MATHEMATICAL MODELS FOR MASS AND HEAT
TRANSPORT IN POROUS MEDIA PART I
Stefan BALINT1 and Agneta M. BALINT2
West University of Timisoara, Timisoara, Romania
Faculty of Mathematics- Computer Science , 2Faculty of Physics
balint@balint.uvt.ro; balint@physics.uvt.ro;
1
ABSTRACT
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.
Keywords: mathematical modeling; ground water flow; diffusion; macro
transport in porous media.
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 Fowler [1]:
“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.”
 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).
74
The third procedure is to use “hypothetical laws” based on quantitative reasoning in the
absence of precise rules (Lotka-Volterra law).
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.
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. (see Fig.1: 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.
Example 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 [2] 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).
75
Figure 2. Periodic porous media
Fluid flow in a porous media
An incompressible viscous fluid moves in the “pore space” of a porous media, according
to the Navier-Stokes equations:
 
u
p
 u  u  
   u  f
t

(3.1)
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.
On the boundary of the fixed solid particles the fluid flow velocity has to satisfy the non
slip condition:
u0
(3.3)
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.
76
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 in [3] 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 and Acrivos in [4] extended the approximation of Hasimoto to calculate the drag
force for higher concentration.
 •Zick and Homsy in [5] 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 (see [6],[7]).
– A general model for the flow through periodic porous media has been advanced by Brenner in
an unpublished manuscript cited in [8] by Adler and in [9] by Brenner. 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.
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·l2/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):
xi   l  xi , P  P0  P, ui   U  ui
(3.4)
with i = 1,2,3.
Two dimensionless parameters would then appear: the length ratio ε = l/L and
the Reynolds number:
Re 
 U  l   Po  l 2 



L
2
(3.5)
which will be assumed to be of order O(ε).
By introducing fast and slow variables, xi and Xi = ε · xi and multiple-scale expansions, it
is then found that the leading order pore pressure p(0) depends only on the global scale (slow
variables), p(0) = p0(Xi).
77
By expressing the solution for ui0 , p1 in the following form:
ui0   kij 
p 1   S j 
where
p
(1)
p (0)
xi
(3.6)
1
p 0 
p
X j
(3.7)
X i  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
xi
S j
xi
0
  2 k ij   ij in 
in Ω
(3.8)
in Ω
(3.9)
with
kij= 0 on Γ
(3.10)
kij, Sj are periodic on ∂Ω
(3.11)
Here Γ and ∂Ω are respectively the fluid-solid interface and the boundary of the standard cell.
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:
 f 
1
f  d
 
(3.12)
f
where Ωf is the fluid volume in the cell.
Then the average of eq.(3.6) gives the law of Darcy :
 ui0    kij  
p 0 
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.
78
For later use, we note that in physical variables (marked by *) the symmetric hydraulic
conductivity tensor is given by:
 k 
1
3

5 1   2
 V 
  s 
 As  l 
2
(3.14)
Comments:
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 the paper of Tartar [10], where it
is shown that for   l / L 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.
In [10] the Darcy’s law is written in the form:
1.
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:
2.
2 p  0
(3.16)
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 in [11]. The computed values for the Wigner-Seitz grain (grain is shaped as a
diamond) are compared with those given by the empirical Kozeny-Carman formula:
 k ij *  k ij  
l2

(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 Navier-Stokes equation.
79
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. Above this surface, the soil is unsaturated, and the “pore space”
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 :
u
k

 p
(4.1)
here: p is the pressure gradient in the groundwater and satisfies the equation:
2 p  0
(4.2)
eq.(4.2) 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.
Table 1 gives typical value of the permeability of several common rock and soil types,
ranging from coarse gravel and sand to finer silt and clay.
Table 1. Rock and Soil permeability
Material
Gravel
Sand
Fractured igneous rock
Sandstone
Silt
Clay
Granite
80
k[m2]
10-8
10-10
10-12
10-13
10-14
10-18
10-20
Eqs. (4.1) (4.2) define the simplest model of the incompressible groundwater flow
through a rigid porous medium.
 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. 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:
u
k p

