Published online 26 May 2006
Published in Vadose Zone J 5:697-705 (2006)
DOI: 10.2136/vzj2005.0084
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Modeling the Relationship between Soil Bulk Density and the Hydraulic Conductivity Function
S. Assouline*
Institute of Soil, Water and Environmental Sciences, A.R.O., the Volcani Center, P.O.B. 6, Bet Dagan 50250, Israel
* Corresponding author (vwshmuel{at}agri.gov.il)
Contribution of the Agricultural Research Organization, Institute of Soil, Water and Environmental Sciences, Bet Dagan, Israel, No. 608/05.
Received 10 July 2005.
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ABSTRACT
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A model is presented that could quantify and predict the effect of an increase in soil bulk density on the hydraulic conductivity function (HCF). Two approaches are formulated to predict the saturated hydraulic conductivity of compacted soils. The first approach is a general expression based on the Kozeny equation that requires only information on the bulk density. The second approach exploits information contained in the water retention curve (WRC). This approach, which relies on Assouline's model for soil HCFs, also provides a basis for a proposed expression to predict the unsaturated HCF of compacted soils. It is shown that a relationship between 
, the power parameter in the expression for the HCF, and 
, the coefficient of variation of the WRC, is applicable to a large number of both compacted and uncompacted soils. The model was verified against measured HCFs for soils at different bulk density values. Both the saturated and unsaturated hydraulic conductivities were well reproduced by the suggested expressions.
Abbreviations: HCF, hydraulic conductivity function • PSD, pore-size distribution • RHC, relative hydraulic conductivity • WRC, water retention curve
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INTRODUCTION
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AN INCREASE in soil bulk density associated with compaction can have a dramatic impact on soil hydraulic properties (Laliberte et al., 1966; Green et al., 2003), and as such can have important effects on vadose zone flow and transport processes. Simulation of flow processes in soils requires expressions for both the WRC and the HCF. Models dealing with the relationship between soil bulk density and the WRC have been suggested (Assouline et al., 1997; Ahuja et al., 1998; Stange and Horn, 2005; Assouline, 2006); however, approaches to model the effect of an increase in the soil bulk density on the HCF are very limited. This study suggests an empirical approach to represent the relationship between soil bulk density and the HCF. The HCF relates the hydraulic conductivity, K, to the soil capillary head, 
, or the volumetric water content, 
. This function comprises two elements: (i) the saturated hydraulic conductivity, Ks, corresponding to the particular case where the soil is saturated with water and = 0 or the water content = s, the saturated water content; and (ii) the unsaturated hydraulic conductivity function K() or K(). Each of these elements will be treated in the following.
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Saturated Hydraulic Conductivity
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The Ks is mostly determined by large pores, which are strongly reduced when the soil bulk density increases (Carter, 1990; Lipiec et al., 1998; Håkansson and Lipiec, 2000). Consequently, drastic reductions in Ks with increasing bulk density have been reported (Laliberte et al., 1966; Dawidowski and Koolen, 1987; D
bicki et al., 1993; Håkansson and Medvedev, 1995). The ratio between the saturated hydraulic conductivity of the compacted soil, Ksc, and that of the initial soil, Ks, can vary by orders of magnitude (Reicosky et al., 1980; Young and Voorhees, 1982; Horton et al., 1994).
Or et al. (2000) selected the Kozeny–Carman relationship for the soil saturated hydraulic conductivity (Carman, 1937) to estimate Ksc/Ks:
 | [1] |
where n and nc are the porosity of the initial and the compacted soils, respectively. Green et al. (2003) relied also on the modified Kozeny–Carman expression for Ks, as was suggested by Ahuja et al. (1984, 1989), to estimate Ksc/Ks.
Laliberte et al. (1966) applied the equation for Ks presented by Wyllie and Sprangler (1952) and the model of Brooks and Corey (1964) for the WRC to express Ksc/Ks:
 | [2] |
where nd is the drainable porosity (calculated as the porosity n minus the residual water content r), a, is the air entry value, 
is the pore-size distribution index, and the subscript c denotes the compacted state.
