Journal ArticleDOI

# A novel one-domain approach for modeling flow in a fluid-porous system including inertia and slip effects

23 Feb 2021-Physics of Fluids (AIP Publishing LLC AIP Publishing)-Vol. 33, Iss: 2, pp 022106

AbstractA new one-domain approach is developed in this work yielding an operational average description of one-phase flow in the classical Beavers and Joseph configuration including a porous medium topped by a fluid channel. The model is derived by considering three distinct regions: the homogeneous part of the porous domain, the inter-region, and the free fluid region. The development is carried out including inertial flow and slip effects at the solid–fluid interfaces. Applying an averaging procedure to the pore-scale equations, a unified macroscopic momentum equation, applicable everywhere in the system and having a Darcy form, is derived. The position-dependent apparent permeability tensor in this model is predicted from the solution of two coupled closure problems in the inter-region and in the homogeneous part of the porous medium. The performance of the model is assessed through in silico validations in different flow situations showing excellent agreement between the average flow fields obtained from direct numerical simulations of the pore-scale equations in the entire system and the prediction of the one-domain approach. Furthermore, validation with experimental data is also presented for creeping flow under no-slip conditions. In addition to the fact that the model is general from the point of view of the flow situations it encompasses, it is also simple and novel, hence providing a practical and interesting alternative to models proposed so far using one- or two-domain approaches.

Topics: Stokes flow (60%), Flow (mathematics) (56%), Porous medium (54%), Inertia (51%)

### Introduction

• The model is derived by considering three distinct regions: the homogeneous part of the porous domain, the inter-region, and the free fluid region.
• The development is carried out including inertial flow and slip effects at the solid–fluid interfaces.
• 1,2 Applications of such a configuration are numerous, ranging from hydrology to chemical engineering.

### B. The g x inter-region

• This region is the transition zone where the average (or macroscopic) velocity experiences abrupt changes.
• Modeling flow in this region has been addressed using a generalized transport equation (or penalization approaches7,8) as explained below, or even by direct numerical simulations.
• Both approaches have strengths and limitations, which are briefly discussed in the following paragraphs.
• In fact, it was shown in this reference that the one-domain approach is necessary in the derivation of the jump conditions using the volume averaging method,18 justifying its archival value.
• The structure of the average models in the three regions is shown to involve a Darcylike momentum equation, which can be condensed into a single one-equation model with a position-dependent apparent permeability tensor.

### II. MICROSCALE MODEL AND DEVELOPMENT OF THE ONE-DOMAIN APPROACH

• The configuration under consideration is the one represented in Fig. 1.
• Before developing the one-domain macroscale model, the underlying boundary value flow problem must be formulated at the microscale.
• Finally, the macroscopic momentum transport equation is derived in each region (Secs. II C–II E).

### B. Average mass balance equation

• The development starts with the derivation of the macroscopic mass balance equation in each region.
• In this domain, it is reasonable to decompose the fluid pressure gradient according to36 rpx ¼ rhpxibx þr~px; (6) ~px being the pressure deviations, and regard rhpxi b x as a constant within the REV.
• These authors solved the closure problem given in Eq. (9) in a variety of flow situations.
• Note that, for the particular problem envisaged by Beavers and Joseph,1 there are no inertial nor slip effects and the tensor Hx is the intrinsic permeability tensor.
• To conclude this section, the average velocity must now be derived by applying the superficial averaging operator to Eq. (19).

### E. Analysis in the fluid-porous medium inter-region (g x region)

• This transition zone is comprised between z ¼ zx (below which the homogeneous porous medium begins) and z ¼ zg (above which the free fluid region commences) as sketched in Fig.
• The solution of the closure problem in the inter-region is the step that is computationally the more demanding.
• The numerical requirements are considerably smaller than those needed for the solution of the pore-scale problem in the entire three-region domain.

