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Transport in Porous Media 45: 321–345, 2001. c 2001 Kluwer Academic Publishers. Printed in the Netherlands.

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Homogenization Modeling and Parametric Study of Moisture Transfer in an Unsaturated Heterogeneous Porous Medium JOLANTA LEWANDOWSKA and JEAN-PAUL LAURENT Laboratoire d’´etude 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 [−].

322 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

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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


325

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)


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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 C2 /C1 > 1, • for 0.05 < θ C2 /C1 > 100, • for θ 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

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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