# Squirt flow in a tight sandstone: an interlaboratory study

Abdulwaheed R Ógúnsàmi, Ian Jackson, Lucas Pimienta, Jérôme Fortin

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## To cite this version:

Abdulwaheed R Ógúnsàmi, Ian Jackson, Lucas Pimienta, Jérôme Fortin. Squirt flow in a tight sandstone: an interlaboratory study. Geophysical Journal International, 2025, 240 (3), pp.1387-1403. 10.1093/gji/ggae451. hal-04919610

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# Squirt flow in a tight sandstone: an interlaboratory study

Abdulwaheed R. Ógúnsàmi,<fnref n="1" />,<fnref n="2" /> Ian Jackson,<fnref n="1" /> Lucas Pimienta,<fnref n="1" />,<fnref n="3" />,<fnref n="4" /> and Jérôme Fortin<fnref n="5" />

Accepted 2024 December 17. Received 2024 December 15; in original form 2024 September 15

## SUMMARY

In the spirit of interlaboratory benchmarking of related techniques, we have re-examined the seismic-frequency mechanical properties of a low-porosity Wilkeson sandstone specimen tested in axial stress oscillation under water-saturated conditions by Pimienta *et al*. The same specimen has been newly tested at periods of 1–1000 s under dry and argon-, water- and glycerine-saturated conditions by torsional and flexural oscillation methods, allowing direct measurement of the shear modulus *G* and Young's modulus *E* and associated strain-energy dissipations. The results show a steady increase of *G* and *E* for the dry specimen with increasing pressure, indicative of progressive closure of the compliant intergranular contacts. Under argon- and water-saturated conditions, the measured moduli differ only marginally from those for dry conditions, without any significant stiffening or dispersion, suggesting that such measurements probe the saturated-isobaric regime. In marked contrast, glycerine saturation results in substantially higher and frequency-dependent moduli, along with frequency-dependent dissipation. We attribute this behaviour to the squirt flow transition with decreasing frequency from the saturated-isolated to the saturated-isobaric regime, modelled with a log-normal distribution of relaxation times (broader than the Debye peak of the standard anelastic solid) superimposed upon a monotonically frequency-dependent background. Although there are differences in detail, these findings corroborate those of Pimienta *et al*. for the same material tested in axial stress oscillation to higher frequencies under water-saturated conditions. Taken together, the two studies thus provide robust support for theoretical models of squirt flow dispersion and dissipation, occurring at frequencies between those of conventional ultrasonic wave-propagation laboratory methods and those of seismic exploration of the shallow crust.

**Key words:** Elasticity and anelasticity; Permeability and porosity; Acoustic properties.

## 1 INTRODUCTION

Squirt flow—defined as the local flow of pore fluid between differentially pressurized parts of a crack-pore network within a porous specimen—remains one of the most widely discussed mechanisms for modulus dispersion and dissipation in fluid saturated rocks (e.g. O’Connell & Budiansky 1977; Mavko & Jizba 1991; Gurevich et al. 2010; Adelinet, Fortin & Guéguen 2011). Such dispersion and dissipation not only limit the reconciliation of ultrasonic mechanical properties of shallow rocks with those obtained from seismic surveys but also preclude the use of seismic attenuation or dissipation as a key seismic attribute—useful in exploration and exploitation of shallow crust resources. Nonetheless, the theoretical understanding of the squirt flow phenomenon is still contended (Müller & Gurevich 2005; Vernik & Kachanov 2012; Sarout 2012; Jackson 2015). While it is broadly accepted that the conventional

macroscopic models of poroelasticity (e.g. Gassmann 1951; Biot 1956) generally fail to account for this mechanism, the laboratory testing of the alternative theoretical models—which have received more attention because they consider the key role played by microstructure—remains a long-standing challenge. For instance, in the classic theory of O’Connell & Budiansky (1977) for frequency-dependent mechanics of fluid-saturated porous media, such squirt fluid flow is responsible for the transition between the saturated-isolated and saturated-isobaric (Gassmann’s undrained) regimes. With decreasing frequency, the theory predicts that three distinct fluid regimes (the saturated-isolated, saturated-isobaric and drained) characterize the mechanics of fluid-saturated porous media. However, only recently has progress been made in experimental delineation of such distinct fluid-flow regimes. (e.g. Sarout 2012; Delle Piane et al. 2015; Pimienta et al. 2017; Sarout et al. 2017; Li et al. 2018; Ógúnsàmi et al. 2020).

