The influence of time domain reflectometry rod induced flow disruption on measured water content during steady state unit gradient flow

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1 WATER RESOURCES RESEARCH, VOL. 42, W08420, doi: /2005wr004604, 2006 The influence of time domain reflectometry rod induced flow disruption on measured water content during steady state unit gradient flow Andrew C. Hinnell, 1 Ty P. A. Ferré, 1 and Art W. Warrick 2 Received 22 September 2005; revised 21 February 2006; accepted 21 April 2006; published 12 August [1] Intrusive measurement techniques require placing a sensor within the sample, possibly changing the conditions under which the measurement is collected and thereby affecting the quality of the measurement. In this study we consider time domain reflectometry (TDR), which is an intrusive water content measurement method. TDR rods are impermeable, and thus water is forced to flow around the rods. In an unsaturated medium this changes the water content distribution in the vicinity of the rods, with the water content increased at the tops of the rods and decreased at the bottoms of the rods. TDR has nonuniform spatial sensitivity, with much higher sensitivity immediately adjacent to rods, in the regions that experience the greatest change in water content due to this flow disruption. Furthermore, the spatial sensitivity of TDR depends on the water content distribution within the sample volume. This raises the possibility that flow disruptions caused by TDR rods may affect the TDR-measured water content. In this study we are specifically interested in the effects of flow disruption due to TDR rods. Therefore we consider steady state unit gradient unsaturated flow in a homogeneous medium to eliminate spatial heterogeneity of water content due to soil heterogeneity and transient flow conditions. For common TDR probe designs in the wide range of soils examined, flow disruption gives rise to a water content measurement error that is less than cm 3 cm 3. This is smaller than the reported accuracy of the TDR method of 0.02 cm 3 cm 3.Asa result, we conclude that it is appropriate to ignore flow disruption caused by commonly used TDR probes when assessing sources of TDR measurement error. Citation: Hinnell, A. C., T. P. A. Ferré, and A. W. Warrick (2006), The influence of time domain reflectometry rod induced flow disruption on measured water content during steady state unit gradient flow, Water Resour. Res., 42, W08420, doi: /2005wr Department of Hydrology and Water Resources, University of Arizona, Tucson, Arizona, USA. 2 Department of Soil Water and Environmental Science, University of Arizona, Tucson, Arizona, USA. Copyright 2006 by the American Geophysical Union /06/2005WR W Introduction [2] Indirect measurement methods can offer rapid, nondestructive, noninvasive monitoring of transient hydrological processes. However, quantitative use of these methods requires that they provide sufficiently accurate measurements without systematic bias. The primary limitation to the utility of indirect methods is their need for calibration. Through calibration, the response of an instrument can be related to the property of interest while accounting for other influences on the instrument response. Some methods, such as time domain reflectometry (TDR) for water content measurement, have minimal need for soil-specific calibration [Topp et al., 1980]. Other methods, such as electrical resistance tomography (ERT), are more subject to calibration uncertainties [e.g., Yeh et al., 2002]. A secondary limitation on the quantitative use of indirect methods is their inherent spatial averaging. Instruments do not necessarily follow arithmetic volume-weighted averaging of the property of interest within their sample volume. Rather, each instrument has a unique spatial weighting that depends upon the underlying physics of the method, the instrument design, and for some instruments, the spatial distribution of the property of interest. Typically, calibrations are performed on homogeneous samples. As a result, if the property of interest varies within the measurement volume of the instrument, the instrument response may not be related to the volume-weighted average property of interest within the sample volume in the manner described by the calibration equation [e.g., Nissen et al., 2003]. A third limitation can apply to the quantitative use of invasive indirect methods; the presence of the instrument may change the spatial distribution of the property of interest within the sample volume. The potential impacts of these limitations on measurement accuracy and bias can be examined through the use of coupled hydrogeophysical models that predict the instrument response as a function of the hydrologic conditions. At a minimum, these models can identify conditions under which indirect measurements can be used with confidence for quantitative hydrologic monitoring. The results may also lead to correction schemes for indirect measurements. If simple measurement corrections are not possible, these coupled hydrogeophysical 1of8

