Dispersion in heterogeneous porous media: One-equation non-equilibrium model

Transport in Porous Media 44: 181-203,200l. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.

181

Dispersion in Heterogeneous Porous Media: One-Equation Non-equilibrium Model

MICHEL QUINTARD1, FABIEN CHERBLANC2 and STEPHEN WHITAKER3

1 Institut de Mecanique des Fluides, Av. Camille Soula, 31400 Toulouse, Cedex France. e-mail: quintard@irriftfr 2LEPT-ENSAM, Esplanade des Arts et Metiers, 33405 Talence cedex, France 3 Department of Chemical Engineering and Material Science, University of California at Davis, 95616 Davis, CA, U.S.A.

(Received: 26 August 1999; in final form: 24 March 2000)

Abstract. In this paper, the method of large-scale averaging is used to develop two different oneequation models describing dispersion in heterogeneous porous media. The first model represents the case of large-scale mass equilibrium, while the second represents the asymptotic behavior of a twoequation model obtained by large-scale averaging. It is shown that a one-equation, non-equilibrium model can be developed even when the intrinsic large-scale averaged concentrations for each region are not equal. The solution of this non-equilibrium model is equivalent to the asymptotic behavior of the two-equation model.

Key words: dispersion, averaging, non-equilibrium, effective properties

Nomenclature

b* f3 cf3

(cf3)

(cf3)~ (Cf3)~ {(cf3)~}W {(Cf3)~ }'l C* f3

Cf3

C~eq

interfacial area of the fJ-a- system contained within the averaging volume V, m2 .

area of the boundary between the T/ and co-region contained in the large-scale averaging volume Voo , m2. vector fields that maps V'C~ onto Cf3, m.

point concentration in the fJ phase, mol m -3. Darcy-scale superficial average concentration, mol m-3.

Darcy-scale intrinsic average concentration in the co-region, mol m-3.

Darcy-scale intrinsic average concentration in the T/-region, mol m-3.

large-scale intrinsic average concentration in the co-region, mol m-3.

large-scale intrinsic average concentration in the T/-region, mol m-3. large-scale average concentration associated with the one-equation non-equilibrium

model, mol m-3.

= (cf3)f3 - C~, large-scale spatial deviation concentration, mol m-3.

large-scale average concentration associated with the one-equation local-equilibrium

model, mol m-3. large-scale average concentration associated with the asymptotic behavior of the

two-equation model, mol m-3. molecular diffusivity, m2/s.

182

D** 00

D** f3

nf3u nl)w U* ~f3 Uf3

V

Vf3 Voo

Vw vI) vf3 (vf3 ) (vf3 )f3

(Vf3)~ (vf3)~ {(vf3) } { (vf3)w}W {(vf3h}1)

MICHEL QUINTARD ET AL.

Darcy-scale dispersion tensor fro the fJ-a system, m2/s.

= Dfi - {Dfi}, large-scale spatial deviation for the dispersion tensor, m2/s.

Darcy-scale dispersion tensor in the w-region, m2/s.

Darcy-scale dispersion tensor in the 1]-region, m2/s.

dominant dispersion tensor for the w-region transport equation, m2/s. dominant dispersion tensor for the 1]-region transport equation, m2/s.

coupling dispersion tensor for the w-region transport equation, m2/s.

coupling dispersion tensor for the 1]-region transport equation, m2/s. large-scale dispersion tensor associated with the one-equation local-equilibrium

model, m2/s. large-scale asymptotic dispersion tensor, m2/s. large-scale dispersion tensor associated with the one-equation non-equilibrium

model, m2/s. unit normal vector pointing from the fJ-phase to the a-phase. unit normal vector directed from the 1]-region towards the w-region. large-scale seepage velocity, mls.

= (v f3)f3 - Ufi, large-scale spatial deviation for the seepage velocity, mls.

local averaging volume for the fJ-a system, m3. volume of the fJ-phase contained in the averaging volume V, m3. large-scale averaging volume for the 1]-W system, m 3. volume of the w-region contained in the averaging volume Voo , m3.

volume of the 1]-region contained in the averaging volume V 00, m3. fluid velocity vector in the fJ-phase, m/s. Darcy-scale superficial average velocity, mls. Darcy-scale intrinsic average velocity (seepage velocity), mls.

Darcy-scale intrinsic velocity in th w-region, m/s.

Darcy-scale intrinsic velocity in th 1]-region, mls. = rpw{ (v f3)w}W + rpl) { (v f3)I)}I), large-scale superficial average velocity, mls. intrinsic regional average velocity in the w-region, mls. intrinsic regional average velocity in the 1]-region, mls.

Greek Symbols a* mass exchange coefficient for the 1]-W system, s-l.

£ f3 porosity for the fJ-a system. £ f3w porosity of the w-region. £ f31) porosity of the 1]-region. rpw volume fraction of the w-region. rpl) volume fraction of the 1]-region.

1. Introduction

In this paper, we examine the transport of a tracer in a heterogeneous porous medium similar to the one represented in Figure 1. Solutes migrating through such natural formations exhibit, in many cases, dispersion that is non-Fickian. Asymmetrical breakthrough curves and tailing are generally observed. This behavior is commonly named 'anomalous' or 'non-ideal' referring to the fact that it cannot be

DISPERSION IN HETEROGENEOUS POROUS MEDIA 183

Figure 1. Two-region model of a heterogeneous porous medium.

represented by the classical advection-dispersion equation valid for homogeneous systems (Koch and Brady, 1987; Cushman and Ginn, 1993; Zhang and Neuman, 1996). This paper is essentially focused on the case of heterogeneous systems made up of two different regions, which are known to exhibit this type of behavior. These systems are generally thought of as a continuous, highly permeable region, that is, the mobile zone, where advection and dispersion are the dominant mechanisms, and a stagnant, low permeability region, that is, immobile zone, where diffusion is the main transport mechanism. In this paper, we are only interested in the large-scale description of dispersion mechanisms, that is, it is assumed that length-scale constraints are such that the Darcy-scale equation corresponds to the classical dispersion equation. A large-scale description of dispersion in the cases

