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 volume-averaged 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

Intrinsic permeability is the proportionality constant in Darcy's law that
describes the interconnectedness of air space in permeable media such as
snow. Darcy's law is a constitutive equation that describes fluid flow
through a permeable medium and can be written as

In this paper, we describe the design, construction, and calibration of an in situ active acoustic device that samples a measurement volume larger than that obtainable with current flow-through devices to acquire a volume-averaged estimate of intrinsic permeability. Acoustically derived permeability measurements are validated with homogenous foam samples of known permeability. Active sampling methods of snow properties have been successfully applied in previous studies. For example, Albert et al. (2007) measured the evolution of an acoustic pulse, generated by firing a pistol blank, and compared the pulse shape with that of a simulated pulse to estimate volume-averaged permeability of an Alaskan snowpack. Kinar and Pomeroy designed (2007) and improved (2015) upon an active acoustic device that infers properties such as snow water equivalent from the differential backscatter at frequencies between 20 Hz and 10 kHz. The two methods that we present are less sophisticated than either the Albert or Kinar methods but have the advantage that no specialized equipment is required. Our design borrows elements from Ishida (1965), Buser (1986), and Moore et al. (1991).

Acoustic permeameter schematic.

We assembled an acoustic permeameter from commonly available parts (Fig. 1).
A Heathkit Model IG-1275 signal generator produced a sine wave of a
specified frequency and amplitude that was split and directed both to
channel 1 of a Tektronix TDS 1001 oscilloscope and to a 4^{™} unidirectional microphone (model 3303038; frequency
range: 50 Hz–15 kHz) and directed to the second input channel on the
oscilloscope. Frequency and amplitude of input and output signals were
stored on the oscilloscope USB drive. We do not have full knowledge of the
characteristics of the microphone transducer. However, the calibration is
empirical and will vary with each microphone, so discrepancies between
microphones are accommodated by the calibration.

The first method used to measure intrinsic permeability involved using an
externally placed microphone (EM), where we placed the test media between
the sound source and the microphone (Fig. 2). The calibration setup for the
EM method had the speaker mounted in an upward orientation and the
microphone mounted directly above the centerline of the PVC tube, facing
downward. Acoustic intensity at a given frequency was measured with the
microphone in open air to establish reference amplitude. Then a foam sample
was placed over the end of the speaker tube and weighted at the edges to
minimize sample vibration. At each applied frequency, attenuation and phase
shift were determined by comparison with the reference measurement. We then
utilized the low-frequency approximation in Moore et al. (1991, hereafter
referred to as M91) to calculate permeability. In M91, estimates of
tortuosity

EM calibration setup with the microphone suspended above a foam sample. Foam samples, the oscilloscope, and the frequency generator are located on the foldup table. Not shown are a ring weight that was placed around the perimeter of the foam sample to minimize vibration and foam sheets that were placed beneath the acoustic tube to dampen amplitude of reflected acoustic energy.

A second permeability measurement method employed the microphone mounted inside the acoustic tube, facing outward, at 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 an open-air value with the open end of the acoustic tube snug against the test media. At this frequency, the wavelength of the emitted frequency is much greater than the length of the acoustic tube. Since the medium represents an acoustic barrier to the emitted waveform, the maximum amplitude of the transmitted waveform was progressively retarded as permeability increased. Simultaneously, the amplitude of the reflected waveform, measured by the internally mounted microphone, increased. From Morse (1952, hereafter referred to as M52), attenuation at a given frequency is a function of flow resistivity. Amplitude of the reflected waveform is inversely proportional to the amplitude of the transmitted waveform, so permeability was computed as a function of the ratio between the amplitude of the imposed waveform and the reflected amplitude.

Flow-through permeameter with three absolute pressure sensors labeled P1, P2, and P ATM. P ATM measures atmospheric pressure. This permeameter design is a modified version of the flow-through permeameter of Hardy and Albert (1993).

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

Regicell foam permeability from Clifton et al. (2008) and FXI foam permeability measured with a flow-through permeameter described in this paper.

