Closed-form solution of fluid flow in and around a crack disk embedded in a 3D porous medium

Abstract:

This paper considers the fluid flow through a porous medium containing intersecting fractures and presents three main analytical findings, namely: (1) mass exchange between fractures and surrounding matrix at the fracture intersection; (2) fluid potential solution (pressure field) within the whole domain under the form of a single singular integral equation; and (3) closed-form solutions of fluid flow in and around a crack disc under a far field pressure gradient. The crack is represented mathematically by a 2D smooth surface (i.e., zero thickness) within a 3D porous medium, while physically by a constant aperture. The fluid flow within the crack obeys Poisseuille's law, while Darcy's law is used to represent the fluid flow in the surrounding matrix. The general solution of pressure field for the general case of multiple intersecting cracks is firstly derived under a singular integral equation form. The mass exchange between the porous matrix and the crack, as well as the mass conservation at the intersection between cracks are the keys to obtaining this general solution. Then, the general solution is written for the case of a single crack. Rigorous derivation of the latter equation allows obtaining a closed-form solution of flow through a single crack. Introducing this solution of flow into the general equation gives the pressure field around the crack. The solution derived in this paper for a crack disk with Poisseuille's flow is slightly different from the well-known Eshelby's solution for the case of flattened inclusion in which the flow obeys Darcy's law.