A low-cost acoustic permeameter

Intrinsic permeability is an important parameter that regulates air exchange through porous media such as snow. Standard methods of measuring snow permeability are inconvenient to perform outdoors, are fraught with sampling errors, and require specialized equipment, while bringing intact samples back to the laboratory is also challenging. To address these issues, we designed, built, and tested a low-cost acoustic permeameter that allows computation of volumeaveraged intrinsic permeability for a homogenous medium. In this paper, we validate acoustically derived permeability of homogenous, reticulated foam samples by comparison with results derived using a standard flow-through permeameter. Acoustic permeameter elements were designed for use in snow, but the measurement methods are not snow-specific. The electronic components – consisting of a signal generator, amplifier, speaker, microphone, and oscilloscope – are inexpensive and easily obtainable. The system is suitable for outdoor use when it is not precipitating, but the electrical components require protection from the elements in inclement weather. The permeameter can be operated with a microphone either internally mounted or buried a known depth in the medium. The calibration method depends on choice of microphone positioning. For an externally located microphone, calibration was based on a low-frequency approximation applied at 500 Hz that provided an estimate of both intrinsic permeability and tortuosity. The low-frequency approximation that we used is valid up to 2 kHz, but we chose 500 Hz because data reproducibility was maximized at this frequency. For an internally mounted microphone, calibration was based on attenuation at 50 Hz and returned only intrinsic permeability. We found that 50 Hz corresponded to a wavelength that minimized resonance frequencies in the acoustic tube and was also within the response limitations of the microphone. We used reticulated foam of known permeability (ranging from 2× 10−7 to 3× 10−9 m2) and estimated tortuosity of 1.05 to validate both methods. For the externally mounted microphone the mean normalized standard deviation was 6 % for permeability and 2 % for tortuosity. The mean relative error from known measurements was 17 % for permeability and 2 % for tortuosity. For the internally mounted microphone the mean normalized standard deviation for permeability was 10 % and the relative error was also 10 %. Permeability determination for an externally mounted microphone is less sensitive to environmental noise than is the internally mounted microphone and is therefore the recommended method. The approximation using the internally mounted microphone was developed as an alternative for circumstances in which placing the microphone in the medium was not feasible. Environmental noise degrades precision of both methods and is recognizable as increased scatter for replicate data points.


Introduction 1
Intrinsic permeability is the proportionality constant in Darcy's Law that describes the interconnectedness of air 2 space in permeable media such as snow. It has long been recognized as an important parameter that regulates air exchange 3 both within the snow pore space and between the atmosphere and the snowpack (Bader, 1939). Since permeability is difficult 4 Fig. 1 but with the speaker mounted in an upward orientation and the microphone mounted directly above the centerline of 11 the PVC tube, facing downward. Acoustic intensity at a given frequency was measured with the microphone in open air to 12 establish reference amplitude. Then a foam sample was placed over the end of the speaker tube and weighted at the edges to 13 minimize sample vibration. At each applied frequency, attenuation and phase shift were determined by comparison with the 14 reference measurement. We then utilized the low-frequency approximation in Moore et al. (1991, hereafter referred to as 15 M91), to calculate permeability. In M91, estimates of tortuosity ( ) and effective flow resistivity ( !" ) were iteratively 16 adjusted until optimal agreement between the measured and modeled value for the propagation constant ( ! ) at a given 17 frequency ( ) was obtained. Snow permeability ( ) can then be computed from flow resistivity given knowledge or 18 assumptions of the snow grain shape. The theoretical basis for the M91 low-frequency approximation is described in Sect. 19 3.1 of this paper. An added benefit of this method relative to a flow-through permeameter is that it produces a measure of 20 tortuosity as well as permeability. 21

Acoustic Permeability Determination -Internal Microphone Method 22
A second permeability measurement method employed the microphone mounted inside the acoustic tube, facing outward, at 23 a fixed position (15 cm) from the open end. We refer to this as the IM method. We measured attenuation at 50 Hz relative to 24 an open-air value with the open end of the acoustic tube snug against the test media. At this frequency, the wavelength of the 25 emitted frequency is much greater than the length of the acoustic tube. Since the medium represents an acoustic barrier to the 26 emitted waveform, the amplitude of the transmitted waveform was progressively retarded as permeability increased. 27 Simultaneously, the amplitude of the reflected waveform, measured by the internally mounted microphone, increased. From 28 Morse (1952, hereafter referred to as M52) attenuation at a given frequency is a function of flow resistivity. Amplitude of the 29 reflected waveform is inversely proportional to the amplitude of the transmitted waveform so permeability was computed as 30 a function of the ratio between the initial amplitude and the reflected amplitude. 31

