This study concerns some asymptotic models used to compute the flow outside and inside fractures in a bidimensional porous medium. The flow is governed by the Darcy law both in the fractures and in the porous matrix with large discontinuities in the permeability tensor. These fractures are supposed to have a small thickness with respect to the macroscopic length scale, so that we can asymptotically reduce them to immersed polygonal fault interfaces and the model finally consists in a coupling between a 2D elliptic problem and a 1D equation on the sharp interfaces modelling the fractures. A cell-centered finite volume scheme on general polygonal meshes fitting the interfaces is derived to solve the set of equations with the additional differential transmission conditions linking both pressure and normal velocity jumps through the interfaces. We prove the convergence of the FV scheme for any set of data and parameters of the models and derive existence and uniqueness of the solution to the asymptotic models proposed. The models are then numerically experimented for highly or partially immersed fractures. Some numerical results are reported showing different kinds of flows in the case of impermeable or partially/highly permeable fractures. The influence of the variation of the aperture of the fractures is also investigated. The numerical solutions of the asymptotic models are validated by comparing them to the solutions of the global Darcy model or to some analytic solutions.

We develop asymptotic modeling for two- or three-dimensional viscous fluid flow and convective transfer at the interface between a fluid and a porous layer. The asymptotic model is based on the fact that the thickness $d$ of the interfacial transition region ${\mathrm{\Omega }}_{\text{fp}}$ of the one-domain representation is very small compared to the macroscopic length scale $L$. The analysis leads to an equivalent two-domain representation where transport phenomena in the transition layer of the one-domain approach are represented by algebraic jump boundary conditions at a fictive dividing interface $\mathrm{\Sigma }$ between the homogeneous fluid and porous regions. These jump conditions are thus stated up to first-order in $O(d/L)$ with $d/L\ll 1$. The originality and relevance of this asymptotic model lies in its general and multidimensional character. Indeed, it is shown that all the jump interface conditions derived for the commonly used 1D-shear flow are recovered by taking the tangential component of the asymptotic model. In that case, the comparison between the present model and the different models available in the literature gives explicit expressions of the effective jump coefficients and their associated scaling. In addition for multi-dimensional flows, the general asymptotic model yields the different components of the jump conditions including a new specific equation for the cross-flow pressure jump on $\mathrm{\Sigma }$.