Inertial eects in channels with periodically varying aperture and impact on solute dispersion a∗

a†

b‡

J. Bouquain , Y. Méheust , D. Bolster , P. Davy

a§

January 21, 2012

(a) Geosciences Rennes (UMR CNRS 6118), Université Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France

(b) Department of Civil Engineering and Geological Sciences, University of Notre Dame, IN, USA

∗ [email protected] † [email protected] ‡ [email protected] § [email protected]

1

1

Introduction

Ever since GI Taylor's seminal work [33] where he identied that the transport of a contaminant in a shear ow could be calculated in only one dimension by using a longitudinal eective dispersion coecient, the concept of eective dispersion has proved enormously popular across a wide variety of elds and applications. This includes micro uidics [27, 34], nutrient transport in bloodow [16, 37], single and multiphase transport in porous media [7, 4, 25, 24, 31, 14, 3] and transport in groundwater systems [10, 11, 15, 23]. The basic idea is simple. At asymptotic times, which are times when the solute has already sampled the full variability of the ow velocities, gradients of solute transverse to the ow direction can be considered negligible. At these times the spreading of the plume only occurs in the longitudinal direction and can be described by a one-dimensional advection-diusion process. The corresponding diusive term features an eective dispersion coecient instead of the molecular diusion coecient, as evidenced and quantied in for the cylindrical geometry by early works [33, 1].

.

At times earlier than these asymptotic

times, the behaviour is more complicated as the rate of spreading of the plume and mixing are not the same [11, 38]. While a variety of works exist studying these preasymptotic times for the cylindrical- or parallel plate- congurations [19, 26, 5], and also when density-driven coupling of ow and transport is present [6], it will not be the focus of our work here. In many real applications the relevant channels do not have constant aperture. However, using slightly more complicated approaches the notion of Taylor dispersion can still be applied [7]. Using these approaches many authors have shown that deviation from parallel smooth boundaries can signicantly alter behaviour, leading to relative increases [22, 32, 36, 18] or even decreases [12, 30, 4] in the eective dispersion. Most studies for ow through porous media assume small Reynolds numbers

Re ∼ o(1).

Thus it is typically reasonable to assume that ow is governed by the

Stokes ow problem where inertial eects can be ignored. While this is very often a reasonable assumption [13], a variety of practical situations exist, the Reynolds number can become of order unity and larger such that inertial eects are no longer negligible. For example, ow through fractures with large apertures can attain Reynolds numbers larger than unity and thus inertial eects can become important [28]. Similarly, such inertial eects may become important for ows where the viscosity can be small (e.g. carbon sequestration where the viscosity of supercritical CO2 can be one or two orders of magnitude smaller than that of water [17]). Increased inertial eects play an interesting role on the structure of the ow [21, 29].In particular they lead to the presence of recirculation zones [28], which represent immobile regions that can have a signicant impact on eective solute transport and in particular on the asymptotic dispersion, both for reactive [13] and inert solute transport [4], depending on the typical mass transfer time scales [20, 8]. In this work, considering an idealised pore geometry, we focus on ow regimes

2

at Reynolds numbers larger than one, where inertial eects become signicant. We do not consider situations where the Reynolds number becomes suciently large for the ow to become turbulent. In particular we consider the evolution in size of the recirculation zones with increasing Reynolds number and investigate what eects it has on the pre-asymptotic transport and ultimately on asymptotic dispersion.

We have performed a two-dimensional nite element analysis of

the ow eld and done numerical random walk simulations.

We describe the

geometry, the mathematical basis and the numerical implementation of our simulations in section 2. The results are presented in section 3 and discussed in section 4. Y:Ici rajouter une description de la structure du papier, en quelques phrases.

2

Methods

2.1 Geometry denition We dene a two dimensional geometry with sinusoidal wall boundary, as described in refs. [23, 4].

¯ − h0 cos h(x) = h where

h

aperture,



2πx L

 (1)

is the half-aperture at horizontal position

0

h

¯ x, h

is the average half-

is the maximum amplitude of the aperture uctuation and

L

the

length of the unit cell ( see Fig. 1). The uid ows from left to right.

h’ h

L Figure 1: aperture,

Geometry of two consecutive unit cells.

0

h

dene the aperture uctuation and

L

¯ h

is the average half-

is the length of the unit cell,

or wavelength of the sinusoidally-varying channel.

