Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol22.Issue1.1998:

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Transport in chemically and mechanically

heterogeneous porous media

V. Two-equation model for solute transport

with adsorption

Azita Ahmadi

, Michel Quintard

a ,

* & Stephen Whitaker

b a

L.E.P.T.-ENSAM (UA CNRS), Esplanade des Arts et Me´tiers, 33405 Talence Cedex, France

b_{Department of Chemical Engineering and Material Science, University of California at Davis, Davis, CA 95616, USA} (Received 30 July 1996; accepted 8 September 1997)

In this paper we develop the two-equation model for solute transport and adsorption in a two-region model of a mechanically and chemically heterogeneous porous medium. The closure problem is derived and the coefficients in both the one- and two-equation models are determined on the basis of the Darcy-scale parameters. Numerical experiments are carried out for a stratified system at the aquifer scale, and the results are compared with the one-equation model presented in Part IV and the two-equation model developed in this paper. Good agreement between the two-equation model and the numerical experiments is obtained. In addition, the two-equation model is used, in conjunction with a moment analysis, to derive a one-equation, non-equilibrium model that is valid in the asymptotic regime. Numerical results are used to identify the asymptotic regime for the one-equation, non-equilibrium model. q_{1998 Elsevier} Science Limited.

Key words:porous media, solute transport, adsorption, mathematical modelling.

NOMENCLATURE

agk ¼Agk=Vj, interfacial area per unit volume, m¹

1 . Agk ¼ area of the g–k interface contained in the

averaging volume,V_j, m2

Abj ¼Ajb, area of theb–jinterface contained in the averaging volume,V, m2

Ahq ¼Aqh, area of the boundary between theh and q-regions contained with the large-scale averag-ing volume,V`, m2

bhh vector field that maps=fhchihgh ontoc˜h, m.

bhq vector field that maps=fhcqiqgqontoc˜h, m.

bqq vector field that maps=fhcqiqgqontoc˜q, m.

bqh vector field that maps=hchihghontoc˜q, m.

cg point concentration in theg-phase, mol m¹3.

hchih Darcy-scale intrinsic average concentration for theb–jsystem in theh-region, mol m¹3

hcqiq Darcy-scale intrinsic average concentration for theb–jsystem in theq-region, mol m¹3

fhchihg h-region superficial average concentration,

mol m¹3 .

fhchihgh ¼J

¹1

h fhchihg,h-region intrinsic average

concen-tration, mol m¹3 .

{hci} ¼ JhfhchihghþJqfhcqiqgq, large-scale intrinsic

average concentration, mol m¹3 . ˜

ch ¼ chih¹ fhchih

, spatial deviation concentra-tion for theh-region, mol m¹3.

fhcqiqg q-region superficial average concentration,

mol m¹3.

fhcqiqgq ¼ Jq¹1fhcqiqg, q-region intrinsic average

concentration, mol m¹3. ˜

cq ¼ cqiq¹ fhcqiq

, spatial deviation concentra-tion for theq-region, mol m¹3.

h dispersion tensor for theb–jsystem in the

h-region, m2s¹1 .

q dispersion tensor for theb–jsystem in the

q-region, m2s¹1 .

Dpp

hh dominant dispersion tensor for theh-region

trans-port equation, m2s¹1 .

PII: S 0 3 0 9 - 1 7 0 8 ( 9 7 ) 0 0 0 3 2 - 8

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Dpp

hq coupling dispersion tensor for theh-region

trans-port equation, m2s¹1 .

Dpp

qq dominant dispersion tensor for theq-region

trans-port equation, m2s¹1 .

Dpp

qh coupling dispersion tensor for theq-region

trans-port equation, m2s¹1 .

Dpp _large-scale, _one-equation _model _dispersion tensor, m2s¹1.

g gravitational acceleration vector, m s¹2.

g magnitude of the gravitational acceleration vector, m s¹2.

I unit tensor

Keq ¼]F=]_c_g¼]F=]hcgig, adsorption equilibrium

coefficient, m.

K ¼ agkKeq/eg, dimensionless adsorption equilib-rium coefficient for thej-region.

K_h ¼ (ejagk)h=ehÿ]F=]hchih

, dimensionless equilib-rium coefficient for theh-region.

K_q ¼ (ejagk)q=eqÿ]F=]hcqiq

, dimensionless equilib-rium coefficient for theq-region.

,_i _i_¼_{1,2,3, lattice vectors, m.} ,_h _{length scale for the}_{h-region, m.} ,_q _{length scale for the}_{q-region, m.}

Lc length scale for the region averaged concentra-tions, m.

L aquifer length scale, m.

LH length scale of the aquifer heterogeneities, m. nhq ¼ ¹nqh, unit normal vector directed from the

h-region towards theq-region.

rj radius of the averaging volume, Vj, for the

j-region, m.

ro radius of the averaging volume,V, for theb–j system, m.

rh scalar that mapsfhcqiqgq¹ fhchihghontoc˜h.

rq scalar that mapsfhcqiqgq¹ fhchihghontoc˜q.