Flow-through Permeability Determination 1
We verified acoustic permeameter measurements with 5-cm-thick reticulated foam samples. The permeability of these foam 2 samples was established with a flow-through permeameter similar to Albert et al. (2000) with the exception that we utilized 3 three high-precision (Paroscientific 216B) absolute pressure sensors rather than two relative pressure sensors (Fig. 2). Using 4 absolute pressure sensors rather than relative pressure sensors required a modification of Albert's method so we briefly 5 describe our alternate method. For each foam type we cut a cylindrical section and spread a layer of petroleum gel around the 6 cut edge. We then slid the cut foam into the double-walled permeameter until the inner face of the foam was flush against the 7 inner wall of the permeameter as shown in Fig. 2. The gel sealed gaps between the foam and vessel wall. We cross-calibrated 8 the three pressure sensors before initiating airflow through the permeameter. The three pressure sensors monitored 9 atmospheric, inner cylinder, and outer cylinder pressures. Following Albert et al. (2000), we acquired the pressure drop 10 across each foam sample at 8-12 flow rates while adjusting a valve to regulate the pressure between the inner and outer 11 vessel to eliminate radial flow. We used the slope of the subsequent flow rate vs. pressure drop curve to derive permeability 12 from Darcy's law valid for Reynolds numbers less than 1: 13 (1) 14 where is intrinsic permeability (m 2 ), is volumetric discharge (m 3 -s -1 ), is the sample surface area (m 2 ), is the sample 15 height (m), Δ is the pressure drop (Pa) and is dynamic air viscosity (kg-m -1 -s -1 ). Our samples were distinct in properties 16 from those published by Clifton et al. (2008) (Table 3.1), suggesting that intrinsic permeability of reticulated foam with the 17 same specified pores per linear inch (PPI) but different manufacturers (here Regicell and FXI Corporation) must be 18 determined experimentally. Permeability measured for the FXI samples in Table 3.1 span expected snow permeability values 19 ranging from lightly compacted snow (low permeability) to depth hoar (high permeability), (Arakawa et al., 2009). 20

Theory 21
In this section, we describe the theoretical basis for the two methods used to acoustically measure snow permeability. 22

Theory for the EM Method 23
The complex propagation constant for sound through permeable media can be written in general terms as: 24 where Δ is the phase shift between a free air measurement and the in-snow measurement and is media thickness. We 1 measure attenuation in terms of a voltage difference between the input ( !" ) and output signal ( !"# ) as: 2 Attenborough (1983) developed a four-parameter acoustic model that describes acoustic attenuation through permeable 4 media. These four parameters: porosity, effective flow resistivity, pore shape factor and grain shape factor collapse to two 5 parameters in the low-frequency approximation given in M91: 6 ! = 0.0079 9.10 + 4 !" The M91 authors used Eq. (6) to calculate the propagation constant for a given tortuosity and flow resistivity and Eqs. (2-5) 8 to calculate the propagation constant from phase shift and attenuation. Values for flow resistivity and tortuosity in Eq. (6)  9 were iteratively adjusted until the difference between the propagation constant determined by these two methods was 10 minimized. Due to environment-dependent nonlinear responses at both low and high frequencies the M91 authors obtained 11 the best fit when they used well chosen point measurements rather than measurements over a range of frequencies. Similarly, 12 our results are based on well-chosen point measurements. But instead of iteratively solving Eq. (6) we separate the real and 13 imaginary parts to independently solve for tortuosity and effective flow resistivity: 14 We then apply empirical calibrations to match derived tortuosity and effective resistivity with known values. 17

Theory for the IM Method 18
With the alternate configuration of the microphone placed inside the acoustic tube, Eqs. (2)(3)(4)(5)(6)(7)(8) are no longer applicable. 19 Instead, we use an alternate theoretical basis to determine intrinsic permeability based on results described in M52. M52 20 found and Ishida (1965) verified a simple relationship between flow resistivity and signal attenuation: 21 where is an attenuation constant, is an empirical constant, is the imposed angular frequency and is flow resistivity. 23 As with the EM method, we fit a sine wave to the input and output signals to eliminate white noise and then determined the 24 phase offset and attenuation between the input and output signals as a function of the input frequency. Instead of measuring 25 the attenuation through the media, we measure the amplitude of the reflected waveform at 50 Hz. We take the amplitude of 26 the reflected acoustic energy as an empirically-derived function of transmitted acoustic energy. Using reticulated foam 27 samples of known permeability, we relate the ratio of the reflected signal amplitude to initial signal amplitude with flow-28 through permeability measurements.

