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Published online 1 May 2008
Published in Soil Sci Soc Am J 72:750-757 (2008)
DOI: 10.2136/sssaj2007.0254
© 2008 Soil Science Society of America
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SOIL PHYSICS

Accounting for Bias and Boundary Condition Effects on Measurements of Saturated Core Hydraulic Conductivity

Matthew D. Madsena,b, David G. Chandlera,c,* and W. Daniel Reynoldsd

a Dep. of Plants, Soils and Climate, Utah State Univ., Logan, UT 84322
b currently at, Dep. of Plant and Animal Sciences, Brigham Young Univ., Provo UT, 84602
c currently at, Dep. of Civil Engineering, Kansas State Univ., Manhattan, KS 66506
d Greenhouse and Processing Crops Res. Centre, Agriculture and Agri-Food Canada, Harrow, ON, Canada N0R 1G0

* Corresponding author (dgc{at}ksu.edu).


   ABSTRACT
TOP
 NOTES
 ABSTRACT
INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Most hydrologic studies require knowledge of saturated soil hydraulic conductivity, Ks. This parameter is often measured using saturated soil cores and a constant applied hydraulic head device. A standard approach to reduce uncertainty in the result is to conduct replicate tests at a single hydraulic head gradient. Low-permeability soils are often tested at large hydraulic head gradients to decrease measurement time. Our objective was to test the common assumptions implicit in calculating Ks from constant-head laboratory tests, i.e., the theoretical linear relationship between head gradient and flux density exists in experimental data, and the relationship passes through the origin. In this study, we used linear regression analysis to test these assumptions and determined Ks from a broad range of head gradients for: 4.5-cm (diameter) by 10-cm (height) intact cores of sandy loam soil; 10-cm (diameter) by 10-cm (height) intact cores of clay loam soil; and repacked sand columns of various sizes. We found nonlinear relationships between hydraulic head gradient (i) and flux density (q) for tests conducted on intact cores of both soils, especially for head gradients greater than unity. When we calculated Ks by linear regression of data from intact cores, we found average values approximately one-third greater than the "standard" method of averaging several replicate tests at a single hydraulic head. The difference between the regression and standard analyses was attributed to experimental bias, which is removed by the linear regression. Although no consistent i or q "thresholds" were identified to predict the onset of nonlinearity in i vs. q data, the intact core results imply that i < 1 and q < 5 x 10–3 cm s–1 may be advisable.

Abbreviations: REV, representative element volume


   INTRODUCTION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Most hydrologic studies require knowledge of soil permeability, which is most simply represented as saturated hydraulic conductivity, Ks (cm s–1). In particular, distributed hydrologic models require information on how Ks is affected by soil texture and land cover at the scale appropriate to the model (Gerke, 2006; Vanderborght et al., 2006). Ideally, Ks would be measured at the scale of the representative element volume (REV) (Bear, 1972). However the REV is often difficult to determine, since it is sensitive to soil texture, soil structure, and the support length scale of the experimental technique (Schulz et al., 2006).

The validity of the Ks measurement also depends on the instrument resolution and scale, the experimental state and boundary conditions, and the correctness of simplifying assumptions (Dane, 1980; Wessolek et al., 1994; Basile et al., 2003). Some advantages of laboratory measurement over field measurement include better control over sample saturation and temperature (Dirksen, 1999), use of well-defined sample sizes (Fuentes and Flury, 2005), and greater measurement precision. Despite the general acknowledgment that hydraulic head gradients greater than unity are uncommon in the saturated zone and may adversely influence test results, few researchers ensure that gradients less than or equal to unity are maintained within test cores (Mitchell, 1993; Lal and Shukla, 2004).

Linearity of the relationship between flux density and hydraulic head gradient is commonly cited as a test for the applicability of Darcy's law (Mitchell, 1993; Hillel, 1998; Lal and Shukla, 2004) but this test is not standard practice. In this study, we developed a simple linear regression approach that tests the assumption of linearity, improves measurement resolution, and identifies appropriate experimental boundary conditions for laboratory measurement of Ks on saturated soil cores. We applied this approach to soil cores of different spatial scales and textures to relate several cases of apparent flow threshold phenomena to experimental design, sample treatment, and flow conditions.

