Anisotropy factor of saturated and unsaturated soils

Transcription

1 WATER RESOURCES RESEARCH, VOL. 42,, doi: /2006wr005001, 2006 Anisotropy factor of saturated and unsaturated soils S. Assouline 1,2 and D. Or 1 Received 28 February 2006; revised 6 July 2006; accepted 14 August 2006; published 7 December [1] Variations in the degree of anisotropy in soil hydraulic conductivity with changes in water saturation (S e ) may adversely impact predictability of flow and transport processes. The conceptual layered cake model was extended to consider effects of bulk density variations within a particular soil type. The anisotropy factor as function of matric potential A(y) exhibits different behavior for different soil textures. For example, A(y) for sand shows complex behavior with a local maximum just before A(y) drops to a minimum where sand becomes isotropic (A(y) = 1) near y = 1.0 bar. Experimentally determined relationships between soil bulk density and hydraulic properties show existence of strong correlation between A(S e ) and the ratio of extreme hydraulic conductivity values K(S e ) max /K(S e ) min for each soil for the entire saturation range and across several soil types. The strong dependency of anisotropy factor on extreme values K max and K min was investigated for four simple and continuous probability density functions of bulk density. Additionally, simple analytical expressions for a binary system of alternating layers with K max and K min only were derived. An upper bound for A(S e ) is obtained with equal weight for K max and K min (w = 0.5). The experimental data and model predictions agreed for certain values of weight assigned to either K max or K min (w = 0.02). Other approximations based on K max and K min provide simple estimates for anisotropy factor that could be related to shape of statistical distribution of hydraulic conductivity. Citation: Assouline, S., and D. Or (2006), Anisotropy factor of saturated and unsaturated soils, Water Resour. Res., 42,, doi: /2006wr Introduction [2] Many soils exhibit a certain degree of anisotropy due to stratification associated with soil forming processes such as sedimentation, illuviation, compaction, and particle orientation. Anisotropic soils are characterized by directional hydraulic conductivity (and other transport properties) that may vary with principal flow direction. Unlike scalar values of hydraulic conductivity in isotropic soils, the hydraulic conductivity of anisotropic soils may require tensorial representation. The accentuated directionality of hydraulic and transport properties in anisotropic soils represents a challenge to characterization and prediction of hydrological and environmental processes. In certain circumstance, however, anisotropy may be exploited to manipulate transport processes in preferred directions (e.g., waste isolation). [3] The degree of anisotropy is commonly expressed by the anisotropy factor, A defined as the ratio of diagonal components of the hydraulic conductivity tensor considering flow in one principal direction. Anisotropy in a simple layered cross section (x-z plane), A = K x /K z = K p /K n, is defined by the ratio between the hydraulic conductivity parallel to bedding (K x = K p ), and that normal to bedding (K z = K n ). 1 Laboratory of Soil and Environmental Physics, School of Architectural, Civil and Environmental Engineering, Ecole Polytechnique Federale de Lausanne, Lausanne, Switzerland. 2 On leave from Institute of Soil, Water and Environmental Sciences, Agricultural Research Organization, Volcani Center, Bet Dagan, Israel. Copyright 2006 by the American Geophysical Union /06/2006WR [4] In contrast to a wealth of theoretical description [Scheidegger, 1956; Bear, 1972; Dullien, 1979] and measurement methods [Dagan, 1967; Rice et al., 1970] for hydraulic anisotropy in saturated soils, anisotropy in unsaturated soils was less studied and Raats et al. [2004, p. 1471] concluded that available models for unsaturated anisotropic soils are either unsatisfactory or not easily implemented. Limited laboratory and field experiments have shown that the extent of anisotropy is strongly related to the water saturation degree [Stephens and Heermann, 1988; McCord et al., 1991; Ursino et al., 2001; Raats et al., 2004]. Despite the importance of quantitative description of such dependency, Zhang et al. [2003, p. 313] commented that available modeling tools for describing this phenomena in natural soils are limited. [5] The simplest conceptual model for hydraulic anisotropy assumes a constant anisotropy factor irrespective of soil saturation degree [Neuman, 1973]. This is tantamount to assuming that anisotropy is a property of the porous medium only, disregarding the role of aqueous phase arrangement under different saturations. Despite doubts concerning the validity of the saturation-independent assumption [Neuman, 1973; Philip, 1987], the model enjoyed wide use in numerical flow model. [6] Other conceptual models proposed for representing the relationships between anisotropy and saturation degree, S e, can be classified by assumptions made concerning the origin of anisotropy: (1) pore-scale anisotropy resulting from particle irregularity and preferred orientation affecting pore structure [Bear et al., 1987; Ursino et al., 2000; Friedman and Jones, 2001]; (2) macroscopic anisotropy 1of11

