Soil Science Society of America Journal 66:744-752 (2002)
© 2002 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Three-region Campbell Model for Unsaturated Hydraulic Conductivity in Undisturbed Soils
T. G. Poulsen*,a,
P. Moldrupa,
B. V. Iversenb and
O. H. Jacobsenb
a Dept. of Environmental Engineering, Institute of Life Sciences, Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
b Danish Institute of Agricultural Sciences, Dept. of Crop Physiology and Soil Science, Research Centre Foulum, P.O. Box 50, DK-8830 Tjele, Denmark
* Corresponding author (tjalfep{at}hotmail.com)
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ABSTRACT
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A three-region Campbell (TRC) type model for predicting undisturbed soil unsaturated hydraulic conductivity from water retention is presented. The model assumes that hydraulic conductivity follows separate Campbell functions within the macropore (matric head 
> -10 cm H2O), the mesopore (-10 
> -350 cm H2O), and the micropore ( 
-350 cm H2O) regions, and that soil water retention and two reference-point values of hydraulic conductivity are known. Conductivity and retention data for 100 undisturbed soils from the UNSODA database and 68 soils from a Danish database were used for model development. Conductivity for both highly structured (three-region) and weakly structured (two-region) soils mostly followed a piecewise linear function (with slope ß) in a Log(conductivity) -Log(water content) plot, supporting the TRC model concept. A unique relationship between the Campbell soil-water retention parameter, b, and the unsaturated conductivity parameter, ß, was found valid for both meso- and micropore regions. It was shown that the values of b in the mesopore and micropore regions are not correlated, making the use of single-region expressions (e.g., the Mualem–van Genuchten type models) questionable and suggests that a multiregion model with noncorrelated retention parameters between pore regions, such as the TRC model, may provide a conceptually more correct description of hydraulic conductivity. The TRC model yielded improved conductivity predictions in loamy and clayey soils whereas predictions for sandy soils were comparable to the single-region Campbell and van Genuchten models. TRC model predictions compared well with independent data for three differently textured soil profiles.
Abbreviations: K, hydraulic conductivity • RMSE, root mean squared error • TRC, three-region Campbell • 
, water content • , matric head
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INTRODUCTION
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UNSATURATED HYDRAULIC CONDUCTIVITY is difficult and time-consuming to measure. Especially, this is the case at low soil-water matric heads where water movement is very slow. Models for predicting unsaturated conductivity from parameters that are simpler and faster to measure are therefore valuable. Unsaturated hydraulic conductivity is strongly dependent upon the distribution of pore sizes in the soil and is often predicted assuming that the pore-size distribution is unimodal, i.e., that it can be described by a single-distribution function. However, in undisturbed natural soils and especially in structured soils pore-size distributions are often multimodal requiring two or more distribution functions for adequate description of the entire pore-size distribution. These soils often have a network of large interaggregate pores or cracks that are significantly larger than the pores within the aggregates or soil matrix. Also the distribution of pores in the matrix region is often better described using multimodal-distribution functions.
A limited number of models for predicting the unsaturated hydraulic conductivity (K) in soils with multimodal pore-size distributions have been presented. Keng and Lin (1982) presented a two-region model for predicting K as a function of matric head (). This model can be used for between 0 and -120 cm H2O. The model is based on two exponential expressions using five input parameters. A similar approach was used by Jarvis and Messing (1995) and Jarvis et al. (1999) who presented a two-region model for predicting K in the near saturated region for between 0 and -20 cm H2O also using five input parameters. A three-region model for predicting K as a function of based on the Mualem (1976) conductivity model was presented by Wilson et al. (1992). This model uses 16 input parameters, most of which are fitted from the soil-water retention curve. Durner (1992) and Ross and Smettem (1993) presented multiregion models based on the Van Genuchten (1980) and Brooks and Corey (1966) relationships. The model by Durner (1992) was later used in solving a two-region solute transport problem by Gerke and Van Genuchten (1993). If applied to a three-region system both these models use 12 input parameters.