Standard Method of Determining Saturated Hydraulic Conductivity
The analysis of Ks from constant-head measurements (Fig. 1 ) is based on Darcy's law (Darcy, 1856), which can be written

Formula 1[1]
where Q is the volumetric flow rate (cm3 s–1), A is the cross-sectional area perpendicular to flow (cm2), {Delta}h is the hydraulic head loss (cm), and L is the distance between the points where {Delta}h is determined (cm).


Figure 1
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  		Fig. 1. Experimental setup for laboratory testing of sand samples, includes (1) Mariotte bottle and (2) test column containing soil core and graduated cylinder, shown in downflow configuration. Hydraulic head loss ({Delta}h) measurements were made for all samples between the air inlet for the Mariotte bottle and the center of the outlet tube, {Delta}hm-o, additional measurements were performed by installing manometers (m1, m2, m3, and m4) to measure {Delta}h within the test column and within the soil core.

 
Laboratory determination of Ks depends on careful measurement of {Delta}h and Q. Head measurement may be made external to the core (e.g., Reynolds and Elrick, 2002; ASTM, 2003) or within the core (e.g., Bootlink and Bouma, 2002). Flow rate may be determined manually or automatically by recording outflow into a graduated cylinder or onto an electronic balance (Borcher et al., 1987), or by recording inflow from a supply (Troyer and Skopp, 2002). The wide variety of methodologies used to determine Ks in different laboratories raises the question of measurement parity, since each methodology may be subject to different degrees of measurement bias associated with sample collection conditions, head losses external to the soil, manometer meniscus readings, and flow rate measurement.

The standard method to obtain Ks from constant-head laboratory tests is to apply Eq. [1] to data from individual experiments (i.e., one measurement of {Delta}h and the corresponding Q) and then calculate a "point" value of Ks. Precision is assessed by averaging replicate measurements made at a single value of {Delta}h or within a range conducive to accurate Q measurement (Reynolds and Elrick, 2002; ASTM, 2003). It is common practice to attribute up to a twofold difference in Ks from replicate samples to natural variability. This approach implicitly assumes: (i) a linear relationship between hydraulic head gradient and flux density; and (ii) that the relationship passes through the origin. Evidence of the failure of both of these assumptions is common in the literature under conditions that give rise to "non-Darcy" behavior.

Nonlinear responses between head gradient and flux density have been attributed to nonlaminar flow for high pore-water velocities in sands (Hubbert, 1956) and apparent "threshold gradients" for initiation of water flux in clays. Proposed explanations for threshold gradients include non-Newtonian behavior of fluids (Miller and Low, 1963), particle migration and pore blockage (Mitchell and Younger, 1967), temperature-related activation energy (Swartzendruber, 1968; Zhang et al., 2003), experimental artifacts such as core end-cap effects (Gupta and Swartzendruber, 1962; Olsen, 1965; Mitchell and Younger, 1967; Chan and Kenney, 1973), inflections in nonlinear gradients at low fluxes (Hansbo, 1960, 2003), and soil consolidation due to swelling (Smiles and Rosenthal, 1968). Measurement bias and apparent "non-Darcy" behavior would not be identified, however, using Eq. [1] and the standard "point calculation" approach to Ks determination.

Linear Regression Method of Determining Saturated Hydraulic Conductivity
Darcy's law can be written to more explicitly represent Ks as the proportionality constant between flux density, q, and hydraulic head gradient, i (e.g., Hillel, 1998):

Formula 2[2]
where

Formula 3[3]
and

Formula 4[4]

The value of Ks can thus be determined as the slope of the linear regression of q vs. i across a range of {Delta}h and Q. In this approach, the regression constant b can be used to determine i for q = 0, henceforth referred to as the apparent "threshold" hydraulic head gradient, it:

Formula 5[5]

For it ≥ 0,

Formula 6[6]

Alternatively, q may be used as the independent variable and the regression equation becomes

Formula 7[7]
where c is the regression constant, and

Formula 8[8]
where {Delta}ht is the apparent "threshold" hydraulic head loss ({Delta}ht ≥ 0) and La is the apparent sample length. This approach removes the dependence of it on Ks (Eq. [5]) and allows assessment of experimental uncertainty due to {Delta}ht > 0. A second advantage is that q can be measured more accurately and precisely than i, and thus standard regression analysis (which minimizes the y axis deviations but not the x axis deviations) is more accurate. Although the concept of using multiple hydraulic head gradients and regression analysis to determine Ks is not new, the regression fitting of Eq. [7] and [8] to identify nonlinearity in i vs. q data and remove experimental bias from the Ks calculation is not evident in standard methods or other published literature.

