Soil Science Society of America Journal 66:1733-1739 (2002)
© 2002 Soil Science Society of America
DIVISION S-1—SOIL PHYSICS
Horizontal Infiltration Method for Determining Brooks-Corey Model Parameters
Quanjiu Wanga,
Robert Horton*,b and
Mingan Shaoc
a Institute of Geographic Sciences and Natural Resources Research, Chinese Academy of Sciences, Building 917, 3 Datun Road, Anwai, Beijing 100101, China
b Dep. of Agronomy, Iowa State Univ., Ames, IA 50011
c State Key Lab. of Soil Erosion and Dryland Farming, Institute of Soil and Water Conservation, Chinese Academy of Sciences and Ministry of Water Resources, Yangling, 712100, China
* Corresponding author (rhorton{at}iastate.edu)
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ABSTRACT
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Mathematical models have been widely used to predict water and solute transfer in soil and ground water. The accuracy of water flow and solute transport predictions obtained with these models depends to a large extent on the reliability and accuracy of the soil hydraulic properties. The main soil hydraulic properties are the water retention curve and hydraulic conductivity function. Most methods for determination of soil hydraulic properties that include flow measurements require elaborate instrumentation and a substantial amount of time. It is important to develop simple flow-based experimental methods to estimate soil hydraulic properties. In this paper, an analytical method was developed to determine Brooks-Corey model parameters. The problem of water absorption into a horizontal soil column was solved, and the relationships of cumulative infiltration, infiltration rate, and infiltration time with distance to the wetting front were obtained. Based upon these relationships, a method to estimate soil hydraulic properties was developed. Numerical simulations of water flow into horizontal soil columns were used to evaluate the analytical method. The analytical model was fitted to the numerical infiltration results with coefficients of determination ranging from 0.97 to 0.99. The results based upon the numerical data indicate that the estimated hydraulic properties agreed closely with the given values. The analytical method provides a simple and quick transient water flow approach to estimate soil hydraulic properties.
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INTRODUCTION
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SOIL HYDRAULIC PROPERTIES are critical for prediction of water and solute transfer in soil and ground water. The accuracy of water flow and solute transport predictions depends to a large extent on the reliability and accuracy of the soil hydraulic properties. The required hydraulic properties are mainly the soil water retention curve and hydraulic conductivity function.
Several methods have been developed to directly measure soil water retention curves and hydraulic conductivity functions (Green et al., 1986; Klute and Dirksen, 1986; Bohne et al., 1993). Most of the methods are clear in concept, but they are time-consuming, expensive, and include uncertainty about their limitations (van Genuchten, 1992). Efforts have been made to develop simple methods to estimate the hydraulic properties based on easily obtained soil properties. For example, soil texture data have been used to estimate soil water retention curves (Puckett et al., 1985; Bouma and van Lanen, 1987; Dane and Puckett, 1992; Arya et al., 1999). Unsaturated hydraulic conductivity has been predicted from soil water retention curves (Brooks and Corey, 1964; Mualem, 1976; van Genuchten, 1980; Zhang and van Genuchten, 1994; Perrir et al., 1996), but parameter uniqueness still limits many of the applications (van Genuchten, 1992).
Shao and Horton (1998) developed an integral method to estimate van Genuchten (1980) model parameters. Their method used an analytical solution to Richards equation for transient horizontal infiltration. To solve Richard's equation analytically they assumed that at any given time the soil water suction distribution was linear from the infiltration boundary to the wetting front. Even with this assumption Shao and Horton (1998) reported good agreement between van Genuchten parameters determined from horizontal infiltration experiments and parameters determined by traditional water retention curve measurements. The Shao and Horton (1998) work is one example of how a simple flow experiment can be used to determine soil hydraulic properties. However, there is a need to develop additional methods for determining parameters for models other than the van Genuchten model. It is also important to develop simple methods that have the least restrictive water flow assumptions so that they can be applied to a wide range of soils.
