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DIVISION S-1—SOIL PHYSICS

New Method for Determining Water-Conducting Macro- and Mesoporosity from Tension Infiltrometer

^a Dep. of Soil Science, Univ. of Saskatchewan, Saskatoon, SK, S7N 5A8 Canada
^b Dep. of Physics and Engineering Physics, Univ. of Saskatchewan, Saskatoon, SK, Canada

^* Corresponding author (sibing{at}duke.usask.ca).

	ABSTRACT

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
Characterization of water-conducting porosity at and near saturation is required in understanding rainfall and snowmelt infiltration and runoff as well as chemical transport in soil. There are methods available to quantify water-conducting porosity in situ, but with serious limitations. The objective of this paper was to present a general equation for water-conducting porosity based on ponded- and tension-infiltration measurements. Some analytical solutions are developed for specific unsaturated hydraulic conductivity functions such as the Gardner's exponential and rational power models, Brooks and Corey model, and van Genuchten–Mualem model. Tension infiltrometer measurements were taken at six different pressure heads between –0.3 to –2.2 kPa and double-ring infiltrometer measurements at a pressure head of 0.35 kPa. The analytical solutions were compared with numerical solutions and existing methods for calculation of water-conducting porosity. Both the analytical and numerical solutions can reliably determine the water-conducting porosity of surface soils in situ within the practical pressure head range of the tension infiltrometer. Our method gave consistent water-conducting porosity, regardless of the width of pressure head ranges. The existing methods overestimated water-conducting macroporosity by a factor of greater than two and overestimated total water-conducting porosity by a factor of >10 for measurements taken at large pressure head intervals compared with that of our method. Combining with hydraulic parameter estimation from tension infiltrometer measurements, our method may reduce the number of tension infiltration measurements required to calculate water-conducting porosity.

Abbreviations: DP, Dunn and Philips approach • WL, Watson and Luxmore approach

	INTRODUCTION

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
THE SIGNIFICANCE OF macropores and mesopores to water flow in soils, particularly to infiltration and rapid movement of water, solutes, and pollutants through soils are well recognized (Beven and Germann, 1982; Luxmoore et al., 1990; Ankeny et al., 1990) and are the subject of considerable research interests. Macroporosity and mesoporosity (the fractions of soil volume comprise of pores with diameters >1 x 10^–3 m and between 1 x 10^–5 and 1 x 10^–3 m, respectively; Luxmoore, 1981) of soil core samples or soil columns can be easily determined in the laboratory (Flint and Flint, 2002). Macro- and mesopores include dead-ended and noncontinuous pores, as well as continuous pores with irregular geometry, but only the continuous or interconnected pores contribute to fast water flow in soil. Furthermore, the equivalent diameter of the water-conducting continuous pores is controlled primarily by the part with the smallest diameter (bottleneck) along its pathway although the bottleneck may only be a small fraction of the total length of the pore (Dunn and Phillips, 1991b). The function of water-conducting pores is also influenced by pore tortuosity, surface roughness, etc. (Skopp, 1981; Bouma, 1982). Therefore, higher soil macro- and mesoporosity does not necessarily imply higher hydraulic conductivity and faster chemical transport. Static measurements of pore characteristics such as total macro- and mesoporosity measurements in the laboratory will not adequately describe the actual contribution of pores to flow of water and solutes in soil (Messing and Jarvis, 1993). Hence, in situ measurement of actual water-conducting porosity is of the utmost importance in understanding movement of water and solutes into and through soils.

