dominated flow through variably saturated heterogeneous formations

WATER RESOURCES RESEARCH, VOL. 39, NO. 9, 1255, doi:10.1029/2002WR001857, 2003

Upscaled conductivity in gravity-dominated flow through variably saturated heterogeneous formations David Russo Department of Environmental Physics and Irrigation, Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel Received 19 November 2002; revised 29 April 2003; accepted 24 June 2003; published 17 September 2003.

[1] We investigated effects of a few characteristics of a heterogeneous, unsaturated

formation on the vertical (Kv) and horizontal (Kh) components of the upscaled conductivity tensor, under conditions of steady state, gravity-dominated, unsaturated flow, using Gardner’s [1958] two-parameter (Ks, a) model for the local unsaturated conductivity. Results suggest that relatively large averaging domains are required in order to obtain asymptotic, effective conductivities devoid of the domain size. This is particularly so in relatively dry, stratified, fine-textured (with appreciable capillary forces) formations and when (1) the variability in the soil parameter a is not small compared with the variability in saturated conductivity, Ks (z
0); (2) the fluctuations of loga are negatively correlated with those of logKs (rfa < 0); and (3) the correlation length scale of loga is small compared to that of logKs (u < 1). Larger averaging domains are required in order to ignore the uncertainty about the upscaled log conductivities, particularly in stratified, coarse-textured formations, and when u > 1. Findings of this study suggest that the asymptotic Kv diminishes in dry, stratified, coarse-textured formations and when z
0; for the physically plausible situation in which rfa > 0, however, the diminishing of the asymptotic Kv is balanced by increasing rfa. On the other hand, the increase in the asymptotic Kh, and, concurrently, in the effective anisotropy ratio, due to increasing stratification and decreasing water saturation, is balanced by decreasing capillary forces, as well as by decreasing z and increasing (positive) rfa, and to a lesser extent, by increasing u. Application of the results of this study for assessment of (1) the minimal domain size for which the concept of effective properties is appropriate; and (2) the maximal size of conductivity cells which preserve the heterogeneous structure of the underlying formation with respect to simulation of flow and transport in the vadose zone, are briefly INDEX TERMS: 1832 Hydrology: Groundwater transport; 1829 Hydrology: Groundwater discussed. hydrology; 1875 Hydrology: Unsaturated zone; 1894 Hydrology: Instruments and techniques Citation: Russo, D., Upscaled conductivity in gravity-dominated flow through variably saturated heterogeneous formations, Water Resour. Res., 39(9), 1255, doi:10.1029/2002WR001857, 2003.

1. Introduction [2] There is a great need to know how to upscale constitutive relations from the scale at which they are typically measured (the mesoscale) to a scale more amenable for practical flow and transport simulations. The upscaling of constitutive relations from the mesoscale to the macroscale is complicated by spatial variations in porous media properties which precludes, in general, the use of simple averaging rules. Moreover, it is necessary to treat spatial variability within the context of a stochastic methodology since a full deterministic description of the heterogeneous geologic media is for practical purposes impossible. [3] A general approach which may be applicable to domains (or blocks) the size of which is comparable with the scale of the medium heterogeneity is known as block-averaging. Most work on block-averaging in surface Copyright 2003 by the American Geophysical Union. 0043-1397/03/2002WR001857

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hydrology has been restricted to the case of steady state, saturated groundwater flow [e.g., Rubin and GomezHernandez, 1990; Indelman and Dagan, 1993a, 1993b, 1993c]. These studies provide methodologies for defining upscaled properties which depend on the distribution of the underlying medium property (log-transmissivity) and the block size, which, in turn, is comparable with the medium heterogeneity scale but is much smaller than that of the flow domain. [4] Application of upscaling procedures to partially saturated heterogeneous formations is rather difficult because of the complexity of the unsaturated flow, which, in turn, stems from the dependence of the relevant flow parameters both on formation properties and on flowcontrolled attributes. An upscaling procedure for partially saturated heterogeneous media was introduced by Russo [1992]. The procedure combines the stochastic theory of Yeh et al. [1985a, 1985b] for steady state unsaturated flow with a block-averaging procedure similar to the one suggested by Rubin and Gomez-Hernandez [1990] for saturated, steady state flow. In this procedure, for blocks

7-1

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7-2

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

of arbitrary volume, the upscaled conductivity is defined as the ratio of the volume-averaged flux to the volumeaveraged head gradient. Because in statistically anisotropic formations the flux gradient vector and the head gradient vector are generally not parallel, however, the upscaling procedure of Rubin and Gomez-Hernandez [1990] adopted by Russo [1992] is restricted to statistically isotropic heterogeneous formations. [5] Indelman and Dagan [1993a, 1993b] developed a general upscaling methodology in which the upscaling permeability is achieved by space averaging of the energy dissipation function defined as the mechanical energy per unit weight of fluid dissipated by viscous friction. The procedure which can be applied to anisotropic formations ensures (1) equality of the global responses of the actual and the upscaled formations; (2) equality of the head gradient and the specific discharge of the two formations at the limit of small block size; (3) equality of the effective permeability of the two formations for any shape or size of block, V; and that (4) for average uniform flow and at the limit of large block size, the random upscaled permeability tends to the deterministic effective one. [6] In the present investigation we combine the upscaling methodology of Indelman and Dagan [1993a, 1993b] with the stochastic theory of Yeh et al. [1985a, 1985b] for steady state unsaturated flow, in order to obtain block properties for partially saturated formations at scales comparable with the formation heterogeneity. We investigate effects of a few characteristics of the partially saturated, heterogeneous formation, on the mean and the covariance of the principal components of the upscaled conductivity tensor, and their tendency to their asymptotic, large-scale effective values, under conditions of steady state, unsaturated flow, with vertical mean gradient, perpendicular to the formation bedding. The first characteristic, is a length-scale ratio, r = Ih/Iv, where Iv and Ih are the length scales of the formation heterogeneity in the vertical and the horizontal directions, respectively, which determines the scale of the formation heterogeneity in a direction perpendicular to the mean flow, relative to its counterpart in the direction of the mean flow; the second one is also a length-scale ratio, h = l/Iv, where l is the macroscopic capillary length scale, which determines the characteristic length scale of the unsaturated flow [Raats, 1976] relative to that of the formation heterogeneity in the direction of the mean flow; while the third one is a flowattributed variable, the mean water saturation, S, which determine the average degree of water saturation of the flow domain.

