally suitable for such purposes. The history match thus becomes a mathematical minimization problem. Reservoir history-matching problems are generally characterized by a very large number of unknown parameters. Consequently, the efficiency of the numerical minimization algorithm is a key concern. A second consideration is that reservoir history-matching problems are typically ill-conditioned. Consequently, many sets of parameter estimates may yield nearly identical matches of the data. To ensure that reliable estimates for all parameters are obtained, it is desirable to restrict the range of investigation to that considered reasonable by the engineers. This can be accomplished by including parameter-inequality constraints. In addition to improving the reliability of the history matches, such constraints can improve efficiency further by restricting the algorithm from searching ranges of parameter values that are not considered realistic. Most early automatic history-matching algorithms are not suitable for applications involving large numbers of unknown parameters. They require the evaluation of sensitivity coefficients, which is prohibitively expensive because of the large dimensionality of the history-matching problem. Chen et al. and Chavent et al. made a major breakthrough by introducing optimal-control theory in automatic history matching. With optimal-control theory, the gradient of the performance index can be evaluated very efficiently, regardless of the dimensionality of the history-matching problems. Therefore, the use of optimal-control theory with first-derivative minimization methods-methods that require only the evaluations of the performance index and gradient-provides an attractive and viable approach for automatic history matching. The first-derivative methods used are the steepest-descent and conjugate-gradient methods. We believe, however, that variable-metric (or quasi-Newton) methods would be better choices. The steepest-descent method has the drawback of progressing very slowly in the vicinity of the minimum, and the conjugate-gradient methods have been shown to be inferior to variable-metric methods in both theoretical aspects and numerical experiments. The need to include parameter-inequality constraints in automatic history matching makes variable-metric methods even more attractive than the steepest-descent and conjugate-gradient methods. To prevent the estimates from being physically unrealistic, artificial bounds may be set on the estimates of unknown rock properties. In most previous implementations, however, these bounds were not considered in the determination of the search direction but merely used in limiting the step length taken along the search direction so that the bounds would not be violated. This box-type strategy often upset the "rhythm" of the minimizating methods and thus would impede or even stop the minimization. In contrast, with variable-metric methods, constraints can be integrated into the minimization process. Search directions are sought only in the subspace of the active constraints and thus the desirable convergence properties are preserved. In short, we believe that with the improved convergence rate and the ability to accommodate constraints efficiently, variable-metric methods are particularly suitable for use with optimal-control theory in automatic history matching. In this work, we tested two variable-metric methods with hypothetical two-phase history-matching problems. Their performances were evaluated and compared with those of the steepest-descent method and a conjugate-gradient method developed by Nazareth. SPERE P. 995^