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DIVISION S-1-SOIL PHYSICS

Measuring Hydraulic Properties Using a Line Source

I. Analytical Expressions

^a School of Geography and Geology, McMaster Univ., Hamilton, ON, Canada L8S 4M1
^b Dep. of Land Resource Science, Univ. of Guelph, Guelph, ON, Canada N1G 2W1
^c Graduate Studies and Research, Univ. of Saskatchewan, Rm 50 Murray Bldg, 3 Campus Drive, Saskatoon, SK, Canada S7N 5A4

gparkin{at}lrs.uoguelph.ca

	ABSTRACT

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
In situ measurement of soil hydraulic properties remains a challenge. This study presents new analytical expressions for estimation of soil hydraulic properties below a surface line source by means of multi-purpose time domain reflectometry (TDR) probes and existing quasi-analytical, steady-state solutions for infiltration from a surface line source. Inverse procedures are used to estimate the inverse macroscopic capillary length scale, , and the hydraulic conductivity at saturation, K_s, from pressure head (), water storage (W), and conservative ionic tracer travel time (T) measured via multi-purpose TDR probes placed at several depths below a line source with constant flux of water. Soil water content at saturation, _s, can also be estimated if prior information is available. The parameter and spatial sensitivities of each solution were calculated by means of sensitivity coefficients. The uniqueness of possible combinations of measurements to estimate , K_s, and _s was tested by means of two-dimensional response surfaces. Significant correlation exists between K_s and _s, and thus it is not possible to estimate both K_s and _s by globally minimizing the objective function. Combination approaches with W (i.e., and W, T and W, or and W and T) give unique estimates of and K_s if either _s is known or prior information on _s is available.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
SOIL HYDRAULIC PROPERTIES accurately measured in situ are necessary for understanding and modeling flow and transport processes in unsaturated soils. Most field methods measure soil hydraulic properties using either one-dimensional (1-D) (e.g., Green et al., 1986; Si et al., 1999) or three-dimensional (3-D) water flow conditions (e.g., Reynolds and Elrick, 1985). A method which measures soil hydraulic properties under two-dimensional (2-D) flow conditions has advantages over the 1-D and the 3-D methods. For 1-D infiltration, the influence of the quasi-linear inverse macroscopic length scale parameter, , is only in the transient data, and only in the slope of the wetting front for the case that the water table is very deep. For 2-D flow, on the other hand, does affect the distribution of steady-state water flux, water content, and pressure head. A 3-D method can only measure the hydraulic properties at one location each time, while a 2-D method can measure both the average and the variability of the hydraulic properties along a line source. The method proposed here uses field measurements of water content, pressure head, and conservative ionic tracer travel time under 2-D, steady-state flow conditions to estimate soil hydraulic properties by an inverse procedure.

Measurements taken under steady-state flow conditions are easier and less prone to errors than those taken under transient conditions. For example, tensiometer measurements of pressure head, , under steady-state conditions are more accurate than under transient conditions because of porous cup equilibration time. It is also possible to repeat pressure head, water content, and flow rate measurements at steady state, whereas for transient conditions, the entire experiment must be repeated.

The inverse procedure of parameter estimation requires the minimization of differences between measured dependent variables and appropriate analytical or numerical solutions that contain the set of parameters to be estimated. The development of inexpensive, high-speed computers has facilitated the use of inverse procedures to estimate soil hydraulic properties. For example, Simunek and van Genuchten (1996), Parkin et al. (1995), and van Dam et al. (1992) used inverse procedures to estimate soil hydraulic properties under various flow conditions.

A new multi-purpose time domain reflectometry (TDR) probe was introduced by Baumgartner et al. (1994). One or all of the wave-guides in the probe consists of a hollow stainless steel tube with a porous tip. The hollow tube is filled with water and it serves as a wave-guide for the TDR signal to measure soil water content (Topp et al., 1980) and conservative ionic tracer transport (Kachanoski et al., 1992), and as a tensiometer to measure pressure head. The multi-purpose TDR probe is thus a convenient and versatile instrument with which to acquire data for parameter estimation by inverse procedures. Si et al. (1999) have recently used multi-purpose TDR probes to measure soil hydraulic parameters under 1-D flow conditions.

