Transcript Mathematical models for mass and heat transport in porous

Mathematical models for mass
and heat transport in porous
media
Stefan Balint
and
Agneta M.Balint
West University of Timisoara, Romania
Faculty of Mathematics- Computer Science
Faculty of Physics
[email protected]; [email protected]
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• The
presentation
is
focused
on
the
mathematical modeling of mass and heat
transport processes in porous media.
• Basic concepts as porous media, mathematical
models and the role of the model in the
investigation of the real phenomena are
discussed.
• Several mathematical models as ground water
flow, diffusion, adsorption, advection, macro
transport in porous media are presented and
the results with respect to available experimental
information are compared.
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TOPICS:
•
•
•
•
•
•
•
•
MATHEMATICAL MODELING
POROUS MEDIA
FLUID FLOW IN A POROUS MEDIA
GROUNDWATER FLOW
MASS TRANSPORT IN POROUS MEDIA
COMPUTATIONAL
RESULTS
TESTED
AGAINST EXPERIMENTAL RESULTS
HEAT TRANSPORT IN POROUS MEDIA
COMPUTED CONDUCTIVITY FOR THE
HEAT TRANSPORT IN POROUS MEDIA
TESTED
AGAINST
EXPERIMENTAL
RESULTS
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1.MATHEMATICAL MODELING
• Mathematical modeling is a concept that is
difficult to define. It is first of all applied
mathematics or more precisely, in physics
applied mathematics.
• According to A.C. Fowler: Mathematical Models in the
Applied Sciences, Cambridge University Press, 1998 :
• “Since there are no rules, and an understanding
of the “right” way to model there are few texts
that approach the subject in a serious way, one
learns to model by practice, by familiarity with a
wealth of examples.”
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• Applied mathematicians have a procedure, almost a philosophy that
they apply when building models.
• First, there is a phenomenon of interest that one wants to describe
and to explain. Observations of the phenomenon lead, sometimes
after a great deal of effort, to a hypothetical mechanism that can
explain the phenomenon. The purpose of a mathematical model has
to be to give a quantitative description of the mechanism.
• Usually the quantitative description is made in terms of a certain
number of variables (called the model variables) and the
mathematical model is a set of equations concerning the
variables.
• In formulating continuous models, there are three main ways of
presenting equations for the model variables.
• The classical procedure is to formulate exact conservation laws.
The laws of mass, momentum and energy conservation in fluid
mechanics are obvious examples of these.
• The second procedure is to formulate constitutive relations
between variables, which may be based on experiment or empirical
reasoning (Hook law).
• The third procedure is to use “hypothetical laws” based on
quantitative reasoning in the absence of precise rules (LotkaVolterra law).
.
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• The analysis of a mathematical model leads to
results that can be tested against the
observations.
• The model also leads to predictions which, if
verified, lend authenticity to the model
• It is important to realize that all models are
idealizations and are limited in their applicability.
In fact, one usually aims to over-simplify; the
idea is that if the model is basically right, then it
can subsequently be made more complicated,
but the analysis of it is facilitated by having
treated a simpler version first.
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2. POROUS MEDIA
•
•
Example 1. The soil
Soil consists of an aggregation of variously sized mineral particles (MP) and
the “pore spaces” (PS) between the particles. ( MP –green, PS – red)
Figure 1.
•
•
•
When the “pore space” is completely full of water, then the soil is saturated.
When the “pore space” contains both water and air, then the soil is
unsaturated.
In exceptional circumstances, soil can become desiccated. But, usually
there is some water present.
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• Exemple 2. The column of the length 2m, filled with a
quasi uniform quartz sand, of mean diameter 1.425 mm
used by Bues and Aachib in the experiment reported in
“Influence of the heterogeneity of the solutions on the parameters of miscible
displacements in saturated porous medium, Experiments in fluids”, 11, Springer
Verlag, 25-32, (1991).
is a porous media.
•
Definition 1. A porous media is an array of a great
number of variously sized fixed solid particles
possessing the property that the volume concentration of
solids is not small. (Soil).
• Often it is characterized by its porosity Φ (i.e. the pore
volume fraction) and its grain size d. The latter
characterizes the “coarseness” of the medium.
•
Definition 2. A periodic porous media is a porous
media having the property that the fixed solid particles
are identical and the whole media is a periodic system of
cells which are replicas of a standard (representative)
cell (experimental column).
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•
Figure 2. Periodic porous media
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3. FLUID FLOW IN A POROUS MEDIA
• An incompressible viscous fluid moves in the “pore
space” of a porous media, according to the NavierStokes equations:
•
(3.1)
u
p
t
 