Ro radius of the averaging volume,V`, for theh–q

system, m. t time, s.

hvbih Darcy-scale, superficial average velocity in the h-region, m s¹1.

fhv_bi_hgh intrinsic regional average velocity in theh-region, m s¹1.

fhv_bi_hg ¼Jhfhvbihgh, superficial regional average

velocity in theh-region, m s¹1. ˜

v_bh ¼hv_bi_h¹ fhv_bi_hgh, h-region spatial deviation velocity, m s¹1.

hvbiq Darcy-scale, superficial average velocity in the q-region, m s¹1

fhv_bi_qgq intrinsic regional average velocity in theq-region, m s¹1.

fhvbiqg ¼Jqfhvbiqgq, superficial regional average

velocity in theq-region, m s¹1. ˜

v_bq ¼hv_bi_q¹ fhv_bi_qgq, q-region spatial deviation velocity, m s¹1

fhv_big ¼JhfhvbihghþJqfhvbiqgq, large-scale,

superficial average velocity, m s¹1.

Vh volume of theh-region contained in the averaging volume,V`, m

3 .

Vq volume of theq-region contained in the averaging volume,V`, m

3 .

V` large-scale averaging volume for theh–qsystem,

m3. Greek symbols

a* mass exchange coefficient for the h–q system, s¹1.

e ¼ eb þ ejeg, total porosity for the b–j system.

eh ¼ ebh þ (ejeg)h, total porosity for the h-region.

eq ¼ebqþ(ejeg)q, total porosity for theq-’“n. {e} ¼JhehþJqeq, large-scale average porosity. {e}ð1þ{K}Þ ¼ehð1þKhÞJhþeqð1þKqÞJq, large-scale

average capacitance factor.

Jh ¼1¹Jq, volume fraction of theh-region. Jq ¼1¹Jh, volume fraction of theq-region.

1 INTRODUCTION

Dispersion in heterogeneous media has received a great deal of attention from a variety of scientists who are con-cerned with mass transport in geological formations. It is commonly accepted that dispersion through natural systems such as aquifers and reservoirs involves many different length scales, from the pore scale to the field scale. If one considers the solute transport in such formations, these multiple scales may lead to anomalous and non-Fickian dispersion at the field scale.1–4Here we need to be precise and note that anomalous dispersionrefers to the interpre-tation of field-scale data that does not fit the response of a field-scale homogeneous representation. Similarly, the existence of multiple scales has been related to the observa-tion that dispersivity is field-scale dependent (see a review by Gelhar et al.5), and the theoretical implications of this idea have been discussed extensively.6,7 Clearly, a field-scale description calls for a representation in terms of a heterogeneous domain, and we adopt this point of view in this paper.

1.1 Hierarchical systems

A schematic representation of the problem under considera-tion is illustrated in Fig. 1. While many intermediate scales could be incorporated into the analysis, this study is limited to four typical scales that can be described as follows:

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1. the macropore scale, in which averaging takes place over the volume V_j;

2. the Darcy scale, in which averaging takes place over the volume V;

3. the local heterogeneity scale, in which averaging takes place over the volumeV`;

4. the reservoir- or aquifer-scale heterogeneities, which have been identified by the length scaleLHin Fig. 1; no averaging volume has been associated with this length scale since the governing equations will be solved numerically at this scale.

As we suggested in Part IV8, many applications will require the addition of a micropore scale when the k-region illustrated in Fig. 1 contains micropores, and many realistic systems may contain other intermediate length scales either within the b–j system or within the heterogeneities associated with the averaging volumeV`.

When these length scales are disparate, the method of volume averaging can be used to carry information about the physical processes from a smaller length scale to a larger one, and eventually to the scale at which the final analysis is performed. When the length scales are not

disparate, one is confronted with the problem of evolving heterogeneities.4

In this study we assume that the macropore scale, the Darcy scale and the local heterogeneity scale are con-veniently separated. This assumption was also imposed on the analysis presented in Part IV8, and there it led to a Darcy-scale representation of the dispersion process. The analysis required, among other constraints, that

k,,gprjp,b,,jprop,h,,q (1)

In the multiple-scale problem under consideration in this paper, Darcy-scale properties are point-dependent, and there is a need for a large-scale description. It is generally assumed9,10 that local heterogeneity-scale permeability variations are ‘stationary’. In other words, gradients of the large-scale averaged quantities, which are characteristic of the regional variations, may be assumed to have negli-gible impact on the change-of-scale problem for character-istic lengths equivalent to the large-scale averaging volume represented by the subscript ` _{in Fig. 1. Based on this} assumption, and provided that the following length-scale constraints are satisfied:

,_h,,_qpRopLH#L (2)

there is some possibility that a large-scale description exists for the large-scale dispersion process. Here, we mention thepossibleexistence of an averaged description to remind the reader that process-dependent scales are involved in the analysis, and this may lead to conditions that do not permit the development of closed-form volume-averaged transport equations.

Within this framework, we indicated in Part IV8how a local heterogeneity-scale equilibrium dispersion equation

Fig. 1.Averaging volumes in a hierarchical porous medium.

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could be derived from the Darcy-scale problem provided that certain length and time scales constraints were fulfilled. In this paper, we remove these latter constraints, and we present an analysis leading to a large-scale, non-equilibrium model for solute dispersion in heterogeneous porous media. The removal of these constraints naturally leads to a better description of the process, and this is clearly demonstrated in our comparison between theory and numerical experiments. The penalty that one pays for this improved description is the increased number of effective coefficients that appear in the two-equation model. If laboratory experiments are required in order to determine these additional coefficients, one is confronted with an extremely difficult task; however, in our theoretical development all the coefficients can be determined on the basis of a single, representative unit cell. This means that all the coef-ficients in the large-scale averaged equations are self-consistent and based on a single model of the local heterogeneities.