2 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 models can be used to incorporate indirect measurements directly into hydrologic analyses. [3] Time domain reflectometry (TDR) is a convenient, rapid, and intrusive geophysical method frequently used to measure volumetric water content in the field and the laboratory [e.g., Robinson et al., 2003]. TDR indirectly measures a nonuniformly spatially weighted average of the dielectric permittivity within the sample volume of the probe. If the water content is nonuniform, the measured dielectric permittivity does not equal the arithmetic average. In fact, if the water content is nonuniform, the spatial weighting depends on the volumetric water content distribution within the sample volume [Ferré et al., 1996]. After applying a calibration relationship, the TDR-measured water content will only equal the arithmetic average water content if the water content is uniform within the sample volume. TDR is an intrusive measurement technique. As a result, the presence of TDR probes may change the water content distribution in the immediate vicinity of the rods. Because the sensitivity of TDR depends on the water content distribution, the presence of the TDR rods may give rise to systematic errors in the TDR-measured water content. Unfortunately, in practice the spatial scale at which these water content changes are occurring precludes direct measurements of the water content distribution around TDR probes. Therefore, in this study, we use a coupled hydrogeophysical model to examine the potential errors in water content measurement that may occur due to flow disruptions caused by TDR probes. We focus specifically on the effects of water content changes that arise due to flow disruptions caused by impermeable TDR probes. To eliminate other complications, such as the effects of soil heterogeneity or of spatially nonuniform water contents during transient flow, we consider steady state unit gradient vertical unsaturated flow through a homogeneous medium. On the basis of our analysis, we then consider the likely relative contributions of flow disruption and other causes of nonuniform water content distributions on water content measurement errors using TDR. 2. Theory [4] Water flow in a homogeneous medium undergoing steady state unit gradient vertical flow produces spatially uniform pressure head and water content regardless of the level of water saturation. Philip et al. [1989] and Warrick and Knight [2002] describe the pressure head distributions in the vertical plane in the presence of an infinitely long impermeable horizontal cylinder. These analytical solutions are based on the linearized form of Richards equation. The pressure head increases above the cylinder relative to the pressure head that would exist in the medium with no cylinder present (the background pressure). The region of elevated pressure head extends from the top of the cylinder downward around the cylinder as two lobes. The pressure head decreases below the cylinder (Figure 1). The magnitude of the change in pressure head and the length of the higher-pressure lobes increase with increased cylinder radius and with decreased capillary length of the medium. [5] Most TDR probes comprise two or three parallel metal rods with circular cross sections. The solutions of Philip et al. [1989] and Warrick and Knight [2002] are linear, so the pressure head distribution around multiple Figure 1. Pressure head distribution around an infinite cylinder (radius = 0.25 m) in soil with a Gardner a of 10 m 1 and a background pressure head of 0.5 m computed using the analytical solution of Philip et al. [1989]. horizontal TDR rods can be computed by superposition of solutions for individual rods. However, these analytical solutions are limited to unsaturated systems. In addition, because they rely on the Gardner soil model, which has no associated pressure head/water content relationship, they do not predict the water content distribution. Because the water content distribution is of primary interest in this study, we employ a numerical solution of Richards equation: r KðyÞrH ¼ ðcðyþþs s ðþ ; where y is pressure head [L], K(y) is the pressure head dependent hydraulic conductivity tensor [L T 1 ], t is time [T], H is hydraulic head [L], C(y) is water capacity [L 1 ], and S s is the specific storage of the porous medium [L 1 ]. The water capacity is defined and is zero for both fully saturated and very dry conditions. The specific storage depends on the volumetric water content (q) [L 3 L 3 ], the compressibility of the medium (a) [L 1 ], and the compressibility of water (b) [L 1 ]: S s ¼ a þ qb: To account for both saturated and unsaturated conditions in a medium with a specified air entry pressure (y ae ), a piecewise continuous form of the hydraulic conductivity function is used: 8 < KðyÞ y < y ae K ¼ : K s : y y ae We use the van Genuchten [1980] and Mualem [1976] models, with zero air entry pressure, to relate pressure head, hydraulic conductivity, and volumetric water content. [6] TDR is based on measurement of the velocity of propagation of an electromagnetic wave along a wave guide through a medium. The wave guide commonly consists of ð1þ ð2þ ð3þ 2of8

