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Advances in Water Resources Vol. 22, No. 6, pp 611±622, 1999
Ó 1999 Elsevier Science Ltd
Printed in Great Britain. All rights reserved
0309-1708/99/$ ± see front matter

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 unsaturated zone with a periodic structure of heterogeneity. The e€ective speci®c discharge and hydrodynamic dispersion coecient are expressible in terms of some
cell functions, whose analytical solutions are sought for the simple case of alternate 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 contrasts on the longitudinal and transverse components of the hydrodynamic dispersion coecient are investigated. Ó 1999 Elsevier Science Limited. All rights
reserved

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

KH
l
l0
M
Md
Ms
Mg
p
Pe
Pe0
Rg
s
t
T1
T2
u
~u

u0
U
x
x0
Z
b
d d ; dD ; dk ; db

1 NOMENCLATURE
C
C0
d
D
D
D0
D
E
fE
fM
fZ

I
k
k
k0
Kd

vapor concentration
characteristic vapor concentration
layer thickness
isotropic bulk molecular di€usion coef®cient
bulk molecular di€usion coecient tensor
macroscopic hydrodynamic dispersion
coecient tensor
macroscopic di€usion coecient tensor
cell function introduced in (47)
function de®ned in (82)
vector function de®ned in (77)
vector function de®ned in (99)
second-order identity tensor
isotropic permeability

permeability tensor
macroscopic permeability tensor
sorption partition coecient

1
Tel.: 00852 2859 2622; fax: 00852 2858 5415; e-mail:
cong@hkucc.hku.hk

611

Henry's law constant
mesoscopic length scale
macroscopic length scale
cell vector function introduced in (47)
dynamic component of M
static component of M
molecular weight
absolute air pressure
mesoscale Peclet number
macroscale Peclet number

universal gas constant
cell vector function introduced in (27)
time
short time scale
long time scale
air speci®c discharge ˆ (u,v,w)
velocity ¯uctuation across the layers
e€ective speci®c discharge
air velocity scale
mesoscale coordinates ˆ …x; y; z†
macroscale coordinates ˆ …x0 ; y 0 ; z0 †
function introduced in (85)
retardation factor
ratios of layer properties as de®ned in
Eq. (111)

612
dij
e
g1 ; g2 ; g3

g03
c1 ; c2 ; c3
r; r0
X
q
qs
hg ; hs ; hw
H
f 1 ; f2 ; f3 ; f4
^
…
†
h
i
…
†…m†

C. -O. Ng
Kronecker delta
small perturbation parameter ˆ l/l0
relations de®ned in (78) and (79)
relation de®ned in (83)
relations de®ned in (100) and (101)

nabla operators
…o=ox; o=oy; o=oz†; …o=ox0 ; o=oy 0 ; o=oz0 †
a unit cell
air density
solid density of soil grain
volume fractions of air, solid and water
in soil
absolute temperature
relations de®ned in (92)±(94) and (96)
a normalized quantity
cell average of a quantity
the mth perturbation expansion of a
quantity

2 INTRODUCTION
The spread of vapor in the unsaturated zone often plays
an important role in the contamination or decontamination 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 depends 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 heterogeneous soils, the focus has been limited to the e€ects due
to a small number of layers or regions12,13,20. The effects 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 contrasting length scales, the macroscopic description of
the problem can then be deduced formally with the
multiple-scale method of homogenization3,25. In contrast to many other averaging techniques, the homogenization 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, include the method of moments by Brenner5, the volume

averaging method by Whitaker27 and Carbonell and
Whitaker6, and the homogenization method by Mauri14,
Mei15 and 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 porous media (see a treatise on this subject by Pan®lov and
Pan®lova21, and the references therein). For instance,
Fannjiang and Papanicolaou8 discussed proofs of convergence 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 unsaturated zone (Fig. 1(a)). We emphasize the direct upscaling 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 heterogeneous 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 variables and soil properties. No calibration or statistical
parameters need to be introduced. In short, the major
contribution of this study lies in the systematic derivation, using a versatile asymptotic method of homogenization, of the macroscale equation for vapor transport

Macroscopic equations for vapor transport in a multi-layered unsaturated zone

613

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

3 ASSUMPTIONS

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
(0 < z < da ) and one layer B (da < z < da + db ).

