Homogenization Modeling and Parametric Study of Moisture

Transcription

Transport in Porous Media 45: 321–345, 2001.
c 2001 Kluwer Academic Publishers. Printed in the Netherlands.
321
Homogenization Modeling and Parametric Study of
Moisture Transfer in an Unsaturated Heterogeneous
Porous Medium
JOLANTA LEWANDOWSKA and JEAN-PAUL LAURENT
Laboratoire d’étude des Transferts en Hydrologie et Environnement (LTHE), UMR 5564
CNRS-INPG-UJF-IRD, BP53, F-38041 Grenoble-Cedex 09, France.
e-mail: [email protected]; [email protected]
(Received: 24 February 2000; in final form 14 August 2001)
Abstract. The classical mass balance equation is usually used to model the transfer of humidity
in unsaturated macroscopically homogeneous porous media. This equation is highly non-linear due
to the pressure-dependence of the hydrodynamic characteristics. The formal homogenization method
by asymptotic expansions is applied to derive the upscaled form of this equation in case of large-scale
heterogeneities of periodic structure. The nature of such heterogeneities may be different, resulting
in locally variable hydrodynamic parameters. The effective capillary capacity and the effective hydraulic conductivity are defined as functions of geometry and local characteristics of the porous
medium. A study of a two-dimensional stone-mortar system is performed. The effect of the second
medium (the mortar), on the global behavior of the system is investigated. Numerical results for the
Brooks and Corey hydrodynamic model are provided. The sensitivity analysis of the parameters of
the model in relation to the effective hydrodynamic parameters of the porous structure is presented.
Key words: homogenization, humidity transfer, unsaturated, macroscopically heterogeneous, Brooks
and Corey model, sensitivity analysis.
Nomenclature
b1 , b2
C
C eff
K
Kref
Ks1 , Ks2
K eff
K11 , K22 , K33
l
L
n1 , n2
N
t
X
x
constants of the Brooks and Corel model for materials 1 and 2 [−].
water retention capacity [L−1 ].
effective water retention capacity [L−1 ].
hydraulic conductivity [LT−1 ].
reference conductivity [LT−1 ].
water conductivity at saturation of the materials 1 and 2 [LT−1 ].
effective conductivity tensor [LT−1 ].
principal components of the effective conductivity tensor [LT−1 ].
characteristic microscopic length [L].
characteristic macroscopic length [L].
volume fractions of the materials 1 and 2 [−].
unit vector normal to .
time [T].
physical space variable [L].
macroscopic space variable [−].
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y
ε
ε1 , ε2
θ1 , θ2
θs1 , θs2
θ 0 ψ
ψe1 , ψe2
ξ
1 , 2
JOLANTA LEWANDOWSKA AND JEAN-PAUL LAURENT
microscopic space variable [−].
homogenization parameter.
porosities of the materials 1 and 2 [−].
water contents at partial saturation in the material 1 and 2 [−].
water content at saturation of the materials 1 and 2 [−].
average water content [−].
suction [L].
air entry potential of the materials 1 and 2 [L].
non-dimensional parameter [−].
period.
domain 1 and 2 in the period .
interface between 1 and 2 .
1. Introduction
Real porous media that are encountered in engineering practice are very often
macroscopically heterogeneous. In civil engineering, for example, a building wall
is a porous material made of blocks assembled with mortar. In hydrology, some
stratified soil are composed of two types of porous media (two different soils),
each of which individually can be treated as homogeneous over a certain scale.
The transfer of humidity in such systems can be studied by means of the formal
homogenization method which makes it possible to capture the effect of two-level
(or double) heterogeneities appearing in the problem: the pore scale and the Darcy
scale. By performing the successive micro-macro passages (the upscaling) we can:
(i) derive the homogeneous description of the phenomenon for an equivalent macroscopic continuum, (ii) define its validity domain, and (iii) determine the effective
parameters characterizing the medium. This approach can be appreciated from two
points of view. First, it makes it possible to predict the hydrodynamic behavior of
the medium in its real environmental conditions. Second, it allows the optimization
of the porous materials to reach the desired hydrodynamic characteristics or meet
new material standards.
An extensive literature exists on modeling of unsaturated flow in heterogeneous
porous media, treating different aspects of the problem. The general context of
these studies is the homogenization upscaling of two-phase flow. See, for example,
the papers by Bourgeat (1984), Saez et al. (1989), Hornung (1991), Amaziane
et al. (1991), Arbogast (1993), and many references therein. One of the important research directions in this general context is the modeling of two-phase flow
in double porosity systems, because of its practical application in petroleum engineering (see, for example, the papers by Hornung (1991), Arbogast (1993)).