In the past, several techniques have been employed to quantify the water-conducting macro- and mesoporosity in soil including: staining or dye tracing (Bouma et al., 1979; Ghodrati and Jury, 1990; Weiler and Naef, 2003), tracers and breakthrough curves (Bouma and Wosten, 1979; Yeh et al., 2000), computer assisted tomography (CAT) scanning (Perret et al., 2000), gas diffusion (Bruckler et al., 1989), and infiltration redistribution pattern (Timlin et al., 1994). These methods, however, required either undisturbed soil cores/columns or are tedious to perform under field conditions. Rapid, simple, and in situ ponded- and tension-infiltration measurements, on the other hand, have become popular recently in characterizing water-conducting macro- and mesoporosity in the surface soils. Watson and Luxmoore (1986) and Wilson and Luxmoore (1988) calculated the water-conducting macro- and mesoporosity from the differences in infiltration rates between two pressure heads by using the minimum equivalent pore radius. Several researchers (Azevedo et al., 1998; Buttle and McDonald, 2000; Cameira et al., 2003) have followed the procedure used by Watson and Luxmoore (1986) (WL) as a means of estimating water conducting soil macro- and mesoporosity. Dunn and Phillips (1991a) modified the approach of WL (Watson and Luxmoore, 1986) by replacing the minimum pore radius with the mean pore radius in the pressure head range. The common place of the approaches of WL and Dunn and Phillips (1991a) (DP) is that they both assumed a single pore size (minimum pore radius for WL and mean pore radius for DP). The assumption is unrealistic and may lead to incorrect water-conducting porosity, an unrealistic parameterization of soil properties, and poor performance of hydrological models. Therefore, there is a need for development of a reliable and convenient method for the estimation of water-conducting macro- and mesoporosity.

The objectives of this study were to present a general equation for water-conducting porosity and to derive analytical solutions for the water-conducting porosity based on ponded- and tension-infiltration measurements in conjunction with four commonly used hydraulic conductivity-pressure head functions: (i) Gardner exponential (1958); (ii) Gardner rational (1965); (iii) Brooks and Corey (1966); and (iv) van Genuchten–Mualem (van Genuchten, 1980). The derived general equation and specific analytical solutions for water-conducting porosity are tested in situ using tension and double-ring infiltrometer measurements and compared with WL (Watson and Luxmoore, 1986) and DP (Dunn and Phillips, 1991a) procedures.

	THEORY

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
The maximum water-filled pore size, r, (L) at a specific water pressure head, h, (L) can be calculated from the capillary rise equation (Bear, 1972)

[1]

where is the surface tension of water (M T^–2), is the contact angle between water and the pore wall (assumed to be zero), is the density of water (M L^–3), and g is the acceleration due to gravity (L T^–2).

We assume that the equivalent pores with radii smaller than r calculated from Eq. [1] are full of water and are responsible for all the flux of water under a given water pressure head, and that the equivalent pores with radii larger than the value calculated from Eq. [1] are not contributing to the water flux. According to capillary theory and Poiseuille's law, the flow rate through a single macropore is given by

[2]

where Q(r) is the flow rate (L³ T^–1) as a function of pore radius r and µ is the dynamic viscosity of water (M L^–1 T^–1). Considering the number of water-conducting pores per unit cross-sectional area (L²) for a given pore size r in soils N(r), the water flux density, I(r) (L T^–1), through pores of radius r is given by

[3]

Substitution of Eq. [2] into Eq. [3] yields

[4]

Therefore,

[5]

The water-conducting porosity _m(r) associated with each pore size equals the number of pores per unit area multiplied by the cross-sectional area of the corresponding size pores:

[6]

Watson and Luxmoore (1986) Approach
Watson and Luxmoore (1986) calculated the number of pores [N(a,b)], between two radii a and b (a b). Assuming pore radius equals to the minimum pore radius, the number of pores resulting in a difference in total soil water flux or hydraulic conductivity between two pressure heads corresponding to the two pore radii, is:

[7]

where I(a, b) (L T^–1) is the soil water flux conducted by pores with radii between a and b.

Then the water conducting porosity due to pores in this range (a,b) can be calculated as

[8]

Watson and Luxmoore (1986) used Eq. [7] and [8] for the estimation of N(a,b) and (a,b), respectively, using the minimum equivalent pore radius in each pressure head range. However, their calculation results in the maximum number of pores and hence maximum water-conducting porosity, because pore radius (a) appears in the denominator of Eq. [7] and [8]. Consequently, their approach results in an overestimation of the number of pores per unit area and the total water-conducting porosity in the pressure head range considered.

