Published online 3 October 2006
Published in Vadose Zone J 5:1119-1128 (2006)
DOI: 10.2136/vzj2005.0146
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
ORIGINAL RESEARCH
Bimodal Probability Law Model for Unified Description of Water Retention, Air and Water Permeability, and Gas Diffusivity in Variably Saturated Soil
Tjalfe G. Poulsena,*,
Per Moldrupa,
Seiko Yoshikawab and
Toshiko Komatsuc
a Section for Environmental Engineering, Dep. of Life Sciences, Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
b Dep. of Hilly Land Agriculture, National Agricultural Research Center for Western Region, Ikano 2575, Zentsuji, Kagawa, 765-0053, Japan
c Dep. of Biological and Environmental Sciences, Graduate School of Science and Engineering, Saitama University, 255, Shimo-okubo, Sakura-ku, Saitama, 338-8570 Japan. T.G. Poulsen, currently: Section for Environmental Engineering, Dep. of Life Sciences, Aalborg University, Sohngaardsholmsvej 57, DK-9000 Aalborg, Denmark
* Corresponding author (tgp{at}bio.aau.dk)
Received 12 December 2005.
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ABSTRACT
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Air and water permeabilities and gas diffusivity as functions of soil fluid phase (air or water) contents are governing chemical transport and fate processes in the vadose zone, and have frequently been identified as the three main transport parameters determining time and efficiency during soil vapor extraction (soil venting) at polluted soil sites. A mathematically flexible function that can accurately describe data for both soil water retention and all three transport parameters as functions of fluid phase contents in undisturbed soil with bi- or multimodal pore structure is required for numerical simulation studies. In this study, a bimodal probability law (BPL) model with a total of six fitting parameters is compared with new and literature data for soil water retention and the three transport parameters measured on undisturbed soils. Saturated and unsaturated hydraulic conductivity (K) at six matric potentials (pF) were measured on four volcanic ash soils (Andisols) where data for soil water retention, air permeability (ka), and relative gas diffusivity (Dp/D0) were already available. The BPL model accurately described data for soil water retention and the three transport parameters (K, ka, Dp/D0) for the four Andisols and other soils with bimodal porosity behavior. BPL model parameters did not correlate well between transport processes, however, an existing model for relating saturated hydraulic conductivity to air permeability at –100 cm H2O matric potential performed well for 10 bimodal Andisols. Results indicate that relations between fluid phase transport parameters at given matric potentials may be valid across soil types and useful in the BPL model.
Abbreviations: BPL, bimodal probability law
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INTRODUCTION
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TRANSPORT OF WATER, dissolved chemicals, and gases in soil is controlled by hydraulic conductivity (K), soil air permeability (ka), and soil gas diffusivity (Dp), and their dependency on fluid phase (water or air) content. An accurate description of hydraulic conductivity, air permeability, and gas diffusivity in variably saturated soil is therefore important, for instance, when modeling scenarios for soil vapor extraction at contaminated sites, when evaluating explosion hazards near old landfills producing methane, when assessing soil aeration and potential for methane oxidation in relation to global greenhouse gas balances, or when evaluating groundwater contamination potential by pesticides used in farming (Poulsen et al., 1998, 2001; Kruse et al., 1996; Farhan et al., 2001).
Hydraulic conductivity, air permeability, and gas diffusivity are closely linked to soil fluid phase (water or air) content and to soil water retention. A large number of models for predicting unsaturated hydraulic conductivity from water content and soil water retention (Brooks and Corey, 1966; Campbell, 1974; van Genuchten, 1980; Kosugi, 1996; Poulsen et al., 2002) and several models for predicting air permeability (Ball et al., 1988; Moldrup et al., 1998; Poulsen et al., 1998) and gas diffusivity (Buckingham, 1904; Millington and Quirk, 1960; Moldrup et al., 2000, 2004) from air-filled porosity alone or in combination with soil water retention have been proposed. These models have generally been developed based on repacked soils or undisturbed soils with unimodal pore-size distributions. Hydraulic conductivity has been modeled using several different approaches, ranging from simple power-law functions (Campbell, 1974) to probability law functions (Kosugi, 1996). Air permeability and gas diffusivity have generally been modeled using simple power-law functions.