The problem of two-phase flow was also treated by using the volume averaging
technique. Quintard and Whitaker (1989) developed the large-scale averaged continuity and momentum equations and a method of closure to predict the large-scale
permeability tensor and large-scale capillary pressure.
In this paper, homogenization method using the formal asymptotic expansion
technique is applied, to derive the upscaled form of the water transfer equation
HOMOGENIZATION MODELING AND PARAMETRIC STUDY OF MOISTURE TRANSFER
323
in case of large-scale heterogeneities of periodic structure. The problem concerns
a particular case of two-phase flow in which the air phase is considered as being at constant atmospheric pressure. This assumption is used in many practical
applications, for example: in hydrological problems.
The heterogeneities are represented by two different porous materials. The difference in the materials is ‘measured’ in terms of the homogenization parameter ε
which defines the scale separation. Special attention is given to the description of
scaling procedure and to the definition of the validity domain of the homogenization modeling. The latter was possible due to the normalization procedure.
The approach adopted is general, in the sense that it can be applied to all kinds of
porous geometry and fluid type, and quasi-rigorous, in the sense that, starting from
the assumed asymptotic expansion we can often prove the existence and uniqueness
of the solutions of each successive problem. No assumption is required concerning
the form of the macroscopic equations. Also, it is possible to define the domain of
validity of the one-equation modeling, meaning the local mechanical equilibrium.
The last point has important implications for practical applications.
In Sections 2–7 the development of the macroscopic model for a heterogeneous system of two porous materials, that have hydrodynamic characteristics of
the same order of magnitude with respect to the homogenization parameter ε, is
presented. Note that the system considered is not a double-porosity type of system.
In Section 8, two examples of the application of the model are presented. The first
one concerns the calculation of the effective parameters in a particular case of a
two-dimensional porous medium, together with the study of the domain of validity
of the modeling. In the second example, the numerical solution to the macroscopic
problem of water infiltration into the stratified soil is proposed. In Section 9 the
parametric study of a two-dimensional anisotropic stone-mortar system, is presented. We focus on the role of the mortar, and particularly its porous structure via
the parameters of a hydrodynamic analytical model, on the global behavior of the
system with respect to the transfer of humidity. This research is oriented towards
the restoration of historical monuments but it can be applied to all macroscopically
heterogeneous porous systems.
2. Heterogeneous Porous Medium
In order to formulate the problem let us begin at the mesoscopic scale that is at the
so-called DARCY scale. Let us consider a porous medium composed of two distinct porous materials distributed in such a way that we can define a ‘Representative
Elementary Volume’ (REV) or a period, if the medium has a periodic structure.
Without loss of generality we further assume the periodicity of the medium and we
denote the period, 1 and 2 the two porous sub domains, see Figure 1. The
condition of periodicity is not restrictive in the sense that it does not influence the
development of the macroscopic model. If an equivalent medium exists, it does not
depend on the internal organization of the medium. As to the determination of the
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Figure 1. A period of a porous heterogeneous medium.
equivalent parameters, the assumption of periodicity is rather restrictive but it is at
present the only rigorous approach available.
3. Transfer of Humidity
Assume that in each of the two porous domains the capillary transfer of humidity
under isothermal conditions can be described by the following equations
C1
∂ψ1
K 1 grad ψ1 ) = 0 in
− div(K
∂t
1 ,
(1)
C2
∂ψ2
K 2 grad ψ2 ) = 0 in
− div(K
∂t
2 ,
(2)
with the continuity conditions on the interface between the two sub domains
expressed in the form
[ψ] = 0,
(3)
K grad ψ) · N] = 0,
[(K
(4)
where ψ [L] is the suction (water pressure head relative to atmospheric pressure)
(ψ 0), K (ψ) > 0 [LT−1 ] is the hydraulic conductivity tensor, C(ψ) [L−1 ] is the
water retention capacity, N is the unit vector normal to , t [s] is the time. Note
that Equations (1) and (2) are strongly non-linear due to the dependence of K and
C on the state variable ψ.
In the formulation (1)–(4), it was assumed that the capillary effects dominate the
flow, so that the influence of gravity can be neglected. This assumption is verified
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by relatively large spectrum of porous media for which the Bond number Bo takes
small values.
4. Water Retention Characteristics
Let us then assume that the distribution of the water phase in the pores of each
material can be characterized by the water retention curve, which is the relation
between water content and suction at equilibrium,
ψ1 = ψ1 (θ1 )
in 1 ,
where
θ1 ∈ (0, ε1 ),
(5)
ψ2 = ψ2 (θ2 )
in 2 ,
where
θ2 ∈ (0, ε2 ),
(6)
where θ1 , θ2 are water contents; ε1 , ε2 are the porosities of the materials 1 and 2,
respectively.
