A discontinuous Galerkin method for a coupled Brinkman–Biot problem

Abstract

In this paper, the Brinkman–Biot model is used to simulate the coupling problem of the Brinkman flow and deformable poroelastic media flow, which need to interact through the interface. By introducing the total pressure to rewrite the poroelastic equations, the possible locking phenomenon of the Biot system is overcome. By taking into account complex permeability coefficient, the strong stiffness caused by the Biot system is solved. A discontinuous Galerkin finite element method is used to solve the problem of complex poroelastic media caused by coupling. Firstly, the space is discretised by the discontinuous Galerkin finite element method, and the time is discretised by the backward Euler method. Then the semi-discretisation scheme and the full discretisation scheme are given. Secondly, in the framework of the Galerkin approximation, the existence and uniqueness of solutions and error estimates of semi-discrete and fully discrete schemes are analysed by means of differential algebraic equation theory and weak compactness demonstration. Finally, through numerical experiments, the theoretical convergence rate of the numerical solution of the model and whether the interface conditions coincide are verified. The channel filtration and the actual hydraulic fracturing fluid flow situation are simulated, and the effectiveness and accuracy of the method are verified.