A Closed-Form Model for Relative Permeability Hysteresis Using Saturation and Connectivity

Abstract This paper presents a closed-form analytical function for the relative permeability of the nonwetting phase that captures nonlinear variations with saturation and phase connectivity (normalized Euler characteristic) via the state function approach. The developed physical state function for relative permeability is unique among prior models in that both imbibition and drainage experimental data, as well as subsequent scans, can be matched simultaneously, while maintaining continuous, single-valued relative permeability and residual saturation predictions. The closed-form model is the first completely robust hysteretic model, ensuring that relative permeability remains within its physical bounds for any scan. The results show that the new function can easily match experimental scanning data with significant hysteresis using only one imbibition and drainage curve and predict subsequent two-phase experimental scanning data with high accuracy. Predicted scans are physically consistent in relative magnitude and degree of hysteresis with the tuned experimental data. Residual saturations, which are a natural output of our model, are defined as the remaining saturations at very low relative permeability, e.g., as k r approaches 0.01. The model matches and accurately predicts residual saturation for other scanning curves, yielding the expected Land-type shape of the initial-residual curve. Unlike all prior models, relative permeabilities are exactly zero only when connectivity or saturation is zero, thereby allowing relative permeability continuity during vaporization or condensation of the residual phase, or other complex processes such as film drainage of a spreading intermediate wetting phase in a three-phase system. Although having at least one drainage and imbibition scan is desired for model tuning, the model can generate hysteresis curves from a single drainage or imbibition curve, provided that the degree of hysteresis is specified.