The objectives of this paper are to: (i) extend the quasi-analytical solutions for 2-D infiltration from a surface line source (Philip, 1971) to predict pressure head, , water storage, W, and tracer travel time, T; and (ii) test the sensitivity and uniqueness of the extended expressions for estimating soil hydraulic parameters by means of inverse procedures and measurements of , W, and T under a line source. A field test of the method is given in Zhang et al. (2000).

	Theory

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Line Source Expressions
The differential equation that describes steady-state water movement from a surface line source was quasilinearized by Philip (1966) using the Kirchhoff transform. The transformed dependent variable known as the matric flux potential (Raats, 1970), , is defined as:

(1)

where is the soil water pressure head, and K() is the unsaturated hydraulic conductivity function given by Gardner (1958) as

(2)

On the basis of Eq. [1] and [2], the solutions for the steady-state distribution of and water flux below a surface line source with constant source strength, q (m³ s^-1 m^-1), in an infinite and homogeneous medium, is given by (Philip, 1971)

(3)

and

(4)

where J_z is the water flux density (m³ s^-1 m^-1) in the vertical direction, ₀ and ₁ are the modified Bessel functions of the second type of order zero and one, respectively (Bowman, 1958), and x and z are the rectangular Cartesian coordinates with z representing the vertical direction (positive downward) and x being the horizontal direction perpendicular to the source direction.

We now combine Eq. [1], [2], and [3] to obtain an expression for the pressure head and develop new expressions based on Eq. [3] and [4] to describe water storage and tracer travel time below the line source. Recall that the multi-purpose TDR probe can simultaneously measure , W, and T (Baumgartner et al., 1994; Kachanoski et al., 1992) under line source conditions. This provides the flexibility to use part or all of the measurements to estimate soil hydraulic properties by inverting the new analytical expressions.

Pressure Head Expression
Substituting Eq. [2] into Eq. [1] and integrating produces an explicit relationship between pressure head and matric flux potential:

(5)

Substituting Eq. [3] into Eq. [5], the expression for pressure head below a surface line source at steady state is:

and

(6)

Water Storage Expression
A compatible relationship between soil water content, , and pressure head, , with K() given by Eq. [2] was proposed by Russo (1988):

(7)

where _r (m³ m^-3) is the residual soil water content, and m is a constant.

The steady-state water storage, W (m), in the soil from depth z = 0 to z = L below a surface line source is given by with being the average soil water content, or using the () relationship given in Eq. [7],

(8)

where (x, z) was given in Eq. [6].

Tracer Travel Time Expression
The TDR technique developed by Kachanoski et al. (1992) measures the front position of a conservative ionic tracer as a function of time along a vertical streamline when soil water content is uniform. The method was also used to measure conservative ionic tracer transport along the vertical streamline during constant flux infiltration from a point source (Kachanoski et al., 1994). For 2-D flow from a line source, only the streamline directly under the source is vertical. The water flux directly along this streamline is obtained by setting x = 0 in Eq. [4]:

(9)

The porewater velocity, v, is calculated by

(10)

where (z) is the soil water content at depth z. Combining Eq. [9] and [10] and integrating, we obtain the tracer travel time, T (s), from depth z = 0 to z = z* along the vertical streamline directly below a surface line source under steady-state water flow conditions as

(11)

Equation [11] can be approximated by

(12)

where is the average soil water content from depth z = 0 to z = L with L being the TDR probe length and . Note that can be measured directly by vertical TDR below the line source. The relative error caused by the approximation is calculated by

(13)

Therefore, T_a(z*) underestimates T(z*) if err(z*) < 0 and overestimates T(z*) if err(z*) > 0.

The method of Kachanoski et al. (1994) can be used to measure steady-state tracer travel times below a line source with multi-purpose TDR probes. Details are given in Zhang et al. (2000). Briefly, at steady state, the line source water is replaced with water containing a non-reactive tracer such as potassium chloride. The vertical movement of the center of mass of the tracer front is measured by means of impedance changes in the vertical TDR probes installed directly below the line source.