We can fully characterize the geometry with two dimensionless numbers , namely the aspect ratio of the cell

 =

3

2h L

(2)

and the relative amplitude of the aperture uctuations,

a

:

h0 2h

a= When

a

(3)

equals 1/2, the channel is closed and pores are disconnected from

a

the network. When

goes to 0, the channel is smooth and goes to the parallel

plate geometry. While this is obviously a simplifed model for a real porous medium, Edwards et al.[14] illustrated that it does a very good job of representing ow and transport in a cylindrically packed porous medium. It has also sometimes been considered as an idealised model for the geometry of a geological fracture [?], although realistic fracture geometries are known to be much more complicated [?]. The channel half mean aperture cinematic viscosity

ν

¯, h

the mean uid velocity

u ¯

and the uid

dene the Reynolds number :

¯ 2h u ν

Re =

(4)

On the pore scale, Reynolds number is typically small ([23, 2]), of the order of

10−4 − 10−1 .

For such Reynolds numbers, ow is described by the linear

Stokes equation. For a slowly varying boundary, i.e.,

1

and small Reynolds

number, Kitanidis and Dykaar [23] derived an analytical solution for the ow velocity using a perturbation expansion in

.

However, their approach is no

longer possible when one cannot neglect inertial eects:no obvious analytical approach then exists to solving the nonlinear governing Navier-Stokes equations. As such we resort to solving the system of governing equations numerically as outlined in the following section.

2.2 Basic equations and numerical simulations 2.2.1

Flow

Direct numerical simulations of the steady state ow through the geometry above were conducted. The ow is assumed to be incompressible. The conservation of mass therefore reads as:

∇. u = 0 where

u

is the velocity eld. The conservation of momentum equation is :

 ρ where

ρ

(5)

∂u + (u · ∇)u ∂t

is the density of the uid,



g

= ρ g − ∇p + η ∇2 u

is the gravity eld and

(6)

η

is the dynamic

viscosity of the uid. The system of joint Eqs. equation (5) and equation (6) are solved using a nite element analysis.The nite elements numerical simulation is conducted with

4

the software (Comsol Multiphysics [9]), in two dimensions and using Lagrangequadratic elements. The ow is solved on a mesh made of triangular elements (up to 100,000) with a maximum side length xed to

L/280.

The mesh is

statistically isotropic. The left and right boundaries are periodic, that is, the ow velocity

u cross-

section is the same at the inlet and at the outlet. A mean ow is imposed on the inlet boundary. The outlet is set to a constant pressure, which allows the solver to adjust the pressure in the geometry to suit the global imposed volumetric ow. The numerical method for ow, applied to a dierent geometry, is described in detail in ref. [6].

2.2.2

Transport

We neglect any possible density-driven retroaction of transport on ow.

The

transport problem is therefore treated once the ow eld has been solved for. It is solved numerically by Lagrangian particle random walk simulations based on the Langevin equation. This approach is chosen for two reasons: (i) because for the periodic domain considered here, one does not have to

a priori

impose

the size of the domain and one can allow transport to occur over as a large a distance/computational domain as desired and (ii) because Lagrangian methods do not suer from the problems of numerical dispersion as Eulerian methods do and as we are trying to quantify a dispersive eect we wish to minimize uncertainties on the results as much as possible. The initial condition that we choose is a line uniformly distributed across the width of the channel, i.e.

c(x, z) = δ(x)

(7)

In discrete time, the equation of motion of the at position

x0

nth

solute particle, located

at initial time, is given by the Fokker-Planck equation:

x(n) (t + ∆t |x0 ) = y (n) (t |x0 ) + u(n) (t |x0 )∆t + η1 y (n) (t + ∆t |x0 ) = y (n) (t |x0 ) + v (n) (t |x0 )∆t + η2 where

√ 2D∆t

(8)

2D∆t

(9)

√

x(n) = (x(n) , y (n) ) denotes the position of the particle and v (n) = (u(n) , v (n) ) ηi (i = 1, ..., d) are independently distributed Gaussian random

its velocity.The

variables with zero mean and unit variance.

This Fokker-Planck equation is

exactly equivalent to the advection-dispersion equation.