3 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 Figure 2. Four time domain reflectometry (TDR) probe configurations and geometries considered in this study: two-rod vertical; two-rod horizontal; three-rod vertical; and three-rod horizontal. two or three stainless steel rods aligned parallel to each other (Figure 2). Assuming low-loss conditions, the velocity of propagation is related to the apparent dielectric permittivity of the soil, which can then be related to the volumetric water content. Ferré et al. [1996] present a linearized form of the Topp et al. [1980] equation relating the apparent dielectric permittivity of a medium to its volumetric water content: q ¼ 0:1181 ffiffiffiffi p e A 0:1841 or ea ¼ q þ 0: : ð4þ 0:1181 [7] If the water content is spatially uniform within the sample volume of a TDR probe, this calibration equation can be used directly to determine the dielectric permittivity that would be measured by TDR. If the water content varies within the sample volume, the TDR-measured apparent dielectric permittivity can be computed following the approach of Knight [1992]. Assuming that wave propagation is in the transverse electromagnetic (TEM) mode, the TDRmeasured dielectric permittivity for each plane perpendicular to the TDR rods can be expressed as a weighted average of the local dielectric permittivity values: ZZ e A ¼ A wx; ð zþe r ðx; zþda ð5þ where w(x, z) is the weighting function in the x-z plane (taking the direction of wave propagation to be y), e r (x, z) is the local dielectric permittivity, and A is the area in the x-z plane where w(x, z) > 0. The TDR-measured dielectric permittivity is the square of the integral of the square roots of the e A values along the length of the rods divided by the length of the rods. The weighting function can be determined by equating the total electric energy of the electromagnetic field in a homogeneous medium (f 0 ) to that in the same medium with a single small perturbation of the dielectric permittivity at one point (f(x, z)) [Knight, 1992]: wx; ð zþ ¼ ZZ jrfðx; zþj : ð6þ jrf 0 ðx; zþj da A [8] For TEM propagation the electrical potential distribution in the plane perpendicular to the direction of propagation is the same as the electrostatic case and can be described by the heterogeneous form of Laplace s equation [Knight, 1992]: rðeðx; zþrfþ ¼ 0; ð7þ where f is the electrical potential (volts) and e is the dielectric permittivity (F m 1 ). Because the electromagnetic energy density in the transverse plane is not spatially uniform, the sensitivity (weighting) of the TDR measurement is spatially variable. TDR has the highest sensitivity close to the TDR rods, with sensitivity diminishing sharply away from the rods. The spatial sensitivity does not depend on the water content if the water content is constant within the sample volume [Knight, 1992]. However, the TDR spatial sensitivity can vary strongly with the spatial distribution of the water content in the sample area if the water content is not spatially uniform [Ferré et al., 1996; Nissen et al., 2003]. [9] Ferré et al. [1998] presented a definition of the sample area of a TDR probe in the transverse plane based on the weighting function distribution. The f % sample area is defined as the region of the porous medium that contributes a defined percent ( f ) of the total probe response and that comprises those regions of highest local sensitivity: 100 Xi¼j w i A i f ¼ ZZ i¼1 ; ð8þ w i da A where w is the vector of weighting factors sorted in decreasing order and A i is the area with a weighting factor w i. This definition of the sample area identifies the smallest area that contributes the chosen fraction of the total instrument response and does not impose a presumed sample area shape. For example, the 75% and 90% sample areas are shown for two- and three-rod probes in a medium with a spatially uniform water content in Figure 3. [10] If the water content at each location in the transverse plane is known, for example, from the solution of a water flow model, equation (4) can be used to define the dielectric permittivity distribution. Then, the steady state electrical flow problem can be solved to determine the electrical potential distribution and equation (6) can be used to determine the weighting factors. The TDR-measured dielectric permittivity can be calculated using equation (5) and the 3of8

4 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 Figure 3. Cross sections through two- and three-rod vertical and horizontal TDR probes with a diameter of m and a separation of 0.05 m, located in 100% sand defined by q r = 0.05, q s = 0.38, a = 3.44 m 1, n = 4.42, K s = ms 1. The solid black lines show the 75% (inner contour) and 90% (outer contour) sample areas determined for a medium with a uniform water content. The color flood shows the water content during steady state vertical unit gradient flow at a rate of ms 1. percent sample areas can be determined from equation (8). Finally, equation (4) can be used to determine the TDRmeasured volumetric water content from the calculated TDR-measured dielectric permittivity [Knight et al., 1997]. In practical applications, further errors arise if the calibration equation (e.g., equation (4)) does not