	Materials and methods

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
To evaluate the proposed analytical expressions (Eq. [6], [8], and [11]) for estimating hydraulic properties by means of an inverse procedure, the parameter sensitivity, spatial sensitivity, and the uniqueness of parameter estimates were analyzed. Five parameters (i.e., , K_s, _s, _r, and m) are included in Eq. [6], [8], and [11]. In general, _r does not vary significantly in the field. Therefore, the 1.5 MPa soil water content determined from repacked soil cores was used as an estimate of _r, and (and thus 2/(2 + m) = 0.8), is used (Russo, 1988). Note that is most accurate for sandy loam soils (Russo, 1988). The estimation of parameters , K_s, and _s was investigated by inverting Eq. [6], [8], and [11] numerically.

Parameter Sensitivity
Sensitivity coefficients were calculated by the following formula (Simunek and van Genuchten, 1996):

(14)

where µ_ß is the sensitivity coefficient, which is the change in the auxiliary variable, , corresponding to a 1% change in any parameter ß (i.e., ). The auxiliary variables are pressure head (), water storage (W), and tracer travel time (T).

The sensitivity coefficients, µ_ß, are not only dependent on the position (x, z) of the measurement prediction in the vertical cross section, but also on the values of soil hydraulic parameters (i.e., , K_s, and _s). Thus, to illustrate the sensitivity of the , W, and T solutions, the sensitivity coefficients were calculated for (i) a fixed measurement location and a range of hydraulic parameters, and (ii) a range of measurement locations (0 x 0.5 m, 0 z 1.0 m) for a fixed set of hydraulic parameters . For the fixed measurement location , sensitivity coefficients for were calculated over the range 4 m^-1 36 m^-1, which covers values typical for soil textures from clay loam to coarse sand (Elrick et al., 1989). A range of 0.001 m q/K_s 0.5 m was used, which covers the practical values of source strengths that will maintain predominantly unsaturated soil conditions. Values of _s ranged from 0.30 _s 0.50.

Uniqueness without Prior Information
The uniqueness of parameters , K_s and _s were tested over a wide range of expected parameter values. The objective function, S(ß), used for parameter estimation is given as (Kool et al., 1987)

(15)

where j represents the different sets of measurements; that is, those involving , W, or T; n_j is the number of measurements in a particular set with n the number of measurement sets for the combination; G_ij are ith measurement for the jth measurement set; _ij(,K_s,_s) are the corresponding model predictions; w_j are the weights associated with a jth measurement set and w_ij the weights associated with measurement points. The inverse problem is to minimize the objective function, S(,K_s,_s), with respect to the parameters, , K_s, and _s. Multiplying by 1/n on the right hand side of Eq. [15], the objective will all converge to about one for the combined measurements. Note that if only one set of measurements are used for parameter estimation, Eq. [15] reduces to

(16)

The weighting coefficient associated with a point, w_ij, is set equal to 1/n_j, if the error within a particular measurement set is constant. The sum of w_ij for a particular measurement set is equal to 1. The weighting coefficients associated with a measurement set, w_j, are determined in accordance with the measurement error. The measurement error of jth measurement set can be represented by the least mean squared error, LMS_j, by optimizing Eq. [16]. Namely, where ,_s, and _s are the best-fit values. The larger the LMS value of a measurement set, the lower the weight that will be assigned to it. Thus, w_j is calculated by

(17)

Since error-free data is used in the uniqueness analysis, the LMS values of each measurement set are zero, which makes it impossible to calculate the weights. This problem was solved by using the average of the S_j(,_s,_s) values around the minima to substitute the zero value of LMS of each measurement set. For example, for the response surfaces in the - K_s plane, the weight associated with pressure head, , where _k, K_sk, and _sk are the true values of the corresponding parameters, _k-1 and K_sk-1 are the and K_s values one calculation step less than the true values, and _k+1 and K_sk+1 are those one calculation step more than the true values. The details of calculation steps regarding to , K_s, and _s, and the calculation of the response surfaces are given below.

The uniqueness of the solutions was examined by means of error-free simulated data. Steady-state , W, and T values from a surface line source for a hypothetical sandy loam soil generated numerically by means of Eq. [6], [8], and [11] are given as an example. The hydraulic parameters were arbitrarily set at , and the source strength . The and W data were calculated at the following soil positions (x, z): (0.1 m, 0.2 m), (0, 0.4 m), (0.1 m, 0.6 m), and (0.2 m, 0.8 m). The T data were determined at and . These soil depths represent realistic multi-purpose TDR probe lengths and separation distances.

