Mathematical and Numerical Study of an Industrial Scheme for Two-phase Flows in Porous Media under Gravity

Enchéry, Guillaume; Masson, Roland; Wolf, Sylvie; Eymard, Robert

doi:10.2478/cmam-2002-0019

Cited by 17 publications

(15 citation statements)

References 21 publications

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“…We prove that

0 ⩽ S_{w, i}^{n + 1} ⩽ 1

by contradiction. This proof is independent of the flux approximation and has been shown analogously for other flux approximations (e.g., ). First, we prove that

0 ⩽ S_{w, i}^{n + 1}

.…”

Section: Numerical Resultssupporting

confidence: 68%

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

Schneider

Flemisch

Helmig

2016

Numerical Methods in Fluids

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Summary This article presents a new nonlinear finite‐volume scheme for the nonisothermal two‐phase two‐component flow equations in porous media. The face fluxes are approximated by a nonlinear two‐point flux approximation, where transmissibilities nonlinearly depend on primary variables. Thereby, we mainly follow the ideas proposed by Le Potier combined with a harmonic averaging point interpolation strategy for the approximation of arbitrary heterogeneous permeability fields on polygonal grids. The behavior of this interpolation strategy is analyzed, and its limitation for highly anisotropic permeability tensors is demonstrated. Moreover, the condition numbers of occurring matrices are compared with linear finite‐volume schemes. Additionally, the convergence behavior of iterative solvers is investigated. Finally, it is shown that the nonlinear scheme is more efficient than its linear counterpart. Copyright © 2016 John Wiley & Sons, Ltd.

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“…We prove that

0 ⩽ S_{w, i}^{n + 1} ⩽ 1

by contradiction. This proof is independent of the flux approximation and has been shown analogously for other flux approximations (e.g., ). First, we prove that

0 ⩽ S_{w, i}^{n + 1}

.…”

Section: Numerical Resultssupporting

confidence: 68%

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

Schneider

Flemisch

Helmig

2016

Numerical Methods in Fluids

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“…Some new problems arise, for example in the proof of the discrete maximum principle, or in the proof of the discrete L 2 (0, T ; H 1 (Ω)) in the multidimensional case (note that new results in this direction have been obtained, in the case of no capillary pressure, in [17] …”

Section: Discussionmentioning

confidence: 99%

Mathematical study of a petroleum-engineering scheme

Eymard¹,

Herbin²,

Michel³

2003

ESAIM: M2AN

116

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Abstract. Models of two phase flows in porous media, used in petroleum engineering, lead to a system of two coupled equations with elliptic and parabolic degenerate terms, and two unknowns, the saturation and the pressure. For the purpose of their approximation, a coupled scheme, consisting in a finite volume method together with a phase-by-phase upstream weighting scheme, is used in the industrial setting. This paper presents a mathematical analysis of this coupled scheme, first showing that it satisfies some a priori estimates: the saturation is shown to remain in a fixed interval, and a discrete L 2 (0, T ; H 1 (Ω)) estimate is proved for both the pressure and a function of the saturation. Thanks to these properties, a subsequence of the sequence of approximate solutions is shown to converge to a weak solution of the continuous equations as the size of the discretization tends to zero.Mathematics Subject Classification. 35K65, 76S05, 65M12.

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“…We follow [8,9,11] and use a topological degree argument [6] for the proof of existence. We follow [8,9,11] and use a topological degree argument [6] for the proof of existence.…”

Section: Saturation Estimatementioning

confidence: 99%

“…Previous analysis on the finitevolume method for coupled flow and transport focused on the two-phase case, and either neglected buoyancy forces [11] or treated the numerical flux with PPU [8,22]. Previous analysis on the finitevolume method for coupled flow and transport focused on the two-phase case, and either neglected buoyancy forces [11] or treated the numerical flux with PPU [8,22].…”

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confidence: 99%

Analysis of Hybrid Upwinding for Fully-Implicit Simulation of Three-Phase Flow with Gravity

Hamon¹

2016

SIAM J. Numer. Anal.

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The systems of algebraic equations arising from implicit (backward-Euler) finitevolume discretization of the conservation laws governing multiphase flow in porous media are quite challenging for nonlinear solvers. In the presence of countercurrent flow due to buoyancy, the numerical flux obtained with single-point Phase-Potential Upwinding (PPU) is not differentiable, which causes convergence difficulties for nonlinear solvers. Recently, [Lee, Efendiev, and Tchelepi, Adv. Water Resour., 82 (2015), pp. 27-38] proposed a hybrid upwinding strategy for two-phase flow that gives rise to a differentiable numerical flux across the entire viscous-gravity parameter space. Here, we first present an Implicit Hybrid Upwinding (IHU) scheme for hyperbolic conservation laws, extending the work of Lee, Efendiev, and Tchelepi to an arbitrary number of fluid phases. We show that the numerical flux obtained with the IHU is consistent, and a monotone function of its own saturation. It is also a differentiable function of the saturations in one spatial dimension. In addition, we generalize the IHU numerical scheme to solve the elliptic-hyperbolic partial differential equations governing coupled flow and transport in multiple dimensions. For this problem, we derive pressure and saturation estimates, and prove the existence of a solution to the scheme. Finally, we apply the IHU scheme to the Buckley-Leverett problem with buoyancy. Our numerical experiments confirm that IHU is nonoscillatory, convergent, and improves upon Newton's method convergence rate for two-and three-phase flow.

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Mathematical and Numerical Study of an Industrial Scheme for Two-phase Flows in Porous Media under Gravity

Cited by 17 publications

References 21 publications

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

Monotone nonlinear finite‐volume method for nonisothermal two‐phase two‐component flow in porous media

Mathematical study of a petroleum-engineering scheme

Analysis of Hybrid Upwinding for Fully-Implicit Simulation of Three-Phase Flow with Gravity

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