In this article, a Brinkman–dual–porosity–poroelasticity model is proposed for the purpose of simulating the fluid flow problem in a filled pipe and dual porosity deformable porous media region. The objective is to elucidate the influence of two poroelastic media, namely the matrix and micro-fractures, on fluid flow behaviour. The model is solved using a discontinuous Galerkin finite element method. These two systems need to interact through a set of interface conditions, including a no-exchange condition. In the theoretical analysis, the dual-porosity poroelasticity system is rewritten by introducing the total pressure to avoid the locking phenomenon that may be caused by the system. The space is discretised using the discontinuous Galerkin method, and the time is discretised using the backward Euler method. The numerical schemes of semi-discretisation and full discretisation are established. The well-posedness of the weak formulation and the semi-discrete scheme is demonstrated, the existence and uniqueness of solutions in the full discrete scheme is analysed, and the error estimators for the semi-discrete and full discrete schemes are derived. For the numerical simulation, the theoretical convergence rate of the numerical solution and the correctness of the interface conditions are verified, as well as the fluid flow situation in actual hydraulic fracturing, are also simulated to verify the validity and precision of the method.
Bian et al. (2026) studied this question.