A Stokes–Dual–Porosity–Poroelasticity Model and Discontinuous Galerkin Method for the Coupled Free Flow and Dual Porosity Poroelastic Medium Problem

In this paper, we introduce and solve a novel model that integrates confined flow within a dual porosity poroelastic medium with free flow in conduits. The model is structured around three distinct but interconnected regions: the matrix, micro-fractures, and conduits. Fluid flow within the dual porosity poroelastic medium is described by a dual-porosity poroelastic model, while fluid flow within the conduits is modeled by the Stokes equations. The integration of these two flow dynamics is achieved through a set of interface conditions, including a novel no-exchange condition. Theoretical achievements include the establishment of the existence and uniqueness of the solution for the weak formulation, alongside stability and error estimates for the semi-discrete continuous-in-time discontinuous Galerkin method. Furthermore, the convergence of the full discretisation using the backward Euler time stepping is thoroughly analysed. Two-dimensional numerical experiments are conducted and highlight the optimal convergence rate of the numerical solution, affirming the relevance and applicability of the model to real-world scenarios.