A new formulation of the dispersion tensor in homogeneous porous media

Abstract

Dispersion is the result of two mass transport processes, namely molecular diffusion, which is a pure mixing effect and hydrodynamic dispersion, which combines mixing and spreading. The identification of each contribution is crucial and is often misinterpreted. Traditionally, under a volume averaging framework, a single closure problem is solved and the resulting fields are substituted into diffusive and dispersive filters. However the diffusive filter (that leads to the effective diffusivity) allows passing information from convection, which leads to an incorrect definition of the effective medium coefficients composing the total dispersion tensor. In this work, we revisit the definitions of the effective diffusivity and hydrodynamic dispersion tensors using the method of volume averaging. Our analysis shows that, in the context of laminar flow with or without inertial effects, two closure problems need to be computed in order to correctly define the corresponding effective medium coefficients. The first closure problem is associated to momentum transport and needs to be solved for a prescribed Reynolds number and flow orientation. The second closure problem is related to mass transport and it is solved first with a zero Péclet number and second with the required Péclet number and flow orientation. All the closure problems are written using closure variables only as required by the upscaling method. The total dispersion tensor is shown to depend on the microstructure, macroscopic flow angles, the cell (or pore) Péclet number and the cell (or pore) Reynolds number. It is non-symmetric in the general case. The condition for quasi-symmetry is highlighted. The functionality of the longitudinal and transverse components of this tensor with the flow angle is investigated for a 2D model porous structure obtaining consistent results with previous studies.