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 $$P_{{\text{c}}} = A + B\tan \left( {\frac{\pi }{2} - \pi S_{e}^{C} } \right)\,{\text{for}}\,0 \le S_{{\text{e}}} \le 1,$$ P c = A + B tan π 2 - π S e C for 0 ≤ S e ≤ 1 , where $$S_{{\text{e}}}$$ S e is the normalized saturation. A indicates the wettability: $$A>0$$ A > 0 is a water-wet medium, $$A<0$$ 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.

Incorporation of Sub‐Resolution Porosity Into Two‐Phase Flow Models With a Multiscale Pore Network for Complex Microporous Rocks

“…Afterward, we decrease the capillary pressure to simulate water injection. We employ a more elaborate model that can accommodate different wettabilities and can explicitly determine the saturation for the prevailing capillary pressure (Foroughi et al., 2022). The Leverett J‐function based on this capillary pressure model is as follows: $$$J\left({S}_{w}\right)=A+B\,\tan \left(\frac{\pi }{2}-\pi {S}_{e}^{C}\right)$$$ with fitting parameters A , B , and C .…”

Incorporation of Sub-Resolution Porosity into Two-Phase Flow Models with a Multiscale Pore Network for Complex Microporous Rocks

“…Afterward, we decrease the capillary pressure to simulate water injection. We employ a more elaborate model that can accommodate different wettabilities and can explicitly determine the saturation for the prevailing capillary pressure [Foroughi et al, 2022]. The Leverett J-function based on this capillary pressure model is as follows:…”

Towards simulation of magnet quench using hp-FEM

“…The van Genuchten relation cannot account for the positive pore-water pressures that are characteristic of WR soil, suggesting the need for an alternative. Foroughi et al, (2022) proposed a new saturation-capillary pressure relationship for porous media of varying wettabilities, which proved to be more flexible and accurate than existing relationships [5]. This relationship can be written as…”