184 MICHEL QUINTARD ET AL.

under consideration in this paper is usually undertaken by using a two-equation model involving two averaged concentrations associated to the mobile and immobile zones (Coats and Smith, 1964; De Smedt and Wierenga, 1979; Goltz and Roberts, 1986; Brusseau et ai., 1989; Correa et aI., 1990). Extensions of such models have been proposed for mobile/mobile systems, that is, systems in which some advection may occur in the low permeable region (Skopp et ai., 1981; Gerke and Van Genuchten, 1993; Ahmadi et al., 1998). In the work of Ahmadi et al. (1998), the two-equation model was derived using the method of large-scale averaging, and a comparison with numerical experiments for stratified systems showed good agreement between theory and experiment. This agreement indicates that we can use the two-equation model with confidence to explore, by numerical computation, some interesting aspects of this physical problem.

In this study, we are particularly interested in the fact that the two-equation model has an asymptotic behavior that can be described in terms of a one equation model (Zanotti and Carbonell, 1984; Ahmadi et al., 1998). A one-equation model was derived previously by imposing the condition of large-scale mass equilibrium, that is, essentially equally averaged concentrations for both regions. This requires a rapid relaxation of the concentration field between the two regions, and constraints were developed (Quintard and Whitaker, 1998) indicating when this condition would occur. On the other hand, the asymptotic behavior of the twoequation model does not correspond to local equilibrium, as will be shown in the next section. This suggests that a one-equation, non-equilibrium theory is available. While this theory is not restricted by the condition of large-scale mass equilibrium, it does contain the limitations associated with the asymptotic behavior. This theory is presented in Section 5. In a subsequent section, we show that the large-scale dispersion coefficient, obtained from the two-equation asymptotic behavior, is not equal to that obtained from the one-equation equilibrium model. In this paper, we will assume that the heterogeneous porous medium can be represented locally by a periodic unit cell, and we will use this feature in the development of the closure problems in the next sections.

To be absolutely clear about our choice of words used to describe solute transport in such heterogeneous porous media, we refer to the two-region system illustrated in Figure 1 and identify three transport models as follows:

·1. The two-equation model consists of separate transport equations for both the wand 1]-regions. The dominant coupling between the two equations is represented by an inter-region flux that depends on an exchange coefficient and the difference between the concentrations in the two regions. Coupling also occurs because the gradient of the 1]-region concentration appears in the w-region transport equation, and the gradient of the w-region concentration appears in the 1]-region transport equation; however, this coupling is not as important as the coupling caused by the inter-region mass flux.

2. The one-equation equilibrium model consists of a single transport equation for both the wand 1]-regions. When the two concentrations in the two regions are

https://www.researchgate.net/publication/222442458_Whitaker_S._Transport_in_chemical_and_mechanical_heterogeneous_porous_media_IV_large-scale_mass_equilibrium_for_solute_transport_with_adsorption._Adv._Water_Resour._22_33-57?el=1_x_8&enrichId=rgreq-e9bd5cac-6d5d-4f55-9312-27d482a7ec51&enrichSource=Y292ZXJQYWdlOzIyNzI1NjQyNztBUzo5OTA2MjAxOTc4ODgxOEAxNDAwNjI5NjM1MjIz

https://www.researchgate.net/publication/244114040_Development_of_Transport_Equations_for_Multiphase_Systems_I-II-III?el=1_x_8&enrichId=rgreq-e9bd5cac-6d5d-4f55-9312-27d482a7ec51&enrichSource=Y292ZXJQYWdlOzIyNzI1NjQyNztBUzo5OTA2MjAxOTc4ODgxOEAxNDAwNjI5NjM1MjIz

https://www.researchgate.net/publication/222305768_Transport_in_chemically_and_mechanically_heterogeneous_porous_media_V._Two-equation_model_for_solute_transport_with_adsorption?el=1_x_8&enrichId=rgreq-e9bd5cac-6d5d-4f55-9312-27d482a7ec51&enrichSource=Y292ZXJQYWdlOzIyNzI1NjQyNztBUzo5OTA2MjAxOTc4ODgxOEAxNDAwNjI5NjM1MjIz



close enough, the transport equations that represent the two-equation model can be added to produce this model. By close enough, we mean that the principle of largescale mass equilibrium is valid and the constraints associated with this condition have been developed by Quintard and Whitaker (1998). To be very clear; we note that the one-equation equilibrium model is obtained directly from the two-equation model by imposing the constraints associated with local mass equilibrium.

3. The one-equation non-equilibrium model consists of a single transport equation for both the wand r]-regions. In this case, the condition of large-scale mass equilibrium is not imposed on the large-scale averaged equations; instead, a longtime constraint is imposed. This long-time constraint can be imposed in two ways. First one can average the Darcy-scale dispersion equation over both the wand r]-regions illustrated in Figure 1 and then impose a long-time constraint on the closure problem. Second, one can begin with the two-equation model, determine the sum of the spatial moments of the two equations, and construct a one-equation model that matches the sum ofthe first three spatial moments in the long-time limit. The second analysis yields exactly the same equation as the first. This situation is comparable to the process of Taylor dispersion (Taylor, 1953) where one finds that the method of moments (Aris, 1956) gives the same result as direct averaging (Carbonell and Whitaker, 1983) together with the use of a closure problem to predict the axial dispersion coefficient.