<footnote n="1">¹Research School of Earth Sciences, Australian National University, Canberra, ACT 2601, Australia. E-mail: ian.jackson@anu.edu.au</footnote>

<footnote n="2">²Woodside Energy, Perth 6000, Western Australia, Australia</footnote>

<footnote n="3">³Université de Pau et des Pays de l'Adour, E2S-UPPA, CRNS, LFCR, 64000 Pau, France</footnote>

<footnote n="4">⁴Rock Physics Laboratory, ISTE, Université de Lausanne, 1015 Lausanne, Switzerland</footnote>

<footnote n="5">⁵Laboratoire de Géologie, Ecole Normale Supérieure/CNRS, UMR8538, PSL Research University, 75005 Paris, France</footnote>
Although there have been several experimental investigations on the frequency-dependent mechanical properties of fluid-saturated porous media (see e.g. review in Müller and Gurevich 2005), more rigorous experimental observations needed for a conclusive validation of such theoretical prediction of squirt flow have been a difficult task due to a few critical technical issues. On one hand, experimental fluid flow boundary conditions are an issue potentially compromising access to the undrained regime during oscillation of either axial or hydrostatic loading. By taking into consideration the dead volume on the experimental fluid drainage system, in one example, Pimienta *et al.* (2015) show that dissipation peaks and modulus dissipation associated with such experimental fluid flow boundary conditions can be identified by their characteristic frequencies that scale with the spatial scale *x* of fluid flow as $f \sim x^{-2}$. Torsional and flexural oscillations involve no perturbation to the total volume of the specimen and accordingly no driving force for fluid exchange between specimen and an external reservoir, allowing access to undrained conditions even at very low frequency (0.001–1 Hz; Li *et al.* 2018; Ógúnsàmi *et al.* 2020). However, for testing in flexural oscillation, lateral fluid flow in response to pore-pressure gradients caused by flexure of a laboratory specimen is an issue.

On the other hand, limited range of the testing frequency is another issue. In the literature, such a problem is commonly addressed in two ways. One approach is by performing broad-band testing, for example, by combining a type of forced oscillation test with ultrasonic measurements (e.g. Adam & Batzle 2008; David *et al.* 2013; Borgomano *et al.* 2017; Pimienta *et al.* 2017) or, more generally, by combining multiple complementary techniques (e.g. Li *et al.* 2018; Ógúnsàmi *et al.* 2020). The other approach is by performing low-frequency measurements with high-viscosity fluid as the saturant (Mikhaltsevich *et al.* 2015; Pimienta *et al.* 2015, 2017; Subramaniyan *et al.* 2015; Spencer & Shine 2016; Chapman *et al.* 2018). The latter approach is very useful because the characteristic time for squirt flow varies inversely with fluid viscosity (Jones 1986; Winkler & Murphy 1995), thereby improving the chance to document such squirt flow transitions over a broadened range of effective frequencies. However, such an approach has so far received less attention than it deserves.

To contribute to the current understanding of such squirt flow transitions and to add to the literature database on experimental studies, we report the experimental testing of the mechanical properties of dry and fluid-saturated sandstone specimen, in an extension of the study of Pimienta *et al.* (2017) on Wilkeson sandstone. Our contribution brings in further interrogation of the seismic properties of the same specimen under different fluid-saturated conditions (using argon and glycerine as pore fluid) and with a different experimental set-up. We used the forced-oscillation apparatus at the Australian National University (ANU), which allows the direct measurement for both the shear and Young's moduli along with the corresponding dissipation, to explore squirt flow within a specimen subjected to both torsional and flexural forced oscillations. Our study provides additional, yet complementary testing data on the specimen, suitable to explain the frequency-dependent seismic properties across the different fluid flow regimes, as predicted by the model of O'Connell & Budiansky (1977).