The multiregion models developed for predicting hydraulic conductivity throughout the entire water-content range, i.e., the models by Durner (1992), Wilson et al. (1992), and Ross and Smettem (1993), require a significant number of input parameters (from 12–16 if a three-region problem is considered). These models are therefore less applicable in cases where retention and conductivity measurements are limited or in cases where the models are to be used in stochastic calculations where low-parameter models generally are desired. Models requiring fewer input parameters are further useful in geographical information systems (GIS) for characterizing soils on a regional scale.
The objective of this paper is therefore to present a low-parameter three-region model for predicting unsaturated hydraulic conductivity in undisturbed soils from soil-water retention properties. The model is applicable in the water content range from saturation to the wilting point ( = -15000 cm H2O) and is developed using measured conductivity data available in the literature.
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Existing Models
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Several one-region models for predicting unsaturated hydraulic conductivity as a function of soil-water content () are available. Some of the most widely used are the Campbell (1974) and the Van Genuchten (1980) type relationships given by
 | [1] |
 | [2] |
where KS is saturated hydraulic conductivity, s and r are saturated and residual soil water contents, respectively, n is the Van Genuchten water-retention parameter, b is the Campbell water-retention parameter (equal to the slope of the soil-water retention curve in a Log - vs. Log system) and A, B and 
are constants related to the pore-size distribution of the soil. The parameter in Eq. [2] equals 0.5 if the Mualem (1976) hydraulic conductivity model is assumed. Campbell (1974) suggested A = 2 and B = 3 based on a derivation from pore-size distribution and adding a pore-connectivity term (increasing the B-value from 2 to 3). Poulsen et al. (1999) suggested A = 2 and B = 10/3 based on measurements from 191 undisturbed soils from the UNSODA database (Leji et al. 1996). The Campbell (1974) model (Eq. [1]) is identical to the model presented by Brooks and Corey (1966) if r equals zero. The parameter r is in essence a fitting parameter in both the Van Genuchten and the Brooks-Corey models.
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Datasets Used
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Measurements of soil water retention and unsaturated hydraulic conductivity for a selected set of undisturbed soils were used in the model development. This set consisted of 100 undisturbed soils (Table 1) selected from the UNSODA database (Leji et al., 1996) and 68 soils from Jacobsen (1989). The 168 soils were selected based on the following criteria: (i) soil-water retention measurements were available down to at least -100 cm H2O soil-water matric head, (ii) hydraulic-conductivity measurements were available down to at least -100 cm H2O, and (iii) there would be at least five conductivity measurements available for each soil. In addition, retention and conductivity measurements for three Danish soils (Lindhardt et al., 2002) not used in model development were used for independently testing the model.
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Table 1. Testure distribution of 100 soils from UNSODA (Leji et al., 1996) and 68 Danish soils (Jacobsen, 1989) used in model development: Sandy (sand), Loamy (loamy sand, sandy loam, loam, silt loam, and silt), and clayey (sandy clay loam, silty clay loam, clay loam, sandy clay, silty clay, and clay) soils. Clay <0.002 mm, silt 0.002 to 0.02 mm, and sand >0.02 mm.
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Model Development
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A close examination of the conductivity measurements for the 168 soils revealed that the relationship between hydraulic conductivity and soil-water content for 0 > > -15000 cm of H2O generally could be approximated by a function that was piecewise linear in a Log - LogK system and consisted of three linear parts each of the same form as the Campbell (1974) relationship (Eq. [1]). The two intercepts were generally between -5 and -15 cm H2O and -200 and -500 cm H2O, respectively.
The Campbell hydraulic conductivity model is appealing because of its simplicity and limited input parameter requirements especially in connection with multipore-region models compared with more complex models such as the Van Genuchten (1980) or Brooks and Corey (1966) relationships. The Campbell model has also been shown to perform well in case of nonstructured soils (Poulsen et al., 1999). This model was therefore selected as basis for the development of a predictive three-region model for unsaturated hydraulic conductivity in undisturbed soils. The Campbell model is assumed valid in each of the three pore-size regions, with different b-values for the three regions.
