Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue7.1999:

Macroscopic equations for vapor transport in a

multi-layered unsaturated zone

Chiu-On Ng

1

Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, People's Republic of China

(Received 18 March 1998; revised 14 August 1998; accepted 14 September 1998)

Using the method of homogenization, we present a systematic derivation of the macroscopic equations for air ¯ow and chemical vapor transport in an unsatu-rated zone with a periodic structure of heterogeneity. The e€ective speci®c dis-charge and hydrodynamic dispersion coecient are expressible in terms of some cell functions, whose analytical solutions are sought for the simple case of al-ternate stacking of two strictly plane layers of di€erent properties. For this kind of bi-layered composite, the e€ects of convection velocity and layer property con-trasts on the longitudinal and transverse components of the hydrodynamic dis-persion coecient are investigated. Ó_{1999 Elsevier Science Limited. All rights} reserved

Key words:vapor transport, layered porous media, dispersion, homogenization theory.

1 NOMENCLATURE

Printed in Great Britain. All rights reserved 0309-1708/99/$ ± see front matter PII: S 0 3 0 9 - 1 7 0 8 ( 9 8 ) 0 0 0 3 4 - 7

C vapor concentration

C0 characteristic vapor concentration

d layer thickness

D isotropic bulk molecular di€usion coef-®cient

D bulk molecular di€usion coecient ten-sor

D0 _{macroscopic hydrodynamic dispersion}

coecient tensor

D _{macroscopic di€usion coecient tensor} E cell function introduced in (47)

fE function de®ned in (82) fM vector function de®ned in (77) fZ vector function de®ned in (99)

I second-order identity tensor

k isotropic permeability k permeability tensor

k0 _{macroscopic permeability tensor} Kd sorption partition coecient

KH Henry's law constant

l _{mesoscopic length scale}

l0 _{macroscopic length scale}

M cell vector function introduced in (47) Md _{dynamic component ofM}

Ms _{static component ofM}

Mg molecular weight

p absolute air pressure Pe mesoscale Peclet number Pe0 _{macroscale Peclet number} Rg universal gas constant

s cell vector function introduced in (27)

t time

T1 short time scale

T2 long time scale

u air speci®c discharge ˆ(u,v,w)

~

u velocity ¯uctuation across the layers u0 e€ective speci®c discharge

U air velocity scale

x mesoscale coordinatesˆ …x;y;z† x0 _{macroscale coordinatesˆ …x}0_;y0_;z0_† Z function introduced in (85)

b retardation factor

dd;dD;dk;db ratios of layer properties as de®ned in

Eq. (111)

1

Tel.: 00852 2859 2622; fax: 00852 2858 5415; e-mail: cong@hkucc.hku.hk

2 INTRODUCTION

The spread of vapor in the unsaturated zone often plays an important role in the contamination or decontami-nation of an aquifer, if the contaminants are volatile enough. Soil vapor extraction, which works simply by creating air ventilation in the contaminated soil, is a proven e€ective method to clean up unsaturated soils contaminated with volatile organic compounds (e.g., see Pedersen and Curtis23). The clean-up rate however de-pends on the soil properties such as permeability which controls the air ¯ow-rate, and retardation factor and dispersion coecient which a€ect the vapor transport rate. Very often, the contaminated zone is strati®ed; layers of soil with di€erent properties are embedded in one-another. For modeling purposes, it is desirable if e€ective equations can be developed for the vapor transport in such a multi-layered heterogeneous system, where advantage is taken of the fact that the layer thickness is usually much smaller than the global length scale for the transport.

So far in studies on vapor transport in heteroge-neous soils, the focus has been limited to the e€ects due to a small number of layers or regions12,13,20. The ef-fects of multi-layered heterogeneity on vapor transport are yet to be investigated in detail. Typically, solute transport in a strati®ed medium can be studied with stochastic models, which invariably involve a number of statistical site characterization parameters such as variances, correlation lengths and distribution coe-cients for various material and chemical properties (e.g., see Dagan7 and Gelhar9 and the references therein).

A di€erent approach which allows more exact mathematical analysis can be taken when an idealized form, namely periodicity, is assumed of the medium

heterogeneity. In the presence of two or more con-trasting length scales, the macroscopic description of the problem can then be deduced formally with the multiple-scale method of homogenization3,25. In con-trast to many other averaging techniques, the ho-mogenization method derives phenomenological equations on the basis of micromechanics, in a general and rigorous manner without any closure hypothesis. Despite the idealization, the resultant e€ective equation can often contain information that lead to a better insight into the mechanisms of the problem. See Mei

et al.18 for a review of some of the applications of this method.

