Published in Soil Sci. Soc. Am. J. 68:713-718 (2004).
© 2004 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
DIVISION S-1—SOIL PHYSICS
A Simple Method for Estimating Water Diffusivity of Unsaturated Soils
Quanjiu Wanga,b,c,
Mingan Shaoc and
Robert Horton*,d
a State Key Laboratory of Soil Erosion and Dryland Farming on the Loess Plateau, Institute of Soil and Water Conservation, Chinese Academy of Sciences, Yangling, 712100
b Institute of Water Resources Research, Xi'an University of Technology, Xi'an 710048, China
c Institute of Geographical Sciences and Natural Resources Research (Beijing 100101) and Institute of Soil and Water Conservation (Yangling, 712100) of Chinese Academy of Sciences, China
d Dep. of Agronomy, Iowa State University, Ames, IA 50011
* Corresponding author (rhorton{at}iastate.edu).
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ABSTRACT
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Numerical models have been extensively used for predicting water and solute transport in saturated and unsaturated soils. Soil hydraulic properties are required for quantitatively describing water and chemical transport processes in soils by the numerical models. Soil water diffusivity is one of the important hydraulic properties. Several approaches have been developed to estimate the soil water diffusivity, however the intensive calculations and time-consuming measurements required by the methods for determining soil water diffusivity limit the application of the methods. It is necessary to develop a method with easily performed experiments for determining soil water diffusivity. In this paper, the problem of water absorption into a horizontal soil column is solved, and the relationship between cumulative infiltration and infiltration rate with the distance of the wetting front are obtained. Based on the relationships, a new and relatively simple method for estimating soil water diffusivity is presented in this paper. Experiments of water absorption into 60-cm long and 25-cm long horizontal soil columns were conducted to evaluate the method. Estimates of soil water diffusivity by the new method were in good agreement with estimates by the Bruce and Klute method. With the new method short (approximately 8 cm) soil columns can be used, and it is possible to use the new method to estimate soil water diffusivity of undisturbed soils. Therefore, the present method provides simplicity for determining soil water diffusivity.
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INTRODUCTION
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MATHEMATICAL MODELS have been used in research and management to predict water and solute transfer in soil and ground water. The accuracy of water flow and solute transport predictions obtained with the models depends to a large extent on the reliability and accuracy of the soil hydraulic properties. The required hydraulic properties are hydraulic conductivity, water diffusivity, and specific water capacity (Bohne et al., 1995). Among the three parameters, only two of them are independent.
In recent years, there have been increased efforts to estimate soil water diffusivities of unsaturated soils, as one of the major soil hydraulic properties. Usually horizontal infiltration experiments have been used to relate soil water diffusivity to the volumetric water content by the method of Bruce and Klute (1956). The method is based on the Boltzmann transformation. The slope of the water content distribution curve along the soil column needs to be measured to estimate the water diffusivity. Kirkham and Powers (1972) described in detail this common method for estimating soil water diffusivity. However, it is difficult to exactly determine the slope of the water content distribution curve, and thus the difficult estimation of the slope of the water content distribution leads to soil water diffusivity estimation error. Cassel et al. (1968) presented a method for estimating soil water diffusivity from time-dependent soil water content distributions in the horizontal redistribution process. Their method requires measuring water content distribution with time and also involves both relatively intensive calculation and time-consuming experiments. Clothier et al. (1983) presented a fitting function chosen from those presented by Philip (1960) to approximate the water distribution curve in the Bruce and Klute (1956) method. This made possible a simple analytical expression of the water diffusivity by avoiding finding the slopes of the soil water distribution curve. However, the fitting function of Clothier et al. (1983) may not apply to all soils. McBride and Horton (1985), based on the Bruce and Klute (1956) method, developed a method of determining the water diffusivity from horizontal infiltration experiments. However the method involves intensive calculations. Warrick (1994) gave a detailed review on soil water diffusivity estimation for fixed water content at the inlet boundary. Shao and Horton (1996) developed a method to estimate the soil water diffusivity by using a nonhysteretic analytical solution to horizontal redistribution based on general similarity theory. The method only requires information on the advance of the wetting front with time to obtain the soil water diffusivity in the process of water redistribution. Shao and Horton (1996) assumed a power function between the soil water diffusivity and the soil water content, however the form of their power function may not apply to all soils. A power function relationship between soil water diffusivity and relative water content may have a more general application to soils than does the Shao and Horton (1996) power function relationship.
