Abstract
A thermodynamic description of porous media must handle the size- and shape-dependence of media properties, in particular on the nano-scale. Such dependencies are typically due to the presence of immiscible phases, contact areas and contact lines. We propose a way to obtain average densities suitable for integration on the course-grained scale, by applying Hill’s thermodynamics of small systems to the subsystems of the medium. We argue that the average densities of the porous medium, when defined in a proper way, obey the Gibbs equation. All contributions are additive or weakly coupled. From the Gibbs equation and the balance equations, we then derive the entropy production in the standard way, for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in boundary pressures, temperatures, and chemical potentials. Linear relations between thermodynamic fluxes and forces follow for the control volume. Fluctuation-dissipation theorems are formulated for the first time, for the fluctuating contributions to fluxes in the porous medium. These give an added possibility for determination of the Onsager conductivity matrix for transport through porous media. Practical possibilities are discussed.
Highlights
Porous media represent a vast and important class of systems; present for instance in biology, geology and in technological applications
We will be able to present fluctuation-dissipation theorems for the porous medium, which can serve as the Green–Kubo relations do in homogeneous systems to determine transport coefficients
We show below why the grand potential X grand canonical ensemble (GC),representative elementary volume (REV) = − pV REV and as a consequence p have configurational contributions, in the case of system control by T, V REV, μn, μw, μr
Summary
Porous media represent a vast and important class of systems; present for instance in biology, geology and in technological applications. We will be able to present fluctuation-dissipation theorems for the porous medium, which can serve as the Green–Kubo relations do in homogeneous systems to determine transport coefficients. The entropy production for the REV will lead us directly to the thermodynamic flux-force relations, and define a basis for the fluctuation-dissipation theorems. Such theorems are, as far as we know, formulated here for the first time for transport through porous media. With the REV and its Gibbs equation in place, we can find the entropy production (Section 6) and the constitutive equations (Section 7) in the standard way
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