We assume that the retention curves for imbibition and drainage phases are known
and that they are uniquely defined that is the hysteresis effect is not addressed
in this study. The water retention capacities C1 (ψ1 ) and C2 (ψ2 ) are written as
follows:
dθ1
,
(7)
C1 (ψ1 ) =
dθψ1
dθ2
.
(8)
C2 (ψ2 ) =
dθψ2
5. Dimensional Analysis
The key point of our analysis is the scale separation condition that is written
l
(9)
ε = 1,
L
where l is a characteristic microscopic length and L is a characteristic macroscopic
length. The characteristic microscopic length is associated to the dimension of
the period or the dimension of the REV. The characteristic macroscopic length is
usually identified with the global dimension of the medium. It holds under steadystate conditions, when a general homogenization analysis is performed. However,
in real problems a characteristic macroscopic length related to the phenomenon
appears (e.g. the wavelength in case of wave propagation). In such cases this length,
different from the global dimensions of the medium, should be taken into account
when checking the scale separation condition (9). In practical cases, ε is a function
of time and space coordinates (Auriault and Lewandowska, 1998). In this paper we
suppose that condition (9) is satisfied.
Let us introduce the following representations of all variables appearing in
Equations (1)–(8):
X
ψ1 = ψ1c ψ1∗ , C1 = C1c C1∗ , K 1 = K1cK ∗1 , x = , t = T t ∗ , (10)
L
326
X
,
(11)
l
where the subscript ‘c’ means the characteristic quantity (constant) and the superscript star denotes the non-dimensional variable. We note the following space
variables:
ψ2 = ψ2c ψ2∗ ,
C2 = C2c C2∗ ,
K 2 = K2cK ∗2 ,
y=
• X (X1 , X2 , X3 ) is the physical (dimensional) space variable,
• y (y1 , y2 , y3 ) is the microscopic (non-dimensional) space variable,
• x (x1 , x2 , x3 ) is the macroscopic (non-dimensional) space variable.
If we substitute (10) and (11) into (1)–(4) and assume ψ1c = O(ψ2c ), then Equations (1)–(4) can be presented in the following non-dimensional form:
∗
∂
C1c l 2 ∗ ∂ψ1∗
∗ ∂ψ1
K1ij
= 0 in 1 ,
C
−
(12)
K1c T 1 ∂t ∗
∂yi
∂yj
∗
∂
C2c l 2 ∗ ∂ψ2∗
∗ ∂ψ2
C
−
(13)
K2ij
= 0 in 2 ,
K2c T 2 ∂t ∗
∂yi
∂yj
with the continuity conditions at ψ1∗ = ψ2∗ ,
∗
∗
K1c
∗ ∂ψ1
∗ ∂ψ2
Ni = K2ij
Ni .
K1ij
K2c
∂yj
∂yj
(14)
(15)
Further, we define a non-dimensional parameter ξ that characterizes the regime of
the transfer of humidity, as follows:
ξ=
Cc l 2
.
Kc T
(16)
To see the interaction between two porous domains, let us assume that the
characteristic values of the hydrodynamic properties of two materials are of the
same order of magnitude
O(ε 1 ) C1c
K1c
=
= O(ε 0 ) O(ε −1 ).
C2c
K2c
(17)
Note that we do not consider the double-porosity systems that have very contrasted
characteristics: K1c /K2c = O(ε 2 ).
The characteristic time T of the phenomenon is sufficiently long so that the ratio
(Cc L2 )/(Kc T ) is of the first order of magnitude
Cc L2
= O(1).
Kc T
(18)
Note that condition (18) is a necessary condition of the homogenizability of diffusion-type transient processes (Auriault and Lewandowska, 1993).
327
Finally, we have
ξ = ξ1 = ξ2 = O(ε 2 ).
(19)
Clearly, we impose the time and length scale constraints that are rather restrictive but, as it will be shown, these assumptions underlie the hypothesis of local
mechanical equilibrium that is interesting from a practical point of view.
6. Formulation of the Problem
Taking into account the estimations (17)–(19), the problem can be put in the nondimensional form
∗
∗
∂
2 ∗ ∂ψ1
∗ ∂ψ1
(20)
K1ij
= 0 in 1 ,
ε C1 ∗ −
∂t
∂yi
∂yj
ε
2
∂ψ ∗
C2∗ ∗2
∂t
∂
−
∂yi
∗
∗ ∂ψ2
K2ij
= 0 in
∂yj
2 ,
(21)
with
ψ1∗ = ψ2∗
∗
K1ij
on
(22)
,
∗
∂ψ1∗
∗ ∂ψ2
Ni = K2ij
Ni
∂yj
∂yj
on
.
