Published in Vadose Zone Journal 4:151-162 (2005)
© 2005 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Uncertainty of Hydraulic Conductivity under Field Conditions and at Fixed Pressure Heads and Water Contents
A. L. B. Hurtado and
Q. de Jong van Lier*
Department of Exact Sciences, University of São Paulo, Piracicaba, SP, Brazil
* Corresponding author (qdjvlier{at}esalq.usp.br)
Received 21 January 2004.
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ABSTRACT
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Unsaturated hydraulic conductivity (K) can be expressed as a function of soil water content (
) or pressure head (h). Methods to determine K often give highly variable results, thus making it difficult to report values for K in a reliable manner. Due to the sensitivity of pore conductivity to pore radius, it can be expected that K expressed as a function of h shows less variability than expressed as a function of . To test this hypothesis, K was determined at several values of and h using the instantaneous profile method at 48 locations in the surface and a subsurface layer of a Typic Hapludox in Brazil. Results were analyzed considering the field values of K at the time of observation (Kt), at fixed values of (K), and at fixed values of h (Kh). Results show that the coefficient of variation of K is higher than that of Kh, especially in the surface layer, which suggests that, if one has the choice, it would be preferable to estimate soil water movement using measured h values, and using K(h), instead of measuring and using K(). We also show that the variability in Kt increases as the soil gets drier, while K and Kh show more variability for wetter soil.
Abbreviations: ZFP, zero flux plane
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INTRODUCTION
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THE UNSATURATED HYDRAULIC CONDUCTIVITY is a key property for modeling water flow in the vadose zone. This property can be measured directly (Klute and Dirksen, 1986), indirectly (van Genuchten and Leij, 1992), or implicitly, by using inverse methods (Hopmans and Simunek, 1999; Butters and Duchateau, 2002). Direct measurements can be performed by means of laboratory or field experiments, using the instantaneous profile method (Hillel et al., 1972; Libardi et al., 1980; Falleiros et al., 1998) or infiltration methods (Reynolds et al., 1985), or in laboratory trials, such as the outflow method (Gardner, 1956). Field and laboratory methods are both known to give highly variable results (Warrick and Nielsen, 1980; Bosch and West, 1998; Bruckler et al., 2002), thus making it difficult to report values for hydraulic conductivity in a reliable manner.
The hydraulic conductivity can be thought of as a composite of capillary conductivities, which reflects its dependency on the water-filled pore-size distribution, the tortuosity, and pore connections. The water-filled pore volume can be described using the water content–pressure head relationship, (h). This correlation led to capillary bundle models (Childs and Collis-George, 1950; Mualem, 1976) and the establishment of functional relationships between (h) and the hydraulic conductivity (van Genuchten, 1980; Schaap and Leij, 2000). The hydraulic conductivity can be expressed as a function of soil water content or pressure head. In the first case, its variability at a fixed water content is roughly due to variations in the pressure head: specific pore geometries lead to different (h) relationships, pressure heads, and diameters of water-containing pores. In the latter case, at a fixed pressure head, variability is caused mainly by differences in water content or in the pore-size distribution of the water saturated pores, although the radius of the largest meniscus is constant. Poiseuille's capillary flow equation shows that the hydraulic conductivity is highly sensitive to the radius of the largest water-containing pores in a soil. It is reasonable to assume that this radius is closely correlated to the radius of the largest meniscus, and hence to the pressure head. For this reason it is to be expected that the hydraulic conductivity expressed as a function of pressure head, correlated to the largest water-containing pores, shows less variability than expressed as a function of water content.
The purpose of this research was to verify the above hypothesis by making field measurements of the unsaturated hydraulic conductivity and water retention characteristics with a high number of replications to allow an appropriate statistical analysis. Confirmation of this hypothesis would suggest that the use of h and K(h) instead of and K() for estimation or modeling soil water flow would lead to less uncertainty.
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MATERIALS AND METHODS
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Experimental Design
The unsaturated hydraulic conductivity was determined using the instantaneous profile method at 48 locations, one per meter, along a 48-m transect in the center of a 70-by 15-m field plot on a Typic Hapludox in São Paulo state, Brazil (22°42' S, 47°38' W), having a clay content of 17% (surface layer) and 23% (subsurface layer). At each location, sequentially numbered from 1 to 48, five tensiometers with mercury manometers were installed at depths of 0.075, 0.225, 0.375, 0.525, and 0.675 m. The plot was kept bare during the experiment.
