Abstract
Thermodynamic equilibrium, which involves mechanical, thermal, and chemical equilibria, in a multiphase porous medium, is defined and discussed, both at the microscopic level, and at the macroscopic one. Conditions are given for equilibrium in the presence of forces between the surface of the solid matrix and the fluid phases. The concept ofapproximate thermodynamic equilibrium is introduced and discussed, employing the definition of athermodynamic potential. This discussion serves as the basis for the methodology of determining the number of degrees of freedom in models of phenomena of transport (of mass, energy, and momentum) in porous media. Equilibrium and nonequilibrium cases are considered. The proposed expressions for the number of degrees of freedom in macroscopic transport models, represent the equivalent ofGibbs phase rule in thermodynamics.
Based on balance considerations and thermodynamic relationships, it is shown that the number of degrees of freedom, NF, in a problem of transport in a deformable porous medium, involving NP fluid phases and NC components, under nonisothermal conditions, with equilibrium among all phases and components, is
Under nonequilibrium conditions among the phases, the rule takes the form
In both cases, when fluid phase velocities are determined by Darcy's law, NF is reduced by NP. When the solid matrix is nondeformable, NF is reduced by 3. The number of degrees of freedom is also determined for conditions of approximate chemical and thermal equilibria, and for conditions of equilibrium that prevail only among some of the phases present in the system. Examples of particular cases are presented to illustrate the proposed methodology.
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Abbreviations
- D α E :
-
Coefficient of dispersion ofE in anα-phase
- D γα :
-
Coefficient of molecular diffusion of a γ-component in anα-phase
- D *γα :
-
Coefficient of molecular diffusion of a γ-component in anα-phase in a porous medium
- e :
-
Density of an extensive quantity,E
- e γα :
-
Density ofe γα (=E per unit volume ofα-phase)
- E γα :
-
An extensive quantity,E, of a γ-component in anα-phase (e.g.,E=m,γ,H).
- ε :
-
Energy
- f Eα→β :
-
Rate of transfer ofE from anα-phase to aβ-one, across their common microscopic interface per unit volume of porous medium
- f γα :
-
Generalized force acting on γ-component inα-phase
- g :
-
Gravity acceleration
- G :
-
Gibbs free energy
- H :
-
Enthalpy
- j H :
-
Microscopic conductive heat flux (=ρu(V H−V))
- j γ :
-
Microscopic diffusive mass flux of a γ-component relative to the mass weighted velocity (=ργ(V γ−V))
- j E :
-
Microscopic diffusive flux ofE
- j γ :
-
Microscopic diffusive mass flux of a γ-component
- J H :
-
Macroscopic conductive heat flux
- J γ :
-
Macroscopic diffusive flux of a γ-component
- J* E :
-
Dispersive flux ofE
- k α :
-
Effective permeability of anα-phase
- m :
-
Mass
- m γ :
-
Mass of a γ-component
- m γα :
-
Mass of a γ-component in anα-phase
- n γα :
-
Mole fraction of a γ-component in anα-phase
- NC:
-
Number of components
- NE:
-
Number of constraints
- NF:
-
Number of degrees of freedom
- NP:
-
Number of phases
- NV:
-
Number of variables
- p :
-
Pressure
- pα :
-
Pressure in anα-phase
- R αβ :
-
Radius of curvature of anα-β-interface
- S :
-
Entropy
- S*:
-
Entropy function of an element
- S α :
-
Saturation of anα-phase
- S αβ :
-
Area of surface of contact ofα-phase with all other phases (denoted byβ) withinU o
- t :
-
Time
- T :
-
Temperature
- u α :
-
Specific internal energy of anα-phase
- u α :
-
Velocity of a surface (e.g., ofS αβ)
- U :
-
Internal energy
- U :
-
Volume
- U o :
-
Volume of domain of REV
- U oα :
-
Volume ofα-phase in REV
- v :
-
Specific volume of mass (=1/ρ)
- V :
-
Mass weighted velocity of a fluid phase
- w :
-
Displacement
- x :
-
Horizontal coordinate
- x :
-
Position vector
- x′ :
-
Position vector of point at the microscopic level
- x o :
-
Position vector of the centroid of an REV
- z :
-
Vertical coordinate (positive upward)
- α :
-
Symbol for anα-phase
- β :
-
Symbol for aβ-phase
- ∈ γα :
-
Symbol denotingE γα per unit mass ofα-phase
- γ :
-
Symbol denoting a γ-component
- γαβ :
-
Surface tension betweenα- andβ-phases
- \(\Gamma ^{E_\alpha ^\gamma } \) :
-
Rate of production ofE γα per unit mass of anα-phase
- δ :
-
Unit tensor
- η :
-
Coefficient of thermoelasticity
- θ α :
-
Volumetric fraction of anα-phase (≡U α/U o)
- λα :
-
Thermal conductivity of anα-phase
- λ *α :
-
Coefficient of thermal conductivity of anα-phase in a porous medium
- Λε :
-
Lagrange multiplier for energy
- Λ mγ :
-
Lagrange multiplier for mass of γ-component
- Λ U :
-
Lagrange multiplier for volume
- μα :
-
Dynamic viscosity of anα-phase
- μ ′α :
-
Bulk viscosity of anα-phase
- μ γα :
-
Chemical potential of a γ-component of anα-phase
- ρb :
-
Bulk mass density of soil
- ρα :
-
Mass density of anα-phase
- α γα :
-
Mass density of a γ-component in anα-phase
- σ :
-
Stress tensor
- σ′ s :
-
Effective stress\( \equiv \overline {\sigma '_s } \)
- τ :
-
Shear stress. Deviatoric stress
- φ :
-
Porosity
- ϕ :
-
Potential energy function
- Φ :
-
Thermodynamic potential
- ω γα :
-
Mass fraction of a γ-component in anα-phase
- a :
-
Air
- g :
-
Gas
- f :
-
Fluid
- ℓ :
-
Liquid
- s :
-
Solid
- w :
-
Water
- α :
-
α-phase
- β :
-
β-phase
- λ :
-
λ-phase
- H :
-
Heat
- γ :
-
γ-component
- \(\overline {(..)} ^\alpha \) :
-
Intrinsic phase average of\(\overline {(..)} ( = \tfrac{1}{{\mathcal{U}_{o\alpha } }})\smallint _{\mathcal{U}_o } (..){\text{ d}}\mathcal{U}).\)
- \(\mathop G\limits^ \circ \) :
-
Deviation ofG from its intrinsic phase average,\(\overline G ^\alpha \), over an REV
- \(\tfrac{{{\text{D}}_E (..)}}{{{\text{D}}t}}\) :
-
Material derivative of (..), as observed by theE-continuum
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Bear, J., Nitao, J.J. On equilibrium and primary variables in transport in porous media. Transp Porous Med 18, 151–184 (1995). https://doi.org/10.1007/BF01064676
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DOI: https://doi.org/10.1007/BF01064676