A Closed-Form Equation for Capillary Pressure in Porous Media for All Wettabilities
Sajjad Foroughi, Branko Bijeljic, Martin J. Blunt
Abstract
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 $$P_{{\text{c}}} = A + B\tan \left( {\frac{\pi }{2} - \pi S_{e}^{C} } \right)\,{\text{for}}\,0 \le S_{{\text{e}}} \le 1,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>P</mml:mi> <mml:mtext>c</mml:mtext> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>B</mml:mi> <mml:mo>tan</mml:mo> <mml:mfenced> <mml:mrow> <mml:mfrac> <mml:mi>π</mml:mi> <mml:mn>2</mml:mn> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mi>π</mml:mi> <mml:msubsup> <mml:mi>S</mml:mi> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mi>C</mml:mi> </mml:msubsup> </mml:mrow> </mml:mfenced> <mml:mspace/> <mml:mtext>for</mml:mtext> <mml:mspace/> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mtext>e</mml:mtext> </mml:msub> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> where $$S_{{\text{e}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>S</mml:mi> <mml:mtext>e</mml:mtext> </mml:msub> </mml:math> is the normalized saturation. A indicates the wettability: $$A>0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is a water-wet medium, $$A<0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo><</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> 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.