Solute dispersion in porous flow with a constant velocity gradient

In a previous paper, we have shown that stepwise changes in the macroscopic flow velocity of a carrier fluid through a porous medium, substantially affects the dispersion of solutes that it carries. This paper extends the work to the case of a continuously changing flow velocity. The stochastic model used to describe pore deflection of the flow path, is reduced by the use of Dynkin' s formula to the formulation of a deterministic differential equation. An exact solution is found for the case of a flow velocity that varies linearly with position, and the resulting integral equation for the solute probability distribution is also solved exactly. This is applied to the case of a solute concentration plume represented by an initially Gaussian concentration peak. The modulated Gaussian that is calculated shows that the time evolution of the plume differs fundamentally from that predicted by the standard diffusive model. Implications of this conclusion for modelling dispersion in a variable flow velocity are further explored