2. Theoretical Considerations 2.1. The Upscaling Methodology [7] For the sake of completeness, we will review briefly the main results obtained in the past with the upscaling methodology of Indelman and Dagan [1993a, 1993b] that are relevant to the present analysis. Their theory is based on the following assumptions: (1) uniform average flow under a constant head gradient prevails in an unbounded heterogeneous domain of stationary random hydraulic conductivity, K, whose statistical moments are known; (2) the flow domain is partitioned into numerical cells V

of given shape and magnitude; (3) the upscaled conduc~ which depends on both K and V, is a symmetrical tivity K, tensor regarded as a continuous random space function (RSF) with principal axes parallel to those of the formation heterogeneity; (4) the necessary and sufficient conditions to ~ are be satisfied by K heðxÞi ¼ heðxÞi and Cee ðxÞ ¼ Cee ðxÞ

ð1Þ

where e(x) is the energy dissipation function given by e(x) = rj  K  rj, x = (x1, x2, x3) is the Cartesian coordinate vector (with x1 directed vertically downward), j = d1ixi y is the hydraulic head, y Ris the pressure head, d1i is the Kronecker delta, e (x) = (1/w) w(x)e(x0)dx0 is the space average of e(x), he(x)i and Cee(x), where x = x0 x00 is the separation vector, are the mean and the covariance of e(x), respectively, and he(x)i and Cee (x) are their counterparts in the upscaled formation of conductivity K ¸ ; and (5) like e(x), e (x) is a stationary RSF characterized at second order by the mean, he(x)i and the covariance, Cee (x), which, in turn, depend on both K and the elements V. [8] Note that the first requirment in equation (1) ensures the equality between the effective conductivities of the ~ efi = upscaled formation and of the actual formation, i.e., hK ii K ef, and, consequently, the equality between the total mean fluxes in both formations, for any partition element, V. Using small perturbation analysis, expanding logconductivity and the pressure head in series of logK perturbation, y, and pressure head perturbation, h, the second-order approximation of the energy dissipation is eðxÞ ¼ Kg

o   1 þ y þ ð1=2Þy2 J2 þ2ð1 þ yÞJrh þ ðrhÞ2 þ . . . ð2Þ

where Kg is the geometric mean of K, J = hrf(x)i is the mean head-gradient vector, and J = jJj. Taking the expected value of equation (2), the mean of the dissipation is ( heðxÞi ¼ Kg

)  X @Cyh ð0Þ X @Chh ð0Þ 1 J 1 þ s2y J 2 þ2 i i i 2 @xi @x2i ð3Þ

Using equations (2) and (3), the covariance of the dissipation is Cee ðxÞ¼ Kg2

( ) X @Cyh ðxÞ X  4 @ 2 Chh ðxÞ 2 J Cyy ðxÞ þ 4J J þ4 JJ i i @x ij i j @x @x i i j ð4Þ

where Chh(x) = hh(x0)h(x00)i, Cyy(x) = hy(x0)y(x00)i and Cyh(x) = hy(x0)h(x00)i are the (cross) covariances between fluctuations of log-conductivity and the pressure head. [9] Employing the aforementioned assumptions, using small perturbation analysis, at a first-order in log-conductivity variance, sy2 = Cyy(0), and for the case of axisymmetric element V, assuming that the axes of symmetry of V are parallel to the principal axes of the actual formation heterogeneity, Indelman and Dagan [1993b] showed that the ~ first two moments the upscaled log-conductivity, Y = log K

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

that satisfy equation (1), are the upscaled geometrical means, Kgi, given by: 1þ K gi ¼ Kg

h

n 3þ2b1 2ðn 1Þ

þ

1þ

Bi s2Y

1 2

1 d1i 1 nb n 1



i s2y

and the upscaled log-conductivity covariance, CYY(x), given by CYY ðxÞ ¼

1 V2

Z

dx0

V

Z

CYY ðx0 x00 Þdx00 ¼

1 V2

Z

CYY ðxÞHðx x0 Þdx0

Vx

ð6aÞ

where the transformation on the right-hand-side (RHS) of equation (6a) is the Gauss-Cauchy algorithm, n is the number of the space dimensions, sY2 = CYY (0) is the upscaled log-conductivity variance given by sY ¼

1 V2

Z

dx0

V

Z

1 V2

Cyy ðx0 x00 Þdx00 ¼

Z Cyy ðxÞHðxÞdx ð6bÞ

V

b1 = C1(0)/s2y, Ci(x) = hy(x0)@h(x00)/@xii is the cross covariance between log-conductivity and head gradient fluctuations, Bi = Gi/sY2, Gi is a regularized cross covariance, which represents the mean value of the cross covariance Ci(x) between two points x0 and x00 independently sweeping over the element V, given by Gi ¼

1 V2

Z V

dx0

Z

Ci ðx0 x00 Þdx00 ¼

1 V2

Z Ci ðxÞHðxÞdx

ð7Þ

7-3

Substitution of equation (5) for a three-dimensional domain (n = 3) in equation (10a), leads to the results ~ ef i ¼ Kvef ¼ hK 11

ð5Þ

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 1þ

   1 b1 s2y 2

  ~ ef i ¼ hK ~ ef i ¼ Kg 1 þ 1 b1 s2y Khef ¼ hK 22 33 2

ð10bÞ

ð10cÞ

which are identical to the well-known results for the effective conductivities of the actual formation [e.g., Dagan, 1989], and therefore lead to the important property. [11] Quantification of the first two moments of the upscaled log-conductivity, therefore, requires the calculations of the (cross) covariances, Ci(x), i = 1, 2, 3, and Cyy(x), which, in turn, for unsaturated flow conditions, requires the calculations of the cross covariances between perturbations of the pressure head and those of the relevant formation properties. 2.2. The Cross Covariances [12] For steady state, unsaturated flow, the (cross-)covariances required for the calculation of the first two moments of the upscaled log-conductivity, can be obtained by a firstorder perturbation approach [Yeh et al., 1985a,1985b]. To do so, the following assumptions are employed: (i) the heterogeneous formation has a three-dimensional structure with axisymmetric anisotropy, and the flow domain is variably saturated and unbounded; (ii) the local steady state unsaturated flow obeys Darcy’s law and continuity, which, if local isotropy is assumed, reads: qi ¼ K ðc; xÞ

@fð xÞ @qi ð xÞ ; ¼ 0; i ¼ 1; 2; 3 @xi @xi

ð11Þ

V

and H(x) is the joint volume of V and its translation by x [Dagan, 1989], having the following properties: Z Hð0Þ ¼ 1

H ðxÞdx ¼ V 2

ð8Þ

Note that equation (6a) is a regularized log-conductivity covariance which represents the mean value of the covariance Cyy(x) when one of the extremities of the vector x describes the element V and the other extremity independently describes the translated element Vx. [10] Considering the upscaled log-conductivity as a multivariate normal (MVN) RSF, at first-order, the principal ~ iii, components of the upscaled mean conductivity tensor, hK (i = 1,2,3, no sum on i) are given by 2 ~ ii i ¼ Kgi exp4 1 hK V

Z

3

  1 Y 0 ð x þ x0 Þdx0 5 ¼ Kgi 1 þ s2Y 2

ð9Þ

where qi is the water flux vector, and the Einstein summation convention is used in (11) and elsewhere; (iii) the local relationships between K and c (considered here as a positive quantity) are isotropic and nonhysteretic and are given by the Gardner-Russo model [Gardner, 1958; Russo, 1988], i.e., Kðc; xÞ ¼ Ks ð xÞ exp½ að xÞc

ðc; xÞ ¼

   2=m0 þ2 1 1 exp að xÞc 1 þ að xÞc 2 2

ð12aÞ

ð12bÞ

where Ks is saturated conductivity, a = l 1 is a parameter of the formation, l is the macroscopic capillary length scale and m0 is a parameter selected here as m0 = 0; (iv) both logKs and loga, are MVN RSFs, ergodic over the region of interest, characterized by constant means, F = E[logKs] and A = E[loga], and by covariances Cff(x) and Caa(x), respectively, and cross-covariance, Cfa(x) given by:

V

Moreover, from the mean of the energy dissipitation function in the upscaled formation, the effective conductivity of the upscaled formation is given [Indelman and Dagan, 1993b] by     ~ iief i ¼ Kgi 1 þ 1 Bi s2Y hK 2

