Non-isothermal Transport of Multi-phase Fluids in Porous Media. The Entropy Production

Abstract

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...

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.…”

“…It should contain a statistically representative collection of pores. We have recently discussed [10] a new scheme to define as basis set of additive variables: the internal energy, entropy, and masses of all the components of the REV. These variables are additive in the sense that they are sums of contributions of all phases, interfaces and contact lines within the REV.…”

“…Using Euler homogeneity of the first kind, we were able to derive the Gibbs equation for the REV. This equation defines the temperature, pressure and chemical potentials of the REV as partial derivatives of the internal energy of the REV [10].…”

Flow-Area Relations in Immiscible Two-Phase Flow in Porous Media

“…A further development somewhat along the same lines, based on non-equilibrium thermodynamics uses Euler homogeneity, more about this later, to define the up-scaled pressure. From this, Kjelstrup et al derive constitutive equations for the flow [10,11].…”

Relative Permeability Scaling From Pore‐Scale Flow Regimes

“…Its structural resemblance with single-phase Darcy and its simplicity led to its widespread use, despite the abundant evidence from both experimental and theoretical analyses of its, often strong, limitations. Multiphase Darcy's law has been rigorously formulated by upscaling the Stoke's equation at the pore scale (e.g., Auriault, 1987;Daly & Roose, 2015;Hassanizadeh & Gray, 1980;Hornung, 1997;Lasseux et al, 2008;Kjelstrup et al, 2018;Whitaker, 1986), and since the microscopic details of the flow are usually neglected, its classical interpretation has been recently questioned. In fact, although the inherent instability of multiphase flows (Ling et al, 2017) may render the analysis of pore-scale distribution of the flowing phases challenging, recurring features in the topology of the nonwetting and wetting phase at the pore scale have been revealed by X-ray microtomography and high-performance computing (Armstrong et al, 2014(Armstrong et al, , 2012Blunt et al, 2013;Berg et al, 2013;Cueto-Felgueroso & Juanes, 2012;Gao et al, 2017;Garing et al, 2017;Li et al, 2005Li et al, , 2018bLin et al, 2018;Prodanovic et al, 2006Prodanovic et al, , 2007Reynolds et al, 2017;Tallakstad et al, 2009;Tahmasebi et al, 2017;Zarikos et al, 2018).…”