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Macroscopic model for unsteady ow in porousmedia
Didier Lasseux, Francisco J. Valdès-Parada, Fabien Bellet
To cite this version:
Didier Lasseux, Francisco J. Valdès-Parada, Fabien Bellet. Macroscopic model for unsteady ow in
porousmedia. Journal of Fluid Mechanics, Cambridge University Press (CUP), 2019, 862, pp.283-311.
�10.1017/jfm.2018.878�. �hal-02006880v2�
J. Fluid Mech. (2019), vol. 862, pp. 283-311. doi:10.1017/jfm.2018.878
1
Macroscopic model for unsteady flow in
porous media
Didier Lasseux
1
†, Francisco J. Valdés-Parada
2
Fabien Bellet
3
1
CNRS, I2M, UMR 5295
Esplanade des Arts et Métiers, 33405 Talence, Cedex, France.
2
Universidad Autónoma Metropolitana-Iztapalapa
Departamento de Ingeniería de Procesos e Hidráulica
Av. San Rafael Atlixco 186, 09340 Ciudad de México, Mexico.
3
Lab oratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay
3, rue Joliot Curie, 91192 Gif-sur-Yvette Cedex, France.
(Accepted 28 October 2018)
The present article reports on a formal derivation of a macroscopic model for unsteady
one-phase incompressible flow in rigid and periodic porous media using an upscaling tech-
nique. The derivation is carried out in the time domain in the general situation where
inertia may have a significant impact. The resulting model is non-local in time and
involves two effective coefficients in the macroscopic filtration law, namely a dynamic
apparent permeability tensor, H
t
, and a vector, α, accounting for the time-decaying in-
fluence of the flow initial condition. This model generalizes previous non-local macroscale
models restricted to creeping flow conditions. Ancillary closure problems are provided,
which allow computing the effective coefficients. Symmetry and positiveness analyses of
H
t
are carried out, evidencing that this tensor is symmetric only in the creeping regime.
The effective coefficients are functions of time, geometry, macroscopic forcings and the
initial flow condition. This is illustrated through numerical solutions of the closure prob-
lems. Predictions are made on a simple periodic structure for a wide range of Reynolds
numbers smaller than the critical value characterizing the first Hopf bifurcation. Finally,
the performance of the macroscopic model for a variety of macroscopic forcing and ini-
tial conditions is examined in several case studies. Validation through comparisons with
direct numerical simulations is performed. It is shown that the purely heuristic classical
model, widely used for unsteady flow, consisting in a Darcy-like model complemented
with an accumulation term on the filtration velocity, is inappropriate.
1. Introduction
Unsteady flow in porous media has been the subject of active research over, at least,
the past sixty years. One of the main interests has been the propagation of acoustic
waves in porous structures with applications in seismic waves, enhanced oil recovery,
ocean bottom interactions and coastal waves, superfluid flow in porous media, among
many others, in addition to the fundamental nature of deriving appropriate physical
models. This was initiated by the pioneer works from Biot (1956a,b) to analyze effects
such as wave speed, attenuation, viscous dissipation and anisotropy. An overview of the
literature on the subject may lead to classify studies into three main groups, namely
studies about elastic media without any fluid external forcing, studies of time-dependent
flow in rigid porous media and fluid flow through elastic media. In the present work,
† Email address for correspondence: didier.lasseux@u-bordeaux.fr
2 D. Lasseux, F. J. Valdés-Parada, F. Bellet
the interest is focused upon incompressible and unsteady single-phase flow through rigid
homogeneous periodic porous media. Existing reported works may be conveniently sum-
marized by distinguishing those carried out in the time-domain from those developed in
the frequency-domain. In the following paragraphs a non-exhaustive literature review of
both branches is presented.