 u u  

   u  f
• satisfying the incompressibility condition:
•
u  0
(3.2)
• where: u  ut, x is the fluid flow velocity; p is the pressure,
ρ is the fluid density, μ is the fluid viscosity and f is the
density of the volume force acting in the fluid.
• It is important to realize, that eqs. (3.1), (3.2) are valid
only in the “pore space” (x belongs to the “pore space”).
They can be obtained from the mass and momentum
conservation
laws
and
constitutive
relations
characterizing viscous fluids.
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• On the boundary of the fixed solid particles the fluid flow velocity has
to satisfy the non slip condition:
•
(3.3)
u0
• Unfortunately, the boundary value problem (3.1), (3.2), (3.3) even in
the case of a very slow stationary flow (Stokes flow), can not be
solved numerically in a real situation due to the great number of the
boundaries of fixed solid particles.
• Consequently, the mathematical model defined by the eqs. (3.1),
(3.2), (3.3) can not be analyzed numerically in a real case and can
not be tested against the observations.
• Several models for flow through porous media are based on a
periodic array of spheres.
•
Hasimoto H. in On the periodic fundamental solution of the Stokes equations and
their application to viscous flow passed a cubic array of spheres”, J.Fluid Mech.,5, 317-328
(1959).
obtained the periodic fundamental solution to the Stokes problem by
Fourier series expansion, and applied the results analytically to a
dilute array of uniform spheres.
•
Sangani A.S., Acrivos A. in “Slow flow through a periodic array of spheres”,
Int.J. Multiphase Flow, 8(4), 343-360 (1982)
extended the approximation of Hasimoto to calculate the drag force
for higher concentration.
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•
•
•
•
•
- Zick A.A. and Homsy G.M. in “Stokes flow through periodic arrays of spheres”, J.Fluid
Mech.,115, 13-26 (1982). use Hasimoto’s fundamental solution to formulate an
integral equation for the force distribution on an array of spheres for
arbitrary concentration. By numerical solution of the integral equation,
results for packed spheres were obtained, for several porosity values.
- Continuous variation of porosity was examined only when the
particles are in suspension.
- Strictly numerical computations have been made earlier, based
on series of trial functions and the Galerkin method, for cubic packing of
spheres in contact :
Snyder L.J. and Stewart W.A. “Velocity and pressure profiles for Newtonian creeping flow in
regular packed beds of spheres”, A.I.Ch.J.,12(1), 167-173 (1966).
Sorensen J.P. and Stewart W.E. “Computation of forced convection in slow flow through
ducts and packed beds. II. Velocity profile in a simple cubic array of spheres, Chem.Engng.Sci., 29,
819-825 (1974).
• A general model for the flow through periodic porous media has been
advanced by Brenner in an unpublished manuscript cited in
• Adler P.M. Porous Media: Geometry and Transports, Butterworth-Heinemann. London (1992)
•
Brenner H. “Dispersion resulting from flow through spatially periodic porous media”,
Phil.Trans.R.Soc. London, 297 A, 81-133 (1980).
•
In fact, Brenner showed how Darcy’s experimental law and the
permeability tensor can in principle, be computed from a canonical
boundary value problem in a standard (representative) cell.
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• In the following we will present briefly this model.
• Consider a periodic porous media which is a union of cells (cubes)
of dimension l which are replicas of a standard (representative) cell.
Let P0 the characteristic variation of the global pressure P* which
may vary significantly over the global size L of the porous media.
Thus the global pressure gradient is of order O(P0 /L). Let the two
size scales be in sharp contrast, so that their ratio is a small
parameter ε = l/L<<1. Limiting to creeping flows, the local gradient
must be comparable to the viscous sheers so that the local velocity
is U=O(P0 ·l 2/μ·L), where μ is the viscosity of the fluid. Denoting
physical and dimensionless variables respectively by symbols with
and without asterisks, the following normalization may be introduced
in the Navier-Stokes equations (3.1), (3.2):
•
(3.4)
xi   l  xi , P  P0  P, ui   U  ui
• with i = 1,2,3.
• Two dimensionless parameters would then appear: the length ratio
ε = l/L and the Reynolds number:
•
(3.5)
 U  l
  Po  l 2 
Re 



2
L
• which will be assumed to be of order O(ε).
• By introducing fast and slow variables, xi and Xi = ε · xi and multiplescale expansions, it is then found that the leading order p(0) pore
pressure depends only on the global scale (slow variables), p(0) =
p0(Xi).
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• By expressing the solution for ui0  , p1 in the following form:
•
p( 0)
0 
ui  kij 
•
•
where p
•
S j
(1)
X i 
xi
1
p0 
1
p  S j 
p
X j
(3.6)
(3.7)
depends on Xi only, the coefficients kij(xi, Xi) and
Sj(xi, Xj) are found to be governed by the following canonical Stokes
problem in the standard (representative) cell Ω:
• kij
in Ω
(3.8)
0
xi
xi
 2 kij   ij
in Ω
with
• kij= 0 on Γ
• kij, Sj are periodic on ∂Ω
(3.9)
(3.10)
(3.11)
• Here Γ and ∂Ω are respectively the fluid-solid interface and the
boundary of the standard cell.
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• Equations (3.8)-(3.11) constitute the first cell problem. For a
chosen granular geometry, the numerical solution of (3.8)-(3.10)
replaced in (3.6), (3.7) gives the local velocity and pressure
fluctuation in terms of the global pressure gradient p0 / xi
• Let the volume average over the standard cell be defined by:
1
•
(3.12)
 f 
f  d
 f
• where Ωf is the fluid volume in the cell.
• Then the average of eq.(3.6) gives the law of Darcy :
•
p0 
0
 ui    kij  
x j
i  1,2,3;
j  1,2,3
(3.13)
• where < kij > is the so called hydraulic conductivity tensor, which is
the permeability tensor < Kij > divided by μ.
• For later use, we note that in physical variables (marked by *) the
symmetric hydraulic conductivity tensor is given by:
•
l2
(3.14)
 kij *  kij  