We propose that Ks calculations from laboratory core tests are commonly subject to error from systematic measurement bias and inappropriate experimental conditions. Hence, the objectives of this study were to: (i) demonstrate the determination of Ks by regression fitting of Eq. [7] and [8] to i vs. q data; and (ii) identify the extent and causes of bias using the regression and standard "point calculation" approaches for a range of soil textures. Our hypothesis was that apparent thresholds to flux are artifacts caused by experimental bias and changes in the soil fabric arising from experimental boundary conditions foreign to typical in situ conditions outside of riparian zones and flood-irrigated fields. The implications for investigations of scale effects were briefly addressed for simple media through comparison of small repacked sand columns, a reanalysis of Darcy's (1856) data from large sand columns, and undisturbed silt loam and clay loam cores of varying lengths.


   MATERIALS AND METHODS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
Undisturbed soil cores were collected from a study site near Moab, UT, where soils have developed from sandstone to form the Rizono-Rock outcrop complex [loamy, mixed (calcareous), mesic Lithic Ustic Torriorthents]. The hydrometer method (Gee and Or, 2002) was used to determine an average soil particle size distribution of 87% sand, 11% silt, and 2% clay. Soil cores were collected manually from bare soil and across the tire tracks of a seismic exploration vehicle in 4.5-cm-diameter thin-wall polyvinyl chloride soil core sampling cylinders (Geoprobe, Salina, KS). To minimize disturbance during sampling, the soil surface was prewetted by misting. The sampling cylinders were then hand pressed into the soil a few centimeters and filled with 15 to 20 mL of water. One hour after the water had infiltrated, the sampling cylinders were hand pressed 10 to 15 cm into the soil and extracted. Excess soil was shaved from the bottom end of each cylinder with a sharp knife, resulting in cores 8 to 12 cm long. The sampling cylinders extended several centimeters above the top of the soil cores. The cores were capped for transportation and stored in the laboratory for 2 wk at ambient summer conditions in Utah (20–35°C). The range of temperatures and moisture contents in the samples were within the ranges typical of field conditions and were therefore not expected to alter the soil within the cores. Soil bulk density within the site ranged from 1.2 to 1.6 g cm–3 (Lebron et al., 2007). Mean organic C by the method of Walkley and Black (1934) for 115 samples from 0- to 10-cm soil depth at a nearby site was 9.4 g kg–1 and was not tested for the Moab samples in this study.

The first set of tests was performed on repacked columns of construction sand, washed construction sand, disturbed Moab site soil, and undisturbed samples from the Moab site. A 4-cm extension was affixed to the bottom of each sample to support the soil core and attach water inlet tubing to the sample column (Fig. 1). The extension was filled level with 6-mm-diameter gravel, separated from the soil with 0.85-mm wire mesh, and sealed to the soil core tube with duct tape. Bored rubber stoppers were used to attach the apparatus to a water inlet and outlet manifold constructed of 6.4-mm i.d. plastic tubing and four plastic clamps. The manifold could be configured for upward or downward flow though the sample and facilitated sample wetting. Before the experiment, the soil cores were slowly saturated with tap water from the bottom up during a period of 24 h or until water began to pond on the surface of the soil. Municipal water in Logan, UT, has been extensively characterized in cross-laboratory comparisons and is unlikely to deflocculate soil aggregates due to a high pH (7.84) and water chemistry dominance of Ca+, Mg2+, and CO32– . A subset of samples was saturated during a period of 4 d, as suggested by Reynolds and Elrick (2002); however, the resulting Ks values were similar regardless of saturation method.