(23)
The problem is formulated as follows: suppose that the physics of humidity transfer
locally in a porous heterogeneous medium at the Darcy scale is governed by Equations (20)–(23), find the macroscopic description of this process for an equivalent
continuous medium.
7. Homogenization
In this study, the method of homogenization by formal asymptotic expansions
is applied (Sanchez-Palencia, 1980; Bensoussan et al., 1987). The formalism of
this method has been extensively developed for the last 10 years, in particular
towards the non-linear problems (Amaziane et al., 1991; Arbogast, 1993; Jikov
et al., 1994). The procedure adopted in this paper is presented in details in Auriault
(1991). The homogenization begins with the postulation that all unknowns φ can
be presented in form of the asymptotic expansions
φ(xx , y , t) = φ 0 (xx , y , t) + εφ 1 (xx , y , t) + ε 2 φ 2 (xx , y , t) + · · · ,
where φ(xx , y , t) stands for ψ1∗ , ψ2∗ , C1∗ , C2∗ , K1∗ , K2∗ , θ1∗ or θ1∗ .
It is assumed that φ i are y -periodic.
(24)
328
Note that due to scale separation the unknowns are functions of three variables
x
(x , y , t), where x = (x1 , x2 , x3 ), y = (y1 , y2 , y3 ) the relation between two scales is
x = ε y and that the derivation operator is
∂
∂
∂
→
+ε .
∂yy
∂yy
∂xx
(25)
The methodology of homogenization involves introducing the expansion (24) into
the problem (20)–(23) and then equating the terms of the same powers of ε. It
yields the successive boundary value problems to be solved on the period.
At the order O(ε 0 ) we obtain
0
∂
0 ∂ψ1
(26)
K1ij
= 0 in 1 ,
−
∂yi
∂yj
∂
−
∂yi
0
0 ∂ψ2
K2ij
=0
∂yj
ψ10 = ψ20
0
K1ij
(27)
in 2 ,
on ,
0
∂ψ10
0 ∂ψ2
Ni = K2ij
Ni
∂yj
∂yj
(28)
on
(29)
,
with ψ10 and ψ20 y -periodic.
We also have
K10 = K1 (ψ10 ) > 0,
(30)
K20 = K2 (ψ20 ) > 0.
(31)
It can be shown that the solution to the problem (26)–(29) is a function ψ 0 ,
which is constant over the period
ψ10 = ψ20 = ψ 0 (xx , t).
(32)
It means that the first order solution is independent of the local variable y and
that we have one water suction field ψ 0 at the first order of approximation. This is
equivalent to the state of local mechanical equilibrium.
At the next order O(ε 1 ), and using the result (32), we get the boundary value
problem for the second order terms ψ11 and ψ21
0
∂
∂ψ11
∂ψ
0
−
+
(33)
K1ij
= 0 in 1 ,
∂yi
∂xj
∂yj
0
∂ψ21
∂ψ
∂
0
+
K2ij
= 0 in
−
∂yi
∂xj
∂yj
2 ,
(34)
ψ11 = ψ21 on ,
0
0
∂ψ11
∂ψ21
∂ψ
∂ψ
0
0
K1ij
+
+
Ni = K2ij
Ni
∂xj
∂yj
∂xj
∂yj
329
(35)
on
,
(36)
were ψ11 and ψ21 are y -periodic.
It can be shown that the solution to the problem (33)–(36) can be put in the form
of a linear function of the macroscopic suction gradient ∂ψ 0 /∂xi as follows
ψ11 = χi
∂ψ 0
1
+ ψ (xx , t),
ψxx i
(37)
ψ21 = χi
∂ψ 0
1
+ ψ (xx , t)
ψxx i
(38)
where χ (yy , ψ 0 ) and χ (yy , ψ 0 ) are y -periodic vectors and their volume average is
zero-valued
χ d + χ d
χ =
χ
1
1
2
||
=0
(39)
1
ψ and ψ are arbitrary functions of x and t.
The demonstration of the solution (37)–(38) is not given. The reader is referred
to the paper (Auriault and Lewandowska, 1997) for a solution to a similar problem.