# 2 METHODS

## 2.1 Specimen description

Two cylindrical specimens of Wilkeson sandstone, of 15 mm diameter and making up 95 mm in combined length, were cored

and machine-ground from a single cylindrical specimen of larger (31 mm) diameter earlier tested in Pimienta *et al.* (2017). Well-sorted and rounded grains with a nearly uniform grain size of around 400 µm (Duda & Renner 2013) characterize Wilkeson sandstone, originally sourced from the vicinity of Wilkeson sandstone quarry, Wilkeson, WA, USA.

It has quartz content of about 50 per cent (Duda & Renner 2013) with a clay content of about 8 per cent—of which 4 per cent is kaolinite and 4 per cent of other clay minerals. Micrographs (e.g. Fig. 1) and pore structure characterization of Wilkeson sandstone can be found elsewhere in which the Wilkeson sandstone has been studied not only for its mechanical properties (Duda & Renner 2013; Ahrens *et al.* 2018) but also for its transport properties (Schepp & Renner 2021). After calculating bulk volume from the precise length and diameters of the specimen, we measured the porosity and grain density under ambient conditions using the water saturation triple weight method and the pressure-dependent permeability using the transient-flow technique.

## 2.2 Experimental method

We used the ANU attenuation apparatus (Jackson & Paterson 1993) to perform forced-oscillation and permeability experiments on the specimens. The apparatus is a high-pressure gas-medium apparatus with pore-fluid delivery that enables *in situ* examination of fluid-flow properties in conjunction with the forced-oscillation test, through an independent application and control of pore-fluid pressure on a cylindrical specimen of 15 mm diameter and 150 mm length (Jackson & Paterson 1993; Lu & Jackson 1996). Fig. 2 illustrates the key components of the apparatus for mechanical testing of rock properties; it comprises of (1) a composite beam or the experimental assembly consisting of a specimen assembly in series with an elastic element, (2) the displacement transducers that provide for the measurement of the twist or flexure of the beam and (3) the electromagnetic drivers that supply the required torque or bending force. Fig. 3 illustrates the associated pore-fluid delivery system on the apparatus.

### 2.2.1 Saturation procedure and permeability measurements

The arrangements on the ANU apparatus facilitate *in situ* fluid saturation without removing the specimen from the assembly. Because it can be quite difficult to saturate specimen of low porosity and permeability, we apply a rigorous approach consisting of key four critical elements: purging, evacuation, fluid charging and bleeding of residual air in the pore-fluid system, as detailed previously (Li 2016; Li *et al.* 2018; Ógúnsàmi *et al.* 2020, 2021). Such an approach guarantees the best possible approach to conditions of full saturation with limited influence of any residual or trapped air. After initial measurements on the specimen, dry and argon-saturated, we sought to saturate the specimen with glycerine. However, because of the high viscosity of glycerine, both charging the system with glycerine and removing it from the pore space of a low-permeability specimen can be very challenging. For an initial attempt at glycerine saturation with the pore-fluid intensifier (upper right-hand corner of Fig. 3), the oil pump driven by compressed air was used. Insufficiently close control of pressurization with the viscous glycerine pore fluid resulted in a local pore pressure transient exceeding the confining pressure and associated bloating of the copper jacket that accordingly required replacement. Torsional oscillation tests conducted dry and argon-saturated, before this first abortive attempt at
<figure><img src="image_3.png" /><figcaption>**Figure 1.** Representative microstructure of the intact Wilkeson sandstone from (a) optical photomicrograph obtained under cross-polarizers and (b) scanning electron microscope, from Ahrens, Duda & Renner (2018).</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>**Figure 2.** Experimental arrangement of the ANU attenuation apparatus showing (a) schematic of the computer control and data acquisition and the key component for forced-oscillation studies under (b) flexure and (c) torsion with alternative electrical connections to the drivers and capacitance transducers (reproduced from Li et al. 2018).</figcaption></figure>