The purpose of the present paper is to develop a method for determining the water diffusivity of unsaturated soils. To obtain an analytical expression of the diffusivity, the problem of water absorption into a horizontal soil column is solved. The relationships among cumulative infiltration and infiltration rate and the wetting front are obtained. With these relationships the soil water diffusivity as a function of relative water content can easily be determined. Three soils were used to evaluate the method. This paper compares analytical values of the soil water diffusivity with those obtained by the method of Bruce and Klute (1956).
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THEORY
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Darcy's equation describing one-dimensional horizontal flow of water in unsaturated soil is
 | [1] |
where q is soil water flux (cm min–1), k(h) is unsaturated hydraulic conductivity (cm min–1), h is soil–water suction (cm), x is the horizontal distance (cm), D(
) is the soil water diffusivity (cm2 min–1), and is the soil water content (cm3 cm–3).
The following equation was used to describe the unsaturated conductivity (Brooks and Corey, 1964)
 | [2] |
where ks is saturated conductivity (cm min–1), hd is air-entry suction (cm), and m is a constant.
The soil water retention curve (Brooks and Corey, 1964) is
 | [3] |
where s is the saturated soil water content (cm3 cm–3), r is the residual water content (cm3 cm–3), n is a parameter, and the left hand ratio is equal to the relative water content, S.
For the water absorption problem of one-dimensional horizontal soil column, the general equation and the initial and boundary conditions are
 | [4] |
where i is the initial soil water content (cm3 cm–3).
Integrating Eq. [4] with respect to gives,
 | [5] |
Parlange (1971) considered that the first term in Eq. [5] is small compared with the rest of the terms in Eq. [5] and may be neglected when soil water content is close to the saturated water content. Eq. [5] reduces to:
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We can let
 | [7] |
Because the right side in Eq. [7], describing the soil water flux at the soil surface, is actually infiltration rate, i, Eq. [6] becomes
 | [8] |
Converting Eq. [8] as
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Then Eq. [9] can be integrated and converted as follows
 | [10] |
where hx is the suction at the x position. If the initial water content is relatively small and hx is relatively large, h1 –mx is very small at the wetting front, x = xf, and xf is the wetting front distance and can be observed visually. Therefore Eq. [10] can be simplified
 | [11] |
and i is expressed as
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Equation [8] actually assumes that the flux is only a function of time, and Eq. [12] expresses the relationship between the infiltration rate and the wetting front distance. The infiltration rate decreases with the advance of the wetting front.
Combining Eq. [10] with [12], a new relation is found
 | [13] |
The suction, hx, at a distance, x, can be calculated by Eq. [13]. Combining Eq. [3] with [13] the soil water content at a distance, x, can also be expressed as
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The cumulative infiltration to the wetting front can be expressed as
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Combining Eq. [14] with [15], the cumulative infiltration is as follows
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If the initial water content (air-dried soil) is small, and the residual water is considered to equal the initial water content, the cumulative infiltration is
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Equation [17] shows that the cumulative infiltration is a linear function of the distance to the wetting front.
The infiltration rate is the differential rate of the cumulative infiltration
 | [18] |
Combining Eq. [18] with [17] and [12]
 | [19] |
Equation [19] shows the relationship between time and the wetting front distance. Combining Eq. [14] with Eq. [19]
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Let
Hence
 | [21] |
The well-known Boltzmann transform equation is x(,t) = 
()t1/2, and Eq. [21] is similar in form to the expression of the Boltzmann transform.
Soil water diffusivity can be expressed as a function of soil water retention curve and unsaturated soil water conductivity curve
 | [22] |
The soil water diffusivity may be described as a power function of relative water content, as follows for a Brooks–Corey (Brooks and Corey, 1964) soil
 | [23] |
where Ds is the diffusivity of saturated soil and L is a parameter.
Combining Eq. [2] and [3] with Eq. [22] gives
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To obtain the related parameters in Eq. [24], Eq. [24] can be rearranged as
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Hence
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 | [27] |
(m – 1)/n in Eq. [26] and [27] can be calculated based on Eq. [17], and kshd/(m –1) in Eq. [26] can be estimated based on Eq. [12]. Therefore, when the relations of the infiltration rate and the cumulative infiltration with the wetting front distance are known, the soil water diffusivity can be easily estimated.
To get even easier parameter expressions, Eq. [12] and [17] are expressed as follows:
 | [28] |
 | [29] |
Where
 | [30] |
 | [31] |
Hence Ds and L can be expressed as follows:
 | [32] |
 | [33] |
When a and b are obtained through the analysis of experimental data, the parameters Ds and L can be estimated with Eq. [32] and [33]. Then the soil water diffusivity can be readily determined by Eq. [24].