Dunn and Phillips (1991a) Approach
Dunn and Phillips (1991a) modified the approach of WL by assuming a uniform (box-car) distribution of pore number. Their calculation results in the mean number of pores per unit area , as if all the pores have the same radius. The is given by

[9]

where a and b are pore radii of the lower and upper limits of the integrals of each sequential water pressure head range.

Neglecting the dependence between N(a,b) and pore cross-sectional area, and assuming constant slope of infiltration rate I as a function of pore radius ,

which means a linear dependence of I(a,b) on r, DP also calculate the mean water-conducting porosity , for pore size between a and b,

[10]

Therefore, for calculation of water-conducting porosity, the DP approach as well as WL approach unrealistically assumed uniform pore-size distribution within the pressure head range, and linear dependence of hydraulic conductivity on pore radius.

A New Approach
We considered the number of pores per unit area (L²) as a function of r. The total number of pores in a given pore-size range, n(r), is the cumulative pore number distribution and is given by

[11]

where P(r) is the number of pores per unit area per unit pore radius. For unit hydraulic gradient, steady infiltration rate at a pressure head equals the hydraulic conductivity K (L T^–1). The hydraulic conductivity at a given pressure head h or pore size r, K(r) can be expressed by

[12]

where r is the upper limit of the integrals determined by the water pressure head. The expression for P(r) can be obtained by taking derivatives of both sides of Eq. [12]

[13]

According to Eq. [6], the water-conducting porosity in a given pressure head range can be expressed as,

[14]

Substitution of Eq. [13] for P(r) in Eq. [14] leads to

[15]

This is a general expression for water conducting porosity.

We can show that Eq. [15] reduces to WL equation. By assuming r constant and equal to minimum pore radius in the pressure head range, Eq. [15] reduces to

[15a]

For unit hydraulic gradient, infiltration rate is equal to hydraulic conductivity. Thus, Eq. [15a] is exactly the WL expression if unit gradient flow is assumed.

Generally, soil hydraulic conductivity is expressed as a function of soil water content or pressure head. Since Eq. [15] involves hydraulic conductivity as a function of pore radius, in the following we revise Eq. [15] and express water-conducting porosity in terms of hydraulic conductivity as a function of pressure head, K(h). For convenience, we use the following variable substitution

[16]

Substituting Eq. [2] for Q(r) and Eq. [1] for r into Eq. [15], leads to

[17]

Integration of Eq. [17] by parts leads to

[18]

Equation [18] is an exact equation for calculation of soil water-conducting porosity for a given range of pore radii or pressure heads. Integration in Eq. [18] may be performed numerically by a software package such as MathCad 2000 (Math Soft, Cambridge). For known hydraulic property functions, like Gardner's exponential (1958) and rational (1965) functions, Brooks and Corey (1966), and van Genuchten-Mualem (van Genuchten, 1980) function, exact analytical solution for Eq. [18] may be obtained by direct integration.

Gardner (1958) Exponential Model
Gardner's (1958) exponential hydraulic conductivity function has the following form,

[19]

where, K(h) is the hydraulic conductivity corresponding to the applied water pressure head h, K_s is the saturated hydraulic conductivity (L T^–1), and _GE is the inverse macroscopic capillary length scale (L^–1).

Substitution of Eq. [19] for K(h) in Eq. [18] yields

[20]

Substitution of Eq. [16] into Eq. [20] and subsequent integration give (Gradshteyn and Ryzhik, 2000, Eq. [2.322], p. 104)

[21]

Brooks and Corey (1966) Model
The Brooks and Corey (1966) model for the unsaturated hydraulic conductivity as a function of applied water pressure head has the following form

[22]

where h_e is the air entry potential (L) and ß_BC is a fitting parameter. Note that h_e H(a) because h_e corresponds to the largest pore size in soil. Substitution of Eq. [22] for K(h) in Eq. [18] yields

[23]

Substitution of Eq. [16] into Eq. [23] and subsequent integration yield

[24]

Gardner (1965) Rational Power Model
Following Gardner (1965), we employ the following unsaturated hydraulic conductivity model to relate the capillary pressure head to the reduction of hydraulic conductivity from its saturated value K_s:

[25]

where _GP and ß_GP are curve fitting parameters.