Solid boundaries are

modeled as elastic reection boundaries in order to account for their impermeability. The mean half aperture coecient

Dm

¯, h

the mean uid velocity

u ¯

and molecular diusion

dene the Péclet number :

Pe =

¯u 2h ¯ Dm

The average position of the solute plume is that of its center of mass :

5

(10)

N 1 X h (n) i x (t) N n=1

xG (t) =

(11)

The velocity of the center of mass is the mean horizontal velocity of the solute plume. We normalize it by the mean advection velocity :

u ˜G (t) =

1 dxG u ¯ dt

.

(12)

The cross-sectionally summed mass is computed according to

ˆ

h(x)

M (x, t) =

C(x, y, t) dy

,

(13)

−h(x) resulting in a mean cross-sectional concentration

¯ t) = C(x,

¯ t) C(x,

in the form

1 M (x, t)xG xG . 2h(x)

(14)

We quantify longitudinal dispersion in this system from calculations of the horizontal spatial moments of the plume as it evolves in time.

The

ith

local

moment is given by summation over the ith power of the particle trajectories of all

N

simulated particles originating from a single

x0

,

N 1 X h (j) ii x (t) J→∞ N j=1

µ(i) (t | x0 ) = lim

The global moments are obtained by summation over all initial positions

M 1 X (i) (m) µ (t | x0 ) M →∞ M m=1

m(i) (t) = lim

(15)

x0 , (16)

The apparent dispersion coecient is then given by

Da (t) =

i 1 d h (2) m (t) − m(1) (t)2 2 dt

(17)

When particles have experienced all the ow lines, the asymptotic apparent dispersion coecient is reached. It is dened as

Da∞ (t) = lim Da (t) t→∞

(18)

As the Taylor-Aris dispersion occurring in a channel of uniform aperture is a well-known limit case to the conguration studied here, we consider dispersion coecients that are normalized by the Taylor-Aris dispersion coecient

DT.A.

([33, 1])

˜ a (t) = Da (t) D DT.A. (t) 6

(19)

and in particular an asymptotic exponent

∞ ˜ a∞ (t) = Da (t) D DT.A. (t)

DT.A. (t)

.

(20)

is dened for a parallel plate fracture with identical mean aperture

as (Wooding [35])


¯ 2 ¯h 2 u DT.A. (t) = Dm + 105 Dm The simulations presented release

N

.

(21)

particles from each initial position. The

line source is represented by distributing them evenly along the cross-section of the channel at horizontal position adimensionnalized time by velocity

u ¯

τ

x = 0.

For all the time dependent studies, we

the time needed for a particle moving at the mean

to cross a single unit cell :

τ= Additionally, For convenience,

u ¯

tu ¯ L

(22)

is xed to 1 in all the simulations.Y: A-t-

on besoin de parler de ça ? J: c'est utile pour évaluer facilement Dm à partir

de Pe, si le lecteur le souhaite

3

Results

3.1 Flow phenomenology The presence of recirculation zones depends on the geometry ( and

a) and on the

Reynolds number. Examples are given in Fig. 2. At small Reynolds numbers, Stokes ow conditions are fullled and the streamline shape is homothetic from that of the walls (Fig. 2A). Note that even in Stokes ow conditions, recirculation zones can appear but



and

a

have to be high enough. None of these conditions

are fulllled in our parameter ranges. At higher Re values, ow lines and vertical velocity maps become asymmetric with respect to the vertical line

x = L/2

(Fig. 2B), showing that the ow is not reversible any more and that inertial eects cannot be neglected. The recirculation zone appear in the widest part of the cell for even higher Reynolds (Re

= 20 in Fig. 2C for which ( = 0.47, a = 0.4)) and its size grows monotonously with Re. The growth is asymmetrical but ultimately leads to a ow shape similar to the one seen with a single fracture with perpendicular dead ends such as in the geometry seen in Lucas[28].Y:Pas sûr de bien comprendre la dernière partie de la phrase. Tu pourrais expliciter STP ?J:C'est une phrase de Diogo. je pense qu'il fait référence à la geometrie dans l'article de Lucas et al., avec un bras mort perpendiculaire, et une zone de recirculation qui le remplit intégralement. J'ai ajouté la référence.

φ VRZ /V .

