Upscaling solute transport in porous media from the pore scale to dual- and multicontinuum formulations

We provide pore to Darcy- scale theoretical upscaling of solute transport in porous media and discuss the key theoretical elements underlying double- and multi- rate mass transfer formulations which are typically adopted to interpret laboratory- and/or field- scale transport experiments. We model pore scale transport by considering advective and diffusive processes. The resulting mass balance equation is subject to volume averaging relying on an unsteady closure. This leads to a nonlocal in time continuum-scale two-equation transport model, which we compare against existing double- and multi- rate mass transfer formulations. Coefficients appearing in our upscaled model are expressed as functions of time and pore scale geometry and velocity distribution. We analyze in detail the scenario associated with two-dimensional flow in a plane channel and discuss the temporal dynamics and the associated asymptotic behavior of the different effective coefficients appearing in the upscaled system of equations. The relative influence of the terms included in the continuum-scale model is quantitatively assessed.