A control volume based finite element method for simulating incompressible two-phase flow in heterogeneous porous media and its application to reservoir engineering

Abstract Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in [7]. Theoretical convergence is proved via a compactness argument. The numerical phase saturation converges strongly to a weak solution in L 2 in space and in time whereas the numerical phase pressures converge strongly to weak solutions in L 2 in space almost everywhere in time. The proof is not straightforward because of the degeneracy of the phase mobilities and the unboundedness of the derivative of the capillary pressure.

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A finite element method for degenerate two-phase flow in porous media. Part I: Well-posedness

Journal of Numerical Mathematics ◽

10.1515/jnma-2020-0004 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Vivette Girault ◽

Beatrice Riviere ◽

Loic Cappanera

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Porous Media ◽

Two Phase Flow ◽

Flow Problem ◽

Flow In Porous Media ◽

Phase Flow ◽

Two Phase ◽

Existence Of A Solution ◽

Element Method

Abstract A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a solution are established.

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Chapter V A Finite Element Method For Incompressible Two-Phase Flow

Mathematical Models and Finite Elements for Reservoir Simulation - Single Phase, Multiphase and Multicomponent Flows through Porous Media - Studies in Mathematics and Its Applications ◽

10.1016/s0168-2024(08)70083-3 ◽

1986 ◽

pp. 311-364

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Two Phase Flow ◽

Phase Flow ◽

Two Phase ◽

Element Method

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Variational multi-scale finite element method for the two-phase flow of polymer melt filling process

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-07-2019-0599 ◽

2019 ◽

Vol 30 (3) ◽

pp. 1407-1426 ◽

Cited By ~ 12

Author(s):

Xuejuan Li ◽

Ji-Huan He

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Numerical Method ◽

Two Phase Flow ◽

Polymer Melt ◽

Phase Flow ◽

Filling Process ◽

Two Phase ◽

Content Type ◽

Element Method

Purpose The purpose of this paper is to develop an effective numerical algorithm for a gas-melt two-phase flow and use it to simulate a polymer melt filling process. Moreover, the suggested algorithm can deal with the moving interface and discontinuities of unknowns across the interface. Design/methodology/approach The algebraic sub-grid scales-variational multi-scale (ASGS-VMS) finite element method is used to solve the polymer melt filling process. Meanwhile, the time is discretized using the Crank–Nicolson-based split fractional step algorithm to reduce the computational time. The improved level set method is used to capture the melt front interface, and the related equations are discretized by the second-order Taylor–Galerkin scheme in space and the third-order total variation diminishing Runge–Kutta scheme in time. Findings The numerical method is validated by the benchmark problem. Moreover, the viscoelastic polymer melt filling process is investigated in a rectangular cavity. The front interface, pressure field and flow-induced stresses of polymer melt during the filling process are predicted. Overall, this paper presents a VMS method for polymer injection molding. The present numerical method is extremely suitable for two free surface problems. Originality/value For the first time ever, the ASGS-VMS finite element method is performed for the two-phase flow of polymer melt filling process, and an effective numerical method is designed to catch the moving surface.

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