Analytical Solutions for Co- and Countercurrent 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. Our analytical solutions are new in two ways: First, we fully account for the effects of both capillary and viscous forces on the transport for arbitrary capillary-hydraulic properties. Second, we fully take hydrodynamic dispersion for the variable two-phase flow field and adsorption effects of the solutes into account. All previously obtained results for component transport in immiscible two-phase systems account for changes in the flow field due to the components’ presence but they ignore dispersive effects. This is surprising given the enormous practical importance of diffusive effects both in hydrological settings and for situations that arise in the context of oil production for chemical flooding, polymer flooding and the transport of wettability altering agents. In both cases, one often faces the situation of fractured media or thin-layer structures where the diffussive transport processes of components dominate over viscous forces. We consider a situation where the components do not affect the flow field and focus on dispersive effects. 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 countercurrent flow and advective transport in one dimension. We show that for both cases the solute front can be located graphically by a modified Welge tangent and that the mathematically obtained solutions correspond to the physical notion that the solute concentrations are functions of saturation only. For the advection-dispersion case, we derive approximate analytical solutions by the method of singular perturbation expansion. We give some illustrative examples and compare the analytical solutions with numerical results.