The application of averaging methods to convection± dispersion transport in porous media has received much attention in the past, but the focus has mainly been on the e€ects of heterogeneity at the microscopic (i.e., pore or grain level) scale. Typical works, among others, in-clude the method of moments by Brenner5, the volume averaging method by Whitaker27 and Carbonell and Whitaker6, and the homogenization method by Mauri14, Mei15and Auriault and Adler1. Clearly it is desirable if similar techniques can be applied to study e€ects due to ®eld scale heterogeneity.

Over the past decade, much theoretical work has been devoted to the development of so called homogenized models for convection±di€usion in heterogeneous po-rous media (see a treatise on this subject by Pan®lov and Pan®lova21, and the references therein). For instance, Fannjiang and Papanicolaou8discussed proofs of con-vergence for linear di€usion in an oscillating velocity ®eld. All such works however emphasize more on mathematical formalism or proofs than on physical applications. There indeed remains work to be done to utilize these theoretical developments to generate direct and explicit formulations readily applicable for speci®c problems.

In this paper, we extend the homogenization method to develop macroscopic equations for the air ¯ow and the transport of chemical vapor in a multi-layered un-saturated zone (Fig. 1(a)). We emphasize the direct up-scaling from the mesoscopic or formation scale (i.e, starting governing equations are already homogenized over the pore scale). To enable the analysis, we shall assume an ideal soil structure in which the heterogeneity is periodic. Macroscopic description of the heteroge-neous system is then obtained upon averaging over a unit cell of periodicity. On further assuming each cell being composed of two horizontal layers each with distinct properties (Fig. 1(b)), we may determine the e€ective coecients which appear in the macroscopic equations analytically in terms of the basic ¯ow vari-ables and soil properties. No calibration or statistical parameters need to be introduced. In short, the major contribution of this study lies in the systematic deriva-tion, using a versatile asymptotic method of homoge-nization, of the macroscale equation for vapor transport

dij Kronecker delta

e small perturbation parameter ˆl_/l0 g1;g2;g3 relations de®ned in (78) and (79)

g0

3 relation de®ned in (83)

c1;c2;c3 relations de®ned in (100) and (101) r;r0 nabla operators

…o₌o_x;o₌o_y;o₌o_z†;o₌o_x0_;o=oy0_;o=oz0_†

X a unit cell

q air density

qs solid density of soil grain

hg;hs;hw volume fractions of air, solid and water in soil

H absolute temperature

f1;f2;f3;f4 relations de®ned in (92)±(94) and (96)

^

…† a normalized quantity hi cell average of a quantity

in a multi-layered porous medium; the results can be of potential great practical interest to water resources en-gineers.

The assumptions and the method of analysis are further described in Section 3, which is followed by a deduction of the perturbation equations in Section 4. The e€ective equations for air ¯ow and vapor transport, up to second-order accuracy, are then found formally in terms of some cell functions in Sections 5 and 6 re-spectively. These equations are generally good for all kinds of periodic heterogeneity. In Section 7, the par-ticular case of periodic bi-layered formation is consid-ered. Solving the cell functions, the macroscopic equations are obtained with coecients explicitly ex-pressed in terms of ¯ow velocity and medium parame-ters. It will be seen that the second-order correction to the speci®c discharge arises because of air compress-ibility, and that the macrodispersion caused by the lay-ers is of Taylor type, i.e., the mechanical display-ersion coecient is proportional to the square of the ¯ow ve-locity. Following a normalization in Section 8, the fac-tors of the longitudinal and transverse components of

the hydrodynamic dispersion coecient are investigated in Section 9.

3 ASSUMPTIONS

A soil system can in general be described by three vastly di€erent length scales7. The microscopic or pore scale is the order of size of solid grains, while the mesoscopic scale measures the formation heterogeneity, and the macroscopic or regional scale is the global dimension to be a€ected by the chemical transport. In the present work, our focus is to study the e€ect of strati®cation on the vapor transport at the regional scale, and we shall start from governing equations already averaged over the microscale. No reference to the microscale will therefore be needed in this study.

For the development in Sections 4±6, it is sucient to assume that the soil heterogeneity is purely periodic, no matter what exact form it is. The special case of alter-nate stacking of two kinds of horizontal layers, in each of which the material is homogeneous and isotropic, will be considered in Section 7. In any case, the thickness of a layer is assumed to be much smaller than the regional length scale.

We consider an unsaturated zone where the moisture is at the residual level. Water with dissolved and sorbed chemicals is held immobilized in the micropores, while air with chemical vapor can ¯ow readily in the macropore space. It is also assumed that residual non-aqueous phase liquid (NAPL) is not present in the soil matrix, or has already been volatilized into vapor phase. Equilibrium chemical phase partitioning on the pore scale is assumed. That is, the local phase exchange among the aqueous, vapor and sorbed phases occurs much faster than the transport on the formation scale. It also means that rate-limiting e€ects such as aqueous di€usion in aggregates are minimal at the microscopic scale. We argue that this local equilibrium assumption is not necessarily a limiting one, since the primary time scale in this study is that of global transport which is usually long enough to render any non-equilibrium phase partitioning on the pore level to appear less signi®cant on the macroscale.

