An extended finite element method for hydraulic fracture propagation in deformable porous media with the cohesive crack model

Abstract

In this paper, a fully coupled numerical model is developed for the modeling of the hydraulic fracture propagation in porous media using the extended finite element method in conjunction with the cohesive crack model. The governing equations, which account for the coupling between various physical phenomena, are derived within the framework of the generalized Biot theory. The fluid flow within the fracture is modeled using the Darcy law, in which the fracture permeability is assumed according to the well-known cubic law. By taking the advantage of the cohesive crack model, the nonlinear fracture processes developing along the fracture process zone are simulated. The spatial discretization using the extended finite element method and the time domain discretization applying the generalized Newmark scheme yield the final system of fully coupled nonlinear equations, which involves the hydro-mechanical coupling between the fracture and the porous medium surrounding the fracture. The fluid leak-off and the length of fracture extension are obtained through the solution of the resulting system of equations, not only leading to the correct estimation of the fracture tip velocity as well as the fluid velocity within the fracture, but also allowing for the eventual formation of the fluid lag. It is illustrated that incorporating the coupled physical processes, i.e. the solid skeleton deformation, the fluid flow in the fracture and in the pore spaces of the surrounding porous medium and the hydraulic fracture propagation, is crucial to provide a correct solution for the problem of the fluid-driven fracture in porous media, where the poroelastic effects are significant.