The objective of this paper is to present an analytical method which makes use of horizontal infiltration data to determine the Brooks and Corey (1964) model parameters. The problem of water absorption into a horizontal soil column is solved using an assumption that at any given time there is a nonlinear distribution of soil water suction from the infiltration boundary to the wetting front. Based upon this assumption, we present relationships between cumulative infiltration, infiltration rate, and infiltration time versus the location of the wetting front. Knowing these relationships, the Brooks-Corey model parameters can be determined.
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THEORY
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The Darcy equation describing one-dimensional horizontal flow of water in unsaturated soil is:
 | [1] |
where q is soil water flux density (cm min-1), k(h) is unsaturated hydraulic conductivity (cm min-1), h is soil-water suction (cm), and x is the horizontal position (cm).
The following equation describes the soil water retention curve (Brooks and Corey, 1964):
 | [2] |
where 
is the soil water content (cm3 cm-3), s is the saturated soil water content (cm3 cm-3), r is the residual water content (cm3 cm-3), Se is effective saturation, hd is air entry suction (cm), and n is a parameter.
The unsaturated hydraulic conductivity can be expressed as (Brooks and Corey, 1964):
 | [3] |
where ks is saturated hydraulic conductivity (cm min-1), and m is a constant.
For the problem of water absorption into a one-dimensional horizontal soil column, the flow equation and the initial and boundary conditions are:
 | [4] |
where i is the initial soil water content (cm3 cm-3).
To solve Eq. [4], we assume that the hd/h distribution along the soil column at any time can be described as:
 | [5] |
where h(x) is the suction distribution with position x, xf is the wetting front distance from the infiltration surface (cm), and a is a coefficient.
Combining Eq. [1] with Eq. [5], the flux at any position, x, is:
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When x = 0, the flux, q(0), is the infiltration rate, i:
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Equation [7] indicates that the infiltration rate is a function of the wetting front distance, and the infiltration rate decreases as the distance to the wetting front increases.
According to Eq. [2], [3], and [5], the right side of the Eq. [4] expresses as:
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And the left side is:
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Combining Eq. [4] with Eq. [8] and [9]:
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Integrating Eq. [10]:
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Let
Equation [11] expresses as:
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Equation [12] indicates that x2f is a linear function of time, t. This is similar to the Boltzmann transformation expression (x2f = 
2t; where is the Boltzmann variable) when xf is the wetting front distance.
The cumulative infiltration to a certain wetting front can be expressed as:
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According to Eq. [2] and [5], soil water content distribution along a soil column may be expressed as:
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Combining Eq. [13] with Eq. [14], the cumulative infiltration is:
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When x = xf and soil water content is the initial water content, the parameter, a, can be expressed as:
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When the initial water content is small, a is close to 1. For simplicity, let
Equation [15] converts as:
 | [17] |
Equation [17] indicates that cumulative infiltration is a linear function of the wetting front distance.
Equations [7], [12], and [17] describe the theoretical functions for unsaturated soil water movement. The required hydraulic properties are the soil water characteristic curve and unsaturated conductivity function. Within Eq. [2] and [3], there are six parameters (r, s, ks, hd, m, and n) that need to be estimated. Three of the parameters (r, s, and ks) should be obtained experimentally.The parameters s, and ks should be measured directly on the horizontal soil column, and r can be estimated as water content of soil approaching air-dry conditions. The residual water content should be less than or equal to i. Three parameters (hd, m, and n) can be estimated with the horizontal infiltration data.
Equations [17], [7], and [12] can be expressed as follows:
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 | [19] |
 | [20] |
where:
Therefore, parameters n, m, and hd can be expressed as:
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 | [22] |
 | [23] |
Equation [21] shows that n is only a function of A1 when s, r, and i are given. When the relationship of cumulative infiltration versus wetting front distance is obtained, the parameter, n, may be estimated with Eq. [21]. The parameter, hd, may be estimated with Eq. [22] when the relationship of infiltration rate versus wetting front distance is measured. Finally, m may be estimated from Eq. [23] when the relationship of wetting front distance versus time is measured.