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

The complex propagation constant for sound through permeable media can be
written in general terms as

With the alternate configuration of the microphone placed inside the
acoustic tube, Eqs. (2)–(8) are no longer applicable. Instead, we use an
alternate theoretical basis to determine intrinsic permeability based on
results described in M52. M52 found and Ishida (1965) verified a simple
relationship between flow resistivity and signal attenuation:

Having calibrated permeability of the reticulated foam samples with the flow-through permeameter (in Fig. 3), we subsequently obtained acoustic measurements by the EM method. Three replicates were acquired for each foam sample at each frequency. After some experimentation, we opted to utilize data acquired at 500 Hz because this frequency lies within the valid range of frequencies (up to 2 kHz) for the low-frequency approximation and because the 500 Hz results were more replicable than those acquired at higher frequencies. The setup for this phase of the experiment is shown in Fig. 2.

In M91 flow resistivity and tortuosity were determined by iteratively
finding the best fit between measurements given in Eqs. (2)–(5) and the
theoretical prediction given in Eq. (6). Since we know a priori the foam tortuosity
and permeability, we compared the known effective flow resistivity with the
effective flow resistivity computed from Eqs. (2)–(8). As in Albert et al. (2007) and
using a pore shape factor ratio (

Comparison of acoustically derived permeability using the M91 method at 500 Hz and three different initial conditions. The three initial conditions correspond to input rms signal voltages of 1.3, 2.3, and 5.6 volts highlighted in blue, black, and red, respectively. Data were acquired outdoors on clear days during periods of light winds. Color-filled regions indicate data within 1 standard deviation of the mean. The asterisks indicate data points that exceeded 1 standard deviation from the mean (before they were excluded) and were thus excluded from subsequent analysis.

In Fig. 4 we plotted the results of three independent data sets, with each data
set corresponding to a different volume setting on the signal generator and
rendered in a different color. For a given data set, the signal generator
volume was established for the first data point and not changed for the
remaining data points. Data corresponding to the blue, black, and red data sets
had initial imposed volumes that were associated with root mean square (rms) signal input voltages of
1.3, 2.3, and 5.6 V, respectively. Plotted in log–log format, data points for each
data set in Fig. 4 are roughly collinear, and the slope of the best-fit line
for a given data set is nearly parallel to the other two data sets, but the
data sets are not coincident (Fig. 5). This result indicates that, as the
amplitude of the imposed sine wave was increased, absolute attenuation of
the signal also increased. Since the acoustically derived permeability is
sensitive to the amplitude of the imposed waveform, it is therefore
important to use the same amplitude waveform in the calibration as in
subsequent permeability measurements. The relationship between

Linear regressions for data in Fig. 4 with the same color for each
data set.

Uncalibrated tortuosity calculated for different foam samples was nearly very similar but unrealistically high with an average of 17.7 and standard deviation of 1.54 between all of the samples. A slight but monotonic decrease in measured tortuosity with decreasing permeability accounted for most of the standard deviation as opposed to random scatter. For comparison, analogous data from Álvarez-Arenas et al. (2006), Kino et al. (2012), and references in Melon and Castagnede (1995) and Doutres and Atalla (2012) note that reticulated foam commonly has a narrow range of tortuosity from 1.03 to 1.06. Tortuosity of the FXI-reticulated foam is not independently measured but shares other specifications with these other sources and likely has tortuosity in the same range.

We empirically compensate for the differences between the theoretical and
experimentally derived values for tortuosity and effective flow resistivity.
Our rationale is that the functional relationships between the theoretically
and derived values for tortuosity and effective flow resistivity are the
same, so the differences in values can be resolved by a first-order
calibration. Calculated tortuosity depended very weakly on media
permeability, so multiplying the calculated tortuosity by an
empirically derived constant renders the theoretical value. On the other
hand, calculated permeability varies linearly (on a log scale) with
effective flow resistivity (as shown in Fig. 5), so a linear calibration is
required to calculate permeability from measured effective flow resistivity.
The multiplicative constant for tortuosity is found by dividing the mean
known tortuosity by the mean derived tortuosity. The linear calibration
relating effective flow resistivity to permeability is taken from the slope
of the line in Fig. 5. In practice, one must calibrate the acoustic
permeameter with media of known permeability and tortuosity to derive these
multiplicative constants. Once these multiplicative constants have been
determined for the acoustic permeameter, the tortuosity and permeability of
other media having similar impedance can be derived without a priori knowledge of
media tortuosity or permeability. For the 5.6 V data set we find the
multiplicative constant as

Flow-through permeability as a function of relative flow
resistivity using the M52 method and Eq. (9) at 50, 100, and 250 Hz.
Three data points were acquired at each frequency and each foam type. The
dashed red line defines a second-order polynomial curve fit for the
measurements with