2 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS resulting from discrete homogeneous layers [Mualem, 1984; Stephens and Heermann, 1988; Green and Freyberg, 1995]; or (3) directional dependence expressed through statistical correlation lengths of hydraulic properties of spatially variable soils [Yeh et al., 1985; Indelman and Dagan, 1993; Russo, 1995; Zhang et al., 2003]. Recently, a tensorial connectivity-tortuosity model for the unsaturated hydraulic conductivity function was applied to study the relationship between anisotropy and saturation [Zhang et al., 2003; Raats et al., 2004]. [7] Pore network models were used to study anisotropy at the pore scale [Bear et al., 1987; Ursino et al., 2000; Friedman and Jones, 2001] yielding different and opposite trends for the relationships between A and S e depending on assumptions and simulations conditions. For example, Bear et al. [1987] found that A(S e ) increases slightly near saturation, then decreases as S e decreases, to values below unity. This suggests that below a certain saturation degree the medium behaves as isotropic followed by reversal of the ratio between the hydraulic conductivities in the two principal directions. Friedman and Jones [2001] have measured the changes in the anisotropy factor of the apparent electrical conductivity during drying of initially saturated pack of platy mica particles. The experiment was simulated using critical path analysis of a simplified three-dimensional pore network model. Friedman and Jones [2001] results show that following an initial moderate increase in anisotropy factor with desaturation, it decreases for low saturations similar to trends proposed by Bear et al. [1987] except A was always greater than unity. They suggested that as saturation approaches the percolation threshold of the system the conducting lattice became isotropic and the anisotropy factor of the entire system tends to the value of the so-called lattice compaction term (a property of anisotropic lattice geometry). [8] Ursino et al. [2000] considered three different anisotropic configurations in Miller-similar porous media: (1) the distribution of pore diameters was anisotropic; (2) only the density of the pores in the flow cross sections was anisotropic; (3) both density and pore size distribution were anisotropic. In the first case, the macroscopic anisotropy factor remained constant and did not depend on the saturation degree. The two other cases led to anisotropy factors that were strongly dependant upon saturation degree. The anisotropy factors switched from values above 1.0 to values below 1.0, in agreement with Bear et al. [1987] findings, and vice versa. In the study of Ursino et al. [2000], the correlation structure was isotropic, a condition that differ significantly from that in studies where the soil anisotropy is generated by an anisotropic correlation structure [Yeh et al., 1985; Russo, 1995; Green and Freyberg, 1995; Zhang et al., 2003] or by the presence of thin layers of different materials [Mualem, 1984; Pruess, 2004]. The results of these studies show that the macroscopic anisotropy is saturation dependent, and that the factor of anisotropy increases as saturation degree decreases. [9] Mualem [1984] proposed a conceptual model for the relationship between anisotropy factor and saturation degree assuming a soil body consisting of many thin layers each with its own hydraulic properties drawn from a uniform probability density function of saturated hydraulic conductivity, K s. The relevant parameters for other soil hydraulic properties were related to K s through regression analyses using data from 45 different soils [Mualem, 1974]. The resulting conceptual anisotropic porous medium is composed of layers representing a wide range of soil textures ranging from coarse sands to clays within the same hypothetical soil volume. Model calculations showed that, as S e decreases, A(S e ) first decreases to a minimum, and then increases rapidly as the soil dries by orders of magnitude relative to its value at saturation. That study has also indicated that the range of the saturated hydraulic conductivity distribution and its lower limit were the two main factors that dominate the soil degree of anisotropy. [10] The hypothetical porous medium in the basis of Mualem s [1984] model encompass a wide range of soil textures ranging from coarse sands to clays within the same unit volume, which may represent an extremely heterogeneous depositional environment unlikely at small scales. We are interested in the behavior of soils in which anisotropy arises within narrow range of textural units due to particle segregation, compaction banding, and internal illuvation that would typically affect porosity or bulk density. Furthermore, linking anisotropy