in a multi-layered porous medium; the results can be of
potential great practical interest to water resources engineers.
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 respectively. These equations are generally good for all
kinds of periodic heterogeneity. In Section 7, the particular case of periodic bi-layered formation is considered. Solving the cell functions, the macroscopic
equations are obtained with coecients explicitly expressed in terms of ¯ow velocity and medium parameters. It will be seen that the second-order correction to
the speci®c discharge arises because of air compressibility, and that the macrodispersion caused by the layers is of Taylor type, i.e., the mechanical dispersion
coecient is proportional to the square of the ¯ow velocity. Following a normalization in Section 8, the factors of the longitudinal and transverse components of

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 alternate 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 ratelimiting 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
e ˆ `=`0  1 will be used as the ordering parameter in
the analysis. The coordinates at the formation and regional scales are denoted by x  …x1 ; x2 ; x3 †  …x; y; z†
and x0  …x01 ; x02 ; x03 †  …x0 ; y 0 ; z0 † respectively. The medium 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
hf i 

1
jXj

Z Z Z

X

f dX:

…1†

614

C. -O. Ng

We further assume that the mesoscale Peclet number
is of order unity:
Pe  U `=D ˆ O…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 important as advection. It also implies that the regional
Peclet number is much greater than unity:
…3†
Pe0  U `0 =D ˆ O…1=e†  1;
or the vapor advection dominates at this upper scale.
Two time scales are therefore required. While the global
advection time scale T1 ˆ `0 =U is the primary time scale
for the transport, the longer time scale T2 ˆ `02 =D is for
the global di€usion/dispersion:
T2 ˆ `02 =D ˆ …U `0 =D†…`0 =U † ˆ Pe0 T1
ÿ1

ˆ O…e T1 †  T1 :

…4†

Let us start from the governing equations for the air
¯ow and vapor transport on the formation scale. Assuming steady air ¯ow, the continuity states that
…5†

where q is the air density, and u is the speci®c discharge
(i.e., ¯ow per unit area of soil) which can be found by
Darcy's law:
u ˆ ÿk
rp;

…6†

where k is the second-order tensor of air conductivity
divided by the air speci®c weight, and p is the absolute
air pressure. In eqn (6), the gravity e€ect has been ignored. Since the pressure change can be appreciable in
soil vapor extraction, we need to consider the corresponding air density change. Modeling air as an ideal
gas, the equation of state is
pMg
ˆq
…7†
Rg H
where Mg is 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 both H and Mg are constant so that the air
density is linearly proportional to the pressure.
For the vapor transport, we invoke the advection±
di€usion equation:
oC
‡ r
…uC† ÿ r
…D
rC† ˆ 0;
…8†
ot
where C is the vapor concentration (i.e, mass of chemical in vapor phase per unit volume of pore air), D is 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
b

b ˆ hg ‡ …hw ‡ Kd hs qs †=KH

…9†

where hg , hw and hs are respectively the air-®lled porosity, water and solid volume fractions, qs is the solid
density, Kd is the sorption partition coecient and KH is
the Henry's law constant.
Let us now introduce the following normalized
quantities (distinguished by a hat):
x ˆ `^
x;

4 BASIC GOVERNING EQUATIONS

r
…qu† ˆ 0;

the mesoscale when eqn (2) is true, as shown by Ng and
Mei19. Therefore D depends only on the soil microstructure (typically given by the product of the tortuosity 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:

t ˆ …`0 =U †^t;

^
C ˆ C0 C;

u ˆ U ^u;

0

^
p ˆ …U ` =k†p;

^
k ˆ k k;

…10†

where C0 is a characteristic vapor concentration and k is
a scale for the air conductivity. In terms of these normalized quantities, the above governing equations can
now be expressed as follows.
r^
…^
p^u† ˆ 0;

…11†

^
r^p^;
e^u ˆ ÿk

…12†

oC^
^
r^C†
^ ÿ r^
…D
^ ˆ 0:
‡ r^
… ^uC†
…13†
o^t
From here on let us return to physical variables but
keep the ordering parameter for identi®cation. We further introduce the following multiple-scale expansions:
eb

f ˆ f …0† ‡ ef …1† ‡ e2 f …2† ‡ e3 f …3† ‡


;

…14†

r ! r ‡ er0

…15†

…i:e:; o=oxi ! o=oxi ‡ eo=ox0i †;

o=ot ! o=ot1 ‡ eo=ot2 ;

…16†

0

where f …x; x † stands for u, p or C. Upon substituting
these expansions, we may obtain the following perturbation equations, which form the basis of subsequent
deductions:
From eqn (11), O…1†:
r
…p…0† u…0† † ˆ 0;

…17†

O…e†:
r0
…p…0† u…0† † ‡ r
…p…0† u…1† ‡ p…1† u…0† † ˆ 0:

…18†

From eqn (12), O…1†:
0 ˆ ÿk
rp…0† ;

