An Indirect Calibration Procedure for Using TDR in Solute Transport Studies

• Abstract
  
• Introduction
  
• Materials and methods
  
• Model 1. The three pathway conductivity model
  
• Model 2. The statistical pore size distribution model
  
• Results and discussion
  
• Summary and conclusions
  
• Notes
  
• References
  

Abstract

The possibility of measuring both soil and water content and salinity for relatively large and undisturbed sampling volumes with one single probe, makes TDR an interesting tool in solute transport studies. Steady state flow experiments in soil instrumented with TDR probes were performed to identify solute transport parameters. The propagation of a tracer mass through a volume of soil was deduced from the attenuation of an electromagnetic wave traveling along parallel transmission lines embedded in the soil volume. One possible way to derive solute transport parameters using TDR is to relate the bulk electric conductivity of the soil, EC_a, to the soil water electrical conductivity, EC_w. In this paper an indirect calibration procedure is presented which utilizes conceptual models to relate EC_a to EC_w, and hence allows TDR to be used in solute transport studies. The performance of this approach, using two different models, i.e. (I) the Three Pathway Conductivity Model (14), and (ii) the Statistical Pore Size Distribution Model (10), was investigated using a controlled laboratory experiment on a repacked sandy soil column. Although the first model provides a better prediction of the experimentally obtained EC_a-EC_w relationship, the application of either model is currently limited by parameters that are very sensitive or very difficult to assess.

Introduction

TDR has become a widely accepted method for determining volumetric soil moisture content, . The travel time of an electromagnetic (EM) wave in a coaxial system in which the soil serves as the dielectric, has been related to the apparent dielectric constant. Empirical and conceptual models (17, 15, amongst many others), have been developed which relate the apparent dielectric constant of wet soil as measured with TDR to the volumetric soil moisture content. Furthermore, the loss of energy of the reflected EM wave is a measure of bulk soil electrical conductivity, EC_a. Distributed circuit analysis has been used to calculate EC_a (1), while refined interpretation of the TDR wave form using more rigorous analysis of the propagation of an EM wave in a porous medium are presented in (18, 25, 26). To overcome multi-reflection interferences caused by impedance discontinuities and to simplify measurements, it is proposed to restrict TDR measurement of EC_a to the signal load at infinity (18, 11) or:

where K_c is the probe constant. K_c is related to the characteristic impedance of the probe, Z₀, and the probe length, L, (4):

It is suggested that a correlation be applied to the impedance at infinity using the resistance of the cable to direct current (EC_a = K_c/(Z - Z_cable)) (4). A direct calibration of the TDR probe in different solutions with known EC_a was used to calculate Z_cable and K_c (4, 2).

The possibility of measuring both soil moisture and bulk electrical conductivity with one single probe, makes TDR attractive for solute transport properties (3, 6, 8, 19, 24). The time history of soil bulk electrical conductivity, EC_a(t), as estimated from the time history of impedance of TDR waveforms at infinity Z(t), is approximately linearly related to electrical conductivity of the soil water, EC_w(t). The latter in turn, is nearly linearly related to the resident concentration, C(t), of a non-reactive tracer in the soil solution (6) or:

If the soil water initially has a low electrical conductivity with concentration C_t=ti, a solution with high EC and concentration C₀ of a non-reactive tracer can be added to the soil to displace the initial soil water. The relative concentration of the solute added to the soil ((C_t-C_t=ti)/(C₀-C_t=ti)=C/C₀, if C_t=ti is sufficiently small), can be calculated according to:

which is a modified version of the original equation developed by (6).

The empirically estimated C(t)/C₀ from the impedance at infinity can be used to estimate solute transport parameters, provided that 1/Z_,0 is known. The 1/Z_,0 value corresponds to the attenuation at infinity along the horizontal axis of the TDR trace, when the soil water in the TDR influence region is completely saturated with solution of concentration C₀. The value of 1/Z_,0 reflects the reduction of electrical conductivity in the porous medium compared to the electrical conductivity of water (EQUATION 3), an effect induced by the tortuosity of the porous system.

Different procedures have been suggested to calibrate TDR for solute transport studies. In a direct way, one can relate independent measures of the soil water concentration to the attenuation of a TDR signal at infinity as described in (2, 24) for disturbed soil samples. By mixing soil with different solution concentrations, one obtains soil water mixtures with varying EC_w and values. The EC_a can be measured using TDR, and a relationship between EC_a-EC_w- can be established. Alternatively, one measures the solute concentration at limited by well selected times, during the transport experiment, using e.g., suction samplers (20). Another approach would be soil coring of the material around the TDR rods. Values of EC_a can then be directly be related to resident concentrations. Time consuming calibration analysis, however, makes the direct approach impractical, especially when dealing with different layered soils showing different physico-chemical properties. Hence, indirect procedures were developed to calibrate TDR for solute transport studies.

