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 λ and α 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 λ and weakens as the distribution of α broadens, and reproduce the behavior of Haines jumps observed in recent experiments on an ordered pore network.
Highlights
Unsaturated porous media are ubiquitous in geophysical and industrial processes
Because the capillary energy of a pore depends upon the saturation state of its connected neighbors, the establishment of a local equilibrium derives from many-body interactions similar to those handled by statistical mechanics [1], such as lattice gases [2], neural networks [3,4], bird flocks [5], and spin glasses [6,7]
To illustrate how the geometry of pores and necks affects hysteresis, we consider generic statistical distributions of these quantities, and we examine the collective behavior of a porous medium created by a dense packings of spheres
Summary
Unsaturated porous media are ubiquitous in geophysical and industrial processes. They include, for example, soils partially filled with water, fuel cells, or oil reservoirs holding several gas and liquid phases. Because the capillary energy of a pore depends upon the saturation state of its connected neighbors, the establishment of a local equilibrium derives from many-body interactions similar to those handled by statistical mechanics [1], such as lattice gases [2], neural networks [3,4], bird flocks [5], and spin glasses [6,7] In this context, we propose a mean-field theory of fluid retention in porous media partially filled with a connected wetting fluid. The percolation framework, while fruitful in describing geometrical domain size and shape, does not directly account for surface energies and displacement work, unlike the approach outlined here Recent experimental techniques, such as x-ray microtomography, have the potential to inform numerical simulations by revealing the internal liquid distribution among pores within a solid matrix in detail [48,49,50,51,52,53]. By identifying viscous forces as the mechanism that absorbs the “latent energy” released in the phase transitions, we predict recent observations of Haines jump by Armstrong and Berg [58] with an ordered pore network
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