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Published online 24 August 2006
Published in Vadose Zone J 5:1005-1016 (2006)
DOI: 10.2136/vzj2005.0128
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

Scaling Approach to Deduce Field Unsaturated Hydraulic Properties and Behavior from Laboratory Measurements on Small Cores

A. Basilea, A. Coppolab,*, R. De Mascellisa and L. Randazzoc

a Institute for Mediterranean Agricultural and Forestry systems (ISAFOM), National Research Council (CNR), Ercolano (NA), Italy
b Department for Agro-Forestry Systems Management (DITEC), University of Basilicata, Potenza, Italy
c Department of Agricultural Engineering and Agronomy (DIAAT), University of Naples "Federico II", Portici (NA), Italy
* Corresponding author (acoppola{at}unibas.it)

Received 31 October 2005.



   ABSTRACT
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Hydraulic properties should be determined at the scale of the process modeled. The methods to hydraulically characterize a soil in situ remain extremely difficult to implement, requiring measurements of water content and pressure head with adequate time–depth resolution. We recently proposed a method that reduced the number of field measurements required for complete field hydraulic characterization. In this paper, we extend the previous method to cases where the only available information consists of laboratory hydraulic properties, along with a measurement of the maximum water content in the field, whose value depends on the way the wetting of the porous medium is performed. Specifically, the field hydraulic parameters were estimated with a scaling procedure accounting for the ratio between the total porosity estimated by the maximum water content in the laboratory and the partial porosity effectively involved in the field, as estimated by the field-measured maximum water content. The scaling method was evaluated with data from four soils (three volcanic sandy loam soils and one silty clay loam soil). Soil hydraulic properties were measured both in situ by a field internal drainage test and in the laboratory in an evaporation experiment (Wind's method). Scaling-based hydraulic properties were also compared with those estimated by applying a simplified method based on the unit-gradient water flow assumption where only water content measurements performed during the internal drainage test were of concern. The hydraulic properties estimated with the scaling method were compared with the measured ones and with those from the unit-gradient method in terms of the Relative Mean Error (RME) and Relative Root Square Mean Error (RRMSE). The scaling method proved to be especially effective when applied to the three sandy loam soils, where scaled retention and hydraulic conductivity curves to a large extent reproduced those measured in the field. For the silty clay loam soil, appropriate results were observed only for the water retention curve, while poorer scaled hydraulic conductivity was obtained.

Key Words: REV • representative elementary volume


   INTRODUCTION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
EFFICIENT MANAGEMENT of soil and water necessitates characterization of soil hydraulic behavior across large areas. This requires that soil water–pressure head, {theta}(h), and hydraulic conductivity–water content, k({theta}), relationships be specified at the scale of concern.

Laboratory measurements are more straightforward than those in the field, but their validity is essentially related to the sample sizes, which have to appropriately represent the heterogeneity of the medium being studied. A large body of literature discusses the validity of laboratory hydraulic characterization in inferring field hydrological behavior (van Genuchten et al., 1999).

The most rigorous field method to determine k({theta}) and {theta}(h) is the instantaneous profile method (Watson, 1966). Indeed, at the field scale these functions are generally obtained by monitoring a redistribution process along a profile of limited dimensions in space (pedon scale) through direct measurements of both soil water content and soil water pressure head profiles.

This technique is cumbersome and time-consuming. Therefore, two main simplifications have been largely applied: (i) considering approximations about the flow regimes during redistribution or (ii) drawing field hydraulic variables partly from lab measurements.

The first condition has produced several approximate methods, mainly based on the assumption of unit hydraulic head gradients, that are useful to characterize hydraulic relationships by measuring only water contents during redistribution (Libardi et al., 1980; Sisson et al., 1980; Chong et al., 1981). The method proposed by Libardi et al. (1980) is additionally based on the assumptions of a linear water content–storage relationship and an exponential k({theta}) relationship. Following the analysis provided by Flühler et al. (1976), Sisson and van Genuchten (1991) assumed the adequacy of unit-gradient water flow models given that the instantaneous profile method in its classical form would suffer from the uncertainties arising from noises, especially those observed in pressure head measurements. Actually, the malfunction of even one tensiometer introduces significant errors into the calculation of hydraulic properties. Their study showed that the data of the instantaneous profile method applications could be used in a Cauchy problem to find hydraulic parameters. However, though attractive, such simplified methods should be applied with caution because the founding assumptions were only rarely validated on critical field measurements. As stated by Kutilek and Nielsen (1994), "the accuracy of field-measured estimates of hydraulic properties will increase as the number of simplifying assumptions decreases."