…19†

O…e†:


u…0† ˆ ÿk
rp…1† ‡ r0 p…0† ;

…20†



u…1† ˆ ÿk
rp…2† ‡ r0 p…1† :

…21†

O…e2 †:

615

Macroscopic equations for vapor transport in a multi-layered unsaturated zone
From eqn (13), O…1†:
…0†

…0†

…0†

r
…u C † ÿ r
…D
rC † ˆ 0;

…22†

O…e†:
oC …0†
‡ r
…u…1† C …0† ‡ u…0† C …1† † ‡ r0
…u…0† C …0† †
b
ot1
ÿ r
‰D
…rC …1† ‡ r0 C …0† †Š ÿ r0
…D
rC …0† † ˆ 0;
…23†
2

O…e †:
 …0†

oC
oC …1†
‡ r
…u…2† C …0† ‡ u…1† C …1†
‡
b
ot2
ot1
ÿ r
‰D
…rC …2† ‡ r0 C …1† †Š
…24†

5 AIR FLOW
16

Similar analysis has been given by Mei and Auriault
who considered incompressible seepage ¯ow in a threescale porous medium. From eqn (19) it is clear that
p…0† ˆ p…0† …x0 †;

…25†

or the leading order pressure is independent of x, and
hence from eqn (17)
r
u…0† ˆ 0:

…26†

By linearity let us put the following formal expression
for p…1† into eqn (20):
p…1† ˆ s
r0 p…0† ‡ p…1† …x0 †;

…27†

where s…x; x0 † is a vector, and get

r0 p…0† ;
u…0† ˆ ÿk

…28†

where
 ˆ k
…rs ‡ I†
k

…29†

in which I is the second-order identity tensor. Further
substituting eqn (28) into eqn (26) will give the cell
governing equation for s:
r
‰k
…rs ‡ I†Š ˆ 0 in X;

…34†
which suggests the following representation for u…1† :
u…1† ˆ K 1
r0 p…0† ‡ K 2 : …r0 p…0† †…r0 p…0† †=p…0†
…35†

where K 1 is a second-order tensor, and K 2 and K 3 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
following X-cell problems:

r
K 1 ˆ r0
k;

…36†

 ‡ k;

r
K 2 ˆ r
…ks†

…37†


r
K 3 ˆ k;

…38†

and K 1 , K 2 and K 3 are X-periodic. On taking X-average
of eqn (35), the O…e† e€ective speci®c discharge is expressible by
hu…1† i ˆ hK 1 i
r0 p…0† ‡ hK 2 i: …r0 p…0† †…r0 p…0† †=p…0†
‡ hK 3 i: r0 …r0 p…0† †:

…39†

Note that the O…e† e€ective speci®c discharge hu…1† i is
identically equal to zero for the particular case that the
medium properties remain constant macroscopically
(i.e., independent of x0 ), and the material is isotropic and
homogeneous within each region of soil. This is because
when the properties do not change with respect to x0 , the
forcing term on the right-hand side of eqn (36) will be
zero, which implies the vanishing of K 1 and the ®rst
term in eqn (39). Further by isotropy and homogeneity
the X-averaged third-order tensors hK 2 i, hK 3 i are proportional to the permutation tensor, whose double-dot
product with a symmetrical second-order tensor is always 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 †.

…30†

which is subject to the boundary condition that s is Xperiodic. For uniqueness, we also require hsi ˆ 0. Finally 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† ˆ 0;

…31†

hu…0† i ˆ ÿk0
r0 p…0† ;

…32†

0

where k is the e€ective conductivity given by
 ˆ hk
…rs ‡ I†i:
k0 ˆ hki

 …r0 p…0† †…r0 p…0† †=p…0† ‡ k:
 r0 …r0 p…0† †;
‡ kŠ:

‡ K 3 : r0 …r0 p…0† †

‡ u…0† C …2† † ‡ r0
…u…1† C …0† ‡ u…0† C …1† †

ÿ r0
‰D
…rC …1† ‡ r0 C …0† †Š ˆ 0:

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

r0 p…0† ‡ ‰r
ks

r
u…1† ˆ …r0
k†

…33†

6 VAPOR TRANSPORT
Let us ®rst show that C …0† is independent of the mesoscale. On multiplying eqn (22) with C …0† and using the
product rule of di€erentiation, we obtain
2

2

…1=2†r
…u…0† C …0† † ‡ …1=2†C …0† r
u…0†
ˆ r
‰…D
rC …0† †C …0† Š ÿ D: rC …0† rC …0† :