When the TDR probes are installed vertically, the indirect calibration reduces to equating the specific mass of tracer applied, to the difference in EC_a before and after the solute is applied (6, 24). Using this approach, the calibration coefficients drop out if reasonable transport models are considered (3). With a vertical configuration however, it is assumed that all solute applied at the soil surface will immediately be detected by the TDR probe. In addition, the indirect calibration procedure for vertically installed probes ignores possible horizontal bypass of solute in a transport experiment. Furthermore, analysis of the TDR trace for moisture content becomes problematic for multilayered soils or deep in a homogeneous soil, due to multiple reflections or complete attenuation of the electromagnetic wave. To overcome this problem and to obtain solute resident concentration data and moisture content data representative for each measurement depth, one can install the TDR probes horizontally. The indirect calibration for horizontally installed TDR probes, however, is more involved.

With horizontally installed probes, on can use a step type boundary condition at the soil surface during a solute displacement experiment. At infinite time, all solute free water will be completely displaced and the calibration constant, 1/Z, 0 (EQUATION 4), can be directly read from the TDR screen. Several disadvantages should be considered with this approach. First, in field conditions, huge amounts of tracer solutions are needed to displace all soil water, especially when measurements extend to considerable depths within the soil profile. Second, physical non-equilibrium effects may increase the time required to distribute the solute uniformly in the soil, to an extent where it becomes practically impossible to continue application of the solute (9).

Alternatively, one can adopt a pulse boundary condition in a displacement experiment. In this way, more accurate estimates of solute transport parameters are obtained (5, 23). When using the pulse type boundary condition, the calibration constant can be obtained by equating the time-integrated TDR signal to the total solute mass added (19, 24). In this approach however, one assumes complete mass recovery and no differences between solute resident and flux concentration. These restrictions make it difficult to assess solute transport parameters especially when the soil is structured and contains micro-scale heterogeneities.

In this present study, the use of conceptual models which relate EC_a to EC_w was evaluated as a method to calibrate TDR theoretically. A solute transport experiment was performed in a repacked sandy soil to investigate the possibility of using two different conceptual models in the calibration procedure, i.e., the Three Pathway Conductivity Model (14) and (ii) the Statistical Pore Size Distribution Model (10).

Materials and methods

The TDR rod configuration consisted of two steel rods, 5 mm in diameter, 25 mm apart and 150 mm long, joined by an epoxy resin and connected to a Tektronix 1502B(*) cable tester using a 50 coaxial cable (7). The TDR probes were calibrated for soil moisture by measuring TDR travel times in soils with known moisture content and comparing these with a conceptual model (15, 17). For the determination of the cell constant, K_c in EQUATION 1, a direct calibration in salt solutions was performed (2). Both electrical conductivity and Z were measured in distilled water and in 6 different KCl solutions. The measurements were made with a commercial 4-electrode conductivity meter. As this meter provided automatic correction for solution temperature, the readings had to be adjusted to the electrical conductivity at the actual solution temperature measured. These measurements were also corrected for the cable resistance (O.5 ) (4), which was inferred from the curvilinearity of the calibration curve when the electrical conductivity became extremely high (2). The K_c factor (EQUATIONS 1, 2, and 3) obtained for the probes was equal to 4.758 m^-1 (2).

A coarse sandy soil (TABLE 1) was sieved and packed to fill two PVC cylinders (0.2 m internal diameter, 0.2 m length). The soil is structureless and shows no immobile water during transport experiments (19). Distinct points of the moisture retention curve of the experimental soil were measured on separate samples, using sand box and pressure plate apparatus. The small columns were equipped with tensiometer cups (diameter of 0.018 m, length of 0.06 m and air entry value of -70 m kPa) installed 1 cm above the bottom of the columns. Bifilary TDR probes were inserted horizontally at the column midplanes and connected to the cable tester. To establish saturated flow conditions, mariotte systems were used to maintain constant water levels of approximately 1.5 cm above the soil surface. The saturated water fluxes were equal to 32 and 38 cm/hr for the two columns, respectively. Unsaturated conditions were then established by placing the soil sample on a 1 m long free draining PVC cylinder uniformly filled with a medium textured sand. A constant flux of water was maintained using a syringe peristaltic pump. A schematic of the experimental setup for unsaturated flow conditions is shown in FIGURE 1.