The large-scale model that results from our analysis features large-scale properties which are point-dependent with a characteristic length scale, LH, describing the regional heterogeneities. These regional heterogeneities are incorporated into any field-scale numerical description. They will certainly contribute to anomalous, non-Fickian field-scale behaviour, but this behaviour will be taken care of by the field-scale calculations and the large-scale averaged transport equations.

1.2 Large-scale averaging

Within this multiple-scale scheme, we focus our attention on the large-scale averaging volume illustrated in Fig. 2 and thus restrict the analysis to a two-region model of a hetero-geneous porous medium. It is important to understand that the general theory is easily extended to systems containing many distinct regions, and an example of this is given by Ahmadi and Quintard11. Systems of the type illustrated in Fig. 2 are characterized by an intense advection in the more permeable region, while a more diffusive process takes place in the less permeable region. Observations of many similar systems, often referred to as systems with stagnant regions or mobile–immobile regions, have been reported in the literature (see reviews12,13). The expected large-scale behaviour is characterized by large-scale dispersion with retardation caused by the exchange of mass between the different zones. Models proposed for describing solute transport in such cases correspond to the introduction of a retardation factor in the dispersion equation, or a two-equation model for the mobile and immobile regions14–21(see also the reviews cited above12,13). Extensions of these models have been proposed for mobile water in both regions (Skoppet al.22, for the case of small interaction between the two regions, and Gerke and van Genuchten23). In the paper by Gerke and van Genuchten, the solute inter-porosity exchange term is related intuitively to the water inter-porosity exchange

term, i.e. in the case of local mechanical non-equilibrium, and to an estimate of the diffusive part that resembles previously proposed estimates in the case of mobile– immobile systems. It should be noted that the model of Gerke and van Genuchten23 accounts for variably saturated porous media, a case that is beyond the scope of this paper.

In this paper, we propose a general formulation of these two-equation models using the method of large-scale averaging. We obtain an explicit relationship between the local scale structure and the large-scale equations, suitable for predictions of large-scale properties, which incorporates both coupled dispersive and diffusive contributions. Finally, this methodology is illustrated in the case of dispersion in a stratified system for which we compare the theory both with numerical experiments and with the non-equilibrium, one-equation model of Marle et al.24

The Darcy-scale process of solute transport with adsorp-tion in theh–qsystem shown in Fig. 2 is given by

eh 1þKh ÿ ] ch

]_t þ=· vb

h ch

¼=_· _Dp

h·= ch

ð3Þ

B:_C:₁ _c_hh¼ cq q_,

atAhq (4)

B:_C:₂ _¹_n

hq· vb

h ch

¹Dp

h·= ch

¼ ¹n_hq· v_b_q cq

¹Dp

q·= cq

ÿ _,

atA_hq ð5Þ

eq 1þKq ÿ ] cq

]_t þ=· vb

q cq

¼=_· _Dp

q·= cq

(6) Here K_h and K_q represent the Darcy-scale equilibrium adsorption coefficients, which may be non-linear functions of the concentrations, hchih and hcqiq. In addition to the solute transport equations, we shall need to make use of the two Darcy-scale continuity equations that take the form

=_·v_b

h¼0 (7a)

=_·_v_b

q¼0 (7b)

along with the boundary condition for the normal compo-nent of the velocity, which is given by25,26

B:_C:₃ _n

hq· vb

h¼nhq· vb

q, atAhq (8)

In Part IV8 the region-average transport equations were developed, and the superficialaverage forms are given by

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h-region:

q-region:

In our study of the one-equation model presented in Part IV, we made use of the single, large-scale continuity equation; however, in the analysis of mass transport processes using the two-equation model, we shall need the regional forms of the two continuity equations. These can be expressed as:

h-region:

=_·fv_b_hg þ 1 V`

Ahqnhq· vb

hdA¼0 (10a)

q-region:

=_·fv_b_qg þ 1 V`

Aqhnqh· vb

qdA¼0 (10b)

Because the regional velocities are not solenoidal, as are the Darcy-scale velocities contained in eqns (7), one must take special care with the various forms of the regional continuity equations.

In eqns (9), we see various large-scale terms such as ]

ch h h_=]

t in eqn (9a) and Jq

v_b

q q

in eqn (9b), and we see other terms such as n_hqc˜h andvbqc˜q

(9b) (9a)

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that involve the spatial deviation quantities. In addition, the inter-region flux is specified entirely in terms of the Darcy-scale variablessuch as vb

hand cq

. In the fol-lowing section we shall develop the closure problem which will allow us to determine the diffusive terms such as

_˜

Dph·=c˜h and the dispersive terms such as v˜bqc˜q

. More importantly, we shall develop a representation for the inter-region flux that is determined entirely by the closure problem. This means that the representation for the inter-region flux is limited by all the simplifications that are made in development of the closure problem. 2 CLOSURE PROBLEM

In the development of a one-equation model, one adds eqns (9a) and (9b) to obtain a single transport equation in which the inter-region flux terms cancel. In that case, the closure problem is used only to determine the effective coefficients associated with diffusion and dispersion. For the two-equation model under consideration here, the closure problem completely determines the functional formof the inter-region flux and the effective coefficients which appear in the representation of that flux. Closure problems can be developed in a relatively general manner; however, the development of a local closure problem requires the use of a spatially periodic model. This means that some very specific simplifications will be imposed on our representation for the inter-region flux and for the large-scale dispersion; however, these simplifications are not imposed on the other terms in eqns (9a) and (9b).

