Theory and Applications of Macroscale Models in Porous Media

Abstract

Systems dominated by heterogeneity over a multiplicity of scales, like porous media, still challenge our modeling efforts. The presence of disparate length- and time-scales that control dynamical processes in porous media hinders not only models predictive capabilities, but also their computational efficiency. Macrosopic models, i.e., averaged representations of pore-scale processes, are computationally efficient alternatives to microscale models in the study of transport phenomena in porous media at the system, field or device scale (i.e., at a scale much larger than a characteristic pore size). We present an overview of common upscaling methods used to formally derive macroscale equations from pore-scale (mass, momentum and energy) conservation laws. This review includes the volume averaging method, mixture theory, thermodynamically constrained averaging, homogenization, and renormalization group techniques. We apply these methods to a number of specific problems ranging from food processing to human bronchial system, and from diffusion to multiphase flow, to demonstrate the methods generality and flexibility in handling different applications. The primary intent of such an overview is not to provide a thorough review of all currently available upscaling techniques, nor a complete mathematical treatment of the ones presented, but rather a primer on some of the tools available for upscaling, the basic principles they are based upon, and their specific advantages and drawbacks, so to guide the reader in the choice of the most appropriate method for particular applications and of the most relevant technical literature.

Appendix: Notation

1.1 Hybrid Mixture Theory

1.2 Latin Symbols

$A^{\alpha _{j}}$ :

Helmholtz free energy of the jth component in the $\alpha$ phase [J/kg]

$A^{\alpha }$ :

Total Helmholtz free energy of the $\alpha$ phase computed in volume V (=$A_{I}^{\alpha } M^{\alpha }$) [J/kg]

$A_{I}^{\alpha }$ :

Inner part of the Helmholtz free energy of the $\alpha$ phase [J/kg]

$B_c$ :

Viscoelastic parameter [m$^ 3$ s/kg]

$b^{\alpha _{j}}$ :

External entropy source for the jth component in the $\alpha$ phase [J/(kg s K)]

$b^{\alpha }$ :

External entropy source for the $\alpha$ phase [J/(kg s K)]

$C^{\alpha _{j}}$ :

Mass concentration of the jth component in the $\alpha$ phase $(=\rho ^{\alpha _{j}}/\rho ^{\alpha })$ [dimensionless]

D :

Coefficient of diffusivity [m$^2$/s]

$d_{kl}^{\alpha }$ :

Rate of deformation tensor of the $\alpha$ phase [s$^{-1}$]

${}^{\beta } {\hat{e}}^{\alpha _{j}}$ :

Net mass transfer from the phase $\beta$ to the jth component in the $\alpha$ phase [kg/(m$^3$ s)]

${}^{\beta } {\hat{e}}^{\alpha }$ :

Net mass transfer from the phase $\beta$ to the phase $\alpha$ [kg/(m$^3$ s)]

$E^{\alpha _{j}}$ :

Internal energy of the jth component in the $\alpha$ phase [J/kg]

$E^{\alpha }$ :

Internal energy of the $\alpha$ phase [J/kg]

${\hat{E}}^{\alpha _{j}}$ :

energy gained by the jth component from other components in the same phase [J/(m$^3$s)]

$E_{MM}^{s}$ :

Trace of Lagrangian strain tensor of the solid phase [dimensionless]

$g_{l}^{\alpha _{j}}$ :

Gravitational force on the jth component in the $\alpha$ phase [m$^2$/s]

$g_{l}^{\alpha }$ :

Gravitational force on the $\alpha$ phase [m$^2$/s]

$G^{w}$ :

Gibbs free energy [J/kg]

G(t):

Stress relaxation function for the viscoelastic biopolymers [Pa]

$G_i$ :

With i from 0 to N, relaxation parameters for the Generalized Maxwell model [Pa]

$h^{\alpha _{j}}$ :

Net external energy source for the jth component in the $\alpha$ phase [J/(kg s)]

$h^{\alpha }$ :

Net external energy source for the $\alpha$ phase [J/(kg s)]

${\hat{i}}_{l}^{\alpha _{j}}$ :

Momentum transfer to the jth component due to interaction with other components in the same phase [N/m$^3$]

$p^{\alpha }$ :

Thermodynamic pressure in the $\alpha$ phase [Pa]

p :

Total thermodynamic pressure for the particle (solid+liquid) [Pa]

$\mathrm{P}^{\alpha }$ :

Classical pressure of the $\alpha$ phase [Pa]

$q_{l}^{\alpha _{j}}$ :

Heat flux vector for the jth component in the $\alpha$ phase [J/(m$^2$ s)]

$q_{l}^{\alpha }$ :

Heat flux vector for the $\alpha$ phase [J/(m$^2$ s)]

$q_{l}$ :

Total heat flux vector for the particle (solid+liquid) [J/(m$^2$ s)]