accurately reflect the relationship between bulk dielectric permittivity and volumetric water content. In this study we assume perfect calibration and compare the TDR-measured water content with and without flow disruptions caused by TDR probes to determine the impact of these flow disruptions on the accuracy and bias of TDR measurements during steady state unit gradient vertical flow through a homogeneous medium. On the basis of these findings, we compare this source of error with other likely sources of uncertainty for water content measurement with TDR. 3. Methodology [11] A variably saturated water flow model was coupled to a TDR response model in the FEMLAB modeling environment (Comsol Inc, Los Angeles, California). FEM- LAB allows full coupling of multiple physical processes and efficient numerical solutions. The code computes the steady-state water content distribution both with and without impermeable TDR rods present. During unit gradient flow in a homogeneous medium (no TDR rods present), the water content is spatially uniform; this water content is referred to here as the background water content (q BKG ). Assuming no calibration errors, this is the water content that TDR would measure if the probe did not change the local water content distribution around the rods. Unit gradient flow through a homogeneous medium was selected specifically for this study because the water content without the rods present is spatially uniform, which allows for clearer identification of the impacts of flow disruption caused by the TDR probes on the TDR response and sample areas. The TDR-measured water content (q TDR ) is computed based on the predicted water content distribution including the effects of flow disruptions. Similarly, this disturbed water content distribution is used to define the sample areas of the TDR probes with consideration of flow disruptions. The water content measurement error (E) is defined as the difference between the TDR-measured water content and the water content of the medium without the TDR rods present, such that a positive error is associated with a measurement bias toward overestimation of the water content: E ¼ q TDR q BKG : [12] The ability to design TDR probes for specific applications has supported the broad use of the method for water ð9þ 4of8

5 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT Table 1. Ranges of Parameter Values Describing Probe Configuration and Geometry, Soil Type, and Applied Flux Considered Parameter Range Number of Values Number of rods 2 or 3 2 Probe orientation vertical or horizontal 2 Soil type 100% sand to 100% silt to 100% clay 1326 Diameter, D m to 0.05 m 15 Separation, two-rod probe 1.5 D to 500 D 27 Separation, three-rod probe 2.5 D to 500 D 27 Water flux optimized to find maximum error variable W08420 content measurement in many fields. A wide range of TDR probes can be described based on their configuration and geometry. We limit our study to standard TDR probes that comprise two or three parallel metal rods with circular cross sections buried with the rods parallel to the soil surface. The configuration of these probes describes the number of rods (two or three) and the orientation of the plane containing the rods (vertical or horizontal). The probe geometry describes the rod diameter and the separation of the outermost rods. Four probe types are considered and are referred to as (1) tworod vertical probes, (2) two-rod horizontal probes, (3) threerod vertical probes, or (4) three-rod horizontal probes (Figure 2). [13] A common domain and finite element mesh are used for the water flow and TDR finite element models. The domain is square and oriented vertically. A TDR probe with the desired configuration and geometry is centered at the midpoint of the domain. For the water flow problem, the lateral boundaries of the domain are zero flux and the pressure head on the upper and lower boundaries are set equal to establish unit gradient flow conditions. The rod surfaces are defined as internal zero water flux boundaries. For the electrical flow problem, all external boundaries are zero electrical flux. The rod surfaces are defined as internal constant potential boundaries: For the two-rod probes, f = 1 volt for the left or upper rod and f = 1 volt for the right or lower rod; for the three-rod probes, f = 1 volt for the center rod and f = 1 volt for the outer rods. A 5.0-m by 5.0-m domain size was shown to be sufficiently large to approximate an infinitely large domain by varying the outer dimensions of the domain until further increases in the domain size did not affect the water content distribution around the TDR probe or the shape or the extent of the 90% sample area. [14] Six parameters that affect the TDR-measured water content were identified: water flux; soil type; number of rods forming the probe; orientation of the plane containing the rods; rod diameter; and rod separation. The total number of possible combinations of these six parameters is too large to examine completely with fine resolution of each parameter value (Table 1). Therefore we initially identified the conditions that lead to the greatest measurement errors using 25 different probe and