In the second approach, the field method requires the direct measurement of either soil water content or soil water pressure head profiles and the indirect determination of the other variables through the separately lab-determined water retention curves. For example, Shouse et al. (1992a, 1992b) estimated the soil water contents in fields by coupling pressure head measured in the field and laboratory {theta}(h). Ahuja and Williams (1991) and Rockhold et al. (1996) estimated {theta}(h) in the field from measurements on soil cores. In the latter methods complications arise because in situ determined hydraulic properties often disagree with those determined on undisturbed samples collected at the same site, in a way that field hydraulic properties cannot be straightforwardly drawn from laboratory ones. As shown in Fig. 10 of the paper by Pachepsky et al. (2004), if one considers the UNSODA database (Leij et al., 1996), at the same pressure heads, field water contents are almost always markedly lower than laboratory values. Actually, what is generally observed is that the field retention curves are significantly lower than laboratory ones (Basile et al., 2003). Basile et al. (2003) also argued that the laboratory desorption curves represent the maximum values that may occur in the field.

Laboratory hydraulic conductivities at h = 0, k0, are generally observed to be different from those measured in the field. It has been amply shown (Eching et al., 1994; Rockhold et al., 1996; Basile, 2004, unpublished data, among others) that the laboratory determined k0 can be one or more orders of magnitude greater than those measured in the field. Consequently, erroneous estimates of field absolute hydraulic conductivities can arise when using the lab k0 to match field relative hydraulic conductivities.

At present, there are still great difficulties in providing quantitative explanations and formalization of differences in laboratory vs. field hydraulic properties. Recent efforts (Basile et al., 2003) aimed to propose criteria to systematically manage the observed dissimilarities between field and laboratory behavior in a way that laboratory information could be converted to the field.

The main practical purpose for which the formulation was developed was to reduce the number of field measurements required for a complete field hydraulic characterization.

The objective of this study was to formulate a scaling approach that allows the method proposed by Basile et al. (2003) to be extended to cases when only laboratory hydraulic properties are available. This objective is crucial since field-scale water flow predictions are frequently made with only laboratory or small-scale data available.

The appropriateness of the proposed approach was evaluated by comparing the scaling-based predictions of hydraulic properties with those from the classical analysis of the instantaneous profile method (Watson, 1966) and those from the unit-gradient simplification proposed by Sisson and van Genuchten (1991). A numerical analysis was also provided to establish the applicability conditions of the method depending on the ratio between field/lab {theta}0 values, as well as on the ratio between field or lab {alpha} values and the laboratory n parameter.


   THEORY
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 THEORY
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
In their study, Basile et al. (2003) determined the hydraulic properties of the soils at the pedon and laboratory scales. Soil water retention was described by the unimodal {theta}(h) relationship proposed by van Genuchten (1980) and expressed here in terms of the effective saturation, Se, as follows:

Formula 1[1]
with Se = ({theta} – {theta}r)/({theta}0 – {theta}r), {theta}r and {theta}0 being the residual water content and the water content at h = 0, respectively, and in which {alpha} (cm–1), n, and m are curve-fitting parameters.

Mualem's expression was used to calculate relative hydraulic conductivity, kr (Mualem, 1976). Assuming m = 1 – 1/n, van Genuchten (1980) obtained a closed-form analytical solution to predict kr at a specified volumetric water content:

Formula 2[2]
in which k0 is the hydraulic conductivity measured at {theta}0, and {tau} is a parameter which accounts for the dependence of the tortuosity on the water content. In this study {tau} was fixed at a value of 0.5.

For all the horizons a discrepancy was found between the water content values at h = 0 obtained with the two methods, with a tendency for the water retention values to be higher for the laboratory curves. The two-parameter vectors differed only in parameter {theta}0 and {alpha}. The latter was higher for the laboratory, as expected given that the larger pores, nearly all filled with water in the laboratory tests, empty at relatively lower pressure head values, h. Once the larger pores have been emptied, at a value of h approximately equal to 1/{alpha}, the fraction of pores that empties in response to a given variation in the pressure head h tends to become the same both for the field and laboratory samples, as indicated by comparable values of parameter n that were observed for the two curves.

