A mathematical model for dispersion with a moving front in groundwater

A new, non‐Fickian, constitutive equation is presented for the mass flux due to macroscopic dispersion in porous media. This constitutive equation contains the classical term of the diffusive type as well as a new term that may be viewed as an inertia term. Combination of the constitutive equation with the mass balance equation for a conservative contaminant yields a set of four first‐order partial differential equations in terms of the three components of the vector of dispersive mass flux and the concentration. For the case of longitudinal dispersion only, this system is reduced to a set of two first‐order partial differential equations in terms of the convective and dispersive mass fluxes as the two unknown functions. The sets of differential equations are hyperbolic and can be used to simulate and predict the movement of the front of a contaminant plume at finite velocity. An expression of the velocity of the front is presented in terms of the Darcian velocity and the ratio of the two dispersion coefficients that enter into the model. The effect of the non‐Fickian terms decreases for increasing times; the model reduces to the classical one as time increases. An exact solution is presented for the case of one‐dimensional flow and compared with the results of simulated column experiments.