Journal of Applied Science and Engineering (Jul 2024)
Analytical and numerical solutions of flow in and around a fracture within a porous media: Application to equivalent permeability estimation
Abstract
Equivalent permeability is the most important parameter to estimate the total flow transported by a fractured geological formation. The inclusion-based effective medium model has been widely used to estimate this parameter. In such an approach, the fracture is assimilated as an elliptical inclusion in which the fluid flow is governed by Darcy’s law. However, experimental investigations have shown that the cubic law is more appropriate to describe the flow in a fracture. This study aims at firstly developing the fluid potential solution of water flow through and around an individual fracture embedded by an infinite geological porous formation under a remote constant pore pressure gradient. Multi-region boundary element method associating to the mass conversation within the fracture is applied to obtain the solution of the water flow through a porous formation containing an individual fracture under a hypersingular integral equation form. Secondly, writing the hypersingular integral equation for the case of superconductive fracture leads to the well-known airfoil equation, and then the closed-form solution of flow within superconductive fracture is rigorously derived. Thirdly, the fluid potential equation is numerically solved for general case of a finite conductive fracture by using boundary integral element method. The comparison between numerical and analytical solution helps to confirm the accuracy of both analytical and numerical development. Fourth, the total flow transported by a fracture is computed and fitted by a function of the fracture conductivity. This numerical solution associating to the self-consistent homogenization scheme engenders an equivalent permeability model, which exhibits a percolation threshold.
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