The authors ascribed such behavior to the nearly complete saturation (i.e., a higher fraction of air removed during wetting) that could be achieved in the laboratory and interpreted such differences as a result of different hysteretic paths being observed, arising in turn from different wetting procedures. This hypothesis was reinforced by observations of nonhysteretic k({theta}) curves when field and laboratory measurements were compared, which, in the van Genuchten–Mualem framework, may only be expected when similar n values are also obtained (see Eq. [2]). In order that hydraulic conductivity curves, k({theta}), almost overlapping, may be derived from kr({theta}) curves arising from different lab–field retention curves, the matching factors for relative conductivity in the field and in the laboratory have to be different. They proposed an expression (Eq. [7] in Basile et al., 2003) to derive the field k0 from the lab one, which for applications can be put in the following form, accounting for the differences in the whole range of the field hydraulic conductivity curve:

Formula 3[3]
where Fkr is derived from laboratory n and by parameters {theta}0 and {alpha} determined in the field in the early phases of the field test. In practice, Eq. [3] permits scaling of the Lk0 values by the ratio between the integrals of the kr functions. In Eq. [3] and in the following, labels L and F are used to signify laboratory and field, respectively.

In this study the field hydraulic properties are deduced from the laboratory ones and from parameter {theta}0 determined in a field test. If the theoretical framework holds, a simple scaling procedure of the laboratory retention curve can be applied to derive the field retention curve. With the {theta}(h) parameters, the k({theta}) field curve is estimated through Mualem's model by using a k0 matching factor, lower than the lab one, obtained as described in Eq. [3]. A schematic view of the proposed scaling procedure is given in Fig. 1 , the formalization of which is given in Eq. [4]:

Formula 4[4]
where hc is a critical pressure head taken on the lab water retention curve and corresponding to the {theta}0 on the field curve, while Se is the effective saturation.


Figure 1
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  		Fig.1. Schematic of the scaling procedure: (A) laboratory (dashed line) and field (solid line) measured water retention curves, (B) effective saturation (Se) of the measured laboratory curve; (C) laboratory-measured (dashed line) and field-scaled (bold solid line) probability density functions (PDFs), (D) laboratory- (dashed line) and field-measured (solid line) water retention curves, and field-scaled retention curve (bold solid line). The critical pressure head, hc, is the value on the laboratory curves corresponding to F{theta}0.

 
The rationale of the proposed procedure relies on the following considerations:
  1. The porous media being compared are the same, a condition in principle applying each time the lab sample is actually a representative elementary volume (REV) for the pedon scale.
  2. Due to the wetting procedure, in the field the porous system is only partially filled with water at h = 0.
  3. The part of the porous system not involved in the field wetting is in the range of larger pores.

Figure 1A shows a typical case of lab–field differences in the retention curve. The Se(h) curve in Fig. 1B corresponds to the {theta}(h) curve in the laboratory (Fig. 1A, dashed line). We recall that the actual saturation, Se, is considered a cumulative distribution function of pore size with a density function f(h) that may be expressed by the following equation (Durner, 1994):

Formula 5[5]

The f(h) curve corresponding to Fig. 1B is depicted in Fig. 1C (dashed line). Assumptions 1, 2, and 3 lead one to consider that the fraction of the total porous system (as detected in the laboratory) actually involved in the field corresponds to the shaded side of the lab pore size distribution in Fig. 1C. Through Eq. [4], the shaded part of the lab distribution transforms to the estimated field distribution (solid line in Fig. 1C) to give a FSe(h) = 1 at a water content F{theta}0 and at h = 0. In practice, this corresponds to shifting the part of the lab curve from LSe(F{theta}0) to LSe = 0 to the left along the h axis of h – hc and upward along the Se axis of LSe(h)/LSe(hc). Finally, the scaled Se(h) converts to the field (lab-scaled) {theta}(h) curve in Fig. 1D (dashed line).

This scaling formulation is something like the approach of the most commonly used hysteresis scaling procedures (Kool and Parker, 1987; Luckner et al., 1989).

Once scaled FSe(h) are estimated according to Eq. [4], the relevant value of {alpha} can be determined by an optimization procedure.


   MATERIALS AND METHODS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
The study concerned four soil profiles whose main soil physical and textural characteristics are reported in Table 1. The soils are located in alluvial plains (Piano Campano and Piana del Volturno, Campania region, Southern Italy). Three of these (site 2, site 3 and site 15) are sandy loam soils including volcanic sediments and ash fall. These volcanic soils are very fertile and intensively cultivated. The VIT soil, located in the Volturno plain, is a silty clay loam soil with vertic characteristics.


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  		Table 1. Physical characteristics and texture (USDA) of the examined horizons.

 
Field Method
The {theta}(h) and k({theta}) relationships were determined in the field by the instantaneous profile method (Watson, 1966). A schematic view of the method is shown in Fig. 2 .