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MATERIALS AND METHODS
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One-dimensional water absorption experiments for horizontal soil columns were performed to evaluate the new method. Experiments were conducted using a horizontal cylindrical column with a length of 60 cm and a diameter of 9 cm to measure soil water content by a 
radiation attenuation method. The relatively large soil column and the radiation attenuation method were used to accommodate the Bruce and Klute method. The new method did not require measurements of soil water content profiles. Three soils from Shaanxi province of China were used as the experimental materials. The soils were Yulin sand, Shuide loam and Xian silt loam, and the related physical properties of the three soils are listed in Table 1. Air-dried soils were passed through a 2-mm screen, and the columns were packed layer by layer with air-dried soils of a given soil bulk density. A Marriotte tube was used to supply water, and a zero water head was maintained at the soil inlet. The volume of infiltrated water was recorded with time. A radiation attenuation method (relative error <2% for volumetric water content) was used to measure soil water content profiles in 1-h intervals, to calculate the diffusivity by the Bruce and Klute (1956) method. The wetting front distance with time was observed visually based on obvious color differences at the interface of the wet and dry soil. The infiltration rate versus time was calculated based on the cumulative infiltration with time. Using the observed data of the cumulative infiltration versus time, the changes of infiltration rate and the wetting front distance with time, soil water diffusivities were calculated by the new analytical relationships presented in the theory section. The diffusivities of the three soils were also calculated with the Bruce and Klute method based on the measured soil water content profile.
It is convenient to use small soil columns to estimate soil water diffusivity. To indicate the effect of soil column size on the new method, a horizontal infiltration experiment with Yulin sand was also conducted using a column with a length of 25 cm and a diameter of 4 cm. A Marriotte tube was used to supply water, and a zero water head was maintained at the soil inlet. The volume of infiltrated water was recorded with time. The wetting front distance with time was observed visually. The new method was used to estimate soil water diffusivity.
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RESULTS
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As mentioned above, to obtain soil water diffusivity of unsaturated soils, the advance of the wetting front with time, cumulative infiltration and infiltration rate are measured in the experiments. Figure 1 and 2
show the measured cumulative infiltration versus the wetting front distance and the infiltration rate versus the inverse wetting front distance, respectively, for each soil. Each data set is treated with Eq. [28] and [29]. The a and b parameters are related to Ds and L in Eq. [23], which describes the soil water diffusivity by a power function of relative water content. The a and b parameters are estimated by least squares regression of fitting the observed data of the cumulative infiltration versus the wetting front distance and the infiltration rate versus the inverse wetting front distance, respectively, for the soils, and the results are shown in Table 2. All of the coefficients of determination values, R2, of the fitting are >0.97. Therefore it may be safe to say that Eq. [28] and [29] describe the relations of cumulative infiltration versus the wetting front distance and the infiltration rate versus the inverse wetting front distance well. The Ds and L in Eq. [23] are calculated by using Eq. [32] and [33], and the results are shown in Table 3. To evaluate the new method, the soil water diffusivity is also determined by the method of Bruce and Klute (1956), and the results are also listed in Table 3.
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Fig. 1. Cumulative infiltration versus the wetting front distance for three soils ([a] Yulin Sand, [b] Shuide Loam, and [c] Xian Silt Loam).
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Fig. 2. Infiltration rate versus the inverse wetting front distance for three soils ([a] Yulin Sand, [b] Shuide Loam, and [c] Xian Silt Loam).
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The soil water diffusivity of the new method is determined by Eq. [32] and [33]. Because the values of Ds and L are dependent on a and b constants in Eq. [28] and [29], it may be interesting to look at the sensitivities of soil water diffusivity to a and b coefficients in Eq. [28] and [29]. The Ds value depends on both a and b, but L only depends on b. When b increases, L also increases. For a given soil, saturated and initial water content are constant (here s is assumed to 0.45 cm3 cm–3, and i is 0.02 cm3 cm–3), a sensitivity analysis shows that L increases about 10 times when b increases from 0.3 to 0.4. The values listed in Table 2 and 3 also show that L increases as b increases. The value of L influences the shape of the curve of soil water diffusivity, thus the shape of the curve of soil water diffusivity is mainly affected by the value of b. We may know from Eq. [32] that Ds linearly increases as parameter a increases for given values of b and initial and saturated water contents. For given values of parameter a, initial and saturated water content, Ds increases as b decreases. In addition, because Ds determines the magnitude of the diffusivity function, the magnitude of the diffusivity is affected by both parameters, a and b.