soil property combinations. On the basis of these simulations we examined in detail those combinations that led to the largest measurement errors. The results of the analysis are presented in terms of the maximum measurement error. We used the local search algorithm fminsearch from the Matlab optimization toolbox (Mathworks, Natick, Massachusetts) to identify the maximum measurement error and the associated flux for each set of soil and probe parameters under consideration. Detailed analyses only considered this maximum-error flux. Preliminary analyses also showed that three-rod horizontal probes consistently resulted in the largest measurement error; therefore this probe type was used when examining the dependence of measurement error on soil parameters. Finally, for a given probe, the largest measurement errors occur in 100% sand, so this medium was used when examining the effects of probe parameters on measurement error. [15] Detailed investigation of the influence of soil type on measurement error was completed using a set of 1326 soil textures defined by varying the sand/silt/clay ratios at 2 percent increments. The hydraulic parameters (residual water content, saturated water content, and saturated hydraulic conductivity), and van Genuchten parameters (a and n) that correspond to these soils were identified using the ROSETTA [Schaap and Leij, 1998] pedotransfer function based on the soil texture (sand/silt/clay percentage). A three-rod horizontal probe with a rod diameter of m and a rod separation of 0.05 m were used because they are similar to commercially available probe geometries (e.g., CS605 TDR probe by Campbell Scientific, Inc., Logan, Utah). The detailed sensitivity analysis of the effect of probe configuration and geometry on the TDR-measured water content was completed using all four configurations and 279 separation diameter pairs in 100% sand. Diameters (D) ranged from to 0.05 m to cover the rod sizes used for most applications. The smallest separation examined was chosen such that the rods were close, but not touching (2.5 D for the three-rod probes and 1.5 D for the two-rod probes). The maximum separation was 500 D, with the further limitation that the separation be less than or equal to 0.5 m. 4. Results and Discussion [16] Flow disruption caused by the TDR probes has a dual impact on the measured water content. First, the disruption causes a change in the water content within the sample volume of the probe. Second, this change in water content distribution can alter the spatial sensitivity of the instrument. The regions experiencing the greatest changes in water content due to flow disruption are located within the 75% and 90% sample areas of a TDR probe (Figure 3). However, for the probes and flow conditions considered in this study, the maximum change in water content is quite small (0.01 cm 3 cm 3 ). The small changes in dielectric permittivity associated with these water content changes have a negligible effect on the spatial weighting. As a result, 5of8

6 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 Figure 4. TDR-measured water content and arithmetic average water content within the sample area as a function of selected percent sample area, f, for a three-rod horizontal probe with a diameter of m and a separation 10 times the diameter in sand with a flux of ms 1 downward past the probe. the sample areas determined for the heterogeneous water content distribution are indistinguishable from those in a homogeneous medium (not shown). Although the spatial sensitivity of a standard TDR probe is not expected to change due to flow disruption, the spatial sensitivity is not uniform. As a result, the TDR measurement error is a function of the absolute changes in water content within the sample volume and the nonuniform spatial sensitivity of TDR. The contribution of nonuniform spatial weighting can be examined by comparing the TDR-weighted average water content and the arithmetic average water content as a function of the choice of the f% sample area (Figure 4). For the boundary conditions examined and domain size used, the arithmetic average water content for the entire domain (100% sample area) is equal to the background water content. The arithmetic average water content for small percentage sample areas is greater than the background water content, reflecting the buildup of water in the regions of highest TDR sensitivity close to the TDR rods. The arithmetic average water content remains higher than the actual water content for percent sample areas as large as 90%. The arithmetic average water content only decreases when the areas of slightly reduced water content that lie in the region of minimal TDR sensitivity are included in the average. The TDR-measured water content for small percentage sample areas varies erratically due to the discrete nature of our water flow and electrical flow solutions. The TDR-measured water content becomes more stable by the 50% sample area, approaching a constant value equal to that of the 100% sample area. Unlike the arithmetic average, the TDR-measured water content is not sensitive to the large area of slightly decreased water content far from the