solely to K s values associated with Mualem s [1984] range of soil textures may not be similar to wide variability observed in K s values within the same soil texture. [11] Recently, Assouline [2006a, 2006b] has presented an empirical approach using changes in soil bulk density to predict concurrent variations in hydraulic properties. These relationships were tested for a large range of soil types and textures with satisfactory results. We thus propose combining such an approach with the conceptual model of Mualem [1984] to enable explicit consideration of anisotropy behavior of a specific soil texture using more realistic soil hydraulic properties. The primary objective of this study is to apply the approach of Assouline [2006a, 2006b] to quantify anisotropy of saturated and unsaturated soils consisting of many thin layers differentiated by their bulk density, as it may occur during sedimentation, compaction, and reconsolidation affecting bulk density but not texture. Additionally, previous studies considered a rigid soil matrix, whereas the new approach may be useful to study effects of shrink-swell behavior on anisotropy of nonrigid soils as a function of hydration status. 2. Conceptual Model for Anisotropy [12] Building on Mualem s [1984] concepts, we consider a soil of a certain texture consisting of a large number of thin, yet distinguishable, layers differentiated by their bulk density (the variations may result in from cycles of deposition, compaction, reconsolidation, or shrinking and swelling). For simplicity, we assume layers are parallel to bedding plane hence we reduce principal flow directions to two only: normal (vertical) and parallel (horizontal) to layering. Each layer is characterized by its own hydraulic conductivity function, K(S e ), relating its hydraulic conductivity, K, to effective saturation degree S e =(q q r )/(q s q r ), with q, q s and q r representing water content, saturated and residual water contents, respectively. The layered structure is expressed by means of a probability density function, f(r), of the bulk density, r, the independent variable that permits estimation of macroscopic hydraulic conductivity in 2of11

3 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS the two principal directions. Parallel to the hypothetical layering, the hydraulic conductivity K p (S e ), is defined by the arithmetic mean of K(S e ) for the layers: Z K p ðs e Þ ¼ f ðrþks ð e ; rþdr ð1þ The hydraulic conductivity normal to the layers, K n (S e ), is given by the harmonic mean of K(S e ) values for the layers: K n ðs e Þ ¼ Z f ðrþ 1 KS ð e ; rþ dr ð2þ The anisotropy factor A(S e ) is thus defined by AS ð e Þ ¼ K pðs e Þ K n ðs e Þ For a uniform probability density function, f, with r ranging from a lower limit r min to a higher one, r max : f ðrþ ¼ ð3þ 1 ðr max r min Þ ; r min r r max ð4þ The corresponding anisotropy factor for saturated soil (S e =1),A s, is thus 1 2 Z rmax A s ¼ ðr max r min Þ r min Z rmax K s ðrþdr K s ðrþ 1 dr r min For unsaturated conditions, the corresponding factor of anisotropy, A(S e ), is defined by AS ð e Þ ¼ 1 2 Z rmax ðr max r min Þ r min KS ð e ; rþdr Z rmax r min ð5þ KS ð e ; rþ 1 dr [13] The effects of bulk density variations on soil water retention curve could be modeled in different degrees of detail [e.g., Or et al., 2000; Stange and Horn, 2005; Assouline, 2006a]. We opted for the approach of Assouline [2006a] in which water retention curve is expressed as h S e ðy; rþ ¼ 1 exp ar ð Þ jyj 1 jy L j 1 i mr ð Þ ; ð6þ 0 jyj jy L j ð7þ where the parameters a and m for a soil of given texture are functions of r. The coefficient of variation, e(r), of S e (y, r) is computed according to a(r) and m(r) [Assouline, 2006a]: n xr ð Þ 2=mðrÞ G 1 þ 2=m r ð ð ÞÞ G ð1 þ 1=mðrÞÞ er ð Þ ¼ xr ð Þ 1=mðrÞ ð 1 þ 1=m ð r ÞÞþ1= jy L j ð8þ where x(r) =a(r) m(r) and G denotes the gamma function. [14] We employ Assouline s [2006b] approach for modeling the relationship between bulk density and hydraulic conductivity. At saturation, the expression relating K s to r is given by 2 1 r 33 r K s ðrþ ¼ K s ðr i Þ s r 7 r ðd 7Þ 5 ð9þ i r i r s where r i is a reference bulk density for which K s (r i )is known, r s is the soil particle density, and d is related to the soil texture. [15] For unsaturated conditions, K(S e, r) was defined by KS ð e ; rþ ¼ K s ðrþ Z Se ðrþ 0 Z y 1 ds 7 y 1 5 ds hr ð Þ ð10þ where y is the matric tension, and the power h is related to the coefficient of variation, e, of the water retention curve S e (y, r) (equation (8)) according to hr ð Þ ¼ 3:04e er ð Þ þ 0:22 ð11þ Readers are referred to Assouline [2006a, 2006b] for additional details. [16] For a prescribed soil texture, and assuming a probability density function, f(r), the relationships presented above may be used to determine S e (y, r), K s (r) and K(S e, r) for the entire range of r, enabling direct computation of the corresponding A(S e ) for the entire range of saturation degree (equations (5) and (6)), including complete saturation where S e = Methodology and Data Sets [17] Calculated A s and