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Dep. of Agricultural Engineering and Agronomy, Univ. of Naples Federico II, 80055 Portici (Naples), Italy
ancoppol{at}unina.it
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ABSTRACT |
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 TOP ABSTRACT INTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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s. Because of the predominant effect of the shape of the retention curve near saturation upon the shape of the whole hydraulic conductivity curve, predictions of hydraulic conductivity often differed from unsaturated conductivity observations by even two orders of magnitude. To the contrary, excellent descriptions of retention data were observed when superposition of two unimodal retention curves was adopted. The first function was either a van Genuchten (VGbim) or a simple one-parameter formulation introduced by Ross and Smettem (RSbim) for describing macroporosity, while the second was in both cases a van Genuchten formulation. The agreement was especially good for higher water content values, leading to values of the coefficient of determination, R2, very close to unity. The laboratory-measured unsaturated conductivity values compared more closely when bimodal approaches were used, and the predictions were frequently well within one order of magnitude of the measurements.
Abbreviations: RSbim, bimodal Ross and Smettem • VGbim, bimodal van Genuchten • VGfix, unimodal van Genuchten fixed s • VGopt, unimodal van Genuchten optimized s
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INTRODUCTION |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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or of the suction head h, as well as the water retention function, (h). Although both properties may be obtained by direct measurements, it is somewhat time-consuming to determine hydraulic conductivity. However, alternative theoretical approaches based on statistical distribution models of pore size allow reasonably accurate estimates of conductivity to be obtained through the use of more easily measured retention data (Mualem, 1986). For use in simulation models, hydraulic properties are often advantageously represented through analytical expressions in which there are parameters to be defined with reference to the soils in question (van Genuchten and Nielsen, 1985; Bruce and Luxmoore, 1986; Mualem, 1986; Santini et al., 1995). The equation proposed by van Genuchten (1980) for the water retention curve is widely adopted and generally combined with Mualem's expression (1976) for the estimation of hydraulic conductivity.
Many studies have discussed the validity of the van Genuchten–Mualem model on the basis of direct comparisons between estimated and measured values of hydraulic conductivity (Michiels et al., 1989; Stephens, 1992), as well as on the basis of theoretical analyses (Mualem, 1986; Alexander and Skaggs, 1986; Vogel and Cislerova, 1988; Sidiropoulos and Yannopoulos, 1988). The model has produced satisfactory results in soils with a unimodal pore-size distribution, normal or log-normal shaped.
Nevertheless, in well-aggregated natural soils the pore system is frequently partitioned into intraaggregate or textural pores and interaggregate or structural pores (Fies, 1992; Tamari, 1994), thus resulting in pore-size distributions that are often bimodal (Bruand and Prost, 1987; Smettem and Kirkby, 1990; Othmer et al., 1991; Durner, 1994; Zurmühl and Durner, 1998; Mohanty et al., 1997), with one maximum in the range of textural pores and another in the range of structural pores. In such soils the independent draining of the inter- and intraaggregate pores frequently results in a steep slope of the retention curve near saturation (Thony et al., 1991), which a single van Genuchten or any unimodal-type function does not reproduce adequately (Smettem and Kirkby, 1990; Othmer et al., 1991; Durner, 1994; Mallants et al., 1997).
Recently, some approaches have been proposed that view these soils as consisting of two pore systems, each of which is characterized by its own retention function (Othmer et al., 1991; Durner, 1992, 1994; Wilson et al., 1992; Ross and Smettem, 1993). The retention function of the whole porous medium has been described by linearly overlapping functions of the same form (e.g., Durner, 1992) or of different forms (e.g., Ross and Smettem, 1993). In the formulation assumed by Durner (1992), retention is described by summing two van Genuchten-type functions, one for the structural component and the other for the textural component of pore space. The larger number of parameters involved increases the probability that two or more are closely correlated. For describing macroporosity, Ross and Smettem (1993) introduced a simple one-parameter function. It results in an expression for retention with fewer parameters, which the authors tested by means of a simultaneous fit of both retention and unsaturated conductivity data, producing slightly different results from those obtained by using a function with two van Genuchten distributions.