### III. RESULTS

• The objective of this section is to present some numerical results on model structures in order to validate the model derived above and illustrate its performance.
• In the following paragraphs, the numerical results are presented for specific flow conditions.
• The numerical tests were performed considering two types of porous structures.
• To this end, the closure problems given in Eqs. (9) and (25) were numerically solved in a coupled manner in a single unit cell for the x-region and in the entire g-x region, respectively.
• It was verified that taking zg ¼ r0 and zx ¼ 11‘c satisfied the above criteria for all the cases reported in the present section.

### A. Creeping flow regime

• The first test is carried out considering the simple situation in which the flow remains in the creeping regime under no-slip conditions (i.e., for n1k ¼ n2k ¼ 0).
• The value of the intrinsic permeability, obtained from Eq. (12) is Hxxx ¼ 0:019 47‘2c , which is in perfect agreement with the estimate from the analytical expression provided by Chai et al.38.
• It is now of interest to investigate the influence of the porous medium geometry and of the averaging domain size, r0, on the predictions of the average velocity in the inter-region.
• The velocity profiles resulting from the ODA and the DNS are reported in Fig. 6, where, once again, excellent agreement is found.
• This is due to the fact that the flow remains unidirectional and non-inertial in the free fluid region for these values of Re.

### C. Creeping slip-flow

• The final case study corresponds to slip flow under non-inertial conditions, i.e., Re¼ 0.
• More specifically, four values of nk are examined, with the largest one remaining Oð0:1Þ in order to avoid using the governing flow equations beyond their range of validity (i.e., out of the slip regime).
• 41,42 The use of the same slip coefficient in both the porous medium and at the top wall of the free-fluid region is justified by the fact that the characteristic length of the latter region is not much larger than the one in the porous medium in the particular configuration considered here.
• Nevertheless, since the boundary condition given in Eq. (1c) may also be conceived as an effective boundary condition over rough surfaces, it is certainly possible to encounter physical situations in which n1k ¼ n2k when the contrast of the characteristic lengths is more pronounced.
• Finally, it is important to mention that the predictions of the average velocity from the DNS and the ODA are in excellent agreement in all the physical situations considered here.

### D. Comparison with experimental data

• This configuration is similar to the one used by Beavers and Joseph with the difference that the porous medium is made of an array of inline micropillars having a square cross section of 240lm 240lm.
• The velocity fields obtained with this instrument are line averages in the y-direction of the sketch shown in Fig. 1 (or equivalently, the z-direction in the work by Terzis et al.).
• These results were subsequently averaged along the x-direction in zones near the entrance, the outlet and in the middle of the system as shown in Fig.
• The experimental results taken at the middle of the system are now compared to the predictions of the model resulting from the ODA presented above.
• Notice that due to the shape of the experimental system, i.e., the wall effects in the y-direction, the velocity profiles tend to reach a constant value in the free fluid region sufficiently far away from the upper wall and the porous medium surface.

### IV. CONCLUSIONS

• A new formulation for describing steady, Newtonian, and incompressible flow between a porous medium and a fluid was derived.
• The predictive capabilities of the model were validated through in silico experiments by comparing the average velocity profiles resulting from the new ODA with pore-scale DNS, finding excellent agreement (i.e., with relative percent error values with respect to the DNS smaller than 0.1%).
• The microstructure plays a key role in the magnitude and shape of the flow field not only in the homogeneous part of the porous medium, but also in the inter-region.
• When the slip condition is imposed only in the porous medium, the velocity in the free fluid region experiences the most drastic changes.
• From the above, it is concluded that the new one-domain approach developed here is quite practical as it represents accurately data from both direct numerical simulations and experiments at a reasonable computational cost.

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A novel one-domain approach for modeling ow in a
uid-porous system including inertia and slip eects
To cite this version:
F. Valdés-Parada, Didier Lasseux. A novel one-domain approach for modeling ow in a uid-porous
system including inertia and slip eects. Physics of Fluids, American Institute of Physics, 2021, 33
(2), pp.022106. �10.1063/5.0036812�. �hal-03150992�

A novel one-domain approach for modeling
flow in a fluid-porous system including
inertia and slip effects
Phys. Fluids 33
, 022106 (2021); doi: 10.1063/5.0036812
F. J. Vald
1
and D. Lasseux
2,a)
AFFILIATIONS
1
Departamento de Ingenier
ıa de Procesos e Hidr
onoma Metropolitana-Iztapalapa, Av. San Rafael Atlixco 186,
exico, Mexico
2
I2M, UMR 5295, CNRS, Univ. Bordeaux, 351 Cours de la Lib
eration, 33405 Talence Cedex, France
a)
Author to whom correspondence should be addressed: didier.lasseux@u-bordeaux.fr
ABSTRACT
A
new one-domain approach is developed in this work yielding an operational average description of one-phase ﬂow in the classical Beavers
and Joseph conﬁguration including a porous medium topped by a ﬂuid channel. The model is derived by considering three distinct regions:
the homogeneous part of the porous domain, the inter-region, and the free ﬂuid region. The development is carried out including inertial
ﬂow and slip effects at the solid–ﬂuid interfaces. Applying an averaging procedure to the pore-scale equations, a uniﬁed macroscopic
momentum equation, applicable everywhere in the system and having a Darcy form, is derived. The position-dependent apparent permeabil-
ity tensor in this model is predicted from the solution of two coupled closure problems in the inter-region and in the homogeneous part of
the porous medium. The performance of the model is assessed through in silico validations in different ﬂow situations showing excellent
agreement between the average ﬂow ﬁelds obtained from direct numerical simulations of the pore-scale equations in the entire system and
the prediction of the one-domain approach. Furthermore, validation with experimental data is also presented for creeping ﬂow under no-slip
conditions. In addition to the fact that the model is general from the point of view of the ﬂow situations it encompasses, it is also simple and
novel, hence providing a practical and interesting alternative to models proposed so far using one- or two-domain approaches.
I. INTRODUCTION
Modeling one-phase ﬂow in a coupled ﬂuid-porous medium sy s-
tem has been the su bject of active research since the pioneering work
of Beavers and Joseph.
1,2
Applications of such a conﬁguration are
numerous, ranging from hydrology to chemical engineering. The
main difﬁculty lies in a physically relevant way of reconciling a macro-
scopic description of the ﬂow with a Darcy equation (or its variants)
applicable in the bulk of the porous medium to the Navi er–Stokes
equation in the bulk ﬂuid. Indeed, such a description requires one to
account for the rapid variat ion of to pology (and of the ﬂow) in the sep-
aration zone betwe en the two media, as evidenced, for inst ance, in th e
experimental works of Goharzadeh et al.,
3
4
or
more recently, by Terzis et al.
5
As a generic conﬁguration envisaged by Beavers and Joseph,
1
consider the system sketched in Fig. 1 consisting of a channel partially
ﬁlled with a porous medium. Let a single incompressible and
Newtonian ﬂuid (the b -phase) ﬂow through and above the porous
medium under steady conditions. This system may be decomposed
into three distinct regions (see Fig. 1)asfollows.
A. The x-region
This region corresponds to the homogeneous part of the porous
medium. For simplicity in the analysis, the r-phase, representing the
solid phase constitutive of the porous matrix, is assumed to be rigid
and homogeneous. Consequently, in this region, the porosity, the seep-
age velocity, and the perm eabil ity are position-independent quantities.
Here, Darcy’s law, or its modiﬁcations, is applicable as a result of
upscaling the microsc ale equations by making use of an averaging
domain representative of the porous structure and physical mecha-
nisms at play. Typically, the averaging volume corresponds to one or
more unit cell(s) if the structure is assimilated to be a periodic one.