Steady state flow was verified by constant tensiometer and TDR moisture content readings. The inlet solution was then switched from tap water to a CaCl₂ concentration with known electrical conductivity which induced an increase in attenuation of the TDR signal. It was assumed that equilibrium had been reached when all soil water was displaced with a solution of known electrical conductivity, and Z/t 0. The displacement experiment was carried out with increasing concentration of CaCl₂ solutions. A new solute concentration step was imposed when ten pore volumes were eluted. The measured EC_a-EC_w relationships, measured at three different moisture contents, were compared with the EC_a-EC_w relation estimated from the two conceptual models.

Model 1: The three pathway conductivity model

This model assumes that the EC_a in soil can be represented by conductance via three parallel pathways: (I) conductance through a continuous soil solution pathway (large pores); (ii) conductance through alternating zones of solid particles and soil solution which envelopes them (fine pores), and (iii) conductance through a continuous solid phase. As the last pathway is considered relatively insignificant, the follow conceptual model is proposed (14):

where EC_ws is electrical conductivity (dS/m) of the soil solution in small pores (the series coupled pathways of solids and pore water), and EC_wc is the conductivity for the pores constituting continuous pathways. _ws and _wc are water contents of the series-coupled pathways and the continuous pathways, the sum being equal to _w, the total volumetric water content. _s is the volumetric content of the solid phase, and EC_s is the electrical conductivity (dS/m) of the solid phase. For the conditions of our experiment, it is reasonable to set EC_ws equal to EC_wc. The electrical conductivity of the soil solid phase, EC_s, can be estimated from the soil clay content (13):

where A = 0.023 (dS/m) and B = 0.0209 (dS/m). This resulted for the studied soil in EC_s values of 0.063 dS/m. The moisture content in the series-coupled pathways, _ws, has been found to correlate well with the total soil moisture content _w,

where a = 0.528 and b = 0.0463 (13). Inserting EQUATIONS 6 and 7 in EQUATION 5 enables the calculation of EC_a for a given EC_w and _w.

Model 2: The statistical pore size distribution model

Mualem and Friedman (10) suggested a model for relating EC_a to EC_w based on the hypothesis that the flow lines of water molecules under a hydraulic gradient in a given soil are similar to the flow lines of electrical current (i.e., the charged ions) in the same soil under electrical potential. The model incorporates a geometry factor, accounting for the tortuosity and connectivity of the pores and reduced water flow in unsaturated soil when compared with water flow in a bundle of straight capillary tubes. The model concept is that this geometry factor can also be used to relate soil water electrical conductivity to the electrical conductivity of the bulk medium. The following model is suggested (10):

where is the soil volumetric moisture content, _r is the residual moisture content, Se is the degree of saturation
(Se = (-_r)/( _s-_r)), _s is the saturated moisture content; is the soil matric potential (cm), and the pore connectivity and tortuosity parameter, which was set equal to 0.5. In this study the van Genuchten model (21) was used to describe the moisture retention of the soil.

where , n, and m are empirical parameters representing the shape of the moisture retention characteristic. Inserting EQUATION 9 in EQUATION 8, and assuming n>2, EQUATION 8 yields (22):

where p₁ = m + 1/n, q₁ = 1-1/n, p₂ = m+2/n = 1-2/n, I the incomplete beta function; and B, the complete beta function. In order to obtain realistic values for this model, the drying curve of the moisture retention characteristic was measured on a subsample taken within the two soil columns. The incomplete and complete beta function were evaluated numerically using continuous fractions (12).

Results and discussion

The time history of the inverse of the TDR reflection magnitude, i.e., 1/Z, during a saturated steady state solute flow experiment is shown in FIGURE 2. The addition of soil solutions with increasing concentrations resulted clearly in steps of decreasing Z. The 1/Z curve typically reached an equilibrium (FIGURE 2, inset). It was assumed that the solute was distributed uniformly in the complete pore space, since there was no clear indication that non-equilibrium flow phenomena were occurring (19). We then increased the soil solution concentration, until a new equilibrium was reached. The estimated Z, corrected for the cable resistance, Z_cable, was inserted in EQUATION 3 with calibration constant K_c = 4.758 M^-1 to obtain the TDR estimate of EC_a.