Moldrup et al. (2003a) found for a range of volcanic ash soils that air permeability in some soils exhibited bi- or multimodal behavior. It was observed that the air permeability at a soil water potential of approximately –1000 cm H2O increased rapidly with air-filled porosity but was almost constant in other regions of air-filled porosity. The main reason is that the pore-size distribution and pore connectivity properties of volcanic soils often are very different from "normal" mineral soils, and apparently a highly connected pore network can develop in volcanic soils under dry conditions. This indicates that while power function models are adequate for predicting air permeability in many normal mineral soils, there are cases for soils with multimodal pore-size distributions where power function models are not sufficient.
Multimodal behavior of hydraulic conductivity has been observed across soil types, and models for predicting multimodal hydraulic conductivity are available (Keng and Lin, 1982; Durner, 1992; Wilson et al., 1992; Gerke and van Genuchten, 1993; Ross and Smettem, 1993; Poulsen et al., 2002). Abenney-Mickson et al. (1996) used a multimodal van Genuchten (1980)–type model to simulate soil water retention in volcanic ash soils. To our knowledge, it has not been investigated whether multimodal models can be used to simultaneously describe hydraulic conductivity and the two gas transport parameters in soils.
The objective of this study is to present a model platform for describing soil water retention, unsaturated hydraulic conductivity, air permeability, and gas diffusivity in soils with bi- or multimodal behavior based on measurements of soil water retention, hydraulic conductivity, air permeability, and gas diffusivity for a data set comprising mainly volcanic soils from Japan. The purpose is to provide a basis for possible unified modeling of these parameters.
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THEORY
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We assume that the relationship between any of the target parameters (soil water potential, unsaturated hydraulic conductivity, air permeability, or gas diffusivity) as a function of fluid phase (water or air) content can be represented as a sum of S-shaped functions, each being similar to the functions used to model soil water retention curves. Using the Brutsaert (1966) probability law and the van Genuchten (1980) closed-form expressions for soil water retention as a basis and assuming zero residual water or air content, the following multimodal expression is suggested:
 | [1] |
where P represents either soil water matric potential, air permeability (ka), relative gas diffusivity (Dp/D0), or hydraulic conductivity (K); 
is volumetric phase content (water- or air-filled porosity); n is the number of pore regions (modalities) considered; and A, B, and C are model variables. Variable A has the same unit as the soil parameter being modeled, B and C are dimensionless. The parameters Dp and D0 are the gas diffusion coefficients in soil and free air, respectively. In case of soil water retention P is set to represent pFmax – pF, where pF = log(–
), the soil water matric potential in cm H2O, and pFmax = 6.9 is the value of pF where the soil contains no more water (Gronenvelt and Grant, 2004). Equation [1] can be used for any value of n. In this study, however, we chose a value of n = 2 for illustration of the concept. This yields
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Parameter Estimation
Determination of parameters A, B, and C in Eq. [2] was done by fitting the equation to the measured data such that the RMSE between measured and calculated values was minimized:
 | [3] |
where N is the number of measurements and P is the property (pF, unsaturated hydraulic conductivity, air permeability, or gas diffusivity) that is modeled. Parameter values were fitted using the automatic optimization procedure (solver) in Microsoft Excel.
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DATA AND MEASUREMENTS
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Data used in this study were taken partly from the literature (Moldrup et al., 2003a, 2003b; Osozawa, 1987, 1998) and partly from measurements conducted during this study. All data were measured on undisturbed samples representing 14 soils from Japan (Fig. 1). The literature data consisted of data for soil water retention, air permeability, and gas diffusivity, whereas data for saturated and unsaturated hydraulic conductivity were measured during this study.
Soil water retention data were available for all 14 soils, namely four soils from Gunma Prefecture (labeled Tsumagoi 3, 4, 6, and 7), one soil from Kanagawa Prefecture (labeled Miura 1), two soils from Ibaraki prefecture (labeled Tsukuba and Alakawa), one soil from Aichi Prefecture (labeled Toyohashi), and one soil from Saitama Prefecture (labeled Kounosu). Air permeability data were available for the four Tsumagoi, the Miura 1, the Tsukuba, Alakawa, and Toyohashi soils. Gas diffusivity was available for the four Tsumagoi, the Kounosu, the Miura 1, and the Tsukuba, Alakawa, and Toyohashi soils. Saturated and unsaturated hydraulic conductivity for the four Tsumagoi and the Miura 1 soils and additional saturated hydraulic conductivity data for one soil from Gunma Prefecture (labeled Tsumagoi 5) and four soils from Kanagawa Prefecture (labeled Miura 2–5) were measured during this study. It is noted that soil labeling was done in accordance with the previous studies where the soils have been presented and used and, generally, is based on the name of the local area where soil sampling was done.
