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Numerical simulations of flow and transport in porous media usually rely onhybrid-dimensional models, i.e., the fracture is considered as objects of alower dimension compared to the embedding matrix. Such models are usuallycombined with non-conforming discretizations as they avoid the inherentdifficulties associated with the generation of meshes that explicitly resolvefractures-matrix interfaces. However, non-conforming discretizations demand amore complicated coupling of different sub-models and may require special careto ensure conservative fluxes. We propose a novel approach for the simulationof flow and transport problems in fractured porous media based on anequi-dimensional representation of the fractures. The major challenge for thesetypes of representation is the creation of meshes which resolve the severalcomplex interfaces between the fractures and the embedding matrix. To overcomethis difficulty, we employ a strategy based on adaptive mesh refinement (AMR).The idea at the base of the proposed AMR is to start from an initially uniformcoarse mesh and refine the elements which have non-empty overlaps with at leastone of the fractures. Iterating this process allows to create non-uniformnon-conforming meshes, which do not resolve the interfaces but can approximatethem with arbitrary accuracy. We demonstrate that low-order finite element (FE)discretizations on adapted meshes are globally and locally conservative and wesuitably adapt an algebraic flux correction technique to ensure the discretemaximum principle. In particular, we show that the notorious conditions onM-matrices have to be adapted to the basis functions defined on non-conformingmeshes. Although the proposed applications come from geophysical applications,the obtained results could be applied to any diffusion and transport problems,on both conforming and non-conforming meshes.

A novel equi-dimensional finite element method for flow and transport in  fractured porous media satisfying discrete maximum principle and conservation  properties