Implementation of a finite-volume method for the determination of effective parameters in fissured porous media

Abstract

Many studies have proposed one-equation models to represent transport processes in heterogeneous porous media. This approach is based on the assumption that dependent variables such as pressure, temperature, or concentration can be expressed in terms of a single large-scale averaged quantity in regions having very different chemical and/or mechanical properties. However, one can also develop large-scale averaged equations that apply to the distinct regions that make up a heterogeneous porous medium. This approach leads to region-averaged equations that contain traditional convective and dispersive terms, in addition to exchange terms that account for the transfer between the different media. In our approach, the fissures represent one region, and the porous media blocks represent the second region. The analysis leads to upscaled equations having a domain of validity that is clearly identified in terms of time and length-scale constraints. Closure problems are developed that lead to the prediction of the effective coefficients that appear in the region averaged equations, and the main purpose of this article is to provide solutions to those closure problems. The method of solution makes use of an unstructured grid and a joint element method in order to take care of the special characteristics of the fissured network. This new numerical method uses the theory developed by Quintard and Whitaker and is applied on considerably more complex geometries than previously published results. It has been tested for several special cases such as stratified systems and ''sugarbox'' media, and we have compared our calculations with other computational methods.

Asymptotic and numerical modelling of flows in fractured porous media

“…we get the first asymptotic model of flow along the fault interface Σ: 12) where, in these equations the jumps and the mean-values of p and v p · ν are now the one obtained from the solution of the asymptotic model of flow inside the approximate porous matrix Ω, that is system (2.4)-(2.7).…”

“…Reviews on such problems can be found in [1,6,7,18], for instance. Some authors neglect the flow in the porous matrix and only concentrate on the study of the flow in fracture networks (see [1,7] and references therein), others propose to treat the specific geometry of the fractures by a specific numerical method (see for instance [12,13] where joint elements are used). In the diphasic case, where a convection-diffusion-dispersion equation has also to be taken into account, we can refer to the recent reference [24] in which the authors solve their model by using a cell-centered finite volume method.…”

A practical model for fluid flow in discrete-fracture porous media by using the numerical manifold method

“…The greatest challenges of modeling fluid flow in discretely fractured porous media are the complex geometry of fractures, the large contrasts in geometries and the hydraulic properties between fractures and the rock matrix. One approach to model such a problem is to use the same dimension for fractures as for the rock matrix in an equi-dimensional formulation [15][16][17][18][19][20]. The authors developed a similar approach for dominant fractures as a heterogeneous medium [21].…”

“…The fractures are assumed to be thin and unfilled, so the distribution of the hydraulic head on each surface of a fracture and within a fracture is uniform in the direction normal to the fracture surfaces. Therefore, we consider these two fracture surfaces to consist of two Dirichlet boundaries, i.e., (1) a Dirichlet boundary with a given hydraulic head h 0 =h r (x, y) for a fracture with an unknown hydraulic head h f (x, y): 11 (16) Combining equations (12) and (14) produces the total work that is associated with normal-to-fracture flow on both surfaces of a fracture: According to the energy-work theorem and the minimization of the total potential energy, we can obtain the final terms in the global equilibrium equations that are associated with normal-to-fracture flow, as shown in Section 3.4. This approach is reasonable because we assume that the fractures are thin and unfilled.…”

Comparison of cell- and vertex-centered discretization methods for flow in a two-dimensional discrete-fracture–matrix system

“…This approach apply when both fractures and matrix play a significant role for the flow and transport processes and the model can not be homogenized. Two model approaches exist for the DFM model: In the DFM lower-dimensional formulation (DFML) [41,44,45,49,58], fractures are modeled as line segments in 2D, and as planar regions in 3D, whereas in the DFM equi-dimensional formulation (DFME) [13,19,46,57,58,66], fractures are modeled in the same dimension as the matrix. The DFML has been more common than the DFME, since modeling fractures as lower-dimensional objects simplifies grid generation and data requirements.…”