probe, leading to an over-prediction of water content even for the 100% sample area. The equivalence of the TDR-measured water contents for the 90% and 100% sample areas for all conditions studied here supports the definition of the 90% sample area as representative of the sample area of a TDR probe in an infinite domain. [17] For two- and three-rod vertical probes, a decreased rod separation causes an increase in water content above and beside the upper rod, and a decreased water content below the lower rod (Figures 5a and 5b). Similarly, horizontal two- and three-rod probes show a larger buildup of water above the rods as the separation is reduced (Figures 5c and 5d). Furthermore, if the rods of a horizontal probe are sufficiently close, the regions of water buildup coalesce (Figure 5d). Generally, the region of increased water content and the magnitude of the water content increase are larger for horizontal probes than for vertical probes with the same separation, especially for probes with small separations (Figures 5b and 5d). Adding a center rod to form a threerod probe amplifies the shadowing effect seen in vertical probes and increases the likelihood of coalescence of wetted regions above horizontal probes. [18] For a given rod diameter, separation of the outermost rods, and soil type, a three-rod horizontal probe causes the greatest flow disruption and a two-rod vertical probe causes the least flow disruption (Figure 3). As a result, the threerod horizontal configuration produces the largest measurement error, and the two-rod vertical configuration produces the smallest measurement error (Figure 6). TDR measurement error increases nonlinearly with water flux (Figure 6) from approximately zero for very dry conditions to a maximum at an intermediate flux, then diminishes for increasingly wet conditions. For fluxes less than the saturated hydraulic conductivity, the TDR-measured water content is larger than the background water content (positive measurement bias). For fluxes slightly greater than the saturated hydraulic conductivity, an unsaturated region may remain below the TDR rods, resulting in a small negative measurement bias. For higher fluxes the medium remains fully saturated and there is zero measurement error. For all probe configurations, the measurement error increases with increasing rod diameter and with decreasing rod separation, with a stronger dependence on rod diameter than rod separation (not shown). [19] The logarithm of the maximum measurement error for each of the 1326 soils considered using a three-rod horizontal probe with a geometry similar to the CS605 TDR probe is plotted on a standard soil triangle in Figure 7. The largest measurement error was cm 3 cm 3 in 100% sand. This is far lower than the method uncertainty of 0.02 cm 3 cm 3 reported by Topp et al. [1980]; in fact, it is too small to measure reliably with any existing methods. Coarse-grained, well-sorted soils (red zones in Figure 7) have the largest maximum measurement errors, and finegrained, poorly sorted soils (blue zones in Figure 7) have the smallest maximum measurement errors. The maximum measurement error consistently occurred at a water flux close the to the water flux associated with the inflection point on the soil characteristic curve. At these fluxes the water content change for a given change in pressure head is greatest. The slope of the characteristic curve for a coarsegrained, well-sorted soil is steeper than the slope of a poorly sorted, fine-grained soil (i.e., the change in water content is larger for a given change in pressure head), resulting in larger changes in the water content around the TDR probe and correspondingly larger measurement errors. [20] For standard TDR probes we only predict measurable errors (E > 0.02 cm 3 cm 3 ) in the 100% sand shown in Figure 7 for a three-rod horizontal probe with a large rod diameter (>0.02 m) and a small separation (<4 D). However, 6of8

7 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 Figure 5. Cross sections through two- and three-rod vertical and horizontal probes during steady state vertical unit gradient flow at a rate of ms 1 in a sand. Both the contour lines and the color flood show the water content distribution. significant measurement errors could also occur in wellsorted, coarse-grained soils with low air entry pressure and a sharp drop to residual water content (not shown). As a result, measurements made in the sand described by Carsel and Parrish [1988] or in other well-sorted coarse materials (e.g., mining waste rock piles or construction debris) may result in measurable errors due to flow disruption. Similarly, more significant measurement errors may occur for TDR probes with larger diameters and/or smaller separations than those examined in this study or for probes attached to impermeable probe bodies. 