Relating wave frequency and pore fluid homogeneity in quasi-saturated sands

. P-wave velocity is regularly proposed as a proxy for water saturation when studying quasi-saturated soils. However, several authors have underlined that this velocity can only be reliably correlated to water saturation if the pore fluid is “homogeneous” . Using bender elements to measure compression wave velocity in quasi-saturated Fontainebleau sand, we show it is highly sensitive to both saturation and frequency. The characteristic length of pore pressure diffusion is used to explain the link between frequency, gas distribution and P-wave velocity. This criterion is shown to be fairly reliable for assessing the homogeneity of a pore fluid distribution in quasi-saturated Fontainebleau sand.


Introduction
Compression waves in soils are very sensitive to water saturation. For this reason, their velocity has been correlated to the water saturation ratio [1], Skempton's value [1][2][3] or even to the liquefaction resistance of quasi-saturated soils [4]. However, it has also been shown * Corresponding author: gfloodpage@gmail.com that is just as sensitive to pore fluid homogeneity as it is to water content [1,5]. As is visible in Fig. 1, while some sets of experimental data follow the theoretical trend predicted for homogeneous fluid distributions (shown in black for Fontainebleau sand), this is not necessarily the case. Fig. 1. Comparison in between / ratio and Skempton's B-value as found in geotechnical literature [1][2][3]5,13] and analytical values derived for Fontainebleau Sand.
The equation relating saturation ratio and P-wave velocity is generally derived within the framework of linear elastic plane wave theory. Considering an undrained P-wave modulus and a density : (1) The undrained P-wave modulus can in turn be related to the undrained bulk modulus . Considering the equations provided by [6] and supposing that solid grains are incompressible ( ≫ and ≫ ), one can express as follows: where is the dry bulk modulus of the soil, is the bulk modulus of the pore fluid, is the bulk modulus of the minerals composing the solid fraction of the soil and is the porosity. It is worth noting that this expression can also be derived from the "Gassman equation" [7], which is more commonly adopted by geophysicists. The dry bulk modulus is estimated via the shear wave velocity , supposing a Poisson ratio . Skempton's B-value is computed as described in [6], considering an incompressible solid phase: While the theoretical expressions written above are perfectly valid in the framework described by [6], their use in the context of acoustic wave propagation can be unreliable. The main issue lies with the definition of the fluid compressibility of a gas-water mixture. This can be understood within the framework of homogenisation theory.
Considering a fluid mixture composed of a water phase (index "w") and a gas phase (index "g"), the bulk modulus of the resulting mixture (index "f") will be some combination of the moduli and weighted by the saturation ratio . The lower bound for the bulk modulus of this mixture can be computed supposing the fluid pressure has the time to equilibrate between stiff waterfilled pores and compliant gas-filled pores. This approach results in the "Reuss" bound for fluid compressibility: This expression is sometimes also named after the works of Wood [8] and Koning [9]. Alternatively, the upper bound of the fluid bulk modulus can be estimated considering that the fluid strain is equal to the strain in each phase: = = . This condition is only satisfied if the fluid pressure does not have the time to equilibrate between more and less compressible regions. This could notably occur for soils with a very heterogeneous gas distribution as the pressure equilibration distance would be considerably greater than for homogeneous gas distributions. This approach leads to the "Voigt" bound for the fluid bulk modulus, sometimes named after Hill [10]: ratio is contained somewhere between the bounds. The main objective of this paper is to provide a quantifiable criterion to assess whether a pore fluid distribution can be described as homogeneous.
2 Laboratory data