Mualem and Assouline (1989) suggested that the hydraulic conductivity at saturation, Ksc, can be estimated by
 | [3] |
where Se is the effective saturation degree defined as ( – r)/(s – r), and the subscript c denotes the compacted state. When the expression of Brooks and Corey (1964) is adopted to represent the WRC of the compacted soil, the resulting relative change in the saturated hydraulic conductivity Ksc/Ks is
 | [4] |
When the porosity is equal to the saturated water content, s, the ratio (sc – rc)/(s – r) in Eq. [4] is similar to ndc/nd in Eq. [2]. Consequently, Eq. [2] and [4] address the same components based on soil parameters but with different relative weights (power values).
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Unsaturated Hydraulic Conductivity Function
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The hydraulic conductivity function (HCF) represents the dependence of K on , , or Se, and plays a predominant role in flow and transport processes under unsaturated conditions. Compaction usually reduces the K values under given soil wetness conditions (Laliberte et al., 1966; Walczak et al., 1993; Horton et al., 1994), although in some compaction ranges and at low water contents, compaction may increase the conductivity (Lipiec and Hatano, 2003). Unfortunately, detailed experimental data on the effect of bulk density on K() are scarce.
Methods to estimate and predict Kc() of compacted soils rely mostly on the various approaches used to model HCFs (Mualem, 1986). Reicosky et al. (1980) and Green et al. (2003) suggested using the method of Campbell (1974); however, most of the studies apply Mualem's (1976) model (Or et al., 2000; Lipiec and Hatano, 2003; Moroizumi and Horino, 2004). According to this model, the expression of K(Se) is
 | [5] |
where s is a dummy variable for integration. Replacing Ks by Ksc and Se() by Sec() leads to the definition of Kc(Sec) that represents the HCF for the compacted state. For some analytical functions of Sec(), Eq. [5] can yield closed-form analytical expressions for Kc(Sec) (see Appendix).
Assouline (2001, 2004) suggested a different model for K(Se):
 | [6] |
where s is a dummy variable of integration and is a parameter that depends on soil structure and texture. In this case, too, replacing Ks by Ksc, Se() by Sec(), and by c, leads to a definition of Kc(Sec) for the compacted state for which closed-form analytical expressions can be derived (see Appendix).
Assouline (2001, 2005) suggested that the power in Eq. [6] is related to the coefficient of variation, , of the WRC. Consequently, Eq. [6] can be used to predict the RHC [relative hydraulic conductivity, K()/Ks] of soils, given their WRC. A method to predict the WRC of compacted soils and its corresponding coefficient of variation, c, has been presented recently in Assouline (2006). It should be interesting and valuable to know if a similar relationship exists also between c and c. Such a relationship would enable one to predict the effect of compaction on the soil HCF. Therefore, the objectives of this study were to investigate the relationship between the bulk density of a soil and its hydraulic conductivity (saturated and unsaturated) to extend the predictive capability of Assouline's (2001) model to compacted soils.