We can rewrite Eq. [18] as follows:

[26]

where

[27]

[28]

Substituting Eq. [16] and [25] into Eq. [27], we get

[29]

After introduction of a new variable V = ^ß_GP, substitution of Eq. [25] into Eq. [28] and subsequent integration yields

[30]

Complete derivation of the analytical solution for the Gardner rational (1965) power model is given in Appendix A.

van Genuchten-Mualem (van Genuchten, 1980) Model
Based on Mualem (1976) predictive hydraulic conductivity model, van Genuchten (1980) expressed hydraulic conductivity in terms of the pressure head as

[31]

Where _VG, n, and m are curve fitting parameters. We can rewrite Eq. [18] as

[32]

where

[33]

[34]

Substituting Eq. [16] and [31] into Eq. [33], and substituting the limit we get

[35]

where m = 1 – (1/n). After introduction of a new variable w(r) = [_VGh(r)]ⁿ, substitution of Eq. [16] and [31] into Eq. [34] leads to

[36]

where V is a dummy integration variable. We can rewrite Eq. [36] as

[37]

where

[38]

[39]

[40]

Complete derivation of the analytical solution for the van Genuchten–Mualem (van Genuchten, 1980) model is given in Appendix B.

	DEMONSTRATIONS

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
To test the above solutions, field experiments using a tension infiltrometer and a double–ring infiltrometer were performed on a farm field in Laura, SK, Canada (51° 52' N lat., 107° 18' W long.). The soil in that area is described as an Elstow association consisting of Dark Brown Chernozems (Typic Ustolls) developed on loamy glacio-lacustrine parent material with silt loam surface soil texture. The lacustrine sediments are underlain by glacial till that is drained by the Tessier aquifer. The water table occurs at approximately 15 m below the surface within a sand layer (Dyck et al., 2003). The site has been under a crop-fallow rotation dominated by wheat (Triticum aestivum L.) with some barley (Hordeum vulgare L.) since 1966.

Before the infiltration measurements commenced, the straw residue was removed from the surface. Any vegetation present was carefully trimmed to the soil surface using a pair of scissors and then removed. A thin layer (5 mm or less) of testing sand was added to the soil surface to ensure a level base and good contact between the infiltrometer disk and the field soil. The testing sand is reported to have an air-entry value slightly higher than –3 kPa pressure head and saturated hydraulic conductivity of 5.3 x 10^–5 m s^–1. The nylon mesh attached to the tension infiltrometer disc had an air-entry value of about –3 to –3.2 kPa pressure head. Infiltration measurements were performed using a tension infiltrometer with a 0.2-m diameter disc (Soil Moisture Measurement Systems, Tuscon, AZ). The tension infiltrometer, preset at –2.2 kPa pressure head (corresponding equivalent pore diameter 1.36 x 10^–4 m), was gently placed on the sand and the amount of water infiltrating into soil, measured by the water level drop in the graduated reservoir tower, was recorded as a function of time. When the amount of water entered into the soil did not change with time for three consecutive measurements taken at 5-min intervals, steady-state flow was assumed and steady-state infiltration was calculated based on the last three measurements. The pressure heads were then set sequentially to –1.7, –1.3, –1.0, –0.6, and –0.3 kPa (corresponding to 1.76 x 10^–4, 2.3 x 10^–4, 3 x 10^–4, 5 x 10^–4, and 1 x 10^–3 m equivalent pore diameter) and the corresponding steady-state infiltration rates were obtained. Generally, steady state was achieved within 20 to 30 min.

Immediately adjacent to the tension infiltrometer, a double-ring infiltrometer with inner and outer rings of 0.2 and 0.3 m in diameter, respectively, was used to determine steady infiltration rate at a constant head of 3.5 x 10^–2 m (Reynolds et al., 2002).