 x
v

k  p
     g 
  y

u v

0
x y
where u  u , v  with boundary conditions that :
(4.5)
(4.6)
v  0 on y  0
p  0, v   
h
h
u
on y  h
t
x
p    g  h0  y  on x  0
81
(4.7)
These conditions describe p  0 on x  L 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 non-dimensionalize the variables by
scaling
as
follows:
k    g  h0
x  L, y  h0 , p    g  h0 , u 
,
L
(4.8)
  h0  
v  kg , t 

kg
all for obtain various obvious balances in the equations and boundary conditions. The DupuitForchheimer approximate solution is obtained when h0 << L. In this case we define  = h0/L and
the equations become:
 p

p
u
v    L 
x

y


(4.5’)
2 p
y 2
with:
 2 
2 p
x 2
0
(4.6’)
p
 1 on y  0
y
h  p 
p h
 
 1   2  
on y  h
t  y 
x x
p  1  y on x  0
p  0,
p0
(4.7’)
on x  1
Since   1 we proceed by
expanding
p  p0    p1  ...etc.
2
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.
However, we also note that if p/y + 1 = O(2) then, h – t = O(d 2) which
suggests that
h  h0   2  h1  ... and h0  constant
82
Alternatively, we realize that ht = O(2) simply indicates that the timescale of relevance
to transient problems is longer than our initial guess O(F·h0·/k··g), so that we rescale t with
(and subsequently omitting the
1 /  2 . Putting t  t 2
over bar) we rewrite the  2  h   p  1   2  p  h kinematical boundary condition
 y 
t
x x


as:
(4.10)
Now
we
seek
p  p0   2  p1  ....
(4.11)
and we find successively:
 2 p1
y 2
and
expansions
h  h0  ....
 h0 xx ,
p0  h0  y
p1
 0 on
y
y0
(4.12)
(4.13)
 h
p1
  y  2o
y
x
2
whence
(4.14)
h0
 h  h 
 h0  20   0 
t
x
 x 
2
so that eq. (4.10) gives:
2
(4.15)
dropping the
equation:
subscript,
we
h   h 

h 
t x  x 
obtain the nonlinear diffusion
(4.16)
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 h  1 on x  1 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:
(4.17)
83
but it is not so clear what the other is. It can be determined by means of the following trick.
Define:
h
U   p  dy
0
and note that the flux q is given by
(4.18)
h
U   p  dy
0
h
h
p
U
 dy  

x
x
0
q   u  dy  
0
Furthermore
U1
at
2
x  0 and U  0 at
x 1
(4.19)
(4.20)
and therefore we have the exact result
1
0 q  dx  1 2
(4.21)
In a steady state, ht  q x  0 , so q is constant, and therefore
h
q  1  h 
2
x
(4.22)
The steady solution (away from x = 1) is therefore
h  1  x 
1
2
(4.23)
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  2 ,
1
x  1   X , h  
84
p  h  y and so we put
1
2
 H,
p 
1
2
Y ,
p 
1
2
P
(4.25)
and we choose
 2
(4.26)
to bring back the x derivatives in Laplace’s equation, we get
2P
x 2
with:

2P
y 2
0
P
 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
(4.27)
(4.28)
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.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
A.C. Fowler: Mathematical Models in the Applied Sciences, Cambridge
University Press, 1998
Bues M.A., Aachib M, “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).
Hasimoto H. “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).
Sangani A.S., Acrivos A. “Slow flow through a periodic array of spheres”, Int.J.
Multiphase Flow, 8(4), 343-360 (1982).
Zick A.A., Homsy G.M. “Stokes flow through periodic arrays of spheres”, J. Fluid
Mech.,115, 13-26 (1982).
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).
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).
85
[10] 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.
[11] 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 661675(1996).
86