Published online 26 January 2006
Published in Vadose Zone J 5:121-124 (2006)
DOI: 10.2136/vzj2005.0066
© 2006 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
NOTES
Closed-Form Expression for the Multi-Modal Unsaturated Conductivity Function
E. Priesack*,a and
W. Durnerb
a Institute of Soil Ecology, GSF-National Research Centre for Environment and Health, P.O. Box 1129 Neuherberg, D-85764 Oberschleissheim, Germany
b Institute of GeoEcology, Braunschweig Technical Univ., Germany
* Corresponding author (priesack{at}gsf.de)
Received 20 May 2005.
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ABSTRACT
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In this note we derive a closed-form expression representing the hydraulic conductivity function for soils with a multi-modal pore size distribution. By combining the multi-modal representation of the retention function of Durner with the conductivity representation model of Mualem and following van Genuchten, we derive a simple analytical expression for the conductivity of soils with heterogeneous pore systems. Examples for the representation of bi-modal and tri-modal retention and corresponding conductivity curves demonstrate the applicability of the simple analytical expressions. It is concluded that the usefulness of the multi-modal representation of retention functions is increased by providing an analytical expression to directly calculate unsaturated hydraulic conductivities.
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INTRODUCTION
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COMPUTER SIMULATION models for water flow and solute transport in the unsaturated zone have become standard tools to better understand and manage agricultural production systems or subsurface pollution problems. The accuracy of such simulations depends strongly on a reliable description of the flow and transport properties for the porous medium being considered. In particular, soil water flow simulations based on numerical solutions of the Richards equation require knowledge of the effective unsaturated soil hydraulic properties. These soil properties are (i) the water retention curve relating the volumetric water content 
to the matric head h and (ii) the hydraulic conductivity curve giving the unsaturated hydraulic conductivity K as a function of or h.
Preferential soil water flow in structured soils or unsaturated fractured rock is often explicitly considered by invoking more complicated pore size distributions, including macro- and mesopore systems. Different approaches have been introduced to represent retention and conductivity curves by means of a piece-wise composition based on subcurves that each describe a homogeneous pore system. Peters and Klavetter (1988) first proposed the superposition of two uni-modal pore systems to represent the hydraulic properties of fractured rock. This approach was used by Othmer et al. (1991) to describe soil hydraulic properties with a distinct secondary pore system. The approach was generalized by Durner (1992), Wilson et al. (1992), and Ross and Smettem (1993) to consider multi-modal pore size distributions. One of the main problems to overcome was the description of macropore effects near saturation because close to saturation uni-modal functions such as the closed-form Mualem–van Genuchten model (van Genuchten, 1980) become inaccurate (Durner (1994) and references therein). This problem was further discussed in detail by Mohanty et al. (1997), who also provided an alternative representation of multi-modal retention and conductivity functions by refining the Fermi-function approach used by Wilson et al. (1992) and the bi-modal function proposed by Smettem and Kirkby (1990). The multi-modal functions were found to perform better in simulations of non-capillary-dominated flow near saturation in field soils, and generally improved the representation of the hydraulic functions of structured soils.
Recently, a more physically based approach to represent the hydraulic properties of structured porous media was derived by Tuller and Or (2002) and Or and Tuller (2000, 1999). They provided a framework to derive relationships between matric potential, liquid retention, and hydraulic conductivity for arbitrary pore space geometry using physical solid–liquid interactions and basic flow behavior. They also provided examples of hydraulic functions for macroporous and aggregated soils, and for fractured tuff from Yucca Mountain. While the derived functions are rather complex, they give estimates of the saturated conductivity and good representations of the hydraulic properties of structured porous media if additional data on pore space geometry are available.
The multi-modal representation of soil water retention introduced by Durner (1992) assumes that the pore size distribution of structured porous media can be described by a linear superposition of pore size subsystems. In combination with Mualem's conductivity prediction model (Mualem, 1976; van Genuchten, 1980), representations of the hydraulic conductivity functions (Durner, 1994) can be obtained by numerical integration that show typical multi-modal shapes. These representations can then be used to model preferential water flow strictly by means of Darcy's Law (Zurmühl and Durner, 1996).
Since the multi-modal representation of the retention curve according to Durner (1994) does not allow one to explicitly express the soil matric head as a closed-form function of the volumetric water content, it was assumed that Mualem's predictive model for the relative hydraulic conductivity function could only be evaluated by numerical integration (Durner, 1994; Zurmühl and Durner, 1996), but, as we show in this note, such an explicit functional form is not needed to evaluate the integral in Mualem's model and a closed-form expression for the multi-modal representation of the hydraulic conductivity curve can be derived.