We dene the volume fraction of the recirculation zone volume of the recirculation zone to that of the cell,

7

as the ratio of the

Max : 0.89

Max : 0.99

A

B 0.6

0.5 0

v u

0

v u

-0.5

-0.6

Min : -0.89

Min : -0.99

Max : 1.05

C

Max : 0.99

D

1

0.6

0.5 0

v u

0

v u

-0.5 -0.6 -1 Min : -1.05 Figure 2:

Min : -0.99

The color maps show the vertical component of the velocity eld,

normalized by the mean horizontal velocity cell geometry is ( In A,

Re = 0.1

= 0.47, a = 0.4),

u.

and 4 Reynolds numbers are considered:

and the ratio between the volume of the recirculation zone and

φ is 0%; in B, Re = 10 and φ = 10%; in C, Re = 20 Re = 100 and φ = 75%. Fluid ows from left to right.

the volume of the cell

φ = 47%; φ

in D,

v,

Flow lines are superimposed. The

and

is a function of the geometry and of the Reynolds number (see Fig. 3).

Maps of

φ

as a function of the relative aperture uctuation

a

and the Reynolds

number are shown in Fig. 3 for two values of the cell aspect ratio isoline

φ = 10%

.

The

is the threshold at which we consider that the presence of a

recirculation zone begins to have a signicative impact on asymptotic transport (i.e. on the asymptotic dispersion coecient). Depending on the geometry, the 10% limit can be reached at a quite low Reynolds number value. For example, for

a = 0.25

, i.e.

when the aperture at the widest

x

position is equal to 3

times the aperture in the channel throat, Reynolds number values as low as 20 are sucient for recirculation zones to signicantly impact transport. But at

a = 0.166

(i.e.

the max/min aperture ratio is 2), transport is unaected by

recirculation zones for

R . 100.

8

ε = 0.2

Re

% 10

100 90 80 70 60 50 40 30 20 10 0 0.1

75%

0.15

0.2

0.25

0.3

0.35

0.4

50%

a ε = 0.55

25%

0%

Re

10%

100 90 80 70 60 50 40 30 20 10 0 0.1

0.15

0.2

0.25

0.3

0.35

0.4

a Figure 3:

Volume of the recirculation zone as a function of the Reynolds

number and the relative aperture uctuation ratio

.

a,

for two values of the cell aspect

The dashed line represent the 10% threshold below wich the transport

is slightly to no impacted by the presence of a recirculation zone.

9

4 5

3.2 Transport phenomenology

2 1.6

2.4

0.7

1

~ u G

3

A

h

C

h − h’

h(xG)

h + h’

0.0

0.8

~

Da

B

0

Figure 4:

1

2

3

τ

4

5

6

Evolution of (A) the normalized horizontal velocity

normalized dispersion coecient

˜a D

u ˜G

, (B) the

and (C) the aperture at the position of the

h(xG ) as a function of the reduced time τ = t u ¯/L. The geometry  = 0.47 and a = 0.4, ow and transport byP e = 100 and Re = 1.

center of mass is dened by

Fig. 4 shows a representative analysis of the observables for the transport in a case where

P e = 100, Re = 1,  = 0.47

and

a = 0.4.

The ow conguration is

very similar to the one shown in Fig. 2A, without any recirculation zone. We expect that once all the particles have experienced all velocities in the domain, the horizontal velocity of the center of mass converge to the mean ow velocity. This behaviour is conrmed in Fig. 4A. This remains true only when there is no reciculation zone.

The normalized dispersion coecient

˜ a = Da /DT.A. D

also tends to an asymptote at a value clearly higher than 1.

It means that

the dispersion coecient value is higher than the one predicted for the uniform aperture case of identical mean aperture. The early time oscillations of the dispersion coecient are directly related to the oscillation of the velocity of the center of mass, due to the aperture spatial variations, the volumetric ow rate being conserved. The shape of the time evolution of the velocity is directly related to the geometry parameters ( and

a),

Reynolds number (Re) and Péclet number (P e). When

a

is increased,

the velocity dierence between the widest and thinest zone becomes higher, the

10

oscillations amplitude also. When

 is increased, the length of the cells is smaller, Re alone

the frequency of the oscillation increases. The eect of an increase of

is that the recirculation zones widen and the ow tends to a case similar to the parallel plate fracture (the oscillation amplitude decreasing dramatically), but with a smaller eective mean aperture. Consequently, the velocity of the plume center of mass also increases as most of the solute is in the non-recirculation zone, leading to a higher oscillation frequency.