2.1 Inter-region flux

In the development of a two-equation model, we need to represent the inter-region flux terms in a useful form, and this means decomposing that flux into large-scale quantities and spatial deviation quantities. Directing our attention to the h-region transport equation, we make use of the decomposition

c_h

¼c_hh h

þc˜_h (11)

in order to express the inter-region flux as 1

Ahqnhq· vb

h ch

¹Dp

h·= ch

h dA ¼ 1 V` Z

Ahqnhq· vb

h ch

¹Dp

h·=f ch

h gh dA þ 1 V` Z

Ahqnhq· vb

hc˜h¹Dph·=c˜h

dA ð12Þ

The second term on the right-hand side of this result is in a convenient form for use with eqn (9a) since the unit cell closure calculations will provide us with values for both

hvbihandDph; however, we need to consider carefully how

we treat the first term. In the derivation of eqn (9a) we made use of the following decomposition for the dispersion

tensor:

h¼ Dph

þD˜p_h (13)

When this decomposition is used with eqn (12), we can express the first term on the right-hand side as

Ahqnhq· hvbihfhchi

ÿ h

¹Dp

h·=fhchihghÞdA¼

Z Ahqnhq·

hv_bi_hfhc_hih_gh_¹

·=_fhc_hih_gh þD˜ph·=fhchihgh

dA ð14Þ

The large-scale averaged quantities can be removed from the first two terms on the right-hand side of eqn (14); how-ever, we shall leave the gradient of the large-scale average concentration inside the third term to obtain

Z Ahq

nhq· hvbihfhchih

ÿ h

¹Dp

h·=fhchihghÞdA

¼ 1

Z Ahq

nhq·hvbihdA

fhchihgh

¹ 1

AhqnhqdA

· Dp

·=fhchihgh

¹ 1

Ahqnhq·D˜ p

h·=fhchihghdA

ð15Þ

One can show that the first term on the right-hand side of this result is zero for a spatially periodic system. This occurs because the periodicity condition for the velocity,

Periodicity : vb

h(rþ,i)¼ vb

h(r), i¼1,2,3

(16) allows us to write

Ahqnhq· vb

hdA¼

Ahqnhq· vb

hdA þ 1 V` Z Ahe

nhe· vb

hdA ð17Þ

in which Aherepresents the area of entrances and exits for theh-region contained in a unit cell of a spatially periodic porous medium. Use of the divergence theorem and eqn (7) allows us to express eqn (17) as

Ahqnhq· vb

hdA¼

Z Vh

=_· _v

hdV¼0 (18)

and use of this result with eqn (15) leads to the form 1

Ahqnhq· vb

h ch

¹Dp

h·= ch

¼ ¹ 1

AhqnhqdA

· Dp

h h ·= ch h h ¹ 1 V` Z Ahqnhq·

˜ Dph·=

ch h h

ð19Þ

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result, we make use of the averaging theorem to obtain

¹ 1

AhqnhqdA

· Dp

·= _c

gh ¼=_J

h· Dph

·= _c

ð20Þ

For a spatially periodic system, =_J_h _{is zero and eqn (20)} allows us to express eqn (19) as

Ahqnhq· vb

h ch

¹Dp

h·= ch

¼ ¹ 1

Ahqnhq·D˜ p

h·= ch

ð21Þ

We are now ready to return to eqn (12) and express that

inter-region flux according to 1

Ahqnhq· vb

h ch

¹Dp

h·= ch

¼ 1

Ahqnhq· vb

hc˜h¹Dph·=c˜h¹D˜

h·= ch

ð22Þ

Substitution of this result into eqn (9a) leads to a form of the large-scale average transport equation that is ready to receive results from the closure problem.

h-region:

Here we should note that every term in this result is either a large-scale average quantity or a spatial deviation quantity except for the Darcy-scale velocity, hvbih. This Darcy-scale quantity has not been decomposed like all the other terms, because it will be available to us directly by solution of the Darcy-scale mass and momentum equations for a unit cell in a spatially periodic model of a heterogeneous porous medium. The analogous result for the q-region can be obtained from eqn (9b) and is given by

q-region:

(23a)

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In order to evaluate the terms in eqns (23a) and (23b) that contain the spatial deviation concentrations, we need to develop the closure problem forc˜h andc˜q. The

govern-ing differential equation forc˜hcan be obtained by

subtract-ing the intrinsic form of eqn (23a) from the Darcy-scale equation forhchi

that is given by eqn (3). We develop the intrinsic form of eqn (23a) by dividing that result by Jh, and this leads to a rather complicated result. However, prior studies27,28clearly indicate that it is an acceptable approx-imation to ignore variations of the volume fraction,Jh,in the development of the closure problem, and this means that the intrinsic form of eqn (23a) can be expressed as

eh 1þKh

ÿ ] ch

]_t þ=· vb

h n oh ch h h h i

¼=_·

· =_c_hh h

þJ

¹1 h V`

Ahqnhqc˜hdA

þ D˜ph·=c˜h

n oh

¹=_· _v_˜

bhc˜h

h ÿ ¹J ¹1 h V` Z Ahqnhq·

v_b

hc˜h¹Dph·=c˜h ¹D˜ph·= ch

dA ð24Þ

Subtraction of this result from eqn (3) leads to eh 1þKh

ÿ ]c˜h

]_t þ=· vb

h ch

¹ v_b h n oh ch h h

¼=_· _Dp

h·= ch

¹ Dp

·=_f _c

ÿ h

¹=_· J ¹1 h Dph

Z Ahq

nhqc˜hdAþ D˜

h·=c˜h

n oh

" #

þ=_· v˜_bhc˜h

h þJ ¹1 h V` Z Ahqnhq·

v_b

hc˜h ¹Dp

h·=c˜h¹D˜

h·= ch

dA ð25Þ

Directing our attention to the convective transport term in eqn (25), we make use of the velocity decomposition given by

h¼ vb

n oh

þv˜bh (26)

to obtain v_b

h ch

¹ v_b h

n oh

¼v_b

hc˜hþv˜bh ch

(27) Within the framework of the closure problem, we can use eqns (10) and (18) to obtain

