INVERSE AND ILL-POSED PROBLEMS IN RESERVOIR SIMULATION

The coupled systems of nonlinear partial differential equations which are used to model the flow of fluids in porous media give rise to several different types of ill-posed and inverse problems. The mathematical models which describe the fluid flow processes contain function parameters which describe properties of the fluids or properties of the medium through which they flow. For multiphase or multicomponent problems, these function parameters can be strongly nonlinear functions, depending upon unknown variables like pressure and fluid saturations. The determination of these parameters is an inverse problem which is highly ill-conditioned and difficult to solve. The enormous size of the physical problem and the paucity of flow data for use in the inverse problem add significantly to the computational complexity. Even if the parameters were known exactly, the nature of the equations leads to other ill-posed problems. The coupled systems of nonlinear transport-dominated partial differential equations are notoriously difficult to solve numerically, due to their ill-posed nature. Finally, the enormous size of the problems leads to huge matrices upon discretization which have extremely large condition numbers and must be treated with great care. In this paper several aspects of ill-posed and inverse problems are surveyed and various promising solution techniques are presented.

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