ð10aÞ

Cpp ðxÞ ¼ s2p* expð x0 Þ

ð13Þ

where f and a are the perturbations of log Ks and loga, respectively, p = f or a, p* = f, a or fa; x0 = (x x00)/Ip0 is the scaled separation vector, x0 = jx0j; sf2 and sa2, and If = (If1, If2, If3) and Ia = (Ia1, Ia2, Ia3) are the respective variances and correlation scales of logKs and loga, sfa2 is the crossvariance between perturbations of logKs and loga, and Ifa =

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RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

2IfIa/(If + Ia). Note that because of the assumption of axisymmetric anisotropy, Ifv = If1, Ifh = If2 = If3 and Iav = Ia1, Iah = Ia2 = Ia3. [13] Use of the aforementioned assumptions under ergodic conditions, by elimination of q from (11), expression of the various parameters and variables on the RHS of (12a) in terms of means and perturbations, i.e., logKs = F + f, loga = A + a, c = H + h, and use of Taylor’s expansion with firstorder terms retained, gives the first-order perturbation approximation of the steady state flow equation as:   @2h @h @f @a ¼ ð2Ji d1i Þ Ji ðJi d1i Þa þ Ji H @xi @xi @xi @x2i ð14Þ

where i = 1, 2, 3, and  = exp(A) is the geometric mean of a. [14] On the assumption that the flow is gravity dominated, i.e., the mean pressure-head gradient is zero so that the mean head-gradient, Ji, is given by Ji = 1i, i = 1, 2, 3, the exact solutions of (14) in terms of the (cross-) spectral relationships between the RSFs of f, a and h, which are obtained by using Fourier-Stieljes integral representations, are:   2 ^ 2 2^ ^ ^ hh ðk Þ ¼ k1 Cff ðk Þ þ  H Caa ðk Þ 2H Cfa ðk Þ C k 4 þ 2 k12

ð15Þ

  ^ ff ðk Þ H C ^ fa ðk Þ k1 ðk1 j0 k 2 Þ C ^ Chf ðk Þ ¼ k 4 þ 2 k12

ð16Þ

  ^ fa ðk Þ H H ^ aa ðk Þ k1 ðk1 j0 k 2 Þ C ^ Cha ðk Þ ¼ k 4 þ 2 k12

ð17Þ

where j0 is the imaginary unit, k = (k1, k2, k3) is the wave number vector, k = jkj, and the inverse Fourier transform of the covariance of the formation properties, C pp (x) (Equation (13) with p = log Ks or loga), is given by: ^ pp ðk Þ ¼ C

s2p Ip1 Ip2 Ip3

 2 2 k2 þ I 2 k2 p2 1 þ2p1 k12 þ Ip2 2 p3 3

ð18Þ

Combining (15) – (17), the cross-spectral relationship between the logK perturbation, y = f Ha h, and the ^ hy(k), is given by: pressure head perturbation, h, C ^ hy ðk Þ ¼ C ^ hf ðk Þ C ^ hh ðk Þ H C ^ ha ðk Þ C

ð19Þ

while the cross-spectral relationship between the logK perturbation and the pressure head gradient perturbation, ^ i(k) (i = 1, 2, 3), is given by: C   ^ ff ðk Þ 2H C ^ fa ðk Þ þ 2 H 2 C ^ aa ðk Þ k k1 k C ^

 Ci ðk Þ ¼ k 4 þ 2 k12 2

ð20Þ

[15] Note that because c = H + h is a function of water ^ mn(k) (m,n = f,a,h) saturation, , the (cross-) spectra C (Equations (15) – (17) and (19) – (21)) depend on ; this dependence, however, is omitted for simplicity of notation. Furthermore, for H!0 (or !0), they reduce to their counterparts associated with steady flow in saturated formations. [16] The (cross-) covariances, Cmn(x) associated with the ^ mn(k) (m, n = f, a, h, i) (Equations (15) (cross-) spectra C through (21)), are then calculated by taking the Fourier ^ mn(k), that is, transform of the respective C Z1 Z1 Z1 Cmn ðxÞ 1

1

^ mn ðk Þdk expð j0 k  xÞC

ð22Þ

1

and are given elsewhere [Russo, 1998].

and the spectrum of the log-conductivity perturbation, ^ yy(k), is given by: C ^ ff ðk Þ þ 2 H 2 C ^ fa ðk Þ þ 2 C ^ yy ðk Þ ¼ C ^ aa ðk Þ 2H C ^ hh ðk Þ C ^ ha ^ hf* ðkÞ þ 2 C * ðk Þ C

Figure 1. Log-conductivity covariance as a function of the scaled separation distance, in the direction of the mean flow (Figures 1a, 1c, 1e), and perpendicular to the mean flow (Figures 1b, 1d, 1f ) for selected values of the mean water saturation, S, and the length scales ratios, r and h (denoted by the numbers labeling the curves), and sf2 = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0).

ð21Þ

^ *hf (k) = C ^ hf(k) + C ^ fh(k) and C ^ *ha (k) = C ^ ha(k) + C ^ ah(k). where C

3. Results and Discussion 3.1. The (Cross) Covariances Cyy(x), C1(x), and C2(x) [17] The (cross) covariances Cyy(x), C1(x) and C2(x) = C3(x) are depicted in Figures 1 and 2 for the direction of the mean flow (Figures 1a, 1c, 1e, 2a, 2c, 2e, 2g 2i, and 2k), and perpendicular to the mean flow (Figures 1b, 1d,

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

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7-5

Figure 2. Cross covariances between perturbations of log-conductivity and the pressure head gradient in the longitudinal, C1 (Figures 2a – 2f), and the transverse, C2 = C3, (Figures 2g– 2l) directions, as functions of the scaled separation distance, in the direction of the mean flow (Figures 2a, 2c, 2e, 2g, 2i, 2k), and perpendicular to the mean flow (Figures 2b, 2d, 2f, 2h, 2j, 2l), for selected values of the mean water saturation, S, and the length scales ratios, r and h (denoted by the numbers labeling the curves), and s2f = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0).

1f, 2b, 2d, 2f, 2h, 2j, 2l), for selected values of the length-scales ratios, r = Ih/Iv and h = l/Iv = (Iv) 1, and mean water saturation, S = h(x)i. To simplify the calculations, it is assumed here that the correlation scales of logKs and loga are identical, i.e., Iv = Ifv = Iav, and Ih = Ifh = Iah. Note that for a given mean pressure head, H, S is obtained by using equation (12b) (with m0 = 0), expressing , y, and loga in terms of means and perturbations, and utilizing Taylor’s expansion, with first-order terms retained, i.e.,   

  S ¼ exp  H þ s2ha =2 1 þ  H þ s2ha =2

ð23Þ

2 where sha = C*ha(0)/2. [18] If not stated otherwise, the following benchmark values were adopted in Figures 1 and 2 and elsewhere, F = 0, Kg = 1cm/h, sf2 = 0.5, sa2 = 0.06, sfa2 = 0 and Iv = Ifv = Iav = 20 cm. Note that for the small limit of r and/or the large limit of h , i.e., r!0 and/or h!/, Cyy(x)!Cww(x), where Cww(x), which is independent of r, is given by:

Cww ðxÞ ¼ Cff ðxÞ þ 2 H2 Caa ðxÞ 2HCfa ðxÞ

ð24Þ

Inasmuch as 2Chh(x) = (1/2)[C*hf (x) 2HC*ha(x)] [Russo, 1995], comparison of equation (24) and the Fourier transform of (21) reveals that Cyy ðxÞ ¼ Cww ðxÞ 2 Chh ðxÞ

