MATHEMATICAL MODELS FOR MASS AND HEAT
TRANSPORT IN POROUS MEDIA. PART II
Agneta M. BALINT1 and Stefan BALINT2
West University of Timisoara, Timisoara, Romania
Faculty of Physics, 2Faculty of Mathematics- Computer Science
balint@physics.uvt.ro; balint@balint.uvt.ro;
1
ABSTRACT
The presentation is focused on the mathematical modeling of mass and heat
transport processes in porous media. Several mathematical models of macro
transport in porous media are presented and the results with respect to
available experimental information are compared.
Keywords: mathematical modeling; porous media; mass transport; heat
transport; black smoker.
Mass transport in porous media
We present the mass transport in porous media as it is described by Auriault and
Lewandowska in [1].
 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 [2].
 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 

t xi


  Dij  c  vi  c   0


x j



c
 N i  Dij 

x j

86

    c

t

on 
(5.1)
(5.2)
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  .
The coefficient  denotes the adsorption parameter (  > 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
u  0
(5.4)
which has been already presented in the earlier sequence, will be directly used.
 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 * 


Dc  tc t * yi


  Dij *  c *  vc  l  vi * c *  0


y j
Dc



c *   c  l
c *
 N i   Dij * 

 *


y j  Dc  tc
t *

In this way three dimensionless numbers appear:
Pel 
Ql 
Pl 
vc  l
Dc
c l
Dc  t c
the Péclet number
the Damköhler number
l2
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.
87
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.
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:
Rd 
c 0
  * c 0   0

Dij 

c   vi0   0
t xi 
x j  xi


(5.8)
where: – the effective diffusion tensor Dij* is defined as:
1
Dij* 
and the vector field


f
 j

Dik   I kj 
y k


 d


(5.9)
 j is the solution of the standard (representative) Ω cell problem:
j
is periodic
  

1

(5.10)
  d  0
f


 
 D jk  I jk  k

yi 
y j


 

 N i  Dij  I jk  k

y j
 

  0



  0 on .


(5.11)
(5.12)
(5.13)
– the coefficient Rd, called the retardation factor, is defined as
Rd 
88
 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 of soil; as  S p /  .
0
– the effective velocity  vi  is given by the Darcy’s law.
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  c 0  0
xi


(5.16)


(5.17)
  1j d
(5.18)
1
c 0
  ** c 0  
Rd 

Dij 

 vi1  c 0   vi0  c  0
t xi 
x j  xi
where: – the macroscopic dispersion tensor is defined as:
1
D 

**
ij
and the vector field
 1j

f

 1j 
p 0 1


Dik I kj 
d 


yk 
xk 

i1
f
is the solution of the following cell problem:


yi
1 
1
 
 D  I   k    v 0   k  1  v 0  v 0
k
k
 ij  jk y j   i yi 

 

 k1 

 N i  Dij I jk 
 0 on 

y j 

 1j
k
is periodic
(5.19)
(5.20)
(5.21)
89
1
 1  


1
d  0
(5.22)
f
– the coefficient Rd is given by (5.14) or (5.15)
0
– the effective velocity  vi  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).
  Rd 
c

 
t
xi

  c   
 vi    c   O( 2 )
  Dij** 




x

x
j 
i

(5.23)
In this equation the dispersive term as well as the transient term is of the order ε.
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:


 0 0
vi  c  0
yi
(5.24)

c 0 
 N i   Dij 
 0 on 

y j 

(5.25)
Eq. (5.24) rewritten as
vi0 
c o
0
yi
(5.26)
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.
iv) Strong diffusion, advection and adsorption
 
 
 
the case: Pl  O  1 , Ql  O  1 , and Pel  O  0 .
90
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:
Rd 


c 0


 vi0  c 0  0
t xi
(5.27)
where Rd is given by (5.14).
The next order approximation of the macroscopic equation is:
Rd 




  c'    c 0  



    i2    
 v1i    c 0    vio    c1  
t
t
xi
xi
(5.28)
 

  c0 

Dij*** 
0
xi
x j
where the symbol < > Γ means
  i2   
1

 i
2
dS
(5.29)

