2LEPT-ENSAM, Esplanade des Arts et Metiers, 33405 Talence cedex, France .... in the large-scale description of dispersion mechanisms, that is, it is assumed ...
Transport in Porous Media 44: 181-203,200l. © 2001 Kluwer Academic Publishers. Printed in the Netherlands.
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Dispersion in Heterogeneous Porous Media: One-Equation Non-equilibrium Model MICHEL QUINTARD 1, FABIEN CHERBLANC 2 and STEPHEN WHITAKER 3 1Institut 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 3Department 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.
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Darcy-scale dispersion tensor fro the fJ-a system, m 2/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.
D** 00 D** f3
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.
nf3u nl)w U*
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.
Uf3 V
= (vf3)f3 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 V00, m3 . fluid velocity vector in the fJ-phase, m/s. Darcy-scale superficial average velocity, mls. Darcy-scale intrinsic average velocity (seepage velocity), mls.
~f3
Vf3
Voo Vw
vI) vf3 (vf3 ) (vf3 )f3
(Vf3)~ (vf3)~ {(vf3) } { (vf3)w}W {(vf3h}1)
Ufi,
Darcy-scale intrinsic velocity in th w-region, m/s. Darcy-scale intrinsic velocity in th 1]-region, mls. = rpw{ (vf3)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 £ 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]-W
system, s-l.
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
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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
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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
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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
(1)
in Vf3 ,
aCf3 at + V . (vf3cf3) = V . (V(:J VC(3)
10
Vf3 ,
(2)
(3)
B.C. 1.
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)
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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
