Thermodynamically consistent non-local damage formulation for fluid-driven fracture in poro-viscoelastic media

We present a novel non-local integral-type damage formulation for hydraulic fracture of poro-viscoelastic media under the framework of irreversible thermodynamics. The poro-viscoelastic material is modeled by a generalized Maxwell model, whose shear modulus is described in terms of Prony series. A bilinear damage law is assumed, which is driven by three equivalent strain invariants. Darcy’s law is employed to describe the fluid flow in the entire domain including the fracture process zone, where the permeability is assumed to be nonlinear and anisotropic. A monolithic two-field (u-p\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\varvec{u}}-p$$\end{document}) mixed finite element method is employed to discretize the coupled hydromechanical system. A Newton–Raphson method is utilized to solve the nonlinear system, and a backward Euler scheme is applied to evolve the system in time. Several numerical examples are presented to investigate the time-dependent deformation response of saturated porous media. In particular, we study the effects of relaxation time and the ratios of anisotropic initial permeability on the strongly coupled processes of the solid deformation, fluid transport and damage evolution of geomaterials. In addition, the different modes of energy dissipation mechanisms including damage and solid viscous response are presented and discussed.