B. The g x inter-region
This region is the transition zone where the average (or macro-
scopic) velocity experiences abrupt changes. For creeping-ﬂow prob-
lems, the extent of this region has been proposed to be, at most, on the
order of 10
c
,
6
c
being the size of a geometrical unit cell representa-
tive of the porous material. Note that, even if the porous medium
structure is assumed to be homogeneous, the velocity, the porosity,
and the permeability exhibit spatial variations in this part of the sys-
tem. Modeling ﬂow in this region has been addressed using a general-
ized transport equati on (or penalization approaches
7,8
)asexplained
below, or even by direct numerical simulations.
9,10
C. The g-region
This region corresponds to the port ion of the free ﬂuid above the
porous medium where the ﬂow is one-dimensional and parallel to
the channel axis. It is limited by th e upper wall (also cons idered as the
r-phase) at z ¼ h. Here, the volume fraction of the ﬂuid phase in the
averaging domain is equal to 1, except near the walls, and ﬂow is
described by the Navier–Stokes equations.
Traditionally, ﬂow and transport in this system have been studied
using a two-domain approac h, in which the average equations in the
g-region are coupled to those in the x-region by me ans of suitable
boundary conditions. Another alternative is the one-domain approach,
in which a single average equation is used to describe transport in the
three reg ions of the system. Both approaches have strengths and limi-
tations, which are brieﬂy discussed in the following paragraphs.
The two-domain approach has rece ived far more attention than
the one-domain approach. This may be expla ined by the fact that the
model invo lves balance equations with constant transport coefﬁcients ,
thus making their solution fairly easy to achieve, which represents an
appealing strength. However, there is no consensus about which equa-
tions should be used in the porous medium and which boundary con-
ditions should be employed. Originally, Beavers and Joseph
1
proposed
to couple Darcy’s law with the Stokes equation by means of a jump
condition in the velocity, which has the st ructure of a Newt on’s
cooling law equation. Late r, Neale and Nader
11
proposed to couple the
Darcy–Brinkman and Stokes equations with conditions of continuity
of both the stress and ve locity. Later, Ochoa-Tapia and Whitaker
12
fol-
lowed a similar path, albeit they proposed to consider a discontinuity
in the viscous stress. The jump conditions proposed by both Beavers
and Joseph
1
and Ochoa-Tapia and Whita ker
12
were written in te rms
of adjustable coefﬁcients that needed to be determined experimentally.
In addition, the position of the dividing surf ace where th ese boundary
conditions are applied cannot be arbitrarily ﬁxed. These issues have
been discussed in many works. Among them is the one by Vald
es-
13
in which an iterative methodology was proposed to
determine the dividing surface position as well as the values of the
jump coefﬁcients in these models. These authors also derived a more
general two-domain model for which there are, in general, discontinu-
ities of both the velocity and stress. Recently, the Beavers and Joseph
boundary condition has been applied to ine rtial ﬂow conditions, and it
was found that the value of the adjustable coefﬁcient increases with the
Reynolds number.
14
Nevertheless, this boundary condition has been
recently found to be unsuitable for arbitrary ﬂow directions.
15
Amore
detailed review and discussion of the Beavers and Joseph boundary
condition are available from Nield
16
and Auriault.
17
In the one-domain approach, the system is regarded as a pseudo-
continuum, and a single equation is used to model the ﬂow at the mac-
roscopic scale everywhere. A heur istic or penalization approach is to
postulate that such a model is the Navier–Stokes equation with a
Darcy term involving a position-dependent permeability. A more rig-
orous approach was proposed by Ochoa-Tapia and Whitaker,
12
who
derived a generalized tra nsport equation resulting from averaging and
not upscaling the Stokes equation. This equation is more complicated
than th e penalization appro ach, and it has archival more than practical
value. In fact, it was shown in this reference that the one-domain
approach is necessary in the derivation of the jump conditions using
the volume averaging method,
18
justifying its archival value.
Furthermore, the methodolog y proposed by Vald
13
showed that it is necess ary to accoun t for the spatial variations of the
effective coefﬁcients present in the one -domain approach in order to
determine the values of the jump coefﬁcients and the dividing surface
position. This difﬁculty in predicting the spatial variations of the
permeability and porosity may explain why the one-domain
approach has been less-frequently used than the two-domain
approach. In addition, the corresponding average model is, in gen-
eral, more complicated than the Darcy–Brinkman equation as orig-
inally derived by Ochoa-Tapia and Whitaker.
12
The few
applications of this modeling alternative use ad hoc coefﬁcient s and
average expressions that are not alwa ys in agreement with the
results obtained from pore-scale simulations (s ee for example Refs.
7, 8,and19), and only an approximate closure scheme has been
proposed to compute the spatial variation s of the permeability.
6
Moreover, existing analyses have been mainly focused on ﬂow in
the creeping regime, in the absence of rarefaction effects .
FIG. 1. Schematic representation of the ﬂow in a channel partially obstructed with a
porous medium, showing the three regions of the system, the volume averaging
0
, and the characteristic lengths L for the macroscopic size of the
porous medium,
b
in the ﬂuid phase, b, and
r
in the solid phase, r, within the
porous medium. The x, g x, and g regions, respectively, correspond to
z z
x
; z
x
z z
g
, and z
g
z h. The unit normal vector at the solid-
ﬂuid interface, directed from the b-phase to the r-phase, is denoted by n.

