Three-dimensional control-volume distributed multi-point flux approximation coupled with a lower-dimensional surface fracture model

Abstract

A novel cell-centred control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulation is presented for discrete fracture-matrix simulations on unstructured grids in three-dimensions (3D). The grid is aligned with fractures and barriers which are then modelled as lower-dimensional surface interfaces located between the matrix cells in the physical domain. The three-dimensional pressure equation is solved in the matrix domain coupled with a two-dimensional (2D) surface pressure equation solved over fracture networks via a novel surface CVD-MPFA formulation. 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 define the transfer function, which is added to the lowerdimensional flow equation and couples the three-dimensional and surface systems. An additional transmission condition is used between matrix cells adjacent to low permeable fractures to couple the velocity and pressure jump across the fractures. Convergence and accuracy of the lower-dimensional fracture model is assessed for highly anisotropic fractures having a range of apertures and permeability tensors. A transport equation for tracer flow is coupled via the Darcy flux for single and intersecting fractures. The lower-dimensional approximation for intersecting fractures avoids the more restrictive CFL condition corresponding to the equi-dimensional approximation with explicit time discretisation. Lower-dimensional fracture model results are compared with equi-dimensional model results. Fractures and barriers are efficiently modelled by lowerdimensional interfaces which yield comparable results to those of the equi-dimensional model. Pressure continuity is built into the model across highly conductive fractures, leading to reduced local degrees of freedom in the CVD-MPFA approximation. The formulation is applied to geologically complex fracture networks in three-dimensions. The effects of the fracture permeability, aperture and grid resolution are also assessed with respect to convergence and computational cost.

Fracture size and transmissivity correlations: Implications for transport simulations in sparse three‐dimensional discrete fracture networks following a truncated power law distribution of fracture size

“…The two main methodologies in use are continuum models, where the rock mass is represented as a porous medium and the hydraulic conductivity is a scale‐dependent correlated stochastic field, and discrete fracture network (DFN) models, where the geometry and properties of individual fractures are explicitly represented. Other models that consider the combined effect of fracture and matrix have also been recently developed [ Ahmed et al ., , ; Roubinet et al ., ; Willmann et al ., ]. Although DFN models can typically represent a wider range of transport phenomena than continuum models [ Painter and Cvetkovic , ; Painter et al ., ], the inclusion of detailed features introduces additional layers of uncertainty because more parameters have to be calibrated [ Neuman , ].…”

Finite-Volume Discretisations for Flow in Fractured Porous Media

“…While beneficial for the gridding, the coupling between the discretization of fractures and matrix is more involved than in the classical FV DFM models using conforming grids that are considered in this paper. Such methods include the vertex-centred method by Reichenberger et al (2006), the cell-centre based one introduced by Karimi-Fard et al (2004) and extended by Sandve et al (2012), and the related method of Ahmed et al (2015) using transfer functions for the fracture-matrix coupling. Other approaches include the vertex approximate gradient and hybrid FV schemes (Brenner, Groza, Guichard, Lebeau, & Masson, 2016) and the two-phase method (Monteagudo & Firoozabadi, 2004), and refer to Geiger & Matthäi (2014) for a review of numerical methods based on the DFM model.…”

Parallel numerical modeling of hybrid-dimensional compositional non-isothermal Darcy flows in fractured porous media

“…Let us denote by R ν (X D ) the vector R ν,i , i ∈ C ∪{e} , and let us rewrite the conservation equations (18), (19), (20) and the closure laws (21) as well as the boundary conditions in vector form defining the following non-linear system at each time step n = 1, 2, ..., N T…”

“…In [4] a cell-centred Finite Volume scheme using a Two Point Flux Approximation (TPFA) is proposed assuming the orthogonality of the mesh and isotropic permeability fields. Cell-centred Finite Volume schemes can be extended to general meshes and anisotropic permeability fields using MultiPoint Flux Approximations (MPFA) (see [15], [16], [17], [18], [19]). MPFA schemes can lack robustness on distorted meshes and large anisotropies due to the non symmetry of the discretization.…”