Figure 2
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  		Fig. 2. Schematic of instantaneous profile method for the (a) infiltration and (b) redistribution stages.

 
A plot of 3 by 3 m was bounded by a 30-cm-high board placed 15 cm above and 15 cm below the soil surface. Five neutron probe access tubes and tensiometers were installed for monitoring depth and time evolutions of volumetric water content and pressure head, respectively. Neutron probe measurements were taken each 10 cm starting from a depth of 20 cm up to 130 cm for all sites. The pressure head values were recorded with a portable needle transducer for Site 15 and mercury manometer for Sites 2, 3 and VIT. The tensiometer cups were placed at different depths for each site, ensuring that two tensiometers were positioned in each horizon. The tensiometers were installed at 15, 35, 50, 70, 90, and 110 cm for Site 2; at 15, 35, 50, and 70 cm for Site 3; at 15, 25, 35, 55, 80, and 100 cm for Site 15; and at 15 and 25 cm for Site VIT. The plot was flooded and a head of 10 cm was kept constant for several days until the pressure head measured at the deepest tensiometer was zero and the soil water content measured at the same depth did not change with time (Fig. 2a). Once the water supply was stopped and evaporation was prevented from the soil surface by covering it with a plastic sheet, the drainage process took place, monitored through time by systematic pressure heads and volumetric water content measurements that began as soon as the ponded water infiltrated the surface (Fig. 2b). The measurements were stopped when water content change with time was too small to detect. The frequency of the measurements was higher during the first 24 h (3 measurements) and was then progressively reduced as the drainage slowed. Measurements were interrupted when the transient flow was so slow that further measurements did not offer any information on it and were therefore redundant.

The water retention curve for each horizon was obtained by coupling the measurements of {theta} and h performed at the same depth and time. The unsaturated hydraulic conductivity curve was obtained by the algorithm suggested by Watson (1966), hereafter summarized following Kutilek and Nielsen (1994).

Water content {theta}(z, t) measurements allow calculation of the water stored Wt at time t in the profile between the soil surface and depth z:

Formula 6[6]

The average flux density during a time interval {Delta}t = t2 – t1 is Formula 6(t) = –{Delta}W/{Delta}t, {Delta}W being the water loss from the soil layer (0, z). Substituting the average flux density in the finite difference form of the Darcy equation, Formula 6 = –k(Formula 6){Delta}H/{Delta}z, yields

Formula 7[7]
where the gradients of the total potential head, H (=h + z), are calculated from h(z,t) measurements, and Formula 7 is related to the mean Formula 7 of {Delta}h in {Delta}H.

Sisson et al. (1980) estimated hydraulic conductivity without the need for calculating hydraulic gradients and water flux densities. Assuming unit-gradient water flow conditions and a relatively homogeneous soil profile, the Richards equation reduces to

Formula 8[8]

For a initial condition of {theta}(z,0) = {theta}in(z), the solution of the Eq. [8], known as a Cauchy problem (Lax, 1972), is

Formula 9[9]
that is, the ratio of the depth to time at which water contents are measured. When the profile is saturated initially, {theta}in equals the water content at saturation.

Sisson and van Genuchten (1991) estimated the parameters in Eq. [2] by equating the experimental values of the derivative of the k({theta}) (Eq. [9]) to the derivative of the Eq. [2]:

Formula 10[10]
with A = 1 – Se1/m, in a parameter optimization procedure. According to the approach proposed by Sisson and van Genuchten (1991), the parameter {alpha} in Eq. [1] (and thus the retention curve) is also estimated by using an experimental {theta}(h) point that, in our study, was taken to be h = –60 cm.

Laboratory Method
In the laboratory, the {theta}(h) and k({theta}) relationships were determined by means of an evaporation experiment (Wind, 1968) according to the calculus procedure proposed by Tamari et al. (1993). Soil samples ({phi} = 86 mm, h = 150 mm) were collected in duplicate at the end of the field experiment in the middle of each horizon (Fig. 2b). In the laboratory, the samples, after preliminary air drying, were saturated from the bottom and the saturated hydraulic conductivity measured by a permeameter (Reynolds and Elrick, 2002). Then, the sample drying process was conducted by starting from hydrostatic equilibrium conditions, setting a zero flux at the bottom surface, and at the same time, at appropriately preset time intervals, monitoring the weight of the whole sample and the pressure heads at two different depths by means of tensiometers. The equipment and data acquisition were completely automated. Measurements were interrupted when air entered the upper tensiometer. In relation to the texture of the different samples, each test lasted between 3 and 5 d.