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DERIVATION OF CLOSED-FORM EXPRESSIONS
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The multi-modal representation of the retention function S = S(h) for soils with heterogeneous pore systems is obtained by assuming a linear superposition of van Genuchten type subcurves Si = Si(h),1 
i k representing the retention functions of the pore size subsystems (Durner, 1992, 1994) that each represent a nonlinear pore size distribution:
 | [1] |
in which the subcurves are given by
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where k is the number of subsystems, and wi are the weighting factors subject to 0 < wi < 1 and 
wi = 1. The van Genuchten parameters for the subcurves are given by 
i, ni, and mi subject to the conditions i > 0, mi > 0, and ni > q for the parameter q > 0 of Eq. [3].
Additionally, in Eq. [1] denotes the actual, s the saturated, and r the residual volumetric water content.
By using Mualem's prediction model (Mualem 1976) in its most generalized form (Hoffmann-Riem et al., 1999) the unsaturated hydraulic conductivity K(S0) can be calculated from relative saturation S0 = S(h0) at the soil matric head h0 by using
 | [3] |
in which the parameters p, q, r (subject to p 
0, q > 0, and r > 1) determine the shape of the hydraulic conductivity function K.
Applying the chain rule of calculus, the integral in Eq. [3] can be converted to an integral in h (Fayer and Simmons, 1995):
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Therefore, using mi = 1 – q/ni and Si,0 = Si(h0), the hydraulic conductivity K(S0) can be written as (van Genuchten, 1980):
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Equation [5] may be expressed in terms of the matric head h; thus, to obtain the following closed-form equation for the relative hydraulic conductivity
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For r = 1, which includes the case of the Burdine model (r = 1, q = 2, p = 2), the multi-modal unsaturated conductivity function can be written as a sum of conductivity subcurves:
 | [7] |
in which 
= 
i = 1kwii. This representation of the hydraulic conductivity curve is similar to that given by Ross and Smettem (1993) and by Mohanty et al. (1997) for the capillary dominated flow domain. Only the subcurves are derived from Mualem's model (r = 2), whereas for Eq. [7], as in Burdine's model, r = 1 is used. However, this may not be a significant difference if the parameters p and q are both optimized to obtain representations for the uni-modal hydraulic conductivity subcurves (Hoffmann-Riem et al., 1999; Kosugi, 1999). The case r = 1 also includes the approach proposed by Othmer et al. (1991), if, as in their example, the same matric potential is assumed for both flow regimes.
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EXAMPLES
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Three examples are given to demonstrate the applicability and flexibility of the multi-modal representation for the soil hydraulic functions. Figures 1
to 3 show the total fitted multi-modal curves (solid lines) and the corresponding subcurves (dashed lines) that add up to the total curve. In case of an aggregated loam soil (Smettem and Kirkby, 1990), the bi-modal representation of the retention curve was fitted to measured data to estimate the shape parameters (Table 1) that best described the curve (Fig. 1a). By using these parameters and choosing p = 0.5, q = 1, and r = 2 according to Mualem (1986), we obtained a representation of the conductivity curve that matched the measured data reasonably well (Fig. 1b); see also Durner (1994) for this example. Figure 2 shows similar results for a tri-modal representation of the retention curve in case of a sandy loam presented by Mallants et al. (1997), given by their Fig. 2c, 8c, and 9b.
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Fig. 1. (a) Fitted water saturation data of an aggregated loam soil, and (b) the related relative hydraulic conductivity for r = 2. Solid lines represent the total retention or conductivity curve, while the dashed lines refer to the subcurves. Data source: Smettem and Kirkby (1990).
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Fig. 3. (a) Fitted water saturation data of a silt loam soil with biological macropores, and (b) the related relative hydraulic conductivity for r = 3. Solid lines represent the total retention conductivity curve, while the dashed lines refer to the subcurves. Data source: Mohanty (1999).
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Fig. 2. (a) Fitted water saturation data of an heterogeneous field soil (Ap horizon), and (b) the related relative hydraulic conductivity for r = 2. Solid lines represent the total retention or conductivity curve, while the dashed lines refer to the subcurves. Data source: Mallants et al. (1997).
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In contrast to the first two examples, the application of a bi-modal representation to a macroporous silt loam (Mohanty, 1999) resulted in an inadequate representation of the conductivity curve, although the fit of the retention data (Table 1) was rather good (Fig. 3a
). However, by adapting the parameter r (i.e., by using r = 3 instead of r = 2 and keeping p = 0.5 and q = 1), the relative conductivity could be brought into much better agreement with the measured data (Fig. 3b).