${}^{\beta } {\hat{Q}}^{\alpha _{j}}$ :

Net heat gained by the jth component in the phase $\alpha$ from the phase $\beta$ [J/(m$^3$ s)]

${}^{\beta } {\hat{Q}}^{\alpha }$ :

Net heat gained by the phase $\alpha$ from the phase $\beta$ [J/(m$^3$ s)]

${\hat{r}}^{\alpha _{j}}$ :

Mass production of the jth component in phase $\alpha$ due to chemical reactions within the phase [kg/(m$^3$ s)]

$S^\alpha$ :

Degree of saturation for the $\alpha$ phase [dimensionless]

$\hbox {REV}$ :

Representative elementary volume

t :

Time [s]

$t_{kl}^{\alpha _{j}}$ :

Stress tensor of the jth component in the phase $\alpha$ [Pa]

$t_{kl}^{\alpha }$ :

Stress tensor of the phase $\alpha$ [Pa]

$t_{kl}$ :

Total stress tensor of the particle (solid+liquid) [Pa]

$T^{\alpha _{j}}$ :

Temperature of the jth component in the phase $\alpha$ [K]

T :

Temperature [K]

${}^{\beta } {\hat{T}}_{l} ^{\alpha _{j}}$ :

Momentum transfer to the jth component in the phase $\alpha$ due to mechanical interactions with the phase $\beta$ [N/$m^3$]

${}^{\beta }{\hat{T}}_{l}^{\alpha }$ :

Momentum transfer to the phase $\alpha$ due to mechanical interactions with the phase $\beta$ [N/m$^3$]

$u_{k}^{\alpha _{j}}$ :

Diffusion velocity of the jth component in the phase $\alpha$ [m/s]

$v_{l}^{\alpha _{j}}$ :

Velocity of the jth component in the phase $\alpha$ [m/s]

$v_{l}^{\alpha }$ :

Velocity of the $\alpha$ phase [m/s]

$x_{k}^{\alpha }$ :

Eulerian coordinate in the $\alpha$ phase [m]

$X_{K}^{\alpha }$ :

Lagrangian coordinate in the phase $\alpha$ [m]

1.3 Greek Symbols

$\delta _{kl}$ :

Kronecker delta function in Eulerian coordinates

$\delta _{KL}$ :

Kronecker delta function in Lagrangian coordinates

$\varepsilon ^{\alpha }$ :

Volume fraction of the phase $\alpha$ [dimensionless]

$\eta ^{\alpha _{j}}$ :

Entropy of the jth component in the $\alpha$ phase [J/(kg K)]

$\eta ^{\alpha }$ :

Entropy of the $\alpha$ phase [J/(kg K)]

${\hat{\eta }}^{\alpha _{j}}$ :

Entropy gained by the jth component in a phase by interaction with other components in the same phase [J/(m$^3$s K)]

$K^\alpha$ :

Permeability of the porous matrix for phase $\alpha$ [m$^2$]

$\varLambda ^{\alpha _{j}}$ :

Net entropy production for the jth component in the $\alpha$ phase [J/(m$^3$ s K)]

$\varLambda ^{\alpha }$ :

Net entropy production in the $\alpha$ phase [J/(m$^3$ s K)]

$\varLambda$ :

Net entropy production in the system at mesoscale [J/(m$^3$ s K)]

$\lambda _i$ :

Relaxation time for ith element of generalized Maxwell model [s]

${\mathcal {M}}$ :

Memory function in the generalized Darcy’s law for biopolymers [m$^5$/(kg s)]

$\rho ^{\alpha _{j}}$ :

Density of the jth component in the $\alpha$ phase [kg/m$^3$]

$\rho ^{\alpha }$ :

Density of the $\alpha$ phase [kg/m$^3$]

$\phi _{l}^{\alpha _{j}}$ :

Entropy flux vector for the jth component in the $\alpha$ phase [J/(m$^2$ s K)]

$\phi _{l}^{\alpha }$ :

Entropy flux vector for the $\alpha$ phase [J/(m$^2$ s K)]

${}^{\beta } {\hat{\phi }}^{\alpha _{j}}$ :

Entropy transfer to the jth component in the $\alpha$ phase from the $\beta$ phase [J/(m$^3$ s K)]

${}^{\beta } {\hat{\phi }}^{\alpha }$ :

Entropy transfer to the $\alpha$ phase from the $\beta$ phase [J/(m$^3$ s K)]

1.4 Subscripts

$k,\; l$ :

Coordinate indices

1.5 Superscripts

$\mathrm{s}$ :

Solid phase

$\mathrm{w}$ :

Water (or liquid) phase

$\mathrm{f}$ :

Fluid phase

$\alpha ,\; \beta$ :

General representation of phases

j :