The measured and fitted moisture retention curve is shown in FIGURE 3. Initial identification trials, using the multi-variate secant algorithm (16), without any restrictions resulted in optimal values for the n-parameter (EQUATION 9 far less than 4. In order to solve the integral in the denominator of EQUATION 8, one needs to invoke the Burdine restriction on the fit of the moisture retention, i.e., n>2. Numerical analysis of the ratio in the right hand side of EQUATION 10 , however, indicated that a stable solution of the integrals in this model can be obtained when n is greater than or equal to 4. A more stable solution could be obtained by setting different constraints to the values of m in EQUATION 9 such that for the numerator in EQUATION 8 m=1-1/n and for the denominator m=1-2/n, keeping the -parameter in EQUATION 8 constant. This however means that two different equations, describing the same moisture retention data have to be used in EQUATION 8, and the model becomes more empirical. Rather than going to more empiricism, the parameters of the moisture retention parameters were identified uniquely, with the restriction of n>4, resulting in a less optimal fit.

The crosses in FIGURE 4 depict the TDR-measured EC_a values for different values of EC_w, given two different moisture contents for the first column. Similar results were obtained for the replicate soil column (results not shown). The measured EC_a-EC_w relationship is linear (R² greater than or equal to 0.97) within the range of four measurements. The model-calculated estimated of the EC_a-EC_w relationship are depicted by the lines (continuous = Model 1; dashed = Model 2). Values of EC_s and were equal to 0.063 dS/m and 0.5, respectively. Model 2 overestimated the slope of the EC_a-EC_w curve. The discrepancies between measured and predicted EC_a increased with the soil moisture content. Model 1, however, matches closely the measured data points especially when the soil water salinity is not too high (EC_w<12 dS/m).

In order to analyze the performance of both models in more detail, a one-dimensional sensitivity analysis was carried out. A good model stands the joint test of being robust with respect to variation in parameters that are difficult to measure, and being strongly influenced by properties that are easy to measure. A sensitivity coefficient, defined as relative changes of the slope of the EC_a-EC_w curve, was calculated for different values of the model input parameters. The results of the sensitivity analysis for both models are depicted in FIGURE 5.

From FIGURE 5 it can be concluded that the slope of the EC_a-EC_w relationship calculated with Model 1 is insensitive to the clay percentage and the A and B parameters of EQUATION 6, and hence to EC_s. The estimate of EC_a from EC_w is very sensitive to the estimate of the a and b parameter of EQUATION 7. These parameters relate the volumetric moisture content in the serial pathways o the total volumetric moisture content. The moisture content in the serial pathways, however, is dependent on the soil structure, and hence, is hard to quantify. It is expected that the identification of this sensitive parameter is subjected to a lot of uncertainty.

From the bottom part of FIGURE 5 it can be concluded that the EC_a-EC_w relationship, as calculated with Model 2, is very sensitive to the estimate of the saturated moisture content, the slope parameter n, the skewness parameter m, and the pore connectivity and tortuosity parameter . It should be kept in mind that to obtain numerical convergence of the integrals in EQUATION 8, the n parameter of EQUATION 9 was restricted to values larger than 4. Moisture retention data of loamy or clayey soils cannot be fit to EQUATION 9 given these restrictions. Hence, Model 2 can only be used when dealing with coarse textured sandy soils. Given these severe restrictions, the poor model performance on the measured data and the height sensitivity of the model parameters to the EC_a-EC_w are of concern.

Summary and conclusions

The performance of an indirect TDR calibration procedure for assessing the EC_a-EC_w relationship from the attenuation of the TDR trace during steady state flow conditions is investigated. In this procedure, the TDR probes are calibrated in different salt solutions in order to obtain the probe constant prior to the installation of the probes in the soil. Next, the electrical conductivity of the soil water is calculated from the TDR estimated bulk electrical conductivity. Subsequently, two conceptual models, relating to EC_a to EC_w were evaluated. The two models, referred to as the Three Pathway Conductivity Model (14) and the Statistical Pore Size Distribution Model (10) conceptualize the relationship between the bulk electrical conductivity in the soil and the electrical conductivity of the soil water as a function of the water content and the pore geometry in two different ways.

Model 1 (14) matched the experimentally determined EC_a-EC_w data very well, while Model 2 (10) overestimated the slope of the EC_a-EC_w curve. Both models however have some limitations which make their use for the calibration of TDR for solute transport studies cumbersome. Model 1 (14) is very sensitive to the estimate of the soil water in the so-called series pathways, which is a soil parameter hard to quantify. Model 2 (10), on the other hand, is very sensitive to the slope parameter of the moisture retention characteristic. In addition, the model in its present form can only be used to coarse textured soils. Considering these uncertainties, more research is needed to appropriately quantify the model parameters for a variety of soils. For assessing solute transport parameters in soils, other indirect calibration procedures, based on mass recovery hypothesis, or direct calibration procedures should be compared with the presented indirect methodology. The calibration procedure has been used successfully in the exploratory phase of an experimental design.

Notes

* Reference to specific products does not imply endorsement by the U.S. Bureau of Mines.

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