…40†

The second term on the left-hand side vanishes by virtue
of eqn (26). Further taking X±average of eqn (40) and
invoking the X-periodicity, we ®nally obtain

616
Z

C. -O. Ng
D: rC …0† rC …0† dX ˆ 0:

…41†

X

Since D, a di€usion coecient, is positive de®nite,
eqn (41) must imply that rC …0† is zero and C …0† is constant with respect to x. Hence we can write that
C …0† ˆ C …0† …x0 ; t1 ; t2 †:

…42†

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

oC …0†
‡ r
…u…1† C …0† ‡ u…0† C …1† † ‡ r0
…u…0† C …0† †
ot1
…43†
ÿ r
‰D
…rC …1† ‡ r0 C …0† †Š ˆ 0:

oC …0†
oC…1†
‡ r0
‰hu…1† iC …0† ‡ hu…0† C …1† iŠ
‡ hbi
ot1
ot2
…51†
ÿ r0
‰hD
rC …1† i ‡ hDi
r0 C …0† Š ˆ 0:

hbi

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


oC …0†
oC…1†
‡ r0
u…1† C …0† ‡hu…0† iC…1†
‡ hbi
hbi
ot1
ot2


…52†
ÿ r0
D0
r0 C …0† ˆ 0;

where

u…1†  hu…1† ÿ D
rE ‡ u…0† Ei

…53†

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

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

oC …0†
‡ r0
…hu…0† iC …0† † ˆ 0:
…44†
ot1
We may now ®nd a representation for C …1† . By
eliminating the unsteady term oC …0† =ot1 from eqns (43)
and (44), we obtain
h
i
u…0†
r0 C …0†
r
‰u…0† C …1† ÿ D
rC …1† Š ˆ r
D ÿ ~
h
i
‡ ÿr
u…1† ÿ r0
u…0† ‡ …b=hbi†r0
hu…0† i C …0† ;

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 equation which is valid up to O…e†:

hbi

…45†

where
~
u…0† ˆ u…0† ÿ …b=hbi†hu…0† i

…46†

is the velocity ¯uctuation on the mesoscale. By linearity,
we may put
C …1† ˆ M
r0 C …0† ‡ EC …0† ‡ C…1† …xj0 ; t1 ; t2 †;

…47†

D0  hD
…I ‡ rM† ÿ u…0† Mi

hbi

oC
‡ r0
…u0 C† ÿ er0
…D0
r0 C† ˆ 0;
ot

…54†

…55†

where
C  C …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 to T2 . Also note
that mechanical dispersion now emerges on the macroscale owing to the mesoscale heterogeneity. The e parameter 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.

where M…x; x0 † is a vector, E…x; x0 † is a scalar. The mesocell problems for these functions follow from eqn (45):
u…0† ;
r
‰u…0† M ÿ D
…I ‡ rM†Š ˆ ÿ~

…48†

r
‰u…0† E ÿ D
rEŠ ˆ ÿr
u…1† ÿ r0
u…0†
‡ …b=hbi†r0
hu…0† i;

…49†

and M and E are X-periodic. For uniqueness, we further
require these functions to satisfy
hbMi ˆ hbEi ˆ 0:

…50†
…1†

…1†

Note that eqn (50) implies hbC i ˆ hbiC ; the rationale is to ultimately combine C…1† with C …0† in the ®nal
e€ective transport equation. Physically eqn (47) gives
corrections to C …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 E will 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):

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 alternately in the vertical direction x3 or z. As shown in
Fig. 1(b), a periodic unit cell X consists 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 ˆ kdij ;

Dij ˆ Ddij

where dij is the Kronecker delta, and

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

…57†

…58†

617

Macroscopic equations for vapor transport in a multi-layered unsaturated zone
where ka , kb , Da , Db , ba and bb 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
2d
3
daZ‡db
Za
…59†
f dz5:
hf i  …da ‡ db †ÿ1 4 f dz ‡
0

da

With dependence on z only, the vector s in eqn (30) has
been found by Mei and Auriault16:
ds3
k ÿ1
ˆ ÿ1 ‡ ÿ1 :
dz
hk i

ds1 ds2
ˆ
ˆ 0;
dz
dz

…60†

Also, the macroscale permeability tensor follows readily
from eqn (33):
ka da ‡ kb db
;
…61†
da ‡ db
 

ds3
ka kb …da ‡ db †
0
ˆ hk ÿ1 iÿ1 ˆ
k33 ˆ k 1 ‡
…62†
dz
ka db ‡ kb da
0
0
ˆ k22
ˆ hki ˆ
k11

where da and db are the thicknesses of layers A and B
respectively. As expected, the equivalent longitudinal
and transverse permeabilities are respectively the arithmetic mean and the harmonic mean of the individual
permeabilities. From eqns (28) and (32), the following
relations are also obtained:
u…0†

op…0†
k …0†
ˆ
ˆ ÿk
hu i;
ox0
hki

…i ˆ 1; 2; 3†

such that
 

d
dMis
ˆ 0;
D d3i ‡
dz
dz


d
dMid
d
d
wMi ÿ D
ˆ ÿ~
ui ÿ …wMis †;
dz
dz
dz

hbMid i ˆ hbMis i ˆ 0:

…66†

dM3s
Dÿ1
ˆ ÿ1 ‡ ÿ1
hD i
dz

ÿdb fMs Layer A;
ˆ
da fMs
Layer B
where
fMs ˆ

…71†

Da ÿ Db
:
Da db ‡ Db da

…72†

On further integrating for separate layers as shown in
Fig. 1(b), and making use of continuity of M s across an
interface and the uniqueness condition eqn (69), we
obtain
M1s …z† ˆ M2s …z† ˆ 0 in both layers

…73†

and
M3s …z†

ˆ



ÿdb fMs z ‡ da db fMs =2;

0 6 z < da ;

da fMs z ÿ da …2da ‡ db †fMs =2;

da 6 z < d a ‡ d b
…74†

Note M3s also satis®es
Z
Z
M3s dz ˆ
Layer A

M3s dz ˆ 0:

…75†

Layer B

With regard to eqn (68), we may use the relations
eqns (63)±(65) and eqns (70) and (71) to write its righthand side as
dMis
RHS…68† ˆ ÿ~
ui ÿ w
dz 



b
k
b
k
ˆ
hui;
hvi;
ÿ
ÿ
hbi hki
hbi hki



b
Dÿ1
ÿ ÿ1 hwi
hbi hD i

Layer A;
db f M
ˆ
ÿda f M Layer B
where
f M ˆ …g1 hui; g2 hvi; g3 hwi†;

…76†

…77†

and
g 1 ˆ g2 ˆ

…67†
g3 ˆ
…68†

…69†

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 component Mis can be found in a manner
similar to s:
dM1s dM2s
ˆ
ˆ0
…70†
dz
dz

…63†

op…0†
k …0†
ˆ
…64†
v…0† ˆ ÿk
hv i;
0
oy
hki


…0†
ds3 op…0†
0 op
…0†
ˆ
ÿk
ˆ hw…0† i …65†
w ˆ ÿk 1 ‡
33
dz
oz0
oz0
where we have denoted …u1 ; u2 ; u3 † by …u; v; w†. Therefore
while the horizontal components of the speci®c discharge 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):
Mi ˆ Mid ‡ Mis

and

ba ÿ b b
ka ÿ kb
;
ÿ
ba da ‡ bb db ka da ‡ kb db

ba ÿ bb
Da ÿ Db
‡
:
ba da ‡ bb db Da db ‡ Db da

Hence the problem for M d can be written as

…78†
…79†

:

618

C. -O. Ng


 
db f M
Layer A;
d
dM d
d
ˆ
hwiM ÿ D
…80†
dz
dz
ÿda f M Layer B
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
0

0

d

…M ; E† ˆ Z…z†  …f M ; fE †;

…85†

and
M…z† ˆ Z…z†f M ‡ …0; 0; M3s †;

…86†

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

 
0 < z < da ;
db
d
dZ
…87†
ˆ
hwiZ ÿ D
dz
dz
ÿda da < z < da ‡ db ;
with boundary conditions derived from the continuity of
concentration and ¯ux at the layer interfaces:
Z…z ˆ

ˆ Z…z ˆ

da‡ †;

Z…z ˆ 0† ˆ Z…z ˆ da ‡ db †;
…88†





dZ
dZ
wZ ÿ Da
ˆ wZ ÿ Db
;
dz zˆdaÿ
dz zˆda‡

…89†

and the uniqueness condition
hbZi ˆ ba

Zda
0

Z dz ‡ bb

daZ‡db

Z dz ˆ 0:

f1 ˆ
ÿ

ÿ1

u ‡ …rs
r p†
uŠp
RHS…49† ˆ ‰r p
~

…81†
Layer A;
db fE
ˆ
ÿda fE Layer B;
where


op
op
op ÿ1
0
fE ˆ ÿ g1 hui 0 ‡ g2 hvi 0 ‡ g3 hwi 0 p
…82†
ox
oy
oz
in which g1 and g2 are given by eqn (78) and


ba ÿ bb
ka ÿ kb
0
:
…83†
g3 ˆ
‡
ba da ‡ bb db ka db ‡ kb da
Hence the problem for E can be written as

