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The method of mixed volume element‐characteristic mixed volume element and its numerical analysis for three‐dimensional slightly compressible two‐phase displacement
Author(s) -
Yuan Yirang,
Sun Tongjun,
Li Changfeng,
Yang Qing
Publication year - 2018
Publication title -
numerical methods for partial differential equations
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.901
H-Index - 61
eISSN - 1098-2426
pISSN - 0749-159X
DOI - 10.1002/num.22220
Subject(s) - mathematics , mathematical analysis , finite element method , compressibility , numerical analysis , partial differential equation , mixed finite element method , mechanics , physics , thermodynamics
Numerical simulation of oil‐water two‐phase displacement is a fundamental problem in energy mathematics. The mathematical model for the compressible case is defined by a nonlinear system of two partial differential equations: (1) a parabolic equation for pressure and (2) a convection‐diffusion equation for saturation. The pressure appears within the saturation equation, and the Darcy velocity controls the saturation. The flow equation is solved by the conservative mixed volume element method. The order of the accuracy is improved by the Darcy velocity. The conservative mixed volume element with characteristics is applied to compute the saturation, that is, the diffusion is discretized by the mixed volume element and convection is computed by the method of characteristics. The method of characteristics has strong computational stability at sharp fronts and avoids numerical dispersion and nonphysical oscillation. Small time truncation error and accuracy are obtained through this method. The mixed volume element simulates diffusion, saturation, and the adjoint vector function simultaneously. By using the theory and technique of a priori estimates of differential equations, convergence of the optimal second order inl 2norm is obtained. Numerical examples are provided to show the effectiveness and viability of this method. This method provides a powerful tool for solving challenging benchmark problems.