Total velocity-based finite volume discretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pressure

Brenner, Konstantin; Droniou, Jérôme; Masson, Roland; Quenjel, El Houssaine

doi:10.1093/imanum/drab018

Cited by 11 publications

(8 citation statements)

References 40 publications

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“…For future researches, we suggest to test the so-called method A on a two-phase flow test and to compare it to the approaches presented in [7]. Moreover, in a forthcoming work, we propose two other methods to really impose the pressure continuity condition at interfaces.…”

Section: Discussionmentioning

confidence: 99%

“…For flows in highly heterogeneous porous media, rigorous mathematical results have been obtained for schemes involving the introduction of additional interface unknowns and Kirchhoff's transforms (see for instance [6,11,12,23]), or under the non-physical assumption that the mobilities are strictly positive [28,29]. It was established very recently in [7] that cell-centered finite-volumes with (hybrid) upwinding also converge for two-phase flows in heterogeneous domains, but with a specific treatment of the interfaces located at the heterogeneities. Here, the novelty lies in the fact that we do not consider any specific treatment of the interface in the design of the scheme.…”

Section: Goal and Positioning Of The Papermentioning

confidence: 99%

“…Moreover, one can still make use of the parametrized cut-Newton method proposed in [4] to compute the solution to the nonlinear system corresponding to the scheme. This method appears to be very efficient, while it avoids the possibly difficult construction of compatible parametrizations at the interfaces as in [7][8][9].…”

Section: Goal and Positioning Of The Papermentioning

confidence: 99%

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Upstream mobility Finite Volumes for the Richards equation in heterogenous domains

Bassetto¹,

Cancès²,

Enchéry³

et al. 2021

Preprint

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This paper is concerned with the Richards equation in a heterogeneous domain, each subdomain of which is homogeneous and represents a rocktype. Our first contribution is to rigorously prove convergence toward a weak solution of cell-centered finite-volume schemes with upstream mobility and without Kirchhoff's transform. Our second contribution is to numerically demonstrate the relevance of locally refining the grid at the interface between subregions, where discontinuities occur, in order to preserve an acceptable accuracy for the results computed with the schemes under consideration.

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Section: Discussionmentioning

confidence: 99%

Section: Goal and Positioning Of The Papermentioning

confidence: 99%

Section: Goal and Positioning Of The Papermentioning

confidence: 99%

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Upstream mobility Finite Volumes for the Richards equation in heterogenous domains

Bassetto¹,

Cancès²,

Enchéry³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Fractional flow formulations are well established for sequential simulations [25,26] and discrete interface conditions [27,28]. Recently, the total velocity formulation -the most common fractional flow formulation -has also been used to enable significant benefits for fully implicit simulations in terms of accuracy [17,28,29] and nonlinear convergence [12,16,[30][31][32]. In this work we focus on the latter, where we exploit a fractional flow formulation to distinguish the parabolic total flow subproblem from the hyperbolic transport subproblem in the context of a fully implicit method.…”

Section: Total Velocity Formulationmentioning

confidence: 99%

Smooth Implicit Hybrid Upwinding for Compositional Multiphase Flow in Porous Media

Bosma,

Hamon,

Mallison

et al. 2021

Preprint

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In subsurface multiphase flow simulations, poor nonlinear solver performance is a significant runtime sink. The system of fully implicit mass balance equations is highly nonlinear and often difficult to solve for the nonlinear solver, generally Newton(-Raphson). Although the physical problem is inherently nonlinear, discretization schemes introduce several strong nonlinearities that can cause Newton iterations to diverge or converge very slowly. This frequently results in time step cuts, leading to wasted iterations and computationally expensive simulations. Much literature has looked into how to improve the nonlinear solver through enhancements or safeguarding updates. In this work, we take a different approach; we aim to improve convergence with a smoother finite volume discretization scheme which is more suitable for the Newton solver.Building on recent work, we propose a novel total velocity hybrid upwinding scheme with weighted average flow mobilities (WA-HU TV) that is unconditionally monotone and extends to compositional multiphase simulations. Analyzing the solution space of a one-cell problem, we demonstrate the improved properties of the scheme and explain how it leverages the advantages of both phase potential upwinding and arithmetic averaging. This results in a flow subproblem that is smooth with respect to changes in the sign of phase fluxes, and is well-behaved when phase velocities are large or when co-current viscous forces dominate. Additionally, we propose a WA-HU scheme with a total mass (WA-HU TM) formulation that includes phase densities in the weighted averaging.The proposed WA-HU TV consistently outperforms existing schemes, yielding benefits from 5% to over 50% reduction in nonlinear iterations. The WA-HU TM scheme also shows promising results; in some cases leading to even more efficiency. However, WA-HU TM can occasionally also lead to convergence issues. Overall, based on the current results, we recommend the adoption of the WA-HU TV scheme as it is highly efficient and robust.

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“…Following [23], the model also includes a layer of damaged rock at matrix fracture interfaces. This additional accumulation term plays a major role in the numerical analysis of the model and also improves the nonlinear convergence at each time step of the simulation [23,12]. It must be kept sufficiently small to maintain the accuracy of the solution (see [23]).…”

Section: Introductionmentioning

confidence: 99%

Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

Bonaldi,

Brenner,

Droniou

et al. 2020

Preprint

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We consider a two-phase Darcy flow in a fractured and deformable porous medium for which the fractures are described as a network of planar surfaces leading to so-called hybrid-dimensional models. The fractures are assumed open and filled by the fluids and small deformations with a linear elastic constitutive law are considered in the matrix. As opposed to [10], the phase pressures are not assumed continuous at matrix fracture interfaces, which raises new challenges in the convergence analysis related to the additional interfacial equations and unknowns for the flow. As shown in [16,2], unlike single phase flow, discontinuous pressure models for two-phase flows provide a better accuracy than continuous pressure models even for highly permeable fractures. This is due to the fact that fractures fully filled by one phase can act as barriers for the other phase, resulting in a pressure discontinuity at the matrix fracture interface.The model is discretized using the gradient discretization method [22], which covers a large class of conforming and non conforming schemes. This framework allows for a generic convergence analysis of the coupled model using a combination of discrete functional tools. In this work, the gradient discretization of [10] is extended to the discontinuous pressure model and the convergence to a weak solution is proved. Numerical solutions provided by the continuous and discontinuous pressure models are compared on gas injection and suction test cases using a Two-Point Flux Approximation (TPFA) finite volume scheme for the flows and P2 finite elements for the mechanics.

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Total velocity-based finite volume discretization of two-phase Darcy flow in highly heterogeneous media with discontinuous capillary pressure

Cited by 11 publications

References 40 publications

Upstream mobility Finite Volumes for the Richards equation in heterogenous domains

Upstream mobility Finite Volumes for the Richards equation in heterogenous domains

Smooth Implicit Hybrid Upwinding for Compositional Multiphase Flow in Porous Media

Gradient discretization of two-phase poro-mechanical models with discontinuous pressures at matrix fracture interfaces

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