A(S e ) values (equations (5) and (6)) rely on the approach of Assouline [2006a, 2006b] that was calibrated and validated with experimental data of Laliberte et al. [1966], Reicovsky et al. [1980] and Smith and Woolhiser [1971]. We thus used the calibrated expressions for S e (y, r), K s (r) and K(S e, r) as presented by Assouline [2006a, 2006b] to compute the various anisotropy factors in what follows. [18] For saturated conditions we used K s (r) estimated for five soils: Columbia sandy loam, Touchet silt loam, and an unconsolidated sand [Laliberte et al., 1966], Barnes loam [Reicovsky et al., 1980], and Pouder fine sand [Smith and Woolhiser, 1971]. For unsaturated conditions, the expressions for S e (y, r) and K(S e, r) corresponding to Columbia sandy loam, Touchet silt loam, and an unconsolidated sand [Laliberte et al., 1966] were used. A summary of the data for these soils is given in Table 1. [19] Four different probability density functions, f(r), were used, including: uniform (UNIF), centered and symmetrical triangular (C-TRI); triangular and skewed to the left (L-TRI); and triangular and skewed to the right (R-TRI). These simple density functions cover a wide range of possible trends in bulk density statistical distributions observed in natural soils [Warrick and Nielsen, 1980]. 3of11

4 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Table 1. Characteristics of the Soils in the Data Set a Soil Type Sand, % Silt, % Clay, % r i,gcm 3 r s,gcm 3 d K s, mmh 1 K smin K smax, mmh 1 Columbia sandy loam Touchet silt loam Unconsolidated sand Barnes loam Poudre fine sand a Characteristics are soil type, mechanical composition, particle density, r s, reference bulk density, r i, for the HCF data, values of the parameter d in equation (9), the saturated hydraulic conductivity, K s, at the reference bulk density, and the K smin and K smax values corresponding to r min = 1.0 g cm 3 and r max = 2.0 g cm 3. [20] For each soil, we considered seven different ranges of bulk densities by varying the lower and upper limits of f, r min and r max, within the maximum range of values found in soils between 1.0 and 2.0 g/cm 3. The resulting (r min, r max, r max /r min ) values are summarized in Table 2. The range of the corresponding K s values for the different soils between 1.0 and 2.0 g/cm 3 is given in Table 1. Additional combinations of r min and r max leading to the same ratios were also considered, but had no effect on the results. The different distributions for r max /r min = 2.0 are depicted in Figure 1a. For each r value, K s (r) at saturation was computed according to equation (9), and the water retention curve S e (y, r), and the hydraulic conductivity function K(S e, r) for unsaturated conditions was determined using equations (7), (9) and (10). Hence for each r min and r max pair, the corresponding values of K min and K max were determined spanning the entire range of S e. We then discretized the interval between r min and r max into small Dr increments using about 40 intervals to ensure accurate numerical integration. The results from the four different probability distribution functions and their corresponding f(k s ) distributions were corrected such that R f(k s )dk s = R f(r)dr =1. The f(k s ) probability density functions corresponding to the f(r) distributions in Figure 1a are shown in Figure 1b for the Columbia soil data. [21] For the uniform distribution, A s, and A(S e ) were estimated numerically using equations (5) and (6), whereas for all other distributions, these were estimated numerically using equation (3) (employing the Simpson rule for numerical integration). [22] As indicated in the introduction, experimental data for validation of proposed A(Se) relationships are scarce. The most complete data sets available include the results of Stephens and Heermann [1988], McCord et al. [1991], Ursino et al. [2001] and Zhang et al. [2003]. [23] Stephens and Heermann s [1988] experiment consisted of point source infiltration into a porous medium composed of alternating layers of equal thickness of fine and coarse sand. The calculated A values and the corresponding measured K values were reproduced from their Figures 5 and 13. The so-called measured A values were not considered following the authors own assessment that errors in unsaturated hydraulic conductivity measurements of the sands may have affected these values. [24] The data of McCord et al. [1991] were obtained from tracer experiments following precipitation on a sandy hillslope which behaved as an anisotropic medium. We considered the entire range of A and K values reported in that study. [25] Ursino et al. [2001] performed a tracer experiments in a laboratory sand tank. Three different sands were randomly packed in layered cubes, leading to two- Table 2. Minimum and Maximum Soil Bulk Density Values Considered and the Corresponding Ratio r max /r min r min r max r max /r min of11 Figure 1. (a) Four probability density functions, f(r), used (for r max /r min = 2.0) and (b) the corresponding f(k s ) distributions for the Columbia soil.

