We derive the entropy production for transport of multi-phase fluids in a non-deformable, porous medium exposed to differences in pressure, temperature, and chemical potentials. Thermodynamic extensive variables on the macro-scale are obtained by integrating over a representative elementary volume (REV). Using Euler homogeneity of the first order, we obtain the Gibbs equation for the REV. From this we define the intensive variables, the temperature, pressure and chemical potentials and, using the balance equations, derive the entropy production for the REV. The entropy production defines sets of independent conjugate thermodynamic fluxes and forces in the standard way. The transport of two-phase flow of immiscible components is used to illustrate the equations.But, unlike what has been done before, we shall seek to reduce drastically the number of variables needed for the description, allowing us still to make use of the systematic theory of non-equilibrium thermodynamics. While the entropy production in the porous medium so far has been written as a combination of contributions from each phase, interface and contact line, we shall write the property for a more limited set of macro-scale variables. This will enable us to describe experiments and connect variables at this scale.The theory of non-equilibrium thermodynamics was set up by Onsager [5,6] and further developed for homogeneous systems during the middle of the last century [7]. It was the favored thermodynamic basis of Hassanizadeh and Gray for their description of porous media. These authors [2,3] discussed also other approaches, e.g the theory of mixtures in macroscopic continuum mechanics, cf. [1,4].The theory of classical non-equilibrium thermodynamics has been extended to deal with a particular case of flow in heterogeneous systems, namely transport along [8] and perpendicular [9] to layered interfaces. A description of heterogeneous systems on the macro-scale has not been given, however. Transport in porous media take place, not only under pressure gradients. Temperature gradients will frequently follow from transport of mass, for instance in heterogeneous catalysis [10], in polymer electrolyte fuel cells, in batteries [9,11], or in capillaries in frozen soils during frost heave [12]. The number of this type of phenomena is enormous. We have chosen to consider only the vectorial driving forces related to changes in pressure, chemical composition and temperature, staying away for the time being from deformations, chemical reactions, or forces leading to stress [13]. The multi-phase flow problem is thus in focus.The development of a general thermodynamic basis for multi-phase flow [2,3] started by introduction of thermodynamic properties for each component in each phase, interface and three-phase contact line. A representative volume element (REV) was introduced, consisting of bulk phases, interfaces and three-phase contact lines. Balance equations were formulated for each phase in the REV, and the total REV entropy production was the sum of the separ...

We define the pressure of a porous medium in terms of the grand potential, and compute its value in a nano-confined or nano-porous medium, meaning a medium where thermodynamic equations need be adjusted for smallness. On the nano-scale, the pressure depends in a crucial way on the size and shape of the pores. According to Hill [1], two pressures are needed to characterize this situation; the integral pressure and the differential pressure. Using Hill's formalism for a nano-porous medium, we derive an expression for the difference between the integral and the differential pressures in a spherical phase α of radius R, p α − p α = γ/R. We recover the law of Young-Laplace for the differential pressure difference across the same curved surface. We discuss the definition of a representative volume element for the nano-porous medium and show that the smallest REV is half a unit cell in the direction of the pore in the fcc lattice. We also show, for the first time, how the pressure profile through a nano-porous medium can be defined and computed away from equilibrium.

We develop constitutive equations for multi-component, multi-phase, macro-scale flow in a porous medium exposed to temperature-, composition-, and pressure -gradients. The porous medium is non-deformable. We define the pressure and the composition of the representative elementary volume (REV) in terms of the volume and surface averaged pressure and the saturation, and the respective driving forces from these variables. New contributions due to varying porosity or surface tension offer explanations for non-Darcy behavior. The interaction of a thermal and mechanical driving forces give thermal osmosis. An experimental program is suggested to verify Onsager symmetry in the transport coefficients.relations between experiments particular for these flows, as derived for instance from the Onsager relations.The procedure that we used to obtain the Gibbs equation for coarse-grained variables [1] assumed that the additive thermodynamic variables of the REV are Euler homogeneous functions of the first order. As it is so central to this work, we will briefly review the procedure in Section 2, highlighting the main points. In the procedure, we regard the REV as a complete thermodynamic system. Hansen and Ramstad [3] suggested this possibility already some time ago. Since then the hypothesis has been supported through measurements on Hele-Shaw cells [4] and through network simulations [5]. The variables of the REV will then fluctuate similar to the variables in a normal thermodynamic state around a mean value.The equations presented here, will allow us to revisit previously published experimental results, and explain in more detail for instance when we can expect deviations from Darcy's law. It appears that the strictly linear theory may cease to hold for small pressure differences; also for single fluids [6][7][8][9]. Observations of deviations from Darcy's law were made for water or water solutions in clay [8,9]. Thresholds and/or deviations from straight lines in plots of flow versus the overall pressure difference, were reported. Boersma et al. [8] found a dependency of the threshold on the average pore radius,r, for flow in a porous medium made of glass-beads. These observations have, as of yet, no unique explanation. When dealing with immiscible fluids, Tallakstad et al. [16] observed a square dependence of the flow rate on the pressure difference under steady-state flow conditions. Sinha and Hansen [18] explained this square dependence by the successive opening of pores due to the mobilization of interfaces when the pressure difference across the sample is increased. This explanation was supported by a mean-field calculation and numerical experiments using a network model. Sinha et al. [17, 18] followed up the original Tallakstad study, originally done in a two-dimensional model porous medium, both experimentally and computationally in threedimensional porous media, with the same result.There is not only a need to better understand deviations from Darcy's law for volume transport. Other driving forces than those related...

