A Mixed-dimensional Discontinuous Galerkin Method for Coupled Flow and Transport in Fractured Porous Media

Fluid flow and solute transport in fractured porous media are phenomena that control key processes in groundwater monitoring, generating thermal energy, earthquake prediction, and biomedical engineering, e.g. model activities of the heart or brain from noninvasive measurements on the chest or scalp.

The mathematical representation of the fluid interaction between fracture and rock matrix domains is not straightforward because these two domains usually possess vastly different flow properties. For instance, fluid conductivity through a system of fractures can be much greater than that of the rock matrix.

To tackle this issue, there are three main approaches, (i) equidimensional, (ii) mixed-dimensional representing fractures as split surfaces, and (iii) mixed-dimensional representing fractures as internal walls [1, 2]. The first method demands excessively refinement of the fracture domain, more computationally burdens, while the second method requires a sophisticated contact model when mechanical deformation is included in the model 3.

Hence, the third method is selected in this study because of its favourable computational cost and straightforward implementation when it is incorporated more phenomena. Researchers have traditionally used the continuous Galerkin (CG) method to solve flow and transport in fractured porous media problems for decades.

However, it is not suitable for solving the transport equation because it may not satisfy mass conservation. Moreover, it cannot represent fractures that act as flow barriers [4]. This study aims to present the advantages of using the discontinuous Galerkin (DG) method for solving coupled flow and transport in fractured porous media.

A mixed DG × CG space is utilised to solve the pressure equation. Subsequently, a velocity field is established using CG or Raviet − T homas function space; then the transport equation is solved on DG space. This procedure provides more accurate solutions in a convection-dominated regime, in which a sharp flood-front of the tracer is established, than those of CG formulation.

The result of this work is compared to relatively new numerical methods, including hybrid-finite-element-finitevolume, lowest order Raviart-Thomas mixed finite elements, and multi-Point flux approximation methods as part of “Verification benchmarks for single-phase flow in three-dimensional fractured porous media [4, 5, 6].”