Abstract
The thermodynamically constrained averaging theory (TCAT) has been used to develop a simplified entropy inequality (SEI) for several major classes of macroscale porous medium models in previous works. These expressions can be used to formulate hierarchies of models of varying sophistication and fidelity. A limitation of the TCAT approach is that the determination of model parameters has not been addressed other than the guidance that an inverse problem must be solved. In this work we show how a previously derived SEI for single-fluid-phase flow and transport in a porous medium system can be reduced for the specific instance of diffusion in a dilute system to guide model closure. We further show how the parameter in this closure relation can be reliably predicted, adapting a Green’s function approach used in the method of volume averaging. Parameters are estimated for a variety of both isotropic and anisotropic media based upon a specified microscale structure. The direct parameter evaluation method is verified by comparing to direct numerical simulation over a unit cell at the microscale. This extension of TCAT constitutes a useful advancement for certain classes of problems amenable to this estimation approach.
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Abbreviations
- b :
-
Entropy source density
- \({\varvec{\mathrm b}}\) :
-
Variable vector that maps \(\nabla C^{Aw}\) to \({\tilde{C}}_{Aw}\)
- C :
-
Species concentration in an entity
- \({\hat{\varvec{\mathsf {D}}}}\) :
-
Effective diffusivity tensor
- \({{\hat{\varvec{\mathsf {D}}}}}^{ABw}\) :
-
Second-rank symmetric closure tensor for a binary system
- \({{\hat{D}}}\) :
-
Diagonal component of the macroscale effective diffusivity tensor
- \({{\hat{D}}}_{Aw}\) :
-
Molecular diffusion coefficient
- \(\varvec{\mathsf {d}}\) :
-
Rate of strain tensor
- \({\overline{E}}\) :
-
Partial mass energy
- \({\mathcal E}_{**}^{{\overline{\overline{w}}}}\) :
-
Particular material derivative form of a macroscale entity total energy conservation equation
- \(\varvec{\mathsf {G}}\) :
-
Geometric orientation tensor
- G :
-
Green’s function associated with the initial and boundary value problem for concentration deviations
- \({\mathcal G}_{**}^{{\overline{\overline{w}}}}\) :
-
Particular material derivative form of a macroscale body force potential balance equation
- \({\overset{{\alpha }\rightarrow {ws}}{{G}_{0}}}\) :
-
Macroscale transfer associated with body force potential from the \(\alpha =w,s\) to the ws interface
- \({\varvec{\mathrm g}}\) :
-
Gravitational acceleration vector
- h :
-
Energy source density
- \({\varvec{\mathsf {I}}}\) :
-
Identity tensor
- \(\mathscr {I}\) :
-
Initial condition source term
- \({{\mathscr {I}}_{s}}\) :
-
Index set of species
- i :
-
Species qualifier
- \(J_s^{ws}\) :
-
Mean curvature of the ws interface
- \(K_{E}\) :
-
Kinetic energy term due to velocity fluctuations
- \(\ell _\mathrm{mi}\) :
-
Microscale length scale
- \(\ell _\mathrm{r}^\mathrm{r}\) :
-
Resolution length scale
- \(\ell ^\mathrm{ma}\) :
-
Macroscopic length scale
- \(\ell ^\mathrm{me}\) :
-
Megascale length scale
- \({\mathcal M}_{**}^{{\overline{\overline{{i}w}}}}\) :
-
Particular material derivative form of a macroscale species mass conservation equation
- \({\overset{{{i}{{\kappa }}}\rightarrow {{{i}{{\alpha }}}}}{M}}\) :
-
Transfer of mass of species \({i}\) from the \({{\kappa }}\) entity to the \({{\alpha }}\) entity per unit volume per unit time
- \(MW_i\) :
-
Molecular weight of species i
- \(MW_w\) :
-
Molecular weight for entity w
- \({{\varvec{\mathrm n}}_{{{\alpha }}}}\) :
-
Outward unit normal vector from entity \({{\alpha }}\)
- p :
-
Fluid pressure
- \({\overset{{\alpha }\rightarrow {ws}}{{Q}_{1}}}\) :
-
Entity-based energy exchange term from entity \(\alpha =w,s\) to the ws interface
- \({\varvec{\mathrm q}}\) :
-
Non-advective energy flux vector
- \({\varvec{\mathrm q}}_{{\varvec{\mathrm g}}0}\) :
-
Non-advective energy flux vector due to the product of fluctuations
- R :
-
Ideal gas constant
- r :
-
Mass production rate density
- \({\varvec{\mathrm r}}\) :
-
Position vector
- \({\mathcal S}_{**}^{{\overline{\overline{w}}}}\) :
-
Particular material derivative form of a macroscale entropy balance
- \({\overset{{\alpha }\rightarrow {ws}}{{{\varvec{\mathrm T}}}_{0}}}\) :
-
Macroscale momentum transfer from entity \(\alpha =w,s\) to the ws interface
- \({\mathcal T}_*^{{\overline{\overline{w}}}}\) :
-
Particular material derivative form of a macroscale differential thermodynamic equation
- \({\mathcal T}_{{\mathcal G}*}^{{\overline{\overline{w}}}}\) :
-
Particular material derivative form of the body source potential equation
- \(\varvec{\mathsf {t}}\) :
-
Stress tensor
- t :
-
Time
- \(t_\mathrm{{mi}}^*\) :
-
Microscale time scale
- \(t_\mathrm{{ma}}^*\) :
-
Macroscale time scale