 
Layer A;
db fE
d
dE
ˆ
hwiE ÿ D
…84†
dz
dz
ÿda fE Layer B
By now it is clear that in eqns (80) and (84) for M 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-dependence for the two functions will be the same, and the
solutions are in the form

daÿ †

where

…90†

da

When hwi is not zero, the solution to the above
problem is8


d
Da
b Db
>
ÿ bbaaDdaa ÿb
f;
> f2 ehwiz=Da ‡ hwib z ÿ d2a ‡ hwi
‡bb db 1
>
>
>

da
hwiz=Db
b Db
>
ÿ bbaaDdaa ÿb
f
e
ÿ
f;
‡
z
ÿ
d
ÿ
a
>
3
‡bb db 1
2
hwi
hwi
>
>
:
da 6 z < da ‡ db ;
…91†

da db
hwi

2

…ehwida =Da ÿ 1†…ehwidb =Db ÿ 1†…Da db ‡ Db da †

 !
;
exp

f2 ˆ ÿ

hwida hwidb
Da ‡ Db

ÿ1 hwi

…ehwidb =Db ÿ 1†…Da db ‡ Db da †
…ehwida =Da ‡hwidb =Db ÿ 1†hwi

2

…92†

3

;

…93†

and
f3 ˆ

eÿhwida =Db …ehwida =Da ÿ 1†…Da db ‡ Db da †

 !
:
exp

hwida hwidb
Da ‡ Db

ÿ1

hwi

…94†

2

In the case when the ¯ow is horizontal, or hwi is equal to
zero, the solution is reduced to
8 db 2 da db
ÿ 2Da z ‡ 2Da ‡ f4 ;
>
>
>
>
< 0 6 z < da ;
 2

…95†
Z…z† ˆ
2
2da ‡da db
d
b†
>
‡
f
;
z ‡ da …d2Da ‡d
> 2Dab z2 ÿ
4
2Db
>
b
>
:
da 6 z < da ‡ db ;
where
f4 ˆ

da db …Da bb db2 ÿ Db ba da2 †
:
12Da Db …ba da ‡ bb db †

…96†

Note also the limit
da2 db2
as hwi ! 0:
…97†
12Da Db
Before further substitutions, let us ®rst obtain the
following cell average:
f1 !

huZ ÿ D
rZi ˆ ÿf Z

…98†

where
f Z ˆ …c1 hui; c2 hvi; c3 hwi†;

…99†

in which
…Da db ‡ Db da †…kb ba ÿ ka bb †
f;
…ba da ‡ bb db †…ka da ‡ kb db † 1


ba Da ÿ bb Db
f1 :
c3 ˆ
ba da ‡ bb db
c1 ˆ c 2 ˆ

…100†
…101†

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


D0 ˆ hD
I ‡ r‰M s ‡ M d Š ÿ u‰M s ‡ M d Ši
ˆ hD
…I ‡ rM s †i ÿ huM d ÿ D
rM d i ÿ huM s i
ˆ D ÿ huZ ÿ D
rZif M
ˆ D ‡ f Z f M ;

…102†

where the last term in the second line is zero because of
eqns (73) and (75), and D is the e€ective di€usion coecient given by

619

Macroscopic equations for vapor transport in a multi-layered unsaturated zone
Da da ‡ Db db
;
da ‡ db

…103†

Da Db …da ‡ db †
;
Da db ‡ Db da

…104†

D11 ˆ D22 ˆ hDi ˆ
ÿ1

D33 ˆ hDÿ1 i

ˆ

Dij ˆ 0 for i 6ˆ j:

…105†

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cient D0 is
the sum of the e€ective molecular di€usion coecient D
and the mechanical dispersion coecient f Z f M which is
proportional to the square of the velocity. Substituting
eqns (77) and (99) Eqn. (102) can be written in the alternative form
D0ij ˆ Dij ‡ ci gj hui ihuj i:

…106†

While it is trivial to see that c1 g2 ˆ c2 g1 , it can also be
shown after some algebra that c1 g3 ˆ c3 g1 and
c2 g3 ˆ c3 g2 . Therefore the dispersion coecient tensor is
symmetric: D0ij ˆ D0ji .
Finally the combined e€ective speci®c discharge
eqn (56) may also be expressed as follows:
u0 ˆ hu…0† i ‡ hu…1† i ‡ hu…0† E ÿ D
rEi
ˆ hu…0† i ‡ fE hu…0† Z ÿ D
rZi