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PRESENTATION OF THE MODELS
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Saturated Hydraulic Conductivity
In this study, two models of the effect of compaction on the saturated hydraulic conductivity were investigated. The first is based on the equation of Kozeny (1927, as cited in Scheidegger, 1960) for the soil saturated hydraulic conductivity. The Kozeny equation suggests a relationship between the soil permeability, k, and the geometrical properties of a porous medium:
 | [7] |
where n is the soil porosity, 
, the tortuosity, S, the surface exposed to the fluid per unit volume, and c the Kozeny constant, which is related to pore shape. The theoretical value of c fluctuates very little, ranging from 0.5 for a circle to 0.66 for a strip (Scheidegger, 1960), although Carman (1937) found that the best agreement with experiments was obtained for c = 0.2. Let us adopt the simple model in which a porous medium is represented as an ensemble of capillaries: j equivalent capillaries of length l, all having the mean radius of the ensemble, rm. Consider an air-dried soil sample of thickness x, surface area A, total mass M, and total volume xA. The bulk density, 
, of the sample is = M/(xA). The tortuosity, , is defined as = l/x, and the surface per unit volume exposed to the fluid, S, is S = j2
rml/(xA). The soil sample is compacted to a new thickness, xc, and bulk density c = M/(xcA). This process is characterized by a reduction of rm to rmc (Or et al., 2000, Assouline, 2006). We assume that this process is also accompanied by an elongation of the equivalent length, lc, so that l/lc = rmc/rm. Consequently, the effect of the increase in bulk density on and S can be expressed as follows:
 | [8] |
 | [9] |
Applying the Kozeny equation (Eq. [7]) for k and kc leads to a definition of the ratio between the values of the soil saturated hydraulic conductivity in the compacted and initial states, which is the first expression for Ksc/Ks suggested in this study:
 | [10] |
The second suggested expression is based on the model of Assouline (2001) for the soil HCF (Eq. [6]). Applying the same approach as in Mualem and Assouline (1989) leads to the following definition for Ksc:
 | [11] |
When the expression of Brooks and Corey (1964) is adopted for the WRCs of the initial and compacted soils, Ksc/Ks can be expressed by
 | [12] |
When the model of Assouline et al. (1998, 2000) for the WRC is used, Ksc/Ks is given by (Assouline and Tartakovsky, 2001)
 | [13] |
where 
and µ are WRC parameters, L is the capillary head corresponding to a very low water content, r, 
denotes the 
function, and the subscript c stands for the compacted state. Equation [12] and [13] are thus two additional suggested expressions for Ksc/Ks that use information from the WRC.
Unsaturated Hydraulic Conductivity Function
The model for the HCF of the compacted soil is based on the model of Assouline (2001) for the HCF (Eq. [6]), where the power was found to be related to the coefficient of variation, , of the WRC (Assouline, 2005):
 | [14] |
The hypothesis of this study is that the () relationship remains valid for the compacted state also. The application of the relationships between soil bulk density and the WRC parameters (Assouline, 2006) enables computation of ac, c, 
c, µc, and c. Therefore, the use of Eq. [2] and [4] for predicting Ksc, and Eq. [A1]
to [A3] for predicting Kc(Sec), is straightforward. Assuming that a relationship () exists and remains valid also during compaction, c can be estimated from c. This provides alternative ways to predict Ksc (Eq. [12]–[13]) and Kc(Sec) (Eq. [A4]–[A6]).
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METHODOLOGY AND DATA
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Experimental data from Laliberte et al. (1966), Reicovsky et al. (1980), and Smith and Woolhiser (1979) were used. Laliberte et al. (1966) measured the Ks and the HCF of three soils (Columbia sandy loam, Touchet silt loam, and an unconsolidated sand) at various bulk densities. They also measured the main drying WRC of these soils at various bulk densities that differed from those for which the hydraulic conductivity was measured. The different densities were obtained by vibrating the soil columns after packing them with air-dried soil. Soltrol C, a light hydrocarbon oil, was used as the wetting fluid to prevent swelling or clay dispersion effects; capillary pressure values hence were corrected to account for the difference between the density of the Soltrol and that of water. Reicovsky et al. (1980) measured the Ks and the main drying WRC of Barnes loam at bulk densities ranging from 0.99 to 1.59 g cm–3. The various degrees of compaction were obtained by means of a standard laboratory press with pistons that pressed moist soil samples slightly from both ends simultaneously. Smith and Woolhiser (1979) measured the hydraulic functions of Pouder fine sand in the laboratory for three different bulk densities.
For each of the soils in the data set, the lowest bulk density was considered to be , with the remaining c values representing the various degrees of compaction. The expressions relating the effective saturation degree, Se = ( – r)/(s – r) to , Se(), according to the models of Brooks and Corey (1964):
 | [15] |
and Assouline et al. (1998):
 | [16] |
were fitted to the measured data. Additional details can be found in Assouline (2006). For Eq. [15], the reported values of a, , and ac and c were used when available. For the respective values of s and sc (corresponding to the initial soil condition and the various degrees of compaction, respectively), we used the reported volumetric water contents at = 0. In all cases, the value of L was set equal to –1.5 MPa, while the values for r and rc were evaluated at = L. The WRC parameters in Eq. [16], and µ for the initial soil, and c and µc corresponding to the various degrees of compaction, were determined by best fit by means of an iterative nonlinear regression procedure based on the Levenberg–Marquardt method (Glantz and Slinker, 1990). The values of were computed by means of
 | [17] |
where
 | [18] |
and
 | [19] |
and where = µ and denotes the function. The values of c were computed by replacing and µ in Eq. [17]
to [19] by c and µc corresponding to the different degrees of compaction.
