Modeling and Scaling Spontaneous Imbibition With Generalized Fractional Flow Theory and Non-Boltzmann Transformation

Summary Spontaneous imbibition (SI) is a process by which liquid is drawn into partially saturated porous media by capillary forces, relevant for subsurface processes, such as underground fluid storage and withdrawal. Accurate modeling and scaling of countercurrent SI have long been challenging. In this study, we proposed a generalized fractional flow theory (GFFT) using the Hausdorff fractal derivative, combined with non-Boltzmann scaling. The model links imbibition distance to time through the power law exponent α/2, where α is the fractal index (0§amp;lt;α§amp;lt;2 in this study). To numerically solve the proposed model, we presented a new mathematical approach by converting the proposed fractional partial differential equation to a pair of first-order ordinary differential equations. We applied the GFFT to various experimental and simulated data sets of both porous and fractured media and showed, for the first time, that α varied with contact angle (of the imbibing fluid). We also demonstrated that α depended on other factors, such as dynamic viscosity, pore structure, and fracture properties. Using concepts of GFFT and non-Boltzmann scaling, for the first time, we developed a scaling approach. By analyzing SI data from sandstones, diatomite, carbonate, and synthetic porous media, we demonstrated that the non-Boltzmann scaling provided a better collapse of the SI data than the traditional Boltzmann approach (α = 1), with α values ranging from 0.88 to 1.54. These deviations illustrate the model’s adaptability to different porous materials. Using the GFFT, we expect to better predict fluid imbibition rates when properties, such as porosity, permeability, initial and maximum saturations, viscosity, and wettability, are known, offering a more accurate alternative to traditional models.