…107†

or, omitting the leading order superscript and using
eqn (99),
u0i ˆ …1 ÿ ci fE †hui i:

…108†

Note that use has been made of hu…1† i ˆ 0 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):


 
oC^
^ 0
r^0 C^ ˆ 0;
‡ r^0
^u0 C^ ÿ Pe0ÿ1
r^0
D
a
o^t

…109†

where
0

g^i ˆ da gi ˆ

…dk ÿ db †…1 ‡ dd †
…1 ‡ db dd †…1 ‡ dk dd †

g^3 ˆ da g3 ˆ

…1 ÿ db dk †…1 ‡ dd †
;
…1 ‡ db dd †…dd ‡ dk †
 2 
U
dd …dk ÿ db †…dd ‡ dD † ^
^
f
ci ˆ
c ˆ
Da da i …1 ‡ dk dd †…1 ‡ db dd † 1

c^3 ˆ



^
u ˆ u =U ;

^ 0 ˆ D0 =Da ;
D

Pe0a

0

ˆ U ` =Da :

…110†

Pe0a

ˆ O…1=e†, or the dispersion is
Note that by eqn (3)
signi®cant over a long time scale comparable to T2 .
We further introduce the following ratios of layer
properties
dd ˆ db =da ;

dk ˆ kb =ka ;

dD ˆ Db =Da ;

db ˆ bb =ba ;
…111†

…112†
…113†

g^03 ˆ da g03 ˆ

…114†
…i ˆ 1; 2†;

U2
dd …1 ÿ db dD † ^
c ˆ
f;
Da da 3
…1 ‡ db dd † 1


…115†
…116†

where

U2
f
da db 1
"
#
^ a
^ b
1
…ehwiPe
ÿ 1†…ehwiPe
ÿ 1†…Pea ‡ Peb †
ˆ
1ÿ
;
^
a ‡Peb † ÿ 1†hwiPe
^ a Peb
…ehwi…Pe
^ 2
hwi

f^1 ˆ



…117†
and
Peb ˆ Udb =Db ˆ Pea dd =dD :

…118†

Note that f^1 is always greater than zero for any ®nite
^ and it tends to a ®nite limit when hwi
^ bevalue of hwi
comes zero:
^ ! 0:
f^1 ! Pea Peb =12 ashwi

…119†

On substituting eqn (82), the normalized form of the
speci®c discharge eqn (108) may now be written as (for
i ˆ 1; 2; 3):
(
"

ÿ1
o^
p
o^
p
0
0
ui 0 ‡ g^2 h^
vi 0
u^i ˆ 1 ‡ Pea p^ c^i g^1 h^
o^
x
o^
y
#)
op^
^ 0
…120†
h^
ui i:
‡ g^03 hwi
o^z
Also, the normalized form of the hydrodynamic dispersion coecient (106) can be written as (for
i; j ˆ 1; 2; 3):
^  ‡ c^i g^j h^
^ 0ij ˆ D
ui ih^
uj i;
D
ij

0

…i ˆ 1; 2†;

…1 ÿ db dD †…1 ‡ dd †
;
…1 ‡ db dd †…dd ‡ dD †

Pea ˆ Uda =Da ;

ˆ hu…0† i ÿ fE f Z ;

hbi

and the following dimensionless variables and parameters:

…121†

^  is the normalized di€usion coecient given by
where D
ij
^  ˆ 1 ‡ dD d d ;
^ ˆ D
D
11
22
1 ‡ dd

…122†

^  ˆ dD …1 ‡ dd † ;
D
33
dd ‡ dD

…123†

^ ij ˆ 0
D

…124†

for i 6ˆ j:

620

C. -O. Ng

9 THE DISPERSION COEFFICIENT

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

We stress that the dispersion coecient given in
eqn (121) is a function of both the convection velocity 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 longitudinal and the transverse components of the dis^ 0 , whose explicit forms
^ 0 and D
persion coecient, D
11
33
are:

Case

2
^ 0 ˆ 1 ‡ dD dd ‡ …dk ÿ db † …1 ‡ dd †…dd ‡ dD †dd
D
11
1 ‡ dd
…1 ‡ db dd †2 …1 ‡ dk dd †2
8h
i 2
^ b
^ a ÿ1†…ehwiPe
ÿ1†…Pea ‡Peb †
h^
ui
< 1 ÿ …ehwiPe
^ 6ˆ 0
for hwi
hwi…Pe
^
‡Pe
†
a
^
hwi
b ÿ1†hwiPe
^ a Peb
…e
;
 ÿ
: Pea Peb
2
^
for
h
wi
ˆ
0
h^
u
i
12