Similarly, for each soil and each bulk density, we fitted Eq. [6] to the measured RHC data, so that the parameter c corresponding to each c value could be determined.
Relevant soil properties, values of the parameters in Eq. [6], [15], and [16], the measured Ks, and the computed values of the coefficients of variation, and c are presented in Table 1. The mechanical composition of Barnes soil was reported by Young (1984).
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RESULTS AND DISCUSSION
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Models for the Saturated Hydraulic Conductivity
Model based on Kozeny's Equation (Eq. [10])
Adopting a capillary model, the radius of a pore, r, is inversely related to the capillary head, . Consequently, the PSD (pore size distribution) corresponding to a given WRC can be derived and expressed in terms of dSe(r)/dr. Therefore, the impact of compaction on the soil PSD can be estimated. This was done for Columbia sandy loam, Touchet silt loam, and the unconsolidated sand. Based on the predicted soil WRCs for the different levels of compaction resulting from the application of Eq. [16] (Assouline, 2006), the PSDs of these soils at their respective bulk densities are depicted in Fig. 1a
through 1c. Application of Eq. [16] to the WRC leads to a continuous Weibull-type PSD (Assouline et al., 1998), which is skewed to the right for the sand and to the left for the more fine-textured soils. Due to compaction, the relative fraction of smaller pores increases at the expense of the larger ones. Consequently, a reduction in the mean pore radius occurs, accompanied by a decrease in variance of the PSD, similar to the trends shown by Or et al. (2000) and by Startsev and McNabb (2001).
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Fig.1. Pore size distributions derived from the predicted water retention curve expressions according to Eq. [16] for uncompacted (dashed lines) and compacted (solid lines) soils at different bulk densities: (a) Columbia sandy loam, (b) Touchet silt loam, and (c) an unconsolidated sand.
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Using the predicted PSDs, it is possible to evaluate the relative change in the mean pore radius rmc/rm in Eq. [10], induced by an increase in bulk density. For all of the soil data we used, the relationship between rmc/rm and c/ is shown in Fig. 2
. The overall trend is well represented by the equation
 | [20] |
Inserting this result into Eq. [10] leads to the following relationship:
 | [21] |
When c/ is known, computing the ratio nc/n is straightforward. Therefore, the only unknown term in Eq. [21] is cc/c. By using the measured Ksc/Ks values from the data set (Table 1), it is possible to compute the relative change in the pore shape factor, cc/c, caused by an increase in . The relationship between the computed cc/c and c/ is depicted in Fig. 3
. It seems that a general equation of the type
 | [22] |
can represent the trend for the different soils quite well, with the value of 
being related to the soil texture. For loamy soils, varies between 2 and 4, and for the sand, varies between 4 and 6. The resulting general expression stemming from the model in Eq. [10] is thus
 | [23] |
Notice that for = 7, Eq. [23] reduces to Ksc/Ks = (nc/n)3, which resembles a relationship suggested by Ahuja et al. (1989). Similarly, for = 5, Ksc/Ks = (nc/n)3(c/)–2, which is an expression based on the Kozeny–Carman equation (Eq. [1]) and used by Or et al. (2000).