The local boundary value problem for determining the vector field


 2
 
 Dij   I jk  k

yi 
y j




 2
 N i   Dij   T jk  k

y j


 2 is the following:
 0  k2
   vi 
 vk0 * vk0


y
i

(5.30)

     vk0 *


(5.31)
on 
 i2 is periodic
(5.32)
 2 
1


2
d  0
(5.33)
f
Remark that  2 depends not only on the advection, as it was in the 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.
The tensor D*** is expressed as:
1
Dij*** 
1


f


 2j

 Dik   I kj 
y k





  v0   2  v0      2  
i
j
j
i
  d

Rd



(5.34)
91
and depends on the adsorption coefficient α too. Therefore D*** may be called the dispersionadsorption 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
Rd* 
c

 
t
xi
 ***   c   
 Dij 

 vi    c   O( 2 )

 xi

x
j


(5.35)
where
Rd*  Rd   


    i2  
xi

(5.36)
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.
vi). Large temporal changes
 
 
the case: Pl  O , Ql  O  2 , and Pel  O  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.
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  =0.41 are reported in [3].
If the characteristic length associated with the pore space in the fluid-solid system is
defined (after Whitaker 1972) as:

1 
(6.1)
0.24
 1.6 10 3
150
(6.2)
l  d p 
then, the small homogenization parameter is:

92
l
L

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 exceed
Pel  O( 0 )  O( 1 ) . In terms of the order of magnitude, this condition can be written as:
.
Pel  O( 1 )  0.6 103
(6.3)
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 approaches 0.6103, then the problem
becomes non-homogenizable and the experimental results are limited to the particular sample
examined.
In the case considered by Neung-Won, Bhakta and Cabanell [3], 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 paper [3] exhibit the feature of nonhomogenizability. For this reason the results obtained can not be extended to size conditions.
Bues and Aachib studied in 1991 [4] 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 [4] 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.
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 300 °C.
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
93
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.
 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 [15],[16] are
given by:
 T f
T f 
  k f  T f
 f c f 
uj 
x f
(7.1)
 t


x
j


 s  cs 


Ts
 k s  Ts
t


x  s

(7.2)


where T f , Ts ,  f ,  s , k f , k s , c f , cs and  f , s 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
T f  Ts
kf 
T f
xk
 nk  k s 
x 
Ts
 nk
xk
(7.3)
x
(7.4)
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 , u 2 , 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:
f 
 s  cs
 s
 f c f
T s


t
0
where:– the macroscopic conductivity tensor K is defined as:
94

 u  T 0   K  T 0

(7.5)
K ij 
–
–
1


 j 
dy


 a y   ij  yi

i, j  1, 2, 3
if y   f
1

a  y   k
s
 k m if y   s

the function a(y) is given
(7.6)
by
(7.7)
the functions Ψj belong to HY defined as:


H Y    H 1     periodic and    y  dy  0



(7.8)
and satisfy

 a( y)   j      a( y)  y j

(7.9)

 
The function space H  which appears in relation (7.8) is the Sobolev
 0 space used in [7].
Equation (7.5) is similar to the equation obtained in [8] and for u  0 H is similar to
the eq. presented in [9].
1
Computed conductivity for the heat transport in
porous media tested against experimental results
In [10] the authors 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 [11] and [12].
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 and Acrivos
in [13] give KT =1.46.
As a check Lee et al.[6] have also calculated the effective conductivities with Φ =0.5and
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 [14].
The experimental data for KL show a growth of (Pe) m , where m has been estimated to
be 1.256 by Levec et al. [11] and 1.4 by Green et al. [12]
95
The discrepancy between theory and experiments must be again attributed to the
difference in packing.
To see the effect of k s/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.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
96
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