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DIVISION S-1-SOIL PHYSICS

Measuring Hydraulic Properties Using a Line Source

II. Field Test

^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 discussions Summary and conclusions REFERENCES

 
We designed and tested a field method to measure unsaturated soil hydraulic properties using multi-purpose time domain reflectometry (TDR) probes below a surface line source with constant flux of water. The surface line source was produced with a moving irrigation system at a field site sheltered from precipitation. Two hundred multi-purpose TDR probes were vertically installed in the soil beneath the line source to measure soil water pressure head (), water storage (W), and tracer travel time (T). The soil hydraulic properties, the inverse macroscopic capillary length (), hydraulic conductivity at saturation (K_s), and soil water content at saturation (_s) were estimated by inverse procedures with new analytical expressions. Five combinations of measurement sets, namely W-only, and W, and T, and W and T, and and W and T were used. Approximate confidence contours in the –K_s plane were calculated to show the precision of the parameter estimates. For comparison, hydraulic properties were also measured by means of the Guelph Permeameter (GP) and the modified Guelph Pressure Infiltrometer (GPI) systems. Hydraulic parameters estimated from only W measurements were similar to those estimated from the combinations of W and , or W and T, or W and and T. The estimated hydraulic parameters were similar to those obtained with three-dimensional (3-D) infiltration measurements by means of the GP and GPI systems.

	INTRODUCTION

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

 
THERE ARE MANY TECHNIQUES AVAILABLE to measure the hydraulic properties of unsaturated soils in the field (e.g., Green et al., 1986; Reynolds and Elrick, 1985). The choice of which technique to use is often governed by soil type, degree of spatial variability, and the end-use of the hydraulic parameters.

Inverse procedures have the benefits of (i) ability to calculate parameter values that produce the best-fit between observed and simulated values, and (ii) quantifying the confidence limits in parameter estimates and the predictions. If the estimation involves two parameters, the parameter confidence range can be expressed in confidence contours (Draper and Smith, 1966.) The confidence interval can be used to give the level of confidence of analytical or numerical models of water flow (Hill, 1989). For example, by using inverse problem methodology, Russo et al. (1991) expressed the validity of estimates of soil hydraulic parameters in terms of confidence intervals.

One of the obstacles to the use of the inverse procedure is the lack of sufficient information to solve the inverse models (Poeter and Hill, 1997). The use of multi-purpose time domain reflectometry (TDR) probes (Baumgartner et al., 1994) provides an efficient way to measure pressure head, water storage, and the travel time of a conservative ionic tracer by means of a single probe. Si et al. (1999) recently measured the field average hydraulic parameters during one-dimensional (1-D) constant flux infiltration with multi-purpose TDR probes to obtain the average K() and () functions for a field site at Camp Borden, Ontario. Zhang et al. (2000) developed three expressions to estimate soil hydraulic properties using a constant flux surface line source. They also analyzed the sensitivity and uniqueness of these expressions.

The objectives of this study were to: (i) conduct a field test of the Zhang et al. (2000) expressions describing the steady state distributions of pressure head (), water storage (W), and tracer travel time (T) for constant flux below a surface line source; and (ii) use inverse procedures and the in situ measured , W, and T distributions to estimate inverse macroscopic capillary length (), saturated hydraulic conductivity (K_s), and saturated volumetric soil water content (_s) of a uniform sandy soil.

	Theory

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

 
Using Philip's (1971) steady-state expressions for a constant flux surface line source and Russo's (1988) relationship between soil water content () and pressure head (), Zhang et al. (2000) derived the following expressions:

(1)

and

(2)

and

(3)

where x is the horizontal distance perpendicular to the line source, z is the vertical distance from the source (positive downward), is the inverse macroscopic capillary length (m^-1), K_s is the hydraulic conductivity at saturation (ms^-1), q is source strength (m³ s^-1 m^-1), (x, z) is the pressure head (m) at soil position (x, z), ₀(z) is the pressure head directly below the source at soil depth z, ₀ and ₁ are modified Bessel functions of the second type of order zero and one, respectively (Bowman, 1958), W(x, L) is the water storage from z = 0 to z = L, _s is soil water content at saturation (m³ m^-3), _r is residual soil water content (m³ m^-3), L is the length of TDR probe, and T(z*) (s) is tracer travel time from the surface to depth z = z*, directly below the line source .

