Universal scaling of spontaneous imbibition for arbitrary petrophysical properties: Water-wet and mixed-wet states and Handy's conjecture

Spontaneous imbibition (SI) is a key process in many petrophysical applications, ranging from the mass transfer in fractured reservoirs during a waterflood to wettability characterization of rock samples, or steam migration in geothermal reservoirs. Scaling groups are an essential tool for upscaling laboratory data and modeling and describing SI. A general form has been debated for over 90 years, and several dozen specific groups have been proposed. Here, we give the first general scaling group for arbitrary wettability state, viscosity ratios, rock type, initial water content, and boundary conditions. The result is obtained by extending recent findings for water-wet systems but otherwise arbitrary properties (Schmid and Geiger, 2012) to the mixed-wet case. The group is based on the only known exact, general solution to Darcy's equation with capillarity, and we show that this solution can be viewed as the capillary analogue to the Buckley-Leverett solution for viscous dominated flow. Our group serves as a 'master equation' that contains many of the previously obtained groups as special cases, and its generality can be used to give the first predictive theory for the validity range of specific groups. Based on the universal group, we show that SI is best characterized by the cumulative inflow of the wetting phase and not by the movement of the wetting front, as has been conjectured. Furthermore, our results give strong evidence that Darcy's equation is suitable for describing SI, contrary to what has been hypothesized. The general correlation can be fitted by an exponential model for mass transfer that closely correlates 45 published water-oil, and water-air SI experiments obtained for widely different petrophysical properties.