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Published online 1 August 2008
Published in Vadose Zone J 7:941-947 (2008)
DOI: 10.2136/vzj2007.0164
© 2008 Soil Science Society of America
677 S. Segoe Rd., Madison, WI 53711 USA
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ORIGINAL RESEARCH

The Effects of Anisotropy on In Situ Air Permeability Measurements

Karletta Chief*, Ty P. A. Ferré and Andrew C. Hinnell

Dep. of Hydrology and Water Resources, Univ. of Arizona, Tucson, AZ 85721
* Corresponding author (karletta.chief{at}dri.edu).

All rights reserved. No part of this periodical may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher.


Received 6 October 2007.



   ABSTRACT
TOP
 ABSTRACT
INTRODUCTION
 Theory
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
Previous research has established a correlation between air permeability (ka) and saturated hydraulic conductivity for agricultural soils based on ex situ air permeability (kex situ). In situ air permeability (kin situ) measurements in nonagricultural soils, however, have shown a decrease in correlation that may be attributed to soil anisotropy. Our objectives were: (i) to examine the effects of anisotropy on kin situ using a three-dimensional air flow model; (ii) to develop a method to identify anisotropy using kin situ and kex situ measurements; and (iii) to determine the sample volume of an air permeameter as a function of the permeameter design and the anisotropy ratio. Numerical results showed that the ka measured in situ in anisotropic media results in some average of the horizontal and vertical permeabilities. The averaging depends on the degree of anisotropy and the ratio of the diameter to the insertion depth of the permeameter. Therefore, a shape factor developed for an isotropic soil can give unreliable results. We determined that paired in situ and ex situ permeability measurements can be used to infer the anisotropy ratio. This approach is more accurate if the vertical permeability, kaz, is higher than the horizontal, kax. The sample volume does not extend outside of the air permeameter for high kax/kaz. It is stretched vertically for low kax/kaz. A field experiment showed qualitative agreement with model predictions, but anisotropy alone was not able to fully explain the difference between kin situ and kex situ.

Abbreviations: SCAP, soil corer air permeameter


   INTRODUCTION
TOP
 ABSTRACT
 INTRODUCTION
Theory
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
AIR PERMEAMETERS can provide rapid measurements of the permeability of near-surface sediments (e.g., Steinbrenner, 1959; van Groenewoud, 1968; Iversen et al., 2001b). Based on the demonstrated log–log correlation between air permeability (ka) and saturated hydraulic conductivity (Ksat) (Loll et al., 1999; Iversen et al., 2001a), these measurements can be used to estimate Ksat without the need to apply water to the soil. The ability to measure this important hydraulic property without the use of more cumbersome, time-consuming direct methods may provide a practical approach to generate more complete data for use in distributed hydrologic models.

Research indicating a correlation between ka and Ksat is based on ex situ ka measurements where boundary conditions are well defined. For agricultural soils, the Loll et al. (1999) regression had 95% confidence intervals, which indicated a prediction accuracy better than a ±0.7 order of magnitude. Iversen et al. (2001a) also found a good relationship between log(ka) and log(Ksat) with a regression that had 95% confidence intervals that indicated an accuracy better than ±1.2 orders of magnitude. Iversen et al. (2003) posed that in situ ka could prove to be a promising tool to estimate Ksat and characterize spatial variability. Chief et al. (2008) tested the applicability of this correlation for in situ ka and laboratory Ksat in nonagricultural soils, resulting in site-specific regressions that had 95% confidence intervals as high as ±3.8 orders of magnitude and as low as ±1.2 orders of magnitude. Chief et al. (2008) hypothesized that this decrease in the accuracy of the ka vs. Ksat relationship in some burned soils may have been due in part to soil anisotropy.

The term anisotropy is used to describe directionally dependent hydraulic conductivity. This can arise from anisotropy at the pore scale and from the effects of directionally dependent heterogeneity at or below the scale of measurement. Examples of the latter include root holes, other preferential pathways, or discrete layers. In this study, we examined the potential impacts of anisotropy on in situ air permeability measurements. In addition, we investigated whether paired in situ and ex situ measurements could be used to identify anisotropy and, in some cases, to infer both the vertical and horizontal permeabilities. One advantage of the numerical modeling approach used in this study is that it could be used to investigate the effects of these smaller scale heterogeneities. The final objective was to examine the spatially variable sensitivity and determine the change in sample volume due to varying anisotropy and permeameter dimensions.