The concept of the TRC model for predicting hydraulic conductivity from soil water content is illustrated in Fig. 1
using data for Soil 4032 from UNSODA. The soil water content range is divided into three separate regions corresponding to > -10 cm H2O (Region I), -10 > -350 cm H2O (Region II), and -350 cm H2O (Region III). The slopes of the retention curve in a Log vs. Log - coordinate system for the three regions are denoted b1, b2, and b3, respectively. Similarly, the slopes of the hydraulic conductivity curve in a Log vs. LogK coordinate system are denoted ß1, ß2, and ß3. Region I represents the macropores with equivalent pore diameter, d > 300 µm, Region II represents the mesopores, 300 > d > 10 µm and Region III represents the micropores, d < 10 µm. The -10 cm H2O soil-water matric head was also proposed as the lower limit of the macropore region by Wilson et al. (1992). Wilson et al. (1992) further proposed = -250 cm H2O as the lower bound for the mesopore region whereas Addiscott and Whitmore (1992) suggested = -350 cm H2O as the lower limit for the region of significant water flow in soils. Here = -350 cm H2O is used as it corresponded well with the matric potentials found for the 168 soils used in the model development. The fitted values of b1, b2, and b3, for Soil 4032 in Fig. 1 are 51.1, 17.1, and 5.2, respectively, and ß1, ß2, and ß3 equal 113.7, 31.6, and 5.9. The corresponding slopes of the Campbell (1974) model (2b + 3) are 105.1, 37.3, and 13.4, respectively. This already implies that very different values of Campbell b in the three pore-size regions may be needed to realistically describe the hydraulic parameters throughout the whole range of soil-water matric potentials.
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Fig. 1. Three-region Campbell (TRC) model concept, (a) soil water retention and (b) hydraulic conductivity as a function of soil-water content. Data from UNSODA (Leij et al., 1996) Soil 4032.
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Concerning very dry soil conditions measurements of soil-water retention for < -15000 cm of H2O (Campbell and Shiozawa 1992) and Schofeld (1935) for seven soils of different texture indicate that the retention curve in this dry region is linear in a vs. Log - coordinate system and approaches zero soil water content at a soil-water matric head of 
-107 cm of H2O (corresponding to oven-dry soil). The soil-water retention curve in the very dry region (could be labeled as a Region IV) therefore does not share the Campbell (1974) form. This also suggests that the residual soil water content used in the models by Van Genuchten (1980) and Brooks and Corey (1966) should always be equal to zero as discussed by Webb (2000). It is noted that the present TRC model does not per se assume zero residual soil water content but merely assumes that any residual water content will not affect the hydraulic conductivity within each of the three separate pore-size regions. The likely explanation for the different shape in the soil-water retention curve for < -15000 cm of H2O is that in this region most of the soil water is sorbed to the soil particles in the form of water films (Petersen et al., 1996) and water retention therefore behaves different in this region. It is likely that the shape of the hydraulic conductivity curve is also different for < -15000 cm of H2O but as there are no data available for characterizing the hydraulic conductivity in this extremely low soil-water matric head region it is not considered in the present study.
It was observed that the slope of the soil-water retention curve (Log - vs. Log) in the mesopore region (b2) and the micropore region (b3) in general were proportional to the corresponding slopes of the hydraulic conductivity curve (LogK vs. Log) in the same regions (ß2 and ß3). In the macropore region, however, there was no correlation between the slope of the retention curve (b1) and the hydraulic conductivity curve (ß1), likely because soil water retention and hydraulic conductivity are affected very differently by soil structure in this region. We therefore suggest that the hydraulic conductivity in the macropore region must be estimated based upon direct measurements of hydraulic conductivity and soil water content at saturation (KS, S) and at = -10 cm H2O (K2, 2) assuming a linear relationship in a LogK vs. Log coordinate system as illustrated in Fig. 1b, i.e., for Region I:
 | [3] |
 | [4] |
Similarly, hydraulic conductivity in the mesopore region is estimated using the known (measured) values of K2 and 2, and the slope of the hydraulic conductivity curve ß2, i.e., for Region II:
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Values of b2 and ß2 for the 168 soils determined from the () and the K() data within Region II were used to establish a relationship between the two parameters. The relation between the two parameters in general followed a linear relationship,
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The hydraulic conductivity as a function of soil water content in the micropore region is calculated using a value of LogK3 = LogK2 - ß2 (Log2 - Log3) in combination with measured or predicted values of 3 and ß3, i.e., for Region III:
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A relation between ß3 and b3 similar to Eq. [6] was established using retention and conductivity data in the micropore region, where available (data were available for a total of 56 out of the 168 soils). Again the relationship was linear and very similar to the relationship between b2 and ß2.
 | [8] |
The two relationships, Eq. [6] and [8], are plotted together with the measured data in Fig. 2
. The relationships yield very similar predictions of b and ß in Regions II and III. The relationship between b and ß for all data in the two regions combined is
 | [9] |
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Fig. 2. Relation between the Campbell soil-water retention parameters (b2, b3) and the TRC hydraulic conductivity parameters (ß2, ß3) in the mesopore and micropore regions.