The aim of this study was to evaluate the capacity of the flexible retention relations proposed by Durner (1992) and Ross and Smettem (1993) to describe the retention data of an aggregated soil and to estimate the corresponding hydraulic conductivity function with a predictive estimation procedure involving only water retention data and using conductivity at saturation, Ks, as a match point. The predictive capability of hydraulic conductivity is independently tested by using a broad set of unsaturated conductivity observations determined through the crust method (Booltink et al., 1991).
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Theory |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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(h) relationship proposed by van Genuchten (1980) is expressed here in terms of effective saturation Se as follows:
 | (1) |
 | (2) |
(cm-1), n, and m are curve-fitting parameters. In particular, -1 corresponds roughly with the so-called air-entry pressure value for low ratios of m/n, while for high m/n values it is roughly equal to the suction head at the inflection point of the curve (van Genuchten and Nielsen, 1985). The product nm determines the slope of the curve at high h values and may thus be seen as a parameter primarily affected by soil texture. s and r represent the water content at saturation and the residual water content, respectively, and may either be fixed or treated as parameters to be optimized.
The actual saturation, Se, is considered as a cumulative distribution function of pore size with a density function f(h), which may be expressed by the equation (Durner, 1994):
 | (3) |
Mualem's expression to calculate relative hydraulic conductivity, Kr, is based on the capillary bundle theory (Childs and Collis-George, 1950; Mualem, 1976) and may be represented as follows:
 | (4) |
is a parameter that accounts for the dependence of the tortuosity and the correlation factors on the water content, being estimated by Mualem to have an optimum average at 
0.5 for the generally disturbed soil samples he used.
Using Mualem's models and assuming
, van Genuchten (1980) obtained a closed-form analytical solution to Eq. [4] to predict Kr at a specified volumetric water content:
 | (5) |
However, restricting m and n sometimes limits the flexibility of Eq. [1] in describing retention data of several soils and supplies reasonably accurate estimates of unsaturated hydraulic properties for n values >1.25 (van Genuchten and Nielsen, 1985).
In natural soils, the presence of aggregates frequently results in a retention function curve having at least two points of inflection. To represent such behavior, a double porosity approach can be used that assumes that the pore space from r to s consists of two fi(h) distributions obtained by Eq. [3], each occupying a fraction 
i of that pore space.
The retention expression proposed by Durner (1992) thus assumes the following form:
 | (6) |
i is the weighting of the total pore space fraction to be attributed to the ith subcurve, and i, ni and mi still represent the fitting parameters for each of the partial curves.
For describing macroporosity, Ross and Smettem (1993) introduced the following simple one-parameter function:
 | (7) |
The consequent retention formula for aggregated soils may therefore be expressed as follows:
 | (8) |
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The parameters of the unimodal and bimodal retention functions were obtained by minimizing the following objective function:
 | (9) |
is the parameter vector, N is the number of (h) data measured, j and 
(hj, P) are the measured and estimated water contents at hj, respectively. Assuming all data values of equal quality, the weights, 
j, for the least squares optimization were set to unity (ordinary least squares). |
Materials and methods |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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600 cm3 (8.5 cm in diameter and 12.0 cm high). A total of 18 samples were collected and were labeled as indicated in Table 1.
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s gravimetrically and hydraulic conductivity Ks at saturation with the falling-head method (Klute and Dirksen, 1986). Measurements of unsaturated hydraulic conductivity were obtained on the same samples by the crust method (Bouma et al., 1983) modified by Booltink et al. (1991). The method determines hydraulic conductivity by measuring the water flux density in an unsaturated soil sample and is particularly suitable for obtaining conductivity values corresponding with low h values. Unsaturated conditions are obtained by supplying water through a crust with a lower hydraulic conductivity than the soil sample.