   Theory
TOP
 ABSTRACT
 INTRODUCTION
 Theory
Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 
Air permeability is defined as the ability of soil to transmit air through interconnected air-filled pores under an imposed air pressure gradient. Like hydraulic conductivity, air permeability is a function of volumetric water content, porosity, pore size distribution, and pore geometry (Roseberg and McCoy, 1990). Unlike water flow, the flow of gases through porous media has been shown to exhibit non-Darcian "slip flow" in that velocities near the pore wall are not zero (Klinkenberg, 1941). Weeks (1978) demonstrated that "slip flow" is only significant for soils with air permeability <0.01 µm2, such as soils with high silt and clay fractions (Springer et al., 1995). For most soils, air permeability can be measured by imposing controlled, isothermal, steady-state, laminar air flow conditions and applying Darcy's law.

Assuming negligible air density, the air flux, q [L T–1], under an applied vertical pressure gradient, dP/dz (dimensionless) with a dynamic air viscosity, {eta} [M L–1 T–1], can be described as

Formula 1[1]
The air permeability, ka [L2], can be determined with ex situ paired measurements of the pressure gradient and the air flow, Q [L3 T–1] at steady state as

Formula 2[2]
where the area perpendicular to flow, As [L2], is the inner area of the air permeameter and the path length, H [L], is the length of the inserted cylinder. The air pressures at the top and bottom of the inserted cylinder are Pi and Po [M L–1 T–2], respectively.

Several designs have been described for surface and insertion air permeameters to measure the air permeability of in situ and ex situ soil samples (Kirkham, 1946; Grover, 1955; Corey, 1957; Steinbrenner, 1959; Fish and Koppi, 1994; Davis et al., 1994; Iversen et al., 2001b; Jalbert and Dane, 2003). The soil corer air permeameter (SCAP) is an alternative air permeameter designed for insertion into desert soils, which are often too hard to allow manual insertion and too gravelly to allow surface measurements (Chief et al., 2006). The SCAP was developed to work with commercially available soil corers, making use of their driving mechanisms and sample retainer cylinders. This design also allows in situ and ex situ measurements of air permeability (Fig. 1 ).


Figure 1
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  		FIG. 1. Air flow lines associated with ex situ and in situ air permeability measurements for insertion air permeameters.

 
For in situ measurements, the air flow through the soil beyond the end of the air permeameter is not one dimensional. The effects of this divergent flow beyond the outflow end of an inserted air permeameter can be addressed by applying a shape factor (Grover, 1955; Liang et al., 1995). Frevert (1948) and Grover (1955) used experimental data from electrolytic models to estimate the shape factor for various diameters and insertion depths of air permeameters. Kirkham et al. (1958) and Boedicker (1972) later modified Grover's (1955) nomograms. Liang et al. (1995) and Jalbert and Dane (2003) used finite-element models to improve shape factor estimates for different air permeameters. These shape factor models apply a single shape factor for in situ air permeability measurements in homogeneous and isotropic soils, which is dependent only on the geometry of the air permeameter.

Assuming zero gauge pressure at the ground surface, the shape factor is used to relate the known applied flow and input pressure to the air permeability by multiplying the inserted length of the air permeameter by a constant unitless factor, {Lambda}. This gives the following expression for the air permeability:

Formula 3[3]
The multiplier can be combined with the height, H [L], and diameter, D [L], of the air permeameter to form a dimensionless shape factor, G:

Formula 4[4]
The air permeability can then be calculated as

Formula 5[5]
Liang et al. (1995) developed a two-dimensional finite-element air flow model for a homogenous and isotropic medium in cylindrical coordinates (using a finite-element method from ANSYS FEA 5.0, Swanson Analysis Systems, Canonsburg, PA, field-size model ANSYS F) to estimate the shape factor. The shape factor is dependent on the diameter and insertion height of the air permeameter and is valid for D/H ratios <10. More recently, Jalbert and Dane (2003) developed the same two-dimensional air flow finite-element model using HYDRUS-2D (Simunek et al., 1996) but with a larger simulation volume and increased triangular density near the bottom insertion edges. They found the following expression for the shape factor as a function of the D/H ratio of the air permeameter:

Formula 6[6]
where D [L] is the diameter of the ring, and H [L] is the height of the ring inserted into the soil. None of the previous studies have considered the possible impacts of anisotropy on the instrument response.


