Semianalytical solutions for cocurrent and countercurrent imbibition and dispersion of solutes in immiscible two-phase flow

We derive a set of semianalytical solutions for the movement of solutes in immiscible two-phase flow. Our solutions are new in two ways: First, we fully account for the effects of 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 into account. The understanding of immiscible two-phase flow and the simultaneous miscible displacement and mixing of components within a phase is important for many applications, including the location of nonaqueous phase liquids in the subsurface, the design of contaminant cleanup procedures, the sequestration of carbon dioxide, and enhanced oil-recovery techniques. For purely advective transport we combine a known exact solution for the description of immiscible two-phase flow with the method of characteristics for the advective transport equations to obtain solutions that describe cocurrent flow and countercurrent spontaneous 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 advective-dispersive solute transport, we derive approximate analytical solutions by the method of singular perturbation expansion. On the basis of this, we obtain analytical expressions for the growth of the dispersive zone for the case with and without the influence of capillary pressure. We show that for the case of spontaneous countercurrent imbibition the order of magnitude of the growth rate is far smaller than that for the viscous limit. We give some illustrative examples and compare the analytical expressions with numerical reference solutions. Copyright 2011 by the American Geophysical Union.