From Eq. [16], it is clear that the value of 0 kPa pressure head cannot be related to a pore size and hence upper limit for the integral in Eq. [18] cannot be defined. However, a small pressure head can be related to a pore size. We assumed that the maximum pore diameter at the site is 5 x 10^–3 m. This is a reasonable assumption as no pores greater than 5 x 10^–3 m in diameter exists at the infiltration measurement location. The pressure head corresponding to 5 x 10^–3 m in diameter is –0.06 kPa. Combining the measurements of tension infiltrometer and double-ring infiltrometer, a plot of steady infiltration rates (geometric means) at +0.35, –0.3, –0.6, –1.0, –1.3, –1.7, and –2.2 kPa pressure head was created, and the infiltration rate at –0.06 kPa was estimated using a spline interpolation. Dunn and Phillips (1991a) also used similar procedure for the estimation of macroporosity in no-till and conventional tillage plots.

The three-dimensional steady infiltration rates obtained at the above-mentioned pressure heads were used to obtain unsaturated hydraulic conductivity using the method proposed by Logsdon and Jaynes (1993). Their method is based on the Wooding's (1968) approximated solution for unconfined steady infiltration rate, q(h) from a shallow circular water source and the Gardner's (1958) exponential hydraulic conductivity function as shown below,

[41]

where r_d is the radius (m) of the disc of tension infiltrometer. Following the Logsdon and Jaynes (1993) method, nonlinear regression of infiltration rates against pressure heads was conducted to obtain the fitting parameter . The fitted value and measured q(h) were then used to calculate K(h) at each applied water pressure head. We did not use the estimated and K_s values to estimate K(h) values based on Gardner (1958) exponential function of hydraulic conductivity, because the function may not best fit the measured data.

Once the hydraulic conductivity at different pressure heads was known, Eq. [8], [10], [21], [24], [26], and [32] were used for the estimation of water-conducting macroporosity in –0.06 to –0.3, and mesoporosity in –0.3 to –0.6, –0.6 to –1.0, –1.0 to –1.3, –1.3 to –1.7, and –1.7 to –2.2, and total water-conducting macro- and mesoporosity in –0.06 to –2.2 kPa pressure head ranges. The water-conducting porosity for the above pressure head ranges were also estimated numerically for the four models based on Eq. [18], and following the WL and DP procedures. The mean differences of water-conducting porosity values obtained from WL, DP, and new procedure for the four models were then compared using SAS PROC UNIVARIATE procedure (Delwiche and Slaughter, 1998).

The K(h) function obtained from field measured steady-state infiltration rate-water pressure head data fitted very closely to the four models (R² > 0.97) over the range of pressure heads studied (Fig. 1) . However, the Brooks and Corey (1966) model underestimated K(h) at h between –0.06 to –1 kPa and overestimated from –1 to –2.2 kPa pressure head. The fitted values for the Gardner exponential (1958) and rational (Gardner, 1965) power models were within the range of typical values described by Elrick and Reynolds (1992) for different textural classes (Table 1). For the van Genuchten–Mualem (van Genuchten, 1980) model the curve fitting parameters were within the range of typical values of K_s, , and n described by Rawls and Brakensiek (1989) for silt loam soil. The estimated parameter ß for the Brooks and Corey (1966) model, however, falls well below the theoretical minimum value reported by Brooks and Corey (1966). This is probably linked to the assumption that we made during curve fitting: the maximum continuous pore diameter at the site is 5 x 10^–3 m. The pressure head corresponding to the pore size of 5 x 10^–3 m in diameter is –0.06 kPa. In addition, presence of a wide range of pore sizes as indicated by the extremely low value of ß and absence of discontinuity in the distribution of pore sizes may also be the reasons for the small ß value (Brooks and Corey, 1966).

View larger version (23K): [in this window] [in a new window]   Fig. 1. Measured (symbol) and fitted (lines) unsaturated hydraulic conductivity of the surface soil for; (a) Gardner rational (1965) and Brooks and Corey (1966) models; (b) van Genuchten—Mualem (van Genuchten, 1980) and Gardner exponential (1958) models.