   Materials and Methods
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
Results and Discussion
 Conclusions
 REFERENCES
 
A radially symmetric model (e.g., Jalbert and Dane, 2003) could be used to investigate vertical and horizontal anisotropy. We chose, however, to develop a three-dimensional model, which could be extended to consider the effects of nonhorizontal layers and heterogeneous property distributions. The three-dimensional, steady-state, finite-element air flow model was developed using FEMLAB 3.0A (Comsol, Los Angeles, CA).

Field Measurements
Chief et al. (2006) measured in situ and ex situ air permeability at the soil surface (A horizon) on four agricultural soils along a 14- to 18-m transect at 2-m intervals. These measurements were made on soils with an average moisture content of 0.10 m3 m–3, ranging from 0.05 to 0.19 m3 m–3 (Table 1 ). We show results for these soils: Anthony very fine sandy loam; Vinton fine sandy loam; Gila very fine sandy loam; and Pima loam. At the subgroup level, according to the U.S. Soil Taxonomy (Soil Survey Staff, 1999), the agricultural soils were classified as Typic Torrifluvents. Anthony and Gila are coarse-loamy, mixed, calcareous, thermic, Typic Torrifluvents. The Vinton soil is a sandy, mixed (calcareous), thermic Typic Torrifluvent and the Pima soil is a fine-silty, mixed (calcareous), thermic Typic Torrifluvent. Table 2 provides detailed information regarding the five sites and taxonomic identifications.


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  		TABLE 1. Mean and standard deviation for soil properties of four agricultural soils in Tucson, AZ.

 

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  		TABLE 2. Site information and taxonomic identification of four agricultural soils in Tucson, AZ.

 
FEMLAB Model
FEMLAB consists of a graphical and MATLAB scripting interfaces that can be used individually or together to build and execute the solver. The applicable partial differential equation, model domain, boundary conditions, initial conditions, material properties, sources, and fluxes are defined within the FEMLAB model.

We solved the steady-state flow problem, which can be expressed as

Formula 7[7]
where P is the pressure, t is time, da is the inverse of the diffusion coefficient, {Gamma} is the spatial flux vector in terms of P, and F is the internal source term. For our conditions, da and F were set to zero. Neumann boundary conditions define the flux normal to the boundary according to

Formula 8[8]
where n is the normal vector and Gf is the flux across the boundary. The value of Gf is zero for impermeable boundaries. The flux through the permeameter for an applied pressure was determined by integrating the normal component of the calculated boundary flux over the soil surface within the permeameter. Within FEMLAB, a Dirichlet boundary condition defines the potential at the boundary as

Formula 9[9]
where R is the specified potential, T is the transpose, and µ is the Langrangian multiplier. In FEMLAB, a constant applied pressure, Pi, for a Dirichlet boundary is defined as

Formula 10[10]
We solved the steady-state air flow problem, expressed as

Formula 11[11]
where Pe is the excess pressure above atmospheric pressure (gauge pressure). Assuming homogeneity and allowing for different air permeabilities in the vertical (kaz) and horizontal (kax and kay) directions, we solved

Formula 12[12]
where kax is the air permeability in the x direction, kay is the air permeability in the y direction, kaz is the air permeability in the z direction, and {eta} is the air viscosity. The principal axes of anisotropy are aligned with the vertical and horizontal directions in the coordinate axes. We assumed that the air permeabilities in the x and y directions, kax and kay, were equal. We related the permeability in the vertical direction to the horizontal permeability as kaz = 10akax. We considered anisotropy ratios (kax/kaz) ranging from 0.06 to 15.8, which are represented by a values ranging from –1.2 to 1.2 at 0.3 increments. An isotropic medium has an a of zero. The internal diameter of the air permeameter was kept constant at 5.3 cm with a wall thickness equal to D/20. For the homogeneous conditions considered, the results apply for any combination of D and H that give a specific D/H ratio. For simplicity, we only varied H to simulate D/H ratios of 0.25, 0.5, 1, 2, 4, 6, 8, and 10.