The objective of this work is to address these issues by proposing a
closed one-domain approach in the three-region system for one-phase
ﬂow including inertial and Knudsen effects in the slip regime. The
model derived with this approach has a simple structure facilitating its
application, nevertheless different from the modiﬁed Darcy–Brinkman
equation used in the literature. The analysis is focused on the Beavers
and Joseph conﬁguration sketched in Fig. 1. The idea is to obtain
average models that are valid in the three regions of the system and
are coupled by continuity conditions using a simpliﬁed version of the
volume averaging method. More speciﬁcally, this is achieved by deriv-
ing macroscale equations in both the x-andg x regions in which
the effective permeability tensors are obtained from coupled closure
problems. The solution in the g-region is obtained analytically since
the ﬂow is one-dimensional along the channel axis (e
x
)inthisregion.
It is also coupled with that in the g x inter-region. The structure of
the average models in the three regions is shown to involve a Darcy-
like momentum equation, which can be condensed into a single
one-equation model with a position-dependent apparent permeability
tensor. The solution of the closure scheme to obtain this tensor is
computationally much less demanding than performing direct
numerical simulations. The uniﬁed model is quite practical and
appealing due to its simplicity. It is developed in a rather general ﬂow
context, in which inertial and/or slip effects at the solid/ﬂuid interfa-
ces may be present, widely extending the scope of the derived model
with respect to the existing works reported so far. At this point, it is
worth mentioning that there are some relevant works including slip
effects in dual-porosity media;
20–22
however, these works do not
include inertial effects and do not account for the spatial variations of
the apparent permeability tensor as it is done here.
With this aim in mind, the article is organized as follows. In
Sec. II, the governing equations at the microscale are pr esented, and
the derivations of the macroscale equations in each region are
reported, yielding the uniﬁed one-domain momentum equation.
Section III is dedicated to a series of illustrative examples showing the
performance of the one -domain approach. This is achieved by com-
parisons of the ﬂow ﬁelds obtained from direct numerical simulations
at the mi croscale with predictions of the ﬂow from the one -domain
approach developed here. It is carried out for creeping ﬂow in the
absence and in the presence of slip effects as well as in the presence of
inertia. In addition, validation is also presented by comp arison with
experimental data unde r creeping ﬂow and no-slip conditions.
Conclusions are drawn in Sec. IV.
II. MICROSCALE MODEL AND DEVELOPMENT OF THE
ONE-DOMAIN APPROACH
The conﬁgur ation under con sideration is the one repre sented in
Fig. 1. Before developing the one-domain macroscale model, the
underlying boundary val ue ﬂow problem must be formulated at the
microscale.ThisisreportedinSec.II A foll owed by a ra pid prese nta-
tion of the derivation of the average mass-equation applicable every-
where in the system (Sec. II B). Finally, the macroscopic momentum
transport equation is derived in each region (Secs. II CII E).
A. Microscale model
The governing equations for mass and momentum transport at
the pore-scale (in the three regions) are formulated in a rather general
version in which inertia and/or slip effects may be present. Assuming
incompressible ﬂow, the mass and momentum balance equations in
the j-region (j ¼ x; gx; g) are given by
rv
j
¼ 0; in the b-phase; (1a)
qv
j
rv
j
¼rp
j
þ lr
2
v
j
; in the b-phase: (1b)
In these two equations, q and l denote the ﬂuid density and dynamic
viscosity, both considered as constant, whereas v
j
and p
j
represe nt the
ﬂuid velocity and pressure in th e j-region. Without loss of generality,
volume forces are not considered in the momentum transport equa-
tion. Taking into account the possible exist en ce of rare faction effects , in
particular if the ﬂowing ﬂuid is a gas, due to the size of the pores and/or
channel in conjunction with the thermodynamic conditions, the inter-
facial boundary condit ion can be formulated as follows:
23–26
v
j
¼n
k
k I nn
ðÞ
n rv
j
þrv
T
j