2. Two-Equation Model

Most of the discussion in this section is similar to material available in Quintard and Whitaker (1998) and Ahmadi et al. (1998). For this reason, we will not present a complete development of the different models discussed here. For the purpose of our discussion, we will use the notation corresponding to the method of volume averaging. The problem of describing dispersion in a heterogeneous porous medium starts with the pore-scale equations, which we list below for the case of a tracer.

v . vf3 = 0 in Vf3 ,

aCf3 at + V . (vf3cf3) = V . (V(:J VC(3) 10 Vf3 ,

B.C. 1.

(1)

(2)

(3)

The velocity field for the ,B-phase is determined by solving Stokes' equations subject to the no-slip condition at the ,B-u interface.

A Darcy-scale description can be developed, provided that certain length-scale constraints are satisfied (Brenner, 1980; Eidsath et al., 1983; Plumb and Whitaker, 1988; Mei, 1992), leading to the classic dispersion equation which we write as

a(cf3(Cf3)(:J) * -----'-a....:..t - + V· (cf3(Vf3)(:J(cf3)f3) = V· (Df3· V(c(:J)f3). (4)


Here (cfJ)fJ is the intrinsic averaged concentration and (VfJ)fJ is the intrinsic average velocity defined respectively by

(5)

(6)

where cfJ is the ,B-phase volume fraction. Equation (1) leads to the following Darcy-scale continuity equation:

(7)

We can now formulate the Darcy-scale problem associated with a two-region porous medium, and this is given by

(8)

B.C. 1 (10)

B.C. 2 (11)

(12)

In this case, the velocity field is determined by a solution of Darcy's law independently from the dispersion problem.

A complete development leading to the two-equation model presented in this section can be found in Ahmadi et al. (1998). Averaged concentrations are defined for each region, and for the r]-region we have

(14)

in which <pry represents the volume fraction of the r]-region defined explicitly by

(15)



With these definitions, the two-equation model is written as

a{ (C,B),B}" <p"c,B" at" + <p,,{(v,B),,}"· V{(C,B)~}"-

- v· [d1J({(c,B)~}" - {(c,B)!}''')] - u",,· V{(C,B)~}" - u"w· V{(C,B)!}W

= V· (D~~. V{(C,B)~}") + V· (D~:. V{(c,B)!}W)-

- a*({ (C,B)~}" - {(c,B)!}w), (16)

a{ (C,B)~}W {() }W V{( ),B}W <Pwc,Bw at + <Pw v,B W . c,B W -

- V· [dw({(c,B)!}W - {(C,B)~}")] - u w,,· V{(C,B)~}1J - UWW · V{(C,B)!}W

= v . (D:; . V{ (C,B)~}") + V . (D:: . V{ (c,B)!}W) -

-a*({(c,B)!}W - {(C,B)~}"). (17)

All coefficients in these equations are given explicitly by a set of three closure problems that can be found in Ahmadi et al. (1998). As discussed in Ahmadi et al. (1998), this model requires some length-scale and time-scale constraints; however, they are less severe than the constraints associated with the condition of largescale mass equilibrium. The model associated with large-scale mass equilibrium is actually best discussed using a non-equilibrium model, and this is done in the next section.

3. One-Equation Model: Large-Scale Equilibrium

An important situation arises from the non-equilibrium model when the exchange coefficient, a*, is such that at any time, or at least after a very short time, the two regional averaged concentrations are essentially equal. This corresponds to the approximation represented by

(18)

and this will be used as a definition of large-scale mass equilibrium. When this condition occurs, it is convenient to introduce the following concentration

(19)

where we have used a specific notation for the average concentration, C~eq' indicating that it is not a simple volume averaged value, but a porosity weighted averaged value. A similar notation was used for saturation in dealing with two-phase flow in heterogeneous porous media in Quintard and Whitaker (1990). We have added the subscript eq to emphasize that this concentration has a special feature associated with the equality of the region averaged concentrations.



The governing equation for C~eq is obtained from the sum of Equations (16) and (17) given by

ac* {StJ} a;e

q + {(VtJ}} . VCfeq = V . (D;; . VCfeq ), (20)

in which we have made use of the notation

D** D** + D** + D** + D** eq - '1'1 '1W W'l ww'

(21)

(22)

(23)

Equation (20) gives the one-equation model corresponding to the condition of large-scale mass equilibrium represented by Equation (18).

4. Two-Equation Model: Asymptotic Behavior

The equilibrium model discussed in the preceding section is not the only oneequation model that can be used to describe transport in a two-region model of a heterogeneous porous media. It has been shown elsewhere (Zanotti and Carbonell, 1984; Ahmadi et ai., 1998) that, as time goes to infinity, the asymptotic behavior of the coupled system of equations can be described by a one-equation model. Considering a flow parallel to the x-axis, this one-equation model can be written as

ac* ac* a2c* {s }~ + {(v )}~ = D** tJ

oo. (24)

tJ at tJ ax 00 ax2

Here the averaged concentration is defined in a manner similar to C~eq' that is, we have used

C* _ qJ'I stJ 'I { (ctJ}~}'I + qJwstJw{ (ctJ}~}W tJoo -

qJ'IstJ'I + qJwE:tJw (25)

In this case there is no reason for an equality between the regional averages, and in general, we will have

(26)

however, the difference between the two regional concentration will generally be constrained by

(27)

The asymptotic dispersion coefficient is given by (Zanotti and Carbonell, 1984; Ahmadi et al., 1998)

D** = D** + D** + D** + D** + 00 '1'1 '1W W'l ww

+ (qJwstJw{ (vtJ}'l} - qJ'lstJ'l{ (VtJ}w})2

a*{stJ}2 (28)

DISPERSION IN HETEROGENEOUS POROUS MEDIA

2.0E-OS .------------------------,

1.SE-OS

1.6E-OS

1.4E-OS

1.2E-OS

~ " 1.0E-OS ~ c:l..