A unique –h relation was determined for each pair of locations. At 24 positions in between the tensiometer locations (i.e., between 1 and 2, 3 and 4, 5 and 6, ..., 47 and 48), three undisturbed soil samples were taken at each of the tensiometer depths. Water retention data were obtained on these samples by standard laboratory techniques (sand boxes and pressure plates) and fitted to the van Genuchten (1980) equation.
A first series of instantaneous profile data was obtained in October and November 2001, after a rainy period (Table 1) that left the soil at relatively high water contents such that water flow was downward in the monitored layers. The surface was protected from evaporation and infiltration using a threefold plastic cover, while the pressure head at each location and depth was monitored at t = 0 h, t = 18 h, and t = 44 h, and approximately every 2 d afterwards, leading to 19 sets of observations during 42 d. We will refer to this experiment as the covered test.
A second series of data was obtained in April 2002 after considerable rainfall in previous days (Table 1). During the following 15 d, no rainfall was observed and pressure heads were measured 12 times. At the time of this experiment, many tensiometers malfunctioned due to obstructions and leakage in the synthetic water-filled manometer tubes, which had been exposed to high temperatures and solar radiation for several months. All tensiometers were then tested for mercury rise; only 23 of the 48 sets of tensiometers were found to be in good conditions and were used subsequently. No plastic cover was used during this second experiment, thus allowing simultaneous evaporation and drainage to produce a zero flux plane (ZFP). This experiment will be referred to as the uncovered test.
Soil water contents at each depth and time were calculated from pressure heads by means of the van Genuchten (h) equation, using empirical parameters obtained from the closest sample point, at 0.5 m from each location. By using individually determined –h relations at each location, we limited as much as possible the introduction of spatial variability in –h in our analysis.
For the uncovered test, the depth of the ZFP at each observation time zZFP,t (m) was estimated by fitting observed hydraulic head Hz,t (m) values as a function of depth z (m) using a second-order polynomial equation with regression parameters at (m), bt, and ct (m–1):
 | [1] |
and hence, for dHz,t/dz = 0:
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Obtained zZFP,t values as a function of time t (h) were fitted using Eq. [3] to allow smoothing and interpolation:
 | [3] |
where p (m h–2), q (m h–1), and r (m) are regression parameters.
Two of the monitored depths, z = 0.075 m and z = 0.375 m, representing the surface layer and subsurface layer of this soil were analyzed for their hydraulic conductivity Kz,t (m h–1) as a function of depth and time. Values for Kz,t were calculated using
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Values for 
/t in Eq. [4] for each depth were calculated by fitting one of the simple equations
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to the estimated soil water contents as a function of time. Of these expressions, the equation whose fit resulted in the highest coefficient of determination (R2) was used. The calculated /t values at any position within a 0.15-m-thick layer were considered to be representative of the observation in the middle of that layer. Values for H/z at each observation time were similarly calculated by fitting Eq. [1] to observed hydraulic heads as a function of depth.
Data Analysis
Experimental data sets of K, h, and obtained for the covered and uncovered tests at two depths at each location were submitted to statistical analysis. Frequency distributions, standard deviations, and coefficients of variation were calculated in three ways:
- By considering the values of K, h, and at the observation times, thus accounting for the actual field variation in all three parameters. We will refer to these hydraulic conductivity values as Kt.
- By considering the values of K at fixed soil water contents. For this analysis, Kt values were logarithmically interpolated between estimated water contents and represented by K. Only those data for which at least five interpolated values were available were considered. The analyzed range for z = 0.075 m was 0.17
0.33 m3 m–3, and for z = 0.0375 m, 0.21 0.30 m3 m–3.
- By considering the values of K at fixed pressure heads (Kh). For this analysis, Kt values were logarithmically interpolated between observed pressure heads. Considering only those h data for which at least five interpolated values were available, the analyzed range for z = 0.075 m was –2 h –0.55 m, and for z = 0.375 m, –1 h –0.40 m.
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RESULTS AND DISCUSSION
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Retention data from the 24 sampling locations in general could be fitted well with the van Genuchten (h) equation. Figure 1
shows frequency distributions of the resulting coefficients of determination (R2) and standard deviations. It can be seen that values of R2 concentrated in the highest class (0.95–1.00), while regression standard deviations concentrated mainly between 0.01 and 0.03 m3 m–3.