ð25Þ

[19] Note that unlike in saturated flow, because of the dependence on Chh(x), Cyy(x) exhibits anisotropy even when the formation is isotropic (r = 1). Relatively large r (i.e., the lateral extent of flow barriers, normal to the mean flow, is large compared to Iv) and small h (i.e., the capillary length scale is small compared to Iv) enhance the anisotropy in Cyy(x). This stems from the effect of r and h on the second term on the RHS of equation (25). Both the magnitude and the anisotropy of Chh(x) increase with increasing r, while for fixed Iv,  is inversely proportional to h. Consequently, combinations of relatively large r and small h, contribute to the relatively small positive values of Cyy(x) in the direction of the mean flow, when x1
0 (Figure 1c), and to the relatively large negative values of Cyy(x), in the direction perpendicular to that of the mean flow, resulting in a ‘‘hole’’ type

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RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

covariance (Figure 1d). Larger soil stratification (larger r, Figures 1a and 1b), and coarser soil’s texture (smaller h, Figures 1c and 1d) also diminish the magnitude of Cyy(x). Furthermore, larger r and smaller h increase the decay rate of Cyy(x) at the small x limit, x!0, and retard its tendency to zero at the large x limit, x!1. Note that because of the assumption that the correlation length scales of the formation properties are identical (i.e., If = Ia), the mean water saturation S only affects the magnitude of Cyy(x) (i.e., increasing S diminishes the magnitude of Cyy(x), Figures 1e and 1f), but does not affect the anisotropy of Cyy(x), nor its behavior at the small and the large x limits. [20] The cross covariances C1(x) and C2(x) = C3(x) (Figure 2) also exhibit anisotropy. Note that in the case of r = 1, C1(x1, 0, 0) = Ci(0, x2, 0) = Ci(0, 0, x3) and C1(0, x2, 0) = C1(0, 0, x3) = Ci(x1, 0, 0), i = 2,3. The magnitude, persistence and anisotropy of Ci(x) (I = 1, 2, 3) increase with increasing h (Figures 2c, 2d, 2i, 2j) and decreasing S (Figures 2e, 2f, 2k, 2l). On the other hand, the magnitude, persistence and anisotropy of C1(x) increase with increasing r (Figures 2a and 2b), while the converse is true for C2(x) = C3(x) (Figures 2g and 2h). [21] The (cross) covariances presented in Figures 1 and 2 are based on fixed ratios between the variances and the correlation length scales of the formation properties, logKs and loga, (i.e., z = sa2/s2f = 0.12 and u = Iav/Ifv = Iah/Ifh = 1, respectively). Furthermore, a lack of cross correlation 2 /(s2f sa2)1/2 = 0) between the two properties (i.e., rfa = sfa was assumed. The effect of the variances ratio, z, on Cyy(x) and Ci(x) (i = 1, 2, 3) is opposite to that of the mean water saturation, S; in other words, the magnitude, persistence and anisotropy of Ci(x) (i = 1, 2, 3), and the magnitude of Cyy(x) increase with increasing z. Positive cross correlation between the formation properties restrains the effect of increasing z or decreasing S on Cyy(x) and Ci(x) (i = 1, 2, 3). This stems from the fact that for given h, r, u, and z, when rfa > 0, Cyy(x) and Ci(x) (i = 1, 2, 3) are nonmomotonic functions of H, which decrease with increasing H, reaching a minimum at Hc  rfa(s2f /sa2)1/2 1 [Russo, 1995], and then increase as H increases further. For given r, z, u, and rfa > 0, the value of Hc increases with increasing h (i.e., as the soil texture is finer, with substantial capillary forces). Furthermore, for given r, h, z and rfa, when the correlation length-scales ratio, u, is smaller than one, the persistence of Cyy(x) and Ci(x) (i = 1, 2, 3) decreases and their magnitude slightly increase with decreasing u, while the converse is true when u > 1. [22] At the large h limit, i.e., h! p1, C2i(0)p (i =2 1, 2,2 3) )/ (1-rp)]/(r -1) reduce to C1(0) = sw2 r2[1 tan 1( (1 rp and C2(0) = C3(0) = sw2 [-1 + r2 tan 1( (1-r2)/ (1-r2)]/ 2(r2 1), which are identical to their conterparts for saturated flow parallel to the formation bedding [e.g., Dagan, 1989], with r 1 and sf2 instead of r and sw2 , i.e., C1(0;r)/sw2 = C3(0;r 1)/sf2, C2(0;r)/sw2 = C3(0;r)/sw2 = C1(0;r 1)/sf2 = C2(0;r 1)/sf2. 3.2. Mean Upscaled Conductivity [23] For a three-dimensional flow domain (n = 3), substituting of equation (5) in (9) yields the mean values of the vertical (longitudinal) and the horizontal (transverse) com-

~ 11i and ponents of the upscaled conductivity tensor, Kv = hK ~ 22i = hK ~ 33i, respectively, given by K h = hK 1þ

Kv ¼ Kg

1þ

1



b1 s2y

12  2 2 B1 sY

1 þ 12 b1 s2y

 1 þ 12 B2 s2Y

K h ¼ Kg

!

1 1 þ s2Y 2



!

1 1 þ s2Y 2

ð26aÞ

 ð26bÞ

These expressions can be compared with those obtained by Russo [1992], based on the upscaling procedure of Rubin and Gomez-Hernandez [1990], i.e., ( K v ¼ Kg

( K ¼ Kg

s1y 1 1 þ þ ½C1 ð0Þ G1  2 J1

s2y 1 1 þ þ ½C2 ð0Þ G2  2 J1

) ð27aÞ

) ð27bÞ

Note that although the upscaling procedure of Rubin and Gomez-Hernandez [1990] is formally restricted to isotropic formations, for J1 = 1, the expressions (26) and (27) provide identical results. [24] At the small V limit, V!0, with scaled block side length, b = b1/Iv  1, H(x)!d(x) where d is the Dirac delta function, irrespective of the values of h, r or S. Then in equations (6a) and (6b) C(x)!Cyy(x) and s2 ! sy2, and it follows from equation (7) that GI = Ci(0) and, consequently, B1 = b1 and B2 = b2; furthermore, for three-dimensional domain, C2(0) = (1/2)[1-C1(0)/sy2] and consequently, b2 = (1/2)(1 b1). Substitution of these results in equations (26) and (27) yields the same results, Kv = Kh = Kg(1 + sy2/2). At the other extreme, the large V limit, V!1, with b
1, Gi!0 and s2 ! 0. Consequently, from equations (26), Kv = Kg {1 + [(1/2) - b1]sy2} and Kh = Kg[1 + (1/2)b1sy2], again, identical to the results obtained from equations (27) for the large V limit, V!1. [25] The effect of the length-scales ratios, r = Ih/Iv and h = l/Iv = (Iv) 1, and mean water saturation, S, on the principal components of the effective conductivity tensor, kii (i = 1, 2, 3), for the large V limit, V!1, is demonstrated in Figure 3 for fixed values of l, u, and rfa. At the small r limit, r!0, which implies heterogeneity in the horizontal direction only, (i.e., a formation made of a collection of different, noninteracting, one-dimensional vertically homogeneous columns), C 1 (0) !0, while C 2 (0) = C 3 (0) = C yy (0)/2 = sw2 /2, independent of h, where sw2 = Cww(0). Consequently, Kv approaches the arithmetic mean of K, i.e., Kv = Kg(1 + sy2/2) = Kg(1 + sw2 /2), while Kh approaches the geometric mean of K, i.e., Kh = Kg, independent of h. The latter results are identical to the results for mean horizontal flow, parallel to the bedding, in a two-dimensional, isotropic, saturated formation [Dagan, 1989]. At the infinite limit, r!1, which implies a perfectly stratified formation (heterogeneity in the vertical direction only), with mean flow perpendicular to the bedding, C1(0)! (2sw2 2sh2)/2 and both C2(0) and C3(0) approach zero. Consequently, Kv = Kg(1 sw2 /2) = Kg[1