Results for the EM Method 2
Having calibrated permeability of the reticulated foam samples with the flow-through permeameter (in Fig. 2) we 3 subsequently obtained acoustic measurements by the EM method. Three replicates were acquired for each foam sample at 4 each frequency. After some experimentation, we opted to utilize data acquired at 500 Hz because this frequency lies within 5 the valid range of frequencies for the low-frequency approximation and because the 500 Hz results were more replicable 6 than those acquired at higher frequencies. 7 In M91 flow resistivity and tortuosity were determined by iteratively finding the best fit between measurements given in 8 Eqs. (2)(3)(4)(5) and the theoretical prediction given in Eq. (6). Since we know a priori the foam tortuosity and permeability we 9 compared the known effective flow resistivity with the effective flow resistivity computed from Eqs. (2)(3)(4)(5)(6)(7)(8). As in Albert 10 (2007) and using a pore shape factor ratio ( ! ) of 1, the effective flow resistivity as derived from permeability is: 11 For the foam samples, we found that a linear fit resolved the difference between computed and known flow resistivity for a 13 given permeability. In short, acoustically derived effective flow resistivity did not match the actual effective flow resistivity 14 but deviated by a linear function from it. Therefore, from Eq. (10), we computed flow resistivity is a linear function of 15 permeability. 16 In Fig. 3 we plotted the results of three independent data sets, each data set corresponding to a different volume setting on 17 the signal generator and rendered in a different color. For a given data set, the signal generator volume was established for 18 the first data point and not changed for the remaining data points. Data corresponding to the blue/black/red data sets had 19 initial imposed volume that associated to RMS signal input voltages of 1.3/2.3/5.6V, respectively. Plotted in log/log format, 20 data points for each data set in Fig. 3 are roughly collinear and the slope of the best-fit line for a given data set is nearly 21 parallel to the other two data sets but the data sets are not coincident (Fig. 4). This result indicates that as the amplitude of 22 the imposed sine wave was increased, absolute attenuation of the signal also increased. Since the acoustically derived 23 permeability is sensitive to the amplitude of the imposed waveform, it is therefore important to use the same amplitude 24 waveform in the calibration as in subsequent permeability measurements. The relationship between !" and reversed at the 25 highest permeability in Fig. 3. The reason for this reversal is possibly attributable to non-linearity in microphone response 26 and represents an upper limit to the permeability that can be measured by this technique within the constraints of our 27 experimental design. Triplicate data points were nearly coincident in most cases; however, the blue and black asterisks in 28 Fig. 3 correspond to points that were more than a standard deviation removed from the other two points in the given 29 triplicate measurement. Differences in the imposed amplitude for these two data points does not account for their deviation 30 from the mean suggesting the source of error was environmental. These data were acquired outdoors in a relatively quiet but 31 acoustically uncontrolled location so we attribute this error is to intermittent external noise. It is therefore also important to 32 acquire multiple data points for each measurement so that one can distinguish the impact of incidental, external noise. Doutres and Atalla (2012) note that reticulated foam commonly has a narrow range of tortuosity from 1.03 to 1.06. 5 Tortuosity of the FXI reticulated foam is not independently measured but shares other specifications with these other sources 6 and likely has tortuosity in the same range. 7 We empirically compensate for the differences between the theoretical and experimentally derived values for tortuosity and 8 effective flow resistivity. Our rationale is that the functional relationships between the theoretically and derived values for 9 tortuosity and effective flow resistivity are the same so the differences in values can be resolved by a first-order calibration. 10 Calculated tortuosity depended very weakly on media permeability so multiplying the calculated tortuosity by an 11 empirically-derived constant renders the theoretical value. On the other hand, calculated permeability varies linearly (on a 12 log scale) with effective flow resistivity (as shown in Fig. 4) so a linear calibration is required to calculate permeability from 13 measured effective flow resistivity. The multiplicative constant for tortuosity is found by dividing the mean known tortuosity 14 by the mean derived tortuosity. The linear calibration relating effective flow resistivity to permeability is taken from the 15 slope of the line in Fig. 4. In practice, one must calibrate the acoustic permeameter with media of known permeability and 16 tortuosity to derive these multiplicative constants. Once these multiplicative constants have been determined for the acoustic 17 permeameter, the tortuosity and permeability of other media having similar impedance can be derived without a priori 18 knowledge of media tortuosity or permeability. For the 5.6V data set we find the multiplicative constant as: = 0.057. For 19 the same data set a linear fit between measured flow resistivity and known permeability is given by: 20 !" = −2.0219 × !" ( !" ) − 0.1764 (11) 21 standard deviation of ±0.03. Percentage normalized standard deviation for permeability measurements ranged from 2.1% for 23 the Z80 foam to 11.6% for the Z30 foam with a mean of 6.2% across all foam samples. We normalize the standard deviation 24 because this error measure does not change when effective flow resistivity is calibrated with a multiplicative constant. The 25 average percentage error between known and derived permeability was 17% by the EM method. for the Z50 sample to 19% for the highest permeability (Z10) sample with a mean of 10% across all foam samples. The 1 average percentage error between known and derived permeability was 10% by the IM method. Since data at 50 Hz show 2 improved precision relative to 100 Hz and 250 Hz data, as given by the wider domain of reflected signal amplitude, we use 3 data at 50 Hz to compute permeability. The piecewise spline fit (black, in Fig. 5) describes the straight-line fit between 4 acoustically derived resistivity and the red dashed line delineates a smooth curve used to calibrate measurements to flow-5 through permeability. There is a measurable phase shift with decreasing permeability of the test media, however, M52 does 6 not offer a relationship for tortuosity so none was attempted with this method. Microphone response limitations precluded 7 measurements below 50 Hz. 8