Soil water-conducting porosity was calculated at each pressure head interval (i.e., –0.06 to –0.3, –0.3 to –0.6, –0.6 to –1.0, –1.0 to –1.3, –1.3 to –1.7, –1.7 to –2.2 kPa) (Table 2). We compared WL and DP approaches to the new procedure. For all four hydraulic functions, the mean differences between the new approach and either WL or DP approaches were significantly different from zero at a 95% confidence level. The highest water-conducting porosity values for the small pressure head ranges were given by WL approach followed by the DP and the new procedure. As well the sum of water-conducting porosity in the pressure head range of –0.06 to –2.2 kPa follows the same trend. This is not unexpected because Poiseuille's law states that soil water flux is proportional to the square of pore radius. One large pore will conduct more water than a number of small pores with the same sum of cross-sectional area, because of the power dependence of flux on pore radius. By assuming smaller pore sizes in calculating porosity from a fixed soil water flux, one will end up with larger pore cross-sectional area, and thus with larger porosity than reality. In addition, hydraulic conductivity as a function of pore radius or pressure head is concave and linear approximation results in higher hydraulic conductivity, and thus more pores than reality.

For different hydraulic conductivity functions (i.e., Brooks and Corey's equation, Gardner's exponential and rational equations, and van Genuchten–Mualem's equation), the water-conducting macroporosity for the pressure head range between –0.06 and –0.3 kPa was nearly the same (Table 2). For the other pressure head ranges, different models gave different water-conducting porosity values for the same set of field data. The Brooks and Corey's (1966) equation underestimated soil hydraulic conductivity at high pressure heads (or large pore sizes) and overestimated hydraulic conductivity at low pressure heads (or small pore sizes), leading to smaller porosity of large pore sizes (Fig. 1). Conversely, the extensively used Gardner's (1958) single exponential equation underestimated the initial decrease in hydraulic conductivity resulting from mesopore drainage when the pressure head decreases from –0.3 to –1 kPa. Similar observations have been made by Wilson and Luxmoore (1988), Ankeny et al. (1991), and Jarvis and Messing (1995). Consequently, the Gardner's (1958) single exponential function yielded the largest functional mesoporosity for the range of pressure heads over which data were collected. In contrast, both the Gardner rational (1965) and van Genuchten–Mualem (van Genuchten, 1980) models, which utilize an additional parameter compared with that of the Gardner (1958) exponential and Brooks and Corey (1966) models, fit the measured data very well resulting in nearly the same water-conducting porosity of surface soils over the entire range of pressure heads studied.

Under the assumption of unit gradient and one-dimensional flow, the infiltration rate can be used in Eq. [18] and consequently, in Eq. [21], [24], [26], and [32] should Eq. [19], [22], [25], and [31] be used to fit the infiltration rate as a function of soil water pressure head. As a matter of fact, most of the applications of tension infiltrometers in calculating water-conducting porosity in the literature used infiltration rate, rather than hydraulic conductivity.

The proposed method assumes macro- and mesopores in soils as smooth, cylindrical capillary tubes and laminar water flow in large macropores. In the field, macro- and mesopores may have irregular shapes and water flow in some macropores (shrink-swell cracks) may be turbulent. However, considering the large spatial and temporal variability of hydraulic properties, errors introduced by these assumptions may be negligible.

Another limitation of the proposed method for calculation of soil water-conducting porosity is that a parametric model of K(h) is required. However, most of the inverse procedures for estimation of soil hydraulic conductivity give hydraulic parameters for certain hydraulic functions, not the actual measurement of hydraulic conductivity. Using the proposed new method, one can considerably reduce the number of hydraulic conductivity measurements for accurate estimation of water-conducting macro- and mesoporosity.

In this paper, comparison of a new method with WL and DP approaches is based on one soil textural class (silt loam), and thus the demonstration is a preliminary assessment. Results from more experiments with soils having widely varying soil texture and macropores characteristics are needed to evaluate the proposed analytical solutions for water conducting porosity. Moreover, the new analytical solutions could be further improved by incorporating a two-domain approach, which considers water-conducting macro- and mesoporosity as made up of laminar and turbulent domains.