Description of unsteady incompressible one-phase flow in porous media has been widely
relying on extensions to the steady version of Darcy’s law, or, when inertia is taken into ac-
count, to the Darcy-Forchheimer corrected form. To the best of our knowledge, one of the
earliest extensions to account for unsteady effects was put forth by Polubarinova-Kochina
(1962). In this work, an acceleration term on the filtration velocity was kept in the
macroscopic momentum equation as obtained from a direct analogy with the Stokes
(or Navier-Stokes) equation in which the point velocity is replaced by the average ve-
locity and the external force by the average friction on the solid surface of the porous
matrix, i.e. the Darcy term. Despite its lack of rigorous formal derivation, this type of
approach has been considered as a valid one and became classical over the past half
century (Rajagopal 2007; Bories et al. 2008; Nield & Bejan 2013). This model will be
referred to as the “heuristic model”. It has b een widely used, for instance, in numerical
simulations (Dogru et al. 1978), for stability analysis of fluid-flow between an imperme-
able plate an a porous wall (Hill & Straughan 2008, 2009) or for turbulence in a similar
configuration (Breugem et al. 2006) or in a confined porous medium (Jin & Kuznetsov
2017) as well as for three-dimensional stability analysis of flow between two parallel
porous walls (Tilton & Cortelezzi 2008); for the analysis of forced or natural convection
in porous media (Kuznetsov & Nield 2006); the transition to chaos in natural convection
(Vadasz 1999), among many other applications. Few formal analyses were dedicated to
tentatively derive the heuristic model and some of them may have been inspired by the
development of the steady macroscopic model of one-phase flow in porous media includ-
ing inertia by Whitaker (1996). In fact, in this reference, the acceleration term was kept
in a large part of the development although it was clearly stated, at the final stage, that
the steady ancillary closure problem used to derive the closed average model was only
compatible with a steady version of this model (see section 2.8 in this reference). How-
ever, the unsteady version of this model was used by Tilton & Cortelezzi (2008) with a
reference to Whitaker (1996). Two other works (Teng & Zhao 2000; Breugem et al. 2006)
proposed a development yielding the unsteady form of the macroscopic model developed
by Whitaker (1996) (equation (2.26)) that, indeed, corresponds to the heuristic model.
However, in these works, the closure procedure is not considered and the time-scale
constraint is not addressed. Nevertheless, in a recent paper, Zhu et al. (2014) further
considered this version of the unsteady model and showed, from comparison with direct
numerical simulation (DNS), that it was inappropriate. With the sake of keeping the
same form of the unsteady model, the acceleration term was modified by conveniently
introducing a time constant obtained by averaging the energy equation, an idea that was
employed by Laushey & Popat (1968) to interpret results obtained on model unconfined
aquifers. Comparisons with DNS results showed agreement. However, this time constant
requires knowledge of the pore-scale flow field featuring a non-closed overall model that
can not be used as a predictive one even under creeping flow conditions.
The approach making use of the heuristic model has been also very popular in wave
dampening models in coastal engineering (Hall et al. 1995; Corvaro et al. 2010). In this
field, however, the lack of accuracy of the approach, compared to experimental data,
led numerous authors to modify the heuristic model by affecting a premultiplying fac-
tor, usually called “inertial coefficient”, to the accumulation term. Without any formal
derivation, this was justified by an analogous concept of an added virtual mass force
Macroscopic model for unsteady flow in porous media 3
used for modelling flow around an isolated obstacle. This concept was first introduced by
Sollitt & Cross (1972) and many different forms of the inertial coefficient were proposed
since then (see a short review in Burcharth & Andersen (1995)). A formal derivation of
this modified version of the heuristic model was attempted ( Abderahmane et al. 2002)
but the development suffers again, at the final stage, from a formal identification of
the macroscopic mo del to be obtained with the microscopic mo del. The misleading use
of the heuristic model was pointed out by Auriault (1999) indicating that the macro-
scopic momentum equation should contain a memory effect expressed by a convolu-
tion product between the filtration velocity and a memory function. The proof of this
form was anticipated by the same author (Auriault 1980), and almost simultaneously
by Lions (1981). It was later reconsidered by Allaire (1992), Mikelić (1994) and more
recently in Mei & Vernescu (2010) (the term “permeability” attributed to the memory
function in the latter references is inadequate as it is dimensionally incorrect). However,
as will be commented in the following sections, the reported developments require to
be completed, either by taking into account the initial condition or by explicitly pro-
viding the closure problems yielding the effective coefficients, in particular in the case
where inertia is significant. Upscaling the Navier-Stokes (or Euler) equations was also
addressed using the homogenization technique (Sanchez-Palencia 1980; Masmoudi 1998,
2002; Lions & Masmoudi 2005). However, as will be further commented in section 3.2,
no complete unsteady macroscopic model was reported with this technique. Some other
derivations were reported in the literature, mainly developed in the Fourier domain.
Regarding the literature ab out unsteady flow modelling in porous media in the fre-
quency domain, it is worth mentioning that one serious drawback of early Biot’s theory
lies in the lack of providing numerical predictions of the effective medium coefficients
involved in the macroscale model. This issue was addressed by Auriault et al. (1985)
who used the homogenization technique to derive a Darcy-law typ e model to describe
unsteady creeping flow in rigid and deformable porous media, assuming the fluid to be at
rest in the porous matrix as the initial condition. Predictions of the model were validated
with experimental results. This study is a continuation of previous works by Lévy (1979)
and Auriault (1980), where the homogenization method was used to study flow through
elastic porous media. In the work by Lévy, the resulting expression is also a Darcy-law
type model in the frequency domain, while the work by Auriault is an extension to
include inertial effects and multiphase flow. This upscaling approach was also used by
Sheng & Zhou (1988) (see also Zhou & Sheng (1989)) to predict the dynamic permeabil-
ity as a function of frequency for a variety of microstructures in the creeping flow regime.