A given component of species

1.6 Special Symbols

$\dfrac{D^{\alpha _{j}}}{Dt}$ :

Material derivative of a function with respect to velocity of jth component in the $\alpha$ phase [s$^{-1}$]

$\dfrac{D^{\alpha }}{Dt}$ :

Material derivative of a function with respect to velocity of the $\alpha$ phase [s$^{-1}$]

$v_{l}^{\alpha ,s}$ :

Velocity of the $\alpha$ phase relative to the solid phase (=$v_{l}^{\alpha }-v_{l}^{s}$) [m/s]

1.7 Thermodynamically Constrained Averaging Theory

1.8 Roman Letters

${{\hat{c}}}^{wn}$ :

Capillary pressure relaxation rate coefficient

$\varvec{\mathsf {d}}$ :

Rate of strain tensor

F :

A generalized additive function

$\varvec{\mathsf {G}}$ :

Geometric orientation tensor

${\varvec{\mathrm g}}$ :

Gravitational acceleration vector

${\varvec{\mathsf {I}}}$ :

Identity tensor

${{\mathscr {I}}}_{\mathrm {c}{{\alpha }}}^+$ :

Index set of entities connected to the ${{\alpha }}$ entity that are of a higher dimension than the ${{\alpha }}$ entity

${{{\mathscr {I}}_{\mathrm{f}}}}$ :

Index set of fluid phases

$J_w^{wn}$ :

Mean curvature of the wn interface

$K_{E}$ :

Kinetic energy term due to velocity fluctuations

${{\hat{k}}}_1^{wn}$ :

Interfacial area relaxation coefficient

${{\varvec{\mathrm n}}_{{{\alpha }}}}$ :

Outward unit normal vector from entity ${{\alpha }}$

p :

Fluid pressure

${{\hat{R}}}$ :

Resistance coefficient

${\overset{{{{\alpha }}}\rightarrow {{{\kappa }}}}{{\varvec{\mathrm T}}}}$ :

Transfer rate of momentum from the ${{\alpha }}$ to the ${{\kappa }}$ entity

$\varvec{\mathsf {t}}$ :

Stress tensor

$\varvec{\mathsf {t}}_s^*$ :

Solid-phase stress tensor corresponding to the action of a common curve on the solid phase

t :

time

${\mathscr {U}}$ :

Set of unknowns

${\varvec{\mathrm v}}$ :

Velocity

${\varvec{\mathrm w}}^{wn}$ :

Velocity of the fluid-fluid interface

1.9 Greek Letters

$\gamma$ :

Interfacial or curvilinear tension

${\epsilon }$ :

Porosity

${{{\epsilon }^{{\overline{\overline{{{\alpha }}}}}}}}$ :

Specific entity measure of the ${{\alpha }}$ entity (volume fraction, specific interfacial area)

${\theta }$ :

Temperature

${{{\kappa }}^{\,{\overline{\overline{{wns}}}}}_{G}}$ :

Geodesic curvature of the common curve

${{{\kappa }}^{\,{\overline{\overline{{wns}}}}}_{N}}$ :

Normal curvature of the common curve

${\varLambda }$ :

Entropy density production rate

$\mu$ :

Chemical potential

$\rho$ :

Mass density

${\chi }^{{\overline{\overline{n}}}}$ :

Euler characteristic of the non-wetting phase

$\chi _s$ :

Wetted fraction of the solid phase

${\varphi }^{{\overline{\overline{{ws},{wn}}}}}$ :

Contact angle between the ${ws}$ and ${wn}$ interfaces

$\psi$ :

body force potential per unit mass (e.g., gravitational potential)

${{\varOmega }_{}}$ :

Spatial domain

1.10 Subscripts and Superscripts

$\mathrm {eq}$ :

An equilibrium state

n :

Index that indicates a non-wetting phase

ns :

Index that indicates the non-wetting–solid phase interface

s :

Index that indicates a solid phase

w :

Entity index corresponding to the wetting phase

wn :

Index that indicates the wetting–non-wetting phase interface

wns :

Index that denotes the common curve

ws :

Entity index corresponding to the wetting–solid interface

${{\alpha }}$ :

General entity index qualifier

${{\kappa }}$ :

General entity index qualifier

1.11 Other Mathematical Symbols

$\overline{\ }$ :

Above a superscript refers to a density weighted macroscale average

$\overline{\overline{\ }}$ :

Above a superscript refers to a uniquely defined macroscale average

$\mathrm {D}^{}_{{}}/\mathrm {D}t$ :

Material derivative

1.12 Abbreviations

$\hbox {EI}$ :

Entropy inequality

$\hbox {REV}$ :

Representative elementary volume

$\hbox {TCAT}$ :

Thermodynamically constrained averaging theory

1.13 Homogenization Theory

${\mathcal {B}}$ :

Pore space domain in the unit cell Y

$|{\mathcal {B}}|$ :