5 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Figure 2. Relationship between the anisotropy factor at saturation, A s, and r max /r min for the five soils for the uniform distribution case. dimensional, random, heterogeneous and anisotropic structure. Three flow rates were imposed at the surface and the solute transport in the structure was observed at steady state at the three corresponding mean saturation degrees. The macroscopic anisotropy of the conductivity was estimated by means of the tracer plumes analysis. Information on the system, the sand hydraulic properties and the estimated A values were taken from Ursino et al. [2001] and Ursino and Gimmi [2004]. Additional information on the histograms of the water content in the three sands at the three rates was provided by T. Gimmi (personal communication, 2006). [26] Zhang et al. [2003] conducted detailed numerical experiments using 16 synthetic soils generated by assigning 4 levels of spatial variability and 4 levels of anisotropy. Anisotropy was induced by assigning different correlations lengths at different directions in the simulation domain. The resulting values of A s for the 16 case studies were provided by Z. F. Zhang (personal communication, 2006). The corresponding range of K s (within 95% of the confidence interval) and of the standard deviation of lnk s, s n, were reported in Table 1 of Zhang et al. [2003]. 4. Results and Discussion 4.1. Anisotropy Factor at Saturation [27] The values of A s for five soils were computed using equation (5) for the different r max /r min values and for the uniform f(r) case and are depicted in Figure 2. For low values of r max /r min, below 1.15, A s remains practically constant near unity irrespective of soil type. With the increase in r max /r min, the values and rate of increase of A s vary with soil type with highest rate for Columbia sandy loam, and the lowest for coarse-textured soils. It is interesting that A s (r max /r min ) for these two coarse sands are identical even though their respective K s (r) functions are significantly different. For the highest r max /r min value considered, A s can reach values as high as 3.5 for the sands and 12.4 for the Columbia sandy loam. [28] The parameter d in equation (9) reflects the effect of soil texture on K s (r) [Assouline, 2006b] as shown in Table 1 for different soils. The computed values of A s for the different r max /r min can thus be expressed as a function of d. For example, the results in Figure 2, suggest that the impact of r max /r min on anisotropy factor at saturation decreases with increasing d. [29] In the model of Mualem [1984], the independent variable for anisotropy was K s. Hence he reported that as K smax /K smin increases, A s increased accordingly, approaching a value of 4.6 for K smax /K smin =10 4, independently of K smin. In this study, the primary variable responsible for hydraulic anisotropy is the soil bulk density, r. Therefore A s is soil type dependent and values higher than 4.6 are obtained for the three fine textured soils for (r max /r min )= Anisotropy Factor for Unsaturated Conditions [30] The calculated A(y) values for three soils used to study unsaturated condition are depicted in Figure 3a, for the uniform f(r) case, and for r max /r min = 2.0. The relationship between the matric potential, y, and anisotropy is soil texture dependent. For the two fine-textured soils, A remains nearly constant and equal to A s for y above the air entry value, followed by a decrease toward a minimum and then increases drastically as the soil dries. This general trend is in agreement with theoretical predictions [Mualem, 1984; Yeh et al., 1985; Zhang et al., 2003] and experimental results [Stephens and Heermann, 1988; McCord et al., 1991; Ursino et al., 2001]. We note that this behavior is opposite to trends observed in some pore network models [Bear et al., 1987; Friedman and Jones, 2001], or the experimentally measured anisotropy in electrical conductivity of partially saturated mica particles [Friedman and Jones, 2001]. [31] Minimum values of A close to A = 1, for the two fine textured soils were attained for y 0.8 bar for Columbia sandy loam and Touchet silt loam. In other words, for this particular value of y, the unsaturated porous media behave as hydraulically isotropic. These results may correspond to the percolation threshold for which anisotropic pore networks approach isotropic behavior as predicted by Lobb et 5of11

6 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Figure 3. (a) Relationship between the anisotropy factor, A, and the matric tension, y, for the three soils for r max /r min = 2.0 and A(y) for two lower values of r max /r min, 1.5 and 1.14, in the case of Columbia sandy loam and the unconsolidated sand (uniform f(r) case). (b) Relationship between K p (hydraulic conductivity parallel to the bedding) and K n (hydraulic conductivity normal to the bedding) and the matric tension, y, for the Columbia sandy loam and the unconsolidated sand. al. [1981], and confirmed for 2-D system by Han et al. [1991]. In contrast, Madadi and Sahimi [2003] reported that anisotropy ratio in their fracture network increases as the network approaches its percolation threshold, a possible explanation for this conflicting result will be discussed next. [32] We observed that the behavior of A(y) for sand was quite different and exhibited some surprising features. The results in Figure 3a (unconsolidated sand) show a slight decrease in A following desaturation, then A increases sharply (about threefold), before decreasing toward a minimum close to unity and sharply increasing with subsequent drying (similar to other soils). The local maximum near y = 0.3 bar was not predicted by the virtual porous medium studied by Mualem [1984] nor observed in experimental studies. Inspection of the respective K p (y) and K n (y) functions of the Columbia sandy loam and the unconsolidated sand (Figure 3b) reveals a particular value of y (y 1.1 bar) at which K p (y) and K n (y) practically merge. This is the point where A(y)! 1.0 (Figure 3a). It is interesting to note that K p (y) and K n (y) do not intersect but rather merge before diverging again at different rates as the soil becomes dryer. The origin of the different A(y) behaviors lies in the difference in dk p (y)/dy and dk n (y)/dy for the two soils within the range of y = 0 to 1.0 bar. The local maximum in A(y) obtained for the sand corresponds to the local increase of the difference between K p (y) and K n (y), 6of11