=_· _v

n o

¼0 (28)

and since we are ignoring variations of Jh, the continuity equation for the intrinsic regional average velocity takes the form

=_· v_b h

n oh

¼0 (29)

This result, along with the continuity equation given by eqn (7a) and the decomposition given by eqn (26), can be used to express the convective transport terms in eqn (25) as

=_· v_b_h ch

¹nv_b_hoh ch

¼=_· v_b_hc˜h

þv˜_bh·= _c

ð30Þ

Use of the decomposition for the dispersion tensor given by eqn (13) leads to the following representation for the two dispersive fluxes:

Dph·= ch

¹ Dph

·= _c

¼Dph·=c˜hþD˜

h·= ch

(31) When eqns (30) and (31) are used in eqn (25), our transport equation for the spatial deviation concentration takes the form

(1)

in which the large-scale dispersion tensor is defined by

D

_¼

_J

D

h

þ

J

D

q

þ

D

p h

h

¹

D

q

V

· Z

Ahq

n

b

d

A

þ

D

˜

p h

· =

b

n

o

þ

D

˜

· =

b

n

o

¹

J

v

˜

b

h

þ

J

v

˜

b

q

ÿ

ð

A2

Þ

In order to derive these two results from the two-equation

model presented in this paper, we first add eqns (58) and

(60) and impose the approximations

c

h i

h

¼

h i

c

q

¼

{

h i

c

}

(A3)

to obtain

e

1 þ

K

ÿ

J

þ

e

1 þ

K

ÿ

J

]

{

h i

c

}

]

_t

þ

=

_·

_J

v

n

oh

þ

J

n

v

oq

h

{

h i

c

}

i

¹

ÿ

u

_hh

þ

u

_hq

þ

u

_qh

þ

u

_qq

· =

_{

_{h i}

_c

_}

¼

=

_·

_D

þ

D

pphq

þ

D

ppqh

þ

D

ppqq

ÿ

· =

_{

h

c

i

}

ð

A4

Þ

Use of the definitions

{

e

} 1

ð

þ

{

K

}

Þ ¼

e

1 þ

K

ÿ

J

þ

e

1 þ

K

ÿ

J

(A5)

v

¼

J

v

n

oh

þ

J

n

v

oq

(A6)

allows us to simplify this result to obtain the traditional

accumulation and convective transport term according to

{

e

} 1

ð

þ

{

K

}

Þ

]

{

h i

c

}

]

_t

þ

=

· ÿ

v

{

h i

c

}

¹

ÿ

u

_hh

þ

u

_hq

þ

u

_qh

þ

u

_qq

· =

{

h i

c

}

¼

=

_·

_D

þ

D

pphq

þ

D

ppqh

þ

D

ppqq

ÿ

· =

{

h i

c

}

ð

A7

Þ

In order to determine the form of the overall dispersion

tensor, we recall the definitions of the four dispersion

tensors in the above equation, given by

D

¼

J

D

· I

þ

=

b

ÿ

¹

v

˜

_bh

b

_hh

h

(A8a)

D

¼

J

D

· =

b

¹

v

˜

b

h

(A8b)

D

¼

J

D

· =

b

¹

v

˜

b

q

(A8c)

D

¼

J

D

· I

þ

=

b

ÿ

¹

v

˜

_bq

b

_qq

q

(A8d)

These representations need to be arranged in a form that

will allow us to extract the relation given by eqn (A2), and

we begin this rearrangement with eqn (A8a) to obtain

D

pphh

¼

J

D

· I

þ

=

b

ÿ

¹

v

˜

b

h

¼

J

D

h

þ

D

p h

· =

b

h

¹

v

˜

_bh

b

_hh

h

¼

J

h

D

p h

h

þ

D

h

· =

_b

_hh

_þ

_D

˜

p_h

_·

=

_b

_hh

n

oh

¹

v

˜

_bh

b

_hh

h

i

ð

A9

Þ

We can remove the regional average from the averaging

process to obtain

D

h

· =

_b

h

¼

D

p h

h

· =

_b

h

(A10)

and then use the averaging theorem in order to express this

term as

=

_b

h

¼

=

_b

h

þ

1 V

Z

Ahq

n

_hq

b

_hh

d

A

(A11)

Here we have ignored variations of

J

, since we are

work-ing with the equations of closure problems I and II above.