It is known that thermodynamic properties of a system change upon confinement. To know how, is important for modelling of porous media. We propose to use Hill’s systematic thermodynamic analysis of confined systems to describe two-phase equilibrium in a nanopore. The integral pressure, as defined by the compression energy of a small volume, is then central. We show that the integral pressure is constant along a slit pore with a liquid and vapor in equilibrium, when Young and Young–Laplace’s laws apply. The integral pressure of a bulk fluid in a slit pore at mechanical equilibrium can be understood as the average tangential pressure inside the pore. The pressure at mechanical equilibrium, now named differential pressure, is the average of the trace of the mechanical pressure tensor divided by three as before. Using molecular dynamics simulations, we computed the integral and differential pressures, p ^ and p, respectively, analysing the data with a growing-core methodology. The value of the bulk pressure was confirmed by Gibbs ensemble Monte Carlo simulations. The pressure difference times the volume, V, is the subdivision potential of Hill, ( p − p ^ ) V = ϵ . The combined simulation results confirm that the integral pressure is constant along the pore, and that ϵ / V scales with the inverse pore width. This scaling law will be useful for prediction of thermodynamic properties of confined systems in more complicated geometries.

We have described for the first time the thermodynamic state of a highly confined single-phase and single-component fluid in a slit pore using Hill’s thermodynamics of small systems. Hill’s theory has been named nanothermodynamics. We started by constructing an ensemble of slit pores for controlled temperature, volume, surface area, and chemical potential. We have presented the integral and differential properties according to Hill, and used them to define the disjoining pressure on the new basis. We identified all thermodynamic pressures by their mechanical counterparts in a consistent manner, and have given evidence that the identification holds true using molecular simulations. We computed the entropy and energy densities, and found in agreement with the literature, that the structures at the wall are of an energetic, not entropic nature. We have shown that the subdivision potential is unequal to zero for small wall surface areas. We have showed how Hill’s method can be used to find new Maxwell relations of a confined fluid, in addition to a scaling relation, which applies when the walls are far enough apart. By this expansion of nanothermodynamics, we have set the stage for further developments of the thermodynamics of confined fluids, a field that is central in nanotechnology.

The accurate description of the behavior of fluids in nanoporous materials is of great importance for numerous industrial applications. Recently, a new approach was reported to calculate the pressure of nanoconfined fluids. In this approach, two different pressures are defined to take into account the smallness of the system: the so-called differential and the integral pressures. Here, the effect of several factors contributing to the confinement of fluids in nanopores are investigated using the definitions of the differential and integral pressures. Monte Carlo (MC) simulations are performed in a variation of the Gibbs ensemble to study the effect of the pore geometry, fluid-wall interactions, and differential pressure of the bulk fluid phase. It is shown that the differential and integral pressure are different for small pores and become equal as the pore size increases. The ratio of the driving forces for mass transport in the bulk and in the confined fluid is also studied. It is found that, for small pore sizes (i.e., < 5 σ fluid ), the ratio of the two driving forces considerably deviates from 1.

We describe the thermodynamic state of a single-phase fluid confined to a porous medium with Hill's thermodynamics of small systems, also known as nanothermodynamics. This way of defining small system thermodynamics, with a separate set of control variables, may be useful for the study of transport in nondeformable porous media, where presently no consensus exists on pressure computations. For a confined fluid, we observe that there are two pressures, the integral and the differential pressures. We use molecular simulations to investigate and confirm the nanothermodynamic relations for a representative elementary volume (REV). For a model system of a single-phase fluid in a face-centered cubic lattice of solid spheres of varying porosity, we calculate the fluid density, fluid-solid surface tension, replica energy, integral pressure, entropy, and internal energy.

Classical capillary theory predicts that a non-neutrally wetting ellipsoidal particle adsorbed at a liquid–vapor interface will deform the interface. The deformation gives rise to anisotropic capillary forces of a quadrupolar nature that induce strong directionality in the particle interactions. Here, we investigate the interactions between nanoparticles with characteristic lengths of 1–5 nm. We show that the near-field interactions are dominated by solvent-mediated forces, which arise from the fluid packing between the nanoparticles and direct nanoparticle–nanoparticle interactions. The solvent-mediated forces are two orders of magnitude larger than the estimated capillary force. We find that interacting ellipsoidal nanoparticles adsorbed at the liquid–vapor interface have a larger repulsion in the depletion region than the nanoparticles submerged in a dense bulk phase and argue that this is because of a negative line tension associated with the three-phase line.