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

Figure 3. Principal components of the asymptotic mean upscaled conductivity tensor in the longitudinal, Kv, (Figures 3a and 3b) and the transverse, Kh, (Figures 3c and 3d) directions, and the effective anisotropy ratio, e = Kh/ Kv, (Figures 3e and 3f ) as functions of the length-scale ratio, r, for selected values of the mean water saturation, S, and the length scales ratio, h (denoted by the numbers labeling the curves), and sf2 = 0.5, Iv = 20 cm, z= 0.12, u = 1, rfa = 0, and J = (1, 0, 0). (sy2 2s2h)/2], while Kh = Kg(1 + sy2/2). Note that at the large limit of h, h!1, sy2!sw2 , and the latter expressions correspond to the harmonic mean and the arithmetic means of K, respectively, in agreement with the results of the firstorder results of Yeh et al. [1985b] for the effective conductivities in steady state, unsaturated flow through stratified formations. [26] Between these two limits of r, for given h and S, Kv is a monotonous decreasing function of r (Figures 3a and 3b); for given S, it decreases with r at a rate which decreases with increasing r and h (Figure 3a). On the other hand, for given S and h > 10, Kh (Figure 3c) is a monotonous increasing functions of r; as h decreases, however, Kh becomes a nonmonotonic function of r, which increases rapidly with r, reaches a maximum at a given r = rm, and then decreases with increasing r at a rate that decreases as r increases further. Note that rm decreases with decreasing h. For r = 1, (isotropic case) C1(0) = C2(0) = C3(0), and Kv = Kh. At the large limit of h, h!1, the results for effective conductivities for saturated flow (H = 0, sy2 = s2f ) in a threedimensional, isotropic formation [Gutjahr et al., 1978], are recovered, i.e., C1(0) = C2(0) = C3(0) = sy2/3, and Kv = Kh = Kg(1 + sy2/6).

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[27] Because of the disparity between the sensitivities of Kv and Kh to variations in r and h (Figures 3a and 3c), for given S, the effective anisotropy of the heterogeneous formation calculated as the ratio, e = Kh/Kv, is a monotonous increasing function of r (Figure 3e); it increases with r at a rate which decreases with increasing r and decreasing h. At the small r limit, e!(1 + sw2 /2) 1, independent of h, while at the large r limit, e![1 + (sw2 + 2sh2)/2]/(1 sw2 /2), which, in turn, increases with increasing h; for the large h limit, e reduces to e!(1 + sw2 /2)/(1 sw2 /2). For the isotropic case (r = 1), e = 1, independent of h. [28] For a given h and when r is relatively small, Kh is essentially independent of S (Figure 3d). As r increases, however, Kh increases with decreasing S at a rate which increases with increasing r. As for Kv (Figure 3b), for given h, p and for the value of r at which C1(0) = Cyy(0)/2, i.e., rc = p/ 3, Kv = Kg, independent of S. Note that for h < 10, rc slightly decreases with decreasing h. For given h and r < rc, Kv increases with decreasing S, while the converse is true for r > rc (Figure 3b). For the isotropic case (r = 1) the effective anisotropy, e = Kh/Kv, is equal to one, independent of S. For given h and r < 1, however, e increases with increasing S, while for given h and r > 1, inasmuch as Kh is more sensitive to the changes in S than Kv, e decreases with increasing S, at a rate which increases with increasing r (Figure 3f ). [29] The behavior of the principal components of the effective conductivity tensor (Figures 3a – 3d) demonstrates the combined influence of the formation heterogeneity, the capillary forces and water saturation on the effective conductivity for unsaturated flow. For fixed Iv, the effective conductivity in the longitudinal direction is diminished in relatively dry (small S) formations in which the lateral extent of flow barriers, normal to the mean flow, is large (large r) and the capillary forces are small (small h). On the other hand, the enhanced effective conductivity in the transverse direction, and, concurrently, the enhanced effective anisotropy, due to increasing r and decreasing S, are balanced by decreasing capillary forces (decreasing h). Note that the effect of mean water saturation, S on the principal components of the effective conductivity tensor (Figure 3b and 3d) is due to its effect on Cyy(0) and Ci(0) (i = 1, 2, 3) (Figures 1 and 2). For a given h, Cyy(0) and jCi(0)j (i = 2, 3) increase with decreasing S at a rate which decreases with increasing rd, while the converse is true for jC1(0)j. [30] As expected (see the aforementioned discussion on the (cross) covariances, Cyy(x) and Ci(x), i = 1, 2, 3, Figures 1 and 2), for given values of h, u, and rfa, and for r < rc, Kv increases with increasing variances ratio, z, while the converse is true for r > rc. On the other hand, for given values of r, h, u, and rfa, Kh increases with increasing z. Consequently, for given values of h, u, rfa and r < 1, the effective anisotropy ratio, e, decreases with increasing z, while the converse is true for r > 1, but to a much larger extent. The opposite is true for the effect of the (positive) cross-correlation coefficient, rfa on Kv, Kh and e. Note that Kv is essentially independent of the length-scale ratio, u = Iav/Ifv. On the other hand, Kh, and, concurrently, e, slightly decrease with increasing, u, particularly when r
1. [31] The effect of r, h, S and the scaled block side length, b = b1/Iv, on the scaled principal components of the upscaled conductivity tensor, K0v = Kv/K1v and K0h = Kh/

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RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

Figure 4. Principal components of the mean upscaled conductivity tensor in the longitudinal and the transverse directions, relative to their respective asymptotic values, K’v (Figures 4a, 4b, 4c) and K’h (Figures 4d, 4e, 4f ), respectively, and the scaled anisotropy ratio, e0 = K0h/K0v, (Figures 4g, 4h, 4i) as functions of the block side length b, for selected values of the mean water saturation, S, and the length scales ratios, r and h (denoted by the numbers labeling the curves), and sf2 = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0). K1h (where K1v and K1h are the asymptotic, effective values of Kv and Kh, respectively, evaluated at the large V limit), for the case where the size of V is comparable with that of Iv, is demonstrated in Figure 4. Note that in Figure 4 and elsewhere, V is a parallelepiped-shaped domain with sides b1, b2 and b3, parallel to the coordinate axes (x1, x2, x3) such that b = b1/Iv = b2/Ih = b3/Ih, (i.e., b2 = b3 = rb1). For this case the overlap function H(x) is given [Journel and Huijbregts, 1978; Indelman and Dagan, 1993c] by     HðxÞ 1 jx1 j jx2 j jx3 j 1 1 ¼ 1 V2 b1 b2 b3 b1 b2 b3 jx1 j < b1 ; jx2 j < b2 ; jx3 j < b3

