Numerical simulation of fracture flow with a mixed-hybrid FEM stochastic discrete fracture network model

Abstract

We introduce a discrete fracture network model of stationary Darcy flow in fractured rocks. We approximate the fractures by a network of planar circle disks, which is generated on the basis of statistical data obtained from field measurements. We then discretize this network into a mesh consisting of triangular elements placed in three-dimensional space. We use geometrical approximations in fracture planes, which allow for a significant simplification of the final triangular meshes. We consider two-dimensional Darcy flow in each fracture. In order to accurately simulate the channeling effect, we assign to each triangle an aperture defining its hydraulic permeability. For the discretization we use the lowest order Raviart-Thomas mixed finite element method. This method gives quite an accurate velocity field, which is computed directly and which satisfies the mass balance on each triangular element. We demonstrate the use of this method on a model problem with a known analytical solution and describe the generation and triangulation of the fracture network and the computation of fracture flow for a particular real situation.

Study of the Scale Effect on Permeability in the Interlayer Shear Weakness Zone Using Sequential Indicator Simulation and Sequential Gaussian Simulation

“…Maryška et al [22] established a stochastic discrete fracture network model, and then adopted a numerical method to simulate fracture flow. Malkovsky's et al [23] and Hanor's et al [24] found that the average permeability of fault zones is two orders of magnitude larger than their minimum permeability.…”

BDDC for mixed‐hybrid formulation of flow in porous media with combined mesh dimensions

“…Assuming δ d bounded and greater than zero, and using the fact that k d corresponds to a symmetric positive definite matrix, we see from (29) that block A in (37) is symmetric positive definite. Block C is symmetric positive semi-definite since…”

“…However, this issue is not as pronounced for Raviart-Thomas elements, for which only one element contributes to the stiffness on the diagonal at an interface degree of freedom, and thus irregularities caused by changing number of elements contributing to an interface weight cannot occur. On the other hand, an advantage of the diagonal stiffness scaling is the fact, that-unlike the ρ-scaling-it takes into account the shape and relative sizes of elements, which vary considerably in engineering applications, as well as the effect of δ d introduced in (29) and (36). Unless stated otherwise, scaling (87) is used in the computations presented in Section 8.…”

“…In addition, the presentation of the BDDC algorithm is driven more by an efficient implementation, while it is more oriented towards underlying theory in [6]. We take advantage of the special structure of the blocks in matrix (1) studied in detail in [26,28,29]. In particular, the nonzero structure of block C resulting from a combination of meshes with different spatial dimensions is considered in [30].…”

A Generalized Mixed Hybrid Mortar Method for Solving Flow in Stochastic Discrete Fracture Networks

“…Specific meshing procedure have been proposed to mesh these networks. In [15], [17], [16], the geometry of the network is changed locally to get a mesh of good quality. As we are doing stochastic simulations, we rather need an automatic mesh generation suitable for any DFN.…”