Five constant-head tests were conducted on each sample across a range of {Delta}h from 2.0 to 12.0 cm. This was achieved by sequentially raising the air inlet tube of a Mariotte device in 2-cm increments between tests (Fig. 1). To avoid measurement uncertainty associated with light refraction in the Mariotte bottle, the applied head, {Delta}hm-o, was calculated by subtracting the length of the Mariotte air tube from the distance between the top of the air tube and the center of the outflow tube (Fig. 1). We estimate that the measurement precision was 2 mm; however, measurement accuracy may have been lower due to offsets caused by surface tension effects at the Mariotte air inlet and the outflow dripper. Outflow was measured manually at 10-min intervals using a graduated cylinder. Saturated hydraulic conductivity was calculated as a point value for each {Delta}h using Eq. [1] and as the inverse slope of the linear regression of Eq. [7] using all measured (q, {Delta}h) data pairs for each sample. Four additional manometers were fitted to an undisturbed sand core to determine head loss effects on Ks throughout the apparatus (Fig. 1). The manometers were constructed by affixing 5-mm i.d. plastic tubing to small barbed fittings that had been sealed into holes drilled 1 cm above (m1) and below the soil core ends (m4), and at a spacing of 5.5 cm within the soil core (m2 and m3) and suspending then vertically parallel to the soil core.

The second set of tests was performed on intact soil cores of Brookston clay loam soil (fine, loamy, mixed, mesic Typic Argiaquoll) obtained from two depths in an experimental plot in Ontario, Canada, under continuous corn (Zea mays L.) production since 1959. Three 10-cm-diameter by 10-cm-long cores were taken from the 10- to 20- and 20- to 30-cm depths. Average composition for the soil samples was 28% sand, 35% silt, and 37% clay; pH was 6.1 to 6.5. Organic C content was 14 to 18 g kg–1, as determined via dry combustion (Skjemstad and Baldock, 2007) in a LECO CN-2000 Carbon Analyzer (LECO Corp., St. Joseph, MI). Average soil bulk density was 1.5 g cm–3 for the 10- to 20-cm samples and 1.6 g cm–3 for the 20- to 30-cm samples. The soil structure is medium-coarse subangular blocky with shrinkage cracks, root channels, and worm holes. Three sets of three replicate constant-head tests were performed on each core within a range of {Delta}h from 4.1 to 111 cm, using the saturated tank constant-head method (Reynolds and Elrick, 2002).

Hydraulic conductivity tests on clay samples have previously been compared with straight lines on graph paper (Mitchell and Younger, 1967), but to our knowledge linear regression has not previously been used as a tool for data analysis. To provide context of scale, we applied the technique to data from Darcy's original experiments on large columns (L = 0.6–1.7 m, A = 0.1 m2), which included tests on unwashed sand (Exp. 1, Series 1 and 2) and washed sand (Exp. 1, Series 3 and 4) from the Seine River, France (Darcy, 1856).


   RESULTS AND DISCUSSION
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
CONCLUSIONS
 REFERENCES
 
Identification of Experimental Bias
At large scale, Darcy's (1856) tests with repacked sand columns were conducted across a wide range of hydraulic head gradients and exhibit very linear responses (Fig. 2 , Table 1 ); however, one exception to linearity in Darcy's trials appears in Series 1. The test result for i = 18.8 falls outside of the 95% confidence interval for the regression of the remaining data, perhaps as evidence of divergence from Darcy flow at this high hydraulic gradient (Fig. 2). Regression of Darcy's data shows a consistent small negative bias in it (Table 1). At this large experimental scale, such bias is insignificant and the regression confidence intervals ({alpha} = 0.05) bracket the origin for all experiments except for Series 1 (Fig. 2, inset). The assumptions of a linear relationship between applied head and flux that intersects the origin are essentially satisfied for Darcy's experiments.


Figure 2
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  		Fig. 2. Hydraulic head gradient (i) vs. flux density (q) data from Darcy's Exp. 1, Series 1 to 4, with linear regression trend lines (solid) and confidence intervals (dotted). Inset shows trend lines and confidence intervals near the origin.

 

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  		Table 1. Saturated hydraulic conductivity, Ks, calculated as the slope of the linear regression (slope of line) and average of several point calculations (point) for constant-head measurements made at different head gradients, i. The apparent threshold head gradient, it, is presented for the linear regression results; P gives the statistical significance level at {alpha} = 0.05 for Ks and it.

 
At small scale, our tests with repacked sand columns were made at hydraulic head gradients and flux densities an order of magnitude less than Darcy's tests (Fig. 3 ) and measurement error is significant. In particular, the relative magnitude of it (Table 1) is greater for this experimental scale than for Darcy's experiments and the emergent bias (it ~ 0.2) is significantly greater than zero ({alpha} = 0.05). Nevertheless, the range of Ks as calculated by linear regression is similar to those from Darcy's experiments with comparable material, despite the order of magnitude difference in experimental scale (Table 1).


