Two-phase Discrete Fracture Matrix models with linear and nonlinear transmission conditions

Aghili, Joubine; Brenner, Konstantin; Hennicker, Julian; Masson, Roland; Trenty, Laurent

doi:10.1007/s13137-019-0118-6

Cited by 36 publications

(67 citation statements)

References 30 publications

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“…there exists a set of control points {x f } f∈F h such that for any f ∈ F h , x f ∈f and, for any pair of neighboring elements facets f, f ∈ F h sharing an edge σ of normal n, it holds that Kn is parallel to the segment joining the control points x f and x f . 2 We refer to the recent paper [20] for details on effective strategies for mesh generation and implementation aspects in the context of Virtual Element approximations of coupled multi-dimensional flow problems.…”

Section: Assumptionmentioning

confidence: 99%

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Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

et al. 2020

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This work deals with the efficient iterative solution of the system of equations stemming from mimetic finite difference discretization of a hybrid-dimensional mixed Darcy problem modeling flow in fractured porous media. We investigate the spectral properties of a mixed discrete formulation based on mimetic finite differences for flow in the bulk matrix and finite volumes for the fractures, and present an approximation of the factors in a set of approximate block factorization preconditioners that accelerates convergence of iterative solvers applied to the resulting discrete system. Numerical tests on significant three-dimensional cases have assessed the properties of the proposed preconditioners.

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Section: Assumptionmentioning

confidence: 99%

“…Several other numerical techniques have been proposed in recent years for fractured media flow, reflecting the importance of the subject for various applications. With no claim of completeness, in addition to the already cited literature we mention here some recent works by several research groups [2,[24][25][26]56].…”

Section: Introductionmentioning

confidence: 99%

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

et al. 2020

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“…These discontinuities play a major role and should be accurately captured in many important processes such as capillary driven imbibition in oil recovery or capillary barrier effects in oil migration and gas storage. This is particularly enhanced in the case of Discrete Fracture Matrix (DFM) models which exhibit highly contrasted permeabilities and capillary pressure curves between the fracture network and the surrounding matrix domains [9,35,42,38,34,12,13,2,31,14,1,3,45,4].…”

Section: Introductionmentioning

confidence: 99%

“…Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models. 1 discretizations like the Mixed Hybrid Finite Element (MHFE) method [34,2] or the Hybrid Finite Volume scheme [1]. The additional asset of nodal discretizations compared with cell centered discretizations is to include d.o.f.…”

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confidence: 99%

Vertex Approximate Gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Brenner

Masson

Quenjel

2020

Journal of Computational Physics

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In this article, a new nodal discretization is proposed for two-phase Darcy flows in heterogeneous porous media. The scheme combines the Vertex Approximate Gradient (VAG) scheme for the approximation of the gradient fluxes with an Hybrid Upwind (HU) approximation of the mobility terms in the saturation equation. The discretization in space incorporates naturally nodal interface degrees of freedom (d.o.f.) allowing to capture the transmission conditions at the interface between different rock types for general capillary pressure curves. It is shown to guarantee the physical bounds for the saturation unknowns as well as a nonnegative lower bound on the capillary energy flux term. Numerical experiments on several test cases exhibit that the scheme is more robust compared with previous approaches allowing the simulation of 3D large Discrete Fracture Matrix (DFM) models. 1 discretizations like the Mixed Hybrid Finite Element (MHFE) method [34,2] or the Hybrid Finite Volume scheme [1]. The additional asset of nodal discretizations compared with cell centered discretizations is to include d.o.f. at rock type interfaces as long as the mesh is conforming with the heterogeneities of the porous medium. These nodal pressures and saturations unknowns will be used to enforce the Darcy normal flux continuity equations combined with the saturation jump condition [16,11,27]. In this work, we will consider the Vertex Approximate Gradient (VAG) discretization developed in [27,12,13] for two-phase Darcy flows in heterogeneous media. The VAG scheme is based on nodal d.o.f. like Control Volume Finite Element (CVFE) methods but it also includes the cell d.o.f. which are eliminated at the linear algebra level at each Newton iteration without any fill-in. These cell d.o.f. provide an additional flexibility in the choice of the control volumes in order to avoid mixing different rock types at nodal control volumes. It has been shown in [27] to be more accurate than usual CVFE discretizations.The time integration will be chosen implicit to avoid severe time step restrictions in high velocity regions such as fractures. It will be also fully coupled to account for the strong coupling between the pressure and saturation unknowns in the transmission conditions at different rock type interfaces. As noticed in [5], the pressure and saturation unknowns cannot be decoupled at such different rock type interfaces to preserve the stability of the discretization.The selected numerical method should also provide a robust nonlinear convergence of the Newton type solver to allow for large time steps, typically at the time scale of the matrix in DFM simulations. Let us refer to [41,37] for alternative linearization schemes to the Newton method which aim to the improvement of the nonlinear solver. In this work, the additional nonlinear solver robustness will be first achieved by extension of the Hybrid Upwind (HU) transport scheme to the VAG discretization framework. The HU transport scheme has been introduced in [24,28] as an alternative to the Phase Pot...

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“…Such models are discussed e.g. in [2]. The present analysis does not cover such cases, but we refer to [39] for the formal derivation of such models in the specific context discussed here.…”

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confidence: 99%

Rigorous Upscaling of Unsaturated Flow in Fractured Porous Media

List¹,

Kumar²,

Pop³

et al. 2020

SIAM J. Math. Anal.

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In this work, we consider a mathematical model for flow in a unsaturated porous medium containing a fracture. In all subdomains (the fracture and the adjacent matrix blocks) the flow is governed by Richards' equation. The submodels are coupled by physical transmission conditions expressing the continuity of the normal fluxes and of the pressures. We start by analyzing the case of a fracture having a fixed width-length ratio, called ε > 0. Then we take the limit ε → 0 and give a rigorous proof for the convergence towards effective models. This is done in different regimes, depending on how the ratio of porosities and permeabilities in the fracture, respectively matrix scale with respect to ε, and leads to a variety of effective models. Numerical simulations confirm the theoretical upscaling results.

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Two-phase Discrete Fracture Matrix models with linear and nonlinear transmission conditions

Cited by 36 publications

References 30 publications

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

Preconditioning Techniques for the Numerical Solution of Flow in Fractured Porous Media

Vertex Approximate Gradient discretization preserving positivity for two-phase Darcy flows in heterogeneous porous media

Rigorous Upscaling of Unsaturated Flow in Fractured Porous Media

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