A Closed-Form Equation for Capillary Pressure in Porous Media for All Wettabilities

Abstract

A saturation–capillary pressure relationship is proposed that is applicable for all wettabilities, including mixed-wet and oil-wet or hydrophobic media. This formulation is more flexible than existing correlations that only match water-wet data, while also allowing saturation to be written as a closed-form function of capillary pressure: we can determine capillary pressure explicitly from saturation, and vice versa. We propose Pc=A+Btanπ2-πSeCfor0≤Se≤1,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$P_{{\ ext{c}}} = A + B\ an \\left( {\\frac{\\pi }{2} - \\pi S_{e}^{C} } \\right)\\,{\ ext{for}}\\,0 \\le S_{{\ ext{e}}} \\le 1,$$\\end{document}where S_{{text{e}}} is the normalized saturation. A indicates the wettability: A>0 is a water-wet medium, A<0 is hydrophobic while small A suggests mixed wettability. B represents the average curvature and pore-size distribution which can be much lower in mixed-wet compared to water-wet media with the same pore structure if the menisci are approximately minimal surfaces. C is an exponent that controls the inflection point in the capillary pressure and the asymptotic behaviour near end points. We match the model accurately to 29 datasets in the literature for water-wet, mixed-wet and hydrophobic media, including rocks, soils, bead and sand packs and fibrous materials with over four orders of magnitude difference in permeability and porosities from 20% to nearly 90%. We apply Leverett J-function scaling to make the expression for capillary pressure dimensionless and discuss the behaviour of analytical solutions for spontaneous imbibition.