Figure 3
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  		Fig. 3. Hydraulic head gradient (i) vs. flux density (q) results for repacked columns of washed sand, Moab field site sand, and construction sand. All samples exhibit an apparent threshold gradient to initiate flux of approximately i = 0.2.

 
The source of the emergent bias in the small sand column experiments was investigated through tests with a core of undisturbed soil, using the approach of Olsen (1965). Results from measurements of manometers above and below the ends of the soil core (m4–m1) and within the soil core (m3–m2) were nearly identical, but the difference in it between the results from m4–m1 and m3–m2, and measurements made between the Mariotte bottle and the drip tube outlet (m–o) are approximately equal to the emergent bias (Fig. 4 ). We consequently attributed the bulk of the differences in it between the system measurement and the internal manometers to measurement uncertainty, such as drip formation and detachment at the outflow dripper, and head loss within the instrument. The bias in all manometer pairs changed, however, when the flow direction was reversed from downward to upward (Fig. 4). Similar "hysteretic" behavior was previously attributed to head loss across an air bubble in a capillary tube manometer (Olsen, 1965) and may indicate incomplete sample saturation in our soil core, but could conceivably also be caused by directional particle straining, the presence of bacteria, or adsorbed water effects (Mitchell, 1993). In any case, the slope of the lines was not affected by the bias or the change in flow direction for any pair of manometers (Fig. 4), and Ks calculated by linear regression remained between 3.31 x 10–3 and 3.40 x 10–3 cm s–1 for all manometer pairs and test conditions, which was nearly an order of magnitude less than the Ks obtained from the repacked column of soil from the same site (Table 1, repacked site soil).


Figure 4
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  		Fig. 4. Simultaneous measurements of hydraulic head loss ({Delta}h) between the Mariotte bottle air inlet and the soil core dripper outlet were made by manometers above and below the ends of the soil core (m4 – m1) and within the soil core (m3 – m2). The effects of flow direction (up or down) and placement of manometers on the apparent threshold head gradient (it) were measured for a single undisturbed core (length = 8.5 cm, area = 13.9 cm2).

 
We therefore contend that it associated with systematic measurement error is inconsequential to the calculation of Ks by the linear regression approach presented here. On the other hand, we found that measurement bias influenced point Ks calculation for all repacked columns (Table 1) and the extent of this influence, relative to regression Ks, increased for decreasing values of i (Table 1). Similarly, the relative impact of experimental bias would be the greatest on cores with low Ks (Eq. [5]). One might conclude that such errors could be reduced by increased applied head or more precise measurement of {Delta}h to allow similar values of Ks to be determined from point calculations and linear regression. A simpler approach, however, might be to simply increase the number of (q, {Delta}h) measurements per sample.

Boundary Condition Effects
The influence of test boundary conditions on apparent deviations from Darcy flow were investigated for several undisturbed soil samples from the Moab sand and Brookston clay loam. These two soils are extremely different in origin, texture, and disturbance history and should bracket most flow phenomena encountered in undisturbed samples. The sand cores were taken as part of an ecohydrology study within a pinyon–juniper woodland (Lebron et al., 2007) where spatial variability in Ks was hypothesized to be predominantly horizontal and related to the effects of disturbance and vegetation. The clay loam cores were taken in a long-term agronomic study where, within a given treatment, vertical changes were expected to dominate variability in Ks due to changes in secondary porosity.