The uniqueness of the inverse problem is evaluated in terms of 2-D response surfaces of the objective function, s(,K_s,_s), of the soil hydraulic parameter pairs, -K_s, -_s, and K_s-_s. Since the parameters have different units, the relative values of parameters (i.e., K_s/K_sk, /_k, and _s/_sk) will be used when plotting the response surfaces. The response surfaces in the three planes were calculated by MathCad PLUS 6.0 (MathSoft, 1996) for the three single measurement sets (i.e., , W and T), for three two-set combinations of measurements (i.e., +W, +T, and W+T), and for one three-set combination of measurements (i.e., +W+T). Note that the addition signs here represent combinations in Eq. [15] and not mathematical summation. Therefore, seven response surfaces were calculated for each parameter pair. The response surfaces were calculated on rectangular grids with ranges in , K_s, and _s of 6 - 18 m^-1, 1.4 x 10^-5 - 4.2 x 10^-5 ms^-1, and 0.28 - 0.52 m³ m^-3, respectively. Each parameter domain was discretized into 21 points, resulting in 441 grid points for each 2-D response surface. If the inverse problem results in a unique set of parameter values, the response surface will be in the form of a series of concentric circles. In this case, the input parameter values will occur at the center of the concentric circles.

Uniqueness with Prior Information for _s
The value of _s is reasonably easy to measure, or a range of _s is assumed as prior information in a manner similar to Si et al. (1999). If prior information is available for _s, then the objective function used to estimate parameters , K_s, and _s is modified by adding a constraint term to Eq. [15] as in Russo et al. (1991) and Si (1998)

(18)

where w_s is the weight associated with _s, and _sp is the value of _s determined on the basis of the prior information. Note that the total number of combination sets is 1+n for this case. The error of _s is represented by its measurement variance. Thus, the weights associated with each measurement set, w_j, and that associated with _s, w_s, are calculated with

(19)

With the prior information for _s included, Eq. [18] guarantees that _s will converge at a point not far from the prior determined value, _sp.

	Results and discussion

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
Parameter Sensitivity
The sensitivity coefficients for the expressions of pressure head, Eq. [6], of water storage, Eq. [8], and of tracer travel time, Eq. [11], to changes in parameters representing different soils are plotted in Fig. 1 . From the definition of the sensitivity coefficient, µ_ß (Eq. [14]), it follows that more accurate parameter estimates can be obtained and the measurement errors have less impact on estimates, if µ_ß has larger values. Note that the pressure head expression is independent of _s and _r.

View larger version (34K): [in this window] [in a new window]   Fig. 1 Sensitivity coefficients, µ and µKs to 1% changes in and q/Ks. (a) and (b): for pressure head, (c) and (d): for water storage, and (e) and (f): for tracer travel time. Other parameters included: horizontal distance perpendicular to the source x = 0, L = 0.4 m, sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

Ks

Ks

Ks

Ks

Except for µ_ws, the sensitivity coefficients for water storage , µ_W, µ_WKs, and µ_Wr, are almost independent of and q/K_s (Fig. 1c, 1d). The soil below the line source approaches saturation for q/K_s > 0.1 m, and these sensitivity coefficients approach zero (Fig. 1d). For q/K_s < 0.1 m, the values of µ_W and µ_WK_s are approximately 0.0001 m and 0.0002 m, respectively. Thus, the water storage expression is twice as sensitive to K_s than . Water storage measured with a TDR probe length of 0.4 m and by the method of Topp et al. (1980), has a precision of about 0.004 m. Thus, a single TDR measurement of water storage can detect no less than a 40% change in and a 20% change in K_s. The soil water storage expression is about 5 to 15 times more sensitive to _s than any other parameter, with µ_ws increasing as q/K_s increases (Fig. 1c, 1d).

With the exception of soils with small values, the tracer travel time solution is more sensitive to than to K_s (Fig. 1e, 1f). The values of µ_T or µ_TKs for are generally greater than 500 s, but depend on probe length. Thus, measurement error should generally have minimal impact when using the tracer travel time solution in inverse procedures to estimate and K_s. Similar to water storage, the tracer travel time expression is more sensitive to _s than any other parameter, and least sensitive to _r.