glycerine saturation, and after ultimately successful saturation with glycerine, provide clear evidence of some additional microcracking caused by the brief pore-pressure transient. The tighter control, critical for the inevitably slow charging of the system with glycerine, and adjustment of its pore pressure, was subsequently achieved by installation of a manual pump (upper right-hand corner of Fig. 3) to drive the pore-fluid intensifier. The subsequent replacement of glycerine by (miscible) water as pore fluid was accomplished by flushing the specimen several times with water until the egress of water from the specimen is several times the pore volume of the specimen and is predominantly water. Mechanical properties of the specimen measured during the repeated stage of charging with water to flush out the glycerine indicate a systematic decrease in stiffening of the specimen—consistent with progressive replacement with a less viscous fluid—namely, water.

We conducted permeability measurements using the transient-flow technique (Brace et al. 1968; Jones et al. 1997). Following from an initial condition of pore pressure equilibrium maintained across the specimen through an upper to the lower reservoir of

uniform pore-fluid pressure, we applied a sudden pore pressure perturbation to the lower reservoir of larger volume and then monitored the consequent exponential evolution of the pressure in the upper reservoir—reflecting re-equilibration by fluid flow through the specimen. We next obtained the rate constant A, of exponential decay of the imposed pore pressure perturbation, from which we calculated the permeability, k, as

$$k = \frac{A\eta L_s}{[K_f A_s (V_u^{-1} - V_d^{-1})]} \quad (1)$$

(Brace et al. 1968). In eq. (1), Ls and As are the length and cross-sectional area of the specimen, and Kf and η are the bulk modulus and viscosity of the pore fluid. Vd ~ 40 cm³ and Vu = 1.14(±0.19) cm³ are, respectively, the volumes of the external reservoirs connected to the lower and upper ends of the rock specimen (Li et al. 2018; Ògúnsàmì et al. 2020). For argon, we applied the pressure-dependent bulk modulus from Stewart & Jacobsen (1989) and pressure-dependent bulk viscosity from Vidal et al. (1979) as
<figure><img src="image_5.png" /><figcaption>Figure 3. Pressure and pore-fluid delivery system for the ANU attenuation apparatus showing arrangements for independent control of the argon confining pressure and either gaseous (argon) or condensed pore-fluid pressure.</figcaption></figure>

used in Li (2016). We obtained similar information for water from the NIST Chemistry WebBook (https://webbook.nist.gov) of the US National Institute of Standards and Technology and those of glycerine from Lyapin et al. (2017) and Herbst et al. (1993), respectively.

### 2.2.2 Measurement of seismic properties

Using the ANU attenuation apparatus introduced above, we investigated the seismic properties of the Wilkeson specimen under torsional and flexural forced oscillation (Jackson & Paterson 1993; Li et al. 2018; Fig. 2). We applied oscillating torque for the torsional forced oscillation test and oscillating bending force for the flexural forced oscillation test (Fig. 2).

**Shear modulus and dissipation.** We obtained the torsional forced oscillation response of the specimen at mHz–Hz frequencies from the analysis of the mechanical response of the composite beam of Fig. 2(c), subjected to an oscillating torque. The composite beam consists of the test specimen that is connected in series to an elastic

element of known mechanical properties. Two pairs of three-plate capacitance transducers measure the associated twist of the beam upon application of an oscillating torque. Such mechanical response is presented as the displacement $d_1$ associated with the distortion of the specimen assembly—comprising the jacketed specimen, connecting rods and the hollow steel members between which they are sandwiched, and distortion $d_2$ of the entire experimental assembly. The difference, $d_{12}$ (i.e. $d_2 - d_1$) between the distortions of the specimen assembly measured by the upper pair of displacement transducers and the distortion of the entire experimental assembly, as measured by the pair of lower displacement transducers provides the distortion of the elastic element located between the upper and lower displacement transducers. From these, we obtained the complex normalized torsional compliance ($S_{NT}^*$) as follows:

