Monolithic Solution of Mixed-dimensional Models in Fractured Porous Media by Multigrid Methods

Monolithic Solution of Mixed-dimensional Models in Fractured Porous Media by Multigrid Methods

The numerical simulation of fluid dynamics in fractured porous media is a challenging task which is getting increasing attention in recent years due to the wide range of applications in which fractures play an essential role, ranging from the field of subsurface hydrology to biomedicine, for example. Fractures can be incorporated to flow models in different ways. Here, we consider geological discontinuities represented by localized networks of faults and macro-fractures, which require the use of discrete fracture models. Based on model reduction techniques, fractures are represented as (n-1)-dimensional interfaces immersed into an n-dimensional porous matrix, giving rise to mixed-dimensional or reduced models. In this work, we consider a two-dimensional porous medium containing a variable number of horizontal and vertical one-dimensional fractures. Within this framework, our work focuses on the efficient and robust solution of the large system of equations resulting after discretization of the problem. In particular, a monolithic multigrid method is proposed for this purpose. Since a mixed-dimensional problem needs to be solved, we combine two-dimensional smoothing and inter-grid transfer operators for the unknowns in the porous matrix with their one-dimensional counterparts within the fracture network, giving rise to a mixed-dimensional multigrid method. The robustness of the proposed monolithic mixed-dimensional multigrid method is demonstrated through different numerical experiments.