When

Pe

alone is decreased,

the higher diusion coecient contributes to reaching the asymptotic regime sooner, and which means that the oscillations are dampened faster. The rst peak of the dispersion coecient in Fig. 4 occurs at

τ ≈ 0.5 (shown

with a dashed grey line) and nearly coincides with the rst peak of the horizontal velocity. A snapshot of the spatial solute distribution at this moment is shown in Fig. 5. A signicative part of the solute mass is in the cell throat where the velocity is at a maximum while the rest of the solute is in a quite slow zone. In this conguration, the plume is highly streched, leading to a high dispersion coecient value.

11

A

h + h'

h

C(x,y,t)

y

Max

h − h'

8

0

x10-1

3

2

C(x, t)

M(x,t)

6

4 1

2

0

Figure 5:

B

0.0

0.2

0.4

0.6

0.8

x/L

1.0

1.2

0

1.4

Snapshot of (A) the concentration eld and (B) vertically-summed

mass and vertically-averaged density at the rst peak time in the longitudinal dispersion coecient evolution shown by the gray dashed line in gure 4 (i.e. at

τ ≈ 0.5) . The geometry is by P e = 100 and Re = 1.

dened by

 = 0.47

and

a = 0.4,

ow and transport

Figures 6 and 7 provide a comparison of the spatial distribution of the con-

τ ≈ 1.25 for two opposite cases; one highly diusive with = 50)(Fig. 6) and one highly advective with a high (P e = 500)(Fig. 7). The Reynolds number is high in both cases

centration at a time

a low Péclet number (P e Péclet number (Re

= 200),

with fully developped recirculation zones, as seen in Fig. 2D. In

Fig. 7, solute barely enters the recirculation zones and most of the mass is in the center of the cells, moving fast. Particles jump by diusion on one of the external ow line of the recirculation zone and then really enter the zone by advection.

As such, they enter the recirculations zone by the right side.

As

diusion coecient is small, the probability to leave the recirculation zones is low. On the contrary, in Fig. 6 a signicant amount of the total mass has been able to enter the recirculation.Compléter

12

la discussion ?

A

h + h'

h

C(x,y,t)

y

Max

h − h'

0

x10-1

4

1.5

B

C(x, t)

M(x,t)

1.0 2

0.5

0

Figure 6:

0.0

0.5

1.0

1.5

2.0

2.5

x/L

3.0

3.5

4.0

0

Snapshot of (A) the concentration eld and (B) vertically-summed

mass and vertically-averaged density in a geometry with at time

4.5

τ ≈ 1.25.

 = 0.19 and a = 0.38 = 50, Re = 200).

The conguration is highly diusive (P e

13

A

h + h'

h

C(x,y,t)

y

Max

h − h'

3

0

x10-1

3

2

1

1

M(x,t)

C(x, t)

2

B

0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0

x/L

Figure 7:

Snapshot of (A) the concentration eld and (B) vertically-summed

 = 0.19 and a = 0.38 τ ≈ 1.25 . The conguration corresponds to a quasi pure advective case (P e = 500, Re = 200). mass and vertically-averaged density in a geometry with

at time

14

3.3 Time evolution of the apparent dispersion coecient As we saw in Fig. 4, the early time oscillations of the dispersion coecient and velocity of the center of mass

vm

seem to be in phase. It turns out that theses

oscillations can be accounted for as the velocity to a certain power normalized by that proper

λ vm

λ.

Once

law, the longitudinal dispersion data exhibits no

uctuations reminiscent of the sinusoidal boundary conditions any more (see inset in Fig. 8); in addition it is well tted by a stretched exponential behavior. In other words , the

Da

data is well tted by a law in the form (Fig. 8):

Da (τ ) = where

Da∞

Da∞

   γ  τ λ vm 1 − exp − τc

is the asymptotic dispersion coecient,

velocity of the center of mass, and ponential.

The tting parameters

γ is the Da∞ , τc

τc

(23)

a characteristic time,the

exponent inside the stretched ex,

γ

and

λ

correspond to distinct

geometrical properties of the curve, and are therefore obtained with a small uncertainty.

Figure 8:

Time evolution of the normalized dispersion coecient and t

comparison. In this case,

P e = 50, Re = 10,  = 0.47

and

a = 0.4.

For most of the simulation, the time evolution of the dispersion coecient oscillates before reaching the asymptote. These oscillations are directly related to the shape of the geometry and to the evolution of the velocity of the center of mass. As seen in Fig. 4, these oscillations dampen over time. The scillation amplitude increases when

Re

decreases or

to dampen sooner with a lower

P e.

a

increases.