The soil samples brought to saturation were placed on a column of medium-textured sand with an inner diameter of 0.15 m. To monitor the suction head within the sample, at a height of 0.03 and 0.06 m from its base, two tensiometers were installed and connected to pressure transducers. A mixture of fine sand and quick-setting hydraulic cement, thoroughly mixed with water, was uniformly applied at the surface of the soil sample until, on drying, it reached a thickness of 1 cm with a higher hydraulic resistance than that of the soil being studied. Subsequently, a cover in perspex was applied at the top of the steel sampler and connected to a Mariotte feeding column. Immediately after the hardening of the crust, water was applied to the top of the crust. The Mariotte column regulated the constant head water supply to the inlet. Due to the higher hydraulic resistance of the crust, in the sample water fluxes formed in unsaturated conditions.
The flux density measured from the outflow discharge and the suction head h measured with the tensiometers allowed determination of hydraulic conductivity by Darcy's law. Theoretically, when steady-state flow is established and only gravitational flow is assumed, as is verified by measuring the flowing volumes and by reading the tensiometers, the flux density equals the unsaturated hydraulic conductivity K(h). Starting at near saturation, four unsaturated conductivity values were determined for each sample. Unlike the classic crust method, different unsaturated conditions were created through vertical shifts to the Mariotte feeding column rather than through the use of crusts with different conductivities (Booltink et al., 1991).
The (h) relationship for desorption was gravimetrically obtained using a silt-kaolin box apparatus on top of which the samples were placed for suction ranging from 0 to 2.0 m of water. The (h) relation was determined on the top 4 cm after reducing the sample so that equilibrium conditions could be achieved in reasonable times. Drainage from saturation took place in seven increments of h starting from zero: 0.02, 0.1, 0.3, 0.5, 0.7, 1.2, and 2.0 m of water. The suction was created by lowering the water level of the sand box apparatus for h values up to 0.7 m. For the 1.2- and 2.0-m suction values a vacuum system was used, with Mariotte towers to regulate the applied suction. The determination of a single desorption point took from 1 d to 1 wk, the equilibration time principally depending on both the variation of the imposed suction and the suction value itself. Subsequently, disturbed soil samples were taken from the core samples on which the water content at h values of 30, 60, and 120 m was determined by using a pressure membrane apparatus. Under the experimental conditions chosen for carrying out the experiments and within the range of measured suctions up to 0.3 m of water, any visible shrinkage cracks were not noticed. In the 0.5- to 2.0-m range a very weak and only slightly observable shrinkage occurred, with negligible detachment from the cylinder walls.
The bulk density 
was measured by oven drying the samples for 36 h at a temperature of 105°C. Bulk density values are given in Table 1.
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Results and discussion |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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. The conductivity function, Eq. [4], for both unimodal and bimodal (h) relations was calculated by means of a trapezoidal approximation of Eq. [4].
Retention Curves and Associated Porous Systems
The graphs in Fig. 1
, which refer to Samples a14 and a82 for Profile A and a60 and a52 for Profile B, report the retention data measured in the pressure range 0 < h < 200 cm, interpolated by using a unimodal representation of van Genuchten relation (h), with s optimized (VGopt) or fixed at the measured value (VGfix), or a bimodal representation of the same relation as proposed by Durner (VGbim), as well as a bimodal representation of the retention curve as proposed by Ross and Smettem (RSbim).
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, which, by contrast, works reasonably well for values of n > 1.25 (van Genuchten and Nielsen, 1985). In order to enhance the flexibility of the unimodal retention function, the parameter m was always allowed to vary, though still obtaining n values close to unity. In the optimization procedure, the condition 10 > n > 1, m > 0, and > 0 was also set. In VGopt and VGfix, r was optimized, though converging almost always toward zero. In VGbim and RSbim, the values of s and r were set at the value measured in the laboratory and at zero, respectively.
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As regards the unimodal approach, in both cases the singular behavior of the retention curve for smaller h values failed to be adequately described. In particular, when s was fixed, the unimodal approach, which assumes gradual variations in water content, ignored the jump that is shown in the retention data near saturation. Instead, parameter optimization produced an appreciable underestimation of retention at saturation and largely excluded the presence of larger pores, even though the description of retention data for higher h values was improved, as suggested by the higher values of the coefficient of determination, R2, in Table 2 for the case VGopt.
