Analytical solutions for co and counter-current imbibition of sorbing, dispersive solutes in immiscible two-phase flow

We derive a set of analytical solutions for the transport of adsorbing solutes in an immiscible, incompressible two-phase system. This work extends recent results for the analytical description for the movement of inert tracers due to capillary and viscous forces and dispersion to the case of adsorbing solutes. We thereby obtain the first known analytical expression for the description of the effect of adsorption, dispersion, capillary forces and viscous forces on solute movement in two-phase flow. For the purely advective transport, we combine a known exact solution for the description of flow with the method of characteristics for the advective transport equations to obtain solutions that describe both co and spontaneous counter-current imbibition and advective transport in one dimension. We show that for both cases, the solute front can be located graphically by a modified Welge tangent. For the dispersion, we derive approximate analytical solutions by the method of singular perturbation expansion. The solutions reveal that the amount of spreading depends on the flow regime and that adsorption diminishes the spreading behavior of the solute. We give some illustrative examples and compare the analytical solutions with numerical results.