The oscillations tend

Oscillation frequency is directly related to

a mean velocity calculated only in the zone outside of the recirculation zones. Thus, it increases when

Re

or



increases. In addition to properly describing

15

the whole time evolution of the longitudinal dispersion, the t provides us with a robust estimate of the asymptotic value for the dispersion coecient. We will focus on this observable hereafter.

3.4 Asymptotic dispersion coecient as a function of ow and geometry parameters 3.4.1

Dependence on the Reynolds and Péclet numbers

For each geometry, the asymptotic dispersion coecient is computed from the t of the time evolution for the apparent longitudinal dispersion coecient. In

Re and P e. Under conditions ˜ a (t) is controlled by the Péclet number. When the D ˜ a (t) rises and then stabilizes to a large Reynolds Reynolds number is ncreased, D Fig. 9A we show how it varies as a function of

of low Reynolds number,

number value; indeed, when a suciently high Reynolds number is reached, the recirculation zone ceases to grow further (or at least grows very slowly). The numerical model is only valid when the ow is still stationary, so we never make simulation with Reynolds higher than 250. With this limit, we can observe the plateau only for a few geometries that have high

a

values. For most of them,

only the low Reynolds plateau and the beginning of the rise are visible.

A B

Figure 9: for several

Normalized asymptotic dispersion coecient as a function of

Pe

Re

values (A). B shows the same data once scaled and the inset in

B shows the t of the scaling coecient parameters are

 = 0.47

and

κ

as a function of

P e.

The geometry

a = 0.4.

Ce paragraphe et le suivant doivent être complétésWe try to nd ˜ a (t) and Re and P e. For a given geometry D ˜ a (t) as a function of Re can be gathered and for P e > 50, all the curves of D together by applying a scaling coecient κ (Fig. 9B). This scaling coecient only depends on P e. As seen on the inset of Fig. 9B, the relation between both

an empirical relation between

is logarithmic :

κ = α ln(P e) + β

16

(24)

3.4.2

Global scaling

˜ a∞ (t, P e, Re) = [α ln(P e) + β] f (Re) D or

Da∞ (t, P e, Re) = DT.A. [α ln(P e) + β] f (Re) where

f (Re)

is a function going to asymptotic values at low and high

Re

values. The erf function seems to be a good guess. As the second plateau is never reached for most of the geometries, it's impossible to dene a general t.

A

B

Figure 10:

Fitting parameters

α

(A) and

17

β

(B) as a function of a.

3.5 Breakthrough curves

Figure 11:

x = 50L) = 50 and P e = 500) and for Re = 200) in a geometry with

Breakthrough curves after fty unit cells (i. e., at

for congurations of low and high diusion (P e low and high Reynolds numbers (Re

 = 0.19

and

a = 0.38.

= 10

and

Additionnaly, two breakthrough curves for a parallel

plate geometry with the same mean aperture (i.e.

P e = 500

a = 0)

at

P e = 50

and

are shown.

Breakthrough curve examples are given in Fig. 11. The observation is done at a position where

x = 50L,

well after the dispersion coecient has reached the

asymptotic value. The time of rst arrival depends on the advection and diusion of a particle traveling mainly at the center of the cell where the horizontal ow is the fastest. As recirculation grows, most of the ow occurs in a smaller volume of the cell and thus the maximum velocity increases.

As such, the higher the Reynolds

number, the lower the time needed for the fastest particles to travel through the cells. Lower

Pe

values leads to smaller asymptotic dispersion coecient and thin-

ner breakthrough.

In the smooth geometry case, the asymptotic dispersion

coecient is even smaller.

4

Conclusion

We have studied numerically the impact of inertial eects in channels of periodically varying aperture on ow and transport.

We have investigated the

conditions under wich the recirculation zones appear by looking at the volume fraction of the recirculation zone as a function of the geometry parameters (ie

18

the aspect ratio of a cell and the relative amplitude of the aperture uctuations). We have also characterized the asymptotic dispersion coecient and found an empirical relation using the geometry parameters, the Péclet number and the Reynolds number. For a number of geometries, Reynolds number values as low as 20 are sucient for recirculation zones to appear and signicantly impact transport. In such geometries, the recirculation zone can occupy up to 75% of the pore volume even with Reynolds number value lower than 50.

References [1] R. Aris. On the Dispersion of a Solute in a Fluid Flowing through a Tube.

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 235(1200):6777, April 1956.

[2] Jacob Bear.

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