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• Comments:
• 1.The Darcy’s law (3.13) gives the global flow field in the periodic
porous media in function of the global pressure field acting on the
media. It is important to realize that this field exists not only in the “pore
space”, but everywhere in the media, i.e. also in the space occupied by
the solid fixed particles. The answer to the question : What represents
this flow in the space occupied by the solid and fixed particles? – can
be found in
• Tartar L. Incompressible Fluid Flow in a Porous Medium. Convergence of the
Homogenization Process in Non-Homogeneous Media and Vibration Theory; Lecture
Notes in Physics, Vol.127, 368-377 Springer Verlag, Berlin 1980.
where it is shown that for tending to zero, the flow field in the “pore
space’ prolonged by zero in the space occupied by the solid and fixed
particles tends to the global flow field given by the Darcy’s law (3.13).
• The Darcy’s law is written in the form:
•
u  k  p
(3.15)
• and it is shown that the flow is incompressible: i.e. u  0 .
• Therefore, if the hydraulic conductivity tensor is constant (constant
permeability), then we have:
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•
2 p  0
(3.16)
• 2. The particularities of the porous media: porosity, shape of the solid
and fixed particles are incorporated in the permeability tensor < Kij >.
Numerical results for permeability were obtained by
• Lee C.K., Sun C.C., Mei C.C.
“Computation of permeability and dispersivities of
solute or heat in a periodic porous media”Int.J.Heat Mass Transfer,39,4 661-675 (1996)
• The computed values for the Wigner-Seitz grain (grain is shaped as a
diamond) are compared with those given by the empirical KozenyCarman formula:
2
3
•
•
•
•
 k 
1


5 1   2
 V 
  s 
 As  l 
(3.17)
which is an extrapolation of measured data. Within the range of porosities
0.37< Φ < 0.68 the computed results are consistent and in trend with. Outside
this range of porosities the deviation increases.
The computed results for uniform spheres of various packing agree
remarkably well with those obtained by Zick and Homsy, when the porosity is
high.
3. The method, used for the deduction of the new model (eqs.(3-15), (3-16)) of
the fluid flow in a porous media is called the method of homogenization.
Basically, the two phase non homogeneous media is substituted by a
homogeneous “fluid”, which flow is not anymore governed by the NavierStokes equation.
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4. GROUNDWATER FLOW
• The groundwater flow is one that has immense practical importance
in the day-to-day management of reservoirs, flood prediction,
description of water table fluctuation.
• Although there are numerous complicating effects of soil physics
and chemistry that can be important in certain cases, the
groundwater flow is conceptually easy to understand.
• Groundwater is water that lies below the surface of the Earth. Below
a piezometric (constant pressure) surface called the “water table”,
the soil is saturated, i.e. the “pore space” is completely full of water.
p surface, the soil is unsaturated, and the “pore space”
Above this
contains both water and air.
• Following precipitation, water infiltrates the subsoil and causes a
local rise in the water table. The excess hydrostatic pressure thus
produced, leads to groundwater flow.
• The flow satisfies the Darcy’s law presented above :
k
•
(4.1)
u   p

p is the pressure gradient in the groundwater and satisfies:
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2 p  0
•
(4.2)
• which is the incompressibility condition in the case of groundwater flow;
• k is the permeability tensor for simplicity has the form:
kij  k  ij
•
(4.3)
• with k > 0. The constant k is called permeability too and has the
dimension of (length)2.
• Typical value of the permeability of several common rock and soil types
Material
K
[m2]
Material
K
[m2]
Gravel
10-8
Sandstone
10-13
Sand
10-10
Silt
10-14
Fractured igneous rock
10-12
Clay
10-18
• Eqs. (4.1) (4.2) define the simplest model of the incompressible
groundwater flow through a rigid porous medium.
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• Consider now the problem of determining the rate of leakage
through an earth fill dam built on an impermeable foundation. The
configuration is as shown in Fig.3 where we have illustrated the
(unrealistic) case of a dam with vertical walls; in reality the cross
section would be trapezoidal.
• Figure 3. Geometry of dam seepage problem
• A reservoir of height h0 abuts a dam of width L. Water flows through
the dam between the base y = 0 and a free surface (called phreatic
surface) y = h, below which the dam is saturated and above which it
is unsaturated. We assume that this free surface provides an upper
limit to the region of groundwater flow.
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• We therefore neglect the flow in the unsaturated region, and the free
boundary must be determined by a kinematics boundary condition,
which expresses the idea that the free surface is defined by the fluid
elements that constitute it, so that the fluid velocity at y = h is the
same as the velocity of the interface itself:
d
h  y   0 , d     u    

dt
dt t 
•
(4.4)
• where d/dt is the material derivative for the fluid flow.
• In the two-dimensional configuration, shown in Fig.3, we therefore

have to solve:
k p
k  p
u 
v        g 
•
(4.5)
 x
  y

•
•
•
•
•
•
where
u  u, v 
u v

0
x y
with boundary conditions that :
(4.6)
v  0 on y  0
p  0, v   
h
h
u
on y  h
t
x
(4.7)
p    g  h0  y  on x  0
p  0 on x  L
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• These conditions describe the impermeable base at y=0, the free
surface at y = h, hydrostatic pressure on x = 0 and atmospheric
pressure at x = L (the seepage face). The free boundary is to be
determined as part of the solution.
• In order to solve the problem (4.5), (4.6), (4.7) we
nondimensionalize the variables by scaling as follows:
•
(4.8)
k    g  h0
x  L, y  h0 , p    g  h0 , u 
,
L
  h0  
v kg , t 