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MATERIALS AND METHODS
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HYDRUS-1D Version 2.0 (Simunek et al., 1998) was used to solve Eq. [4] for one-dimensional horizontal water movement to produce soil water movement data required to examine the analytical method. Three numerical soils were used in the study. Table 1 shows the values of the physical parameters for the three soils. Following Brooks and Corey (1964) the parameter m was taken as 2 + 3n. When the parameter values listed in Table 1 were used in the HYDRUS-1D model, the cumulative infiltration versus time and the soil suction distribution along the soil column at different times were calculated. For the numerical calculations the suction at the infiltration boundary, x = 0, was fixed at h = hd. The infiltration rate versus time was calculated based on the cumulative infiltration versus time. The wetting front distance versus time also was determined from the calculated soil suction distribution at different times. Using the numerical data of the cumulative infiltration versus time, the infiltration rate versus time, and the wetting front distance versus time, the analytical relationships between cumulative infiltration and the wetting front distance and between infiltration rate and time were determined. First, A1, A2, and A3 were determined by linear regression, then n, hd, and m were estimated for each numerical soil with Eq. [21], [22], and [23]. Once the hydraulic parameters were estimated, water characteristic curve and hydraulic conductivity function for each numerical soil were calculated and compared with the water characteristic curve and hydraulic conductivity function used in the numerical model.
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RESULTS
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Figures 1 and 2
show the cumulative infiltration versus the wetting front distance, the infiltration rate versus the inverse wetting front distance, and the wetting front distance squared versus time, respectively, for each of the numerical soils. Each numerical data set was evaluated with Eq. [18], [19], or [20] via least squares regression. Regression slopes and coefficients of determination are listed in Table 2. The coefficients of determination range from 0.97 to 0.99 indicating the obvious linear relationships between the cumulative infiltration and the wetting front distance, between the infiltration rate and the inverse wetting front distance, and between the wetting front distance squared and time.
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Fig. 1. Cumulative infiltration and infiltration rate versus wetting front distance for three numerical conditions: (a) and (d) are for numerical Soil 1, (b) and (e) are for numerical Soil 2, and (c) and (f) are for numerical Soil 3.
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Fig. 2. Wetting front distance squared versus time for three numerical conditions: (a) is for numerical Soil 1, (b) is for numerical Soil 2, and (c) is for numerical Soil 3.
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Table 2. Results of linear regression for important infiltration relationships (Eq. [18], [19], and [20]) for the three numerical soils. A1 and R21 are slope and coefficient of determination for the cumulative infiltration versus the wetting front distance, A2 and R22 are slope and coefficient of determination for the infiltration rate versus the inverse wetting front distance, and A3 and R23 are slope and coefficient of determination for the wetting front distance squared versus time.
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Using the regression parameters, A1, A2, and A3, the parameters hd, n, and m were estimated with Eq. [21], [22], and [23], in which the values of s, r, and ks were considered to be the same as the original values that were used to produce the numerical soil water movement data. The parameter estimates for all three numerical soils are given in Table 3.
The estimated values of hd are very similar to the given values of hd for all three numerical soils. The maximum absolute difference between estimated and actual hd is 2 cm. The estimated values of n are very similar to the given values of n for all three numerical soils. The maximum absolute difference between estimated and actual n is 0.08. The estimated values of m are similar in magnitude to the given values of m, however, the estimated values of m do not differ as much among the three soils as do the actual m values. The actual m values were determined as m = 3n + 2, but the estimated m values were determined independent from this formula. For Soil 1 to Soil 3 given values of m decrease (Table 1), while the estimated values of m are very similar for all three soils. According to Eq. [3], the ratio m/n is important. Both estimated and given values of the ratio m/n show similar trends by increasing from Soil 1 to Soil 3.
