Universal scaling of spontaneous imbibition for water-wet systems

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.