Cindy Guichard

Vertex-centred discretization of multiphase compositional Darcy flows on general meshes

This paper concerns the discretisation on general 3D meshes of multiphase compositional Darcy flows in heterogeneous anisotropic porous media. Extending Coats’ formulation [15] to an arbitrary number of phases, the model accounts for the coupling of the mass balance of each component with the pore volume conservation and the thermodynamical equilibrium and dynamically manages phase appearance and disappearance. The spatial discretisation of the multiphase compositional Darcy flows is based on a generalisation of the Vertex Approximate Gradient scheme, already introduced for single-phase diffusive problems in [24]. It leads to an unconditionally coercive scheme for arbitrary meshes and permeability tensors. The stencil of this vertex-centred scheme typically comprises 27 points on topologically Cartesian meshes, and the number of unknowns on tetrahedral meshes is considerably reduced, compared with the usual cell-centred approaches. The efficiency of our approach is exhibited on several examples, including the nearwell injection of miscible CO2 in a saline aquifer taking into account the vaporisation of H2O in the gas phase as well as the precipitation of salt.

Gradient discretization of hybrid dimensional Darcy flows in fractured porous media

This article deals with the discretization of hybrid dimensional Darcy flows in fractured porous media. These models couple the flow in the fractures represented as surfaces of codimension one with the flow in the surrounding matrix. The convergence analysis is carried out in the framework of gradient schemes which accounts for a large family of conforming and nonconforming discretizations. The vertex approximate gradient scheme and the hybrid finite volume scheme are extended to such models and are shown to verify the gradient scheme framework. Our theoretical results are confirmed by numerical experiments performed on tetrahedral, Cartesian and hexahedral meshes in heterogeneous isotropic and anisotropic porous media.

Convergence of a nonlinear entropy diminishing Control Volume Finite Element scheme for solving anisotropic degenerate parabolic equations

In this paper, we propose and analyze a Control Volume Finite Elements (CVFE) scheme for solving possibly degenerated parabolic equations. This scheme does not require the introduction of the so-called Kirchhoff transform in its definition. We prove that the discrete solution obtained \emph{via} the scheme remains in the physical range, and that the natural entropy of the problem decreases with time. The convergence of the method is proved as the discretization steps tend to

The gradient discretisation method

0

Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure

. Finally, numerical examples illustrate the efficiency of the method.

Benchmark 3D: the VAG scheme

This monograph is dedicated to the presentation of the gradient discretisation method (GDM) and to some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of partial differential equations. The GDM is a framework which contains classical and recent discretisation schemes for diffusion problems of different kinds: linear or non-linear, steady-state or time-dependent. The schemes may be conforming or non-conforming, low or high order, and may be built on very general meshes. In this monograph, the core properties that are required to prove the convergence of a GDM are stressed, and the analysis of the method is performed on a series of elliptic and parabolic problems. As a result, for these models, any scheme entering the GDM framework is known to converge. A key feature of this monograph is the presentation of techniques and results which enable a complete convergence analysis of the GDM on fully non-linear, and sometimes degenerate, models. The scope of some of these techniques and results goes beyond the GDM, and makes them potentially applicable to numerical schemes not (yet) known to fit into this framework. Appropriate tools are also provided to easily check whether a given scheme satisfies the core properties of a GDM. Using these tools, it is shown that a number of methods are GDMs; some of these methods are classical, such as the conforming finite elements, the non-conforming finite elements, and the mixed finite elements. Others are more recent, such as the discontinuous Galerkin methods, the hybrid mimetic mixed or nodal mimetic finite differences methods, some discrete duality finite volume schemes, and some multi-point flux approximation schemes.

TP or not TP, that is the question

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.

Uniform-in-time convergence of numerical schemes for Richards' and Stefan's models

Let Ωbe a bounded open domain of \(\mathbb{R}^3\) let\(f\epsilon\rm L^2(\Omega)\) and let\(\Lambda\) be a measurable function from Ω to the set \(m^3(\mathbb{R)} of 3 \times 3\) matrices, such that for a.e.

Entropy-Diminishing CVFE Scheme for Solving Anisotropic Degenerate Diffusion Equations

We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The second class contains the so-called approximate gradient schemes, which include finite elements with mass lumping, mixed finite elements, and mimetic finite differences. Both (classes of) schemes are nonconforming and can be expressed using discrete function and gradient reconstructions within a variational formulation. Each class has its specific advantages and drawbacks: monotony properties are natural with the two-point finite volume scheme, but meshes are restricted due to consistency issues; on the contrary, gradient schemes can be used on general meshes, but monotony properties are difficult to obtain.

Multiphase Flow in Porous Media Using the VAG Scheme

We prove that all Gradient Schemes—which include Finite Element, Mixed Finite Element, Finite Volume methods—converge uniformly in time when applied to a family of nonlinear parabolic equations which contains Richards and Stefan’s models. We also provide numerical results to confirm our theoretical analysis.