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DISCUSSION AND CONCLUSIONS
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The derived closed-form expression for the multi-modal representation of the relative unsaturated conductivity provides a simple method to calculate the soil hydraulic conductivities of structured soils. Because of its simple form, the analytical expression is easy to implement into computer codes and also avoids numerical integration errors. The derived formulas presented by Eq. [5] and [6] also give better insight into the relations between the retention parameters (wi, i, ni, q) and the parameters p, q, and r that provide additional degrees of freedom for the conductivity curve. Moreover, we showed that the representation of the retention curve by a superposition of uni-modal subcurves implies that the multi-modal hydraulic conductivity function as predicted with Mualem's model is a superposition of conductivity subcurves. This superposition of conductivity subcurves is linear when the parameter r becomes 1.
The closed-form expression thus improves the applicability of multi-modal representations of the hydraulic curves to simulate water flow in structured soils with the equivalent continuum approach (Simunek et al., 2003; Köhne and Mohanty, 2005) using the one-dimensional Richards equation. The multi-modal functions hence also facilitate the estimation of hydraulic properties of structured soils using inverse methods (Durner et al., 1999; Hopmans et al., 2002).
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REFERENCES
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- Durner, W. 1992. Predicting the unsaturated hydraulic conductivity using multi-porosity water retention curves. p. 185–202. In M.Th. van Genuchten et al. (ed.) Proceedings of the International Workshop on Indirect Methods for Estimating the Hydraulic Properties of Unsaturated Soils. University of California, Riverside.
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- Durner, W., E. Priesack, H.-J. Vogel, and T. Zurmühl. 1999. Determination of parameters for flexible hydraulic functions by inverse modelling. p. 817–829. In M.Th. van Genuchten et al. (ed.) Proceedings of the International Workshop on Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media. Vol. 2. University of California, Riverside.
- Fayer, M.J., and C.S. Simmons. 1995. Modified soil water retention functions for all matric suctions. Water Resour. Res. 31:1233–1238.[CrossRef]
- Hoffmann-Riem, H., M.Th. van Genuchten, and H. Flühler. 1999. General model for the hydraulic conductivity of unsaturated soils. p. 31–42. In M.Th. van Genuchten et al. (ed.) Proceedings of the International Workshop on Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media. Vol. 1. University of California, Riverside.
- Hopmans, J.W., J. Simunek, N. Romano, and W. Durner. 2002. Simultaneous determination of water transmission and retention properties. Inverse methods. p. 963–1008. In J.H. Dane and G.C. Topp (ed.) Methods of soil analysis. Part 4. SSSA Book Ser. 5. SSSA, Madison WI.
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- Kosugi, K. 1999. Lognormal distribution model for soil hydraulic properties. p. 19–30. In M.Th. van Genuchten et al. (ed.) Proceedings of the International Workshop on Characterization and Measurement of the Hydraulic Properties of Unsaturated Porous Media. Vol. 1. University of California, Riverside.
- Mallants, D., P.-H. Tseng, N. Toride, A. Timmerman, and J. Feyen. 1997. Evaluation of multimodal hydraulic functions in characterizing a heterogeneous field soil. J. Hydrol. (Amsterdam) 195:172–199.
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- Mualem, Y. 1976. A new model for predicting the hydraulic conductivity of unsaturated porous media. Water Resour. Res. 12:513–522.[CrossRef]
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- Or, D., and M. Tuller. 1999. Liquid retention and interfacial area in variably saturated porous media: Upscaling from single-pore to sample-scale model. Water Resour. Res. 35:3591–3605.[CrossRef]
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- Othmer, H., B. Diekkrüger, and M. Kutilek. 1991. Bimodal porosity and unsaturated hydraulic conductivity. Soil Sci. 152:139–150.
- Peters, R.R., and E.A. Klavetter. 1988. A continuum model for water movement in an unsaturated fractured rock mass. Water Resour. Res. 24:416–430.
- Ross, P.J., and R.J. Smettem. 1993. Describing soil hydraulic properties with sums of simple functions. Soil Sci. Soc. Am. J. 57:26–29.
- Simunek, J., N. Jarvis, M.Th. van Genuchten, and A. Gärdenäs. 2003. Review and comparison of models for describing non-equilibrium and preferential flow and transport in the vadose zone. J. Hydrol. (Amsterdam) 272:14–35.
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- Tuller, M., and D. Or. 2002. Unsaturated hydraulic conductivity of structured porous media: A review of liquid configuration-based models. Available at www.vadosezonejournal.org. Vadose Zone J. 1:14–37.[Abstract/Free Full Text]
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- Zurmühl, T., and W. Durner. 1996. Modeling transient water and solute transport in a biporous soil. Water Resour. Res. 32:819–829.
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ABSTRACT
INTRODUCTION