kg
• all for obtain various obvious balances in the equations and
boundary conditions. The Dupuit-Forchheimer approximate solution
is obtained when h0 << L.
h
• In this case we define   0 L and the equations become:
u
p
x
 p

v    L 
 y

2
2 p

p
2



0
2
2
y
x
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• with:
•
p
 1 on y  0
y
(4.7’)
h  p 
p h
   1   2  
on y  h
t  y 
x x
p  1  y on x  0
p  0,
p0
on x  1
• Since we proceed by expanding
p  p0   2  p1  ...etc.
The leading order approximation for p is just
p h y
•
(4.9)
• This fails to satisfy the condition at x = 1, where the boundary layer is
necessary to bring back the x derivatives of p, unless there is no
seepage face, that is h(L) = 0. p
h
2



1

O

 O 2  , which
• However, we also note that if
, then
y
t
suggests that
•
h  h0   2  h1  ... and h0  constant
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 
• Alternatively, we realize that h t  O  2
simply indicates that the
timescale of relevance to transient problems is longer than our initial
 , so that we rescale t with
guess O   h0  
k   g


1

2
.
Putting t  t  2
(and subsequently omitting the over bar) we rewrite
the kinematical boundary condition as:
 p 
•
(4.10)
2 h
2 p h
•
 
   1    
t
x x
 y 
• Now we seek expansions
p  p0
•
  2  p1  ....
h  h0  ....
• and we find successively:
p0  h0  y
•
• and
 2 p1
 h0 xx ,
•
2
y
(4.11)
(4.12)
p1
 0 on y  0
y
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(4.13)
24
• whence
•
• so that eq. (4.10) gives:
•
•
•
•
•
•
•
•
•
•
p1
 2 ho
 y  2
y
x
(4.14)
h0
 2h0  h0 
 h0  2  

t
x
 x 
(4.15)
dropping the subscript, we obtain the nonlinear diffusion equation:
h   h 
 h  
(4.16)
t x  x 
Notice, that this equation is not valid to x = 1, because we require p = 0
at x = 1, in contradiction to eq. (4.12). We therefore expect a boundary
layer there, where p changes rapidly.
Eq. (4.16) is a second order equation, requiring two boundary
conditions. One is that:
h  1 on x  1
(4.17)
but it is not so clear what the other is. It can be determined by means of
h
the following trick.
U   p  dy
Define:
(4.18)
and note that the flux q is given by 0
h
h
p
U
(4.19)
2
q   u  dy  
0
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0
x
 dy  
x
25
• Furthermore
1
U

at x  0 and U  0 at x  1
•
2
• and therefore we have the exact result
1
•
q  dx  1

0
(4.20)
(4.21)
2
• In a steady state, ht  qx  0, so q is constant, and therefore
•
(4.22)
h
q  1  h 
2
x
• The steady solution (away from x = 1) is therefore
1
•
(4.23)
h  1  x  2
• And there is (to leading order) no seepage face at x = 1.
• In fact, the derivation of eq. (4.22) applies for unsteady problem also. If
we suppose that q does not jump rapidly near x = 1, then we can use
Dupuit-Forchheimer approximation q  h  hx in eq. (4.21) and an
integration yields:
h  0 at x  1
•
(4.24)
• as the general condition.
• The boundary layer structure near x = 1 can be described as follows:
• near x = 1 we have h  1  x 12 , p  h  y and so we put
Summer University, Vrnjacka
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26
•
x 1   X , h  
1
2
 H,
p 
1
2
Y,
p 
1
2
P
• and we choose
2



•
• to bring back the x derivatives in Laplace’s equation, we get
•
2P 2P
x 2

y 2
0
(4.25)
(4.26)
(4.27)
• with:
•
P
(4.28)
 1 on Y  0
y
P P H
P  0,


  L on Y  H
y x x
P  H Y, H  X 1
as x  
2
P  0 on X  0
• Exact solutions of this problem can be found using complex
variables, but for many purpose the D-F approximation is sufficient,
together with a consistently scaled boundary layer problem.
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27
5. MASS TRANSPORT IN POROUS
MEDIA
•
We present the mass transport in porous media as it is described by
•
Auriault I.L. and Lewandowska J. in “Diffusion, adsorption, advection,
macrotransport in soils”, Eur.J.Mech. A/Solids 15,4, 681-704, 1996.
•
•
The pollutant transport in soils can be studied by means of a model in which the
real heterogeneous medium is replaced by the macroscopic equivalent (effective
continuum) like in the case of the fluid flow. The advantage of this approach is
the “elimination” of the microscopic scale (the pore scale), over which the
variables such as velocity or the concentration are measured.
In order to develop the macroscopic model the homogenization technique of
periodic media may be employed. Although the assumption of the periodic
structure of the soil is not realistic in many practical applications, it was found
reasonably model to real situations. It can be stated that this assumption is
equivalent to the existence of an elementary representative volume in a non
periodic medium, containing a large number of heterogeneities. Both cases lead
to identical macroscopic models as presented in:
•
Auriault I.L., “Heterogeneous medium, Is an equivalent macroscopic description
possible?” Int.J.Engn,Sci.,29,7,785-795, 1995.
Summer University, Vrnjacka
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28
• The physical processes of molecular diffusion with advection in pore
space and adsorption of the pollutant on the fixed solid particles surface
can be described by the following mass balance equation:

c  
c
•
(5.1)