~ S.OE-09

6.0E-09

4.0E-09

2.0E-09

x=66.S m · . · . · . · . · .

O.OE+OO L-__ <f!...-.,"--'--___ ---'c......::."----= ______ ---I

O.OOE+OO 1.00E+OS 2.00E+OS 3.00E+OS 4.00E+OS

t (s)

Figure 2. Asymptotic behavior of the different large-scale models.

Table I. Properties of the stratified system

I)-region w-region

Thickness (m) 0.5 0.5

Permeability (m2) 10-12 10- 13

(Dilb (m2/s) 30010-9 3010-9

(Dilbl (Dil)yy 0.1 0.1

(vtJ) (rnIs) 310-7 0.310-7

189

This expression incorporates the effect of mass exchange between the different regions. By comparing Equations (22) and (28), one sees clearly that

D** -I- D** 00 1 eg' (29)

Indeed, D~ can be much greater than D;; as it is illustrated by the numerical examples obtained for the case of a stratified system in Ahmadi et al. (1998). One example of these computations is shown in Figure 2. This corresponds to the injection at a given concentration of the tracer in a two-strata medium initially at a zero concentration. The flow is parallel to the strata. The curves in Figure 2 represents the time evolution of oC;o%t and oC;eg/ot at a given location x in the porous system. The physical characteristics of the system are summarized in Table I. The distance from the inlet has been chosen large enough so asymptotic behavior was approximately achieved. In addition, Darcy's scale numerical experiments were used to verify that the results provided by the two-equation


model were in good agreement with the actual behavior (Ahmadi et al., 1998). The results in Figure 2 illustrates the dramatic difference that can exist between the local-equilibrium dispersivity, and the actual dispersivity taking into account the crossflow mechanisms.

The result given by Equations (24) and (27) has the same characteristics as the Taylor-Aris theory (Taylor, 1953; Aris, 1956) of dispersion in capillary tubes which is an asymptotic theory valid for times such that

JVf3t/r~» 1. (30)

If one thinks of the different streamlines for laminar flow between two flat plates as being different regions having different concentrations, it seems plausible that a one-equation non-equilibrium model for longitudinal dispersion could be developed for parallel flow in a stratified system. The situation may become more appealing if one imagines that the strata become very thin and the permeabilities are adjusted so that a parabolic velocity profile for (vf3) is created. This analogy between the problem under consideration and Taylor's dispersion has been explored for stratified systems by Lake and Hirasaki (1981). Based on an approximate solution of the concentration field, the authors were able to propose a longitudinal effective dispersion coefficient for multilayered systems. Besides the theoretical background, the proposed analysis provides a general result that can be applied to systems different from stratified media.

The long-time constraint required by the Taylor-Aris theory of dispersion eventually (Whitaker, 1999) leads to a constraint on the concentration given by

cf3 - (cf3)f3 « (Cf3)f3. (31)

This indicates that the concentration must be nearly constant at any cross section of a Taylor-Aris dispersion process. The analogous equilibrium and non-equilibrium models for mass transport in capillary tubes are examined in Appendix B.

At this point, we are in the following situation. In general, a real system will exhibit a non-Fickian behavior, and this behavior can be captured approximately by a two-equation model in the case of a two-region system. However, one-equation behavior can also be observed for two rather different conditions. The first of these corresponds to an equilibrium model, and requires fast exchange between the different regions in order to achieve the condition expressed by Equation (18). The second condition corresponds to the asymptotic behavior of the two-equation model, and is described by the dispersion coefficient given by Equation (28). This latter condition exhibits a much larger dispersivity, and it can provide a better representation of actual behavior than the one-equation equilibrium model. However, the calculation of D~ requires solving three closure problems, and this represents an excessive effort unless there is a particular interest in all the effective properties associated with the two-equation model. In the next section, we introduce a new closure scheme that will give a one-equation non-equilibrium model having the same dispersivity as the asymptotic dispersivity.


5. Large-Scale Non-equilibrium One-Equation Model

We begin this study with the Darcy-scale equations for the ,B-phase, and in this section we will not restrict the analysis to a two-region model of a heterogeneous system. This means that the porosity and permeability depend on position. The velocity field will be affected by the heterogeneities in the system, but it can be determined independently from the dispersion problem. The dispersion tensor in Equation (4) is thus position-dependent, that is, the function expressing the dependence on the velocity depends on the position. Large-scale averages are defined over a large-scale averaging volume V 00 according to

{Std = _1_ [ sfJ dV, Voo lvCXJ

(32)

(33)

and this latter result defines the large-scale averaged concentration in a manner similar to Equation (25). We use a similar definition for the averaged interstitial velocity given by

(34)

We introduce the following large-scale spatial deviation concentration and velocity in terms of the following decompositions:

Using Equations (33) and (35), we have

{sfJ(cfJ)fJ} = {SfJ(Cf; + CfJ)} = {sfJ}Cf;,

(35)

(36)

(37)

(38)

provided that C;; can be removed from the volume integral indicated by { }. This type of approximation has been discussed in detail elsewhere, and the length-scale constraints associated with this simplification are given by (Quintard and Whitaker, 1994a,b,c,d,e; Whitaker, 1999). Similarly, we can use Equations (34) and (36) to obtain

(39)

(40)


Taking the large-scale average of Equation (7) leads to

\7 . ({cp}U~) = o. (41)

We now direct our attention to Equation (4) and form the average to obtain

(42)

In all three of the terms in this result we can interchange integration and differentiation leading to

a{cp(cp)p} at + \7. {cp(Vp)p(cp)p} = \7. {D~. \7(cp)p}. (43)

At this point we use the definitions of the large-scale average properties and the decompositions in order to obtain an unclosed macroscopic equation that is similar in form to the average of Equation (2).

a({cp}c~t) ~ ~ at + \7 . ({cp}U~c~) + \7 . {cpU pCp}

= \7 . ({D~} . \7C~ + {D~ . \7Cp}).

