  Numerical simulations of flow and transport in porous media usually rely on
hybrid-dimensional models, i.e., the fracture is considered as objects of a
lower dimension compared to the embedding matrix. Such models are usually
combined with non-conforming discretizations as they avoid the inherent
difficulties associated with the generation of meshes that explicitly resolve
fractures-matrix interfaces. However, non-conforming discretizations demand a
more complicated coupling of different sub-models and may require special care
to ensure conservative fluxes. We propose a novel approach for the simulation
of flow and transport problems in fractured porous media based on an
equi-dimensional representation of the fractures. The major challenge for these
types of representation is the creation of meshes which resolve the several
complex interfaces between the fractures and the embedding matrix. To overcome
this 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 uniform
coarse mesh and refine the elements which have non-empty overlaps with at least
one of the fractures. Iterating this process allows to create non-uniform
non-conforming meshes, which do not resolve the interfaces but can approximate
them with arbitrary accuracy. We demonstrate that low-order finite element (FE)
discretizations on adapted meshes are globally and locally conservative and we
suitably adapt an algebraic flux correction technique to ensure the discrete
maximum principle. In particular, we show that the notorious conditions on
M-matrices have to be adapted to the basis functions defined on non-conforming
meshes. 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.