$$S_{NT}^* = \frac{d_1^*}{d_{12}^*} = \frac{d_1^*}{d_2^* - d_1^*} = |S_{NT}| \exp(-i\delta_T), \quad (2)$$
where $\delta_T$ (rad) is the loss angle, which is the phase lag of $d_1^*$ relative to $d_{12}^*$, associated with strain energy dissipation. To obtain the shear modulus and associated dissipation in the specimen itself from the normalized torsional compliance, a similar test on a reference assembly containing an elastic control specimen (uncracked glass) was required (e.g. Jackson et al. 2011). We next obtained the shear modulus and dissipation ($Q_G^{-1}$) for the test specimen from the complex compliance differential between the specimen and reference assemblies, along with the known shear moduli for the control specimen and enclosing copper jacket. The uncertainties in the measurements of ±3 per cent in shear modulus and 0.05 log units in associated dissipation are estimated from a posteriori assessment of scatter among the measured values.

**Young's modulus and associated dissipation.** With the arrangement described above but with the electromagnetic drivers switched to supply a bending force, rather than torque, and the appropriate alternative parallel arrangement of the displacement transducers within each pair (Fig. 2b), we probed the specimen's mechanical behaviour under flexural oscillation (Jackson et al. 2011). The raw data from such flexural oscillation measurements are obtained, as for torsional oscillation, as a complex normalized flexural compliance ($S_{\text{NF}}^*$) of the entire specimen assembly given by

$$S_{\text{NF}}^* = \frac{d_1^*}{d_{12}^*} = \frac{d_1^*}{d_2^* - d_1^*} = |S_{\text{NF}}| \exp(-i\delta_F), \quad (3)$$

where, $\delta_F$ (rad) is the loss angle that is the phase lag of $d_1^*$ relative to $d_{12}^*$, associated with strain energy dissipation. We used a finite-difference filament-elongation model (Jackson et al. 2011; Cline & Jackson 2016) to simulate the complex flexural response of the assembly under the same conditions. Trial values of the Young's modulus and associated dissipation for the specimen were varied within the model to match the measured complex normalized flexural compliance $S_{\text{NF}}^*$. For this purpose, we set the shear modulus of the copper jacket material at room temperature to be 38.4 GPa in accord with the results of torsional oscillation measurements for copper. The spiral steel vent tube that makes a compliant connection between the lower end of the flexural assembly and the wall of the pressure vessel was replaced during the course of this experiment. It was therefore necessary to redetermine the effective outer diameter of the model vent tube. The value of 0.5537 mm was required for a period-averaged value of 20 GPa for the Young's modulus of dry Wilkeson sandstone at a pressure of 15 MPa. The difference between this value and that (~34 GPa) of Pimienta et al. (2017) might be partly attributable to stress-induced anisotropy under static axial load in the previous study. However, the impact of microcracking during inadvertent transient overpressurization of the pore space during the first attempt at glycerine saturation is probably the main cause of the difference.

**Strain amplitudes.** The shear strain amplitudes realized during the torsional oscillation tests increase linearly with radial position from zero on the cylindrical axis to a maximum value of ~5 × 10⁻⁶ at the cylindrical surface of the specimen. In flexure, the axial strain amplitudes vary linearly with perpendicular distance from the neutral plane between maximum/minimum values of ±5 × 10⁻⁶ at the radius of the specimen.

## 2.3 Specimen assembly and test protocol

We used an annealed copper tube of 15 mm internal diameter and 0.25 mm wall thickness as a jacket to isolate our specimen from the argon confining pressure medium. The pressure $P_f$ of the pore fluid

(argon, glycerine, water and argon, successively) was taken into consideration along with the confining pressure $P_c$, to enable the test at various differential pressures $P_d = P_c - P_f$. Prior pore pressure equilibration tests ensured conditions of uniform pore pressure throughout the specimen interior. We limited our test to conditions under differential pressures ≤10 MPa to minimize pressure-induced modification of the specimen's microstructure. We conducted forced oscillation tests of longer period (up to 1000 s) for mechanical testing on the fluid-saturated conditions, whereas the default range of oscillation period (1–100 s) was used for dry conditions.