HðxÞ ¼0 V2

otherwise

ð28aÞ

ð28bÞ

[32] For given r, h and S, both K0v and K0h (Figures 4a – 4f ) decrease monotonically with increasing block size approaching unity when the block side length exceeds few correlation scales of logK. For given h and S, both the magnitude and persistence of K0v (Figure 4a) are shown to

increase with increasing r. This stems from the fact that both the magnitude and the persistence of the cross covariance C1(x), and, concurrently, of the mean block cross covariance G1 increase with increasing r. [33] On the other hand, inasmuch as the magnitudes of the cross covariances C2(x) and G2 decrease, while their persistence increase with increasing r, also the magnitude of K0h decreases while its persistence increases with increasing r (Figure 4d). Consequently, for given h and S, the scaled effective anisotropy ratio, e0 = K0h/K0v, (Figure 4g), may increase (when r > 1) or decrease (when r < 1) with increasing b, while for r = 1, e0 = 1, independent of b. The persistence of e0 increases with increasing r, while the absolute deviation of e0 from 1 increases (when r > 1), or decreases (when r < 1) with increasing r. [34] For given r and S, both the magnitude and persistence of K0v and K0h increase with increasing h (Figures 4b and 4e). This stems from the fact that both the magnitude and the persistence of the covariance Cyy(x) (Figure 1), the cross covariances, Ci(x) (Figure 2), and, concurrently, the mean block cross covariances, Gi, (i = 1, 2, 3), increase with increasing h. Because of the disparity between the magnitudes of C1(x), G1, and K0v and those of their counterparts associated with the transverse directions, for given r and S,

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

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both the persistence of the scaled effective anisotropy ratio, e0 and its absolute deviation from 1 (Figure 4h) increase with increasing h. [35] In as much as for given r and h, both the magnitude and persistence of Cyy(x), Ci(x) and, concurrently, of Gi, (i = 1, 2, 3) increase with decreasing S, both the magnitude of K0v and K0h and their persistence increase with decreasing S (Figures 4c and 4f ). Again, because of the disparity between the magnitudes of C1(x), G1, and K0v, and those of their counterparts associated with the transverse direction, for given r and h, both the persistence of the scaled effective anisotropy ratio, e0 and its deviation from 1 (Figure 4i) decreases with increasing S. As expected (see the aforementioned discussion on the (cross covariances) covariances, Cyy(x) and Ci(x), i = 1, 2, 3, Figures 1 and 2), the deviation of K0v, K0h and e0 from 1 and their persistence increase with increasing variance ratio, z = sa2/sf2, while the converse is true for the (positive) cross-correlation coefficient, rfa. To a lesser extent, both the persistence of K0v, K0h and e0 and their deviation from 1, decrease with increasing length-scale ratio, u = Iav/Ifv. 3.3. Upscaled Log-Conductivity Covariance [36] For given values of r, h, S, z, u, and rfa, the covariance of the upscaled log-conductivity for any two blocks separated by x = x0 x00, CYY(x), is calculated from equation (6a) using the Gauss-Cauchy algorithm. Note that unlike Kv and Kh which depend on the respective (cross covariances) covariances C1(x), C2(x) = C3(x) and Cyy(x), CYY(x) depends only on the later. Furthermore, if the upscaled log-conductivity is a MVN RSF, at firstorder, the upscaled conductivity covariance, CK~ K~ (x) is given by CK~ K~ (x) = KgCYY(x). Substitution of equation (6a) in this relataionships leads to an expression for CK~ K~ (x) identical to one obtained by Russo [1992], based on the upscaling procedure of Rubin and Gomez-Hernandez [1990]. The scaled block log-conductivity covariance, CYY(x)/sy2, is plotted in Figure 5 for selected values of h, r, S and the scaled block side-length, b. Note that at the small V limit, V!0, CYY(x) approaches Cyy(x), while at the large V limit, V!1, CYY(x)!0. Similar to Cyy(x), the upscaled log-covariance, CYY(x) exhibits anisotropy; furthermore, it approaches zero after separation distances greater than few logK correlation length scale. As expected, since CYY(x) is a regularization of the log-conductivity covariance, Cyy(x), for given S, h, r, z, u, and rfa, the upscaling procedure smooths the spatial heterogeneities in the underlying formation properties. This smoothing effect increases with the block size as is shown by the behavior of the CYY(x) functions near the origin, x!0, in Figure 5. For given S, z, u, and rfa, larger stratification (larger r) and smaller capillary forces (smaller h) increase both the decay rate of the upscaled log-conductivity variance, sY2 = CYY(0), and the rate of growth of the correlation length scale of the upscaled log-conductivity covariance, IY = [0R 1CYY(x1, 0, 0) dx1]/sY2, with increasing block size at relatively small values of V, and retard its tendency to zero at the large V limit, V!1, when r > 1. Note, however, that when r  1, s2Y!0 as V!1 faster as the capillary forces decrease (smaller h). Note also, that for given S, h, r, z, and rfa, smaller correlation length-scale ratio, u = Iav/Ifav, increases both the decay rate of sY2 and the rate of growth of I Y

Figure 5. Scaled block log-conductivity covariance, independent of mean saturation, as a function of the scaled separation between centroids of two blocks of equal scaled side length, b, in the direction of the mean flow (Figures 5a, 5c, 5e), and perpendicular to the mean flow (Figures 5b, 5d, 5f ) for selected values of b (denoted by the numbers labeling the curves), and h, and for r = 4, sf2 = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0). with increasing V in the vicinity of the small V limit, and accelerates the tendency of sY2 to zero at the large V limit. [ 37 ] The upscaled log-conductivity variance, sY2 = CYY(0), scaled by sy2, is depicted in Figure 6 as a function of the scaled block side length, b = b1/Iv, for selected values of h, u, and r  1. Note that by definition, at the small V limit, V!0, sY2/s2y!1. Furthermore, because of the scaling by sy2, sY2/sy2 is independent of S, sfa2 and z. As expected, for given values of h and r, sY2 /sy2 decreases with increasing b, at a rate which decreases with increasing b, and vanishes at the large V limit. Increasing stratification (larger r) and decreasing capillary forces (smaller h) increase the decay rate of sY2/sy2 in the vicinity of the small V limit, and retard its tendency to zero at the large V limit. Furthermore, smaller length-scale ratio, u = Iav/Ifv, increases the decay rate of sY2/sy2 in the vicinity of the small V limit, and accelerates its tendency to zero at the large V limit. [38] Comparison of Figures 6 and 4 suggests that for given r, h, S, z, u, and rfa, the rate at which K0v, K0h, and, concurrently, e0 = K0h/K0v, approach their asymptotic values is faster than the rate at which s2Y!0. Furthermore, Figure 6 suggests that relatively large averaging domains (i.e., b
10) required in order to ignore the uncertainty about the

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RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

particular, larger n increases the decay rate of s2 /sy2 in the vicinity of the small V limit, V!0, accelerates its tendency to zero at the large V limit, V!1, and increase the rate of increase of the upscaled log-conductivity covariance correlation length scale, I, with increasing block size in the vicinity of the small V limit. The effect of the dimensionality of the averaging domain on CYY(x)/sy2 is particularly significant in coarse-textured (small h), stratified (large r) formations, associated with enhanced anisotropy in the logconductivity covariance, Cyy(x) (Figures 1c and 1d).