Applying Eq. (9) to results obtained with the microphone placed inside the acoustic tube at 50, 100, and 250 Hz yielded results shown in Fig. 6. Since we are measuring reflected acoustic energy rather than transmitted energy, the result shown in Fig. 6 is an indirect (relative) measure of flow resistivity. Triplicates were acquired for each foam sample, and these data were highly reproducible as shown by the close grouping of each triplicate. Data at these three frequencies were acquired on different days, each with a slightly different amplitude baseline (free-air) setting, accounting for the horizontal displacement between curve fits. The normalized standard deviation percentage for 50 Hz measurements ranged from 0.45 % for the Z50 sample to 19 % for the highest-permeability (Z10) sample, with a mean of 10 % across all foam samples. The average percentage error between known and derived permeability was 10 % by the IM method. Since data at 50 Hz show improved precision relative to 100 and 250 Hz data, as given by the wider domain of reflected signal amplitude, we use data at 50 Hz to compute permeability. The piecewise spline fit (black, in Fig. 6) describes the straight-line fit between acoustically derived resistivity, and the red dashed line delineates a smooth curve used to calibrate measurements to flow-through permeability. There is a measurable phase shift with decreasing permeability of the test media; however, M52 does not offer a relationship for tortuosity, so none was attempted with this method. Microphone response limitations precluded measurements below 50 Hz.

Derived permeability using the EM and IM methods is compared with the flow-through (known) permeability in Fig. 7. Intrinsic permeability for each foam sample is given in Table 1. Ideally, all measurements would fall on the diagonal line in Fig. 7, meaning that acoustically measured permeability equals known permeability for each foam sample. We find satisfactory agreement with known permeability for both acoustic methods. The average percentage error for the EM method was 17 and 10 % for the IM method. However, the curve fit for the IM method is nonlinear, so a small error in effective flow resistivity for a high-permeability medium has a greater impact than the same error for a low-permeability medium.

Comparing flow-through (known) permeability with the EM method at
500 Hz and the IM method at 50 Hz for foam samples given in Table 1. Z10
permeability was not computed by the EM method because

The EM and IM methods for determining permeability each have advantages and disadvantages. As mentioned, one advantage of the EM method is that the function relating effective flow resistivity to permeability is linear, unlike the more complicated relationship for the IM method. An advantage of a linear function is that a small error in measuring the independent variable yields an error that is invariant with relative magnitude when determining the dependent variable, regardless of the magnitude of the measurement. This behavior is not true for nonlinear functions such as the more complicated function employed in the IM method. A disadvantage of the EM method is that it is not valid for high-permeability media over the range of parameters that we tested (e.g., 5 cm media thickness). The IM method suffers decreasing precision at high permeability because the slope of the curve is increasing, but, unlike the EM method, the curve fit remains valid. Data acquired using the IM method were highly reproducible for all of our samples, which to some extent compensates for the slope-induced decrease in precision at high permeability. Another potential disadvantage of the EM method is that the microphone must be placed in the media to be measured. Depending on the circumstance, emplacement in the media may be difficult or impractical. Finally, a disadvantage of the IM method is that it is more empirical than the EM method, which may limit its applicability to different media types. Due to the linear functional form of the EM method, we suggest that it is the preferred method as long as the media to be measured have permeability that is within the valid range of measurements given by the calibration.

The results presented in this paper apply directly to reticulated foam, which has similar microphysical characteristics to snow (Schneebeli and Sokratov, 2004; Clifton et al., 2008). Nevertheless, foam is not snow, and we have yet to establish how differences in characteristic impedance between reticulated foam and snow will affect signal attenuation measurements. The next step is to validate acoustically derived measurements in snow by comparison with other methods. Given that characteristic impedance of snow varies linearly with snow density (Marco et al., 1998), we anticipate that the calibration curves for both the EM and IM methods will have similar shape to the curves in Figs. 5 and 6 but with a modified slope for snow.

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

Data and data file descriptions are available online through
the Scholars Archive hosted by Oregon State University with reference

The authors declare that they have no conflict of interest.

We thank FXI Corporation for providing the reticulated foam samples that we used to perform the permeability calibrations. Thanks also to Ziru Liu and Rebecca Hochreutener for their assistance in acquiring data. Edited by: C. Waldmann Reviewed by: N. J. Kinar and one anonymous referee