In the subsequent analysis, only reference to the formation (mesoscopic) scale (`) and the regional (macroscopic) scale (`0_{) will be required. The small ratio}

eˆ`=`0_{1 will be used as the ordering parameter in}

the analysis. The coordinates at the formation and re-gional scales are denoted by x …x1;x2;x3† …x;y;z† and x0 …x01;x02;x03† …x0;y0;z0† respectively. The medi-um heterogeneity is periodic so that it can be divided into unit cells X. The average over X of a quantity

f…x;x0_{†is de®ned by}

hfi 1 jXj

Z Z Z

X

f dX: …1†

Fig. 1. (a) A multi-layered unsaturated zone contaminated with a volatile organic chemical; (b) Blow-up of the periodic structure showing a unit cell X containing one Layer A

We further assume that the mesoscale Peclet number is of order unity:

PeU`=DˆO…1†; …2† where U and D are respectively the characteristic air velocity and bulk molecular di€usion coecient of chemical vapor in pore air. Physically eqn (2) means that over the formation scale vapor di€usion is as im-portant as advection. It also implies that the regional Peclet number is much greater than unity:

Pe0U`0_{=DˆO…1=e† 1; …3†}

or the vapor advection dominates at this upper scale. Two time scales are therefore required. While the global advection time scaleT1ˆ`0=U is the primary time scale for the transport, the longer time scaleT2ˆ`02=Dis for the global di€usion/dispersion:

T2ˆ`02=Dˆ …U`0=D†…`0=U† ˆPe0T1

ˆO…eÿ1T1† T1: …4†

4 BASIC GOVERNING EQUATIONS

Let us start from the governing equations for the air ¯ow and vapor transport on the formation scale. As-suming steady air ¯ow, the continuity states that

r …qu† ˆ0; …5†

whereqis the air density, anduis the speci®c discharge (i.e., ¯ow per unit area of soil) which can be found by Darcy's law:

uˆ ÿk rp; …6†

where k is the second-order tensor of air conductivity divided by the air speci®c weight, and pis the absolute air pressure. In eqn (6), the gravity e€ect has been ig-nored. Since the pressure change can be appreciable in soil vapor extraction, we need to consider the corre-sponding air density change. Modeling air as an ideal gas, the equation of state is

pMg

RgH

ˆq …7†

whereMgis the molecular weight of the air mixture,Rg is the universal gas constant and H is the absolute temperature. While the air ¯ow can be compressible, we assume that bothHandMgare constant so that the air density is linearly proportional to the pressure.

For the vapor transport, we invoke the advection± di€usion equation:

boC

o_t ‡ r …uC† ÿ r …D rC† ˆ0; …8† whereC is the vapor concentration (i.e, mass of chem-ical in vapor phase per unit volume of pore air),Dis the second-order tensor of bulk molecular di€usion coe-cient in the pore air, and b is the retardation factor. Note that by virtue of the high vapor di€usivity in air, the mechanical dispersion is relatively unimportant over

the mesoscale when eqn (2) is true, as shown by Ng and Mei19. Therefore D depends only on the soil micro-structure (typically given by the product of the tortuo-sity tensor and the molecular di€usion coecient in pure air), but not the ¯ow kinematics. Also, the retardation e€ect arises from the local equilibrium partitioning among the vapor, aqueous and sorbed phases. Hence, the retardation factor, a function of soil and chemical parameters, may be expressed as follows:

bˆhg‡…hw‡Kdhsqs†=KH …9†

where hg, hw and hs are respectively the air-®lled po-rosity, water and solid volume fractions, qs is the solid density,Kd is the sorption partition coecient andKHis the Henry's law constant.

Let us now introduce the following normalized quantities (distinguished by a hat):

xˆ`^x; tˆ …`0_{=U†^t; uˆU^u;} CˆC0C^; pˆ …U`0=k†p^; kˆk^k;

…10†

whereC0is a characteristic vapor concentration andkis a scale for the air conductivity. In terms of these nor-malized quantities, the above governing equations can now be expressed as follows.