4. Applications

Figure 6. Scaled block log-conductivity variance, independent of mean saturation, as a function of the block scaled side length, b, for selected values of the length scale ratios r and h (denoted by the numbers labeling the curves), and for sf2 = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0). The tick lines represent the saturated case (h!1). upscaled conductivity (expressed in terms sY2). This is particularly so in stratified, coarse-textured formations. 3.4. Effect of the Averaging Domain [39] The subdomain, V, whose centroid is at x, can be a segment of length b1, parallel to the coordinate axis x1 (onedimensional averaging in the direction of the mean flow), a rectangle with sides b2 and b3, parallel to the coordinate axes (x2, x3) (two-dimensional averaging over an horizontal plane, perpendicular to the direction of the mean flow), or a rectangular parallelepiped with sides b1, b2, and b3, parallel to the coordinate axes (x1, x2, x3) (three-dimensional averaging). We compared the scaled principal components of the upscaled conductivity tensor, K0v and K0h, obtained by one-, two-, and three-dimensional integration. As expected, the persistence of both K0v and K0h decreases with increasing dimensionality of the averaging domain, n. Furthermore, the sensitivity of K0h to n is larger than that of K0v, particularly when r is relatively small (r < 1). [40] Consequently, the absolute deviation of e0 = K0h/K0v from 1 increases with decreasing n, particularly when r is relatively small (r < 1). This stems from the fact that the cross covariances Ci(x) (Figure 2), and, concurrently, Gi (i = 1, 2, 3) are anisotropic (even when the formation is statistically isotropic). In particular, both the magnitude of C2(x) and the gap between the persistence of C2(x1, 0, 0) and that of C2(0, x2, 0) (or C2(0, 0, x3)) increase with decreasing r (Figure 2). [41] Scaled block log-conductivity covariances, CYY(x)/ sy2, obtained by one-, two-, and three-dimensional integration, were also compared. As expected, for given values of h, r, S, and b, the persistence of C(x)/sy2 decreases with increasing dimensionality of the averaging domain, n. In

[42] The upscaled log-conductivity covariance, CYY(x), can be used to assess few aspects pertinent to the discretization of the flow domain in numerical simulations of flow and transport in partially saturated, heterogeneous formations. In the design of numerical simulations of flow in heterogeneous formations, one of the objectives is to preserve the underlying heterogeneity of the formation properties by using a sufficiently dense grid. For example, Ababou [1988] suggested the criterion of at least four grid points per correlation scale of logK. The sY2 /sy2 curves in Figures 7a and 7b suggest that for a predetermined, acceptable reduction in sy2, r1 = (1 sY2/sy2), the required number of conductivity cells per log-conductivity length scale, i.e., b 1 = Iv/b1, should increase in coarse-textured (smaller h), stratified (larger r) formations. For example, for r1=0.1 (i.e., a reduction of 10% in sy2 ) and r = 4 (Figure 7a), b 1 = 5.7, 6.0, 6.5, 7.8, and 10.4 for h = 10, 5, 2.5, 1.25, and 0.625, respectively, while for r1 = 0.1 and h = 1.25 (Figure 7b), b 1 = 6.1, 6.8, 7.8, 8.4, and 8.8 for r = 1, 2, 4, 6, and 8, respectively. These values are larger than the value of b 1 = 5.8 associated with saturated flow conditions. Note also that the values of b 1 decrease with

Figure 7. Upscaled log-conductivity variance (Figures 7a, 7b), and the product Is2 (Figures 7c, 7d), relative to their respective ‘‘point values,’’ sy2 and Ivsy2, as functions of the number of conductivity cells per log-conductivity correlation length scale, b 1, for selected values of the length scales ratios, r and h (denoted by the numbers labeling the curves), and sf2 = 0.5, Iv = 20 cm, z = 0.12, u = 1, rfa = 0, and J = (1, 0, 0). The tick lines represent the saturated case (h!1).

RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

increasing correlation length-scale ratio, u = Iav/Ifv. For example, for r = 4, h = 5 and r1 = 0.1, b 1 = 7.44, 5.97, and 5.40 for u = 0.5, 1, and 2. Furthermore, the values of b 1 increase with increasing dimensionality of the integration domain. For example, for r = 4, h = 5, and r1 = 0.1, b 1 = 3.06, 4.78, and 5.97 for one-, two-, and threedimensional domains, respectively. [43] In the design of numerical studies of transport in heterogeneous, partially saturated formations, one of the objectives is to preserve the macrodispersion associated with the underlying heterogeneity of the formation properties. Using first-order analysis based on a stochastic continuum presentation of the flow and a general Lagrangian description of the transport, Russo [1998] obtained expressions for the principal components of the macrodispersivity tensor, lij(t) (i, j = 1, 2, 3). For the case in which a unit mean head gradient exists only in the longitudinal (vertical) direction, the asymptotic value of the longitudinal component of the macrodispersivity tensor, l11, is

l11

i 0h 1 8H þ 5ðH Þ2 s2a ð8 þ 4H Þs2fa AIyv þ s2 Iyv ¼ ðH Þ2 @ y 4ð2 þ H Þ2 ð29Þ

where Iyv = [0R 1Cyy(x1, 0, 0)dx1]/sy2. [44] Analysis of equation (29) suggests [Russo, 1998] that for a considerable range of mean water saturation, S, the leading term of (29) is the product Iyvsy2. Note that at the large S limit, S!1 (and H!0), the first term on the righthand side of (29) vanishes and (29) degenerates to l11 = Ifvsf2, identical to the expression obtained for steady state, saturated flow [e.g., Gelhar and Axness, 1983; Dagan, 1984]. [45] With respect to block averaging, the question is how this product is affected by the averaging procedure. In other words, if the aim (for given boundary conditions) is to minimize the discrepancy between the spread of the solute body in the heterogeneous formation associated with the underlying formation properties and the one obtained by using block properties, one has to assess the size of the averaging blocks (i.e., the conductivity cells) which might meet this aim. Figures 7c and 7d suggest that for a prescribed reduction in the product of Iyvsy2, i.e., r2 = (1 s2 I/sy2 Iyv), the required number of conductivity cells per log-conductivity length scale, b 1 = Iv/b1, should be increased in coarse-textured (smaller h), stratified (larger r) formations. For example, for r = 4, and r2 = 0.05 (Figure 7c), b 1 = 2.7, 3.1, 3.6, 4.6, and 6.3 for h = 10, 5, 2.5, 1.25, and 0.625, respectively, while for r2 = 0.05 and h = 1.25 (Figure 7d), b 1 = 2.9, 3.3, 4.6, 6.6, and 8.8 for r = 1, 2, 4, 6, and 8, respectively. Again, these values are larger than the value of b 1 = 2.5 associated with saturated flow conditions. Note, however, that the values of b 1 decrease with increasing correlation length scale ratio, u = Iav/Ifv. For example, for r = 4, h = 5, and r2 = 0.05, b 1 = 3.47, 3.12, and 2.41 for u = 0.5, 1, and 2. Furthermore, the values of b 1 increase with increasing dimensionality of the integration domain, n. For example, for r = 4, h = 5, and r2 = 0.05, b 1 = 2.0 and 3.1, for two- and three-dimensional domains, respectively. Note that for one-dimensional domain, s2 I/

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sy2 Iyv = 1; this stems from the general properties of the upscaled log-conductivity covariance, CYY(x), [Dagan, 1989], i.e., (IY/Iyv) = (sy2/sy2)1/n. [46] Figure 7 suggests that for given r and h, the scaled block product, IYsy2/Iyvsy2, is more robust to variations in the scaled blockside length, b, as compared to the scaled block log-onductivity variance sY2/sy2. This means that the increase in IY with increasing b, compensate in part for the reduction in sy2 with increasing b; this is particularly so when the dimensionality of the averaging domain decreases (n!1).