7 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Figure 4. Relationship between the anisotropy factor, A, and the saturation degree, S e, for the three soils for r max /r min = 2.0 and A(S e ) for two lower values of r max /r min, 1.5 and 1.14, in the case of Columbia sandy loam (uniform f(r) case). while for the sandy loam, this difference steadily decreases until a merging point. A similar behavior may be responsible for the findings of Madadi and Sahimi [2003] regarding the apparent increase in anisotropy as their network approached the percolation threshold. [33] The effect of different r max /r min ranges on A(y) is also shown in Figure 3a for Columbia sandy loam and the sand. Lower values of r max /r min resulted lower A values for each y, and smaller influence of y on A. The main characteristics of the A(y) curves for the different soil types are maintained for the different r max /r min ranges, including the local maximum for the sand. However, the values of y at which the different maxima and minima occur are related to the specific r max /r min range. [34] The calculated anisotropy factor for the three soils are depicted in Figure 4 for each soil and for r max /r min =2.0 as a function of the saturation degree S e (using equation (6)). The relationship between the degree of saturation and anisotropy is dependent on soil type with the general trend is that A decreases slightly toward a minimum and then increases back as the soil dries and S e decreases. When expressed in terms of S e, the behavior of A(S e ) for the sand is similar to that for the fine textured soils. Minimum values of A in Figure 4 were attained for S e close to 0.5 for sandy loam and sand, and close to 0.7 for more fine-textured Touchet silt loam. Unlike for the A(y) curves (Figure 3), A(S e ) rising limb toward very low S e values, was more accentuated in fine-textured soils than in the sand. [35] The effect of different r max /r min ranges on A(S e )is also shown in Figure 4 for Columbia sandy loam. As noted for the A(y) representation (Figure 3a), lower values of r max /r min resulted in lower A values for each S e, and smaller influence of S e on A. From the results in Figures 3a and 4, it appears that for low values of r max /r min (below 1.20), r- induced anisotropy may be independent of saturation degree. [36] Many studies dealing with anisotropy of unsaturated soils implicitly assume that the soil solid matrix is rigid, i.e., no changes in pore spaces occur as S e decreases. The potential for shrinking during drying could affect initial r max /r min range, and may lead to different trends with respect to A(S e ). For example, Friedman and Jones [2001] noted shrinkage of mica particles during desaturation followed by reswelling at very low water contents (see Figure A1 in their study). Assuming commensurate changes in r max /r min the hypothetical behavior of A(S e )of Friedman and Jones [2001] would have been similar to the trend in Figure 3, namely, an increase of A to a maximum value followed by a decrease as S e decreased. [37] Pooling all A(S e ) values for the four f(r) distributions and the three soils plotted vs. K(S e ) max /K(S e ) min corresponding to the r max /r min values for the entire range of S e (including S e = 1.0) are shown in Figure 5a. Also included are the A s values for the additional soils that were not part of the unsaturated conditions tests. The results show a strong relationship between A(S e ) and K(S e ) max /K(S e ) min, independent of soil type and range of r max /r min. The relationships are affected by the shape of f(r). The results obtained for the uniform and the symmetrical triangulartype probability density functions of r (Figure 1a) were very similar, but significantly different relationships were obtained for the skewed distribution (R-TRI and L-TRI). The relationships may be expressed as AS ð e Þ ¼ 8 KS ð eþ w max þ ð1 8Þ ð12þ KS ð e Þ min where A(S e )! 1.0 when [K(S e ) max /K(S e ) min ]! 1.0. [38] The parameters for the uniform f(r) are8 = 0.135, and w = For the C-TRI distribution type, 8 = 0.10, and w = For the L-TRI distribution, 8 = 0.01, and w = 0.77, while for R-TRI type, 8 = and w = [39] Mualem [1984] pointed out the important role of K smin on the value of A. The effect of lowest hydraulic conductivity, K(S e ) min values on A is depicted in Figure 5b for Columbia soil with r max /r min = 2.0 and for four hypothetical statistical distributions of r (Figure 1a). The complex interactions between K(S e ) min and A show strong 7of11