^

r …p^^u† ˆ0; …11†

e^uˆ ÿk^_r^_{p^; …12†}

eboC^

o^_t ‡r …^ ^uC^† ÿr …^ D^ r^C^† ˆ0: …13†

From here on let us return to physical variables but keep the ordering parameter for identi®cation. We fur-ther introduce the following multiple-scale expansions:

f ˆf…0†‡ef…1†‡e2

f…2†‡e3

f…3†‡ ; …14†

r ! r ‡er0 …i:e:;o₌o_xi!o₌o_xi‡eo₌o_x0

i†; …15†

o₌o_t!o₌o_t1‡eo₌o_{t2; …16†}

where f…x;x0_{†stands for u, p or C. Upon substituting}

these expansions, we may obtain the following pertur-bation equations, which form the basis of subsequent deductions:

From eqn (11), O…1†:

r …p…0†_u…0†_{† ˆ0; …17†}

O…e†:

r0 _p…0†_u…0†_{† ‡ r …p}…0†_u…1†_‡p…1†_u…0†_{† ˆ0: …18†}

From eqn (12), O…1†:

0ˆ ÿk rp…0†; …19†

O…e†:

u…0†ˆ ÿk r p…1†‡ r0_p…0†_{; …20†}

O…e2_†:

From eqn (13), O…1†:

r …u…0†_C…0†_{† ÿ r …D rC}…0†_{† ˆ0; …22†}

O…e†:

boC

…0†

o_t1 ‡ r …u

…1†_C…0†_‡u…0†_C…1†_{† ‡ r}0 _u…0†_C…0†_†

ÿ r ‰D …rC…1†‡ r0_C…0†_{†Š ÿ r}0 _{D rC}…0†_{† ˆ0;}

…23† O…e2_†:

b oC

…0†

o_t2 ‡ o_C…1†

o_t1

‡ r …u…2†_C…0†_‡u…1†_C…1†

‡u…0†C…2†† ‡ r0 _u…1†_C…0†_‡u…0†_C…1†_†

ÿ r ‰D …rC…2†‡ r0_C…1†_†Š

ÿ r0 ‰D …rC…1†‡ r0C…0††Š ˆ0: …24†

5 AIR FLOW

Similar analysis has been given by Mei and Auriault16 who considered incompressible seepage ¯ow in a three-scale porous medium. From eqn (19) it is clear that

p…0†ˆp…0†…x0_{†; …25†}

or the leading order pressure is independent of x, and hence from eqn (17)

r u…0†_{ˆ0: …26†}

By linearity let us put the following formal expression forp…1†_{into eqn (20):}

p…1†ˆs r0p…0†‡p…1†…x0_{†; …27†}

wheres…x;x0_{†is a vector, and get}

u…0†_{ˆ ÿk r}0

p…0†; …28†

where

kˆk …rs‡I† …29†

in whichI is the second-order identity tensor. Further substituting eqn (28) into eqn (26) will give the cell governing equation fors:

r ‰k …rs‡I†Š ˆ0 inX; …30† which is subject to the boundary condition thatsis X -periodic. For uniqueness, we also require hsi ˆ0. Fi-nally we obtain the leading order macroscale equations for the continuity and speci®c discharge by taking X± average of eqns (18) and (28) respectively:

r0 _p…0†_hu…0†_{i† ˆ0; …31†}

hu…0†_{i ˆ ÿk}0 _r0_p…0†_{; …32†}

wherek0is the e€ective conductivity given by

k0ˆ hki ˆ hk …rs‡I†i: …33†

For the O…e†air ¯ow problem, we put eqns (28) and (27) into eqn (18) to get

r u…1†ˆ …r0_{k† r}0_p…0†_{‡ ‰r ks}

‡kŠ:…r0_p…0†_†…r0_p…0†_†=p…0†_‡k:r0_r0_p…0†_†;

…34†

which suggests the following representation for u…1†_:

u…1†ˆK1 r0p…0†‡K2:…r0p…0††…r0p…0††=p…0†

‡K3:r0…r0p…0†† …35† where K1 is a second-order tensor, and K2 and K3 are third-order tensors. Note that eqn (35) gives nonlinear corrections to Darcy's law. Plugging eqn (35) into eqn (34) and matching the coecients, we obtain the followingX-cell problems:

r K1ˆ r0k; …36†

r K2ˆ r …ks† ‡ k; …37†

r K3ˆk; …38†

andK1,K2andK3areX-periodic. On takingX-average of eqn (35), the O…e† e€ective speci®c discharge is ex-pressible by

hu…1†_{i ˆ hK}

1i r0p…0†‡ hK2i:…r0p…0††…r0p…0††=p…0† ‡ hK3i:r0…r0p…0††: …39† Note that the O…e†e€ective speci®c dischargehu…1†_iis

identically equal to zero for the particular case that the medium properties remain constant macroscopically (i.e., independent ofx0_{), and the material is isotropic and}

homogeneous within each region of soil. This is because when the properties do not change with respect tox0, the forcing term on the right-hand side of eqn (36) will be zero, which implies the vanishing of K1 and the ®rst term in eqn (39). Further by isotropy and homogeneity the X-averaged third-order tensors hK2i, hK3iare pro-portional to the permutation tensor, whose double-dot product with a symmetrical second-order tensor is al-ways equal to zero17. Consequently the second and the third terms in eqn (39) will also vanish. Therefore in such case the correction to Darcy's law is of O…e2_†.

