Control-volume distributed multi-point flux approximation coupled with a lower-dimensional fracture model

Abstract

A cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-matrix simulations. The grid is aligned with the fractures and barriers which are then modeled as lower-dimensional interfaces located between the matrix cells in the physical domain. The nD pressure equation is solved in the matrix domain coupled with an (n-1)D pressure equation solved in the fractures. The CVD-MPFA formulation naturally handles fractures with anisotropic permeabilities on unstructured grids. Matrix-fracture fluxes are expressed in terms of matrix and fracture pressures, and must be added to the lower-dimensional flow equation (called the transfer function). An additional transmission condition is used between matrix cells adjacent to low permeable fractures to link the velocity and pressure jump across the fractures. Numerical tests serve to assess the convergence and accuracy of the lower-dimensional fracture model for highly anisotropic fractures having different apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approach for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretization. Lower-dimensional fracture model results are compared with hybrid-grid and equi-dimensional model results. Fractures and barriers are efficiently modeled by lower-dimensional interfaces which yield comparable results to those of the equi-dimensional model. Highly conductive fractures are modeled as lower-dimensional entities without the use of locally refined grids that are required by the equi-dimensional model, while pressure continuity across fractures is built into the model, without depending on the extra degrees of freedom which must be added locally by the hybrid-grid method.The lower-dimensional fracture model also yields improved results when compared to those of the hybrid-grid model for fractures with low-permeability in the normal direction to the fracture. In addition, a transient pressure simulation involving geologically representative complex fracture networks is presented.

Gradient Discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media

“…In [19] a cell-centered Finite Volume scheme using a Two Point Flux Approximation (TPFA) is proposed assuming the orthogonality of the mesh and isotropic permeability fields. Cell-centered Finite Volume schemes can be extended to general meshes and anisotropic permeability fields using MultiPoint Flux Approximations (MPFA) following the ideas introduced in [30], [28], and [2] for discontinuous pressure models. In [3], a Mixed Finite Element (MFE) method is proposed, and Control Volume Finite Element Methods (CVFE) using nodal unknowns have been introduced for such models in [25] and [24].…”

Stochastic modeling of flow and conservative transport in threedimensional discrete fracture networks

“…Successful characterizations of flow and contaminant transport in fractured geologic formations depend on adequate descriptions of complex geometrical structures, which comprise a wide variety of fractures and their connections (Ahmed et al, 2015;Pichot et al, 2012;Weng et al, 2014). The fracture characteristics can be quantified by using various statistical parameters, including the fracture orientation, length, shape, and permeability alongside the fracture intensity and 25 connectivity (Bonnet et al, 2001;Botros et al, 2008;Bour et al, 2002;Koike et al, 2015;Stephens et al, 2015).…”

Flow simulations in porous media with immersed intersecting fractures

“…Suitable interface conditions are derived and applied in order to close the problem. This approach is usually denoted by reduced Darcy model and is followed for example in [5,2,6,7,8,9,10,11]. Another possible approach is based, e.g., on hybridgrid models, where fractures are n−1-dimensional only in the geometric grid, whereas an n-dimensional grid taking into account fracture aperture is then used for computations, as done, among others, in [12,13].…”

“…It is also available a vast literature on finite volumes, two point flux approximation (TPFA) or multi-point flux approximation (MPFA) [31], (see e.g. [32,6,33,13,8,34,35,36]), also recently studied in the general framework of gradient schemes [11].…”