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Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model

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Abstract

The purpose of this article is to derive a macroscopic model for a certain class of inertial two-phase, incompressible, Newtonian fluid flow through homogenous porous media. Starting from the continuity and Navier–Stokes equations in each phase β and γ, the method of volume averaging is employed subjected to constraints that are explicitly provided to obtain the macroscopic mass and momentum balance equations. These constraints are on the length- and time-scales, as well as, on some quantities involving capillary, Weber and Reynolds numbers that define the class of two-phase flow under consideration. The resulting macroscopic momentum equation relates the phase-averaged pressure gradient \(\nabla \langle p_{\alpha}\rangle^{\alpha}\) to the filtration or Darcy velocity \(\langle {\mathbf{v}}_{\alpha}\rangle\) in a coupled nonlinear form explicitly given by

$$\langle {\mathbf{v}}_{\alpha}\rangle =-\frac{{\mathbf{K}}_{\alpha \alpha}^{{\mathbf{\ast}}}}{\mu _{\alpha}} \,\cdot\, (\nabla \langle p_{\alpha}\rangle^{\alpha}-\rho _{\alpha}{\mathbf{g}})-{\mathbf{F}}_{\alpha \alpha} \,\cdot\, \langle {\mathbf{v}}_{\alpha} \rangle -\frac{{\mathbf{K}}_{\alpha \kappa}^{{\mathbf{\ast}}}}{\mu_{\kappa}} \,\cdot\, (\nabla \langle p_{\kappa}\rangle^{\kappa}-\rho_{\kappa}{\mathbf{g}})-{\mathbf{F}}_{\alpha \kappa}\,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle \qquad\alpha ,\kappa =\beta, \gamma \quad\alpha \neq \kappa$$

or equivalently

$$\langle {\mathbf{v}}_{\alpha}\rangle =-\frac{{\mathbf{K}}_{\alpha}}{\mu _{\alpha}}\,\cdot\, (\nabla \langle p_{\alpha} \rangle^{\alpha}-\rho _{\alpha}{\mathbf{g}})-{\mathbf{F}}_{\alpha \alpha} \,\cdot\, \langle {\mathbf{v}}_{\alpha}\rangle +\,{\mathbf{K}}_{\alpha \kappa}\,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle - {\mathbf{F}}_{\alpha \kappa} \,\cdot\, \langle {\mathbf{v}}_{\kappa}\rangle\qquad \alpha ,\kappa =\beta ,\gamma \quad\alpha \neq \kappa$$

In these equations, \({\mathbf{F}}_{\alpha \alpha}\) and \({\mathbf{F}}_{\alpha \kappa}\) are the inertial and coupling inertial correction tensors that are functions of flow-rates. The dominant and coupling permeability tensors \({\mathbf{K}}_{\alpha \alpha}^{{\mathbf{\ast}}}\) and \({\mathbf{K}}_{\alpha \kappa}^{{\mathbf{\ast}}}\) and the permeability and viscous drag tensors \({\mathbf{K}}_{\alpha}\) and \({\mathbf{K}}_{\alpha \kappa}\) are intrinsic and are those defined in the conventional manner as in (Whitaker, Chem Eng Sci 49:765–780, 1994) and (Lasseux et al., Transport Porous Media 24(1):107–137, 1996). All these tensors can be determined from closure problems that are to be solved using a spatially periodic model of a porous medium. The practical procedure to compute these tensors is provided.

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Abbreviations

\({\mathbf{a}}_{\alpha \kappa}\) :

A vector that maps \(\mu _{\alpha}\langle {\mathbf{v}}_{\kappa}\rangle^{\kappa}\) onto \(\tilde{p}_{\alpha}\) , m−1

\({\mathbf{a}}_{\alpha \kappa}\) :

A tensor that maps \(\langle {\mathbf{v}} _{\kappa}\rangle^{\kappa}\) onto \(\tilde{{\mathbf{v}}}_{\alpha}\)

A αe :

Area of α-phase entrances and exits associated with the macroscopic region, m2

A ακ :

Area of α–κ interface contained within the averaging volume (=A κα ), m2

Ca α :

Capillary number associated to the α-phase \((= \frac{\mu _{\alpha}\left\Vert \langle {\mathbf{v}}_{\alpha}\rangle^{\alpha}\right\Vert}{\sigma})\)

\({\mathbf{F}}_{\alpha}\) :

Inertial correction tensor for the α-phase

\({\mathbf{F}}_{\alpha \kappa}\) :

Coupling inertial correction tensor that maps \(\langle {\mathbf{v}}_{\kappa}\rangle\) onto \(\langle {\mathbf{v}} _{\alpha} \rangle\)

g :

Gravitational acceleration, m s−2

H :

Mean curvature, m−1

\(\langle H\rangle _{\beta \gamma}\) :