To further evaluate the analytical method, the soil water characteristic curves and unsaturated hydraulic conductivity functions were calculated with the parameters listed in Table 1 and 3. Figure 3 shows the given values of the hydraulic properties as well as the predicted values of hydraulic properties for the three numerical soils. For all three numerical soils the predicted values were in good agreement with the given values.
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Fig. 3. Given and estimated water retention curves and hydraulic conductivity functions for three numerical conditions: (a) and (d) are for numerical Soil 1, (b) and (e) are for numerical Soil 2, and (c) and (f) are for numerical Soil 3.
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To solve Eq. [4], an assumption of hd/h(x) distribution along soil columns was given in Eq. [5]. This assumption was critical to the analytical method and therefore must be evaluated. Values of hd/h(x) versus x for the three soils were used to evaluate the assumption. The values of hd/h(x) versus x for the three soils were calculated with the HYDRUS-1D model, and the distribution of hd/h(x) versus x for the three soils were also estimated with Eq. [5]. Figure 4
shows a calculated and an estimated hd/h(x) distribution for each numerical soil. For all three numerical soils, the hd/h(x) distributions varied nonlinearly with x. The analytical hd/h(x) distributions from Eq. [5] were in good agreement with the HYDRUS-1D numerical distributions. These results indicate that the assumed hd/h(x) distribution from Eq. [5] was valid for these soil conditions, and Eq. [5] was a reasonable assumption to use in solving Richards equation for horizontal infiltration.
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Fig. 4. Numerically calculated and assumed distributions of hd/h for three numerical soil conditions: (a) is for numerical Soil 1, (b) is for numerical Soil 2, and (c) is for numerical Soil 3.
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The numerical results reported here represent a first step in establishing the usefulness of the analytical method for determining soil hydraulic properties. One limitation of the numerical investigation is that known hd was used for the infiltration boundary. In the case of actual infiltration into horizontal soil columns, hd is unknown. For an actual horizontal infiltration experiment, a convenient boundary suction is h = 0. One approach to satisfy the requirement of the Brooks-Corey model is to consider that a thin saturated layer as a filter layer forms at the infiltrating boundary. In this way, the wetting front distance equals the measured wetting front distance minus the thin filter layer thickness, and the cumulative infiltration equals the measured amount minus the amount stored in the filter layer. Of course it is difficult to accurately determine the thickness of the filter layer. However, the analytical method described here for estimating soil hydraulic properties is based upon the entire wetted soil water flow properties, such as the infiltration rate versus time, the cumulative infiltration versus time and the wetting front distance versus time. Therefore, a small error in the thickness of the thin layer should not lead to a large error in soil hydraulic parameters like m and n.
In a following paper, the analytical method described in this paper will be applied to measured infiltration data to ascertain how well hd, m, and n can be determined.
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CONCLUSIONS
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In this paper, an analytical method was developed to determine soil hydraulic properties. The problem of water absorption into a horizontal soil column was solved based upon the assumption of a nonlinear soil water suction distribution from the infiltrating boundary to the wetting front. Numerically calculated infiltration data were used to evaluate the method. Numerical data were generated with the HYDRUS-1D computer model. The numerical data were analyzed using linear regression with the simple analytical method to determine hydraulic properties. The results indicate that the assumed nonlinear distributions of soil water suction along soil columns were similar to the numerically calculated distributions. The hydraulic properties determined with the analytical method were in good agreement with the given hydraulic properties used to produce the numerical data. The analytical method provides a new and simple transient water flow approach to determine soil hydraulic properties.
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NOTES
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Journal Paper no. J-19542 of the Iowa Agriculture and Home Economics Experiment Station, Ames, IA 50011. Project No. 3287, and supported in part by Hatch Act and State of Iowa funds and the Huo Yingdong Education fund.
Received for publication September 12, 2001.
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REFERENCES
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ABSTRACT
INTRODUCTION