6 VAPOR TRANSPORT

Let us ®rst show that C…0† _{is independent of the}

meso-scale. On multiplying eqn (22) with C…0† _{and using the}

product rule of di€erentiation, we obtain

…1=2†r …u…0†_C…0†2

† ‡ …1=2†C…0†2r u…0†

Z

X

D:rC…0†rC…0†dXˆ0: …41† Since D, a di€usion coecient, is positive de®nite, eqn (41) must imply that rC…0†_{is zero and C}…0† _is

con-stant with respect tox. Hence we can write that

C…0†ˆC…0†…x0_;t

1;t2†: …42†

We next derive the leading order e€ective transport equation. With eqns (42) and (23) reduces to

boC

…0†

o_t1 ‡ r …u

…1†_C…0†_‡u…0†_C…1†_{† ‡ r}0 _u…0†_C…0†_†

ÿ r ‰D …rC…1†‡ r0_C…0†_{†Š ˆ0: …43†}

On taking X-average of eqn (43) and using Gauss the-orem, we obtain the macroscopic transport equation at the leading order:

hbioC

…0†

o_t1 ‡ r

0 _hu…0†_iC…0†_{† ˆ0: …44†}

We may now ®nd a representation for C…1†_{. By}

eliminating the unsteady term o_C…0†_=ot

1 from eqns (43) and (44), we obtain

r ‰u…0†

C…1†ÿD rC…1†Š ˆ r h Dÿ~u…0†i _r0 C…0†

‡hÿr u…1†ÿ r0_u…0†_{‡ …b=hbi†r}0 _hu…0†_ii_C…0†_;

…45† where

~

u…0†_ˆu…0†_{ÿ …b=hbi†hu}…0†_{i …46†}

is the velocity ¯uctuation on the mesoscale. By linearity, we may put

C…1†ˆM r0_C…0†_‡EC…0†_‡C…1†_xj0_;t

1;t2†; …47†

whereM…x;x0_{†is a vector, E…x;x}0_{†is a scalar. The}

me-socell problems for these functions follow from eqn (45):

r ‰u…0†_{MÿD …I‡ rM†Š ˆ ÿ~u}…0†_{; …48†}

r ‰u…0†_{EÿD rEŠ ˆ ÿr u}…1†_{ÿ r}0_u…0†

‡ …b=hbi†r0 hu…0†_{i; …49†}

andMandEareX-periodic. For uniqueness, we further require these functions to satisfy

hbMi ˆ hbEi ˆ0: …50† Note that eqn (50) implies hbC…1†_{i ˆ hbiC}…1†_{; the}

ratio-nale is to ultimately combine C…1† with C…0† _{in the ®nal}

e€ective transport equation. Physically eqn (47) gives corrections toC…0†owing to mesoscale spatial variation in advection velocity as in the ®rst term, and in air compressibility as in the second term. Obviously for incompressible ¯ows Ewill be identically zero.

Finally we may derive the O…e† e€ective transport equation as follows. Using Gauss theorem and the uniqueness condition (50), we obtain the X-average of eqn (24):

hbioC

…0†

o_t2 ‡ hbi o_C…1†

o_t1 ‡ r

0 _‰hu…1†_iC…0†_{‡ hu}…0†_C…1†_iŠ

ÿ r0 ‰hD rC…1†i ‡ hDi r0C…0†Š ˆ0: …51†

On further substituting eqn (47), the above equation becomes

hbioC

…0†

o_t2 ‡ hbi o_C…1†

o_t1 ‡ r

0

u…1†C…0†

‡hu…0†_iC…1†

ÿ r0_D0 _r0_C…0†_{ˆ0; …52†}

where

u…1† hu…1†_{ÿD rE‡u}…0†_{Ei …53†}

is the O…e†correction to the speci®c discharge, and

D0 hD …I‡ rM† ÿu…0†Mi …54† is the e€ective hydrodynamic dispersion coecient.

Finally we may add eqn (52) multiplied by e to eqn (44) to get the macroscale e€ective transport equa-tion which is valid up to O…e†:

hbioC o_t ‡ r

0 _u0_{C† ÿer}0 _D0 _r0_{C† ˆ0; …55†}

where

CC…0†‡eC…1†; u0 hu…0†_{i ‡eu}…1† _56†

are the combined concentration and e€ective speci®c discharge respectively. Note that while advection is dominant at the leading order, dispersion is signi®cant only over a long time scale comparable toT2. Also note that mechanical dispersion now emerges on the macro-scale owing to the mesomacro-scale heterogeneity. The e pa-rameter in eqns (55) and (56) serves only to indicate the order of the associated term, and can be discarded when the physical terms are computed.

