A time-decoupling scheme for mixed-dimensional poroelastic models with fractures

Abstract

We propose a locking-free decoupling method for a mixed-dimensional poroelasticity model with fractures. By introducing the total pressure, the fractured Biot system is reformulated as a four-field formulation involving the displacement, total pressure, matrix pressure, and fracture pressure. We establish an energy dissipation law for the continuous model, which shows its consistency with the second law of thermodynamics. Based on this formulation, a time-decoupled scheme is developed. At the initial time step, a fully coupled scheme is employed, while for subsequent time steps, the flow problem is solved first, followed by the mechanics problem. A stabilization term is incorporated into the mechanical equation to help reduce the restrictions imposed on the model parameters in the stability analysis. Energy stability is established for the semi-discrete scheme. For the spatial discretization, the displacement and total pressure are approximated by the Taylor--Hood element, while Lagrange finite elements are used for the matrix and fracture pressures. A fully discrete decoupled scheme is then constructed. Energy stability and error estimates are derived for the fully discrete scheme, and the method is shown to be locking-free. Numerical experiments are presented to support the theoretical results.