Furthermore, substituting (37)–(38) into (33)–(36), gives the following -problem
(the local boundary value problem) (Quintard and Whitaker, 1989; Saez et al.,
1989)
∂χk
∂
0
(40)
K1ij Ij k +
= 0 in 1 ,
−
∂yi
∂yj
∂χk
∂
0
(41)
K2ij
Ij k +
= 0 in 2 ,
−
∂yi
∂yj
χk = χk on ,
∂χk
∂χk
0
0
Ni = K2ij Ij k +
Ni
K1ij Ij k +
∂yj
∂yj
(42)
on
,
(43)
where I is the identity matrix. Note that (40)–(43) is a linear problem. In a general
case, it can be solved by a numerical method for each pair of the corresponding
values (K10 , K20 ). At the O(ε 2 ) order we get
1
0
0
∂
∂ψ12
∂ψ11
∂ψ1
∂ψ1
0 ∂ψ1
0
1
−
+
+
K1ij
+ K1ij
+
C1
∂t
∂yi
∂xj
∂yj
∂xj
∂yj
0
∂ψ11
∂ψ1
∂
0
+
(44)
K1ij
= 0 in 1 ,
+
∂xi
∂xj
∂yj
330
C20
∂ψ20
∂t
1
0
∂
∂ψ22
∂ψ21
∂ψ2
∂ψ2
0
1
−
+
+
K2ij
+ K2ij
+
∂yi
∂xj
∂yj
∂xj
∂yj
0
∂ψ21
∂ψ2
∂
0
+
K2ij
= 0 in 2 ,
+
∂xi
∂xj
∂yj
ψ12 = ψ22
on ,
(45)
(46)
0
∂ψ11
∂ψ11 ∂ψ12
∂ψ1
1
+
+
+ K1ij
Ni
∂xj
∂yj
∂xj
∂yj
1
0
∂ψ22
∂ψ21
∂ψ2
∂ψ2
0
1
= K2ij
+
+
+ K2ij
Ni
∂xj
∂yj
∂xj
∂yj
0
K1ij
on
.
(47)
Finally, we obtain the first order effective equation for ψ 0 by taking the average of (44) and (45). If these two equations are added, we get the one-equation
macroscopic transport model in the form
0
0
∂
eff ∂ψ
eff ∂ψ
+
Kij
= O(ε),
(48)
C
∂t
∂xi
∂xj
where the effective water conductivity tensor K eff is defined as
1
∂χj
∂χj
0
0
K1ik
K2ik
Ikj +
d +
Ikj +
d (49)
Kijeff =
|| 1
∂yk
∂yk
2
and the effective water retention capacity C eff is written
C eff = n1 C10 + n2 C20 = n1
dθ10
dθ20
+
n
,
2
dψ 0
dψ 0
(50)
n1 and n2 are the volume fractions of medium 1 and 2
n1 =
|1 |
,
||
n2 =
|2 |
,
||
n1 + n2 = 1.
Lets us now turn to the problem of humidity distribution in the pores. We have
ψ10 (xx , t) = ψ1 (θ10 ) in
1 ,
(51)
ψ20 (xx , t) = ψ2 (θ20 ) in
2 .
(52)
331
Taking into account the solution (32), we can write
θ10 = ϕ1 (θ 0 )
in 1 ,
θ20 = ϕ2 (θ 0 )
in 2 .
Thus, we have
ψ1 (ϕ1 (θ 0 )) = ψ2 (ϕ2 (θ 0 )) = ψ(θ 0 ),
(53)
where θ 0 is the average water content
θ 0 = n1 θ10 + n2 θ20 .
(54)
We also have
C eff =
dθ 0 .
dψ 0
(55)
As it can be seen, we obtain one suction field ψ 0 and two water content fields θ10
and θ20 . Note that θ10 and θ20 are linked by relation (32).
8. Applications
The modeling of humidity transport in a heterogeneous porous medium by homogenization leads to the macroscopic governing Equation (48). If we want to treat a
practical problem, Equation (48), completed by appropriate boundary conditions,
should be solved within the domain considered. Another practical result of the
modeling is the procedure of the determination of the effective transfer parameters
K eff , Equation (49), and C eff , Equation (50), for any local geometry and any set of
local hydrodynamic parameters.
In the following, two numerical examples of the application of the proposed
modeling are presented. The first one concerns the calculation of the effective
parameters in a particular case of a two-dimensional porous medium. In the second
one, the numerical solution to the macroscopic problem of water infiltration into
the stratified soil is presented.
8.1. DETERMINATION OF THE EFFECTIVE PARAMETERS
8.1.1. Input Data and Results
In order to determine the effective transfer parameters K eff and C eff for an equivalent homogeneous medium the following data concerning the two components of
the porous medium are needed
Medium 1.
ψ1 = ψ1 (θ1 ),
K1 = K1 (ψ1 ),
ε1 ,
n1 .
332
Medium 2.
ψ2 = ψ2 (θ2 ),
K2 = K2 (ψ2 ),
ε2 ,
n2 .
Since the effective parameters depend on suction ψ, the iterative procedure presented in Figure 2 is applied. The effective parameters are calculated at each value of
ψ and then the non-linear curves K eff (ψ) and C eff (ψ) are traced. In Figures 4
and 5 the resulting curves for a stone-mortar system (Figure 3) are presented, as an
Figure 2. Numerical solution procedure.