7 LAYERED FORMATION

To obtain analytical solutions for the cell functions, we further consider the simple yet practically important composite meso-structure; the medium is composed of two types of horizontal layers, A and B, stacking alter-nately in the vertical direction x3 or z. As shown in Fig. 1(b), a periodic unit cellXconsists of one Layer A (0<z<da) and one Layer B (da <z<da +db) where

da and db are the respective layer thicknesses. Within each layer the medium is isotropic and homogeneous, and the material properties are known. Speci®cally, the conductivity and di€usion tensors are

kij ˆkdij; DijˆDdij …57†

wheredij is the Kronecker delta, and

…k;D;b† ˆ …ka;Da;ba† in Layer A; …kb;Db;bb† in Layer B;

whereka,kb, Da,Db,ba andbb are all constants. Since strictly horizontal layers are considered, the dependence on the horizontal coordinates (x,y) on the mesoscale can be omitted. The de®nition of an X-average (1) is now reduced to

With dependence onzonly, the vectorsin eqn (30) has been found by Mei and Auriault16:

ds1

Also, the macroscale permeability tensor follows readily from eqn (33):

whereda and db are the thicknesses of layers A and B respectively. As expected, the equivalent longitudinal and transverse permeabilities are respectively the arith-metic mean and the harmonic mean of the individual permeabilities. From eqns (28) and (32), the following relations are also obtained:

u…0†ˆ ÿkop while the horizontal components of the speci®c dis-charge are discontinuous across the layers, the vertical component is continuous and equal to that of the macroscale. Unless stated otherwise, we shall from here on omit the leading order superscript, and use subscripts a and b to distinguish medium properties in layers A and B respectively.

Let us now consider the problem eqn (48) for M…z†. We ®rst decompose the function into static and dynamic components (i.e., the former is not, but the latter is a function of convection velocity):

MiˆMid‡M We anticipate that the static component in eqn (66) is giving rise to the bulk e€ective di€usion coecient, while the dynamic component will lead to the mechanical dispersion coecient of vapor in the layered medium.

The static componentMs

i can be found in a manner

On further integrating for separate layers as shown in Fig. 1(b), and making use of continuity ofMsacross an interface and the uniqueness condition eqn (69), we obtain

With regard to eqn (68), we may use the relations eqns (63)±(65) and eqns (70) and (71) to write its right-hand side as

d

Let us next consider the E problem eqn (49). Using the relationships eqns (18), (27) and (31), we may also write the right-hand side as

RHS…49† ˆ ‰r0_{p~u‡ …rs r}0_{p† uŠp}ÿ1

Hence the problem for Ecan be written as

d and E, the same spatial operations are applied on the left-hand sides, while the right-hand sides di€er only by a factor which is independent of z. Therefore the z -de-pendence for the two functions will be the same, and the solutions are in the form

…Md_;

whereZ…z†is to be found from the following boundary value problem:

with boundary conditions derived from the continuity of concentration and ¯ux at the layer interfaces:

Z…zˆdaÿ† ˆZ…zˆda‡†; Z…zˆ0† ˆZ…zˆda‡db†;

and the uniqueness condition

hbZi ˆba

When hwi is not zero, the solution to the above problem is zero, the solution is reduced to

Z…z† ˆ

Note also the limit

f1!

Before further substitutions, let us ®rst obtain the following cell average:

On decomposingMaccording to eqn (66), and using eqns (85) and (98), the hydrodynamic dispersion tensor (54) may now be developed as follows.

D0ˆ hD I‡ r‰Ms_‡

D₁₁ˆD₂₂ˆ hDi ˆDada‡Dbdb

Similar to the e€ective conductivity, the longitudinal and the transverse components of the e€ective di€usion coecient are respectively given by the arithmetic mean and the harmonic mean of the layer values. Obviously in eqn (102) the hydrodynamic dispersion coecientD0is the sum of the e€ective molecular di€usion coecientD and the mechanical dispersion coecientf_Z f_Mwhich is proportional to the square of the velocity. Substituting eqns (77) and (99) Eqn. (102) can be written in the al-ternative form

Finally the combined e€ective speci®c discharge eqn (56) may also be expressed as follows:

u0ˆ hu…0†_{i ‡ hu}…1†_{i ‡ hu}…0†

or, omitting the leading order superscript and using eqn (99),

u0_iˆ …1ÿcifE†huii: …108†

Note that use has been made ofhu…1†_{i ˆ0 which is true}

for the present particular case (see the discussion in the last paragraph of Section 5).

8 NORMALIZED TRANSPORT EQUATION

Let us normalize the e€ective transport equation using the normalization introduced in eqn (10):

hbioC^ signi®cant over a long time scale comparable toT2.

