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.

Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method

“…This case is the same already studied in the isotropic case because the dispersion tensor in the fissure D * η is isotropic. So, these results can also be found in [6]. Here we study the approximation of the flux in a fissured zone (figure 9).…”

“…These cases occur when a redirection of the flux is requested in the fissure ie at the intersection nodes. These cases have been treated by Caillabet et al [6].…”

“…In order to avoid the need for a very fine mesh in the fissure network, we adopt the joint element strategy which proved to be very efficient in solving the one-phase flow case. This is detailed in [6], and here we only recall what is a joint element.…”

“…The normal component can be expressed in a similar way as in the isotropic case which is detailed in [6]. The transversal flux needs two points called γ 2 and γ 1 on the interface to get a first order discretization given below,…”

Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method

“…This case is the same already studied in the isotropic case because the dispersion tensor in the fissure D * η is isotropic. So, these results can also be found in [6]. Here we study the approximation of the flux in a fissured zone (figure 9).…”

“…These cases occur when a redirection of the flux is requested in the fissure ie at the intersection nodes. These cases have been treated by Caillabet et al [6].…”

“…In order to avoid the need for a very fine mesh in the fissure network, we adopt the joint element strategy which proved to be very efficient in solving the one-phase flow case. This is detailed in [6], and here we only recall what is a joint element.…”

“…The normal component can be expressed in a similar way as in the isotropic case which is detailed in [6]. The transversal flux needs two points called γ 2 and γ 1 on the interface to get a first order discretization given below,…”

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).…”

Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method

“…This case is the same already studied in the isotropic case because the dispersion tensor in the fissure D * η is isotropic. So, these results can also be found in [6]. Here we study the approximation of the flux in a fissured zone (figure 9).…”

“…These cases occur when a redirection of the flux is requested in the fissure ie at the intersection nodes. These cases have been treated by Caillabet et al [6].…”

“…In order to avoid the need for a very fine mesh in the fissure network, we adopt the joint element strategy which proved to be very efficient in solving the one-phase flow case. This is detailed in [6], and here we only recall what is a joint element.…”

“…The normal component can be expressed in a similar way as in the isotropic case which is detailed in [6]. The transversal flux needs two points called γ 2 and γ 1 on the interface to get a first order discretization given below,…”

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).…”

Computation of large-scale parameters for dispersion in fissured porous medium using finite-volume method

“…This case is the same already studied in the isotropic case because the dispersion tensor in the fissure D * η is isotropic. So, these results can also be found in [6]. Here we study the approximation of the flux in a fissured zone (figure 9).…”

“…These cases occur when a redirection of the flux is requested in the fissure ie at the intersection nodes. These cases have been treated by Caillabet et al [6].…”

“…In order to avoid the need for a very fine mesh in the fissure network, we adopt the joint element strategy which proved to be very efficient in solving the one-phase flow case. This is detailed in [6], and here we only recall what is a joint element.…”

“…The normal component can be expressed in a similar way as in the isotropic case which is detailed in [6]. The transversal flux needs two points called γ 2 and γ 1 on the interface to get a first order discretization given below,…”

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).…”