Article: Xu, Jin; Louge, Michel Y.; (2015) “Statistical Mechanics of Unsaturated Porous Media”, Physical Review E, 92 (6)
Abstract: We explore a mean-field theory of fluid imbibition and drainage through permeable porous solids. In the limit of vanishing inertial and viscous forces, the theory predicts the hysteretic “retention curves” relating the capillary pressure applied across a connected domain to its degree of saturation in wetting fluid in terms of known surface energies and void space geometry. To avoid complicated calculations, we adopt the simplest statistical mechanics, in which a pore interacts with its neighbors through narrow openings called “necks,” while being either full or empty of wetting fluid. We show how the main retention curves can be calculated from the statistical distribution of two dimensionless parameters. and a measuring the specific areas of, respectively, neck cross section and wettable pore surface relative to pore volume. The theory attributes hysteresis of these curves to collective first-order phase transitions. We illustrate predictions with a porous domain consisting of a random packing of spheres, show that hysteresis strength grows with. and weakens as the distribution of a broadens, and reproduce the behavior of Haines jumps observed in recent experiments on an ordered pore network.
Funding Acknowledgement: NPRP Grant from the Qatar National Research Fund [6-059-2-023]
Funding Text: We are grateful to Patrick Richard for supplying the random packing of spheres shown in Fig. 4, to Lili Gu for helping reduce the corresponding geometrical data, to Simon Salager, Thies Solling, and Thomas McKay for providing essential context by sharing x-ray CT scan observations, to Philippe Baveye and an anonymous reviewer for their constructive reading of the manuscript, to Valerie Pot for sharing the x-ray CT image in Fig. 1, and to Jean-Philippe Gras, Dani Or, Shmuel Assouline, Thierry Ruiz, Jean-Yves Delenne, Abraham Stroock, Steven Lantz, Olivier Vincent, Erik Huber, Jean-Yves Parlange, Tammo Steenhuis, Patrick Perre, Xavier Frank, James Sethna, Eric Clement, Alexandre Valance, Luc Oger, Renaud Delannay, Yves Meheust, and Leonard Susskind for illuminating conversations. This paper was made possible by the support of NPRP Grant No. 6-059-2-023 from the Qatar National Research Fund.