One-Phase and Two-Phase Flow in Highly Permeable Porous Media

ABSTRACT

Many industrial and natural processes involve flow in highly permeable media, such as exchangers, canopies, urban canyons. Traditional assumptions used for modeling flow equations in low permeability structures may not hold for these systems with very large pores. Reynolds numbers may be too large so that Darcy’s law is no longer valid. Large Capillary and Bond numbers may also invalidate the quasistatic assumptions implicit in many empirical formulations and upscaling results. In this paper, we review several approaches developed to handle such cases, basing our analysis on new experimental data and results from upscaling methods. For one-phase flow this has led to various formulations of macro-scale momentum transport including generalized Forchheimer equations and macro-scale turbulent models. For two-phase flows, we discuss possible ways toward deriving macro-scale models from the pore-scale equations and introduce several macro-scale models: generalized Darcy’s laws, models with cross terms accounting for the viscous interaction between the flowing phases, formulations capturing inertial, or dynamic effects. Models suitable for describing flow in structured media like chemical exchangers containing structured packings are also introduced. Finally, we present hybrid representations that couple approaches at two different scales, for instance, a meso-scale network approach coupled with dynamic rules obtained from pore-scale numerical simulations or experiments. This approach proved useful in describing the diffusion of impinging jets in packed beds, which is not described properly by capillary diffusion.

Nomenclature

aβγ	=	specific interfacial area, m− 1
Aβσ	=	interface between the fluid and solid phases
B	=	mapping variable
b	=	mapping variable, m− 1
Bo	=	Bond number
Ca	=	capillary number
CFD	=	Computational fluid dynamics
d	=	particle diameter, m
DNS	=	Direct numerical solution
FSακ	=	inertia multiphase effective properties
FSβγ	=	interfacial interaction, Pa · m− 1
F	=	inertia term, Pa · s · m− 2
g	=	gravity acceleration, m · s− 2
hη	=	passability correction factor
hk	=	permeability correction factor
Hβγ	=	interface curvature, m− 1
K	=	intrinsic permeability tensor, m2
Kακ	=	cross term
K*ακ	=	modified multiphase permeability tensor, m2
krα	=	α-phase relative permeability
L	=	large-scale characteristic length, m
L1	=	phenomenological parameter, Pa · s
lσ	=	pore-scale characteristic length, m
lp	=	pore-scale characteristic length, m
$\dot{m}$	=	mass exchange term, kg · m− 3 · s− 1
M	=	Navier tensor, m
M*β	=	new effective property, Eq. (52(52) $$\begin{array}{ccc}\hfill {\mathbf{V}}_{\beta }& =& -\frac{1}{{\mu }_{\beta }}{\mathbf{K}}_{\beta }^{*}·\left(\nabla P_{\beta }-{\rho }_{\beta }\mathbf{g}\right)-{\mathbf{u}}_{\beta }^{*}\frac{\partial \epsilon S_{\beta }}{\partial t}\hfill \\ & & -{\mathbf{U}}_{\beta }^{*}·\nabla \frac{\partial \epsilon S_{\beta }}{\partial t}-\frac{1}{{\mu }_{\beta }}{\mathbf{M}}_{\beta }^{*}:\nabla \nabla P_{\beta }\hfill \\ & & -\frac{1}{{\mu }_{\beta }}{\Phi }_{\beta }-\frac{1}{{\mu }_{\beta }}{\mathbf{R}}_{\beta }^{*}:\nabla {\Phi }_{\beta }\hfill \end{array}$$(52) ), m3 · s− 2
V	=	averaging volume
nβσ	=	normal to Aβσ
${\tilde{p}}_{\beta }$	=	pressure deviation, Pa
pc	=	capillary pressure, Pa
Pβ	=	intrinsic phase average pressure, Pa
pβ	=	β-phase pressure, N · m− 2
R*β	=	new effective property, Eq. (52(52) $$\begin{array}{ccc}\hfill {\mathbf{V}}_{\beta }& =& -\frac{1}{{\mu }_{\beta }}{\mathbf{K}}_{\beta }^{*}·\left(\nabla P_{\beta }-{\rho }_{\beta }\mathbf{g}\right)-{\mathbf{u}}_{\beta }^{*}\frac{\partial \epsilon S_{\beta }}{\partial t}\hfill \\ & & -{\mathbf{U}}_{\beta }^{*}·\nabla \frac{\partial \epsilon S_{\beta }}{\partial t}-\frac{1}{{\mu }_{\beta }}{\mathbf{M}}_{\beta }^{*}:\nabla \nabla P_{\beta }\hfill \\ & & -\frac{1}{{\mu }_{\beta }}{\Phi }_{\beta }-\frac{1}{{\mu }_{\beta }}{\mathbf{R}}_{\beta }^{*}:\nabla {\Phi }_{\beta }\hfill \end{array}$$(52) ), m
r0	=	averaging volume characteristic length, m
RANS	=	Reynolds-averaged Navier–Stokes equations
Re	=	Reynolds number
Sβ	=	β-phase saturation
t	=	time, s
Uβ	=	β-phase intrinsic velocity, m · s− 1
U*β	=	new effective property, Eq. (52(52) $$\begin{array}{ccc}\hfill {\mathbf{V}}_{\beta }& =& -\frac{1}{{\mu }_{\beta }}{\mathbf{K}}_{\beta }^{*}·\left(\nabla P_{\beta }-{\rho }_{\beta }\mathbf{g}\right)-{\mathbf{u}}_{\beta }^{*}\frac{\partial \epsilon S_{\beta }}{\partial t}\hfill \\ & & -{\mathbf{U}}_{\beta }^{*}·\nabla \frac{\partial \epsilon S_{\beta }}{\partial t}-\frac{1}{{\mu }_{\beta }}{\mathbf{M}}_{\beta }^{*}:\nabla \nabla P_{\beta }\hfill \\ & & -\frac{1}{{\mu }_{\beta }}{\Phi }_{\beta }-\frac{1}{{\mu }_{\beta }}{\mathbf{R}}_{\beta }^{*}:\nabla {\Phi }_{\beta }\hfill \end{array}$$(52) )
u*β	=	new effective property, Eq. (52(52) $$\begin{array}{ccc}\hfill {\mathbf{V}}_{\beta }& =& -\frac{1}{{\mu }_{\beta }}{\mathbf{K}}_{\beta }^{*}·\left(\nabla P_{\beta }-{\rho }_{\beta }\mathbf{g}\right)-{\mathbf{u}}_{\beta }^{*}\frac{\partial \epsilon S_{\beta }}{\partial t}\hfill \\ & & -{\mathbf{U}}_{\beta }^{*}·\nabla \frac{\partial \epsilon S_{\beta }}{\partial t}-\frac{1}{{\mu }_{\beta }}{\mathbf{M}}_{\beta }^{*}:\nabla \nabla P_{\beta }\hfill \\ & & -\frac{1}{{\mu }_{\beta }}{\Phi }_{\beta }-\frac{1}{{\mu }_{\beta }}{\mathbf{R}}_{\beta }^{*}:\nabla {\Phi }_{\beta }\hfill \end{array}$$(52) ), m
Ur	=	reference velocity, m · s− 1
Vβ	=	β-phase filtration velocity, m · s− 1
vβ	=	β-phase velocity, m · s− 1
vs	=	velocity on effective surface, m · s− 1
${\mathcal{V}}_{\beta }$	=	β-phase volume
${\tilde{\mathbf{v}}}_{\beta }$	=	velocity deviation, m · s− 1
x	=	current point, m
y	=	position relative to the centroid of the averaging volume, m