; at A
brj
: (1c)
Here, I and n, respectively, denote the identity tensor and the unit nor-
mal vector to the solid-ﬂuid interface, A
brj
,inthej-region, directed
from the b-phase to the r-phase. In addition, k ¼ 1forj ¼ x; gx
(i.e., at the solid–ﬂuid interface inside the porous material) and k ¼ 2
when j ¼ g (i.e., at the top surface of the channel). The mean free
path of the ﬂuid molecules is denoted by k while n
k
¼ð2 r
k
Þ=r
k
is
a coefcient taking into account the reﬂection process of the molecules
at the solid wall related to the tangential accommodation coefﬁcient,
r
k
. At this point, it is worth mentioning that the structure of the
boundary condition given in Eq. (1c) is a Navier-type slip, and it is
expressed in its complete form, i.e., including the transpose of the
velocity gradient, which is different from previous works.
27,28
Furthermore, this type of boundary condition has been pr oposed to
study other physical mechanisms such as ﬂow over rough surfaces.
29,30
In this way, th e above bound ary cond ition is quite rich as it allows
studying physical situations beyond only rarefaction effects. In addi-
tion, ma croscopic boundary cond itions at the inlet and outlet of the
system (in the x-direction) should be provided to complete the state-
ment of th e pore-scale problem. However, they will not be used for the
derivations that follow and are hence left unspeciﬁed here.
In order to carry out a macroscale description, an averaging
domain, V,ofmeasureV and characteristic size r
0
, is deﬁned. Two
averaging operators are considered, namely, the superﬁcial and
intrinsic averages. For a piece-wise smooth function, w,denedinthe
ﬂuid phase within th e j-region, they are, respectively, given by
hwi
j
¼
1
V
ð
V
bj
w dV; (2a)
hwi
b
j
¼
1
V
bj
ð
V
bj
w dV; (2b)
with j ¼ x; gx; g.Intheaboveequations,V
bj
(of measure V
bj
)
denotes the portion o f the averaging domain occupied by the ﬂuid
phase in the j-region. Both averaging operators are related according to
hwi
j
¼
V
bj
V
hwi
b
j
: (2c)
The ratio V
bj
=V denotes the volume fraction of the ﬂuid phase within
the averaging domain and it is, in general, a function of position. In
the porous medium bulk, this ratio is a constant corresponding to the
porosity, e.

