Analysis on general meshes of a discrete duality finite volume method for subsurface flow problems (original) (raw)
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Journal of Applied Mathematics, 2016
A Discrete Duality Finite Volume (DDFV) method to solve on unstructured meshes the flow problems in anisotropic nonhomogeneous porous media with full Neumann boundary conditions is proposed in the present work. We start with the derivation of the discrete problem. A result of existence and uniqueness of a solution for that problem is given thanks to the properties of its associated matrix combined with adequate assumptions on data. Their theoretical properties, namely, stability and error estimates (in discrete energy norms and 2-norm), are investigated. Numerical test is provided.
This paper deals with a finite volume analysis for diffusion phenomena, with prescribed periodic boundary conditions. The problem addressed here is the so-called local problem that arises in multi-scale physics such as fluid-flows in porous materials with spatially periodic microstructures. The so-called local problem is discretized with a finite volume method of new generation, namely Discrete Duality Finite Volumes (DDFV for short). In view to deal with stability analysis of the DDFV solution, an adequate discrete H 1 0 −norm is defined and a discrete version of Poincaré inequality involving this discrete norm is introduced. The DDFV problem displays the same main features as the continuous one: (i) A compatibility condition is required from the discrete source-term for existence of discrete solutions; (ii) Two orthogonality conditions are required from DDFV solution for uniqueness. Note that only one such condition is required from classical cell-centered finite volumes dealing with isotropic models. Error estimate results are obtained for L 2 − norm and for a discrete H 1 −norm. Numerical tests confirm our theoretical results. Note that these results are in accordance with those from the fifth international conference on Finite Volumes for Complex Applications (FVCA5) held in France in 2008.
The Discrete Duality Finite Volume Method for Convection-diffusion Problems
SIAM Journal on Numerical Analysis, 2010
In this paper we extend the Discrete Duality Finite Volume (DDFV) formulation to the steady convection-diffusion equation. The discrete gradients defined in DDFV are used to define a cellbased gradient for the control volumes of both the primal and dual meshes, in order to achieve a higher-order accurate numerical flux for the convection term. A priori analysis is carried out to show convergence of the approximation and a global first-order convergence rate is derived. The theoretical results are confirmed by some numerical experiments.
Unstructured mesh based discretization techniques can offer advantages relative to standard finite difference approaches which are still largely used to model flux flow through porous media (e.g. petroleum reservoir simulation), due to their flexibility to model complex geological features and due to their capacity to incorporate mesh adaptation techniques. In this paper we consider an unstructured edge-based finite volume formulation (FV) which is used to solve the elliptic pressure equation and the non-linear hyperbolic equation that arises in reservoir problems when IMPES (IMplicit Pressure Explicit Saturation) techniques are used together with a global pressure approach. The IMPES method is a segregated type method in which the flow equations are manipulated in order to produce an elliptic pressure equation solved implicitly and a hyperbolic type saturation equation which is solved explicitly. The numerical formulation includes an introduction of an adaptive artificial dissipative term capable to deal with the "non-viscous" terms of the Bucklet-Leverett (saturation) equation. In order to improve the quality of the solution we have used two different mesh adaptation tools, mesh embedding and remeshing. A posteriori error estimator based on gradient recovery is used to control the adaptive procedure. Some simple two dimensional model examples are solved in order to show the potentiality of the formulation presented.
Convergence of a Mixed Finite Element: Finite Volume Method for the Two Phase Flow in Porous Media
1997
As a model problem for the miscible and immiscible two phase ow we consider the following system of partial di erential equations: div u(x; t) = 0 u(x; t) = a(c; x) (rp(x; t) + (c; x)) @tc(x; t) + div (u(x; t)c(x; t)) div (rc(x; t)) = f(x; t); with (x; t) 2 (0; T). Here u denotes the Darcy velocity, p the pressure and c the concentration of one phase of the uid. Considering density driven ow or immiscible ow of water and oil in a reservoir the convection of the concentration is dominant to the di usion. Thus we have to look at this system of partial di erential equations as a singular perturbed problem in. For small di usions (0 < << 1) standard Galerkin Finite Element approximations do not produce stable solutions. Therefore we propose a combined Mixed Finite Element-Finite Volume discretisation, speci cally to handle this convection dominated di usion problem. Taking into account the dependence on the di usion parameter we prove a convergence result for this rst order scheme on triangular meshes.
Convergence analysis of an MPFA method for flow problems in anisotropic heterogeneous porous media
International Journal on Finite Volumes, 5, pp 17-56 (2008), 2008
Our purpose in this paper is to present the theoretical analysis of a Multi-Point Flux Approximation method (MPFA method). We start with the derivation of the discrete problem, and then we give a result of existence and uniqueness of a solution for that problem. As in finite element theory, Lagrange interpolation is used to define three classes of continuous and locally polynomial approximate solutions. For analyzing the convergence of these different classes of solutions, the notions of weak and weak-star MPFA approximate solutions are introduced. Their theoretical properties, namely stability and error estimates (in discrete energy norms, L 2 − norm and L ∞ − norm), are investigated. These properties play a key role in the analysis (in terms of error estimates for diverse norms) of different classes of continuous and locally polynomial approximate solutions mentioned above.
Combined Mixed Finite Element and Finite Volume for flow and transport in porous media
This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation, and a conservative finite volume scheme for the concentration equation. A series of numerical examples demonstrates the effectiveness of the methodology for a coupled system which includes an elliptic equation and a diffusion-convection equation arising in modeling of flow and transport in heterogeneous porous media.
Advances in Water Resources, 2013
Transport problems occurring in porous media and including convection, diffusion and chemical reactions, can be well represented by systems of Partial Differential Equations. In this paper, a numerical procedure is proposed for the fast and robust solution of flow and transport problems in 2D heterogeneous saturated media. The governing equations are spatially discretized with unstructured triangular meshes that must satisfy the Delaunay condition. The solution of the flow problem is split from the solution of the transport problem and it is obtained with an approach similar to the Mixed Hybrid Finite Elements method, that always guarantees the M-property of the resulting linear system. The transport problem is solved applying a prediction/correction procedure. The prediction step analytically solves the convective/reactive components in the context of a MAST Finite Volume scheme. The correction step computes the anisotropic diffusive components in the context of a recently proposed Finite Elements scheme. Massa balance is locally and globally satisfied in all the solution steps. Convergence order and computational costs are investigated and model results are compared with literature ones.
Defect and Diffusion Forum, 2010
This paper addresses the key issue of calculating fluxes at the control-volume interfaces when finite-volumes are employed for the solution of partial differential equations. This calculation becomes even more significant when unstructured grids are used, since the flux approximation involving only two grid points is no longer correct. Two finite volume methods with the ability in dealing with unstructured grids, the EbFVM-Element-based Finite Volume Method and the MPFA-Multi-Point Flux Approximation are presented, pointing out the way the fluxes are numerically evaluated. The methods are applied to a porous media flow with full permeability tensors and non-orthogonal grids and the results are compared with analytical solutions. The results can be extended to any diffusion operator, like heat and mass diffusion, in anisotropic media.