In order to obtain this result, we need to impose the restrictions given by

{EpU~Cp} « {cpUpCp} and {EpUpC~} « {cpUpCp}.

(44)

(45)

When averaged quantities can be removed from the average, these restrictions are automatically satisfied, and we have already made use of this condition to obtain Equations (37) and (39). However, a detailed study of the analogous terms in Equations (45) for Darcy-scale dispersion (Whitaker, 1999) failed to provide a rigorous set of constraints, and the terms that we have discarded i~ o~er to obtain Equation (44) need to be considered in detail. The average of cpU pCp represents a large-scale convective transport that is traditionally represented by a dispersion model, while the average of Dft . \7 C p represents a higher order contribution to the large-scale dispersive transport. This term will eventually lead to a third order derivative in the large-scale transport equation and this will generate skewness in any pulsed system.

The Darcy-scale equation for the concentration deviation, C p, represents the governing equation in the closure problem. We obtain this equation by introducing the decomposition given by Equation (35) into Equation (4), to obtain

ag a~ ~ cp- + cp- + \7. (cp(vp)PC~ + cp(vp)PCp)

at at ~

= \7 . (D~ . \7C~) + \7 . (D~ . \7Cp). (46)

At this point, the analysis proposed in Chella et al. (1998) followed previous studies of dispersion in heterogeneous porous media (Gelhar and Axness, 1983;


Dagan, 1984; Gtiven and Molz, 1986) and variations in the Darcy-scale porosity were ignored. However, it was found that this simplification is often too drastic, and should not be made in a precise development.

Assuming that the large-scale averaged porosity {E'ti} is constant, and using Equation (44) in order to simplify Equation (4), we obtain

aCti E'ti * * * ~ E'ti at + {E'ti} V . ({Dti } . VCti + {Dti . VCti }) -

E'ti ~ ~ ti ~ - -({E'ti}U; . VC; - V· {E'tiUtiCti}) + E'ti(vti) . V(C; + Cti ) {E'ti}

= V . (D; . VC;) + V . (D; . VCti ).

Equation (47) can be written as

aCti ti ~ E'ti ~ ~ E'ti- + E'ti(vti) . V(C; + Cti ) - -({E'ti}U; . VC; + V· {E'tiUtiCti})

at {E'ti}

= V . (D; . VC;) + V . (D; . VCti ) -

(47)

- ~V. ({D;} . Vc; + {D; . VCti}), (48) {E'ti}

or

where we have adopted the notation

Dti = D; - {D;}. (50)

Equations (44) and (49) must be solved simultaneously in order to provide a complete solution ofthe problem, and this represents an extremely difficult computational effort. However, assuming that the different length-scale and time-scale associated with the Darcy-scale and the large-scale problems are conveniently separated, the time derivative in Equation (49) can be discarded, as well as terms corresponding to derivatives involving only large-scale averages. Taking this into account, Equation (49) becomes

E'tiUti . VC; + E'ti(vti)ti . VCti = V· (D; . VCti) + V· (Dti . VC;) (51)

and a representation of the concentration deviation can be written as

Cti = b; . VC; + ... , (52)


where the dots are a reminder that second order terms have been neglected. In this equation the mapping vector b~ has to be chosen in order to cancel all first order terms with respect to VC~ in Equation (51). Therefore, it must obey the following 'closure' problem

Large-scale closure problem

Cf3(Vf3)f3 . Vbfi + cf3Uf3 = V· (Dfi . Vbfi) + V· D f3 ,

bfi(r + Id = bfi(r),

{cf3bfi} = O.

(53)

(54)

(55)

In these equations, two source terms will generate large-scale dispersion effects. The first one is associated with the velocity fluctuations while the second one is associated with the variations of the dispersion tensor. The last equation, Equation (55), corresponds to the fact that the average of the deviation is zero, while Equation (54) assumes that the medium can be described locally by a periodic system. This last assumption has been discussed thoroughly in the literature and has been proved to be useful even for disordered systems (see Ahmadi and Quintard (1996) for an illustration). Introducing the decomposition in Equation (44) leads to the following large-scale dispersion equation

a({cf3}C~) *} D** * at + V . (V f3 { C f3 ) = V . ( f3 . V C f3)

in which the large-scale dispersion tensor is given by

D~* = {Dfi} + {Dfi . Vbfi} - {Cf3Uf3bfi}·

(56)

(57)

This expression seems quite different from the result given for D;;; in Equation (28). However, we will see in one example in the next section that these two dispersion coefficients are equal, and this represents an important theoretical result.

Before proceeding to this comparison, we give below some indication on the form of the 'closure' problem in the case of a two-region heterogeneous system. In this case, we have equations similar to Equations (53) and (54) for each region. Equation (55) remains unchanged, and the boundary condition at the interface is

llwry . (Dfiw . Vbfiw) + llwry . Df3w

= llwry . (Dfiry . Vbfiry) + llwry . Df3ry at Awry, (58)

in which we have made use of the continuity of the b~-field and of the normal filtration velocity.