In summary, for error-free measurements directly below the line source, the pressure head expression is more suitable for fine textured, low soils. The expressions of water storage and tracer travel time are equally suitable for a wider range of soils. For best estimates of K_s, the q/K_s ratio should be kept in the range of 0.01 to 0.1 m. For example, the value of q for a typical sandy soil should be between 10^-6 and 10^-5 m³ s^-1 per meter of line source. Generally, the pressure head and tracer travel time expressions are more sensitive to than to K_s, whereas the water storage expression is more sensitive to K_s than to . Both the water storage and travel time expressions are very sensitive to _s, but least sensitive to _r.

Spatial Sensitivity
Figure 2 (a, b, and c) shows the distribution of sensitivity coefficients, µ, µ_WKs, and µ_W, in the x–z plane. The value µ_Ks is equal to 0.0008 m regardless of the position of measurement (not shown) since (x, z) in Eq.[5] is not a function of K_s. The value of µ increases slightly with both soil depth and horizontal distance from the source (Fig. 2a).

View larger version (51K): [in this window] [in a new window]   Fig. 2 Spatial sensitivity of pressure head, water storage and tracer travel time solutions. (a) contours of µ in the x-z plane, (b) contours of µWKs in the x-z plane, (c) contours of µW in the x–plane, and (d) µT and µTKs vs. soil depth. Other parameters included: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

W

W

W

T

Overall, the sensitivities of , W, and T increase with measurement depth (Fig. 2). However, the measurement depth, L, can not be larger than the maximum length of TDR probes. For large measurement depths, a very long time may be required to reach steady state since wetting front velocity is approximately inversely proportional to L². In addition, the measurement of T requires the tracer front to completely pass depth L, which also requires a time proportional to L².

The zero sensitivity contour line in µ_W (Fig. 2c) indicates that water storage measured at that position is independent of the changes in . In other words, for the example given vertical TDR probes placed at approximately would measure the same water storage value at steady-state for soils of different values if all the other parameters are fixed. An increase in the value of results in less horizontal water flow and more vertical water flow. Thus, away from the line source, at shallow depths will decrease and at greater depths will increase, as increases. Along the zero contour line of µ_W, the decrease in at the surface is compensated by the increase in at depth, and the overall change in water storage, W, is zero along the TDR measurement length.

Uniqueness
The response surface of the W expression in the K_s–_s plane is shown in Fig. 3a . The open-ended valley form of the plot indicates that these two parameters are highly correlated and cannot be uniquely determined by the inverse procedure. Because the same function, (, K_s, _s), is included in both the W and T expressions, the response surface of the T expression in the K_s–_s plane (Fig. 3b) is the same as that of the W expression. Since _s is not involved in the expression, any combination of the expressions of , W, and T will not estimate both K_s and _s uniquely. To solve the problem, the objective function with prior information for _s, Eq. [18], can be used in the inverse procedure. The second term in Eq. [18] guarantees the convergence of _s at a value close to _sp.

View larger version (54K): [in this window] [in a new window]   Fig. 3 Response surfaces in the s-Ks plane obtained by using a single set of error-free measurements of (a) water storage, and (b) tracer travel time. Other parameters include: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

View larger version (31K): [in this window] [in a new window]   Fig. 4 Response surfaces in the –Ks plane obtained by means of a single set of error-free measurements of (a) pressure head, (b) water storage, and (c) tracer travel time. Other parameters include: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

View larger version (26K): [in this window] [in a new window]   Fig. 5 Response surfaces in the -Ks plane obtained by the two-set combination approaches: (a) +W, (b) +T, and (c) W+T. Other parameters include: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

View larger version (34K): [in this window] [in a new window]   Fig. 6 Response surface in the -Ks plane obtained by the three-set combination approach. Other parameters include: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

The sensitivity and uniqueness analyses given above have assumed one location along the line source for measurement of , W, and T. However, it is possible to also install many standard or multi-purpose TDR probes along the line source. Measurements of W, , and T from all the probes can be used to estimate the mean values of soil hydraulic parameters. The use of many probes reduces the error of estimates, which is inversely proportional to N^1/2, with N being the total number of measurements. We can also use the method to simultaneously estimate the local values of hydraulic parameters at different positions along the line source. Thus, the spatial variability and the average values of the hydraulic parameters along the line source can be determined in one single experiment. This would avoid any changes in soil hydraulic properties due to temporal variability arising from wetting and drying events, for example.