8 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Figure 5. (a) Relationship between the anisotropy factor, A, and K(S e ) max /K(S e ) min for the saturated and the unsaturated data for the four f(r) distribution types and the fitted functions (equation (12)). (b) Relationship between the anisotropy factor, A, and K(S e ) min for the r max /r min = 2.0 case for the Columbia soil and the four distribution types. dependency on the type (shape) of the f(k s ) distribution (Figure 1b). Additionally, for a given distribution, A decreases with K(S e ) min for the high S e values, and then increases steadily as K(S e ) min becomes smaller. [40] The results in Figures 5a and 5b indicate that, the anisotropy factor in such layered systems can be directly estimated from the extreme hydraulic conductivity values in the sample (i.e., K(S e ) max and K(S e ) min ) for both saturated and unsaturated conditions. The relationship, however, is sensitive to the skewness of the probability density function of the hydraulic conductivity values. We will show that the limits of these dependencies may be determined from simple binary representation of the hydraulic conductivity population considering its end-members only (K(S e ) max and K(S e ) min ) Comparisons of Anisotropy Factor Predictions With Experimental Results [41] The results from studies by Stephens and Heermann [1988], McCord et al. [1991], Ursino et al. [2001] and Zhang et al. [2003] were expressed in terms of A(K max / K min ) and used to verify equation (12) for all the four f(k s ) probability density functions (Figure 6). The data of Stephens and Heermann [1988] correspond to the low end of the K max /K min range and close to equation (12) representing the R-TRI distribution type. The expressions of equation (12) corresponding to the uniform and the symmetrical triangular distributions corresponds quite well to the McCord et al. [1991] results, whereas the results of Zhang et al. [2003] are contained between this curve and the one for the L-TRI distribution. The results of Zhang et al. [2003] were generated by a factorial study of 4 levels of spatial variability and 4 levels of anisotropy levels. The four points corresponding to the absence of anisotropy are denoted by empty triangles in Figure 6. Irrespective of the degree of variability, the points are aligned along A = 1 line, confirming isotropic behavior of these uncorrelated synthetic data. 8of11

9 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS Figure 6. Relationship between the anisotropy factor, A, and K(S e ) max /K(S e ) min (equations (12) (14)) for the four distribution types and the data of Stephens and Heermann [1988], McCord et al. [1991], Ursino et al. [2001], and Zhang et al. [2003] (plus indicates the range of McCord et al. s [1991] results) Determining Anisotropy Factor From Extreme Values of Hydraulic Conductivity [42] The strong relationship between A(Se) and K max /K min in equation (12) suggests that the anisotropy factor is dominated by the upper and the lower limits of the distribution of K s or K(Se). We thus explored the possibility of representing anisotropy behavior by considering a hypothetical two-layer binary system composed by the two endmembers of the hydraulic conductivities K max and K min. These end-members of the distribution may be weighted by their relative abundance (e.g., relative layer thickness), or approximating a certain distribution of K, using the weight w, fork max, and (1 w) fork min. The expression for A, based on equation (3), is A ¼ wð1 wþ K max þ K min þ 2w 2 2w þ 1 K min K max ð13þ For w =0orw = 1, equation (13) yields A = 1, namely an isotropic system with a scalar value of K. Interestingly, the resulting A for a given w value is identical to a system in which K max is weighed by the complementary (1 w) value. Recently, Warrick [2005] has studied aspects of effective unsaturated hydraulic conductivities for onedimensional structured heterogeneity. Considering a binary system similar to that postulated in equation (13), Warrick [2005] found that the effective conductivity approaches the value of hydraulic conductivity of the sublayer with greatest relative thickness, or following our terminology, the K of the layer with the higher w. This explains the symmetry observed in anisotropy calculations for binary systems where A is identical irrespective of whether a sublayer is represented with a relative weight of w or (1 w). [43] The curves corresponding to w = 0.5 (similar weights for K max and K min ), and for w = (or the complementary 1 w = which results in an identical curve) are depicted in Figure 6. The upper bound for A(K max /K min ) obtained with w = 0.5 fits well the data of Stephens and Heermann [1988] obtained from 40 layers of equal thickness with only two alternating sand types (very close to the conditions assumed in equation (13) with w = 0.5). [44] The data of McCord et al. [1991], Ursino et al. [2001], and Zhang et al. [2003] were generated from more complex distributions of K values requiring different values of w in equation (13), typically between 0.01 and (corresponding also to 0.99 and 0.999). It is interesting to note that heterogeneous systems corresponding to low (or high) w values resemble isotropic behavior for low K max / K min ratios. [45] Equation (13), with w = 0.02 (or 0.98), fits the empirical relationship derived in this study (equation (12)) for the uniform and the symmetrical triangular distributions. This result highlights the dominance of the combination between extreme values of K and their relative weight. Interestingly, symmetry arguments show that the same anisotropy factor is obtained with application of the low weight (e.g., w = 0.02) to the K max or to K min in equation (13). [46] Motivated by the simplicity attained with w = 0.5, we capitalize on anecdotal evidence showing that the effective unsaturated hydraulic conductivity perpendicular to bedding in unsaturated porous media, can be better approximated by the geometric mean of K [Yeh and Harvey, 1990]. This may also reflect the increased importance of flow across interfaces between layers over flow through the alternating layers. In practice, it can be considered as an exact effective conductivity value in two dimensions [Renard et al., 2000], and it seems to provide an appropriate approximation based 9of11

