Non-isothermal Transport of Multi-phase Fluids in Porous Media. Constitutive Equations

Abstract

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...

Pressures Inside a Nano-Porous Medium. The Case of a Single Phase Fluid

“…The work can be seen as a continuation of our earlier works [10,11]. The work so far considered transport processes in micro-porous, not nano-porous media.…”

“…The grand potential is equal to minus the contribution to the internal energy from the pressure-volume term, k B T ln Z g = Υ = −pV , which we will from now on refer to as the compressional energy. For a single fluid f in a porous medium r, the result was [10,11]…”

Two-Phase Equilibrium Conditions in Nanopores

“…In a porous medium with two fluid phases, we need more than one type of REV, one for each phase. A REV needs, as the name says, to be representative for all molecular interactions in the system [17]. It should be large enough to contain a statistically representative amount of the fluids and solids.…”

“…It should be large enough to contain a statistically representative amount of the fluids and solids. The purpose of the REV is to define a volume element, to be used to define equilibrium, but also to define local equilibrium in a large system where fluid transports take place [17].…”

“…This situation calls for a robust definition of the representative elementary volume (REV) of the small system, to serve as a basis for a definition of pressure. Kjelstrup et al [17] offered a definition of the REV-pressure based on the compression energy of the REV. Galteland et al [18] showed, by evaluating this property for a single component in a nanoporous system, that the REV was not sufficiently described by the bulk fluid pressure.…”