We further introduce the following ratios of layer properties

ddˆdb=da; dkˆkb=ka; dDˆDb=Da; dbˆbb=ba; …111†

and the following dimensionless variables and parame-ters:

Note that f^1 is always greater than zero for any ®nite value ofhw^iand it tends to a ®nite limit when hw^i be-comes zero:

^

f1!PeaPeb=12 ashw^i !0: …119† On substituting eqn (82), the normalized form of the speci®c discharge eqn (108) may now be written as (for

iˆ1;2;3):

Also, the normalized form of the hydrodynamic dis-persion coecient (106) can be written as (for

i;jˆ1;2;3):

^

D_ij0 ˆD^_ij‡^c_ig^_jh^uiihu^ji; …121† whereD^_ijis the normalized di€usion coecient given by

9 THE DISPERSION COEFFICIENT

We stress that the dispersion coecient given in eqn (121) is a function of both the convection ve-locity and the layer properties. The vapor dispersion is enhanced at the macroscopic scale because of the layered heterogeneity. In this section the factors of the hydrodynamic dispersion are examined in further detail. For this purpose, we concentrate on the lon-gitudinal and the transverse components of the dis-persion coecient, D^011 and D^033, whose explicit forms are:

^

D0₁₁ˆ1‡dDdd 1‡dd

‡…dkÿdb† 2

…1‡dd†…dd‡dD†dd …1‡dbdd†2…1‡dkdd†2

1ÿ

…ehwi^Pea_ÿ1†…ehwi^Pebÿ1†…Pea‡Peb† …ehwi…^Pea‡Peb†_ÿ1†hw^iPe

aPeb

h i

hu^i hw^i

2

forhw^i 6ˆ0 PeaPeb

12

ÿ

hu^i2 forhw^i ˆ0

8

<

:

;

…125†

^

D0₃₃ˆdD…1‡dd†

dd‡dD

‡…1ÿdbdD† 2

…1‡dd†dd …1‡dbdd†2…dd‡dD†

1ÿ

…ehwi^Pea_ÿ1†…ehwi^_Pebÿ1†…Pe a‡Peb† …ehwi…^Pea‡_Peb†

ÿ1†hw^iPeaPeb

h i

forhw^i 6ˆ0;

0 forhw^i ˆ0: 8

<

:

…126†

The second (mechanical dispersion) terms in the above expressions are always non-negative; they are zero only when the corresponding velocity component vanishes or when dk ˆdb for D^011, dbdDˆ1 for D^033. Therefore, as expected, in the absence of heteroge-neity the mechanical dispersion of vapor in a porous medium becomes subdominant. Note also that these two dispersion components are even functions of hw^i, i.e., D^0

11…hw^i† ˆD^011…ÿhw^i† and D^033…hw^i† ˆD^033…ÿhw^i†. It can readily be seen fromD^0₁₁ that the longitudinal mechanical dispersion is of the magnitude U2_d2

b=Db, which is the form of Taylor dispersion in a capillary tube2,26. This is reasonable since in our model the dis-persion is essentially caused by the e€ect of velocity variation across the layers interacting with molecular di€usion that promotes lateral transfer between layers with di€erent retardation factor and ¯ow velocity. Similar results were also obtained for the special case of structured media composed of mobile and immobile regions4,11,22,24. The di€usive exchange between these two regions can e€ectively increase the dispersion coef-®cient by the amount proportional toU2

mdim2=Dim, where

Um is the velocity in the mobile region,dimis a charac-teristic dimension of the immobile region, andDimis the molecular di€usion coecient in the immobile region. It suggests that the longitudinal dispersion has a strong dependence on the velocity along the more permeable layer, and the di€usion time scale across the less per-meable layer.

We also remark that the transverse mechanical dis-persion vanishes when the ¯ow is strictly horizontal. This is consistent with the analysis by statistical means which yields a zero transverse asymptotic macro-dispersivity when the heterogeneous structure of the medium does not create transverse dispersion10. In the present case, transverse mechanical dispersion occurs only in the presence of transverse ¯ow, but with a rather weak dependence on it, as will be seen shortly.

To facilitate further discussions, we assume that the vapor transport takes place slower in layer B than layer A. Physically, this can be realized when the material in layer B is more ®ne-grained (therefore smaller conduc-tivity and di€usivity) and has a larger organic matter content (therefore a larger retardation factor) than that in layer A. The material contrasts are thereforedk<1,

dD<1 anddb>1. Speci®cally four cases of parameter

values given in Table 1 are considered on studying their e€ects on the dispersion coecient components. In each case, Peais taken to be unity. By comparing with Case I, the e€ects of a larger contrast in conductivity and re-tardation factor can be examined in Case II. Also by comparing with Case II, the e€ects of a larger contrast in layer di€usivity can be examined in Case III. Again by comparing with Case III the e€ects of a thicker Layer B can be studied in Case IV.

