A simple embedded discrete fracture–matrix model for a coupled flow and transport problem in porous media

Abstract

Accurate simulation of fluid flow and transport in fractured porous media is a key challenge in subsurface reservoir engineering. Due to the high ratio between its length and width, fractures can be modeled as lower dimensional interfaces embedded in the porous rock. We apply a recently developed embedded finite element method (EFEM) for the Darcy problem. This method allows for general fracture geometry, and the fractures may cut the finite element mesh arbitrarily. We present here a velocity model for EFEM and couple the Darcy problem to a transport problem for a passive solute. The main novelties of this work is a locally conservative velocity approximation derived from the EFEM solution, and the development of a lowest order upwind finite volume method for the transport problem. This numerical model is compatible with EFEM in the sense that the same computational mesh may be applied, so that we retain the same flexibility with respect to fracture geometry and meshing. Hence, our coupled solution strategy represents a simple approach in terms of formulation, implementation and meshing. We demonstrate our model by some numerical examples on both synthetic and realistic problems, including a benchmark study for single-phase flow. Despite the simplicity of the method, the results are promising.

Fully-automated adaptive mesh refinement for media embedding complex heterogeneities: application to poroelastic fluid pressure diffusion

“…In order to prove the uniqueness of solution of problem (46), we first rewrite it in an algebraic form by introducing the basis functions {Φ j ∈ V T } and {φ l ∈ Q T } of V T and Q T , respectively. Expanding the two components of the solution with respect to the basis functions, we obtain the following block linear system…”

“…In [38], AMR is used to generated meshes in a domain with discrete fracture networks together with constrained Delaunay triangulations, thus resulting in a conforming mesh that is denser at fracture locations. In [46], an initially uniform mesh of the background medium is adaptively refined in the vicinity of the lower-dimensional fractures. This approach results in a non-conforming mesh, which allows for a more adequate representation of flow and transport phenomena between the background and the fractures.…”

“…In previous works [29,30], we presented initial applications of a novel method based on AMR, which enables fully automated meshing of complex heterogeneous structures, such as stochastic fracture networks. Inspired by the adaptive schemes proposed in [4,15,46], our method does not create meshes which explicitly resolve the heterogeneities. Instead, it creates sequences of nonconforming quadrilateral or hexahedral structured meshes, which approximate the heterogeneities with increasing accuracy.…”

Verification benchmarks for single-phase flow in three-dimensional fractured porous media

“…The study exposed the relative strengths and weaknesses between the participating methods, both in terms of accuracy and computational cost. After the publication of the results, these benchmark cases have been widely applied by the scientific community in testing numerical methods and new simulation tools ( Arrarás et al, 2019;Budisa and Hu, 2019;Köppel et al, 2019a;2019b;Odsaeter et al, 2019;Schädle et al, 2019 ).…”

Numerical upscaling for heterogeneous materials in fractured domains

“…In [4], a simple finite element method (SFEM) is developed for simulation of Darcy flows in fractured media, which is also applied to a coupled flow and transport problem [17]. With the bilinear form (2.6), the SFEM can be written as:…”