	Materials and methods

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

 
Field Experiment Procedure
The experiments were carried out at the same sandy soil site as that of Si et al. (1999) in Camp Borden, Ontario. A description of the field site, experimental design including TDR probe placement, and water delivery system has already been given by Si et al. (1999), so only a brief summary will be repeated here. Fifty nests of multi-purpose TDR probes were installed under the line source with a between-nest spacing of 0.15 m. Each nest included a linear four-probe array with probe lengths of 0.2, 0.4, 0.6, and 0.8 m, and a between-probe spacing of 0.1 m. The probe arrays were installed perpendicular to the line source with the 0.4-m-long probe placed directly under the source (Fig. 1) . The moving sprinkler irrigation system described by Si et al. (1999) was modified to create a line source by adding a metal trough with an open bottom (Fig. 1) and by using one nozzle instead of multiple nozzles. The trough directed a constant flux of water from the nozzle system along a line (0.04 m wide and 8.8 m long) directly above the 0.4-m-long TDR probes (Fig. 1). The flux of water depends on the speed of the moving sprayer system, the time delay at the end of each spray cycle, and the spraying pressure, all of which can be programmed.

View larger version (52K): [in this window] [in a new window]   Fig. 1 Cross-sectional view of the experiment layout

Soil water pressure head was measured with a tensimeter (Soil Measurement Systems, Tucson, AZ) and the water-filled waveguide. Soil dielectric was measured with TDR (Tektronix Cable Tester, Model 1502C, Tektronix, Beaverton, OR) and was converted to water content by the equation given by Topp et al. (1980).

The method of Kachanoski et al. (1992) was used to measure the tracer travel time, T(z*), versus depth of the center of mass of a step pulse of a conservative ionic tracer applied after steady-state water content conditions were achieved. Kachanoski et al. (1992) used impedance, R, measurements at long reflection times from vertically installed TDR probes for 1-D, steady-state flow conditions. Kachanoski et al. (1994) extended this method to measure steady-state conservative ionic tracer movement in the vertical flow line directly under a point source with 3-D flow. By means of the same procedure, the relative mass, M_R(t) of a conservative ionic tracer from the soil surface to the measurement depth, z, along the vertical flow line directly below a line source for a step pulse application is:

(4)

where M_R(t) is the total relative mass of applied tracer at time t, R(t) is the measured impedance load of the TDR rods at time t at a fixed distance along the TDR oscilloscope trace after multiple reflections, t is the time since the addition of the conservative ionic tracer, R_i is the initial measured impedance load before tracer addition, and R_f is the final constant TDR impedance load measured after the dispersed tracer front has passed the ends of the TDR rods.

On the basis of the assumption that the depth-averaged soil water content doesn't vary appreciably [i.e., (z) _{L z}] along the vertical streamline (Philip, 1984; Revol et al., 1991), the relative specific conservative ionic tracer mass can be expressed as the ratio of the depth of center of mass of the advancing tracer front to probe length (Kachanoski et al., 1994). The relative specific tracer mass under two-dimensional (2-D) water movement conditions is:

(5)

where z*(t) is the center of mass of the conservative ionic tracer front at time, t, and L is the length of the TDR probe. Rearranging Eqs. [4] and [5], the location, z*, of the tracer front at time T is given as

(6)

Thus, the location of the center of mass of the tracer front along the vertical streamline below a constant flux line source of water, as a function of time, can be non-destructively determined by measuring only the relative changes in TDR line impedance. The tracer travel time as a function of vertical distance, T(z*), is obtained by inverting z*(T).