5. Summary and Conclusions [47] We combine a stochastic continuum presentation of the steadyßstate, unsaturated flow [Yeh et al., 1985a, 1985b] with the upscaling procedure of Indelman and Dagan [1993a, 1993b] in order to investigate the effects of a few characteristics of a heterogeneous, partially saturated formation, on the mean and the covariance of the upscaled logconductivity, under steady state, unsaturated flow conditions. The present investigation focused on the situation in which the mean head gradient vector is vertical, perpendicular to the formation bedding. The analysis can be extended to the general case in which the mean gradient vector is inclined to the principal axes of the formation heterogeneity at arbitrary angles [Russo, 1995]. However, we did not pursue this line in the present study. The mean and the covariance of the resultant upscaled log-conductivity are identical to those obtained by Russo [1992] based on the upscaling procedure of Rubin and Gomez-Hernandez [1990], although the former is formally restricted to isotropic formations. [48] It should be emphasized here that although their underlying physics are similar, groundwater flow and vadose-zone flow do differ with respect to scale of the flow domain, the flow regime, the number of relevant formation properties, and the direction of the mean flow relative to the formation bedding. Using a simplified description of a steady state, unsaturated flow, the relevant flow entity the hydraulic conductivity, K, given by (12a), depends locally on two formation properties (i.e., logKs and loga) and on the dependent flow variable, the pressure head, y. Consequently, description of the steady state flow, and concurrently, the derivation of block average properties in heterogeneous partially saturated formations, require the mean values of logKs, loga and c, and the (cross-) covariances between their respective fluctuations. One of the distinctive features of unsaturated flow is the anisotropy behavior of the log-conductivity covariance Cyy(x) which stems from its dependence on the pressure head covariance, Chh(x). Because Cyy(x) depends also on the geometric mean of a, its anisotropy increases in stratified, coarse-texture formations. [49] In this study, for given statistics of logKs (F, sf2, Ifv, Ifh) and loga (A, s2a, Iav, Iah) and their cross-correlation (sfa2 , Ifav, Ifah), we concentrate on effects on upscaled log-conductivity, of the mean water saturation, S, degree of stratification (expressed in terms of the length scale ratio, r = Ih/ Iv), and capillary forces (or soil texture, expressed in terms of the length scale ratio, h = l/Iv). We also discussed briefly effects on upscaled log-conductivity of the variances ratio, z = sa2/sa2, the correlation length-scale ratios, u = Iav/Ifv, and

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RUSSO: UPSCALE CONDUCTIVITY IN GRAVITY-DOMINATED FLOW

the cross-correlation coefficient between fluctuations of logKs and loga, rfa. [50] Results of the present analysis show that, in general, relatively large averaging domains (i.e., b
10) are required in order to obtained asymptotic, effective conductivities devoid of any notion of domain size. This is particularly so in relatively dry, stratified, fine-textured formations, and when the variability in the soil parameter loga is not small compared with the variability in logKs (n = sa2/sf2
0), when the fluctuations in loga are negatively correlated with those of logKs (rfa < 0), and when the correlation length-scale of loga is small compared to that of logKs (u = Iav/Ifv < 1). Larger averaging domains are required in order to ignore the uncertainty about the upscaled log-conductivity (expressed in term of the upscaled log-conductivity covariance, s2), particularly in stratified, coarse-textured formations, and when the length-scale of loga is large compared to that of logKs (u = Iav/Ifv > 1). [51] One of the findings of the this study suggests that the magnitude of the asymptotic Kv diminishes in dry, stratified, coarse-textured (with small capillary forces) formations and when the variability in loga is not small compared to that of logKs (z
0). For the physically plausible situation in which fluctuations of loga are positively correlated with those of logKs, (i.e., rfa > 0 [e.g., White and Sully, 1992; Russo et al., 1997]), however, the diminishing of the asymptotic Kv is balanced by increasing rfa. On the other hand, the increase in the asymptotic Kh, and, concurrently, in the effective anisotropy ratio, due to increasing stratification and decreasing saturation, is balanced by decreasing capillary forces, as well as by decreasing z and increasing (positive) rfa, and to a lesser extent by increasing lengthscale ratio, u = Iav/Ifv. [52] The results of this study are of significance with respect to important issues pertinent to flow and transport through unsaturated heterogeneous porous formations, as they allow assessment of (i) the minimal domain size for which the concept of effective properties is appropriate; and (ii) the maximal size of conductivity cells which preserve the heterogeneous structure of the underlying formation with respect to simulation of flow and transport in the vadose zone. Results of this study suggest that for the physically plausible situation in which the length scale of the formation heterogeneity in the horizontal direction is larger than its counterpart in the vertical direction, (i.e., r > 1 [e.g., Byers and Stephans, 1983; Russo and Bouton, 1992], the size of the averaging domain required in order to reduce the uncertainty about the upscaled log-conductivity to an acceptable, predetermined level, should increase with increasing stratification and decreasing capillary forces (coarser soil texture); the required averaging domain should increase further in formations in which the correlation length-scale of loga is large compared to that of logKs (u = Iav/Ifv > 1). For a predetermined, acceptable reduction in the underlying log-conductivity variance, sy2, the required number of conductivity cells per log-conductivity correlation length scale, b 1=Iv/b1, should increase in stratified, coarse-textured formations; b 1 should increase further when the correlation length-scale of loga is smaller than that of logKs (u < 1). [53] Before concluding, we would like to emphasize the limitations of the present study. One of the limitations arises

from the small-perturbation, first-order approximations of the (cross variances) covariances, Cyy(x) and Ci(x), (i = 1, 2, 3), and formally limits the results to formations with log-unsaturated conductivity variance smaller than unity. Another limitation stems from the fact that the calculations rely upon the ergodicity assumptions. Furthermore, the calculations are restricted to conditions of an unbounded flow domain with a constant mean-head gradient. The simplifying assumptions regarding the structure of the heterogeneous formation (characterized by an exponential covariance with the axisymmetric anisotropy), the statistics of the relevant formation properties and the flow-controlled attributes (statistically homogeneous), the local flow (steady state flow), the local constitutive relationships for unsaturated flow (GardnerRusso model), might also limit the applicability of the results of the present study to real-world scenarios. A rigorous analysis of these assumptions, however, is beyond the limited scope of this study. With these limitations in mind, we believe, however, that results of the present investigation are sufficiently reliable so as to indicate appropriate trends. [54] Acknowledgments. This is contribution 1248-E, 2002 series, from the Agricultural Research Organization, The Volcani Center, Bet Dagan, Israel. The research was supported in part by a grant from the United States-Israel Binational Agricultural Research and Development Fund (BARD). The author is grateful to Asher Laufer for his technical assistance during this study.

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D. Russo, Department of Environmental Physics and Irrigation, Institute of Soils, Water and Environmental Sciences, Agricultural Research Organization, The Volcani Center, Bet Dagan 50250, Israel. (vwrosd@ agri.gov.il)