Mathematical analysis of a discrete fracture model coupling Darcy flow in the matrix with Darcy-Forchheimer flow in the fracture

We consider a model for flow in a porous medium with a fracture in which the flow in the fracture is governed by the Darcy-Forchheimer law while that in the surrounding matrix is governed by Darcy’s law. We give an appropriate mixed, variational formulation and show existence and uniqueness of the solution. To show existence we give an analogous formulation for the model in which the Darcy-Forchheimer law is the governing equation throughout the domain. We show existence and uniqueness of the solution and show that the solution for the model with Darcy’s law in the matrix is the weak limit of solutions of the model with the Darcy-Forchheimer law in the entire domain when the Forchheimer coefficient in the matrix tends toward zero. Key-words: flow in porous media, fractures, Darcy-Forchheimer flow, solvability, regularization, monotone operators This work was supported by the Franco-German cooporation program PHC Procope ∗ University of Erlangen-Nuremberg, Department of Mathematics, Cauerstr. 11, D-91058 Erlangen, Germany, e-mail: [email protected] † Inria Paris-Rocquencourt, B.P. 105, F-78153, Le Chesnay, France, email: [email protected] Analyse mathématique d’un modèle discret de fractures couplant un écoulement de Darcy dans la matrice avec un écoulement de Darcy-Forchheimer dans la fracture Résumé : Nous nous intéressons à un modèle d’écoulement dans un milieu poreux avec une fracture. Dans ce modèle l’écoulement dans la fracture est gouverné par la loi de Darcy-Forchheimer alors que l’écoulement dans la matrice est gouverné par la loi de Darcy. Nous proposons une formulation variationelle, mixte pour ce modèle et nous démontrons l’existence et l’unicité de la solution. Pour montrer l’existence nous proposons aussi une formulation analogue pour un modèle basé sur un écoulement Darcy-Forchheimer dans tout le domaine. Nous montrons l’existence et l’unicité de la solution pour ce deuxième modèle et montrons que la solution pour le premièr modèle est la limite faible de celle du deuxième modèle quand le coefficient de Forchheimer dans la matrice tend vers zéro. Mots-clés : écoulement en milieu poreux, fractures, écoulement de Darcy-Forchheimer , solvabilité, régularisation, opérateurs monotones Coupling Darcy and Darcy-Forchheimer flow in a fractured porous medium 3 Introduction Numerical modeling of fluid flow in a porous medium, even single-phase, incompressible fluid flow, is complicated because the permeability coefficient characterizing the medium may vary over several orders of magnitude within a region quite small in comparison to the dimensions of the domain. This is in particular the case when fractures are present in the medium. Fractures have at least one dimension that is very small, much smaller than a reasonable discretization parameter given the size of the domain, but are much more permeable (or possibly, due to crystalization , much less permeable) than the surrounding medium. They thus have a very significant influence on the fluid flow but adapting a standard finite element or finite volume mesh to handle flow in the fractures poses obvious problems. Many models have been developed to study fluid flow in porous media with fractures. Models may employ a continuum representation of fractures as in the double porosity models derived by homogenization or they may be discrete fracture models. Among the discrete fracture models are models of discrete fracture networks in which only the flow in the fractures is considered. The more complex discrete fracture models couple flow in the fractures or in fracture networks with flow in the surrounding medium. This later type model is the type considered here. An alternative to the possibility of using a very fine grid in the fracture and a necessarily much coarser grid away from the fracture is the possibility of treating the fracture as an (n − 1)−dimensional hypersurface in the n−dimensional porous medium. This is the idea that was developed in [2] for highly permeable fractures and in [16] for fractures that may be highly permeable or nearly impermeable. Similar models have also been studied in [11, 6, 17]. These articles were all concerned with the case of single-phase, incompressible flow governed by Darcy’s law and the law of mass conservation. In [14] a model was derived in which Darcy’s law was replaced by the Darcy-Forchheimer law for the flow in the fracture, while Darcy’s law was maintained for flow in the rest of the medium. The model was approximated numerically with mixed finite elements and some numerical experiments were carried out. The use of the linear Darcy law as the constitutive law for fluid flow in porous media, together with the continuity equation, is well established. For medium-ranged velocities it fits well with experiments [8, Chapter 5] and can be derived rigorously (on simpler periodic media) by homogenization starting from Stokes’s equation [19, 3, 4]. However, for high velocities experiments show deviations which indicate the need for a nonlinear correction term, [12], [8, Chapter 5]. The simplest proposed is a term quadratic in velocity, the Forchheimer correction. In fractured media, the permeability (or hydraulic conductivity) in the fractures is generally much greater than in the surrounding medium so that the total flow process in the limit is dominated by the fracture flow. This indicates that a modeling different from Darcy’s model is necessary and leads us to investigate models combining Darcy and Darcy-Forchheimer flow. In this paper we consider existence and uniqueness of the solution of corresponding stationery problems. Assumptions on coefficients should be weak so as not to prevent the use of the results in more complex real life situations. Therefore we aim at weak solutions of an appropriate variational formulation, where we prefer a mixed variational formulation, due to the structure of the problems and a further use of mixed finite element techniques. For a simple d-dimensional domain Ω and for the linear Darcy flow the results are well known (c.f. [9]) and rely on the coercivity of the operator A coming from Darcy’s equation on the kernel of the divergence operator B coming from the continuity equation and the functional setting in H(div,Ω) for the flux and L(Ω) for the pressure. For the nonlinear Darcy-Forchheimer flow the functional setting has to be changed to W (div,Ω) (see Appendix A.1) for the flux so that A will remain (strictly) monotone and to L 3 2 (Ω) for the pressure. This makes it possible to extend the reasoning for the linear case to the homogeneous Darcy-Forchheimer problem and via regularization, using the Browder-Minty theorem for maximal monotone operators, also to prove unique existence in the inhomogeneous case. This work is carried out in the thesis [18]; see also [15, 10, 5] for related results. Here we extend this reasoning to the situation of two subdomains of the matrix separated by a fracture with various choices of the constitutive laws in domains and fractures. One would expect that the Darcy-Forchheimer law is more accurate than Darcy’s law (and this will be partially made rigorous); therefore, (and