Concerning the bimodal approach, the larger number of parameters appearing in relations VGbim and RSbim, lending greater flexibility, allowed the detailed description of the behavior of curve (h) near saturation shown in Fig. 1. High values of parameter n1 of the function VGbim were observed, which gave a more satisfactory prediction of the slope of the retention curve near saturation. The same direction also appeared to be taken by the significantly high values of parameter 1, theoretically matched by a desaturation of larger pores in response to relatively small variations of h. In RSbim, the one-parameter relation also appeared suitable for representing the retention behavior observed at minimum h values, leading to values of the coefficient of determination R2 comparable with those obtained in VGbim.
Using Eq. [3] the composition of the porous system is represented graphically in Fig. 2
, which shows the distributions of pore size as a function of the logarithm of h for the samples reported in Fig. 1. The position of the distribution peak was defined by the value assumed by i, while the width of the distribution toward the fine and large pore sizes depended on the value of parameters ni and mi. As the value of in VGopt was on average lower than that in VGfix, there was a shift in the distribution peak. Clearly, since the value of s produced by optimization is always appreciably lower than that measured, in VGopt the existence of large pores is effectively ignored. The value assumed by 1 in VGbim and RSbim gave an indication of the different proportion of the interaggregate component of the pore system, which occurred in all the cases examined in a range of potential 0 < h < 0.1 m. The peak of the intraaggregate component of pore spaces fell in a range of h between 0.1 and 10 m. Intraaggregate pore density distributions appeared to substantially overlap, while interaggregate distribution in VGbim was broader than in RSbim.
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In Fig. 3a and 3b
, a graphic comparison is reported between measured and estimated water contents, which refers to the total of 18 soil samples examined according to the four different approaches described above. The accuracy of the estimate is evaluated by comparison with a 1:1 line drawn to represent no error. Irrespective of the approach adopted, in the range of water content 0.150 < < 0.350 we always observed close agreement between measurements and estimates. For higher values of , by contrast, VGopt and VGfix appeared biased, systematically under- or overestimating, respectively, the water contents measured. As expected, the points referring to relations VGbim and RSbim always fell on the 1:1 line, thus suggesting that the measured and fitted values were in close agreement. What is more, the estimates were excellent, especially for higher values.
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(h). From the differentiation of Eq. [1] and [7] the slope of the respective curves
 | (10) |
 | (11) |
0. In the first case this condition occurred for values of n > 1. Nevertheless, parameter n in Eq. [1] gave the curve greater flexibility, allowing more or less marked variations in slope in relation to the value assumed by the same parameter. Thus, the high values of parameter n1 in Eq. [6], determining a steep slope in the curve according to the abrupt variation in experimental water content values, resulted in the same slope approaching zero more rapidly. By contrast, in Eq. [7] and hence in Eq. [8], the slope approached zero more gradually. Since the trend in the function near saturation determines the shape of the interaggregate pore distribution, it is evident that although this may apparently be deduced correctly in both cases, its actual shape remains uncertain. The reliability of hydraulic conductivity estimates depends primarily on the accurate prediction of the pore-size distribution toward the large pores. As discussed by Durner (1994), the formulation itself of Mualem's conductivity model or, more generally, of the models based on the Hagen-Poiseuille law, and which integrate the reciprocal of h to obtain hydraulic conductivity, assumes that the contribution of variations in water content to the integral of conductivity is higher for suction values close to zero. To be more precise, the value of the integrand at the numerator of Eq. [4] depends on how rapidly Se' approaches zero as h approaches zero. This makes the model particularly sensitive to the slope of the retention curve near saturation. This subject has received considerable theoretical and experimental treatment that shows that relatively small variations in water content at lower h values may be amplified by the algorithm for determining hydraulic conductivity (van Genuchten and Nielsen, 1985; Vogel and Cislerova, 1988).
From the graphs of Fig. 4 , which again refer to samples reported in Fig. 1, it is possible to verify to what extent the shape of the retention curve toward saturation is directly reflected in that of the conductivity curve. The symbols in the graphs denote the laboratory-measured unsaturated K values, which are scaled in relation to the conductivity value at saturation Ks to obtain relative conductivity values Kr to be compared with the estimated curves. The values of Ks for the four samples examined here are reported in Table 2. The relative conductivity curves Kr(h) are those estimated using Mualem's conductivity expression combined alternatively with the relations VGopt, VGfix, VGbim, and RSbim.
