Identification of spatially varying parameters in distributed parameter systems.

Abstract

The problem of identifying spatially-varying parameters in distributed parameter systems arises in the description of petroleum reservoirs and subsurface aquifers. The major difficulty in developing successful solution techniques lies in the ill-posedness in the sense that small errors in the data may cause large deviations of the solutions. In the conventional least-squares approach, this ill-posedness manifests itself as numerical instability causing anomalous, physically meaningless oscillations in the estimates. The goal of this thesis is to develop numerically stable identification algorithms on the basis of the regularization approach for classes of parabolic systems. In Chapter Two, the regularization identification theory is rigorously developed for quasilinear parabolic systems. This serves as a theoretical basis of an automatic history-matching algorithm for dry gas reservoirs developed in Chapter Four. A systematic procedure is presented for constructing the algorithm in three steps: formulation, discretization and optimization. In Chapter Three, a novel piecewise regularization approach is rigorously developed for the identification of spatially discontinuous parameters in second-order parabolic systems. The results are applied to history-matching of one-dimensional fractured reservoirs. In Chapter Five, a novel regularization history-matching algorithm is developed which is capable of incorporating a priori point estimates of unknown parameters. A multi-objective index is formulated and optimized in a stepwise estimation process; simple rules-of-thumb are presented for determining the optimal weights. Extensive numerical experiments are performed in each phase of the work. The results clearly show that the regularization is an effective method for obtaining a well-behaved approximate estimate for an ill-posed identification problem and that incorporation of available a priori information can enhance the accuracy of the estimate.