The same experimental methods were applied on all soils. Soil water retention was measured by the draining curve. Air permeability was measured on 100-cm3 samples by the method of Grover (1955), also described in Ball and Schjønning (2002). Gas diffusivity was measured on the same 100-cm3 samples by the method of Currie (1960), also described in Rolston and Moldrup (2002). Saturated and unsaturated hydraulic conductivity was measured on the same 100-cm3 samples using the constant head method (Reynolds et al., 2002) and the one-step method by Doering (1965), respectively. Measurements of air permeability and soil water retention were generally performed at soil water potentials of –10, –30, –60, –100, –320, –1000, –3160, and –15 000 cm H2O. For the Tsumagoi 3, 4, 6, and 7 soils, additional air permeability and retention data were measured at –20 cm H2O. Gas diffusivity was generally measured at potentials of –30, –100, –320, –1000, –3160, –15 000 cm H2O, as well as at air-dry conditions (pF = 6.0). For the Kounosu soil, measurements were also performed at oven-dry conditions and for the Tsukuba, Alakawa, and Toyohashi soils, additional measurements were done at –20 and –60 cm H2O. Unsaturated hydraulic conductivity was measured at –10, –20, –30, –60, –100, –320 cm H2O. For Tsumagoi 7, an additional conductivity measurement was done at –1000 cm H2O. An overview of the data used and selected soil characteristics for the soils are presented in Tables 1 and 2.
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Table 2. Bulk density ( b), total porosity ( ), and saturated hydraulic conductivity (KS) data for five additional Andisols used in this study (Fig. 10).
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Fig. 10. Saturated hydraulic conductivity for 10 soils as a function of air permeability at –100 cm soil water potential. Indicated also is the empirical model for predicting KS from ka100 by Loll et al. (1999). Also shown are measurements of relative gas diffusivity for eight soils as a function of ka100.
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RESULTS AND DISCUSSION
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Simple Power-Law Model Performance
Figure 2 shows measured values of pF and K as functions of volumetric soil water content and ka and Dp/Do as functions of volumetric soil air content for the Tsumagoi 3 and 7 soils, where measurements for all four parameters were available. To illustrate problems with applying simple power-law functions to multimodal soils, the Campbell (1974) constitutive parameter model was used to predict pF, K, and ka from soil fluid phase (air or water) content:
 | [4] |
where P is the parameter (pF, K, ka), is the volumetric phase content (water content, 
, or air content, 
), and 
= H1b + H2, where b is the Campbell (1974) soil water retention parameter and H1 and H2 are constants. The * indicates a reference point value. In a log(P/P*)–log(/*) coordinate system, Eq. [4] will yield a straight line with slope . For pF as a function of , Campbell (1974) suggested H1 = 1 and H2 = 0. This expression is used to determine b for the two soils (Fig. 2a and 2b). For predicting K from , Campbell (1974) suggested H1 = 2 and H2 = 3. Alexander and Skaggs (1986) proposed H1 = 1 and H2 = 3. Predictions of K by these two models are plotted in Fig. 2c and 2d for (P*, *) = (KS, S) and (K10, 10), respectively, where subscripts S and 10 refer to saturation and –10 cm H2O soil water potential. The –10 cm H2O was suggested as a reference point value in the study by Poulsen et al. (2002). Moldrup et al. (1998) and Moldrup et al. (2001) suggested 0.05 < H1 < 0.25 when predicting ka from . Predictions of ka using these two values and the largest measured value of ka as reference point are shown in Fig. 2e and 2f. Moldrup et al. (1999) proposed a similar power-law model for predicting relative gas diffusivity from volumetric air content. This model is given as
 | [5] |
where is soil total porosity. This relationship is plotted in Fig. 2g and 2h.
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Fig. 2. Soil properties: (a, b) soil water retention, (c, d) hydraulic conductivity, (e, f) air permeability, and (g, h) gas diffusivity as functions of fluid phase (air or water) content for the Tsumagoi 3 and 7 soils. Also shown are Campbell (1974)–type power function models used to predict these parameters. Curves in Fig. 2c and 2d indicate the Campbell model using saturated hydraulic conductivity and hydraulic conductivity at –10 cm H2O soil water potential as reference point, respectively.
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It is seen that although the Campbell retention model closely matches the measured retention data (Fig. 2a and 2b), the curve shape of the different power-law function models does not match the measurements for the three transport parameters. This is especially the case for hydraulic conductivity and air permeability, whereas predictions are much closer for gas diffusivity.
