A unified non-local damage model for hydraulic fracture in porous media

We present a unified non-local damage model for modeling hydraulic fracture processes in porous media, in which damage evolves as a function of fluid pressure. This setup allows for a non-local damage model that resembles gradient-type models without the need for additional degrees of freedom. In other words, we propose a non-local damage formulation at the same cost of a local damage approach. Nonlinear anisotropic permeability is employed to distinguish between the fluid flow velocity in the damage zone and the intact porous media. The permeability evolves as a function of an equivalent strain measure, where its anisotropic evolution behavior is controlled by the direction of principle strain. The length scale of the proposed model is analytically derived as a function of material point variables and is shown to be dependent on the pressure rate. A mixed finite element method is proposed to monolithically solve the coupled displacement-pressure system. The nonlinear system is linearized and solved using Newton's method with analytically derived consistent Jacobian matrix and residual vector, and the evolution of the system in time is performed by a backward Euler scheme. Numerical examples of 1D and 2D hydraulic fracture problems are presented and discussed. The numerical results show that the proposed model is insensitive to the mesh size as well as the time step size and can well capture the features of hydraulic fracture in porous media.