Fitting Eq. [1], [3], and [5] to observed data resulted in high coefficients of determination, generally around 0.9. The depths of the ZFP during the uncovered experiment increased in time according to Eq. [3], which showed maximum values between 0.40 and 0.50 m at most locations. Figures 2 and 3
show an example of the fitting procedure for Location 12. Although fitting of a three-parameter model to five data points may seem questionable, results are apparently coherent (Fig. [2]), with the estimated zZFP values showing an increase in time (Fig. [3]). In addition, Fig. [2] shows that, except for the surface layer, H (and hence ) varied very little and remained at about –1.30 m during the last six measurements, thus indicating that no significant drainage occurred during the final two-thirds of the experiment. Under these circumstances, precise determination of zZFP is of little importance for estimation of K since d/dt in Eq. [4] is essentially equal to 0 for greater depths such that variations in zZFP within this region will affect 
zZFPz
|tdz only very slightly.
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Fig. 2. Equation [1] fitted to H–z data from the uncovered experiment at Location 12 for each observation time: (a) 0 h, (b) 17.9 h, (c) 47.7 h, (d) 65.1 h, (e) 88.5 h, (f) 114.4 h, (g) 137.8 h, (h) 163.8 h, (i) 186.8 h, (j) 211.5 h, (k) 282.6 h, and (l) 401.4 h.
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A comparison of K values obtained from the covered test with those from the uncovered test at 0.075 m shows good consistency. Figure 4
shows these values for three randomly selected locations (17, 22, and 43) at z = 0.075 m. Some overlap in and h values occurred between the covered and uncovered experiments. In these cases values from the covered test were preferred for further analysis and those from the uncovered test remained unused. At 0.375 m (Fig. 5)
, ranges from the covered and uncovered tests overlapped in all shown cases. Only in a few cases did inclusion of the uncovered test data increase the range of water contents or pressure heads evaluated. We fitted the data with one of the most commonly adopted models for K() relations; that is (Libardi et al., 1980)
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in which 0 is an arbitrarily chosen water content and K0 is the respective hydraulic conductivity. Results are also shown in Fig. 4 and 5. Notice that the fits were reasonably close in most cases, with values of 
that range from 10 to 30 at 0.075 m, and from 30 to 50 at 0.375 m.
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Fig. 4. Experimentally obtained values of the hydraulic conductivity K as a function of the water content from both covered and uncovered experiments, at three locations at the 0.075-m depth, together with regression lines.
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Fig. 5. Experimentally obtained values of the hydraulic conductivity K as a function of the water content from both the covered and uncovered experiments, at three locations at the 0.375-m depth, together with regression lines.
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Tables 2 and 3 show descriptive statistics of the water content, pressure head, and hydraulic conductivity obtained at all locations, both for the covered and uncovered experiments at 0.075 and 0.375 m. At 0.075 m, the mean water content decreased from 0.308 to 0.238 during the covered test, and from 0.244 to 0.183 during the uncovered test, thus showing some overlap and good complementary values. At 0.375 m, however, the observed range in and h did not materially increase by adding the data from the uncovered experiment.
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Table 2. Descriptive statistics of observed water content , pressure head h, hydraulic conductivity (Kt) and lognormal-transformed hydraulic conductivity (lnKt) at all 48 locations at the 0.075-m depth during the covered and uncovered tests.
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Table 3. Descriptive statistics of observed water content , pressure head h, hydraulic conductivity (Kt) and lognormal-transformed hydraulic conductivity (lnKt) at all 48 locations at the 0.375-m depth during the covered and uncovered tests.
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Coefficients of variation of the water content at 0.075 m (12–18%) were about twice as high as those at 0.375 m (6–10%), which is a reflection of the higher variability near the surface. Pressure heads showed a similar tendency. Additionally, the 0.075-m pressure heads and water contents showed significant negative correlation with their own coefficient of variation, as shown by Pearson r values in Table 2. This indicates that variability in estimated water contents and pressure heads increased during drying of the soil.
Observed hydraulic conductivities were in the order of 0.002 to 0.02 cm h–1. Frequency distributions of the hydraulic conductivity were highly asymmetric and non-normal in most cases, as can be verified from the mean/median, asymmetry, and Shapiro–Wilk test values in Tables 2 and 3. Asymmetry was especially high for the uncovered test. Logarithmic transformation of Kt values of this experiment led to normalization, as indicated by higher Shapiro–Wilk values and lower asymmetries. Hydraulic conductivity values are known to have lognormal frequency distributions (e.g., Bosch and West, 1998; Shouse and Mohanty, 1998; Lu and Zhang, 2002). Coefficients of variation for the untransformed hydraulic conductivity values at the 0.075-m depth were between 64 and 143% for the covered experiment. For the uncovered experiment, under drier conditions, values were much higher, ranging from 100 to 340%. These values are within ranges previously reported (e.g., Nielsen et al., 1973; Warrick and Nielsen, 1980; Bosch and West, 1998).