 Dij 
 vi  c   0

t xi 
x j

•
(5.2)



c

c
  
 N i  Dij 
on 



x j 
t
• where c is the concentration (mass of pollutant per unit volume of fluid),
Dij is the molecular diffusion tensor, t is the time variable v is the flow
field and N is the unit vector normal to Γ. The coefficient α denotes the
adsorption parameter (α > 0). For simplicity it is assumed that the
adsorption is instantaneous, reversible and linear.
• The advective motion (the flow) is independent of the diffusion and
adsorption. Therefore the flow model (Darcy’s law and the
incompressibility condition)
•
(5.3)
u  k  p
•
(5.4)
u  0
• which has been already presented in the earlier sequence, will be
directly used.
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• The derivation of the macroscopic model is accomplished by the
application of homogenization method using the double scale asymptotic
developments. In the process of homogenization all the variables are
normalized with respect to the characteristic length l of the periodic cell.
The representation of all the dimensional variables, appearing in eqs.
(5.1) and (5.2) versus the non-dimensional variables is
c  cc  c *; X  l  y ; t  tc  t *; v  vc  v *;
D  Dc  D *;    c   *
• where the subscript “c” means the characteristic quantity (constant) and
the superscript “*” denotes the non-dimensional variable.
• Introducing the above set of variables into eqs. (5.1)-(5.2) we get the
following dimensionless equations:

l2
c *  
c * vc  l


 Dij * 

 vi * c *  0


Dc  tc t * yi 
y j
Dc


c *   c  l
c *

 N i  Dij * 

 *

y j  Dc  tc
t *

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30
• In this way three dimensionless numbers appear:
v l
•
the Péclet number
Pel  c
Dc
•
•
•
•
•
•
Ql 
c  l
Dc  tc
the Damköhler number
l2
Pl 
Dc  tc
The Péclet number measures the convection/diffusion ratio in the pores.
The Damköhler number is the adsorption/diffusion ratio at the pore
surface.
Pl represents the time gradient of concentration in relation to diffusion in
the pores.
In practice, Pel and Ql are commonly used to characterize the regime of
a particular problem under consideration.
In the homogenization process their order of magnitude must be
evaluated with respect to the powers of the small parameter   l L .
• Each combination of the orders of magnitude of the parameters Ql , Pl ,
and Pel corresponds to a phenomenon dominating the processes that
take place at micro scale and different regime governing migration at the
macroscopic scale.
Summer University, Vrnjacka
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31
i). Moderate diffusion, advection and adsorption
– the case of:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  1 .
• The process of homogenization leads to the traditional
phenomenological dispersion equation for an adsorptive solute:
c 0
  * c 0   0
•
(5.8)

Rd 

Dij 

c   vi0    0
t xi 
x j  xi
• where: -the effective diffusion tensor Dij* is defined as:
 j 

1
*
•
(5.9)


Dij 
D

I

d

ik
kj

yk 
 f

• and the vector field  j is the solution of the standard (representative)
Ω cell problem:
 j is periodic
•
(5.10)
1
•
(5.11)
  
  d  0

•
•
f
 
 k 

D
I

 jk  jk
0

y j 
 
 



k
  0 on .
 Ni  Dij  I jk 

y j 
 
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

yi
Banja, October 2007
(5.12)
(5.13)
32
• -the coefficient Rd, called the retardation factor, is
defined as
•
•
•
•
•
•
•
•
Rd 
 f   Sp

(5.14)
with  = the total volume of the periodic cell
 f = the volume of the fluid in the cell
Sp = the surface of the solid in the cell
In terms of soil mechanics
Rd      as
(5.15)
with Φ = the porosity
as = the specific surface of the porous medium
defined as the global surface of grains in a unit volume
as.  S p / 
of soil;
0
• -the effective velocity  vi  is given by the Darcy’s law.
Summer University, Vrnjacka
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33
ii) Moderate diffusion and adsorption, strong advection
• the case:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  0 .
• The process of homogenization leads to two macroscopic governing
equations that give succeeding order of approximations of real pollutant

behavior.
 vi0  c0  0
xi
•
(5.16)
0
0
1
c
  ** c  
Rd 

Dij 

 vi1  c 0   vi0  c  0
t xi 
x j  xi
•
(5.17)
• where: - the macroscopic dispersion tensor is defined as:
1
0




1

p
1
j
•
(5.18)
**
d 
D 
D I 

k   1 d




ij
•
•
•


f
ik


and the vector field 
kj
1
j
yk 
xk


i1
j
f
is the solution of the following cell problem:
  
 k1   0  k1 1
(5.19)

Dij I jk 
 vi 
  vk0  vk0
yi  
y j  
yi 



1 



 N i  Dij  I jk  k   0 on 

y j 

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(5.20)
34
•
•
•
•
•
 1j is periodic
 1  
1

