[1] Spontaneous, counter-current imbibition (SI) is a key mechanism in many multiphase flow processes, such as cleanup of nonaqueous phase liquids (NAPLs), bioremediation, or CO 2 storage. For interpreting and upscaling laboratory SI data, and modeling and prediction purposes, scaling groups are an essential tool. The question of how to formulate a general scaling group has been debated for over 90 years. Here we propose the first scaling group that incorporates the influence of all parameters on SI that are present in the two-phase Darcy model. The group is derived rigorously from the only known exact analytical solution for spontaneous imbibition by relating the cumulative water phase imbibed to the normalized pore volume. We show the validity of the group by applying it to 42 published SI studies for water-oil and water-air experiments, for a wide range of viscosity ratios, different materials, different initial water saturations, and different length-scales. In all cases, water was the wetting phase. Our group serves as a ''master equation'' whose generality allows the rigorous prediction of the validity of a large number of specialized scaling groups proposed during the last 90 years. Furthermore, our results give strong evidence that the Darcy model is suitable for describing SI, and that including dynamic effects in capillary pressure is not necessary for counter-current SI, contrary to what has been hypothesized. Two key applications of the group are discussed: First, the group can serve as the long sought after general transfer rate for imbibition used in dual-porosity models. Second, it is the so far missing proportionality constant in imbibition-germination models for plant seeds.Citation: Schmid, K. S., and S. Geiger (2012), Universal scaling of spontaneous imbibition for water-wet systems, Water Resour. Res., 48, W03507,

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[1] 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 oilrecovery 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.Citation: Schmid, K. S., S. Geiger, and K. S. Sorbie (2011), Semianalytical solutions for cocurrent and countercurrent imbibition and dispersion of solutes in immiscible two-phase flow, Water Resour.

We present analytical solutions for capillary-controlled displacement in one dimension by use of fractional-flow theory. We show how to construct solutions with a spreadsheet that can be used for the analysis of experiments as well as matrix-block-scale recovery in field settings. The solutions can be understood as the capillary analog to the classical Buckley-Leverett solution (Buckley and Leverett 1942) for viscous-dominated flow, and are valid for cocurrent and countercurrent spontaneous imbibition (SI), as well as for arbitrary capillary pressure and relative permeability curves. They can be used to study the influence of wettability, predicting saturation profiles and production rates characteristic for water-wet and mixed-wet conditions. We compare our results with in-situ measurements of saturation profiles for SI in a waterwet medium. We show that the characteristic shape of the saturation profile is consistent with the expected form of the relative permeabilities. We discuss how measurements of imbibition profiles, in combination with other measurements, could be used to determine relative permeability and capillary pressure.

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