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s at the measured value, underestimates of hydraulic conductivity tended to be obtained. By contrast, overestimates were obtained when s is optimized. In neither case did the estimated curves adequately describe the steep decrease in measured hydraulic conductivity values generally expected in soils with a secondary system of large pores that desaturate rapidly with the application of relatively low suction head. By contrast, the laboratory-measured unsaturated conductivity values compared well when bimodal approaches, VGbim and RSbim, were used. The predictions were well within one order of magnitude of the measurements for the illustrated samples. However, such behavior was observed more or less strikingly for all the 18 samples examined. This was only to be expected, given the more accurate description of experimental retention values obtained using relations VGbim and RSbim. Comparison between the conductivity curves VGbim and RSbim showed that, although the fit of the retention data is satisfactory in both cases, the different approach at saturation has effects on conductivity that are amplified and extended to the shape of the whole curve.
Moreover, the use of single van Genuchten expressions (VGopt and VGfix), produced a considerable conductivity drop at "infinite" pore sizes. Such unrealistic behavior, to be ascribed to n values close to unity, was avoided when bimodal functions were used, the conductivity drop being shifted to a physically more realistic region.
Overall, evaluation of predictive capability of the various approaches with reference to all 18 samples examined may be effectively summarized using the scatterplots shown in Fig. 5a and 5b . This scheme allows good visual interpretation of the differences between observed and estimated values in terms of bias and point scatter. Moreover, the graphic results were quantified by calculating the mean residual error, ME, and the mean squared residual error, MSE, which supply a measurement of bias and of scatter, respectively, around the 1:1 line. In calculating the indices, conductivity values were log transformed to give approximately equal weight to values in the whole range observed.
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, though with a less marked scatter, as indicated by the lower value of the corresponding MSE (0.64). In both cases the deviation between measurements and estimates may even reach two orders of magnitude. The unsaturated hydraulic conductivity plot referring to relations VGbim and RSbim is represented in Fig. 5b. Also in this case, the regression lines are used to evaluate graphically the trend between the estimated and measured values. In both cases, comparison shows a more favorable fit and less scatter of the data around the 1:1 line. The coefficient of determination, R2, for the respective regression lines assumes values that in the two cases do not deviate significantly, indicating a satisfactory fit. Similar values of respective MSEs also indicate negligible differences in point scatter. However, the trend in the regression line and the ME value for RSbim suggest a bias, since most of the points are found systematically below the 1:1 line. By contrast, in VGbim the points are more regularly distributed around 1:1 and the slope of the corresponding regression line appears very close to the same 1:1. The results show better fits obtained with a bimodal representation of the pore system and a substantial improvement in the predictive capability of Mualem's expression for hydraulic conductivity. Within the framework of the bimodal approach, the differences observed between relations VGbim and RSbim generally appear minor for most of the horizons in the soil profiles analyzed.
Model Parameterization and Parameter Uncertainty
It should be noted that, if on the one hand the greater flexibility of the VGbim relation allows a slightly more detailed description of the sharp trend in retention data at low suction head and, to a certain extent in those of conductivity, on the other hand, it may lead to ill-determined parameters. The great number of fitted parameters increases the probability that one or more are highly correlated, with the result that neither can be determined accurately. Information concerning the uncertainty of parameter estimation is contained in the variance–covariance matrix of the parameters. As an example, Table 3
illustrates the results of parameter optimization with reference to Sample a14. Comparable results were obtained for all the 18 samples examined. The table reports the parameters estimated by using relations VGbim and RSbim. The correlation matrix was obtained directly from that of variance–covariance. In our analysis, the 95% confidence interval value
is also used to quantify the uncertainty in parameter estimates. Obviously the analysis excludes the cases that refer to relations VGopt and VGfix for which discussion of the results of parameter estimation would be of dubious utility, having verified that they fail to adequately represent the system behavior.