(5.21)
1

 d  0
(5.22)
f
-the coefficient Rd is given by (5.14) or (5.15)
-the effective velocity  vi0  is given by the Darcy’s law.
• In order to derive the differential equation governing the
average concentration < c >, equation (5.16) is added to
equation (5.17) multiplied by ε. after transformations the
final form of the dispersion equation is obtained that gives
the macroscopic model approximation within an error of
O(ε2).
 **   c   


c


•

 vi    c   O( 2 ) (5.23)
  Rd 
 
  Dij 
•
t
xi 
x j  xi
• In this equation the dispersive term as well as the transient
term is of the order ε.
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35
iii) Very strong advection
• the case:
 
 
 
Pl  O  2 , Ql  O  2 , and Pel  O  1 .
• The process of homogenization applied to this problem leads to the
following formulation obtained at ε -1 order:
•
(5.24)
 0 0

v
y
i
•
•

c  0
i
0 


c
  0 on 
 N i   Dij 

y j 

Eq. (5.24) rewritten as
(5.25)
o

c
vi0 
0
•
(5.26)
yi
• shows that there is no gradient of concentration c0 along the
streamlines. This means that the concentration in the bulk of the
porous medium depends directly on its value on the external
boundary V of the medium. Therefore, the rigorous macroscopic
description, that would be intrinsic to the porous medium and the
phenomena considered, does not exist. Hence, the problem can not
be homogenized. This particular case will be illustrated when
analyzing the experimental data.
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36
iv) Strong diffusion, advection and
adsorption
 
 
 
• the case: Pl  O  , Ql  O  , and Pel  O  .
•
The homogenization procedure applied to this problem gives
for the first order approximation the macroscopic governing equation
which does not contain the diffusive term. Indeed, it consists of the
transient term related to the microscopic transient term as well as
the adsorption and the advection
terms:
0
c

•
(5.27)
R 

 v0  c0  0
1
d
1
t
0

x
i

i
• where Rd is given by (5.14).
•
The next order approximation of the macroscopic equation is:
  c'    c 0  
 1
2 
0
o
1
R










v



c



v



c

i

i
i
• d
t
t xi
xi




(5.28)
 ***   c 0 
 Dij 
0
xi
x j
 
Summer University, Vrnjacka
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37
• where the symbol < > Γ means
1
2
•
  i  

2

 i dS

(5.29)
2
• The local boundary value problem for determining the vector field
is the following:
2 
2

 




0
k
k
  vi 

 vk0 * vk0
 Dij   I jk 
yi 
y j 
yi
•
(5.30)

•
•
2 




 Ni   Dij   T jk  k      vk0 * on 

y j 


i2
is periodic
 2 
1
2

d  0

 f
(5.31)
(5.32)
•
(5.33)
2
• Remark that  depends not only on the advection, as it was in the
1
case of  , but also on the adsorption phenomenon. Moreover, in
this case the pollutant is transported with the velocity <v*> equal to
the effective fluid velocity divided by the retardation factor.
•
Summer University, Vrnjacka
38
Banja, October 2007
• The tensor D*** is expressed as:
2





•
1

j
***


Dij 


f



v



v





Dik   I kj 
  d


y
R


k 
d



0
i
2
j
0
j

2
i
(5.34)
• and depends on the adsorption coefficient α too. Therefore D*** may
be called the dispersion-adsorption coefficient.
• Remark that the second term in (5.27) represents the additional
adsorption contribution defined as the interaction between the
temporal changes of the averaged concentration field < c0> and the
surface integral of the macroscopic vector field <  2 >Γ .
• Finally, the equation governing the averaged concentration < c > can
be found by adding eq.(5.27) to eq.(5.28) multiplied by ε.
•
c
  ***   c   

R 
 
Dij 

 vi    c    O( 2 )
t
xi 
x j  xi
*
d
(5.35)

• where
*
2
R

R







d
d
i 
•
(5.36)
xi
• If eq. (5.35) is compared with eq.(5.23) it can be concluded that the
increase by one in the order of magnitude of parameters Pl and Ql
causes that the transient term Rd*    c  t  in the macroscopic
equation becomes of the order one.

Summer University, Vrnjacka
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
39
v). Large temporal changes
• the case:
 
 
Pl  O , Ql  O  , and Pel  O  .
2
1
• This is also a non-homogenizable case and in this case the rigorous
macroscopic description, that would be intrinsic to the porous
medium and the phenomena considered, does not exist.
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40
6. COMPUTATIONAL RESULTS TESTED
AGAINST EXPERIMENTAL RESULTS
• Experimental results obtained when the sample length is L=150 cm,
the solid particle diameter is dp=0.35 cm and the porosity Φ=0.41
are reported in:
•
Auriault J.L., “Heterogeneous medium, Is an equivalent macroscopic description possible?”
Int.J.Engn,Sci.,29,7,785-795, 1995.
• If the characteristic length associated with the pore space in the
fluid-solid system is defined (after Whitaker 1972) as:
•
(6.1)

l  d p 
1 
• then, the small homogenization parameter is:
l 0.24
•
(6.2)
 