The size of the cylindrical model domain was chosen by increasing the height and width of the domain until the domain was large enough that the pressure distribution near the permeameter was insensitive to the domain size. This analysis confirmed that the domain size used by Jalbert and Dane (2003) was sufficiently large. For consistency, we adopted the same domain size as Jalbert and Dane (2003), fixing the lateral extent to be the larger of 25D and 20H and the vertical extent to be the larger of 15D and 10H. A predefined triangular mesh size was set to fine with a finer density within and around the air permeameter (Fig. 2 ). The number of elements ranged from 72,000 to 125,000. The linear system solver converged to a solution when the relative error or the weighted Euclidian norm was <1 x 10–6 for a maximum iterations of 10,000. The outside boundaries of the simulation volume (excluding the soil surface) and the SCAP walls were set to no-flow Neumann boundaries. The soil surface within the SCAP was set to a Dirichlet boundary with inlet pressure equal to 264.4 Pa (2.7 cm H2O). The soil surface outside the SCAP was defined as a Dirichlet boundary equal to 0 gauge pressure. The properties of the soil, air, and SCAP are defined in Table 3 .


Figure 2
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  		FIG. 2. Three-dimensional finite element mesh for entire model domain and the region near the permeameter walls (shown in inset).

 

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  		TABLE 3. Properties of the soil, air, and soil corer air permeameter (SCAP).

 
Sample Volume
Most measurement methods have spatially variable sensitivities. This means that the change in the instrument response to a given change in the property of interest depends on the location of that change relative to the location of the instrument. One accepted approach to define spatial sensitivity is through a perturbation analysis. In this approach, a small change is applied to the property of interest at a large number of locations in the domain, one at a time. A weighting function is defined to characterize the relative change in the computed instrument response as a function of the location of the perturbation. This defines the spatial sensitivity of the instrument. The weighting function is determined by equating the total potential energy in a homogeneous medium ({phi}o) to that in the same medium with a single perturbation at one point [{phi}(x,y)] (Knight, 1992; Hinnell et al., 2006) as

Formula 13[13]
where {omega}(x,y) is the weighting function in the x–y plane and A is the area in the x–y plane where {omega}(x,y) > 0. Ferré et al. (1998) described a method to use these local measurement sensitivities to define unique spatial sample areas (volumes in three dimensions) by integrating the local weighting factors of the model cells in order of decreasing local sensitivity until a defined percentage or fraction, f, of the total sensitivity is reached, according to

Formula 14[14]
where {omega} is the vector of weighting factors sorted in decreasing order, {omega}j is the highest value and {omega}i is the lowest value of the weighting function whose sum defines a specified fraction that contributes to the total response, and Ai is the area with a weighting factor {omega}i. This approach uniquely defines the smallest sample area within which the spatial integral of the local sensitivity adds to a fixed percentage of the sensitivity in the entire domain. The sample volume for an ex situ sample is simply the volume of soil in the measuring device. For in situ measurements, however, the sample volume will depend on the distribution of the pressure head within and around the permeameter.


   Results and Discussion
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Discussion
Conclusions
 REFERENCES
 
Model Validation
Jalbert and Dane (2003) developed an algebraic expression for the shape factor based on the results from a two-dimensional radially symmetric model (Eq. [6]). To validate the FEMLAB numerical model, the shape factor calculated from our results was compared with the Jalbert and Dane (2003) shape factor model (Fig. 3 ). The shape factor, G, was determined using a rearranged Eq. [5]:

Formula 15[15]
The applied pressure is defined by the boundary condition at the ground surface within the air permeameter. The flow is calculated as a boundary integral of the flux across the area at the soil surface within the air permeameter. Shape factors were calculated for a range of permeameter D/H ratios. Figure 3 illustrates the numerical results of the FEMLAB model and shows an increasing shape factor for increasing D/H ratio. The numerical results agreed closely with the findings of Jalbert and Dane (2003) and had a mean squared error of 0.005 m2.