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This relationship is also shown in Fig. 2 together with the 95% prediction interval. The accuracy of Eq. [9] is identical to that of Eq. [6] and higher than Eq. [8]. The explanation is that the number of data points in Region III is smaller than that in Region II and they therefore have a minor effect on overall prediction accuracy. Because Eq. [6], [8], and [9] are all very similar we suggest that ß in each Region (II, III) be predicted from b in each Region (II, III) using the general Eq. [9]. Interestingly, although the value of Campbell b is typically very different in the mesopore (II) and the micropore (III) regions (e.g., Fig. 2), there seems to exist a unique relationship between the Campbell pore-size distribution (water retention) parameter, b, and unsaturated hydraulic conductivity parameter, ß, which spans both regions. Soil water retention and hydraulic conductivity is often calculated assuming implicitly that soil-water retention properties in the mesopore region are related to those of the micropore region. Examples are the retention and conductivity models by Mualem (1976) and Van Genuchten (1980). These models use the same set of parameters to predict retention and conductivity across the entire pore-size distribution. The calculated values of b2 and b3 for the 56 soils were therefore compared to investigate possible relations. The results shown in Fig. 3
indicate that there is no relation between the two parameters for the 56 soils investigated (r2 = 0.01). Conceptually, this makes the use of closed-form expressions such as the Van Genuchten (1980) model questionable, although they can often near-perfectly fit the measured data because of the many fitting parameters. The suggested TRC model (Eq. [3]–[ 5], [7], and [9]) is conceptually appealing as it does not per se assume any correlation between the soil-water characteristics in the different pore-size regions.
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Fig. 3. Relation between soil-water retention parameters b2 and b3 for 100 soils from UNSODA. Dotted lines indicate 95% prediction intervals.
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Model Validation
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Figure 4
shows soil water retention and hydraulic conductivity curves for four soils from the set of 168 soils used in the model development. The retention data clearly illustrate the lack of correlation between b2 and b3 as soils have similar b3 but widely differing b2. The very steep decrease of several orders of magnitude in hydraulic conductivity with soil water content in the macropore region for the soils in Fig. 4 suggests that they have a significant amount of macropores (likely structure related cracks or pores in large soil grains). Predictions of hydraulic conductivity were calculated using the new model (Eq. [3]–[ 5], [7], and [9]) in combination with the measured values of KS, K2, S, 2, 3. For comparison predictions by the Campbell (1974) (Eq. [1]) and the Van Genuchten (1980) (Eq. [2]) models are also shown. Predictions by Eq. [1] were calculated using b = b2 and predictions by Eq. [2] were calculated based on the Van Genuchten parameters fitted from soil-water retention data across all three pore regions. Two sets of predictions by Eq. [2] are shown; (i) using values of KS, S together with the fitted Van Genuchten retention parameters, and (ii) using KS = K2, and S = 2 together with the fitted retention parameters, i.e., predicting only the hydraulic conductivity in the meso and micropore regions and using (K2, 2) as reference-point in the Van Genucthen hydraulic conductivity model. The new TRC model predicts hydraulic conductivity well for all four soils whereas the Campbell (1974) model gives a fair prediction of hydraulic conductivity curve in the macro and mesopore regions for three out of four soils. It is, however, not able to predict conductivity in the micropore region for any of the soils in Fig. 4. The Van Genuchten (1980) model using (KS, S) as reference-point has the lowest prediction accuracy for the four structured soils and is only able to give a fair prediction of hydraulic conductivity for the sandiest soil (Soil 4650). Replacing KS and S with K2 and 2 in the Van Genuchten model, i.e., predicting K in the meso and micropore regions only, greatly improves prediction accuracy especially for the more fine-textured soils. This indicates that soil macroporosity have a significant impact on hydraulic conductivity in the near-saturated region and must be taken into account to achieve accurate predictions of hydraulic conductivity. Figure 4 also illustrates the problem of using a residual soil water content (Soil 4650) where Log - is significantly overpredicted by the Van Genuchten model for the lowest values of .