Parameter Identification by Global Minimization
The hydraulic parameters can be estimated by globally minimizing the objective function, which may include one, two, or three sets of measurements. A unique set of all parameters (i.e., , K_s, and _s) cannot be obtained by minimizing the objective function, Eq. [15]. Therefore, we have to either reduce the number of parameters to be estimated, or use the objective function with prior information included, Eq. [18]. If prior information for _s is available, a unique set of , K_s, and _s can be obtained from either W, or W+T, or W+, or +T, or +W+T.

Parameter Identification Using Sequential Method
Another approach to estimating soil hydraulic parameters is to solve T_a, , and W sequentially. Only one parameter, , is involved in the approximate expression for tracer travel time, Eq. [12], and thus can be obtained by optimizing Eq. [12] directly. Next, by substituting the obtained value of into the pressure head expression, Eq. [6], K_s can be optimized. Finally, by substituting both the obtained and K_s values into the water storage expression, Eq. [8], _s can be estimated.

One possible concern of this method is the error of the approximate expression of tracer travel time. Figure 7 compares the exact expression, Eq. [11], and the approximate expression, Eq. [12], for tracer travel time at different depths in a typical sandy soil. The probe length for the calculation is L = 0.4 m. equation [12] approximates Eq. [11] very well except for very shallow soil depths (Fig. 8) . If the measurement depth is between 10 and 20 cm, from Eq. [13], T_a(z*) underestimates T(z*) by about 37% (Fig. 8). If the measurement depth is between 20 and 40 cm, T_a(z*) will give an error of about ±3% (Fig. 8). Therefore, it is acceptable to use Eq. [12] to approximate Eq. [11].

View larger version (18K): [in this window] [in a new window]   Fig. 7 Comparison of the exact solution and the approximate solution of travel time in a sandy soil. sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

View larger version (14K): [in this window] [in a new window]   Fig. 8 The relative error in the approximate solution calculated at different soil depths. Parameters: sk = 0.4, rk = 0.05, k = 12 m-1, Ksk = 2.8 x 10-5 m s-1, and q = 1 x 10-6 m2 s-1. Subscript k stands for the true value of the corresponding parameter

	Summary and conclusions

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 
An existing analytical solution (Philip, 1971) for steady-state infiltration from a surface line source was extended to obtain expressions for pressure head (Eq.[6]), water storage (Eq.[8]), and tracer travel time (Eq.[11]). The new expressions can be used to estimate soil hydraulic properties by means of inverse procedures with, for example, measurements from multi-purpose TDR probes. Sensitivity analyses indicated that Eq.[6], [8], and [11] are all sensitive enough to estimate hydraulic parameters. The pressure head expression (Eq.[6]) is more suitable for fine textured soils, while the water storage (Eq.[8]) and tracer travel time (Eq.[11]) expressions are suitable for a range of soils. For estimating K_s, the q/K_s ratio should be within the range of 0.01 to 0.1 m. The sensitivity of all three expressions to changes in and K_s increases with soil depth. However, the sensitivity to K_s with horizontal distance from the line source does not change. The sensitivity to changes in increases with horizontal distance from the line source for the pressure head solution only. For water storage, the sensitivity to changes in decreases to zero and then increases with horizontal distance from the source.

Measurements of only pressure head or only tracer travel time will not give unique estimates of and K_s. However, the use of measurements of water storage only can give unique estimates, but is less well determined than K_s. Combining measurements of water storage with pressure head and/or tracer travel time gives unique estimates of and K_s. If water storage and pressure head measurements are used, accuracy of parameter estimates can be increased by increasing the distance of the measurements to the source either vertically or horizontally. If travel time measurements are used, TDR probes must be installed directly below the source and parameter accuracy can be increased by increasing the probe length.MathSoft 1996

	ACKNOWLEDGMENTS

 
This research was supported by the Ontario Ministry of Agriculture, Food and Rural Affairs / University of Guelph Environmental program and the Natural Sciences and Engineering Research Council of Canada. Many thanks to Peter von Bertoldi for his help both in the lab and in the field.

Received for publication May 12, 1999.

	REFERENCES

TOP ABSTRACT INTRODUCTION Theory Materials and methods Results and discussion Summary and conclusions REFERENCES

 

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