An iterative procedure was applied for estimating the water retention curve. We started with a "guess" water retention curve, with initial estimates of parameters in Eq. [1], to convert pressure head data at any one time into estimated water contents at the tensiometer depths., The average water content of the soil sample was then calculated and compared with the average measured water content obtained from the sample weight. If significant differences between calculated and measured average water content were observed, a new parameter set was estimated. The water retention curve was optimized using an iterative procedure, by minimizing the squared differences between calculated and measured average water content. After convergence of the algorithm, water contents at the tensiometer depths were calculated from the measured pressure heads and the estimated water retention curve. The distributions of the measured pressure heads, h(z, t), and of the estimated water contents, {theta}(z, t), were used to calculate the hydraulic conductivity according to the instantaneous profile method previously described.

Moreover, three points—at 3, 6, and 12 bars for Site 2 and Site 15—and four points—at 2, 4, 12, and 15 bars for Site VIT—of the water retention curve were determined by means of the pressure plate apparatus.


   RESULTS AND DISCUSSION
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
In this section, details will be given on (i) comparisons between field and laboratory measurements of hydraulic properties, to evaluate the scaling framework; (ii) comparison between field measured hydraulic properties and those deduced by applying the scaling procedure. The goodness of the scaling procedure will be evaluated by comparing relevant estimates with those derived by the unit-gradient method; and (iii) numerical analysis of the applicability conditions of the scaling procedure in terms of van Genuchten model parameters, to quantify the extent of the ratios between field and laboratory measured hydraulic parameters for the scaling procedure to be expected to be valid.

Hereafter, labels WR and HC will be used to signify water retention and hydraulic conductivity, respectively, with the superscripts F for field curves and L for laboratory curves. Moreover, subscripts m, s, and ug will be used to identify measured, scaled, and unit-gradient curves and parameters, respectively.

Comparing Measured Field and Laboratory Hydraulic Properties
The water retention values from field (circles) and laboratory (triangles) experiments for the Ap horizons of the investigated sites are plotted in Fig. 3 . They were interpolated by applying the Eq. [1] with m = 1 – 1/n. Relevant parameters are shown in Table 2, along with parameters pertaining to the other horizons not shown herein. The graph shows that for all horizons there is a discrepancy between the water content values at h = 0 obtained with the two methods, with water retention values being higher for the laboratory curves. For lower pressure head values the curves tend to converge progressively. The gradual convergence of the laboratory and field retention curves past the inflection point is always reflected in similar values of parameter n. Analogous considerations can be made for the other horizons on the basis of the value of the parameters shown in Table 2.


Figure 3
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  		Fig. 3. Soil water retention curves of the Ap horizons. Triangles and circles refer to laboratory and field experimental data, respectively; lines are for relevant curve-fittings by Eq. [1].

 

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  		Table 2. Parameters of the water retention curve (Eq. [1]). The field parameters were obtained by fixing n at the laboratory value, {theta}0 at the measured value in the field and setting m = 1 – 1/n. Scaled and unit gradient parameters were obtained according to the procedures described in the text.

 
The experimental values k({theta}), both for the field and laboratory, are shown in Fig. 4 . Despite the differences observed in the retention curves at high h values, field and laboratory measurements of hydraulic conductivity differ less than one order of magnitude. Note also the adequate description of lab k({theta}) curves when Mualem's model is applied to estimate laboratory hydraulic conductivities. Moreover, the curves seem to be similar in shape for Sites 2, 3, and 15, while for Site VIT, different slopes were observed.


Figure 4
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  		Fig. 4. Hydraulic conductivity curves of the Ap horizons. Triangles and circles refer to laboratory and field experimental data, respectively; dashed lines refer to the application of Mualem's model (Eq. [2]) to laboratory water retention data.

 
As discussed above, comparable n and overlapping of the k({theta}) curves become fully justifiable when the lab–field retention curves are assumed to be two drying curves of the same porous medium. Overall, it can be argued that the two retention curves respond to different conditions at the beginning of the experiment. The different wetting methods result in a different initial saturation of the porous medium. Consequently, drying then occurs along a curve, which may be likened to the main drying curve in the laboratory and to a secondary drying curve in the field (for a comprehensive discussion of the issue see Basile et al., 2003).