To obtain K and Kh, values were interpolated from Kt within the available ranges. Figure 6
shows for each location of the covered and uncovered experiment plots of and h values for which hydraulic conductivity was obtained. This figure shows that pressure heads exhibit less variability between locations than water contents, which should be expected since at a given depth soil water will move to equilibrate toward the same pressure head, but not the same water content unless the soil is homogeneous. Figure 6 also shows that the uncovered test did not lead to an increase in observed values at the 0.375-m depth. We calculated coefficients of variation of h at fixed values of (CVh) and coefficients of variation of at fixed values of h (CV) from the 24 available (h) relationships. Results show CVh to be of the order of 100%, while CV is of the order of only 10% (Fig. 7) , which is obviously due to the fact that h varies over a much larger range than . The same figure shows the tendency of CVh and CV to be higher in the drier soil. As a consequence, when interpreting K data, it should be remembered that these data generally refer to a relatively wide range in h, whereas Kh data usually refer to a much narrower range in . For the same reason, CVh sometimes will show more erratic behavior with (e.g., for = 0.23 m3 m–3 at z = 0.075 m in Fig. 7).
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Fig. 6. Values of and h for which hydraulic conductivity was obtained in the covered and uncovered experiment, at each location.
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Box plots of interpolated K and Kh values for both depths show considerable normalization of the frequency distributions after lognormal transformation (Fig. 8 and 9)
, as well as increasing dispersion in the values as and h increase, especially for the 0.075-m layer. The same tendency can be observed in Fig. 10 , which shows coefficients of variation of Kh and K at both depths as a function of h or for the untransformed and lognormal-transformed data. The higher variability in the hydraulic conductivity as a function of h or in the wetter soil is opposite to the variability in Kt (Tables 2 and 3), thus showing the higher variability in Kt for the drier soil is mostly due to an increase in the variability in the water content and pressure head, and not to in increase in the variation in hydraulic conductivity at fixed or h.
Mean values of the coefficients of variation of Kh and K using the number of available K values as a weighing factor show a considerable increase when K is expressed as a function of instead of h. At 0.075 m, CV values increased 49% (from 61.9 to 91.9%); at 0.375 m, the increase was 11% (from 60.7 to 67.1%). Using lognormal-transformed data, the increases in CV values were 43 and 16% at 0.075 and 0.375 m, respectively. These results confirm the hypothesis put forward in the introduction that the hydraulic conductivity expressed as a function of pressure head shows less variability than expressed as a function of water content.
When interpreting these results, one should be aware that errors in h propagate in if the van Genuchten (1980) equation is used and in the dH/dz regression equations (based on 5 observations h–z), while errors in propagate in the d/dt regressions based on 20 to 30 –t observations. Both regressions and associated errors propagate in turn into the Kt calculations, and consequently also in the K and Kh interpolations. This clearly shows that errors in measurements of h as well as of propagate in both K and Kh. Nevertheless, we believe that our statistical analysis is sound because of the large number of data points (5 depths at 48 locations) and because of the many measurements in time (20–30 per location); a complete analysis of the error propagation is pathway beyond the scope of this paper and could be a focus of future investigation.
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CONCLUSIONS
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The experimental data we analyzed indicate that the coefficient of variation of the field hydraulic conductivity increases with decreasing water content, while the hydraulic conductivity at a fixed water content or pressure head shows the reverse. This suggests that variability in the field hydraulic conductivity is mainly due to variability in the water content or pressure head. Our analysis also shows that Kh values exhibit less dispersion than K values. This suggests that it would be preferable, if it is an option, to estimate or model unsaturated flow using measured h values and K(h) instead of data and K(). This result is corroborated by the fact that the pressure head is less variable than the soil water content. Moreover, for field studies, h can be measured directly by means of tensiometers, while field measurements of often involve indirect measurements and associated calibration curves.
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REFERENCES
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ABSTRACT
INTRODUCTION
. Numbers indicate the number of locations with an observation of K at a particular h or