Discussion 9
Derived permeability using the EM and IM methods are compared with the flow-through (known) permeability in Fig. 6. 10 Intrinsic permeability for each foam sample is given in Table 3.1. Ideally, all measurements would fall on the diagonal line 11 in Fig. 6 meaning that acoustically measured permeability equals known permeability for each foam sample. We find 12 satisfactory agreement with known permeability for both acoustic methods. The average percentage error for the EM method 13 was 17% and 10% for the IM method. However, the curve fit for the IM method is nonlinear so a small error in effective 14 flow resistivity for a high permeability medium has a greater impact than the same error for a low permeability medium. 15 The EM and IM methods for determining permeability each have advantages and disadvantages. One advantage of the EM 16 method is that the function relating effective flow resistivity to permeability is linear, unlike the more complicated 17 relationship for the IM method. An advantage of a linear function is that a small error in measuring the x-coordinate yields 18 an error that is invariant with relative magnitude when determining the y-coordinate, regardless of the magnitude of the 19 measurement. This behavior is not true for non-linear functions such as the more complicated function employed in the IM 20 method. A disadvantage of the EM method is that it is not valid for high permeability media over the range of parameters 21 that we tested (e.g. 5 cm media thickness). The IM method suffers decreasing precision at high permeability because the 22 slope of the curve is increasing but, unlike the EM method, the curve fit remains valid. Data acquired using the IM method 23 was highly reproducible for all of our samples, which to some extent compensates for the slope-induced decrease in 24 precision at high permeability. Another potential disadvantage of the EM method is that the microphone must be placed in 25 the media to be measured. Depending on the circumstance, emplacement in the media may be difficult or impractical. 26 Finally, a disadvantage of the IM method is that it is more empirical than the EM method, which may limit its applicability 27 to different media types. Due to the linear functional form of the EM method, we suggest that it is the preferred method as 28 long as the media to be measured has a permeability that is within the valid range of measurements given by the calibration. 29 The results presented in this paper apply directly to reticulated foam, which has similar microphysical characteristics as snow 30 (Schneebeli and Sokratov, 2004;Clifton et al., 2008). Nevertheless, foam is not snow and we have yet to establish how 31 differences in characteristic impedance between reticulated foam and snow will affect signal attenuation measurements. The 32 next step is to validate acoustically derived measurements in snow by comparison with other methods. Given that 33

Conclusions 5
We conclude that it is feasible to derive intrinsic permeability with a low-cost acoustic permeameter by either the EM 6 method or the IM method. This system returns an unambiguous, volume-averaged permeability consistent with snow 7 densities ranging from lightly compacted snow to depth hoar. The EM method returned intrinsic permeability within 17% of 8 flow-through measurements and within 2% of comparable tortuosity measurements. The IM method returned intrinsic 9 permeability within 10% of flow-through measurements. Measurement reproducibility by the IM method is very high such 10 that the standard deviation between measurements is dwarfed by environmental factors such as background noise or 11 inhomogeneities in the media of interest. These methods are relevant in studies for which an expedient volume average 12 trumps a more detailed but time-consuming vertical profile that requires measuring multiple samples using a flow-through 13 permeameter. One significant advantage of using these methods is that a particular frequency is measured rather than a broad 14 spectrum. This feature obviates the need for specialized equipment used by other methods, such as a random noise generator 15 and a spectral recorder/analyzer. Disadvantages of this system are that measurements are affected by environmental 16 conditions such as background noise and wind. A future task is to validate this system by comparison with other measures of 17 intrinsic permeability over a range of snow habits and densities.