Three steady-state experiments, each with a different line source strength, were carried out. During each experiment, a flow meter (Neptune Meters Ltd., Canada) measured the total volume of water applied to the 8.8-m-long line source. The source strength was computed by dividing the volume of water by the total elapsed time and the length of the line source. The line source strengths for the three experiments were 0.67 x 10^-6, 0.90 x 10^-6, and 1.49 x 10^-6 m³ s^-1 m^-1, respectively.

For each source strength, water was applied until soil water storage measurements indicated steady-state conditions (approximately 1–2 d depending on source strength). At steady state, measurements of soil water storage and pressure head were taken. The line source water was then replaced with a 0.1% (w/v) KCl solution, and the movement of the center of mass of tracer (Cl^-) was determined by the method of Kachanoski et al. (1992). The strength of the line source was held constant before and during Cl^- application. Before the next experiment, Cl^- from the previous experiment was leached below the measuring zone by applying water to the whole site to induce deep drainage.

Inverse Estimates of , K_s and _s
Five parameters (i.e., , K_s, _s, _r, and m) are included in Eq. [1], [2], and [3]. In general, _r does not vary significantly in the field. Thus, the 1.5 MPa soil water content determined from repacked soil cores was used as an estimate of _r, which is 0.052 m³ m^-3. The soil at the Borden site is fine sand and the reported saturated hydraulic conductivity values is (Si et al., 1999). The soil is reasonably similar to the hypothetical sandy loam soil of Kool et al. (1985), and consequently Russo's (1988) value for was assumed. The inverse procedure of Zhang et al. (2000) was used to estimate , K_s and _s. The general inverse model used to estimate these three parameters is

(7)

where G_i is a measurement, which is either pressure head (), water storage (W), or tracer travel time (T), _i (, K_s, _s) is the model prediction, _i is measurement error, and N is the number of measurements. Since the inversion model (Eq. [7]) is nonlinear, the confidence level of the estimates will be approximate. Details are given below.

Combinations of different sets of measurements (i.e., involving , W, or T) are used in Eq. [7]. Since , W, and T have nonlinear relationships with , K_s, and _s, a nonlinear least squares fitting procedure is used with the assumptions of normality and independence of the errors, (Ratkowsky, 1983; Draper and Smith, 1966). The modified objective function, S_p(, K_s, _s), is the sum of squared error and a constraint term for the nonlinear model (Zhang et al., 2000, Eq. [18]):

(8)

where n = 1, 2, or 3, is the number of combining sets of measurements used, n_j is the number of measurements in a particular set, w_j are the weights associated with a particular measurement set, w_ij is the weights associated with a single measurement, w_s is the weight associated with _s, and _sp is the value of _s determined using prior information. An estimate of _sp = 0.42 is used based on the work by Si (1998) conducted on the same site. The weight w_s will be determined on the basis of the variance of _s. If it is assumed that the 95% confidence interval can be approximated by , then , where is the standard error of _s. The estimated variance of _s, Var_s, is then equal to .

The values of the weighting coefficients in Eq. [8] are determined by the method given by Zhang et al. (2000). Briefly, the weighting coefficient associated with a point, w_ij, is set equal to 1/n_j with n_j being the number of points in a particular measurement set, assuming that the error within a particular measurement set is uniform. By this approach, the summation of w_ij for a particular measurement set is equal to 1. The weighting coefficients associated with each measurement set, w_j, and with _s, w_s, are calculated by

(9)

where LMS_j is the least mean squared error obtained by assigning all the weights to a single set of measurements (Zhang et al., 2000).

If in Eq. [7] is normally distributed and has a mean of zero, the least squares estimate of the parameters (, K_s, and _s) which minimize the objective function, Eq. [8], also gives the maximum likelihood estimate of the three parameters (Draper and Smith, 1966). The calculation of Eq. [8] was performed with MathCad PLUS 6.0 (MathSoft, Inc., 1996). Finding the minimum value of S(, K_s, _s) was accomplished by calculating S(, K_s, _s) on a cubic grid around the initial guess values for and K_s, and the prior determined _s. Once the point with the least error was found, the calculation was repeated by setting a smaller calculation step and a suitable range of and K_s on the basis of the convergence point values. In the final calculation, the step in was set equal to 0.25 m^-1, that of K_s to 2.5 x 10^-7 ms^-1, and that of _s to 0.001 m³ m^-3. The reason we used a cubic grid instead of a search routine is that the former can guarantee the global convergence regardless of calculation errors.

