A reduced model for Darcy’s problem in networks of fractures

Abstract

Subsurface flows are influenced by the presence of faults and large fractures which act as preferential paths or barriers for the flow. In literature models were proposed to handle fractures in a porous medium as objects of codimension 1. In this work we consider the case of a network of intersecting fractures, with the aim of deriving physically consistent and effective interface conditions to impose at the intersection between fractures. This new model accounts for the angle between fractures at the intersections and allows for jumps of pressure across intersections. This fact permits to describe the flow when fractures are characterized by different properties more accurately with respect to other models that impose pressure continuity. The main mathematical properties of the model, derived in the two-dimensional setting, are analyzed. As concerns the numerical discretization we allow the grids of the fractures to be independent, thus in general non-matching at the intersection, by means of the extended finite element method (XFEM). This increases the flexibility of the method in the case of complex geometries characterized by a high number of fractures.

Gradient Discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media

“…defined by the vector space X 0 D from (20), the discrete gradient operators ∇ Dm from (21) and ∇ D f from (22), and the function reconstruction operators Π M , Π F from (26). From Lemma 5.3 and Lemma 4.1 of [13] and Lemma 1.51 of [11], one has the following discrete Poincaré estimates…”

“…It corresponds to a high ratio assumption between the permeability at the fracture intersections and the width of the fracture compared with the ratio between the tangential permeability of each fracture and its length. We refer to [20] for a more general reduced model taking into account discontinuous pressures at fracture intersections in dimension d = 2.…”

The Virtual Element Method for Discrete Fracture Network Flow and Transport Simulations

“…In this work we recall some results concerning the application of the Virtual Element Method [4,3,2] to the steady state simulation of the flow in DFNs [1,22,27,30,24,32,23,15,16,17,18,19,9,10,8,13]. In this approach we can exploit the flexibility of VEM in order to tackle the geometrical complexity.…”

An Efficient XFEM Approximation of Darcy Flows in Arbitrarily Fractured Porous Media

“…Realistic simulations of intersecting faults in a three dimensional domain are presented in [8], where the continuity of pressure and mass conservation are enforced at the intersections. More general coupling conditions are introduced and discussed in [9] to account for different properties of the fractures allowing for pressure and velocity jumps at the intersection, similarly to the conditions derived in [2] for the matrix-fracture system. The aforementioned work [9] considers the case of a network isolated by the porous matrix, in the limit case where the matrix can be regarded as impermeable with respect to the fractures.…”

“…More general coupling conditions are introduced and discussed in [9] to account for different properties of the fractures allowing for pressure and velocity jumps at the intersection, similarly to the conditions derived in [2] for the matrix-fracture system. The aforementioned work [9] considers the case of a network isolated by the porous matrix, in the limit case where the matrix can be regarded as impermeable with respect to the fractures. The case of an isolated fracture network in an impermeable matrix is also tackled in [9] by means of the XFEM method and an optimization strategy to enforce the coupling conditions at the intersections.…”