DISPERSION IN HETEROGENEOUS POROUS MEDIA

6. Comparison Between Two-Equation Model Asymptotic Behavior and One-Equation Non-equilibrium Model

195

In Appendix A we show that the solution of the closure problem in the case of a stratified system leads to the following dispersion coefficient

D ** -p -* * (lry + lw)2

<pry (Dpry)xx + <Pw(Dpw)xx + 12 x

x [ <Pry + <Pw ] (<Pwcpw{ (vp)ry} - <pryCpry{ (Vp)w})2

(Dfiry)yy (Dfiw)yy {cp}2 (59)

This must be compared to the value of D: for the same system. This question was developed in Ahmadi et al. (1998) (Appendix B), and we simply list the result as

with

* 12 (Dfiry)yy(Dfiw)yy a = ----::- -----''-'------'----

(lry + lw)2 <Pw(Dfiry)yy + <pry (Dfiw)yy

U sing these values we find the fundamental result

D ** = D** p 00·

(60)

(61)

(62)

This indicates that the one-equation non-equilibrium model corresponds to the asymptotic behavior of the two-equation model. It should be possible to show that the three closure problems that are necessary to calculate 28, can be reduced to the single closure problem given in the preceding section; however, this represents a very complicated task that is beyond the scope of this paper.

Incidentally, it is interesting to compare the effective dispersion coefficient provided by Equation (59) with the one given by Lake and Hirasaki (1981) in the case of a stratified system ((Lake and Hirasaki, 1981), Equation (18)). With the appropriate nomenclature translation we found that they are equal. The authors intuition was remarkable, and it is interesting that their result fits into the proposed theory for this particular case. In addition, it was proved in Ahmadi et al. (1998) that the asymptotic dispersion coefficient obtained from the two-equation model was exactly equal to the one obtained from a direct, that is, Darcy-scale, solution of the problem (Marle et al., 1967).

7. Conclusion

In this paper, we demonstrated that two different one-equation models can be derived to describe dispersion in heterogeneous media. The first corresponds to the


case of large-scale mass equilibrium. The second corresponds to the asymptotic behavior of a two-equation model classically used in the case of two-region heterogeneous systems. A one-equation, non-equilibrium model has been developed, assuming that the large-scale region averaged concentrations are not equal. It was found that the solution of this one-equation non-equilibrium model is equivalent to the asymptotic behavior of the two-equation model. This fundamental result has important practical implications, since the one-equation, non-equilibrium model is much simpler than the two-equation model that involves three closure problems. Of course, the one-equation non-equilibrium model is only useful if one is interested in long-time behavior, and when this is not the case one must revert to the original two-equation model.

Acknowledgements

Financial support from CNRSIINSUIPNRH is gratefully acknowledged.

Appendix A. Results for a Stratified System

We consider a unidirectional flow in the x-direction parallel to the stratification in a horizontal perfectly stratified system of constant thickness (Figure A.l). All properties (permeability, dispersion, porosity, ... ) depend only on the vertical coordinate y. The two regions are homogeneous, and the characteristics associated with the regions are denoted with the subscript 17 or w. The flow problem is trivial since the velocities are constant in each strata. We use the following notation for the velocities:

((V,B)ry)x = cf3ryUf3ry,

((vf3)",)x = cf3w Uf3w'

Figure A.i. Stratified model of a heterogeneous porous medium.

(A.I)

(A.2)

•


The periodic unit cell representative of this system is one-dimensional along the y-direction, and in this case the closure variable b~ depends only on the y-coordinate. It should be noticed that the periodic unit cell associated to the flow in a stratified system bounded by impervious boundaries would have strata twice as thick as the original medium.

The closure problem associated with the one-equation non-equilibrium model (Equation 53) simplifies to the form

(A3)

for the x-component of the closure variable, denoted b'f;. In the 1]-region, the solution of this partial differential equation is

(A4)

where

(A5)

A similar solution is found in the (V-region. The boundary conditions Equation (54) are expressed by

(A.6)

(A.7)

In order to specify the additional boundary condition needed at the interface AI)"" we take Equation (53) in the sense of distribution (see Quintard and Whitaker (1993) or Gray et al. (1993)) to obtain

(A8)

Then, using the average condition given by Equation (55), we have

(A9)


With the four equations (Equations (A6) through (A9)), we are able to determine the four unknowns (A~, B~, Au" Bw). We obtain

l~ + lw cfi~C{J1)cfiwC{Jw (Ufiw - Ufi~) A1) = (A 10)

2 {cfi} (D~1))YY

l1) + lw cfi~C{J1)cfiwC{Jw (Ufiw - Ufi~)

2 {cfi} (D~w)Yy (All)

B1) Bw = (l1) + lw)2 cfi1)C{J1)cfiwC{Jw (U _ U ) x 12 {cfi}2 fiw fi1)

x [Cfi~ C{J~ - cfiw C{J~ ]. (D~1))YY (D~w)Yy

(A12)

The closure variables are now fully determined, it allows us to calculate explicitly the large-scale dispersion coefficient. In the case of a stratified medium, Equation (57) takes the form

Dr {D~} - {cfifJf3b~} (A.13)

(A.14)

~ {y=o ~ e=lry - cfiwUfiw }y=-lw b~w dy - cfi1)Ufi~ }y=o b~~ dy. (A15)

Substituting the closure variables b~ and bw in terms of their representations leads to

D~* = C{J~(D~~)xx + C{Jw(D~w)xx + (A16)

+ (l~ + lw)2 (cfi~C{J1)cfiwC{Jw)2 [ C{J1) + C{Jw ] x

12 {cfi}2 (D~1))yy (D~w)\'Y

x (Ufi1) - Ufiw)2 (A.I7)

and this result can be formulated as

D;* = C{J~(D~1)Lx + C{Jw(D~w)xx + (A18)

(A19)

Appendix B. One-Equation Equilihrium/Non-Equilihrium Models

The concepts associated with equilibrium and non-equilibrium, one-equation models can be clarified in terms ofthe classic Taylor-Aris dispersion process (reference

•


Taylor (1953) and Aris (1956)). For diffusion and convection in a capillary tube that is subjected to a pulse of dilute chemical species, the governing differential equation and boundary conditions are given by

CfJ = F(z), t = 0,

CfJ = 0, z = ±oo.