The measured Ksc/Ks values and those estimated according to the expressions suggested by Or et al. (2000) (Eq. [1]), Laliberte et al. (1966) (Eq. [2]), Mualem and Assouline (1989) (Eq. [4]), and the new model (Eq. [23]) are compared in Fig. 4
. It seems that Eq. [2] (Laliberte et al., 1966) and Eq. [23] with = 4 provide the best overall agreement with the measured data. The Kozeny–Carman-based equation (Eq. [1], Or et al., 2000) somewhat overestimates the Ks of this data set. This equation is equivalent to Eq. [23] with = 6, which holds better for coarse-textured soils. On the contrary, the model of Mualem and Assouline (1989, Eq. [4]) slightly underestimates Ksc/Ks for this data set, and is equivalent to Eq. [23] with = 2 (which holds more for the loamy soils).
Model based on Assouline's Expression for the Hydraulic Conductivity Function (Eq. [11])
The premise of this model is that a relationship between c and c exists. The method presented in Assouline (2006) to predict the WRC of a soil at a higher bulk density is used here to estimate the c and µc WRC parameters, and to compute the corresponding c values (Eq. [17]–[19]) for the bulk densities for which Ksc was measured (Table 1). The fitted c values for the various soils are plotted in Fig. 5
against the corresponding c values for the various c values, along with the (, ) points representing a wide range of uncompacted soil types (Assouline, 2005). Considering all (, ) and (c, c) points as one data set, an expression slightly different from Eq. [14] fits the data better:
 | [24] |
This relationship is depicted in Fig. 5, along with its 95% confidence interval and with Eq. [14]. All points (those for the uncompacted and the compacted soils used in this study) as well as the curve representing Eq. [14] are contained within the 95% confidence limits. Consequently, Eq. [24] can be considered as an expression of the () relationship that can be used also for soils at different bulk densities.
By using the estimated parameters ac, c, c and µc, and the c values resulting from Eq. [24], the ratio Ksc/Ks can be computed for each c of each soil according to Eq. [12] and [13]. A comparison between the predicted and the measured Ksc/Ks values for the various soils represented in the data set is shown in Fig. 6
, along with the curve representing Eq. [23] with = 4. This curve provides the best agreement with the data; however, the results from Eq. [12] and [13] also reproduce the measured data relatively well. While Eq. [13] slightly underestimates Ksc/Ks, Eq. [12] tends to overestimate the data. Equation [13] overall performs better than Eq. [12]. This result is a first validation of the hypothesis that () (Eq. [24]) remains appropriate for the c(c) relationship for compacted soils. Further validation is provided below that deals with the HCF.
Model for the Unsaturated Hydraulic Conductivity Function
A direct result of the proposed method to predict the effect of compaction on the WRC (Assouline, 2006) is the ability to predict the HCF of compacted soils by applying Mualem's (1976) model (Eq. [5]) (see Appendix, Eq. [A1]–[A3]). Here the ability of Eq. [6] to predict the HCF of compacted soils is investigated. The expression for the HCF that results from Eq. [6] depends on the model used to describe the WRC (see Appendix, Eq. [A4]–[A6]). In the following, the expressions given in Eq. [A4] and [A6] will be considered, along with the measured HCFs for different c values for Touchet silt loam, Columbia sandy loam, and unconsolidated sand (Laliberte et al., 1966). The K() functions of the different soils at the different levels of compaction are predicted by means of Eq. [23] for saturation ( = s), and by Eq. [A4] and [A6] with Eq. [24] for unsaturated conditions. The predictive capability of the different equations is presented in Fig. 7a
and 7b, where the predicted K() functions and the measured (K, ) values for different c values of the three soils are compared. The predicted K() curves according to Eq. [A6] are depicted in Fig. 7a while those resulting from Eq. [A4] are presented in Fig. 7b. Both expressions produce satisfactory results. Results are slightly better for Eq. [A6], especially for Touchet silt loam (Fig. 7a); however, overall, the predicted functions reproduce the measured data well, while the effect of bulk density on K() is expressed well by the suggested models across the entire range of values, including s, for all three soil types. This is an additional validation of the hypothesis that () (Eq. [24]) can be extended to compacted soils.