Moldrup et al. (2001) used the Campbell (1974) constitutive parameter model (Eq. [4]) to relate the average slopes of the hydraulic conductivity, air permeability and gas diffusivity to b. Figure 3a, 3b, and 3c illustrate how b is found from soil water retention and is found for K, ka, and Dp/D0 for the Tsumagoi 3 soil. Figure 3d shows values of as a function of b for K, ka, and Dp/D0 for Tsumagoi 3, 4, 6, and 7, the four soils where data for all four parameters were available. Also plotted are the four relationships for predicting from b given earlier. Values of for hydraulic conductivity for the Tsumagoi soils are placed between the Campbell (1974) and the Alexander and Skaggs (1986) slopes, as also found by Poulsen et al. (1999). The values of for air permeability are lower than for hydraulic conductivity, as also predicted by the Moldrup et al. (1998) and (2001) models, but the slope values are much more scattered compared with hydraulic conductivity. On the other hand, gas diffusivity slopes are fairly close to the values predicted by the Moldrup et al. (1999) model ( = 2 + 3/b, dotted line in Fig. 3d.). The data in Fig. 3d indicate that although the soils are strongly multimodal, (b) values are similar to those found for unimodal soils.
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Fig. 3. Campbell constitutive model slope for (a) soil water retention, (b) hydraulic conductivity, (c) air permeability and relative gas diffusivity, and (d) values of constitutive Campbell model slopes for the Tsumagoi 3, 4, 6, and 7 soils for hydraulic conductivity, air permeability and relative gas diffusivity. Also shown in Fig. (d) are the corresponding Campbell slopes proposed in literature (solid and dotted curves).
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BPL Model Performance
Data for soil water retention ( as a function of pFmax – pF) are shown for nine soils in Fig. 4. Most of the soils do not show strong bimodal behavior, with the exception being Miura 1 (Fig. 4e). For all soils the BPL model is able to fit the measured data very well. The saturated water content is often regarded as a fitting parameter (van Genuchten and Nielsen, 1985), and retention curves have often been fitted in the literature without using the measured value. For simplicity and because the modeling presented here is done to illustrate the BPL concept, the measurement at saturated conditions (open symbol in Fig. 4) was not included in the fitting. In reality the data may exhibit trimodal behavior, and adding one additional term to Eq. [1] could easily accommodate the saturated measurement.
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Fig. 4. Measured and bimodal probability law (BPL) model (Eq. [2]) fitted values of soil water content as a function of minimum soil water potential given as pF (pFmax = 6.9) minus actual soil water potential measured as pF for nine volcanic ash soils. Curves indicate fitted BPL model (Eq. [2]) and Terms 1 and 2 in the BPL model, respectively.
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Hydraulic conductivity is plotted as a function of volumetric soil water content in Fig. 5. One of the four soils (Tsumagoi 3) shows strong bimodal behavior, although this was not the case for soil water retention. Figure 6 shows measurements of air permeability as a function of volumetric air content. Here several of the eight soils show strong bimodal behavior (Tsumagoi 3, 4, and 6 and Miura 1). Relative gas diffusivity as function of soil volumetric air content is shown in Fig. 7. In this case, only the Kounosu soil shows a tendency for bimodal behavior. In all cases the BPL model can be fitted to closely describe the measured parameter values including any bimodal behavior in Fig. 5
through 7. Because the number of measurements in some cases is relatively low, the results in Fig. 5 through 7 cannot be taken as final proof of the accuracy of the BPL model used in this study. However, the results still demonstrate the ability of the BPL model concept to describe the multimodal behavior of the soils. The six BPL model parameters (A1–2, B1–2, and C1–2) fitted for each soil are presented in Table 3. The set of constants associated with the term in Eq. [2] that reaches 50% of its A value first is labeled "1", and remaining constants are labeled "2".
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Fig. 5. Measured and bimodal probability law (BPL) model (Eq. [2]) fitted values of hydraulic conductivity as a function of soil water content for four volcanic ash soils. Curves indicate fitted BPL model (Eq. [2]) and Terms 1 and 2 in the BPL model, respectively.
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Fig. 6. Measured and bimodal probability law (BPL) model (Eq. [2]) fitted values of air permeability as a function of air-filled porosity for eight volcanic ash soils. Curves indicate fitted BPL model (Eq. [2]) and Terms 1 and 2 in the BPL model, respectively.