These authors proposed to scale the predicted dynamic permeability, κ(ω), by its static
value, κ
0
, in order to produce a universal curve independent of the microstructure when
plotted against a scaled frequency (ω
c
) that is particular of the microscale geometry and
flow properties. In this way, these authors proposed the following empirical relationship
κ(ω)
κ
0
= f
ω
ω
c
(1.1)
with f being a so-called universal structure function independent of the microstructure.
Later on, Charlaix et al. (1988) reported experimental measurements of the dynamic
permeability on capillary tubes and model porous media made of fused glass beads and
crushed glass of different sizes for conditions in which the flow was in the transition
between the creeping and inertial regimes. These authors found that their experimen-
tal measurements were in agreement with the relationship proposed by Sheng & Zhou
(1988). However, their experiments were performed on samples featuring a rather nar-
row range of topology varieties. A few time later, Johnson (1989) prop osed an ana-
4 D. Lasseux, F. J. Valdés-Parada, F. Bellet
lytical expression for f, which is given not only in terms of ω
c
, but also of a parameter
M = 8α
τ
κ
0
/εΛ
2
, with α
τ
, ε and Λ being the tortuosity factor, the porosity and a charac-
teristic length that was taken to be twice the pore volume to surface ratio (Johnson et al.
1987), resp e ctively.
Advances in numerical capabilities made possible predictions of the dynamics of the
permeability in more complex geometries than those used before. In this regard, Chapman & Higdon
(1992) solved the unsteady version of the Stokes problem in several three-dimensional
periodic unit cells. The resulting average velocity was used in the unsteady version of
Darcy’s law in the frequency domain to predict the dynamic permeability. In order to em-
phasize porosity and frequency effects, the permeability dependence up on frequency was
not represented in the universal curve suggested above. In the same year, Smeulders et al.
(1992) reported numerical simulations and experimental measurements that corrobo-
rated the universal relationship proposed by Sheng & Zhou (1988) when more parame-
ters are considered in the structure function. In addition, these authors rigorously derived
the analytical relationship proposed by Johnson et al. (1987) using the homogenization
technique. Departures from the relationship given in equation (1.1) were reported by
Achdou & Avellaneda (1992) for microgeometries consisting of corrugated tubes. These
authors observed a slower convergence of the dynamic permeability towards its steady
state value than that predicted by the empirical relationship. This issue was later ad-
dressed by Cortis et al. (2003), who used direct numerical simulations to show that the
predictions from the relationship in equation (1.1) are justified for microchannels with
corrugated, and even wedge-shaped, walls. In the present work, the issue of the univer-
sality of the above mentioned empirical relation is not going to be further discussed.
The purpose of this article is to carry out a careful derivation of the macroscopic un-
steady model for one-phase flow in rigid and periodic porous media including inertial
effects and taking into account the influence of the initial flow condition. This is achieved
by upscaling the unsteady solution of the initial boundary value problem operating at
the pore-scale using a short-cut version of the volume averaging technique, which has the
nice feature to lead to a closure scheme for the prediction of the corresponding effective
medium coefficients. The developments detailed hereafter are organized as follows. After
recalling the pore-scale model in section 2, the upscaling procedure is detailed in section
3. The development is performed in the time-domain yielding the unsteady macroscopic
model which involves the time rate of change of the convolution product between the
dynamic apparent permeability tensor, H
t
, and the macroscopic pressure gradient, as
well as an effective vectorial term, α, which accounts for the effect of the initial con-
dition. The two effective coefficients H
t
and α can be computed from the solution of
two time-dependent closure problems that are explicitly provided. This general model
encompasses the special case of creeping flow. Symmetry and positiveness properties of
the dynamic apparent permeability tensor are investigated. In addition, illustrative ex-
amples of the dynamics of the effective coefficients are provided. Section 4 is dedicated
to results obtained for a model periodic porous structure involving four stiff case studies,
which serve as tests of the performance of the upscaled and heuristic models with respect
to direct numerical simulations. Concluding remarks are presented in section 5.
2. Pore-scale model
The development starts with the classical mass and momentum Navier-Stokes equa-
tions describing flow of a single Newtonian and incompressible fluid phase β that satu-
rates the void space of a porous medium whose skeleton is made of a non deformable solid
phase σ such as the one sketched in figure 1a. At any p oint in the p ore-space occupied