 1.6  103
L 150
• According to the theoretical analysis presented in sequence 5, a
rigorous macroscopic model exists if the Péclet number, which
characterizes the flow regime, does not exceeds Pel  O( 0 )  O( 1 )
• In terms of the order of magnitude, this condition can be written as:
Pel  O( 1 )  0.6 103
Summer University, Vrnjacka
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(6.3)
41
•
Therefore a dispersion test through a sample of the length
L=150 cm (dp=0.35 cm) is “correct” from the point of view of the
homogenization approach, provided the maximum Péclet number is
much less than 0.6 103 . If the Péclet number approaches0.6 103 ,
then the problem becomes non-homogenizable and the
experimental results are limited to the particular sample examined.
• In the case considered by
•
Neung -Wou H., Bhakta J, Carbonell R.G. “Longitudinal and lateral
dispersion in packed beds; effect of column length and particle size
distribution”, AICHE Journal, 31,2,277-288 (1985)
• the range of the Péclet number was 102 -104 which is practically
beyond the range of the homogenizability.
•
In order to make the problem homogenizable, the flow regime
should be changed, namely the Péclet number should be
decreased. If however, we want the Péclet number to be, for
example Pe = 103, then the sample length L should be greater than
240 cm. Moreover, almost all the previous experimental
measurements quoted in the above paper exhibit the feature of nonhomogenizability. For this reason the results obtained can not be
extended to size conditions.
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42
•
•
Bues M.A. and Aachib M. studied in 1991 in the paper
“Influence of the heterogeneity of the solutions on the parameters of
miscible displacements in saturated porous medium, Experiments in fluids”,
11, Springer Verlag, 25-32, (1991).
•
the dispersion coefficient in a column of length 2 m, filled with a quasi
uniform quartz sand of mean diameter 1.425 mm. The investigated range of
the local Péclet number was 102-104. Concentrations were measured at
intervals of 20 cm along the length of the column. The corresponding
parameter ε (ratio of the mean grain diameter to the position x) for each
position was: 1.36·10-2; 4.67·10-3; 2.8·10-3 ; 2.02·10-3; 1.57·10-3 ; 1.29·10-3 ;
1.09·10-3 ; 1.01·10-3 ; 8.9·10-4 ; 7.9·10-4 respectively. The order of
magnitude O(ε-1) corresponds to 75; 214; 357; 495; 636; 775; 917; 990;
1123; 1266 respectively.
It can be seen that the condition Pel <<O(ε-1) is roughly fulfilled at the end
of the column when the flow regime is Pel = 240.
The experimental data presented in the above paper show the asymptotic
behavior of the dispersion coefficient that reaches its constant value for:
x = 180.5 cm.
Thus, one can conclude that the required sample length for the
determination of the dispersion parameter in this sand at Pel = 200 is at
least 2 m.
•
•
•
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43
7. HEAT TRANSPORT IN POROUS
MEDIA
• An interesting example of heat transport in porous
media by convection and conduction represents the
relatively recent discovered “black smokers” on the
ocean floor. They are observed at mid-ocean ridges,
where upwelling in the mantle below leads to the
partial melting of rock and the existence of magma
chambers. The rock between this chambers and the
ocean floor is extensively fractured, permeated by
seawater, and strongly heated by magma below.
Consequently, a thermal convection occurs, and the
water passing nearest to the magma chamber
dissolves sulphides and other minerals with ease,
hence the often black color. The upwelling water is
concentrated into fracture zones, where it rises
rapidly. Measured temperatures of the ejected fluids
are up to 3000C.
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44
Summer University, Vrnjacka
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45
•
Another striking example of heat
transport in porous media is offered by
geysers, such as those in Yellowstone
National
Park.
Here
meteoric
groundwater is heated by subterranean
magma chamber, leading to thermal
convection concentrated on the way up
into fissures. The ocean hydrostatic
pressure
prevents
boiling
from
occurring, but this is not the case for
geysers, and boiling of water causes
the periodic eruption of steam and
water that is familiar to tourists.
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•
FAMOUS GEYSERS
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• The heat transport in soil can be studied by means of
a model in which the real heterogeneous medium is
replaced by the macroscopic equivalent (effective
continuum) like in the case of mass transport.
• In order to develop the macroscopic model the
homogenization technique of periodic media may be
employed. It can be shown that the assumption of
“periodic media” is equivalent to the existence of an
elementary representative volume in a non periodic
medium, containing a large number of heterogeneities.
• The starting basic equations for diffusion and
convection of heat according to
•
Mei C.C. “Heat dispersion in porous media by homogenization method, Multiphase
Transport in Porous Media”, ASME Winter Meeting, FED vol.122/HTD vol.186 11-16
(1991).
Lee C.K., Sun C.C., Mei C.C. “Computation of permeability and dispersivities of
solute or heat in periodic porous media” Int.J.Heat and Mass Transfer 19,4 p.661-675
(1996).
• are given by:
•
•
 T
T 
 f  c f   f  u j  f   k f  T f
x j 
 t
 s  cs 
Ts
 k s  Ts
t
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xf
x  s
(7.1)
(7.2)
52
•
•
•
•
•
•
•
•







Tf ,Ts ,  f , s , k f , ks , c f , cs and  f , s
where
denote
respectively
the
temperatures,
densities,
thermal
conductivities, specific heats and partial volumes of the fluid
and solid in the Ω standard (representative) cell. On the solid
fluid interface Γ, the temperatures and heat flux must be
continuous
Tf  Ts
x 
(7.3)
T f
Ts
(7.4)
 nk
x
xk
xk
where nk represent the components of the unit normal vector
pointing out of the fluid. In eqs. (7.1) and (7.2) energy
dissipation by viscous stress has been neglected, which is
justifiable for low Reynolds numbers.
It was assumed that the flow is independent of the
temperature. Therefore, u  u1, u2 , u3  in eq. (7.1) represents the
Darcy’s flow.
The derivation of the macroscopic model is accomplished by
the application of homogenization method.
The macroscopic model is defined by the equation:
kf 
f 
 nk  ks 
 s  cs
 s
 f  cf
T s


t
0

 u  T   K  T
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0
0

(7.5)
53
• where:-the macroscopic conductivity tensor K is:
 j 

1
•
dy
K ij 
a y     ij 
i, j  1,2,3


y
 
i 

•
-the function a(y) is given by
•
if y  
1

a y   k
s

 km
(7.6)
(7.7)
f
if
y  s
-the functions Ψj belong to HY defined as:


(7.8)
1
HY    H    periodicand   y dy  0



• and satisfy
(7.9)

 a( y)   j     a( y)  y j
• The function space which appears in relation (7.8) is the
Sobolev space used in :
•
•
•
Sanchez-Palencia E., Lecture Notes on Physics.,vol.127, Springer, Berlin, 1980
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.
54
• Equation (7.5) is similar to the equation obtained in:
•
Prasad V., Convective Heat and Mass Transfer in Porous Media, Kluwer Academic
Publishers, Dodrecht, 1991, p.563
• and for
•
0
u  0 is similar to the eq. presented in:
.,
Mei C.C., Auriault J.L
Ng C.O. Advances in Applied Mechanics vol.32,
Academic Press, New York, 1996 p.309.
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8. COMPUTED CONDUCTIVITY FOR THE HEAT
TRANSPORT IN POROUS MEDIA TESTED
AGAINST EXPERIMENTAL RESULTS
•
Lee C.C., Sun C.C. and Mei C.C. “Computation of permeability and dispersivities of solute or
heat in periodic porous media” Int.J.Heat Mass Transfer col.39,4 p.661-676 (1996).
•
compute and compare conductivity for heat transport in porous media with
experimental results. In the following we will present these results.
• With the mean flow directed along the x-axis, the longitudinal and
transverse conductivities KL and KT for heat were computed for
Péclet numbers Pe up to 300 for two porosities Φ = 0.38 and Φ =
0.5; the thermal properties for fluid and solid phases were assumed
to be equal kf = ks and ρs · cs= ρf · cf. They were compared with
some experimental results for randomly packed uniform glass
spheres in water with roughly comparable thermal properties
reported in:
•
•
Levec J. and Carbonell R.G. “Longitudinal and lateral thermal dispersion in packed beds. II.
Comparison between theory and experiment” A.I.Ch.E.J. 31, 591-602 (1985)
Green D.W., Perry R.H. and Babcock R.E. “Longitudinal dispersion of thermal energy through
porous media with a flowing fluid” A.I.Ch.E.J. 10,5, 645-651 (1960).
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• In the limit Pe = 0, both (KL, KT) approach unity because the
composite medium is homogeneous and there is no distinction
between Ωf and Ωs for pure diffusion.
• For a simple cubic packing of spheres with Φ = 0.48 and ks= kf = 2:
•
Sangani A.S. and Acrivos A. “The effective conductivity of a periodic array of
spheres” Proc.R.Soc. Lond. A.386, 262-275 (1983)
• give KT =1.46.
• As a check Lee et al. have also calculated the effective
conductivities with Φ = 0.5 and the same ratio of conductivities ks, kf
and obtain KT =1.458. The small discrepancy is again due to
different grain geometries.
• Computation in the relatively high Pe region show that the
dispersivities KL, KT increase with decreasing porosity as in the case
of passive solute. This is again due to increased micro scale mixing
in the pore space caused by increased velocity gradient for smaller
porosity value. The same trend has been observed for 2D array of
cylinders in:
•
Sahrani M. and Kavary M., “Slip and no slip temperature boundary conditions at the
interface of porous media: convection”, Int.J.Heat Mass Transfer 37, 1029-1044
(1994)
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•
The experimental data for KL show a growth of (Pe)
estimated to be 1.256 by:
•
Levec J. and Carbonell R.G. “Longitudinal and lateral thermal dispersion in packed
beds. II. Comparison between theory and experiment” A.I.Ch.E.J. 31, 591-602 (1985)
•
and 1.4 by:
•
Green D.W., Perry R.H. and Babcock R.E. “Longitudinal dispersion of thermal
energy through porous media with a flowing fluid” A.I.Ch.E.J. 10,5, 645-651 (1960).
•
The discrepancy between theory and experiments must be again attributed
to the difference in packing.
•
To see the effect of ks/kf, were calculated KL and KT for two porosities:
Φ = 0.38 and Φ =0.5 and two conductivity ratios, ks/kf = 0 and 1. At the
higher Péclet number, the longitudinal conductivity KL is greater, although
the difference is small. This increase is due to heat diffusion in the solid
phase. When the thermal gradient is in the direction of the mean flow,
diffusion through the solid phase augments dispersion Kxx in the fluid when
ks/kf ≠ 0. But for Kyy which is associated with the thermal gradient normal to
the flow, transverse dispersion is weakened by the loss of heat into solid.
Quantitatively, the effect of ks/kf=1 on either KL and KT appears to be
significant only at relatively low Péclet number. This result is reasonable
since for high Pe dispersion by convection must be dominated and diffusion
in the solid must become immaterial.
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m
, where m has been
58
Thank you for
your attention
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