B. Average mass balance equation
The development starts with the derivation of the macroscopic
mass balance equation in each region. Application of the superﬁcial
averaging operator to the continuity equation (1a) and use of the spa-
tial averagin g theorem
31
hr ai
j
¼rhai
j
þ
1
V
ð
A
brj
n a dA; j ¼ x; gx; g (3)
(a denoting a vector ﬁeld deﬁned in the b-phase), together with the
interfacial boundary cond ition [Eq. (1c)], leads to the macrosc opic
mass conservation equation which takes the same form in the three-
regions. It is given by
rhv
j
i
j
¼ 0; in the j region; j ¼ x; gx; g: (4)
The attention must now be focused on the average momentum equa-
tion in each region and this is the purpose of Secs. II CII E.
C. Analysis in the homogeneous part of the porous
medium (x-region)
This portion of the syste m has received cons iderable attention in
the literature; hence, the derivations provided here are presented in a
brief manner. More details can be found in references dedicated to the
averaging of one-phase ﬂow in a porous medium in the presenc e of
inertia
32,33
and to the de rivation of a macroscopic model for one-
phase ﬂow when slip is present at the solid–ﬂuid interfaces.
26,34
For
the sa ke of convenience, the str ucture of the porous medium is mod-
eled as a periodic array of solid inclusions as illustrated in Fig. 2.
Hence, the averaging domain, V, may be deﬁned as a ﬁnite arra y of
unit cells (each of size
c
), with the following constraint for its charac-
teristic size, r
0
:
r
0
L: (5)
Here, represents the largest characteristic size at the pore-scale [i.e.,
¼ maxð
b
;‘
r
Þ], while L is the smallest characteristic size of the
porous medium at the macroscale (i.e., the smallest dimension of the
porous medium in the three directions of space).
In order to derive the average momentum transport equation, it
is convenient to follow the short-cut approach suggested by Barre`re
et al.,
35
in which Eqs. (1) are considered in th e avera gin g do main, con-
ceived as a Representative Elementary Volume (REV), ma de of one or
more periodic unit cell(s) as sketched in Fig. 2. In this doma in, it is re a-
sonable to decompose the ﬂuid pressure gradient according to
36
rp
x
¼rhp
x
i
b
x
þr
~
p
x
; (6)
~
p
x
being the pressure deviations, and regard rhp
x
i
b
x
as a consta nt
within the REV. This assumption is justiﬁed on the basis of the
separation of length scales given in (5). In this way, the pressure devia-
tions and the ﬂuid velocity can be assumed to be periodic at the inlets
and outlets of the REV, and the pore-scale model can be written as
follows:
rv
x
¼ 0; in V
bx
; (7a)
qv
x
rv
x
¼r
~
p
x
þ lr
2
v
x
rhp
x
i
b
x
; in V
bx
; (7b)
v
x
¼n
1
k I nn
ðÞ
n rv
x
þrv
T
x

; at A
brx
; (7c)
wðr þ l
i
Þ¼wðrÞ; i ¼ 1; 2; 3; w ¼ v
x
;
~
p
x
; (7d)
h
~
p
x
i
b
x
¼ 0: (7e)
The last equation is an average constraint bounding the ﬁeld of the
pressure deviations that is compliant with the assumption that hp
x
i
b
x
is constant within the REV, and it is necessary for the ﬂow problem to
be well-pose d. In Eq. (7d), l
i
(i ¼ 1, 2, 3) repr esents the periodic lattice
vectors of the REV.
The formal solution of th i s problem can be written as
v
x
¼
F
l
rhp
x
i
b
x
; in V
bx
; (8a)
~
p
x
¼f rhp
x
i
b
x
; in V
bx
; (8b)
where F and f are closure variables that resu lt from solving the follo w-
ing problem in the (perio dic) REV
rF ¼ 0; in V
bx
; (9a)
q
l
2
jjrhp
x
i
b
x
jje
p
F
T
rF ¼rf þr
2
F þ I; in V
bx
; (9b)
F ¼n
1
k I nn
ðÞ
n rF þrF
ðÞ
T1

; at A
brx
; (9c)
wðr þ l
i
Þ¼wðrÞ; i ¼ 1; 2; 3; w ¼ F; f ; (9d)
hfi
b
x
¼ 0: (9e)
In Eq. (9b), e
p
is a unit vector in the direction of the macroscopic pres-
sure gradient. It is deﬁned as
e
p
¼
rhp
x
i
b
x
jjrhp
x
i
b
x
jj
: (10)
In addition, in Eq. (9c), the sup erscript T1 denotes the transpose oper-
ator that permutes the ﬁrst and second indices of a third order tensor
T,namely,T
T1
ijk
¼ T
jik
.
FIG. 2. Sketch of the averaging domain of size r
0
and a periodic two-dimensional
unit cell (of size
c
) for the analysis of the ﬂow problem in the homogeneous part of
the porous medium. Note that, in general, r
0
c
. The solid phase, r,is
represented as a random distribution of circular inclusions embedded in the ﬂuid
phase, b.

