Forchheimer's correction in modelling flow and transport in fractured porous media

The scope of this manuscript is to investigate the role of the Forchheimer correction in the description of variable-density flow in fractured porous media. A fractured porous medium, which shall be also referred to as "the embedding medium", represents a flow region that is made macroscopically heterogeneous by the presence of fractures. Fractures are assumed to be filled with a porous medium characterized by flow properties that differ appreciably from those of the embedding medium. The fluid, which is free to move in the pore space of the entire flow region, is a mixture of water and brine. Flow is assumed to be a consequence of the variability of the fluid mass density in response to the generally nonuniform distribution of brine, which is subject to diffusion and convection. The fractures are assumed to be thin in comparison with the characteristic sizes of the embedding medium. Within this framework, some benchmark problems are solved by adopting two approaches: (i) the fractures are treated as thin but d-dimensional flow subregions, with d being the geometric dimension of the embedding medium; (ii) the fractures are regarded as (d-1) -dimensional manifolds. In the first approach, the equations of variable-density flow are written in the same, d-dimensional form both in the fractures and in the embedding medium. In the second approach, instead, new equations are obtained by averaging the d-dimensional ones over the fracture width. The reliability of the second approach is discussed by comparing the results of the (d - 1) -dimensional numerical simulations of the selected benchmark problems with those obtained by using the d-dimensional approach. Moreover, the deviations of the results determined by accounting for the Forchheimer correction to flow velocity are compared with those predicted by Darcy's law.