- \({{{\varvec{\mathrm u}}}^{\,{}}_{}}\) :
-
Species deviation velocity vector
- \({\varvec{\mathrm v}}\) :
-
Velocity
- \({\varvec{\mathrm x}}\) :
-
Position vector locating the centroid of the REV
- x :
-
Mole fraction of a species in an entity
- \(\Gamma \) :
-
Boundary of REV
- \({{\Gamma }_{w \mathrm e}}\) :
-
External boundary of the w phase on the boundary of an REV
- \({{\Gamma }_{w \mathrm i}}\) :
-
Internal boundary of the w phase within an REV
- \(\gamma ^{ws}\) :
-
Macroscale interfacial tension of the ws interface
- \({{\hat{{\gamma }}}}\) :
-
Macroscale activity coefficient
- \({{{\epsilon }^{{\overline{\overline{{{\alpha }}}}}}}}\) :
-
Specific entity measure of the \({{\alpha }}\) entity (volume fraction, specific interfacial area)
- \(\eta \) :
-
Entropy density
- \({\theta }\) :
-
Temperature
- \({\Lambda }\) :
-
Entropy production rate density
- \({\lambda }_{{\mathcal G}}^w\) :
-
Lagrange multiplier for potential energy balance equation
- \({\lambda }_{{\mathcal M}}^{{i}w}\) :
-
Lagrange multiplier for mass conservation equation
- \({\lambda }_{{\mathcal M}_{i}}\) :
-
Constant related to the sum of potentials
- \({\lambda }_{{\mathcal T}}^w\) :
-
Lagrange multiplier for thermodynamic equation
- \({\lambda }_{{\mathcal T}{\mathcal G}}^w\) :
-
Lagrange multiplier for derivative of potential energy equation
- \(\mu \) :
-
Chemical potential
- \(\mu _0\) :
-
Reference chemical potential
- \({\varvec{\mathrm \xi }}\) :
-
Microscale position vector relative to the centroid of the REV
- \(\rho \) :
-
Mass density
- \({\varvec{\mathrm \varphi }}\) :
-
Entropy density flux vector
- \(\psi \) :
-
Body force potential per unit mass (e.g., gravitational potential)
- \({{\Omega }_{}}\) :
-
Spatial domain
- \({\Omega }^0\) :
-
Megascale spatial domain
- \({{\omega }_{{}{}}}\) :
-
Mass fraction of a species in an entity
- A :
-
Species qualifier (subscript, superscript)
- B :
-
Species qualifier (subscript, superscript)
- \({\mathcal E}\) :
-
Energy equation qualifier (subscript)
- \({\mathcal G}\) :
-
Potential equation qualifier (subscript)
- \({i}\) :
-
General index denoting a species (subscript, superscript)
- l :
-
Length qualifier for regularly shaped domain (subscript)
- \({\mathcal M}\) :
-
Mass equation qualifier (subscript)
- N :
-
Number of chemical species (superscrpt)
- \({\varvec{\mathrm r}}\) :
-
Associated with the position vector \({\varvec{\mathrm r}}\) (subscript)
- s :
-
Index that indicates a solid phase (subscript, superscript)
- \({\mathcal T}\) :
-
Thermodynamic equation qualifier (subscript)
- \({\mathcal T}{\mathcal G}\) :
-
Fluid potential energy identity qualifier (subscript)
- w :
-
Entity index corresponding to the wetting phase (subscript, superscript)
- ws :
-
Entity index corresponding to the wetting-solid interface (superscript)
- \({\varvec{\mathrm x}}\) :
-
Associated with the position vector \({\varvec{\mathrm x}}\) (subscript)
- \({\varvec{\mathrm \xi }}\) :
-
Associated with the position vector \({\varvec{\mathrm \xi }}\) (subscript)
- −:
-
Above a superscript refers to a density weighted macroscale average
- \(=\) :
-
Above a superscript refers to a uniquely defined macroscale average
- \(\tilde{}\) :
-
Deviation of a microscale quantity from the macroscale average
- \(\otimes \) :
-
Outer vector product
- \({{{\left\langle f \right\rangle _{{{\Omega }_{{{\alpha }}}},{{\Omega }_{}},w}}}}\) :
-
Averaging operator, \(\int _{{\Omega }_{{{\alpha }}}}wf\, \mathrm {d}\mathfrak {r}/\int _{{{\Omega }_{}}} w\, \mathrm {d}\mathfrak {r}\)
- \(\mathrm {D}^{}_{{}}/\mathrm {D}t\) :
-
Material derivative
- CEI:
-
Constrained entropy inequality
- DNS:
-
Direct numerical simulation
- EI:
-
Entropy inequality
- MVA:
-
Method of volume averaging
- REV:
-
Representative elementary volume
- SEI:
-
Simplified entropy inequality
- TCAT:
-
Thermodynamically constrained averaging theory
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Acknowledgements
The work of CTM was supported by Army Research Office Grant W911NF-14-1-02877, and National Science Foundation Grant 1619767. The research contributions of BDW and SO were supported by the National Science Foundation, Grant EAR-1521441. The efforts of W. G. Gray to review and comment on various versions of this work are appreciated. FJVP gratefully acknowledges the support of CTM, which enabled the presentation of this work at the 2017 fall meeting of the American Geophysical Union.
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Miller, C.T., Valdés-Parada, F.J., Ostvar, S. et al. A Priori Parameter Estimation for the Thermodynamically Constrained Averaging Theory: Species Transport in a Saturated Porous Medium. Transp Porous Med 122, 611–632 (2018). https://doi.org/10.1007/s11242-018-1010-9
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DOI: https://doi.org/10.1007/s11242-018-1010-9