The values of soil water diffusivity estimated by the simple approach are compared with those obtained by the method of Bruce and Klute (1956), and the results are shown in Fig. 3
. The results indicate that the water diffusivity estimated by the simple method is very close to the values obtained by the method of the Bruce and Klute (1956). The Root Mean Square Error (RMSE) between the values estimated by the simple method and those obtained by the method of Bruce and Klute (1956) are analyzed. The values of the RMSE for three soils, from Yuling sand, Shuide loam, and Xian silt loam of Shaanxi in China, are 0.54, 0.18, and 0.13, respectively. The small RMSE values indicate that the simple method can be effectively used to estimate soil water diffusivity.
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Fig. 3. Comparison of the simple method with the Bruce and Klute method ([a] Yulin Sand, [b] Shuide Loam, and [c] Xian Silt Loam; the solid points represent the Bruce and Klute method, the solid curves represent the new method for large soil columns, and the diamonds the new method for a small soil column).
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To analyze the effect of soil column size on the new method, the infiltration data observed for the small soil column are used to estimate the soil water diffusivity. The parameters a and b for Yulin sand are 1.00 and 0.28, respectively; and the Ds and L, estimated by the new method, are 12.5 and 3.3, respectively, when the wetting front distance is 15 cm. Figure 3a shows that the soil water diffusivity estimated by the small and large soil columns are very similar. The results indicate that the new method is applicable to relatively small soil columns.
To analyze the influence of the wetting front distance on the soil water diffusivity, the soil water diffusivity is also calculated with different wetting front distance, as shown in Fig. 4
. The results indicate that the soil water diffusivity calculated at a wetting front distance of 10 cm is nearly identical to that for a wetting front distance of 15 cm. Likewise, soil water diffusivity calculated for a wetting front distance of 8 cm is similar to those for wetting front distances of 10 and 15 cm. When the wetting front is <8 cm, soil water diffusivity differs from the 15-cm wetting front values of soil water diffusivity. This analysis implies that the new method may be ap-plied to soil columns having a minimum wetting front length of 8 cm.
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CONCLUSIONS
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Soil water diffusivity is one of the important hydraulic properties affecting water and solute transport rates in soils. It is necessary to develop a simple method to determine soil water diffusivity because the current methods have limitations. In this paper a simple method has been developed to determine the water diffusivity of unsaturated soils. The problem of water absorption into a horizontal soil column is solved based on the assumption of Parlange (1971), and the relationships among cumulative infiltration with the wetting front distance and infiltration rate with the wetting front distance are analytically obtained. When the relationships are available, the soil water diffusivity can be readily estimated. Estimates of soil water diffusivity by the simple method are in good agreement with estimates by the Bruce and Klute method. The new method can be applied to disturbed or undisturbed soil columns with 8 cm or more wetting front length. The present method provides simplicity for determining soil water diffusivity. The method is applicable to laboratory determination of soil water diffusivity of unsaturated soils.
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ACKNOWLEDGMENTS
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This work was funded in part by the China National Natural Science Foundation (Project numbers: 90102012 and 40025106) and the key project of resources, ecological and environmental research of the Chinese Academy of Sciences (Project No: KZCX2-411), and this journal paper of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA, Project No. 3287, was supported in part by Hatch Act and State of Iowa funds.
Received for publication February 26, 2003.
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REFERENCES
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- Bruce, R.R., and A. Klute. 1956. The measurement of soil moisture diffusivity. Soil Sci. Soc. Am. Proc. 20:458–462.
- Cassel, D.K., A.W. Warrick, D.R. Nielsen, and J.W. Bigger. 1968. Soil-water diffusivity values based upon time dependent soil-water content distributions. Soil Sci. Soc. Am. Proc. 32:774–777.
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- Kirkham, D., and W.L. Powers. 1972. Advanced soil physics. Wiley Interscience, New York.
- McBride, J. and R. Horton. 1985. An empirical function to describe measured water distributions from horizontal infiltration experiments. Water Resour. Res. 21:1539–1544.
- Philip, J.R. 1960. General method of exact solutions of the concentration-dependent diffusion equation. Aust. J. Phys. 13:1–12.
- Parlange, J.-Y. 1971. Theory of water movement in soils: 1. One-dimensional absorption. Soil Sci. 111:134–137.
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ABSTRACT
INTRODUCTION
