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Implicit finite element methods for two‐phase flow in oil reservoirs
Author(s) -
Langtangen Hans Petter
Publication year - 1990
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650100605
Subject(s) - discretization , mathematics , finite element method , conjugate gradient method , galerkin method , compressibility , porous medium , flow (mathematics) , mathematical analysis , geometry , mathematical optimization , mechanics , physics , materials science , porosity , composite material , thermodynamics
The equations governing immiscible, incompressible, two‐phase, porous media flow are discretized by generalized streamline diffusion Petrov–Galerkin methods in space and by implicit differences in time. Systems of non‐linear algebraic equations are solved by Newton–Raphson iteration employing ILU‐preconditioned conjugate‐gradient‐like methods to the non‐symmetric matrix system in each iteration. The resulting solution methods are robust, enable complex grids with irregular nodal orderings and allow capillary effects. Several numerical formulations are tested and compared for one‐, two‐ and three‐dimensional flow cases, with emphasis on problems involving saturation shocks, heterogeneous media and curved boundaries. For reservoirs consisting of multiple rock types with differing capillary pressure properties, it is shown that traditional Bubnov‐Galerkin methods give poor results and the new Petrov–Galerkin formulations are required. Investigations regarding the behaviour of several preconditioned conjugate‐gradient‐like methods in these type of problems are also reported.