Fig. 2 shows for the four cases the longitudinal dis-persion coecientD^0

11 as a function of the longitudinal speci®c discharge hu^i when there is no transverse ¯ow (i.e.,hw^i ˆ0). According to eqn (125)D^0₁₁increases with

Table 1. Four cases of values ofk, ,D andd on studying their e€ects on the dispersion coecient components

Case dk db dD dd

I 0.5 10 0.5 1

II 0.05 100 0.5 1

III 0.05 100 0.05 1

IV 0.05 100 0.05 2

Fig. 2. The longitudinal hydrodynamic dispersion coecient ^

the square ofhu^i. The increase is relatively mild in Case I whenhu^iis between 0 and 1. A ten-time increase in the contrast of the conductivity and the retardation factor (Case II) only slightly increases the e€ect of the velocity on dispersion. However a ten-time increase in the dif-fusion coecient contrast (Case III) can appreciably enhance the increase of the dispersion with velocity. The enhancement is even more dramatic when the thickness of layer B is doubled (Case IV). The results con®rm our earlier statement that the longitudinal dispersion has a strong relationship with both the longitudinal velocity and the e€ective di€usion time scale across the less permeable layer.

Fig. 3 shows similar plots as in Fig. 2, but for hw^i ˆ0:5. By comparing the two ®gures, it is clear that the vertical ¯ow suppresses the increase of the longitu-dinal dispersion coecient with the horizontal velocity, more in Cases III and IV than Cases I and II. This is reasonable because the transverse ¯ow can reduce the rate of transport along the layers and therefore lower the extent of longitudinal dispersion.

The variations between the transverse dispersion co-ecient and the transverse speci®c discharge are shown in Fig. 4. Over the range of hw^i from 0 to 5, D^0₃₃ in-creases slightly in Cases I and II, but remains practically unchanged in Cases III and IV. In fact the transverse dispersion coecient does not as strongly depend on the velocity as the longitudinal dispersion coecient does. It is clear from eqn (126) that _D^0

33 does not increase with the square of the vertical velocity, but only with the value inside the square brackets which has a maximum of unity when hw^ibecomes in®nite. In addition, the ef-fect of the velocity disappears when the product dbdD happens equal to unity. Therefore the enhancement of the transverse mechanical dispersion by the seepage velocity is in general very limited. The transverse dis-persion is more dominated by the e€ective molecular di€usion normal to the layers which is controlled bydD or the di€usivity in the less permeable layer.

10 CONCLUDING REMARKS

In this paper we have presented a mathematical deri-vation of the macroscopic equations for the transport of chemical vapor in a multi-layered unsaturated zone. Based on the assumptions of cell periodicity and a sharp contrast in length scales, the method of homogenization is applied in order to identify the range of validity of the theory. The e€ective transport eqn (55), in which the speci®c discharge and dispersion coecient are expres-sed in terms of some cell functions, is good for all forms of periodic heterogeneity with a characteristic dimension much smaller than the global length scale. Analytical solutions to the cell functions have been obtained for the simple case of bi-layered unit cell. In terms of ¯ow ve-locity and medium parameters, the speci®c discharge and the hydrodynamic dispersion coecient are given respectively by eqns (108) and (106), or in dimensionless form, eqns (120) and (121).

For this periodic bi-layered composite, the factors of the dispersion components have been examined in some detail. The longitudinal dispersion is found to be caused by Taylor mechanism. That is, the coecient varies with the square of the longitudinal velocity, and linearly with the di€usion time across the less permeable layer. A larger contrast in the retardation factor and conductiv-ity can only modestly increase the dependence of the dispersion on the ¯ow velocity. Also, an increase in the transverse ¯ow velocity will lower the value of longitu-dinal dispersion coecient. On the other hand, owing to the ideal layering structure, the transverse mechanical dispersion does not depend on the longitudinal ¯ow, and only weakly depends on the transverse ¯ow velocity. Extensions of the present theory are desirable for the cases of: (i) strongly kinetic phase exchange on the pore scale so that a source term may appear in the transport eqn (8); (ii) more complex periodic layering structures;

Fig. 3. The longitudinal hydrodynamic dispersion coecient ^

D0

11as a function of the longitudinal speci®c dischargehu^ifor

the four cases; the transverse speci®c dischargehw^i ˆ0:5.

Fig. 4.The transverse hydrodynamic dispersion coecient_D^0

33

(iii) a larger contrast in the layer properties, and (iv) vapor transport in¯uenced by gravity.

ACKNOWLEDGEMENTS

Funding for this research was made available from CRCG research grant 335/064/0082 awarded by the University of Hong Kong. The author is grateful to Prof.Chiang C. Mei of the Department of Civil and Environmental Engineering at the Massachusetts Insti-tute of Technology for his inspiration and guidance when the work was initiated. Thanks also go to Prof.Mikhail Pan®lov of the Russian Academy of Sci-ences for drawing the author's attention to some of his pertinent work.

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