The area averaged form of Equation (B.1) is given by

o(CfJ)fJ + (v )fJ o (cfJ)fJ = VfJ o2(cfJ)fJ _ ~(CfJv)fJ ot z OZ OZ2 OZ z

in which the spatial deviation quantities are defined according to

C(J = cfJ - (cfJ)fJ,

~ _ fJ Vz - V z - (vz ) .

(B.1)

(B.2)

(B.3)

(BA)

(B.5)

(B.6)

(B.7)

Equation (B.5) is analogous to the one-equation, non-equilibrium model given by Equation (79) of Quintard and Whitaker (1998). The analogy results from the fact that Equation (B.5) represents a governing differential equation for a single concentration, (CfJ)fJ, and it contains a term involving the spatial deviation, C(J, from that single concentration. For the case of transport in heterogeneous porous media, we say that large-scale mass equilibrium is valid when the terms involving deviations from the large-scale average concentration are negligible. The analogous condition for mass transport in a capillary tube requires that the last term in Equation (B.5) be negligible and we express this idea as

o o2(c )fJ OZ (C(JVz)fJ «VfJ o~ (B.8)

This restriction is analogous to the three restrictions given by Equations (81) of Quintard and Whitaker (1998). Since the length scales for the two quantities in Equation (B.8) are identical, we can express that result as

(B.9)

At this point one normally needs estimates for both terms in order to develop a constraint (Whitaker, 1988); however, in this case we only need to estimate C(J on


the basis of the closure problem for this quantity (Carbonell and Whitaker, 1983). This leads to

~ ((Vz)f3 r;;d(Cf3)f3) Cf3 =0

'Df3 dZ (B.IO)

and use of this estimate, along with the estimate for the velocity deviation given by

(B.ll)

allows us to express Equation (B.8) as

[«(vz )f3)2 r;; d(Cf3)f3] d(Cf3)f3 ----"- --- «'Df3 --.

'Df3 dZ dZ (B.12)

This can be arranged in the form of a constraint

Pe2 « 1, (B.l3)

in which Pe represents the Peclet number defined by

(vz )f3 ro Pe = .

'Df3 (B.I4)

When the constraint given by Equation (B.l3) is valid, the term involving the concentration deviation in Equation (B.S) can be discarded and we obtain the analog to the one-equation equilibrium model for transport in heterogeneous porous media. For mass transport in a capillary tube, this model takes the following form.

B.I. EQUILIBRIUM MODEL

_d (_C_f3 )_f3 + (V ) f3 _d (_C_f3 )_f3 _ 'Df3 _d2_( C-,-f3 )_f3

dt z dZ - dZ2 ' Pe « 1. (B.IS)

This result should be compared to the one-equation equilibrium model given by Equation (75) in Quintard and Whitaker (1998).

If we avoid imposing the simplification indicated by Equation (B.8), we can develop the analog of the one-equation non-equilibrium model. This requires the closure problem for e;; which is given by (Whitaker, 1999, Prob. 3-8)

de;; de;; d(Vze;;)f3 ~ d(Cf3)f3 [I d ( de;;) d2e;;] -+vz-- +vz--='Df3 -- r- +-- ,

dt dZ dZ dZ r dr dr dZ2

(B.I6)

e;; = 0, t = 0, (B.17)

de;; (B.18) -=0 r = ro ,

dr '


C/J = 0, z = ±oo. (B.19)

In addition to the governing differential equation, the initial condition, and boundary conditions, we also have the following constraint on the average of the spatial deviation concentration:

(C/J)/3 = o. (B.20)

While this boundary value problem for C/J is more complex than the original boundary value problem, a careful analysis indicates that it can be simplified to eventually produce the solution given by (Whitaker, 1999, Prob. 3-8)

[~ (!...)2 _ ~ (!...)4 _ ~] (vz )/3 r;; J(c/3)/3 4 ro 8 ro 12 D/3 Jz

(B.2l)

This result is the origin of the estimate given by Equation (B.l 0) and the single constraint that must be imposed in order that this result can be extracted from Equations (B.16) through (B.20) is the time-scale constraint given by

(B.22)

Use of Equation (B.2l) in Equation (B.5), along with the expression for the velocity deviation given by

(B.23)

leads to the classic result (Aris, 1956) given by the following.

B.2. NON-EQUILIBRIUM MODEL

(B.24)

It is important to recognize that the equilibrium model is obtained by imposing a constraint on the concentration deviation contained in the averaged equation, while the non-equilibrium model makes use of a constraint imposed on the closure problem for the concentration deviation. The difference in the two models results from the fact that the equilibrium model is constrained by Equation (B.13) while the non-equilibrium model is constrained by Equation (B.22).

In this simple study of equilibrium and non-equilibrium models, the difference between the two models is clearly established by the different constraints that are


imposed. The situation for heterogeneous porous media is analogous but much more complex. In the case of heterogeneous porous media, one can have flow parallel to stratified systems, flow orthogonal to stratified systems, or flow in an arbitrary complex arrangement of the distinct regions (Ahmadi et ai., 1998). Constraints that are analogous to Equation (B.13) have been developed (Quintard and Whitaker, 1998), but only limited results are available in which theory and experiment are compared.

References

Ahmadi. A. and Quintard, M.: 1996, Large-scale properties for two-phase flow in random porous media, 1. Hydrol. 183, 69-99.

Ahmadi, A., Quintard, M. and Whitaker, S.: 1998, Transport in chemically and mechanically heterogeneous porous media V: Two-equation model for solute transport with adsorption, Adv. Water ResoUl: 22, 59-86.

Aris, R.: 1956, On the dispersion of a solute in a fluid flowing through a tube, Proc. Roy Soc. (London, A) 235,67-77.

Brenner, H.: 1980, Dispersion resulting from flow through spatially periodic porous media, Trans. R. Soc. 297(1430), 81-133.