Figure 3
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  		FIG. 3. The dimensionless shape factor, G, for insertion air permeameters for Jalbert and Dane (2003) model and FEMLAB model for increasing diameter/height (D/H) ratios.

 
Anisotropic Effects
The validated FEMLAB model was used to investigate the effects of anisotropy on in situ air permeability measurements. The value of kaz was held constant while kax and kay were varied as described above. To examine the potential error in interpretation that may occur due to neglecting anisotropy, we used the FEMLAB-calculated flux that would be measured in situ and applied the shape factor for an isotropic soil to determine an apparent permeability, kin situ. The permeability measured ex situ, kex situ, should be equal to kaz. The ratio of the in situ to ex situ permeability (kin situ/kex situ) defines the error that would occur if the measurements made in an anisotropic medium were interpreted using the shape factor determined for an isotropic soil. This error can be as large as a factor of 2 for large D/H ratios and strongly anisotropic media (Fig. 4A ). For smaller D/H ratios and less anisotropic media, anisotropic effects were the least significant.


Figure 4
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  		FIG. 4. (A) Nomograph to determine soil anisotropy as the ratio of the horizontal to the vertical air permeability (kax/kaz), based on paired in situ and ex situ air permeability measurements (kin situ and kex situ, respectively) using the specific response curve for the diameter/height (D/H) ratio of an air permeameter; and (B) the average kin situ/kex situ ratios for agricultural soils plotted on the nomograph for the soil corer air permeameter D/H ratio of 0.5 to determine soil anisotropy.

 
Given that distributed hydraulic conductivity values can often only be inferred within an order of magnitude, the impact of anisotropy on the measured air permeability may not be critical for larger scale studies. For detailed studies, however, anisotropy can have important impacts on water flow and solute transport (Beckwith et al., 2003a,b).

For these conditions, Fig. 4A can also be used as a nomograph to estimate the anisotropy ratio from paired in situ and ex situ air permeability measurements. First, the shape factor for an isotropic medium is calculated for the D/H ratio of the instrument using Eq. [6]. Then, this isotropic shape factor is applied to determine the in situ air permeability. The ratio kin situ/kex situ is plotted on the y axis along the response curve for the specific D/H ratio. The intersection of the kin situ/kex situ ratio and the response curve defines the point along the x axis or the anisotropy ratio, kax/kaz. The sensitivity of the measurement of anisotropy can be defined as the change in the instrument response (the ratio of in situ to ex situ permeability) as a function of the change in the property of interest (anisotropy). This suggests that the method will be more sensitive when kax is less than kaz and, in particular, it is more sensitive for larger D/H ratios.

This variable sensitivity can be explained by considering each flow path to be divided into a region within the permeameter that experiences vertical flow, a short region at the base of the permeameter that experiences predominantly horizontal flow, and a third section of predominantly vertical flow returning to the ground surface. These sections can be considered to be placed in series. Therefore, the equivalent permeability of each flow tube will be dominated by the section with the lowest permeability. That is, it will be more sensitive to kax if the horizontal permeability is less than the vertical permeability, and vice versa. As the D/H ratio increases, the length of the horizontal flow path section becomes longer relative to that of the vertical flow path sections, which magnifies this effect.

The SCAP measurements of kin situ and kex situ were made in four agricultural soils and are illustrated in Fig. 5 . A line is included to show the expected relation between in situ and ex situ measurements for an isotropic soil. All of the soils except for the Vinton soil were visibly structured (Table 2). Table 4 lists the kin situ/kex situ ratios and there is considerable scatter in the data that cannot be explained by anisotropy alone. (There was no visual evidence of layering or root holes in the field, although that doesn't preclude them.) The scatter may also be due to anisotropy at the scale of the sample volume of the permeameter or of the experiment.


Figure 5
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  		FIG. 5. In situ vs. ex situ air permeability measurements (kin situ and kex situ, respectively) for four agricultural soils in Tucson, AZ, (n = 40) using the Jalbert and Dane (2003) shape factor.

 

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  		TABLE 4. Inferred anisotropy ratios using average in situ and ex situ air permeability (kin situ and kex situ, respectively) measurements in four agricultural soils for an air permeameter diameter/height ratio of 0.5.