…125†

2

^ 0 ˆ dD …1 ‡ dd † ‡ …1 ÿ db dD † …1 ‡ dd †dd
D
33
dd ‡ dD
…1 ‡ db dd †2 …dd ‡ dD †
8h
i
^ b
^ a ÿ1†…ehwiPe
ÿ1†…Pea ‡Peb †
< 1 ÿ …ehwiPe
^ 6ˆ 0;
for hwi
^
hwi…Pe
‡Peb †
a
^ a Peb
ÿ1†hwiPe
…e

:
^ ˆ 0:
0
for hwi

…126†

The second (mechanical dispersion) terms in the
above expressions are always non-negative; they are
zero only when the corresponding velocity component
^ 0 , db dD ˆ 1 for D
^0 .
vanishes or when dk ˆ db for D
11
33
Therefore, as expected, in the absence of heterogeneity the mechanical dispersion of vapor in a porous
medium becomes subdominant. Note also that these
^
two dispersion components are even functions of hwi,
^ 0 …ÿhwi†
^ 0 …hwi†
^ 0 …ÿhwi†.
^ 0 …hwi†
^
^
^
^
ˆ
D
and
D
ˆ
D
i.e., D
11
11
33
33
^ 0 that the longitudinal
It can readily be seen from D
11
mechanical dispersion is of the magnitude U 2 db2 =Db ,
which is the form of Taylor dispersion in a capillary
tube2,26. This is reasonable since in our model the dispersion 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 coef2
=Dim , where
®cient by the amount proportional to Um2 dim
Um is the velocity in the mobile region, dim is a characteristic dimension of the immobile region, and Dim is 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 permeable layer.

I
II
III
IV

dk

db

dD

dd

0.5
0.05
0.05
0.05

10
100
100
100

0.5
0.5
0.05
0.05

1
1
1
2

We also remark that the transverse mechanical dispersion vanishes when the ¯ow is strictly horizontal.
This is consistent with the analysis by statistical means
which yields a zero transverse asymptotic macrodispersivity 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 conductivity and di€usivity) and has a larger organic matter
content (therefore a larger retardation factor) than that
in layer A. The material contrasts are therefore dk < 1,
dD < 1 and db > 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, Pea is taken to be unity. By comparing with Case I,
the e€ects of a larger contrast in conductivity and retardation 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^ 011 as a function of the longitudinal
persion coecient D
speci®c discharge h^
ui when there is no transverse ¯ow
^ 0 increases with
^ ˆ 0). According to eqn (125) D
(i.e., hwi
11

Fig. 2. The longitudinal hydrodynamic dispersion coecient
^ 011 as a function of the longitudinal speci®c discharge h^
ui for
D
^ ˆ 0.
the four cases; the transverse speci®c discharge hwi

Macroscopic equations for vapor transport in a multi-layered unsaturated zone

Fig. 3. The longitudinal hydrodynamic dispersion coecient
^ 011 as a function of the longitudinal speci®c discharge h^
ui for
D
^ ˆ 0:5.
the four cases; the transverse speci®c discharge hwi

the square of h^
ui. The increase is relatively mild in Case I
when h^
ui is 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 diffusion 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
^ ˆ 0:5. By comparing the two ®gures, it is clear that
hwi
the vertical ¯ow suppresses the increase of the longitudinal 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.

^ 033
Fig. 4. The transverse hydrodynamic dispersion coecient D
^ for the
as a function of the transverse speci®c discharge hwi
four cases.

621

The variations between the transverse dispersion coecient and the transverse speci®c discharge are shown
^ 0 in^ from 0 to 5, D
in Fig. 4. Over the range of hwi
33
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
^ 033 does not increase with
is clear from eqn (126) that D
the square of the vertical velocity, but only with the
value inside the square brackets which has a maximum
^ becomes in®nite. In addition, the efof unity when hwi
fect of the velocity disappears when the product db dD
happens equal to unity. Therefore the enhancement of
the transverse mechanical dispersion by the seepage
velocity is in general very limited. The transverse dispersion is more dominated by the e€ective molecular
di€usion normal to the layers which is controlled by dD
or the di€usivity in the less permeable layer.

10 CONCLUDING REMARKS
In this paper we have presented a mathematical derivation 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 expressed 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 velocity 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 conductivity 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 longitudinal 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;

622

C. -O. Ng

(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 Institute 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 Sciences for drawing the author's attention to some of his
pertinent work.
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