For the Moab sand, we present results from intact cores collected across the path made by seismic exploration vehicles (n = 17) and near an ephedra bush (n = 8) and bunchgrasses (n = 3). The data from across the vehicle track are presented in Fig. 5 as open diamonds for sample points within a tire track and closed boxes for samples points clearly outside of a tire track. Each sample is labeled by position in centimeters along a linear transect. The i vs. q relationships for the trafficked area tended to be more variable and of greater slope (lower Ks) than for outside the trafficked area. The relationship between i and q is highly linear for most samples and we used R2 < 0.995 to identify nonlinearity. Within the more linear results (Fig. 5a) the average Ks for the trafficked area (3.3 x 10–3 cm s–1) was about one-half that of the undisturbed area tracks (7.0 x 10–3 cm s–1) and significantly different (P < 0.001). Among the seven samples identified as nonlinear, six demonstrated an increasing slope for increasing i, especially for i > 1 (Fig. 5b). This response is consistent with the hypothesis that particle straining is a common cause for non-Darcy flow, as proposed for compacted clays (Mitchell, 1993), but occurs at substantially lower i for intact sand samples, as proposed by Lal and Shukla (2004). Exclusion of data for i > 1 from the nonlinear samples left Core 360 as the only sample with a nonlinear regression. Core 360 was obtained from the middle of a well-traveled footpath and as a result had dramatically lower Ks (1.0–1.4 x 10–3 cm s–1). Regrouping the Ks results for i < 1 for the samples in Fig. 5b (except Core 360) with the results from the highly linear samples (Fig. 5a) increased the mean Ks for the trafficked (3.8 x 10–3 cm s–1) and undisturbed (7.3 x 10–3 cm s–1) areas slightly but the difference remained significant (P < 0.001).


Figure 5
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  		Fig. 5. Hydraulic head gradient (i) vs. flux density (q) results from intact sand cores collected across the path made by seismic exploration vehicles at the Moab field site. Each sample is labeled by position in centimeters along a linear transect and identified by data marker type as either within a tire track (open diamonds) or clearly outside of a tire track (solid squares). Data series were separated into (a) highly linear and (b) nonlinear with a criterion of R2 = 0.995 to discriminate for linearity. Among the seven samples identified as nonlinear, six demonstrated an increase in the slope of i vs. q for increasing i, especially for i > 1.

 
The results for samples taken near vascular plants at the Moab site were mixed. Cores from under bunchgrass all show highly nonlinear relationships between i and q, but with a decrease in slope for increasing i (Fig. 6a ), which could be attributed to a greater relative flow through the secondary porosity at i > 1 or by progressive dissolution of entrapped air for sequential tests from low to high head gradients (Sato et al., 2005). For i < 1, the average Ks (4.0 x 10–4 cm s–1) for these samples is an order of magnitude lower than the tire-compacted soils and about one-third that of the footpath. This low Ks is supported by the following observations we made for these samples: (i) the soil texture under bunchgrass is somewhat finer than the texture of the other samples; (ii) soil water repellency is common in association with vegetation at the site; and (iii) the bunchgrass samples were difficult to saturate and retained an uncertain proportion of entrapped air during the test. Results from samples taken near ephedra are highly linear (all R2 > 0.995) up to i > 2 (Fig. 6b). Six of seven tests group tightly (Fig. 6b), with an average Ks of 2.3 x 10–3 cm s–1, once again lower than the soils in the vehicle tracks. The Ks for the remaining sample was nearly three times greater (7.1 x 10–3 cm s–1) and similar to the undisturbed soils in Fig. 5a.


Figure 6
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  		Fig. 6. Hydraulic head gradient (i) vs. flux density (q) results for samples taken from (a) under bunchgrass and (b) near an ephedra bush. The bunchgrass samples show a decrease in the slope of i vs. q for increasing i, especially for i > 1. The ephedra results are highly linear (all R2 > 0.995) up to i > 2.

 
The undisturbed clay loam cores had a wider range of permeability than the undisturbed sand cores and required a similarly greater range in {Delta}h for testing (Fig. 7 ). The samples from the 10- to 20-cm depth (174, 215, and 322) had greater secondary soil structure (e.g., worm holes, root channels, and interpedal cracks) than the samples from 20- to 30-cm depth (217, 241, and 352); as a result, sufficient flux for the test was achieved at much lower i for the shallow samples than for the deep samples (Fig. 7a and 7b). Whereas two out of three of the less permeable samples (from 20–30-cm depth) show a significant positive it, the more permeable samples (from 10–20-cm depth) do not (Table 1). This result does not support the common premise that increasing {Delta}h in low-permeability samples is an effective means to improve the accuracy of the point method for calculating Ks. Regardless of statistical significance, Ks calculated using the point method underestimates the slope method (Eq. [7]) when the gradient-axis intercept (Eq. [8]) is positive (this also occurred with the sand samples) and it overestimates the slope method when the gradient-axis intercept is negative.


Figure 7
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  		Fig. 7. Hydraulic head gradient (i) vs. flux density (q) results from undisturbed clay loam soil cores. Samples from (a) 10- to 20-cm depth had greater secondary soil structure than samples from (b) 20- to 30-cm depth, and thus a much lower i was required to achieve adequate flux for the shallow depth than for the deeper depth; (c) sediment was observed in the outflow from two samples that also showed a stepwise reduction in the linear slope of i vs. q for increasing i.