Experimental setup
In order to propose a criterion for assessing the homogeneity at specimen scale of a gas-water mixture, a series of experiments were led on Fontainebleau sand. Compression and shear wave velocities were measured using a set of bender-extender elements (length ≈ 3.5±0.5 mm). The signal was generated with a TG1010A function generator and visualised with an Agilent Technologies InfiniiVision DSO-X-2004A digital oscilloscope.
The sand specimens are built by dry pluviation or dry tamping methods. Their diameter is 100 mm while the height varies between 100 and 200 mm. Once constructed, they are flushed with carbon dioxide for over twenty minutes, before being saturated with deaired water. Subsequently, air and water are mixed together in a closed cell at controlled pressure for at least twelve hours. A magnetic stirrer is used to accelerate the dissolution of air in the water (see Fig. 2). This "new" pore fluid is then injected into the specimen in order to replace the deaired pore fluid by one containing greater quantities of dissolved gas. The saturation state is changed by lowering the pore pressure to allow gas to exsolve. This desaturation method ensures that gas bubbles are mostly contained within the pores for quasi-saturated soils.
Skempton's B-value is measured by subjecting the specimen to an undrained pressure change and measuring the resulting difference in pore pressure. This process is repeated at least six times for each measurement of . The applied pressure changes are around 20 kPa.  Table 1 contains a list of all specimens prepared in the context of this study on quasi-saturated Fontainebleau sand. The name of each specimen indicates its density index and height H. The number at the end distinguishes different specimens with the same density. The saturation pressure is an estimation of the water pressure at which the saturation ratio begins to decrease during each experiment.

Properties of Fontainebleau sand
In order to interpret these experiments, the following data was considered for the median grain size , uniformity coefficient , minimum and maximum void ratios min and max , and grain density for Fontainebleau sand. The data is taken from [11]. Small-strain moduli were determined on dry Fontainebleau sand specimens. The detail of this process is explained in [12]. Fitting the experimental data provides the following estimations of the small-strain shear modulus and a Poisson ratio of 0.18 (with the effective stress and atmospheric pressure): In order to interpret measurements, the small-strain moduli mentioned previously did not seem relevant as the strain range is completely different. Therefore, drained isotropic consolidation tests were carried out on saturated Fontainebleau sand. The volume changes were measured using the pressure-volume controller (see Fig. 2). Using this method, separate fitting curves were obtained for different stress states. As all experiments were performed on overconsolidated soils, only the overconsolidated fit is provided here. A slight discrepancy was noted between the overconsolidated bulk moduli calculated during unloading and reloading phases. As a first approach, this difference was not taken into account as the resulting error is small. The equation given below corresponds to the drained bulk modulus of overconsolidated Fontainebleau sand during a reloading phase: 3 Experimental analysis

Methodology
As shown by [13], detecting P-waves in gassy soils using bender elements is generally quite complex as signals can contain different components propagating at varying velocities. This renders the frequency-velocity interpretation of these signals fairly imprecise.
To limit this error, measurements are made at over twenty different input frequencies ranging in between 2 and 200 kHz. After eliminating poor quality signals, the arrival time is determined using the SLA time domain method described in [12]. This automated interpretation method is based on a statistical tool named "Akaike Information Criterion" and gives similar results to the "First Arrival" method. The frequency of the actual signal is then estimated by Fourier transform over a 1 millisecond fraction of signal after the arrival. This limits the influence of any other existing wave components on the calculated frequency. Finally, data points are grouped by frequency range. The size of the frequency ranges is chosen in a way to ensure there are always a few signals per range. It also gives an indication of the precision of the frequency estimation. The considered P-wave velocity for each frequency range is its median velocity. The shear velocity is also measured at different frequencies for each stage of the experiment. As is not frequency dependant, the adopted value is the median of all calculated shear wave velocities.  We observe a clear dispersion of the P-wave velocity with frequency. As a rule, the lowest frequencies behave in a manner fairly consistent with velocities computed using the Reuss bound for equivalent fluid compressibility (black lines plotted in Fig. 1). However, as frequency increases, the experimental data progressively departs from this prediction.

Experimental observations
One can also notice that this frequency dispersion is only truly observable for "gassy" specimens ( <0.95). While there is a slight spread for "saturated" data points, this is mostly due to measurement error and temperature variations in the laboratory (in between 15 and 23°C).
Therefore, we can conclude that the dispersion observed in this study and in Fig. 1, is not only a matter of gas distribution as proposed by [1,5], but also a function of frequency.