Brusseau, M., Jessup, R. and Rao, R.: 1989, Modeling the transport of solutes influenced by multiprocess nonequilibrium, Water Resour. Res. 25, 1971-1988.

Carbonell, R. G. and Whitaker, S.: 1983, Dispersion in pulsed systems II: Theoretical developments for passive dispersion in porous media, Chern. Engng Sci. 38, 1795-1802.

Chella, R., Lasseux, D. and Quintard, M.: 1998, Multiphase, multi component fluid flow in homogeneous and heterogeneous porous media, Revue Institut Fram;ais du Petrole 53(3), 335-346.

Coats, K. and Smith, B.: 1964, Dead-end pore volume and dispersion in porous media, SPE 1. 4, 73-84.

Correa, A., Pande, K., Ramey, H. and Brigham, W.: 1990, Computation and interpretation of miscible displacement perfonnance in heterogeneous porous media, SPE Reserv. Engng 69-78.

Cushman, J. and Ginn, T.: 1993, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transport in Porous Media 13, 123-138.

Dagan, G.: 1984, Solute transport in heterogeneous porous formations, 1. Fluid Mech. 145, 151-177. De Smedt, F. and Wierenga, P.: 1979, Mass transfer in porous media with immobile water, 1. Hydrol.

41,59-67. Eidsath, A, Carbonell, R., Whitaker, S. and Hennann, L.: 1983, Dispersion in pulsed systems-III,

Chern. Engng Sci. 38, 1803-1816. Gelhar, L. and Axness, c.: 1983, Three-dimensional stochastic analysis of macrodispersion in

aquifers, Water Resour. Res. 19, 161-180. Gerke, H. and Van Genuchten, M.: 1993, A dual-porosity model for simulating the preferential

movement of water and solutes in structured porous media, Water Resour. Res. 29, 305-319. Goltz, M. and Roberts, P.: 1986, Three-dimensional solutions for solute transport in an infinite

medium with mobile and immobile zones, Water Resour. Res. 22, 1139-~48. Gray, W. G., Leijnse, A, Kolar, R. L. and Blain, C. A: 1993, Mathematical Tools for Changing

Spatial Scales in the Analysis of Physical Systems, CRC Press, Boca Raton, FL. Giiven, O. and Molz, F. J.: 1986, Detenninistic and stochastic analyses of dispersion in an unbounded

stratified porous medium, Water Resour. Res. 22(11), 1565-1574. Koch, D. and Brady, J.: 1987, A non-local description of advection-diffusion with application to

dispersion in porous media, 1. Fluid Mech. 180, 387-403.

..


Lake, L. W. and Hirasaki, G. J.: 1981, Taylor's dispersion in stratified porous media, Soc. Petrol. Engn. 1. 459-468.

Marie, C., Simandoux, P., Pacsirszky, J. and Gaulier, C.: 1967, Etude du deplacement de fluides miscibles en milieu poreux stratifie, Revue de l']nstitut Francais du Ptitrole 22, 272-294.

Mei, C. C.: 1992, Method of homogenization applied to dispersion in porous media, Transport in Porous Media 29,261-274.

Plumb, O. and Whitaker, S.: 1988, Dispersion in heterogeneous porous media I: Local volume averaging and large-scale averaging, Water Resour. Res. 24, 913-926.

Quintard, M. and Whitaker, S.: 1990, Two-phase flow in heterogeneous porous media I: the influence of large spatial and temporal gradients, Transport in Porous Media 5,341-379.

Quintard, M. and Whitaker, S.: 1993, One- and two-equation models for transient diffusion processes in two-phase systems, in: Advances in Heat Transfer, Vol. 23. Academic Press, New York, pp. 369-464 .

Quintard, M. and Whitaker, S.: 1994a, Transport in ordered and disordered porous media I: The cellular average and the use of weighting functions, Transport in Porous Media 14, 163-177.

Quintard, M. and Whitaker, S.: 1994b, Transport in ordered and disordered porous media II: Generalized volume averaging, Transport in Porous Media 14, 179-206.

Quintard, M. and Whitaker, S.: 1994c, Transport in ordered and disordered porous media III: Closure and comparison between theory and experiment, Transport in Porous Media 15, 31-49.

Quintard, M. and Whitaker, S.: 1994d, Transport in ordered and disordered porous media IV: Computer generated porous media for three-dimensional systems, Transport in Porous Media 15, 51-70.

Quintard, M. and Whitaker, S.: 1994e, Transport in ordered and disordered porous media V: Geometrical results for two-dimensional systems, Transport in Porous Media 15, 183-196.

Quintard, M. and Whitaker, S.: 1998, Transport in chemically and mechanically heterogeneous porous media IV: Large-scale mass equilibrium for solute transport with adsorption, Adv. Water Resour. 22(1), 33-57.

Skopp, J., Gardner, W. and Tyler, E.: 1981, Solute movement in structured soils: Two-region model with small interaction, Soil Sci. Soc. Arner. 1. 45, 837-842.

Taylor, S. G.: 1953, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. Roy. Soc. London 219, 186-203.

Whitaker, S.: 1988, Levels of simplification: The use of assumptions, restrictions, and constraints in engineering analysis, Chern. Eng. Ed. 22, 104-108.

Whitaker, S.: 1999, The Method ofVolurne Averaging, Kluwer Academic Publishers, Dordrecht. Zanotti, F. and Carbonell, R.: 1984, Development of transport equations for multiphase systems I:

General development for two-phase systems, Chern. Engng Sci. 39, 263-278. Zhang, D. and Neuman, S.: 1996, Effect of local dispersion on solute transport in randomly

heterogeneous media, Water Resour. Res. 32, 2715-2723.

Documents

Dispersion in heterogeneous porous media: One-equation non-equilibrium model