A Stokes–Darcy–Darcy model and its discontinuous Galerkin method on polytopic grids

Abstract

In this paper, we propose and numerically solve a model considering confined flow in fractured porous media coupled with free flow in conduits. Such a situation arises, for example, for fluid flows in fractured vuggy carbonate reservoirs and caves/conduits, where a matrix and fractures are the main spaces for fluid storage and flow. The flow in caves and conduits is governed by the Stokes equation; the flow in fractured porous media, which consists of both a matrix and fractures, is described by a coupled Darcy–Darcy (bulk–fracture) model. Then, the two models are coupled through suitable (physically consistent) interface conditions on the interface between fractured porous media and caves/conduits, playing a key role in a physically faithful simulation that achieves high accuracy. The construction of all the interface conditions of the coupled model is based on the fundamental properties of the traditional bulk–fracture model and the well-known Stokes–Darcy model. The weak formulation is derived for the proposed model, and the well-posedness of the model is theoretically analyzed. We also propose a polyhedral discontinuous Galerkin finite element approximation for the Stokes flow in conduits and Darcy flow in the porous matrix, which is coupled with a conforming finite element scheme for the flow in the fracture. Optimal error estimates in a suitable energy norm are obtained. Two-dimensional numerical experiments are conducted to validate the proposed model and to demonstrate the features of both the model and the numerical method, such as the optimal convergence rate of the numerical solution; the detail flow characteristics around fractures, the matrix, and conduits; and the applicability to real-world problems.