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(h) expressions identified as VGbim and RSbim are sufficiently accurate, the problem of parameter estimation is thus reduced in minimizing parameter uncertainty. Therefore, in order to reach a correct solution, the choice of initial parameter values that are reasonably close to their actual value becomes important. According to Durner et al. (1999), optimization relative to the same sample was conducted by using different combinations of starting values. The global minimum was detected in correspondence with the parameter vector on which the majority of runs converged. In all cases, the value of r was set at zero and that of s equal to the laboratory-measured value.
The value of the coefficient of determination R2 in VGbim was only slightly better than that obtained in RSbim. However, the latter gave improved parameter estimation with narrower confidence intervals and much more rapid convergence toward the global minimum, which is better identified also with starting values somewhat distant from true values. This confirms an overall reduction in parameter uncertainty with reduction in the number of unknown parameters. However, it should be noted that in VGbim greater uncertainty concerns the part of the curve referring to larger pores, whose shape appears somewhat arbitrary. By contrast, the parameters that define the shape of the curve at lower water content values, where there are more observations, appeared relatively better defined. This would suggest that the particularly wide confidence intervals associated with the estimation of parameters 1, n1, m1, and 1 may be attributed largely to the lack of observations near saturation, which are decisive in defining the actual shape of the curve in that section. The number of parameters that may be identified for a given situation thus depends on the quantity and quality of the data. Frequent accurate measurements at low suction, though not necessarily improving the fit, would allow a reduction in parameter uncertainty and, also for the reasons discussed above, more reliable conductivity estimates. However, both in VGbim and RSbim there is still a close correlation between parameters n2 and m2.
As discussed by Durner et al. (1998), the choice of the most appropriate relation for the purposes of application depends on the objective of the analysis, such that a more detailed description of the retention curve may be found preferable to a more simplified representation with better identified parameters and vice versa. The aim of the interpolation could be merely to find a reasonable description of the shape of the retention curve. In this case the parameters should be seen as curve shape coefficients like those of any alternative interpolation function (Durner, 1994, p. 215).
Nevertheless, if a comparison between two different parametric expressions yields comparable descriptions, the choice of which is more appropriate should be governed by parsimony. The high degree of interdependence among parameters 1, n1, and m1 observed in VGbim would suggest that it be reformulated by fixing or even excluding the correlated parameters. Seemingly, the VGbim relation involves overparametrization to describe behavior effectively expressed by only parameter 1 in RSbim, thus implying parameter redundancy in VGbim. From this point of view, the RSbim relation could be seen as a representation of VGbim simplified to reduce collinearity problems.
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Conclusions |
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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A plausible physical interpretation of the effects of the existence of a secondary system of pores upon hydraulic properties may require modeling that explicitly takes account of a partition of the porous medium into intra- and interaggregate components. As regards van Genuchten's unimodal approach, the improvement in fits obtained by using water retention relations that assume heterogeneity of the porous system and apply simple commonly used functions to different sections of the curve (VGbim and RSbim) seemed to support such a view.
Within the framework of such a bimodal approach, the greater flexibility of the VGbim relation allowed, compared with RSbim, slightly more detailed description of the steep slope indicated by the retention data at low suctions. There is still doubt with regard to the large number of parameters involved in optimization, which are always closely correlated and uncertain, especially those that refer to the curve's first section. In contrast, by using a more simplified representation of the macropore system, the relation RSbim reduced the number of parameters involved in the estimation procedure, while still satisfactorily describing the retention curve. As a result, convergences were much more rapid and parameters better defined.
Albeit with uncertainties in defining the true shape of the curve toward saturation, the satisfactory quality of the fits produced, for both relations, hydraulic conductivity estimates that agreed more closely with direct measurements of conductivity obtained through the crust method. The predictive capability of the RSbim relation was comparable with that obtained using VGbim, with the advantage of having fewer parameters involved.
Finally, the predominant effect of the shape of the retention curve near saturation upon hydraulic conductivity estimates, supported by comparison between conductivity estimates obtained by using various (h) relations, suggests the need for a more accurate and detailed experimental description of the retention curve for high water content values, which would allow less uncertain identification of the parameters involved.
Received for publication June 21, 1999.
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TOPABSTRACTINTRODUCTIONTheoryMaterials and methodsResults and discussionConclusionsREFERENCES |
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