 
Physical interactions between sample and test conditions can be inferred from the undisturbed cores of loamy sand and clay loam soils. For example, two tests on the clay loam soil (Cores 41 and 65) gave "piecewise linear" i vs. q relationships with two distinct regions—one "high-slope" region at "low" flux density, and one "low-slope" region at "high" flux density (Fig. 7c). In addition, sediment was observed in the outflow at the high flux densities. Given that the cores were visibly structured and thus had "bimodal" pore size distributions, we interpret these results as indicating that flow was concentrated in large pores for the high-flux tests and that the appearance of sediment in the outflow was due to progressive erosion of the large pores. Hence, the high-flux tests appeared to be modifying the soil fabric by producing pore water velocities in large pores that exceeded the soil's shear strength. Together, data from cores of loamy sand soil from the disturbed track (Fig. 5b) and from under bunchgrass are nonlinear across an order of magnitude in q (Fig. 6a). There is no clear threshold in q to nonlinear response, but the trends appear linear for both sets of data for i < 1. Separate calculation of Ks for data taken at relatively high and low q for the bunchgrass samples and for clay loam Cores 41 and 65 generally differ by greater than twofold (Table 2 ). On the other hand, several of the undisturbed samples did not exhibit nonlinear response under any of the test conditions. This variability in nonlinear response does not support tests using high outflow rates, but it does support tests conducted at multiple values of i < 1.


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  		Table 2. Saturated hydraulic conductivity, Ks, and apparent threshold head gradient, it, calculated by linear regression for fluxes, q, above (high q) and below (low q) inflections in the laboratory test data trends.

 

   CONCLUSIONS
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
REFERENCES
 
In this study, we demonstrated that linear regression of hydraulic head gradient against flux density is a robust technique to determine Ks independently of systematic error. For repacked columns on the scale of Darcy's original work (Darcy, 1856), this is not an important consideration, but the error component becomes significant at the scale of intact soil cores typical of contemporary studies. We propose that separating error or bias in {Delta}h into the regression constant b allows highly precise measurement of Ks from small cores. The degree of linearity indicated by the regression correlation coefficient is also a useful indicator of data quality and allows the investigator to identify samples or head gradients for which the test is invalid.

We found nonlinear i vs. q relationships for tests conducted on both sandy loam and clay loam soils. We infer that an increasing slope of i vs. q with increasing i for disturbed or compacted soil samples is the result of flow restriction caused by progressive blocking of pores by migrating soil particles, especially at i > 1. On the other hand, we infer that locally high flow rates can progressively erode low-resistance conduits (large pores), causing a decrease in the slope of i vs. q with increasing i. Although no consistent i or q "threshold" values were identified to predict the onset of nonlinearity in i vs. q measurements, the results from the intact sandy loam and clay loam cores imply that i < 1 and q < 5 x 10–3 cm s–1 may be advisable. When used within these boundary condition guidelines, the linear regression approach to calculating Ks should allow sufficiently precise determination of Ks to identify significant differences in bulk hydraulic properties within the generally accepted range of "natural variability."

The advantage of the regression approach based on Eq. [7] and [8] lies in its ability to readily identify experimental artifacts (nonlinearity in i vs. q) and correct for bias (threshold hydraulic gradients), which would otherwise remain hidden when using the standard "point calculation" method. Hence, adoption of the regression approach will serve to improve the quality and intercomparability of Ks values determined with different apparatus and at different scales. This will be an important step toward determining the REV relevant to saturated flow through heterogeneous geologic porous media.


   ACKNOWLEDGMENTS
 
This work was supported by the Utah Agricultural Experiment Station, Utah State University, Logan, and approved as Journal Paper 7854. Additional support was provided by the Bureau of Land Management Contract no. JSA041003 and the Kansas State University Targeted Excellence Program.


   NOTES
TOP
 NOTES
ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 
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Received for publication July 9, 2007.


   REFERENCES
TOP
 NOTES
 ABSTRACT
 INTRODUCTION
 MATERIALS AND METHODS
 RESULTS AND DISCUSSION
 CONCLUSIONS
 REFERENCES
 





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