Volume of ${\mathcal {B}}$ [L$^3$]

${\mathcal {B}}^\varepsilon$ :

Pore space domain in the porous medium $\varOmega$

$c_\varepsilon$ :

Dimensionless pore-scale aqueous concentration of species C [–]

${{\hat{c}}}_\varepsilon$ :

Pore-scale aqueous concentration of species C [mol L$^{-3}$]

$\langle c_\varepsilon \rangle$ :

Average of $c_\varepsilon$ over the unit cell Y

$\langle c_\varepsilon \rangle _{{\mathcal {B}}}$ :

Average of $c_\varepsilon$ over the pore volume ${\mathcal {B}}$

$\langle c_\varepsilon \rangle _\varGamma$ :

Average of the quantity $c_\varepsilon$ over the solid-liquid interface $\varGamma$

${{\overline{c}}}$ :

$:=\root n \of {k_d/k}$, threshold aqueous concentration of species C [mol L$^{-3}$]

${{\overline{c}}}^n$ :

$:=$ solubility product of species C

${\mathbf {D}}$ :

dimensionless molecular diffusion coefficient defined by (119) [–]

${\mathcal {D}}$ :

Molecular diffusion coefficient [L$^2~$T$^{-1}$]

${\mathcal {D}}_0$ :

Characteristic value of ${\mathcal {D}}$, [L$^2$ T$^{-1}$]

${\mathbf {D}}^*$ :

Dimensionless dispersion tensor defined by (137)

Da :

$:={{\hat{t}}}_{\mathrm d}/{{\hat{t}}}_{\mathrm {r}}$, Damköhler number defined by (121)

${\mathcal {G}}$ :

Solid matrix domain in the unit cell Y

k :

Reaction rate of the forward heterogeneous reaction $C \rightarrow S$

$k_{c}$ :

reaction rate of the backward homogeneous reaction $A+B \leftarrow C$.

$k_{d}$ :

Reaction rate of the backward heterogeneous reaction $C \leftarrow S$

${\mathbf {k}}$ :

Closure variable defined by (??)

${\mathbf {K}}$ :

$:=\langle {\mathbf {k}}({\mathbf {y}})\rangle$, permeability tensor

${\mathcal {K}}^\star$ :

Effective reaction rate constant defined by (??)

$\ell$ :

Characteristic length of the periodic unit cell Y

L :

Characteristic length of macroscopic porous medium domain $\varOmega$

n :

Heterogeneous reaction order, [-]

${{\hat{p}}}$ :

Fluid dynamic pressure, [ML$^{-1}$T$^{-2}$]

Pe :

Péclet number defined by (121)

t :

$:={{\hat{t}}}/{{\hat{t}}}_{\mathrm d}$, dimensionless time [–]

${{\hat{t}}}_{\mathrm a}$ :

Advection timescale [T]

${{\hat{t}}}_{\mathrm d}$ :

Diffusion timescale [T]

${{\hat{t}}}_{\mathrm {rj}}$ :

$j=\{1,2,3\}$, reaction timescales [T]

U :

Characteristic velocity associated with $\hat{{\mathbf {v}}}_\varepsilon$ [TL$^{-1}$]

${\mathbf {v}}_\varepsilon$ :

Dimensionless pore-scale fluid velocity [–]

$\hat{{\mathbf {v}}}_\varepsilon$ :

Pore-scale fluid velocity [T L$^{-1}$]

${\mathbf {x}}$ :

Spatial coordinate of the pore space ${\mathcal {B}}_\varepsilon$

${\mathbf {y}}$ :

Spatial coordinate of the unit cell Y

Y :

Spatially periodic unit cell

|Y|:

Volume of the spatially periodic unit cell [L$^3$]

$\alpha$ :

Coefficient for the definition of $Pe = \varepsilon ^{-\alpha }$

$\beta$ :

Coefficient for the definition of $Da = \varepsilon ^\beta$

$\varepsilon$ :

$:= \ell / L$, scale separation coefficient [–]

$\hat{\phi }$ :

Unit cell porosity [–]

$\varGamma$ :

Solid–liquid interface in the unit cell Y

$\varGamma ^\varepsilon$ :

Solid–liquid interface in the porous medium $\varOmega$

$\mu$ :

Fluid dynamic viscosity [M L$^{-1}$ T$^{-1}$]

$\varOmega$ :

Porous medium domain

$\psi$ :

General dimensionless pore-scale function

$\psi _m$ :

Y-periodic terms of order m for the multiple-scale expansion of $\psi$

$\psi _\varepsilon$ :

$\psi$ defined in ${\mathcal {B}}_{\varepsilon }$

${\hat{\psi }}_\varepsilon$ :

Dimensional value of $\psi$ in ${\mathcal {B}}_{\varepsilon }$