According to Zhang et al. (2000), unique parameters cannot be obtained by the measurements of pressure head only or by tracer travel time only in Eq. [8]. Thus, for estimating parameters, the W-only approach, three two-set combination approaches (i.e., + W, + T, and W + T), and the three-set combination approach ( + W + T) were used. Note that the addition signs here represent combinations in Eq. [8] and not mathematical summation.

Confidence Contours and Confidence Intervals
With prior information for _s available, the results of the estimates of and K_s can be expressed as confidence contours in the -K_s plane. The approximate 100(1 - )% confidence contours can be calculated by finding points (, K_s, _s) which satisfy (Draper and Smith, 1966)

(10)

where p = 3 is the number of parameters to be estimated, N is the total number of observations, is the confidence level, (,_s,_s) are the best-fit values of , K_s, and _s, S_p(,_s,_s) is the minimum residual sum of squares occurring at (,_s,_s), and S_p(, K_s, _s) is the residual sum of squares occurring at any other value of (, K_s,_s). The is obtained from a table of the statistical F-distribution corresponding to significance level and numerator degrees of freedom p and denominator degrees of freedom N - p.

When the error of the nonlinear model (Eq. [7]) is assumed to be normally distributed, _s is no longer normally distributed, and the sample variance, , is no longer an unbiased estimate of the population variance, ², (Draper and Smith, 1966). Although confidence regions can still be defined by Eq. [10], they are approximate 100(1 - )% confidence contours for (, K_s, _s) (Draper and Smith, 1966). The approximate confidence interval for each parameter is obtained by finding the minimum and the maximum parameter values inside the corresponding confidence contour.

Guelph Permeameter and Guelph Pressure Infiltrometer Methods
After all the experiments using the multi-purpose TDR probes were completed, the hydraulic properties of the site were measured by the GP and GPI systems at 1-m intervals along the original line source. The auger holes for GP measurements had 0.2-m depth and 0.05-m diameter. Steady-state infiltration rates under two ponded heights, 0.05 and 0.1 m, were measured to calculate and K_s. In addition, the modified GPI system of Fallow and Elrick (1996) was used to estimate by measuring the soil water entry pressure heads with a water manometer attached to the air tube of the GPI as:

(11)

where _we is water entry pressure head. The GPI ring diameter was 0.1 m and the insertion depth was 0.05 m. In total six measurements were taken by means of the GP and another six by the GPI system.

	Results and discussions

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

 
Estimating the Average Values of , K_s, and _s
By setting and using the appropriate number of observations (200 for , 200 for W, and 150 for T), the confidence contours can be estimated by Eq. [10]. The approximate 95% confidence contours in the –K_s plane, obtained by means of W-only, two-set combinations, and the three-set combination with source strength are shown in Fig. 2 . Note that the relative parameter values on the axes in Fig. 2 were scaled by the corresponding best-fit values obtained by means of the three-set combination (i.e., ). The size of the contours determines the precision of each approach. The smaller the contour, the more precise the parameter estimates. The three-set combination approach gives the smallest 95% confidence region. The W + and W + T approaches give similar estimates of and K_s and similar confidence regions. The W-only approach gives the largest range in and K_s. The overlapping of the confidence regions indicates the consistency of the estimates obtained by different approaches. However, there is still about 5% chance that the true value is outside of the confidence region. This is why the 95% confidence region obtained by the + T approach does not overlap with the others.