From eqn (49) we have

b

_hh

h

¼

0 (A12)

and so eqn (A10) takes the form

D

h

· =

b

_hh

h

¼

D

p h

h

·

1 V

Z

Ahq

n

_hq

b

_hh

d

A

(A13)

Use of this result in eqn (A9) allows us to express

D

in

the form

D

pp hh

¼

J

"

D

h

þ

D

p h

h

·

1 V

Z

Ahq

n

_hq

b

_hh

d

A

þ

D

˜

· =

b

n

oh

¹

v

˜

_bh

b

_hh

h

#

ð

A14

Þ

which is more conveniently written as

D

¼

J

D

h

þ

D

p h

h

V

· Z

Ahq

n

b

d

A

þ

J

D

˜

· =

b

n

oh

¹

J

v

˜

_bh

b

_hh

h

ð

A15a

Þ

If we repeat this procedure with eqns (A8b), (A8c) and

(A8d), we obtain

D

pp hq

¼

D

p h

h

V

· Z

Ahq

n

_hq

b

_hq

d

A

þ

J

D

˜

· =

b

n

oh

¹

J

v

˜

b

h

ð

A15b

Þ

D

pp qh

¼

D

p q

q

V

· Z

Aqh

n

_qh

b

_qh

d

A

þ

J

D

˜

p q

· =

b

n

oq

¹

J

v

˜

b

q

ð

A15c

Þ

D

¼

J

D

q

þ

D

p q

q

V

· Z

Aqh

n

_qh

b

_qq

d

A

þ

J

D

˜

p q

· =

b

n

oq

¹

J

v

˜

b

q

ð

A15d

Þ

If we sum these four equations, we begin to obtain

some-thing that resembles the definition given by eqn (A2):

D

þ

D

pphq

þ

D

ppqh

þ

D

ppqq

¼

J

D

h

þ

J

D

q

þ

D

p h

h

V

· Z

Ahq

n

_hq

ÿ

b

_hh

þ

b

_hq

d

A

þ

D

p q

q

V

Z

Aqh

n

b

þ

b

ÿ

d

A

þ

J

D

˜

· =

b

þ

b

ÿ

n

oh

þ

J

D

˜

· =

b

þ

b

ÿ

n

o

¹

J

v

˜

b

þ

b

ÿ

h

¹

J

v

˜

b

þ

b

ÿ

q

(2)

and the resemblance becomes clearer when we make use of

the combined closure variables defined by

b

¼

b

_hh

þ

b

_hq

,

b

¼

b

_qh

þ

b

_qq

(A17)

Use of these relations in eqn (A16) leads to

D

þ

D

pphq

þ

D

ppqh

þ

D

ppqq

¼

J

D

h

þ

J

D

q

þ

D

p h

h

V

· Z

Ahq

n

_hq

b

d

A

þ

D

p q

q

V

· Z

Aqh

n

_qh

b

d

A

þ

J

D

˜

p h

· =

b

n

oh

þ

J

D

˜

p q

· =

b

n

o

¹

J

v

˜

b

h

¹

J

v

˜

b

q

ð

A18

Þ

At this point one need only recognize that

n

¼ ¹n

and

make use of the two boundary conditions given by eqns (49)

and (51) to conclude that

D

þ

D

pphq

þ

D

ppqh

þ

D

ppqq

¼

D

¼

J

D

h

þ

J

D

q

þ

D

p h

h

¹

D

p q

q

V

· Z

Ahq

n

_hq

b

d

A

þ

D

˜

· =

b

n

o

þ

D

˜

· =

b

n

o

¹

J

v

˜

b

h

þ

J

v

˜

b

q

ÿ

ð

A19

Þ

Use of this result with eqn (A7) provides the more compact

form of the one-equation model given by

{

e

}

ÿ

1 þ

{

K

}

]

{

h i

c

}

]

_t

þ

=

· ÿ

v

{

h i

c

}

¹

ÿ

u

_hh

þ

u

_hq

þ

u

_qh

þ

u

_qq

· =

_{

_{h i}

_c

_}

_¼

=

_·

ÿ

D

_·

₌

_{

_{h i}

_c

_}

ð

A20

Þ

In order to calculate values of the dispersion tensor,

D

,

we need the closure problem that produces the closure

variables

b

and

b

. On the basis of the definitions given

by eqn (A17) and the following definitions for the constants

in the two-equation model closure problems:

c

¼

c

_hh

þ

c

_hq

,

c

¼

c

_qh

þ

c

_qq

(A21)

we can add eqns (49) and (51) to obtain

=

_·

_v

b

þ

v

˜

_bh

¼

=

_·

_D

p h

· =

b

ÿ

þ

=

_·

_D

˜

p h

¹

J

¹1 h

c

(A22a)

B

:

_C

:

₁

b

¼

b

at

A

(A22b)

B

:

_C

:

₂

_n

· D

· =

b

¼

n

· D

· =

b

þ

n

_hq

· D

p q

¹

D

ÿ

at

A

ð

A22c

Þ

=

_·

_v

b

ÿ

þ

v

˜

_bq

¼

=

_·

_D

p q

· =

b

ÿ

þ

=

_·

_D

˜

p q

¹

J

¹1 q

c

(A22d)

Periodicity :

b

(r

þ

,

)

¼

b

(r),

b

(r

þ

,

)

¼

b

(r),

i

¼

1 ,

2 ,

3 ð

A22e

Þ

Average :

b

h

¼

0 ,

b

q

¼

0 (A22f)