Comparison of Measured, Scaled and Unit-Gradient Hydraulic Properties
Figure 5 shows the results of applying the scaling procedure proposed (Eq. [4]) to derive the field retention curves from the lab ones. In the graph the scaled field retention curves (FWRs, solid lines) are compared with the field-measured curves (FWRm, circles). The relevant parameters are given in Table 2. Note the satisfactory, sometime very precise, overlapping of the reconstructed curves on the retention curves actually measured in the field, even for site VIT.


Figure 5
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  		Fig. 5. Field soil water retention curves of the Ap horizons. Solid lines and circles refer to scaled and measured field water retention curve, respectively. Dashed lines refer to the application of the unit-gradient method.

 
The graphs in Fig. 6 illustrate, for the horizons shown herein, the experimental field k({theta}) points (circles), along with the relevant curves estimated by the Mualem model. The latter curves were deduced by applying Eq. [2] to the laboratory retention curve scaled by Eq. [4] and by matching it to the Fk0 estimated by Eq. [3]. To also have information on the goodness of the k0 predictions obtainable from the scaled field retention curve, the Fk0 value was estimated by considering at the denominator of Eq. [3] the kr({theta}) curve predicted either from the measured field retention curve (FHCs(1) solid lines in Fig. 6) or the field retention curve obtained from the scaling procedure (FHCs(2), dashed lines in Fig. 6). The corresponding Fk0 (Fk0s(1) or Fk0s(2)) are given in Table 2. Note the very satisfactory predictions for all the horizons examined when scaled retention curves are used. Obviously, the small discrepancies observed between the two curves have to be attributed to the different {alpha} characterizing the two retention curves, which, in turn, result in two slightly different kr({theta}) curves.


Figure 6
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  		Fig. 6. Field hydraulic conductivity curves of the Ap horizons. Circles refer to field measured data. Solid (FHCs(1)) and dashed (FHCs(2)) lines refer to the scaling method with kr({theta}) determined from measured and scaled water retention curves, respectively. Dotted lines refer to the application of the unit-gradient method.

 
It is also important to stress that the Fk0 values estimated by Eq. [3] and reported in Table 2 are always less than Lk0 since the ratio in Eq. [3] is always less than unity because the Fkr curve always lies above its laboratory counterpart. This, in turn, provides the physical basis for explaining the frequently observed lab k0 higher than the field one (Eching et al., 1994; Rockhold et al., 1996; Basile, 2004, unpublished data) and a procedure to manage these differences.

These results seem to confirm that the hysteresis scheme hypothesized holds for explaining differences between lab and field measurements, as only in this case does the schematization leading to Eq. [3] and [4] hold. This also implies that the laboratory samples are actually REVs for the corresponding horizons at the pedon scale. In fact, the main condition for which the scaling procedure we proposed held is that we are dealing with the same porous medium.

In the same graphs (Fig. 5 and 6), the unit-gradient water retention and hydraulic conductivity curves are also shown (FWRug in Fig. 5 and FHCug in Fig. 6). The relevant parameters are given in Table 2.

Although the Sisson et al. (1980) unit-gradient approach holds strictly only for uniform profiles (Sites 3 and VIT), it was applied (less rigorously) also to situations where the profile exhibited some weak layering (Sites 2 and 15).

What should be noted is that the scaling procedure always yielded results comparable to those from the unit-gradient approach, although in practice calling only for laboratory measurements. For Site 15, the FWRug behavior at lower h has to be attributed to the lack of information in the dry range during the field drainage experiment.

Difference-Based Statistics
By reasoning from analogy, similar results can be drawn by observing the difference-based statistics provided in Table 3, both for the water retention and hydraulic conductivity, for all the horizons investigated.


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  		Table 3. Relative mean error and relative root square mean error indexes.

 
The applied indexes, namely the relative mean error, RME, (Williams et al., 1992) and the relative root mean square error, RRMSE, (Jørgensen et al., 1986) are:

Formula 11[11]

Formula 12[12]
where N is the number of data points, Ei and Mi are the estimated and measured water contents or conductivities, respectively, and Formula 12 is the average value of the measured data. Depending on the goal of the comparison and on the curves to be compared, the values and meanings of the estimated (Ei) and the measured (Mi) water contents will change according to the scheme reported in Table 3. The Ei values are calculated from the relevant parameters at the same values of pressure head and water content of the Mi, for the water retention and hydraulic conductivity curve, respectively.

Relative mean error percentages reflect the average deviation between estimated and measured hydraulic data and were calculated to give information on the relative size of systematic errors as their positive (negative) values indicate the percentage of overestimation (underestimation) on the average.