5. Conclusions [21] Unsaturated water flow around impermeable inclusions (e.g., the rods forming a TDR probe) results in a water content distribution that is different than would exist without the inclusions present. For some measurement methods this may also change the spatial sensitivity distribution of the method. We use a coupled hydrogeophysical numerical model to examine a simple example: the impact of standard two- or three-rod TDR probes on the water content during steady state unit gradient flow through a homogeneous medium. Results show that the TDR rods do change the water content within the sample volume of the TDR probe. However, for standard TDR rods in typical soils, the water content changes caused by flow disruption are too small to have a significant impact on the spatial sensitivity or the measurement accuracy of TDR probes. As a result, for most applications of TDR the effects of flow disruption due to the presence of the rods can be ignored; other sources of spatial nonuniformity of water content (e.g., transient flow con- Figure 6. TDR measurement error versus flux for four configurations with D = m and S = 0.05 m and the rods located in coarse grained sand (q r = 0.05, q s = 0.38, a = 3.44 m 1, n = 4.42, K s = ms 1 ). 7of8

8 W08420 HINNELL ET AL.: EFFECT OF FLOW DISRUPTION ON WATER CONTENT W08420 Figure 7. Logarithm of the TDR measurement error for different soil textures. Soil properties were defined using a pedotransfer function [Schaap and Leij, 1998] based on percent sand, silt, and clay. A three-rod horizontal TDR probe was used with D = m and S = 0.05 m. ditions and small-scale soil heterogeneity) and calibration errors are likely to be more significant than flow disruption. Measurement errors are largest for three-rod probes placed in the horizontal plane. In addition, larger-diameter rods and smaller rod separations give larger measurement errors. Finally, measurement errors are higher in coarse-grained, well-sorted soils than in fine-grained, poorly sorted soils. Therefore, although we demonstrate that no measurable errors are expected due to flow disruption for typical applications of TDR, custom probes that use large rods with small separations, probes that are attached to impermeable probe bodies, or measurements made in very coarse materials may lead to more significant errors. The modeling approach described in this study is an efficient tool for identifying and, possibly, correcting these errors. [22] Acknowledgments. The authors would like to thank the anonymous reviewers for their detailed constructive comments. This project was supported by the National Research Initiative of the USDA Cooperative State Research, Education and Extension Service, grant number References Carsel, R. F., and R. S. Parrish (1988), Developing joint probability distribution of soil water retention characteristics, Water Resour. Res., 24(5), Ferré, P. A., D. L. Rudolph, and R. G. Kachanoski (1996), Spatial averaging of water content by time domain reflectometry: Implications for twin rod probes with and without dielectric coatings, Water Resour. Res., 32(2), Ferré, P. A., J. H. Knight, D. L. Rudolph, and R. G. Kachanoski (1998), The sample areas of conventional and alternative time domain reflectometry probes, Water Resour. Res., 34(11), Knight, J. H. (1992), Sensitivity of time domain reflectometry measurements to lateral variations in soil water content, Water Resour. Res., 28(9), Knight, J. H., P. A. Ferré, D. L. Rudolph, and R. G. Kachanoski (1997), A numerical analysis of the effects of coatings and gaps upon relative dielectric permittivity measurement with time domain reflectometry, Water Resour. Res., 33(6), Mualem, Y. (1976), New model for predicting hydraulic conductivity of unsaturated porous media, Water Resour. Res., 12(3), Nissen, H. H., P. A. Ferré, and P. Moldrup (2003), Sample area of two- and three-rod time domain reflectometry probes, Water Resour. Res., 39(10), 1289, doi: /2002wr Philip, J. R., J. H. Knight, and R. T. Waechter (1989), Unsaturated seepage and subterranean holes: Conspectus, and the exclusion problem for circular cylindrical cavities, Water Resour. Res., 25(1), Robinson, D. A., S. B. Jones, J. M. Wraith, D. Or, and S. P. Friedman (2003), A review of advances in dielectric and electrical conductivity measurements in soils using time domain reflectometry, Vadose Zone J., 2, Schaap, M. G., and F. J. Leij (1998), Database-related accuracy and uncertainty of pedotransfer functions, Soil Sci., 163(10), Topp, G. C., J. L. Davis, and A. P. Annan (1980), Electromagnetic determination of soil water content: Measurements in coaxial transmission lines, Water Resour. Res., 16(3), van Genuchten, M. T. (1980), A closed-form equation for predicting the hydraulic conductivity of unsaturated soils, Soil Sci. Soc. Am. J., 44(5), Warrick, A. W., and J. H. Knight (2002), Two-dimensional unsaturated flow though a circular inclusion, Water Resour. Res., 38(7), 1113, doi: / 2001WR Yeh, T. C. J., S. Liu, R. J. Glass, K. Baker, J. R. Brainard, D. Alumbaugh, and D. LaBrecque (2002), A geostatistically based inverse model for electrical resistivity surveys and its applications to vadose zone hydrology, Water Resour. Res., 38(12), 1278, doi: /2001wr T. P. A. Ferré and A. C. Hinnell, Department of Hydrology and Water Resources, University of Arizona, P.O. Box , Tucson, AZ 85721, USA. (andrew@hwr.arizona.edu) A. W. Warrick, Department of Soil Water and Environmental Science, University of Arizona, P.O. Box , Tucson, AZ 85721, USA. 8of8