At this point we are ready to move on to the non-traditional

convective transport terms in eqn (A20), and from eqns (59)

and (61) we have

u

_hh

¼ ¹

1 V

Z

Ahq

n

_hq

· v

b

_hh

¹

D

· =

b

¹

D

˜

p h

d

A

(A23a)

u

¼ ¹

1 V`

Z

Ahq

n

· v

b

¹

D

· =

b

d

A

(A23b)

u

_qh

¼ ¹

1 V`

Z

Aqh

n

_qh

· v

b

_qh

¹

D

p q

· =

b

ÿ

d

A

(A23c)

u

_qq

¼ ¹

1 V

Z

Aqh

n

_qh

· v

b

¹

D

· =

b

¹

D

˜

p q

d

A

(A23d)

Use of the definitions of the closure variables given by eqn

(A17) allows us to add pairs of these equations to obtain

u

_hh

þu

_hq

¼ ¹

1 V

Z

Ahq

n

_hq

· v

b

¹

D

· =

b

¹

D

˜

p h

d

A

(A24a)

u

_qh

þ

u

_qq

¼ ¹

1 V`

Z

Aqh

n

_qh

· v

b

¹

D

· =

b

¹

D

˜

p q

d

A

(A24b)

On the basis of the boundary condition given by eqn (8):

B

:

_C

:

₃

_n

· v

¼

n

· v

,

at

A

(A25)

we can add eqns (A24a) and (A24b) to obtain

u

_hh

þ

u

_hq

þ

u

_qh

þ

u

_qq

¼

1 V

Z

Ahq

n

_hq

· D

· =

b

þ

D

˜

p h

d

A

þ

1 V

Z

Aqh

n

_qh

· D

· =

b

þ

D

˜

p q

d

A

ð

A26

Þ

Integrating eqn (A22c) over the area

A

indicates that the

two integrals in this result sum to zero:

u

þ

u

þ

u

þ

u

¼

0 (A27)

so eqn (A20) simplifies to

{

e

}

ÿ

1 þ

{

K

}

]

{

h i

c

}

]

_t

þ

=

· v

{

h i

c

}

¼

=

_·

ÿ

_D

_·

₌

_{

_{h i}

_c

_}

ÿ

(3)

We refer to this form as the one-equation equilibrium

model, since it is based on the condition of large-scale

equilibrium.

APPENDIX B: MOMENT ANALYSIS OF THE

TWO-EQUATION MODEL

A complete analysis of the three-dimensional moments

associated with the two-equation model can be found in

Zanotti and Carbonell

. In this Appendix, we present a

similar analysis with the emphasis on the asymptotic

behaviour of the system as a whole, i.e. the average

con-centration for the two regions, in order to compare our

work with that of Marle

et al

.

We consider a one-dimensional, large-scale flow

des-cribed by the two-equation model given by

e

J

]

_t

c

h

þ

v

n

o

]

_x

c

h

¼

D

pphh

ÿ

]

_x

c

h

þ

D

pphq

ÿ

]

_x

c

q

¹

a

_c

h

¹

c

q

ÿ

ð

B1a

Þ

e

J

]

_t

c

q

þ

v

]

_x

c

q

¼

D

ppqq

ÿ

]

_x

c

q

þ

D

ppqh

ÿ

]

_x

c

h

¹

a

_c

q

¹

c

h

ÿ

ð

B1b

Þ

We consider the special case of an infinite medium with the

following boundary condition:

lim

x→6`

c

h

¼

0 and lim

x→6`

c

q

¼

0 (B2)

along with a similar condition for all derivatives of the

concentrations. We adopt the following change of

variable:

X

¼

x

¹

V

t

=

{

e

}

(B3)

so that eqns (B1) takes the form

e

J

]

_t

c

h

þ

v

n

o

¹

e

J

V

{

e

}

_]

]

_X

c

h

¼

D

pp hh

ÿ

]

_X

c

h

þ

D

pp hq

ÿ

]

_X

c

q

¹

a

_c

h

¹

c

q

ÿ

ð

B4a

Þ

e

J

]

_t

c

q

þ

v

¹

e

J

V

{

e

}

_]

]

_X

c

q

¼

D

ppqq

ÿ

]

_X

c

q

þ

D

ppqh

ÿ

]

_X

c

h

¹

a

_c

q

¹

c

h

ÿ

ð

B4b

Þ

We define the spatial moments associated with the

concentration fields as

m

_a,_n

¼

Z

þ`

¹`

X

c

a

d

X,

a

¼

h

,

q

(B5)

Multiplying eqns (B4) by

X

and integrating by parts, we

obtain the following set of governing equations for the

moments:

e

J

]

_m

_h,_n

]

_t

¼

n

v

n

o

¹

e

J

V

{

e

}

m

_h,n¹1

þ

n(n

¹

1 )

D

pp hh

ÿ

m

h,n¹2

þ

n(n

¹

1 )

D

pphq

ÿ

m

q,n¹2

¹

a

m

h,n

¹

m

q,n

ÿ

ð

B6a

Þ

e

J

]

_m

_q, n

]

_t

¼

n

v

¹

e

J

V

{

e

}

m

_q,n¹1

þ

n(n

¹

1 )

D

pp qq

ÿ

m

q,n¹2

þ

n(n

¹

1 )

D

pp qh

ÿ

m

h,n¹2

¹

a

_m

q,n

¹

m

h,n

ÿ

ð

B6b

Þ

These equations can be solved sequentially, starting with

moments of order zero. All calculations presented below

have been

performed within SCIENTIFIC

WORK-PLACE

y

_{using the MAPLE}

y

_library.