The RRMSE is used to describe accuracy encompassing both random and systematic errors. It can be especially useful to provide information about the relative size of the differences between estimated and measured water retentions and hydraulic conductivities by normalizing the differences by the mean of measurements.

Every RME index of the LWRm/FWRm comparison is positive, confirming for all the horizons investigated that the laboratory soil water retention curves are positioned above the field ones according to the framework reported. The other comparisons by means of the RME index for the water retentions do not show any trend. The scaled method for the hydraulic conductivity, FHCs seems to be site-dependent, overestimating at Sites 2 and 3 and underestimating at Site 15. Instead, the unit-gradient method, FHCug, underestimates for all soils except for the Ap horizon of Site 2.

Figure 7 illustrates graphs of the RRMSE for the LWRm–FWRm, FWRs–FWRm, FWRug–FWRm for water retention series, and for the LHCm–FHCm, FHCs(2)–FHCm, FHCug–FHCm for hydraulic conductivity series.


Figure 7
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  		Fig. 7. Relative Root Square Mean Error (RRMSE) indexes for water retention and hydraulic conductivity.

 
First, concerning Sites 2, 3, 15 (Site VIT will be discussed separately), the LHCm–FHCm comparison provides the most information on the applicability of the scaling schematization. In fact, for all the horizons of Sites 2, 3, and 15, despite the discrepancies between water retention curves [i.e., differences in {theta}(h) curves in terms of RRMSE ranging from 4.2 to 26.0%], relatively narrow (RRMSE ranging from 1.9 to 15.0%) mean differences between lab and field k({theta}) curves were observed. Such differences are largely consistent with the theoretical representation leading to Eq. [3], especially considering the variability that is generally found for hydraulic conductivity (CV > 50%) (Warrick, 1998).

As for the Fs–Fm series, what should first be noted is that the scaling procedure provides estimated water retention curves differing by <10% from the measured ones (slightly higher for horizons Ap and 2Ab of Site 15) and frequently improved when compared to the unit-gradient method (see Fug–Fm indexes) (Site 2 A, Site 2 Bw, Site 3 Bw, Site 15 A, Site 15 2Ab). Concerning the corresponding hydraulic conductivity estimations, differences are frequently very low, with values higher than 10% only for Site 3. Also considering the hydraulic conductivity predictions, the Fs series proved to be in a number of cases better than the Fug series (Site 2 Ap, Site 2 A, Site 15 A, Site 15 2Ab). However, one should note that enhanced water retention predictions do not directly mean better predictions of hydraulic conductivities.

Overall, the scaling procedure provides results comparable to those from the unit-gradient method, with the difference that the former just requires measurement in the field of the water content at h = 0, while the latter calls for field water content profile measurements over time and at least one {theta}(h) experimental point for estimating the relevant {alpha} (see Table 4 for the comparison of the main requirements of each applied method).


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  		Table 4. Main requirements and assumptions of the compared methods for estimating field hydraulic properties.

 
In any case, looking at the relatively small RRMSE values, both the scaling procedure and the unit-gradient method proved to be effective in reproducing field hydraulic conductivities.

Both in the tables and in the graphs, difference-based statistics are also shown for the Ap horizon of Site VIT. It is worth noting that Site VIT is a structured silty-clay loam soil, with a larger REV for hydraulic properties (at least 103 cm3) than that required for well-sorted soils, especially for hydraulic conductivity (Kutilek and Nielsen, 1994). For such a soil, the samples to be analyzed in the laboratory, given their sizes, were not expected to represent the REV of the soil horizon at the pedon scale. This is the classic case where the schematization we proposed does not apply because the responses observed in the lab and in the field refer to two different porous media, each with its own hydraulic properties. From the RRMSE index graphs, larger discrepancies between estimated and measured field hydraulic conductivities emerged (see also Fig. 6),although with acceptable water retention curves (see Fig. 5).

It could also be argued that acceptable water retentions and poor hydraulic conductivity estimations are largely to be attributed to the sample size we used, which would suffice as a representative volume for water retention but would be inadequate as a REV for hydraulic conductivity.

However, a better estimation of hydraulic conductivity was obtained by the unit-gradient method, since it is in any case based on water content measured directly in the field.

Numerical Analysis on the Applicability Conditions of the Scaling Procedure
Figures 8a , 8b, and 8c convey information on the applicability conditions of the proposed scaling procedure in terms of van Genuchten parameters. The graph shows trends of the F{alpha}/L{alpha} ratio as related to the F{theta}0/L{theta}0 ratio for different values of shape parameters L{alpha} and n and for a starting value of L{theta}0 = 0.5, although it should be pointed out that the plots are independent of the starting value of L{theta}0. Each line of any plot provides the value of F{alpha}/L{alpha} for a given F{theta}0/L{theta}0 ratio and n value so that the scaling procedure holds. In a sense, it might be said that when the F{alpha}/L{alpha} ratio adjusts according to the plots as those in Fig. 5, the selected lab sample size may well represent the REV for horizon at the pedon scale.