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Fig. 7. Measured and bimodal probability law (BPL) model (Eq. [2]) fitted values of relative gas diffusivity as a function of air-filled porosity for nine volcanic ash soils. Curves indicate fitted BPL model (Eq. [2]) and Terms 1 and 2 in the BPL model, respectively.
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Table 3. Fitted values of the bimodal probability law (BPL) model parameters for soil-water retention, air permeability (mm2), hydraulic conductivity (cm d–1), and relative gas diffusivity using Eq. [2] for nine soils from Japan.
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Figure 8 shows the slope of the BPL model predictions (
P/) of hydraulic conductivity, air permeability, and gas diffusivity as functions of fluid phase content. Curves are shown for Tsumagoi 3, 4, 6, and 7, for which data for all three parameters were available. All show distinct peaks that coincide with the steep jumps in the BPL model predictions shown in Fig. 5
through 7. However, the peaks occur at different fluid phase contents and, thus, different soil water potentials. Also, the relative location of the peaks with respect to one another is not constant. This indicates that although all three parameters can be fitted by the BPL model, the values of the model constants A, B, and C for the three parameters are not necessarily related.
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Fig. 8. Rate of change in hydraulic conductivity, air permeability, and relative gas diffusivity with volumetric phase (air or water) content for the Tsumagoi 3 soil calculated by the fitted bimodal probability law (BPL) model as a function of volumetric phase content.
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This was further confirmed by a comparison of A, B, and C values for soil water retention, hydraulic conductivity, air permeability, and gas diffusivity that revealed no correlation, with the exception of the B1 values for soil water retention and hydraulic conductivity. The value of B1 is proportional to the fluid phase content at which the first term in Eq. [2] reaches 50% of its maximum value. Figure 9 shows B1 values for hydraulic conductivity, air permeability, and gas diffusivity as a function of B1 for soil water retention. A decreasing trend is observed between B1 for hydraulic conductivity and B1 for retention, but no correlation is observed for air permeability and gas diffusivity. There was no correlation between BPL water retention constants and BPL constants for air permeability and gas diffusivity, indicating that it is not possible to directly predict BPL constants for gas transport parameters in multimodal soil from soil water retention properties.
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Fig. 9. The bimodal probability law (BPL) model constant B1 for hydraulic conductivity, air permeability and relative gas diffusivity as a function of B1 for soil water retention for the Tsumagoi 3, 4, 6, 7; Miura 1; Tsukuba; Alakawa; and Toyohashi soils.
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Predicting Saturated Hydraulic Conductivity
Loll et al. (1999) developed an empirical model for predicting KS from ka at –100 cm soil water potential (ka100, µm2) in nonvolcanic soils:
 | [6] |
where KS is given in centimeters per day. Values of KS are plotted against ka100 together with predictions by Eq. [6] in Fig. 10. All values of KS fall well within the 95% prediction interval proposed for Eq. [6] by Loll et al. (1999). This confirms that even though air and water permeabilities in multimodal soils behave differently from most unimodal soils, relations between these parameters and other soil properties may still be predicted using predictive models developed for nonvolcanic and unimodal soils. Figure 10 shows measured values of D100/D0, where D100 is the value of D at –100 cm soil water potential (pF = 2). It is seen that gas diffusivity also shows an increasing trend, as was the case for hydraulic conductivity. This suggests that it may be possible to predict gas permeability or hydraulic conductivity at given matric potentials from gas diffusivity.
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CONCLUSIONS
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A model concept for unified description of soil water retention, air permeability, relative gas diffusivity, and unsaturated hydraulic conductivity in variably saturated soils with multimodal behavior is proposed. The concept is based on a multimodal probability law (BPL) model. Six coefficients are used for modeling soil water retention, air permeability, gas diffusivity, or hydraulic conductivity in two pore-size regions. The new BPL model concept was applied to measured water retention, air permeability, gas diffusivity, and hydraulic conductivity data for several soils with multimodal behavior from Japan, and the BPL model was able to closely describe the measured data for all parameters and soils demonstrating the high flexibility of the concept. The BPL concept can easily be modified to allow for modeling transport parameters in tri- or higher modal soils. The BPL constitutive parameter model is easily applicable in numerical simulation models for simultaneous calculation of water, solute, air, and gas transport in the vadose zone. The values of the six model coefficients are likely functions of the soil pore-size distribution and the magnitude of the parameter being modeled. However, as the data available for identifying such functions are very limited, more measurements are required before attempts to predict the coefficients can be made.
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