Simplified thermodynamic model for equilibrium capillary pressure in a fractal porous medium

Abstract

Defining a relation for equilibrium pressure in a porous medium has been difficult to do in terms of readily measurable parameters. We present a simplified analysis of this problem using the first law of thermodynamics combined with a fractal description of a porous system. The results show that the variation in fluid interfacial area with fluid volume, and the respective interfacial surface tensions, are dominant factors determining equilibrium capillary pressure. Departures from equilibrium are seen to occur when fluid-solid contact lines are in movement. By describing the pore space as fractal we are able to obtain an expression for the change in fluid interfacial area with respect to its volume, and the resulting model shows a strong fit to pressure data obtained from a capillary rise experiment conducted in a coarse-grained SiO2 sand.

An Analytic Approach to Capillary Pressure in Tree-Shaped Networks

“…Therefore, it is also a function of network's saturation. These results are in agreement with several studies (see for example [12,15,18,28]), which recognize that the interfaces play a key role in describing the multiphase fluid flow.…”

“…Under these conditions, it is reasonable to consider that the internal energy and the entropy of the fluid are constant [15]. Therefore, as 0 V V According to [15,18] the first right-hand term of Eq. (10) is related to the contact line velocity and the volume of wetting fluid.…”

“…Equation (12) represents the so-called equilibrium capillary pressure [15,18] and shows that this pressure depends both on the interfacial energy of the wetting fluid and on the variation in fluid interfacial areas with respect to fluid volume. Therefore, it is also a function of network's saturation.…”

“…The simulation of fluid flow through these structures requires an accurately knowledge of the *Address correspondence to this author at the Department of Physics and Geophysics Centre of Évora, University of Évora, Portugal; E-mail: afm@uevora.pt pressure distribution of a fluid through the structure. Studies performed in porous media point out the dependence of capillary pressure on the rate change of moisture inside the media, suggesting that departures from equilibrium are dependent on the movement of fluid interfaces (see for example, Beliaeve and Hassanizadeh [12], Deinert et al [15] and Fan et al [18]). Therefore, capillary pressure is an essential quantity that deserves further analysis.…”

“…Based on experiments, Cheng et al [14] showed a connection between the capillary pressure and the fluid interfacial area in porous media. Other studies [15] extended the equilibrium capillary pressure based on the first law of thermodynamics to a fractal porous media. The dynamic capillary rise due to hydrostatic effects was studied by [16].…”