The TRC model is in general able to predict LogK with an average deviation in predictions of 0.28 decades for the four soils in Fig. 4. Prediction accuracy is similar for all three regions (Fig. 5)
with average deviations of 0.23, 0.33, and 0.28 for the macro, meso, and micropore regions, respectively. In comparison, the Van Genuchten (1980) using K2 (the most accurate of the existing models tested) has an average deviation of 0.47 decades for the meso and micropore regions for all soils together. It is noted that the Van Genuchten model is not able to predict conductivity throughout the micropore region for Soil 4650 as part of this region lies below the residual water content used in the model.
Figure 6
shows measured retention and hydraulic conductivity curves for two unstructured soils from UNSODA together with predictions by the Campbell (1974), the Van Genuchten (1980) and the new TRC model. For these soils, showing only two-region behavior, all three K() models are able to predict hydraulic conductivity equally well. The data in Fig. 4 and 6 indicate that the Van Genuchten model is adequate in case of nonstructured soils such as sand. In case of structured soils, it is only possible to use the Van Genuchten model for Region II and III, whereas including Region I requires a multiregion modeling approach. For example the Van Genuchten model used for Regions II and III could be combined with the simple Campbell based model for Region I given in Eq. [3] and [4]. In this case, however, the simple Campbell-based TRC concept with noncorrelated model parameters in Regions I, II and III seems more appealing.
Prediction accuracy for the Campbell (1974), the Poulsen et al. (1999), and the present TRC model was evaluated using the measured hydraulic conductivity data for the 100 soils from UNSODA. The root mean square error (RMSE) was used as the measure of prediction accuracy,
 | [10] |
where X denotes the parameter value and N is the number of measurements. Calculated values of the RMSE for the 100 UNSODA soils are listed in Table 2 for all three models and measured and predicted values of LogK for the Campbell (1974) and the TRC models are shown in Fig. 7
. Also shown are the best-fit regression lines for the data. For all soils combined the Campbell (1974) and the Poulsen et al. (1999) models yield less accurate predictions compared with the TRC model. This is because of poor prediction accuracy in the micropore region where both models significantly under predicts the measured hydraulic conductivity values. In case of the sandy soils, all three models yield similar results (Table 2). The TRC model gives the best results in terms of the RMSE. The Campbell (1974) and the Poulsen et al. (1999) models yield similar RMSE values as also discussed by Poulsen et al. (1999).
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Fig. 7. Measured and predicted values of hydraulic conductivity for 100 soils from UNSODA using the Campbell (1974) model with b = b2, and the TRC model for sandy soils, loamy and clayey soils and all soils combined.
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The best-fit lines in Fig. 7 show that the Campbell model has almost no bias in the predictions for the sandy soils (Fig. 7a) but significantly underestimates hydraulic conductivity for the loamy and clayey soils (Fig. 7b). The TRC model has low bias for both soil texture groups (Fig. 7d,e). The apparent overestimation seen from the data points in Fig. 7d and 7e for the TRC model is because in part of a few soils where Region II starts at a matric head significantly below -10 cm. Measured and predicted values of K are shown for four of these soils in Fig. 8
. It is seen that K is significantly overestimated for all soils but improved predictions could be achieved if proper values for (K2, 2) could be found for these soils. Here the locations of the break-points (K2, 2, and K3, 3) were selected based on values proposed in the literature. Alternative methods for predicting the break-points could be developed using for instance fractal methods such as described by Nimmo (1997) or Hunt et al. (2001). Development of theoretical methods for predicting the break-points, however, is beyond the scope of this work. However, the effects of choosing a different location for the break-point between the meso and micropore regions on prediction accuracy will strongly depend upon the actual soil type. Another source of discrepancy is the sometimes curved shape of the LogK - Log relationship in Region II that causes overestimation of K as seen for Soil 4033 in Fig. 4.