##### Citations
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Journal ArticleDOI
Abstract: This work addresses the macroscopic modeling of flow near porous media boundaries. This includes the vicinity with a fluid channel (i.e., a fracture), another rigid porous medium, or an impervious non-deformable solid. The analysis is carried out for one-phase, steady, incompressible, inertial, and isothermal flow of a Newtonian fluid, considering slip effects at the solid-fluid interfaces. A one-domain approach is proposed, employing a simplified version of the volume averaging method, while conceiving the system as two homogeneous regions separated by an inter-region. The upscaling procedure yields a closed macroscopic model including a divergence-free average (filtration) velocity for the mass balance equation and a unique momentum equation having a Darcy structure. The latter involves apparent permeability tensors that are constant in the homogeneous regions and position-dependent in the inter-region. All the permeability tensors are determined from the solution of coupled closure problems that are part of the developments. The derived model is validated by comparisons with direct numerical simulations in several two-dimensional configurations, namely, two porous media of contrasted properties in direct contact or separated by a fracture, the boundaries being either flat or wavy and a porous medium in contact with a flat or corrugated solid wall or separated from the latter by a fluid layer. The simplicity and versatility of the derived model make it an interesting alternative to existing one-and twodomain approaches developed so far.

1 citations

##### References
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Journal ArticleDOI
Abstract: Experiments giving the mass efflux of a Poiseuille flow over a naturally permeable block are reported. The efflux is greatly enhanced over the value it would have if the block were impermeable, indicating the presence of a boundary layer in the block. The velocity presumably changes across this layer from its (statistically average) Darcy value to some slip value immediately outside the permeable block. A simple theory based on replacing the effect of the boundary layer with a slip velocity proportional to the exterior velocity gradient is proposed and shown to be in reasonable agreement with experimental results.

2,645 citations

Book
01 Jan 1999
Abstract: 1 Diffusion and Heterogeneous Reaction in Porous Media 2 Transient Heat Conduction in Two-Phase Systems 3 Dispersion in Porous Media 4 Single-Phase Flow in Homogeneous Porous Media: Darcy's Law 5 Single-Phase Flow in Heterogeneous Porous Media Appendix Nomenclature References Index

1,240 citations

Journal ArticleDOI
J. Alberto Ochoa-Tapia
Abstract: The momentum transfer condition that applies at the boundary between a porous medium and a homogeneous fluid is developed as a jump condition based on the non-local form of the volume averaged momentum equation. Outside the boundary region this non-local form reduces to the classic transport equations, i.e. Darcy's law and Stokes' equations. The structure of the theory is comparable to that used to develop jump conditions at phase interfaces, thus experimental measurements are required to determine the coefficient that appears in the jump condition. The development presented in this work differs from previous studies in that the jump condition is constructed to join Darcy's law with the Brinkman correction to Stokos' equations. This approach produces a jump in the stress but not in the velocity, and this has important consequences for heat transfer processes since it allows the convective transport to be continuous at the boundary between a porous medium and a homogeneous fluid.

765 citations

Book ChapterDOI
Abstract: The no-slip boundary condition at a solid-liquid interface is at the center of our understanding of fluid mechanics. However, this condition is an assumption that cannot be derived from first principles and could, in theory, be violated. In this chapter, we present a review of recent experimental, numerical and theoretical investigations on the subject. The physical picture that emerges is that of a complex behavior at a liquid/solid interface, involving an interplay of many physico-chemical parameters, including wetting, shear rate, pressure, surface charge, surface roughness, impurities and dissolved gas.

584 citations

Journal ArticleDOI
Abstract: In this paper we illustrate how the method of volume averaging can be used to derive Darcy's law with the Forchheimer correction for homogeneous porous media. Beginning with the Navier-Stokes equations, we find the volume averaged momentum equation to be given by $$\langle v_\beta \rangle = - \frac{K}{{\mu _\beta }} \cdot ( abla \langle p_\beta \rangle ^\beta - \rho _\beta g) - F\cdot \langle v_\beta \rangle .$$ The Darcy's law permeability tensor, K, and the Forchheimer correction tensor, F, are determined by closure problems that must be solved using a spatially periodic model of a porous medium. When the Reynolds number is small compared to one, the closure problem can be used to prove that F is a linear function of the velocity, and order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

568 citations