333
Figure 3. Two-dimensional periodic porous medium: a stone-mortar system.
example. The numerical calculations, that is, the solution to the problem (39)–(43),
were performed using the computer program mono3D of Michel Quintard (1997).
8.1.2. Domain of Validity
In order to determine the domain of validity of the modeling, the ratios K2 /K1 and
C2 /C1 as functions of suction ψ in the case considered in Part 8.1 were calculated
(Figure 6 and 7). It can be seen that K2 /K1 remains constant and equal to ≈ 34
during the process, while the variations of C2 /C1 are as follows
• for 0.2 < θ <0.47
100 > C2 /C1 > 1,
• for 0.05 < θ <0.2
105 > C2 /C1 > 100,
• for θ <0.05
C2 /C1 > 105 .
If Equation (17) is to be verified, the parameter ε should take very small values.
Therefore, it can be stated that the criterion (17) is satisfied, if we have, for example
ε = 10−5 , since the following inequalities hold
K2
= 34 O(ε −1 ) ≈ 105
O(ε 1 ) ≈ 10−5 K1
and
C2
= 1 to 100 O(ε −1 ) ≈ 105
O(ε 1 ) ≈ 10−5 C1
334
Figure 4. Example: suction curves for a stone-mortar system.
b1 = 3, ψe1 = −1.24 m, Ks1 = 3 × 10−7 m/s, ε1 = 0.47, n1 = 0.87,
b2 = 3, ψe2 = −4 m, Ks2 = 3 × 10−7 m/s, ε2 = 0.47, n2 = 0.13.
except for the case when the medium is relatively dry that is θ <0.2. In the latter
case the ratio C2 /C1 takes high values and condition (17) no more holds. From a
purely theoretical point of view we can imagine the extremely small values of ε
that would verify (17) but in practice such a situation is not very likely to happen.
8.2. EXAMPLE OF THE SOLUTION TO THE MACROSCOPIC BOUNDARY VALUE
PROBLEM
The macroscopic problem concerns the one-dimensional water infiltration into the
stratified soil which is initially dry. The hydrodynamic characteristics of the two
335
Figure 5. Example: effective conductivity curves for a stone-mortar system.
b1 = 3, ψe1 = −1.24 m, Ks1 = 3 × 10−7 m/s, ε1 = 0.47, n1 = 0.87,
b2 = 3, ψe2 = −4 m, Ks2 = 3 × 10−7 m/s, ε2 = 0.47, n2 = 0.13.
types of soil are the following: the van Genuchten model for the suction curve
θ(h) and the van Genuchten model with Mualem condition for the conductivity
curve K(h) (van Genuchten and Nielsen, 1985). The parameters of the models are
taken from the GRIZZLY database (http://www.lthe.hmg.inpg.fr) concerning 660
different soils from all over the world.:
Medium 1:
θs = 0.323,
n = 5.021,
Ks = 0.53174 m/ h,
n1 = 0.5.
α = 2.46/m,
336
Figure 6. The ratio K2 /K1 as a function of suction ψ for a stone-mortar system.
Figure 7. The ratio C2 /C1 as a function of suction ψ for a stone-mortar system.
337
Figure 8. Numerical solution to the infiltration problem: comparison between the homogenized solution and the fine-scale heterogeneous solutions.
Medium 2:
θs = 0.559,
n = 2.087,
Ks = 0.00013 m/ h,
n2 = 0.5.
α = 0.89/m,
The numerical calculations were performed for a soil column of 0.40 m in two
cases: 10 soil layers and 20 soil layers. The initial and boundary conditions are as
follows:
ψ(z, t = 0) = −10 m,
ψ(z = 0 m, t) = −0.75 m,
ψ(z = 0.40 m, t) = −10 m.
The effective parameters were calculated according to the iterative procedure
presented in Figure 2. The numerical solution to the macroscopic problem, Equation (48), subjected to the above initial and boundary conditions, was found using
the own computer program (method of lines). In Figure 8 the homogenized solution is compared to the fine-scale heterogeneous solutions (10 soil layers
and 20 soil layers). It can be seen that there exists relatively good agreement
between the homogenized and the heterogeneous solutions. The discrepancy decreases with the increasing number of heterogeneities, that corresponds to better
scale separation.
338
9. Parametric Study of a Two-dimensional Stone-Mortar System
Let us consider a two-dimensional periodic system consisting of blocks of dimensions 40 cm × 20 cm separated by mortar of the thickness 2 cm. In Figure 3 the
assemblage of the blocks in a period is presented. The period consists of material
A: a tuffeau stone, which occupies the domain 1 and material B-the mortar in the
domain 2 . Within each subdomain the medium is isotropic and homogeneous,
and the material properties are known. In the direction X3 the medium is infinite.