Area average over A βγ of the mean curvature, m−1

I :

Unit tensor

\({\mathbf{K}}_{\alpha}\) :

Permeability tensor for the α-phase, m2

\({\mathbf{K}}_{\alpha \kappa}\) :

Viscous drag tensor that maps \(\langle{\mathbf{v}}_{\kappa}\rangle\) onto \(\langle {\mathbf{v}}_{\alpha}\rangle\)

\({\mathbf{K}}_{\alpha \alpha}^{\ast}\) :

Dominant permeability tensor that maps \((\nabla\langle p_{\alpha}\rangle^{\alpha}-\rho _{\alpha}{\mathbf{g}})/\mu_{\alpha})\) onto \(\langle {\mathbf{v}}_{\alpha}\rangle\) , m2

\({\mathbf{K}}_{\alpha \kappa}^{\ast}\) :

Dominant permeability tensor that maps \((\nabla \langle p_{\kappa}\rangle^{\kappa}-\rho _{\kappa}{\mathbf{g}} )/\mu _{\kappa})\) onto \(\langle {\mathbf{v}}_{\alpha}\rangle\) , m2

l α :

Characteristic length for the α-phase, m

\({\mathbf{l}}_{i}\) :

i = 1, 2, 3, lattice vectors, m

l p :

Small length scale representation of the mean pore diameter, m

L :

Characteristic length associated with volume averaged quantities, m

\({\mathbf{n}}_{\alpha \kappa}\) :

Unit normal vector pointing from the α-phase towards the κ-phase \((=-{\mathbf{n}}_{\kappa \alpha})\)

\({\mathbf{n}}_{\beta}\) :

Unit normal vector representing both \({\mathbf{n}}_{\beta \gamma}\) and \({\mathbf{n}}_{\beta \sigma}\)

\({\mathbf{n}}_{\gamma}\) :

Unit normal vector representing both \({\mathbf{ n}}_{\gamma \beta}\) and \({\mathbf{n}}_{\gamma \sigma}\)

p α :

Pressure in the α−phase, Pa

\(\langle p_{\alpha}\rangle\) :

Superficial average pressure in the α-phase, Pa

\(\langle p_{\alpha}\rangle^{\alpha}\) :

Intrinsic average pressure in the α-phase, Pa

\(\widetilde{p}_{\alpha}\) :

Pressure deviation in the α-phase \((=p_{\alpha}-\langle p_{\alpha}\rangle^{\alpha})\) , Pa

\(P_{\alpha}^0\) :

Reference pressure in the α-phase, Pa

r 0 :

Radius of the averaging volume V, m

r :

Position vector, m

Re α :

Reynolds number associated to the α-phase, \(( =\frac{\rho _{\alpha}\left\Vert \langle {\mathbf{v}}_{\alpha} \rangle \right\Vert l_p}{\mu _{\alpha}})\)

t :

Time, s

t * :

Characteristic process time, s

\({\mathbf{v}}_{\alpha}\) :

Velocity in the α-phase, m s−1

\(\langle {\mathbf{v}}_{\alpha}\rangle\) :

Superficial average velocity in the α-phase, m s−1

\(\langle {\mathbf{v}}_{\alpha}\rangle^{\alpha}\) :

Intrinsic average velocity in the α-phase, m s−1

\(\widetilde{{\mathbf{v}}}_{\alpha}\) :

Velocity deviation in the α-phase \((={\mathbf{v}}_{\alpha}-\langle {\mathbf{v}}_{\alpha}\rangle^{\alpha})\) , m s−1

V α :

Volume of the α-phase, contained within the averaging volume, m3

V :

Averaging volume, m3

We α :

Weber number associated to the α-phase \((=\frac{ \rho _{\alpha}\left\Vert \langle {\mathbf{v}}_{\alpha}\rangle^{\alpha}\right\Vert^{2}l_p}{\sigma})\)

\({\mathbf{y}}_{\alpha}\) :

Position of a point in the α-phase relative to the centroid of V, m

δ ακ :

Kronecker delta function (δ ακ  = 1 if α = κ, 0 otherwise)

ε α :

Volume fraction of the α-phase, (= V α /V)

μ α :

Viscosity of the α-phase, Pa s

ρ α :

Density of the α-phase, kg/m3

σ :

Interfacial tension between the β- and γ-phase N/m

\({\mathbf{T}}_{v\alpha}\) :

Viscous stress tensor in the α-phase, Pa

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Lasseux, D., Ahmadi, A. & Arani, A.A.A. Two-Phase Inertial Flow in Homogeneous Porous Media: A Theoretical Derivation of a Macroscopic Model. Transp Porous Med 75, 371–400 (2008). https://doi.org/10.1007/s11242-008-9231-y

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