	CONCLUSIONS

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
We derived a general equation for water-conducting porosity based on tension infiltrometer measurements. Further, for soils with their unsaturated hydraulic conductivity characterized by Gardner's (1958) exponential, Brooks and Corey (1966), Gardner rational (1965), or van Genuchten–Mualem (van Genuchten, 1980) models, we obtained analytical solutions for the water-conducting macro- and mesoporosity in terms of relationships among the capillary pressure head and the parameters defining the hydraulic conductivity-pressure head relationships. In our derivation, no additional assumptions were made besides the well-known capillary equation and Poiseulle's law. Field experiments demonstrated that our method could adequately and efficiently characterize the water-conducting porosity of surface soils in situ, regardless of the size of pressure head ranges used in the calculation.

	APPENDIX A

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
Gardner Rational (Gardner, 1965) Power Model
We can rewrite Eq. [18] as follows:

[A1]

where

[A2]

[A3]

Substituting Eq. [16] and [25] into Eq. [A2], we get

[A4]

Substitution of Eq. [25] into Eq. [A3] yields

[A5]

Introducing a new variable, v = ^ß_GP we can rewrite the Eq. [A5] as

[A6]

Following Gradshteyn and Ryzhik (2000)(Eq. [3.194], p. 313), Eq. [A6] is integrated to give

[A7]

where ₂F₁ is a hypergeometric function. Transformation of Eq. [A7] leads to (Oberhettinger, 1972, Eq. [15.3.4], p. 559)

[A8]

The function [A8] may be expressed in a series form

[A9]

where the terms in the series can be calculated in the following recursive way (Seaborn, 1991):

[A10]

	APPENDIX B

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 
van Genuchten–Mualem (van Genuchten, 1980) Model
We can rewrite Eq. [18] as

[A11]

where

[A12]

[A13]

substituting Eq. [16] and [31] into Eq. [A12], and substituting the limit we get

[A14]

where m = 1 – (1/n). After introduction of a new variable w(r) = [_VGh(r)]ⁿ, substitution of Eq. [16] and [31] into Eq. [A13] leads to

[A15]

where V is a dummy integration variable.

We can rewrite Eq. [A15] as

[A16]

where

[A17]

[A18]

[A19]

Following Gradshteyn and Ryzhik (2000)(Eq. [3.194], p. 313), Eq. [A17] is integrated to give

[A20]

Transformation of Eq. [A20] leads to (Oberhettinger, 1972, Eq. [15.3.4], p. 559)

[A21]

The function [A21] may be expressed in a series form

[A22]

where the terms in the series can be calculated in the recursive way as shown in Eq. [A10].

Following Gradshteyn and Ryzhik (2000)(Eq. [3.194], p. 313), Eq. [A18] is integrated to give

[A23]

Transformation of Eq. [A23] leads to (Oberhettinger, 1972, Eq. [15.3.4], p. 559)

[A24]

The function [A24] may be expressed in a series form

[A25]

where the terms in the series can be calculated in the recursive way as shown in Eq. [A10].

Following Gradshteyn and Ryzhik (2000)(Eq. [3.194], p. 313), Eq. [A19] is integrated to give

[A26]

Transformation of Eq. [A26] results (Oberhettinger, 1972, Eq. [15.3.4], p. 559)

[A27]

The function [A27] may be expressed in a series form

[A28]

where the terms in the series can be calculated in the recursive way as shown in Eq. [A10].

	ACKNOWLEDGMENTS

 
Funding for this project was provided by the National Science and Engineering Research Council of Canada (NSERC) and the University of Saskatchewan through a graduate student scholarship.

	REFERENCES

TOP ABSTRACT INTRODUCTION THEORY DEMONSTRATIONS CONCLUSIONS APPENDIX A APPENDIX B REFERENCES

 

• Ankeny, M.D., M. Ahmed, T.C. Kaspar, and R. Horton. 1991. Simple field method for determining unsaturated hydraulic conductivity. Soil Sci. Soc. Am. J. 55:467–470.[Abstract/Free Full Text]
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