 
To demonstrate the use of Fig. 4A as a nomograph, we applied it to the paired kin situ and kex situ SCAP measurements. The SCAP has a D/H ratio of 0.5; the corresponding response curve is highlighted in red. The intersection of each measured kin situ/kex situ ratio with the response curve is shown as a red circle. The anisotropy ratio can be found from the x coordinate of each point; these values are reported in Table 4. Three of the soils were anisotropic, with kax less than kaz; one soil was isotropic.

Sample Volume as a Function of Anisotropy and Diameter/Height Ratio
In addition to examining the dependence of the air permeameter response on the D/H ratio and the anisotropy, we also investigated the change in the sample volume. For D/H equal to 0.50 in isotropic soils, the 90% sample volume is concentrated within SCAP (Fig. 6A ), which is consistent with the results of Jalbert and Dane (2003). (Note that we calculated the sample volume in three dimensions, but for the homogeneous and laterally isotropic conditions considered here, the sample volume can be shown as a vertical cross-section through the center of the permeameter.) For anisotropic soils with relatively high kaz (a < 0), the 90% sample volume becomes increasingly elongated vertically such that 50% of the total volume is outside of the air permeameter. For positive a values, the 90% volume contracts within the permeameter body. The restriction of the sample volume to the region within the permeameter, where flow is forced to be vertical by the permeameter walls, helps to explain the low sensitivity of the air permeameter measurements to the horizontal permeability when kax < kaz (the ratio of kin situ to kex situ is far smaller than the anisotropy ratio). The sample volume only extends vertically into the underlying medium when kaz is greater than kax. As D/H increases, the absolute maximum extent of the 90% sample volume beneath the permeameter decreases. For instance, for a equal to –1.2, this depth decreases from 16.9 to 12.35 cm as the D/H ratio increases from 0.5 to 2; however, the fraction of the sample volume that is located outside of the permeameter body increases as D/H increases because the volume of the medium within the air permeameter decreases with decreasing insertion depth (Fig. 6).


Figure 6
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  		FIG. 6. FEMLAB results for 90% soil sample volume for varying anisotropic media (graphical anisotropy exponent a = –1.2, –0.9, –0.6, –0.3, 0, 0.3, 0.6, 0.9, and 1.2, where a is the logarithm of the ratio of vertical to horizontal air permeability) for permeameter diameter/height (D/H) ratios of (A) 0.50 and (B) 2.0.

 

   Conclusions
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Discussion
 Conclusions
REFERENCES
 
Field methods to rapidly and accurately map the saturated hydraulic conductivity could improve the accuracy of distributed hydrologic models. Water-based methods are cumbersome and time consuming, while texture-based methods lack accuracy. Air permeability has been shown to be a viable alternative. We show that anisotropic conditions can introduce an error as high as a factor of 2 in air permeability estimates, especially for air permeameters with high D/H ratios, such as surface permeameters. This error, however, will always be much smaller than the anisotropy ratio of the soil, and the error is minimized for permeameters with low D/H ratios, such as the SCAP. For many applications, this error may be considered to be negligible. If anisotropy is important to characterize for a specific application, however, we have shown that paired measurements of in situ and ex situ air permeability can be used to infer the anisotropy ratio. This approach is likely to be more sensitive for larger D/H ratios. Unfortunately, for any given permeameter, the approach will be more sensitive if the horizontal permeability is lower than the vertical permeability, which is not common under field conditions. Applying the method to a field soil showed qualitative agreement with the expectations based on visual inspection, but it appeared that anisotropy alone was not able to fully explain the difference between the in situ and ex situ permeability measurements.


   ACKNOWLEDGMENTS
 
This research and publication was made possible through funding from the National Science Foundation Science and Technology Center (NSF STC) of Sustainability of Semi-Arid Hydrology and Riparian Areas (SAHRA) Research Center and the University of Arizona Marshall Foundation Dissertation Fellowship.


   REFERENCES
TOP
 ABSTRACT
 INTRODUCTION
 Theory
 Materials and Methods
 Results and Discussion
 Conclusions
 REFERENCES
 





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