Our aim was to investigate the influence of the characteristics of the mortar on the
overall behavior of the structure. This is made by varying the parameters of an
analytical hydrodynamic model ψ(θ) − K(θ) for the mortar that is by modifying
the porous structure of the medium B.
9.1. MATERIAL PROPERTIES
9.1.1. Material A: Tuffeau Stone
The hydrodynamic properties of the tuffeau stone have been extensively studied
by Dessandier et Gaboriau (1996) using various laboratory and field techniques.
In the paper a simplified hydrodynamic characterization technique based on the
MIP (Mercury Intrusion Porosity) was used (Laurent, 1996). The suction and the
conductivity curves were represented by the analytical functions proposed Brooks
and Corey in the form (Brooks and Corey, 1964; Campbell, 1994):
−b1
θ1
drainage phase ψ1 [m],
(56)
ψ1 (θ1 ) = ψe1
θs1
K1 (ψ1 ) = Ks1
ψe1
ψ1
2+(3/b1)
,
K1 [m/s],
(57)
where
ψe1
b1
is the air entry potential (or bubbling pressure), ψe1 = −1.24 m,
is a constant, b1 = 3,
θs1 is the water content at saturation. It is supposed equal to porosity ε1 ; θs1
= 0.47,
Ks1
ε1
is the water conductivity at saturation; Ks1 = 3 × 10−7 m/s,
is the porosity; ε1 = 0.47.
The volume fraction of the material A for the geometry of the period presented in
Figure 3 is n1 = 0.87.
339
9.1.2. Material B: Mortar
For mortar the Brooks and Corey representation was also adopted. This means
that the hydrodynamic properties were defined by three parameters b2 , ψe2 , ε2 as
follows:
−b2
θ2
drainage phase ψ2 [m],
(58)
ψ2 (θ2 ) = ψe2
θs2
K2 (ψ2 ) = Ks2
ψe2
ψ2
2+(3/b2)
,
K2 [m/s].
(59)
Each case considered only one of three parameters varied, the two others being
fixed and equal to those of the tuffeau (the reference medium). We noticed a lack
of complete hydrodynamic data concerning the mortar published in the literature.
This is understandable, since mortar, especially in historical monuments, can be of
highly variable composition. Therefore, the ranges of possible (realistic) variations
of its parameters were determined from the soil data given in (Campbell, 1994)
and from the GRIZZLY database. The following three series of calculations were
carried out.
Series 1: variations of b2
ψe2 = −1.24 m,
b2 = 2, 3, 4, 6, 8, 12, 16, 20, 24,
−7
θs2 = ε2 = 0.47, n2 = 0.13.
Ks2 = 3 × 10 m/s,
Series 2: variations of ψe2
ψe2 = −0.05, 0.1, 0.5, 1.24, 2, 4, 6, 8 m,
θs2 = ε2 = 0.47,
Ks2 = 3 × 10−7 m/s,
b2 = 3,
n2 = 0.13.
Series 3: variations of ε2
ψe2 = −1.24 m,
b2 = 3,
θs2 = ε2 = 0.17, 0.27, 0.37, 0.47,
n2 = 0.13.
In this case, it was considered that the variation of porosity of the porous medium
modifies its hydraulic conductivity at saturation, according to Kozeny-Carman formula (Bear, 1972). Hence,we have
3 ε2
1 − ε1 2
.
(60)
Ks2 = Ks1
ε1
1 − ε2
Note that in the literature it is recognized that the parameters b2 , ψe2 , ε2 are
correlated to the distribution of pores of the porous medium. For example, in
soils it is expected that ψe decreases (becomes more negative), as the mean pore
diameter becomes smaller and b increases as the standard deviation of pore size
340
increases. So, clays have larger absolute values of ψe and larger values of b, than
coarse textured soils (Campbell, 1994). Therefore, it is believed that the analysis
that follows could help to optimize the choice of the mortar (its composition, its
porous structure) to obtain the desired effective parameters.
9.2. RESULTS
In Figures 9, 10 and 11 the effective conductivity curves as functions of the parameters b2 , ψe2 , ε2 at three different values of the suction −ψ = 5, 50, 500 m are
presented. Because of the symmetry of the structure, the conductivity tensor has
three components K11 , K22 , K33 in the three principal directions X1 , X2 and X3 . It
can be generally concluded that the increase (decrease) in the parameters b2 , ψe2 ,
Figure 9. Effective conductivity as a function of the ratio b2 /b1 (Brooks and Corey model).