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Fig. 8. Three soils from UNSODA where Region II starts at a soil-water matric head significantly below -10 cm H2O causing over-prediction of unsaturated hydraulic conductivity by the TRC model.
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The TRC model was tested against retention and hydraulic conductivity data for three undisturbed Danish soils (Lindhardt et al., 2002; see Table 3) not used in the model development. Hydraulic conductivity for the three soils was measured down to = -100 cm H2O using an automated drip infiltrometer (van den Elsen et al., 1999) that measures the unsaturated hydraulic conductivity at steady-state water flow conditions in the soil at different matric heads. An initially saturated soil sample is placed on a sandbox and five ceramic cups connected to transducers are placed in the sample. When steady-state water flow is reached a measurement is conducted and the infiltration is continued at a lower matric head. The infiltrometer is able to measure unsaturated hydraulic conductivity in the near saturated region very closely. Soil water retention was measured in triplicate and the data used for model verification are arithmetic averages of three measurements. Because no measurements were available for Region III, it was only possible to test the TRC model against data in Region I and II for the three soils.
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Table 3. Texture and retention properties of three Danish soils (Lindhardt et al., 2002) used in testing the TRC model. Numbers in parenthesis give the observed range. Also shown are root mean square error (RMSE) and bias values for predictions of Log (hydraulic conductivity) by the TRC model and the Campbell (1974) model.
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Figure 9
shows measured conductivity data for three selected locations one for each soil. Also shown are the TRC model predictions. In all three cases, the TRC model is able to well predict the measured data. Measured and predicted values of LogK and the corresponding 95% prediction interval for all data from the three soils are plotted in Fig. 10
. The RMSE and bias calculated for each soil using the TRC model is given in Table 3. For comparison RMSE and bias for the Campbell (1974) model are also given. For the nonstructured Jyndevad and Tylstrup soils the Campbell model is only slightly less accurate, whereas for the very structured Fårdrup soil, the Campbell model significantly overpredicts the measured data. This is consistent with the findings in Table 2 as Fårdrup has higher content of clay and silt compared with the two other soils. The TRC model is generally able to predict hydraulic conductivity to within 0.75 orders of magnitude (Fig. 10). These results support the use of the TRC model to predict unsaturated hydraulic conductivity in undisturbed soils.
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Fig. 9. Soil water retention, hydraulic conductivity, and predicted hydraulic conductivity by the TRC model for three selected sampling locations in three undisturbed Danish soils from (data from Lindhardt et al., 2002).
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CONCLUSIONS
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A soil-type dependent, TRC model for predicting unsaturated hydraulic conductivity in undisturbed soils in the macropore ( > -10 cm H2O), the mesopore (-10 > > -350 cm H2O), and the micropore (-350 > > -15000 cm H2O) was developed using soil water retention and hydraulic conductivity measurements for 168 soils with a broad range of textures.
The model requires knowledge of the soil-water retention curve and uses seven input parameters: hydraulic conductivity at = 0 and = -10 cm H2O, the soil water contents at = 0 cm H2O, = -10 cm H2O and = -350 cm H2O, and the Campbell b values for the soil-water retention curve in the meso and micropore regions.
Although the value of Campbell b was often very different and noncorrelated in the mesopore (II) and the micropore (III) regions, a unique relationship between the Campbell pore-size distribution (water retention) parameter, b, and unsaturated hydraulic conductivity parameter, ß, seems to exist spanning both regions.
Hydraulic conductivity predictions by the new TRC model were compared with existing models for 100 soils from UNSODA and improved prediction accuracy was found in the case of the more structured loamy and clayey soils. For the typically less structured sandy soils the existing models and the TRC model yielded similar prediction accuracy.
The TRC model is appealing for predicting unsaturated hydraulic conductivity, both conceptually (noncorrelated model parameters in the three pore-size regions) and in practice (low parameter requirements).
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ACKNOWLEDGMENTS
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This work was supported by the Danish Research Council, Research Talent Project entitled New methods for measuring and predicting liquid and gaseous phase transport properties in undisturbed soils.
Received for publication March 22, 2001.
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ABSTRACT
INTRODUCTION