The relative impact of compaction on the HCF is in accordance with the corresponding effect on the PSD (Fig. 1a–1c). Soil aggregates during compaction move, deform, or collapse as a result of the applied stress. These processes affect mostly the larger pores by reducing their relative proportion in the PSD (Fig. 1a–1c). Consequently, Ks and the K values close to s are the most affected by increases in . This is more pronounced for coarse-textured (sandy) soils, which usually contain much larger pores. The effects of compaction on the HCF of these soils can be modeled solely in terms of the effect on Ks. For soils containing fine particles, the relative effect of compaction on the smaller pores is still noticeable; however, this effect decreases with the increase in the initial bulk density.
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SUMMARY AND CONCLUSIONS
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An increase in bulk density is an indicator of soil compaction, which can substantially affect the soil hydraulic properties. The bulk density change due to compaction is a variable that integrates information about the total change in the volume of voids of a soil; however, the bulk density cannot account for changes in the volume distribution of voids, the tortuosity, or the connectivity that may happened during compaction, especially during elastic deformation, which can significantly impact the soil hydraulic properties (Lenhard, 1986). But bulk density is a soil physical parameter that is easily measurable, and is generally available as basic information in most agricultural or hydrological systems. In a previous study (Assouline, 2006), we showed that changes in the soil bulk density can account for the major effect of compaction on the WRC, and that this variable can be used to predict the WRC of compacted soils. In this study, we show that the bulk density can be used to quantify and predict the main effect of compaction on the soil HCF.
Two approaches are suggested for predicting the saturated hydraulic conductivity of compacted soils, Ksc. The first approach leads to a general expression stemming from the Kozeny equation; this approach requires only information on the soil bulk density. The power parameter, , in this expression (Eq. [23]) appears to be related to soil texture. For the data set used in this study, = 4 provided good agreement between the computed and measured Ksc values. The second approach takes into account information contained in the WRC and exploits a close correlation between the power parameter, , and the coefficient of variation of the WRC, , which has been found to hold for a wide variety of uncompacted soil types as well as compacted soil data.
The applicability of the () relationship to compacted soils enables application of Assouline's (2001) model to also predict the unsaturated HCF of compacted soils. The performance of this model was good, with the predicted K() functions of three soils at various bulk densities reproducing the measured data relatively well.
Few published data exist on the HCFs of soils at different degrees of compaction; our validation of the suggested approach hence relied on a restricted data set; however, the results indicate that the approach is promising. Additional efforts should be invested in strengthening the theoretical basis of the approach and improving its performance.
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APPENDIX
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When the model of Mualem (1976) is used (Eq. [5]), the resulting closed-form analytical expressions for the HCF depend on the model used to describe the WRC. Using the model of Brooks and Corey (1964) for Sec(), the Kc(Sec) expression resulting from Eq. [5] is
 | [A1] |
where c denotes the pore-size distribution index for the compacted soil. When the two-parameter version of the van Genuchten (1980) model is chosen to represent Sec(), Kc(Sec) becomes
 | [A2] |
where 
c is the parameter corresponding to the WRC of the compacted soil.
Following Assouline and Tartakovsky (2001), application of the Se() model of Assouline et al. (1998, 2000) to Kc(Sec) leads to
 | [A3] |
where c and µc are parameters in the expression of the WRC for the compacted soil, ßc = (||–1 – |L|–1), and (u,v) and (u) denote the incomplete and the complete Gamma functions, respectively.
When the model of Assouline (2001, 2004) is used (i.e., Eq. [6]), the closed-form analytical expressions for the HCF,
 | [A4] |
 | [A5] |
and
 | [A6] |
result from the use of the WRC expressions of Brooks and Corey (1964), van Genuchten (1980), and Assouline et al. (1998), respectively.
The parameters in Eq. [A1] through [A6] retain their previous definitions, with the subscript c denoting a compacted state.
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ACKNOWLEDGMENTS
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Thanks are due to L. Abramovitz for his technical assistance. I thank also Robert Lenhard for his constructive comments and Rien van Genuchten for improving the writing style. This research was supported by Research Grant no. US-3393-03 R from BARD, the United States–Israel Binational Agricultural Research and Development Fund; this support is gratefully acknowledged.
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ABSTRACT
INTRODUCTION
