Semi-analytical solutions to one-dimensional advection-diffusion equations with variable diffusion coefficient and variable flow velocity

Diffusion/dispersion coefficient and flow velocity in an advection-diffusion mass-transfer process have both been commonly considered as constant mean values in most previous studies, which may be reasonable in incompressible fluid flow with a constant viscosity. However, this assumption may not be applicable for some cyclic processes, such as huff-n-puff CO2 injection and cyclic solvent injection in heavy oil enhanced oil recovery (EOR) processes, in which both the diffusion coefficient and flow velocity vary with time and space.This paper develops two novel one-dimensional (1D) advection-diffusion mathematical models: one model considers a constant diffusion coefficient and a variable flow velocity, and the other one considers both parameters as variables. Semi-analytical solutions to both new models are developed through the Laplace transformation and a special approximation scheme to the variable diffusion coefficient and flow velocity. The semi-analytical results are validated by an analytical solution to a special advection-diffusion case as well as the numerical solutions.It is found that the concentration distribution for the constant and variable diffusion coefficients has quite different shapes. The flow velocity can play a much larger role than the diffusion coefficient does in the crude oil-solvent mass-transfer process, implying the pressure gradient between the solvent chamber and the crude oil zone can greatly enhance the solvent dissolution into the crude oil. Gravity force might hinder the mixing process of crude oil and solvent in some cases of solvent-based EOR methods.