Figure 8
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  		Fig. 8. Trends of F{alpha}/L{alpha} ratio as related to the F{theta}0/L{theta}0 ratio for different values of shape parameters L{alpha} and n and for a starting value of L{theta}0 = 0.5 (Plots a, b, c). Plot d is the same as Plot b with Fn = Ln x 1.1.

 
Importantly, the smaller the n value the smaller the F{alpha}/L{alpha} ratio for a given F{theta}0/L{theta}0. Moreover, this is more noticeable as the starting value of L{alpha} decreases. By contrast, the higher the n value the less sensitive the F{alpha}/L{alpha} ratio becomes to the F{theta}0/L{theta}0 ratio.

Additionally, to give an example of the errors one can make by applying the proposed scaling procedure when the hysteresis framework does not perfectly hold and, consequently, the Ln parameter does not represent the Fn, Fig. 8d describes analogous relationships for a case of Fn = Ln x 1.1. In other words, Fig. 8d provides the F{alpha}/L{alpha} ratios necessary for the laboratory and the scaled field retention curves to overlap, for the case of Fn values 10% higher than Ln. Differences between Fn–Ln might occur either due to the inability of the lab sample size to encompass the REV for the horizon or due to measurement errors, or might even be the result of the Ln being estimated by a pedotransfer function. Of course, this circumstance is only of speculative concern, as Fn should be known in order for such a graph to be of practical interest, a condition that the proposed scaling procedure aims above all to overcome. As an example, while applying the scaling procedure we proposed, for Ln = 1.3 and for a F{theta}0/L{theta}0 ratio equal to 0.8, one would select an F{alpha}/L{alpha} ratio of 0.37 from Fig. 8b. By contrast, to predict the correct field retention curve for a real Fn 10% higher than the Ln, one should use a F{alpha}/L{alpha} ratio of 0.20 from Fig. 8d, thus with an error in estimating F{alpha} of about 100%. From the same graph in Fig. 7 one might easily note that the higher the Ln value, the smaller the error in estimating F{alpha}, thus suggesting that the robustness of the proposed scaling procedure increases with the Ln values. Analogous considerations can be made for k0 predictions (results not shown).


   CONCLUSIONS
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
The procedure we proposed, derived in the framework of a hysteretic interpretation of the differences between lab and field hydraulic properties, allows reproduction of both the field-measured retention and hydraulic conductivity with adequate approximation for many purposes. Despite the expected differences between the measured and scaling-based field hydraulic properties, the scaling procedure provided hydraulic property predictions whose accuracy was comparable to that obtained by the unit-gradient approximation, although there existed some cases where either the {theta}(h) or the k({theta}) curve was better described by the unit-gradient method. However, it should be noted that the unit-gradient method requires measurements over time of the soil water content along the profile while our scaling method needs just one soil water content measurement in the field at h = 0 and laboratory characterization. Moreover, at least in principle, there are no restrictions in the applicability of the proposed scaling method (i.e., a priori assumptions on the gradient or the type of process). The only constraint is that the soil sample has to be large enough to be the REV for the horizon at the pedon scale.

Within the scaling framework proposed, it is important to match the lab and field experimental hydraulic conductivity by different matching points, depending on the initial and boundary conditions applied. The field k0, which is lower than its lab counterpart, can be estimated from the Fk0 formula (Eq. [3]) derived in the theoretical framework. When this value is used to match Mualem's relative hydraulic conductivity estimated from the retention curve scaled according to the proposed procedure, field-estimated hydraulic conductivities roughly reproduce the measured ones.

The concept that different initial and boundary conditions result in different hysteresis curves being measured may also be extended given that, in practice, different measurement techniques may result in different initial and boundary conditions being applied, such that the hydraulic behavior we actually measure depends on the technique used, rather than being an intrinsic attribute of soils.

The proposed procedure provides a systematic technique for scaling large laboratory data sets to the field, either deriving from direct measurements or estimated by pedotransfer functions, which too frequently are incorrectly used for field applications.


   REFERENCES
TOP
 EXECUTIVE SUMMARY
 ABSTRACT
 INTRODUCTION
 THEORY
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 





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