View larger version (12K): [in this window] [in a new window]   Fig. 2 The approximate 95% confidence contours for (, Ks) estimated by five different approaches (q = 0.90 x 10-6 m3 s-1 m-1). The confidence regions are correct but the confidence level is approximate. Note: k = 13.9 m-1, Ksk = 4.16 x 10-5 ms-1

The best-fit values of and K_s and the 95% confidence intervals indicate that the three experiments with different source strengths gave nearly the same results (Table 1) . The use of all the data from the three experiments leads to an increase in the precision of parameter estimates, as shown by narrower confidence regions. However, the average values of the parameter estimates remain nearly the same regardless of the number of data sets used.

For the GP method, the average K_s was 5.0 x 10^-5 ms^-1 with the range (minimum and maximum values) of (1.1 x 10^-5, 1.0 x 10^-4) ms^-1, and the average value was 6.5 m^-1 with the range of (2.2, 16.6) m^-1. For the GPI method, the average K_s was 9.2 x 10^-5 ms^-1 with the range of (5.2 x 10^-5, 1.4 x 10^-4) ms^-1, and the average value was 12.3 m^-1 with the range of (8.3, 14.3) m^-1. Although the GP and GPI estimates are more variable, they nonetheless compare well with the estimates obtained by means of the 2-D line source and inverse estimation (Table 1).

Comparisons of the predicted and measured averages of , W, and T of each experiment using the parameter values estimated by the + W + T approach are given in Tables 2 to 4 . The model, Eq. [1], predicts pressure head with error 0.033 m at the measurement depth 0.4 m with only one exception (Table 2). The model underestimates the absolute values of pressure head 0.078 0.106 m at the 0.2-m depth. equation [2] predicts water storage very well, with error 0.003 m (Table 3). In the case of soil water content, the prediction error is 0.01 m³ m^-3. equation [3] predicts tracer travel time with the error of -0.67 0.89 h (Table 4). The model underestimates tracer travel time at the measurement depth 0.2 m, but overestimates it at the measurement depth 0.3 m. Recall the assumption that the depth-averaged soil water content does not vary appreciably along the vertical streamline in determining the location of the conservative ionic tracer front by Eq. [6]. Therefore the measured tracer travel time is approximate making it difficult to distinguish the model error from the measurement error of tracer travel time.

Estimating Local Values of , K_s, and _s

View larger version (21K): [in this window] [in a new window]   Fig. 3 The local values of (a) , (b) ln(Ks), and (c) s and their trends

View this table: [in this window] [in a new window]   Table 5 Statistical summary of Ks and as measured along a 7.5-m long line source

View larger version (30K): [in this window] [in a new window]   Fig. 4 Histograms of (a) , (b) ln(Ks), and (c) s

If we are interested in K_s only, one single W measurement can be used because W is more sensitive to changes in K_s than to . That is, W has nearly zero sensitivity to changes in if the horizontal distance to the source (Zhang et al., 2000). Table 6 shows the results of K_s obtained by one W measurement. Even if the selected value changes from 6 to 36 m^-1, the estimates of K_s do not change very much. Compared with the results in Table 1, all the values of K_s in Table 6 are acceptable.

View this table: [in this window] [in a new window]   Table 6 The mean and standard deviation of saturated hydraulic conductivity obtained by water storage only.

View this table: [in this window] [in a new window]   Table 7 The mean and standard deviation of and Ks using the minimum information.

	Summary and conclusions

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

	ACKNOWLEDGMENTS

 
The 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 discussions Summary and conclusions REFERENCES

 

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This Article Abstract Figures Only Full Text (PDF) Free Alert me when this article is cited Alert me if a correction is posted Services Similar articles in this journal Similar articles in ISI Web of Science Alert me to new issues of the journal Download to citation manager Citing Articles Citing Articles via HighWire Citing Articles via ISI Web of Science (2) Citing Articles via Google Scholar Google Scholar Articles by Zhang, Z.F. Articles by Si, B. Search for Related Content PubMed Articles by Zhang, Z.F. Articles by Si, B. GeoRef GeoRef Citation Agricola Articles by Zhang, Z.F. Articles by Si, B.