341
Figure 10. Effective conductivity as a function of the ratio ψe2 /ψe1 (Brooks and Corey
model).
ε2 of the medium 2 causes the increase (decrease) in the effective conductivity of
the whole structure with respect to the reference homogeneous medium composed
entirely of the tuffeau Kref . Despite the low fraction of the second medium the
modification of the effective parameters of the structure in certain cases is considerable, especially at higher values of suction, when the medium is relatively
dry.
The effective conductivities observed within the range of the investigated variations of the parameters b2 , ψe2 , ε2 are presented in Tables I, II and III. It can be
seen that the air entry potential ψe2 of the medium 2 has the greatest influence on
the effective conductivity, especially when ψe2 ψe1 , Figure 10. Moreover, with
342
Figure 11. Effective conductivity as a function of the ratio ε2 /ε1 (Brooks and Corey model).
Table I. Variations of the effective conductivity with the parameter b2 at selected
values of suction ψ[m]
b2
ψ[m]
Kxx
Kref
Kyy
Kref
Kzz
Kref
2
−5
−50
0.92
1.25
0.9
1.17
0.94
1.32
24
−5
−50
0.75
3.28
0.64
1.96
0.9
4.2
343
Table II. Variations of the effective conductivity with the parameter ψe2 [m] at selected
values of suction ψ[m]
ψe2 [m]
ψ[m]
Kxx
Kref
Kyy
Kref
Kzz
Kref
−0.05
−5
−50
2 × 10−3
7
7 × 10−4
3
0.87
10
−8
−5
−50
2 × 10−3
26
7 × 10−4
10
0.86
37
Table III. Variations of the effective conductivity with the parameter ε2
at selected values of suction ψ[m]
ε2
ψ[m]
Kxx
Kref
Kyy
Kref
Kzz
Kref
0.47
0.17
−5
−50
1
0.35
1
0.17
1
0.87
the increase (decrease) of the parameters b2 , ψe2 , ε2 the anisotropy of the effective
medium appears and becomes important at higher values of suction. Clearly, this
anisotropy is induced by the geometry of the medium, for example by the fact
that the domain of medium 1 is not connected or that domain 2 is more favorably
arranged with respect to water flow in the direction X1 than X2 . In particular, when
ψe2 varies between the values ψe2 = −1.24 m and ψe2 = −0.05 m the ratio of the
three components of the effective conductivity tensor K11 : K22 : K33 is of the order
1000 : 1000 : 1, approximately. This is already observed at relatively small values
of ψ, example :−ψ = 5 m. Practically, it can be said that when ψe2 = −0.1 m
the medium conducts very little in the directions X1 and X2 with respect to the
direction X3 . However, in this case the question arises, concerning the satisfaction
of the Condition (17). When this condition is not fulfilled, a macroscopic equivalent model does not exist. This particular case shows the limit of applicability of
effective medium methods.
10. Conclusions
The homogenization analysis showed that the moisture transfer in an unsaturated
heterogeneous porous medium can be described by a one-equation model for suction. The one-equation model is valid provided the local mechanical equilibrium is
assured, which means that three conditions are satisfied, concerning: (i) the scale
separation, (ii) the characteristic time scale of the phenomenon, and (iii) the ratio of
the hydrodynamic characteristics of the two components of the medium. It should
be noted that despite the fact that local ‘physics’ of the phenomenon is described
344
by a non-linear equation (supposed known and valid at every instant of the process
and everywhere in the medium), the homogenization yields linear equations for the
effective parameters for each value of ψ. The effective moisture transfer parameters
of an equivalent homogeneous medium can be calculated for any geometry of the
period (or the REV) and any set of ‘local’ hydrodynamic data, unless there exists
a very large contrast between the parameters of the two porous media.
The application of the model to capture (quantitatively) the effect of macroscopic heterogeneities on the hydrodynamic behavior of the porous structure is
presented. The parametric study performed in a two-dimensional case using the
Brooks and Corey hydrodynamic model showed that the effective parameters can
be considerably modified, even if the volume fraction is weak. It was also shown
that the geometrical arrangement can dramatically affect the effective conductivity tensor (by the induced anisotropy) while it has no effect on the macroscopic
capillary pressure.
Acknowledgements
We would like to thank Professor Michel Quintard for his kind permission to utilize his computer program mono3D to solve the local boundary value problem.
The valuable discussions on the subject are also acknowledged. This work was
partially funded by the French ‘Geomaterials’ Program of the INSU-CNRS and by
the PNRH Program ‘Transfert complexes en milieu poreux et ressources en eau’ of
the INSU-CNRS.
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Homogenization Modeling and Parametric Study of Moisture

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