A criterion for gas homogeneity
The topic of P-wave dispersion in the presence of partly saturated porous media has previously been explored in the field of petroleum geophysics. Norris [14] uses a characteristic pore pressure diffusion length to assess the homogeneity of gas/water mixtures in Berea sandstone. , the intrinsic permeability , the fluid dynamic viscosity , the porosity and the angular frequency of the wave: For Fontainebleau sand, this diffusion length is plotted in Fig. 5. Despite being given for a density index of 0.54, the graph also holds for denser sands as the impact of porosity is negligible when compared to the variations in fluid bulk modulus. We consider a hydraulic conductivity of approximately 5.10 -5 m/s (taken from [15]), and fluid bulk moduli varying from 2.10 3 (saturated) to 2 MPa (low saturation: <0.1).
To ensure that the Reuss average is valid, local excess pore pressure generated at the peak of a compression wave must have enough time to dissipate before the next peak arrives. Therefore, if the characteristic diffusion length is much greater than the scale of saturation heterogeneities (deduced here from the pore size distribution), the Reuss bound for should be valid. Looking at Fig. 5, this condition is potentially met for the lowest tested frequencies (under 10 kHz). For any higher frequencies, there does not seem to be an order of magnitude between the patch sizes and the diffusion length. For this reason, we cannot assume that pore pressures are at equilibrium Fig 6. Comparing Reuss and Voigt theoretical bounds with experimental values of / ratio and value for frequencies above 10 kHz. Conversely, for all frequencies tested here, the typical scale of saturation heterogeneities is not an order of magnitude greater than the diffusion length. For this reason, the strain field is not expected to be homogeneous for any frequency ( ≠ ). All in all, this diffusion length helps explain why the Reuss bound is potentially valid only for the lowest tested frequencies, while the Voigt bound always predicts Pwave velocities higher than those measured in this experiment. . Returning to Fig. 1, experimental data from [1] (where CO2 flushing is used) and [2] seems to follow similar trends to what was observed here for low frequency signals, whereas data from [1] (no CO2 flushing) and [3,5,13] does not. This is understandable as the first set of experiments use desaturation methods that would ensure a fairly homogeneous gas distribution combined with relatively low frequencies. By contrast, the other sets of data either use higher frequencies or desaturation methods that create more heterogeneous gas distributions. Bearing in mind the considerable error in the estimation of both the frequency and P-wave velocity, our data seems to indicate that the Reuss average is fairly accurate for frequencies below 10 kHz. This is consistent with the predictions of the diffusion length shown in Fig. 5, indicating that local excess pore pressure does have time to dissipate for these frequencies. Therefore, the characteristic diffusion length seems to be a reliable criterion for assessing the validity of the Reuss bound.

Assessing the validity of the Reuss bound for low frequency waves
Concerning frequencies higher than 10 kHz, the experimental data shows that neither the Reuss nor the Voigt bound is accurate. Again this is compatible with the proposed criterion as the diffusion length is of the same order of magnitude as the gas pockets. Unfortunately, we were unable to generate input signals with frequencies higher than 200 kHz and therefore cannot verify that the experimental data does tend towards the Voigt bound for very high frequencies.

Conclusion
Through an experimental study of the impact of saturation and frequency on P and S wave velocities, we have shown that, for Fontainebleau sand, the pore pressure diffusion length proposed by [14] seems to be an accurate criterion for assessing the homogeneity of a gas-water mixture.
So long as another dispersion mechanism is not encountered, this criterion should therefore be reliable when assessing the validity of the Reuss approximation. In the context of in-situ geophysics, frequencies rarely exceed several hundred Hertz. Consequently, for Fontainebleau sand, we can infer that the Reuss bound could reliably be used to estimate the saturation ratio from P and S wave velocities if the average scale of saturation heterogeneities is smaller than a centimetre. However, the same is not necessarily true for other types of soils. For instance, in the case of clays, both permeability and gas pocket sizes are likely to be much smaller. Therefore, further analysis is required to ensure the validity of the Reuss approximation for interpreting in-situ geophysical data in soils other than sands.