Three-dimensional flow in fractured porous media: A potential solution based on singular integral equations

Abstract

Governing equations for flow in three-dimensional heterogeneous and anisotropic porous media containing fractures or cracks with infinite transverse permeability are described. Fractures are modeled as zero thickness curve surfaces with the possibility of multiple intersections. It is assumed that flow obeys to an anisotropic Darcy’s law in the porous matrix and to a Poiseuille type law in fractures. The mass exchange relations at fractures intersections are carefully investigated as to establish a complete mathematical formulation for the flow problem in a fractured porous body. A general potential solution, based on singular integral equations, is established for steady state flow in an infinite fractured body with uniform and isotropic matrix permeability. The main unknown variable in the equations is the pressure field on the crack surfaces, reducing thus from three to two the dimension of the numerical problem. A general transformation lemma is then given that allows extending the solution to matrices with anisotropic permeability. The results lead to a simple and efficient numerical method for modeling flow in three-dimensional fractured porous bodies.

Introduction

Modelling the flow in fractured porous media has a great interest for applications to various industrial problems such as petroleum reservoirs engineering, radioactive waste disposal, CO₂ geological storage, water resources management, etc. Different aspects of flow in fractured porous media, especially in the contexts of petroleum reservoirs, have been investigated by different numerical methods such as Finite Elements [1], [2], [3], Finite Volume [4], [5], [6] and Discrete Fracture Network [7], [8] among many others. In many problems related to flow in fractured geological formations or in micro-cracked porous rocks, one is interested in the steady state flow that takes place under given farfield conditions. This occurs especially when the effective permeability of fractured reservoirs or of microcracked rocks is investigated [4], [9], [10], [11], [12]. For this purpose, the flow has to be determined in and around the cracks embedded in an infinite matrix with uniform permeability. To model steady state flow in an infinite fractured matrix, singular integral equations provide a powerful method that allows developing simple numerical models, general potential solutions and, in some cases, analytical solutions.

In potential solutions, the pressure field in the whole body is built as a function of the infiltration in the fractures. This reduces the dimension of the unknown variable field from three (pressure field) to two (infiltration on crack surfaces) and thus simplifies numerical modelling. For 2D plane flow in cracked bodies, potential solutions based on singular integral equations was introduced first by Liolios and Exadaktylos [13], by using complex number variables. Their solution was restricted to isotropic matrix permeability and, also, excluded crack intersections. The cracks intersections constitute, in fact, a difficulty that mathematical formulations and numerical methods have to handle. A direct formulation of two-dimensional flow in terms of singular integral equations allowed Pouya and Ghabezloo [14] extending the previous solutions to anisotropic matrices. Furthermore, the integration of mass balance relation at crack intersections allowed them to extend their solution to intersecting cracks. The initial objective of the present work is the extension of these results to three-dimensional flow. However, we realized that this extension faces the problem of mass balance conditions on fractures intersection lines that has not been sufficiently well studied in the literature.

As a matter of fact, in presence of fractures, the formulation of the flow problem has to integrate mass balance conditions concerning matrix–fracture exchanges along fracture surfaces as well as matrix–fracture and fracture–fracture exchanges along fracture intersection lines. The relations governing matrix–fracture exchanges along fracture surfaces have been first expressed by Barenblatt et al. [15] in a context of double porosity concept, and then widely used in subsequent work [4], [6], [16], [17]. Their extension to curved fracture surfaces, however, should be examined carefully as it will be shown in this paper. But a review of different numerical modelling works shows that, at least in the context of three-dimensional flow, mass balance conditions at fracture intersections have not been explicitly and clearly formulated. Probably, the concept of double porosity in which the flow in the fracture network and in the matrix are described separately by a doublet of governing equations has made the problem more complicated [18]. It can be noticed that in many works fractures intersections have been simply excluded to avoid conceptual and numerical difficulties. In some works, dealing with 2D flow, the mass balance at fracture intersection points is expressed only in the numerical model: Granet et al. [5] impose some relations on nodal velocities in elements adjacent to an intersection point in order to assure the mass balance. But concerning the 3D intersecting fractures, we could not find any expression of mass balance conditions on intersection lines and points. However, whatever the numerical method used, a rigorous and complete mathematical formulation of the flow problem integrating these conditions is a prerequisite to establishment of relevant numerical models.

In this work, the fractures represent physically the limits of vanishingly thin fractures or very permeable layers in which the fluid is in equilibrium with the pores fluid, and so, the pressure is continuous at the matrix–fracture interface. Also, there is no pressure jump between the two opposite sides of the fracture, corresponding to an assumption of infinite transverse permeability for the fracture. These assumptions exclude the cases of non-equilibrium flow with different pressures in the fracture and the matrix [19] or of fractures, such as some shear faults, acting as impermeable membranes [20]. Geometrically, the fractures are represented by curved surfaces that may have multiple intersections along different lines and points. In this framework, the paper focuses on the mathematical formulation of the flow problem. First the mass balance equations at intersection lines and points are investigated. Then, a potential solution for the steady state flow is established in an infinite fractured body with uniform matrix permeability. The potential solution allows reducing the 3D problem to a 2D problem involving only the flow on fracture surfaces and so making it possible to solve the problem by using 2D numerical methods. The equations for 2D problem are written by supposing a Poiseuille type law (linear relation between infiltration and pressure gradient) for the tangent flow in the fracture, but they can be easily extended to non linear laws. The potential solution is first established for isotropic matrix permeability and then extended to anisotropic permeability by using a linear transformation method that will be introduced beforehand.

In the sequel, light-face (Greek or Latin) letters denote scalars; underlined letters denote vectors, bold-face letters designate second rank tensors or double-index matrices. The scalar product of two vectors a and b is labelled as a · b. For second rank tensors, the tensor transposed from A is denoted A^T, the matrix product is labelled as AB and the determinant as ∣A∣. The operation of A on a is labelled as A · a.

The Greek indices (α, β, …) take the values {1, 2}, and the convention of summation on repeated indices is used implicitly for them. This convention is not used for Latin indices (i, j, k, …) that are used to number surfaces, lines, etc., and are noted indifferently as subscript or upperscript. ∇ represents the gradient and Δ the Laplace operator for a scalar field and (∇). the divergence for a vector field.