Description of Free Energy for Immiscible Two‐Fluid Flow in Porous Media by Integral Geometry and Thermodynamics

Abstract

In integral geometry, intrinsic volumes are a set of geometrical variables to characterize spatial structures, for example, distribution of fluids in two‐fluid flow in porous media. McClure et al. (2018, https://doi.org/10.1103/PhysRevFluids.3.084306) utilized this principle and proposed a geometric state function based on the intrinsic volumes. In a similar approach, we find a geometrical description for free energy of a porous system with two fluids. This is also an extension of the work by Mecke (2000, https://doi.org/10.1007/3-540-45043-2_6) for energy of a single fluid. Several geometrical sets of spatial objects were defined, including bulk of the two fluids, interfaces, and three‐phase contact lines. We have simplified the description of free energy by showing how the intrinsic volumes of these sets are geometrically related. We obtain a description for energy as a function of seven microscopic geometrically independent variables. In addition, using a thermodynamic approach, we find an approximation for the free energy as a function of macroscopic parameters of saturation and pressure under quasi‐static conditions. The combination of the two energy descriptions, by integral geometry and thermodynamics, completes the relation between the associated variables and enables us to find the unknown coefficients of the intrinsic volumes and to calculate the amount of dissipated energy in drainage and imbibition processes. We show that the theory is consistent with a set of experiments performed by Schlüter et al. (2016a, https://doi.org/10.1002/2015WR018254, 2017a, https://doi.org/10.1002/2016WR019815). However, in order to be more conclusive, it needs to be tested with larger data sets.

Geometrically derived efficiency of slow immiscible displacement in porous media

“…[33], while significantly different from the thermodynamically based wettability calculation in Ref. [9] giving a contact angle of 43 degrees. Excluding the efficiency we obtain a contact angle of 46 degrees.…”

“…Note that we do not associate any energy to the three-phase contact line nor the curvature, in contrast to, e.g., Ref. [9]. In Fig.…”

“…Additionally, we have one image of sintered soda lime beads used in Refs. [9,33], which we will return to in the next section.…”

“…If one assumes that the Hadwiger theorem holds for free energy in porous media, then the free energy is described as a linear combination of intrinsic volumes of the fluid and solid phases [7,8]. This gives a free-energy description by fully 27 variables, fortunately geometrical dependencies reduce the number to seven independent geometrical descriptors consisting of the interfacial areas, three-phase contact lines, interface curvatures and the Euler characteristic [9]. However, the interfacial energies are often considered dominant in porous media with μm-size pores.…”

“…When one incompressible phase is entering the porous medium, displacing another incompressible phase leaving the medium, the pressure difference between the entering and produced fluids times the volume change of the fluids yields the work applied to the system. When equating this applied work to the changes in interfacial energies, several authors have estimated the wettability from measurements of applied work and changes in interfaces [9,12,13].…”