High-order bound-preserving local discontinuous Galerkin methods for incompressible and immiscible two-phase flows in porous media. (English) Zbl 1541.65098
Summary: In this paper, we develop high-order bound-preserving (BP) local discontinuous Galerkin methods for incompressible and immiscible two-phase flows in porous media, and employ implicit pressure explicit saturation (IMPES) methods for time discretization, which is locally mass conservative for both phases. Physically, the saturations of the two phases, \(S_w\) and \(S_n\), should belong to the range of \([0, 1]\). Nonphysical numerical approximations may result in instability of the simulation. Therefore, it is necessary to construct a BP technique to obtain physically relevant numerical approximations. However, the saturation does not satisfy the maximum principle, so most of the existing BP techniques cannot be applied directly. The main idea is to apply the positivity-preserving techniques to both \(S_w\) and \(S_n\), respectively, and enforce \(S_w +S_n =1\) simultaneously. Numerical examples are given to demonstrate the high-order accuracy of the scheme and effectiveness of the BP technique.
MSC:
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 76S05 | Flows in porous media; filtration; seepage |
| 76T06 | Liquid-liquid two component flows |
| 76D45 | Capillarity (surface tension) for incompressible viscous fluids |
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76M20 | Finite difference methods applied to problems in fluid mechanics |
| 35Q35 | PDEs in connection with fluid mechanics |
Keywords:
two-phase flow; local discontinuous Galerkin method; bound-preserving; capillary pressure; local mass conservationReferences:
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