Moments of order

The set of differential equations to be solved is

e

J

]

_m

_h,0

]

_t

¼ ¹

a

_m

h,0

¹

m

q,0

ÿ

(B7a)

e

J

]

m

q,0

]

_t

¼ ¹

a

_m

q,0

¹

m

_h,0

ÿ

(B7b)

and the general solution is given by

{

e

}

m

_h,0

¼

e

J

g

h,0

þ

e

J

g

q,0

þ

exp

¹

a

_t

{

e

}

e

J

e

J

3 _e

_J

_g

_h,0

¹

e

J

g

q,0

ÿ

ð

B8a

Þ

{

e

}

m

_q,0

¼

e

J

g

h,0

þ

e

J

g

q,0

þ

exp

¹

a

_t

{

e

}

e

J

e

J

3 _e

J

g

q,0

¹

e

J

g

h,0

ÿ

ð

B8b

Þ

where

g

h,0

and

g

q,0

are the initial values. Adding eqns (B8),

we obtain the following result:

{

e

}

m

¼

e

J

m

h,0

þ

e

J

m

q,0

(4)

In addition, we get

lim

t→`

m

h,0

¼

lim

t→`

m

q,0

¼

e

J

m

h,0

þ

e

J

m

q,0

{

e

}

¼

m

(B10)

Moments of order 1

From eqns (B6) we get

e

J

]

_m

_h,₁

]

_t

¼

v

n

o

¹

e

J

V

{

e

}

m

_h,0

¹

a

_m

h,1

¹

m

q,1

ÿ

ð

B11a

Þ

e

J

]

_m

_q,1

]

_t

¼

v

¹

e

J

V

{

e

}

m

_q,0

¹

a

_m

q,1

¹

m

h,1

ÿ

ð

B11b

Þ

Adding these two equations, we obtain

{

e

}

]

m

]

_t

¼

e

J

]

_m

_h,1

]

_t

þ

e

J

]

_m

_q,1

]

_t

¼

v

n

o

¹

e

J

V

{

e

}

m

_h,0

þ

v

¹

e

J

V

{

e

}

m

_q,0

ð

B12

Þ

The asymptotic behaviour is such that

lim

t→`

{

e

}

]

_m

₁

]

_t

¼

v

n

o

¹

e

J

V

{

e

}

m

þ

v

¹

e

J

V

{

e

}

m

ð

B13

Þ

The reference velocity that makes the right-hand side of

this equation equal to zero is

V

¼

v

n

o

þ

v

(B14)

and we shall use this value for

Vr

in the following

paragraphs.

Using symbolic calculus, we were able to obtain the

following limits:

lim

t→`

m

h,1

¼

e

J

f

þ

e

J

f

{

e

}

þ

e

J

m

e

J

v

n

o

¹

e

J

v

a

_{

_e

_}

ð

B15a

Þ

lim

t→`

m

q,1

¼

e

J

f

þ

e

J

f

{

e

}

¹

e

J

m

e

J

v

n

o

¹

e

J

v

a

_{

_e

_}

ð

B15b

Þ

where

f

¼

m

a,1

(t

¼

0 ),

a

¼

h

,

q

(B15c)

Moments of order 2

The governing equations for the moments of order 2 are

e

J

]

_m

_h,2

]

_t

¼

2 v

n

o

¹

e

J

V

{

e

}

m

_h,1

þ

2 D

pp hh

ÿ

m

h,0

þ

2 D

pphq

ÿ

m

q,0

¹

a

_m

h,2

¹

m

q,2

ÿ

ð

B16a

Þ

e

J

]

_m

_q,₂

]

_t

¼

2 v

¹

e

J

V

{

e

}

m

q,1

þ

2 D

pp qq

ÿ

m

q,0

þ

2 D

ppqh

ÿ

m

h,0

¹

a

_m

q,2

¹

m

h,2

ÿ

ð

B16b

Þ

When these two equations are added, we obtain

{

e

}

]

m

]

_t

¼

e

J

]

_m

_h,2

]

_t

þ

e

J

]

_m

_q,₂

]

_t

¼

2 D

ÿ

þ

D

ppqh

ÿ

m

_h,0

þ

2 D

pp qq

ÿ

þ

D

pphq

ÿ

m

_q,0

þ

2 v

n

o

¹

e

J

V

{

e

}

m

_h,1

þ

2 v

¹

e

J

V

{

e

}

m

_q,1

ð

B17

Þ

The asymptotic limit is obtained by taking the limit of this

equation and using eqns (B15) and (B10) to obtain

{

e

}

]

m

]

_t

¼

2 D

pp hh

ÿ

þ

D

ppqh

ÿ

þ

D

pphq

ÿ

þ

D

ppqq

ÿ

mo

þ

2 e

J

v

n

o

¹

e

J

v

2 a

_{

_e

_}

m

ð

B18

Þ

We are now in a position to conclude that the asymptotic

behaviour of the two-equation model can be represented by

a dispersion equation of the form

{

e

}

]

{

h i

c

}

]

_t

þ

v

]

{

h i

c

}

]

_x

¼

D

pp `

ÿ

]

{

h i

c

}

]

_x

(B19)

where the asymptotic dispersion coefficient is given by

D

pp `

ÿ

¼

D

pphh

ÿ

þ

D

ppqh

ÿ

þ

D

pphq

ÿ

þ

D

ppqq

ÿ

þ

e

J

v

n

o

¹

e

J

v

2 a

_{

_e

_}

ð

B20

Þ

One should note that this development

does not

make the

assumption that the initial conditions are similar for both

regions.

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