Greek Symbols
βi	=	refers to the split β-phase
η	=	passability, m
ηβ	=	relative passability
γβ	=	β-phase indicator
⟨ψβ⟩	=	β-phase average
⟨ψβ⟩β	=	β-phase intrinsic average
μβ	=	β-phase dynamic viscosity, Pa · s
$\overline{{\psi }_{\beta }}$	=	time average
ψβ	=	β-phase variable
ρβ	=	β-phase density, kg · m− 3
σ	=	interfacial tension, N · m− 1
ϵ	=	porosity
ϵβ	=	β-phase volume fraction

Subscripts
β	=	β-phase property
βγ	=	refers to βγ interface
γ	=	γ-phase property
σ	=	σ-phase property
p	=	pore-scale property

Additional information

Notes on contributors

Yohan Davit

Yohan Davit is Chargé de Recherche CNRS at the Institut de Mécanique des Fluides in Toulouse, France. Prior to joining CNRS, he was a postdoctoral researcher at the Mathematical Institute at the University of Oxford. He received his Ph.D. degree, for which he was awarded the prize Léopold Escande, in 2011 from the University of Toulouse. He studied at the Institut National Polytechnique de Grenoble and the Université Joseph Fourier, where he obtained his MS degree. His research interests include the ecology of biofilms, transport phenomena in biological and porous media, and multiscale approaches for heterogenous systems.

Michel Quintard

Michel Quintard is Directeur de Recherche CNRS at the Institut de Mécanique des Fluides in Toulouse, France. Engineer from Top School “Ecole Nationale Supérieure des Arts et Métiers,” he received his Ph.D. in fluid mechanics from the University of Bordeaux (1979). His research interests focus on transport phenomena in porous media with fundamental objectives such as the development of macro-scale models through the use of averaging techniques, applied to different scales and different displacement mechanisms (multiphase, multicomponent, phase change, chemical reaction, biodegradation, dissolution, superfluid,…), with applications in petroleum engineering, environmental hydrogeology, chemical engineering, nuclear safety, aerospace industry, etc. M. Quintard has co-authored about 190 papers in archival journals. He was awarded a Bronze Medal from CNRS in 1984, the Prize Coron-Thévenet from Académie des Sciences de Paris in 1994, and became Chevalier of the Légion d’Honneur in 2009 for his action as President of the Scientific Council of IRSN (French Nuclear Safety and Radioprotection Institute).