Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

Cao, Xiulei; Mitra, Koondanibha

doi:10.1016/j.cam.2018.12.022

Cited by 7 publications

(8 citation statements)

References 43 publications

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“…Since then the dynamic capillary term has been measured experimentally [12,33] and it was used successfully to explain overshoots [18,43,55,62,[67][68][69]. Also the well-posedness of the dynamic capillarity model has been proved [7,16,17,41] and numerical methods have been investigated [13,14,23,34,35].…”

Section: (Right)mentioning

confidence: 99%

“…This expression is obtained by considering sign in (13) and (20) as a function that can be eliminated. With τ = 0 in (13), this results in (22). Since the fraction (22) is bounded by 0 and 1 for (S, p) ∈ H, we have lim S 0 k rα (S, p) = k…”

Section: Relative Permeabilitymentioning

confidence: 99%

“…Typical curves for f (j) and h (j) are shown in Figure 3. The equations (11), (13) and ( 23) are the complete set of equations for our model.…”

Section: Relative Permeabilitymentioning

confidence: 99%

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Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

et al. 2020

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In this work, we study the behavior of saturation fronts for two-phase flow through a long homogeneous porous column . In particular, the model includes hysteresis and dynamic effects in the capillary pressure and hysteresis in the permeabilities. The analysis uses traveling wave approximation. Entropy solutions are derived for Riemann problems that are arising in this context. These solutions belong to a much broader class compared to the standard Oleinik solutions, where hysteresis and dynamic effects are neglected. The relevant cases are examined and the corresponding solutions are categorized. They include nonmonotone profiles, multiple shocks, and self-developing stable saturation plateaus. Numerical results are presented that illustrate the mathematical analysis. Finally, we discuss the implication of our findings in the context of available experimental results.

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Section: (Right)mentioning

confidence: 99%

Section: Relative Permeabilitymentioning

confidence: 99%

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Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

et al. 2020

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“…Many works [9][10][11][12] are devoted to the numerical implementation of the two-phase fluid flow model with the nonequilibrium law from [4]. For example, in [9], a second-order numerical scheme for both spatial and temporal variables is proposed using a mixed finite element method with the lowest order Thomas-Raviar elements and an implicit Euler scheme.…”

Section: Introductionmentioning

confidence: 99%

Analysis of a finite volume element scheme for solving the model two-phase nonequilibrium flow problem

Omariyeva

Yergaliyev

2022

JMMCS

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The paper proposes a hybrid numerical method for solving a model problem of two-phase nonequilibrium flow of an incompressible fluid in a porous medium. This problem is relevant in the modern theory of the motion of multiphase fluids in porous media and has many applications. The studied model is based on the assumption that the relative phase permeabilities and capillary pressure depend not only on saturation, but also on its time derivative. The saturation equation in this problem refers to the type of convection-diffusion with a predominance of convection, which also includes a third-order term to account for the nonequilibrium effects. Due to the hyperbolic nature of the equation, its solution is accompanied by a number of difficulties that lead to the need for an appropriate choice of the solution method. In contrast to previous works, this paper uses a finite volume element method for solving the problem, the construction of which is based on integral balance equations, and an approximate solution is chosen from the finite element space. To discretize the problem, two different dual grids are used based on the main triangulation. In this paper, a number of a priori estimates are obtained which yields the unconditional stability of the scheme as well as its convergence with the second order. The advantages of the approach used include the local conservatism of the scheme, as well as the comparative simplicity of the software implementation of the method. These results are confirmed by a numerical test carried out on the example of a model problem.

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“…In terms of modeling, this is achieved by including the so-called dynamic effects [Beliaev et al-2001, Gray et al-1998, Mikelic-2010. Numerous papers investigate the existence of a solution for systems including such effects, among them we cite [Cao et al-2016, Koch et al-2012, Milisic -2018. Furthermore, the problem has been already studied numerically in, e.g., [Abreu et al-217, Abreu et al-2020, Cao et al-2019a.…”

Section: Introductionmentioning

confidence: 99%

Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media

Illiano¹,

Both²,

Pop³

et al. 2020

Preprint

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We study several iterative methods for fully coupled flow and reactive transport in porous media. The resulting mathematical model is a coupled, nonlinear evolution system. The flow model component builds on the Richards equation, modified to incorporate nonstandard effects like dynamic capillarity and hysteresis, and a reactive transport equation for the solute. The two model components are strongly coupled. On one hand, the flow affects the concentration of the solute; on the other hand, the surface tension is a function of the solute, which impacts the capillary pressure and, consequently, the flow. After applying an Euler implicit scheme, we consider a set of iterative linearization schemes to solve the resulting nonlinear equations, including both monolithic and two splitting strategies. The latter include a canonical nonlinear splitting and an alternate linearized splitting, which appears to be overall faster in terms of numbers of iterations, based on our numerical studies. The (time discrete) system being nonlinear, we investigate different linearization methods. We consider the linearly convergent L-scheme, which converges unconditionally, and the Newton method, converging quadratically but subject to restrictions on the initial guess. Whenever hysteresis effects are included, the Newton method fails to converge. The L-scheme converges; nevertheless, it may require many iterations. This aspect is improved by using the Anderson acceleration. A thorough comparison of the different solving strategies is presented in five numerical examples, implemented in MRST, a toolbox based on MATLAB.

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Error estimates for a mixed finite element discretization of a two-phase porous media flow model with dynamic capillarity

Cited by 7 publications

References 43 publications

Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

Fronts in two‐phase porous media flow problems: The effects of hysteresis and dynamic capillarity

Analysis of a finite volume element scheme for solving the model two-phase nonequilibrium flow problem

Efficient Solvers for Nonstandard Models for Flow and Transport in Unsaturated Porous Media

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