10 ASSOULINE AND OR: ANISOTROPY FACTOR OF SOILS on experimental results of McCord et al. [1991], Ursino et al. [2001], and Zhang et al. [2003]. Replacing the harmonic mean in equation (3) by the geometric one modifies the expression in equation (13) to sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K max K min ½wK max þ ð1 wþk min Š A ¼ wk min þ ð1 w ÞK max ð14þ The resulting relationship (equation (14)) with w = 0.5 provides a good match with the empirical relationship (equation (12)), and with theory [Renard et al., 2000], hence offers a simple first-order approximation for the relationships between extreme values of hydraulic conductivity in anisotropic soil and saturation-dependent anisotropy factor: h i A ¼ 0:5ðK max =K min Þ 0:5 þ 0:5ðK min =K max Þ 0:5 ð15þ The present study offers certain generalizations that might be useful for inclusion of anisotropy into predictive models, such as the relationships between K values and A that hold seamlessly for saturated (easy to determine) and unsaturated conditions using a single expression. Such compact formulation would be easy to implement in any numerical modeling code to assess potential impact of anisotropy on transport processes with relatively modest changes in existing codes. The other aspect is the shape of the density distribution function that requires additional information but may help establish trends and importance of anisotropy at the outset. 5. Summary and Conclusions [47] Anisotropy in hydraulic conductivity based on the layered cake model was described by means of variations in bulk density of the layers with four different probability density functions. Combined with previously established relationships between bulk density and hydraulic properties [Assouline, 2006a, 2006b] we estimated saturated and unsaturated anisotropy factor within a soil type without the prior constraint of a single universal soil type implicitly assumed by Mualem [1984]. [48] Our results confirm characteristics of calculated A(y) similar to previously reported for fine textured soils, however, the results for sand revealed complex behavior going through a local maximum anisotropy factor before attaining isotropic behavior at y 1.0 bar. The behavior of A(S e ) was affected by soil texture, especially for fine textured soils. A values decrease with S e to a minimum followed by a slight increase beyond a minimum value of S e. This is a trend found by most macroscopic anisotropy models [Mualem, 1984; Yeh et al., 1985; Zhang et al., 2003]. In contrast, some studies based on pore-scale anisotropy using pore networks as simulators of porous media [Bear et al., 1987; Friedman and Jones, 2001; Ursino et al., 2000] may result in different and even opposite trends. [49] An important result of this study was the strong dependency of A(S e ) function on the ratio of K(S e ) max / K(S e ) min (equation (12)) for all values of saturation degree. The relationships in equation (12) offers a simple means for quantifying and represent anisotropy in numerical models with seamless transition from saturation to unsaturated conditions. The relationships in equation (12) are affected by the shape of the anisotropy-generating probability density function. The different empirical relationships corresponding to the uniform, symmetrical and skewed triangular distributions used herein match relatively well available experimental data [Stephens and Heermann, 1988; Ursino et al., 2001; McCord et al., 1991] and results from extensive synthetic anisotropy numerical studies [Zhang et al., 2003]. The strong dependency of anisotropy factor on extreme values of hydraulic conductivity were further exploited by simple analytical expression for the anisotropy factor considering alternating layers with two values only K max and K min. This simple expression provide upper bound for A(S e ) function when w = 0.5, which is close to the simulated results for the R-TRI distribution. With w = 0.02, it represents available experimental data and the strong correlation in equation (12) for the uniform and the C-TRI distribution cases. The potential for replacing the harmonic mean by its geometric counterpart provides a good first approximation for most available data based on an extremely simple expression (equation (15)). [50] The extension of the results beyond the layered cake anisotropic systems to consideration of particle orientation and inclusions of various shapes in multi dimensions, and for anisotropic size and length distributions in pore networks, is possible through the use of concepts from percolation theory combined with renormalization group techniques such as described by Lobb et al. [1981], King [1989], and Renard et al. [2000]. The primary constraint to application of these techniques to flow in porous media is lack of definitive data for model testing. [51] Acknowledgments. This study was performed during the first author s visit to EPFL (Switzerland). The first author thanks EPFL/ENAC, M. Parlange, and the EFLUM team for their generous hospitality. The authors gratefully acknowledge Fred Zhang and Thomas Gimmi for kindly providing the additional data from their respective studies and Shmulik Friedman for his insightful comments. This research was partly supported by research grant IS R from BARD, the United States-Israel Binational Agricultural Research and Development Fund; this support is gratefully acknowledged. References Assouline, S. (2006a), Modeling the relationship between soil bulk density and water retention curve, Vadose Zone J., 5, Assouline, S. (2006b), Modeling the relationship between soil bulk density and the hydraulic conductivity function, Vadose Zone J., 5, Bear, J. 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