## 2.4 Modelling the modulus dispersion and dissipation

Several analytical/theoretical models of modulus dispersion and dissipation related to squirt flow can be found in the literature (e.g. Mavko & Jizba 1991; Endres & Knight 1997; Gurevich et al. 2010; Adelinet et al. 2011). Each of these models involves some broad assumptions regarding the microstructure of the specimen. As an example, the model of Gurevich et al. (2010) required a single aspect ratio for the soft pores, obtained from pressure-dependent mechanical properties of the dry specimen based on the theory of Shapiro (2003). For this study, such models appear unduly restrictive for two reasons. First, the microstructure is often too complex to be described with a single aspect ratio. Secondly, if a spectrum of aspect ratios were to be included as in the model of Sun & Gurevich (2020), obtaining the required pressure-dependent behaviours of the soft pores requires ideally ultrasonic measurement particularly to high confining pressure. Such measurements are not available for our specimen.

Instead, we employ a phenomenological model to describe the results of our measurements. Such a model enables a mathematical description of the frequency-dependent mechanical behaviour of a viscoelastic material (Zener 1940; Nowick & Berry 1972). The popular Zener model (here referred to as the standard anelastic solid), for instance, could potentially be used to describe the observed frequency dependence of the modulus and dissipation across the transition from the saturated-isolated to saturated-isobaric regime. If, however, the dispersion and associated dissipation extend across several orders of magnitude in frequency, the simple three-element Zener model with its unique anelastic relaxation time will be inadequate. Cole & Cole (1941), in addressing the deficiency of the Zener model in describing experimental observations of complex dielectric behaviour of polymers, generalized the model to include a distribution of relaxation times rather than a single relaxation time. Such generalized Zener model has been applied in the rock physics literature (e.g. in Spencer 1981; Jones 1986; Batzle et al. 2001; Spencer & Shine 2016).

It often proves desirable to have the flexibility to specify separate distributions of relaxation time associated with monotonically frequency-dependent dissipation and a superimposed broad dissipation peak, along with related modulus dispersion. Accordingly, we use an extended Burgers model of linear viscoelasticity (Jackson & Faul 2010) without the series dashpot—which is essentially an extended standard anelastic solid (ESAS) with a distribution of relaxation times sufficiently flexible to describe both the smoothly monotonic background dissipation and any superimposed dissipation peak (Jackson 2015).

The time-dependent response (deformation), to application of a unit step-function stress at time $t = 0$, is given by the creep function $J(t)$ (e.g. Nowick & Berry 1972; Faul & Jackson 2015). For the ESAS model, with a distribution $D(\ln\tau)$ of relaxation times, the
creep function is specified as

$$J(t) = J_U \left\{ 1 + \Delta \int_{\ln\tau_L}^{\ln\tau_H} D(\ln\tau) \left[ \left(1 - e^{-\frac{t}{\tau}}\right) \right] d\ln\tau \right\}. \quad (4)$$

The response of such an ESAS model to an oscillating time-varying stress is the complex dynamic compliance calculated as the Laplace transform of the creep function (eq. 5, Jackson 2015). The real and negative imaginary parts of the dynamic compliance are combined in expressions (eqs 10 and 11, Jackson 2015) for the modulus and dissipation as functions of angular frequency, $\omega = 2\pi/T_o$, or oscillation period, $T_o$.

For the monotonic background dissipation, we applied a normalized distribution of relaxation times $D_B(\ln\tau)$, introduced by Anderson & Minster (1979) as follows (e.g. Faul & Jackson 2015):

$$D_B(\ln\tau) = \frac{\alpha \tau^\alpha}{\tau_H^\alpha - \tau_L^\alpha}, \quad (5)$$

with $0 < \alpha < 1$ and $\tau_L < \tau < \tau_H$, so that the contribution from $D_B(\ln\tau)$ applies only between the bounds ($\tau_L$, $\tau_H$). From the experimental observations, the terms $\alpha$ and $\tau_L$, $\tau_H$ (respectively, the frequency dependence, and lower and upper bounds of the distribution of relaxation times) and the associated relaxation strength $\Delta_B$, can be determined.

In order to model any dissipation peak superimposed upon such background, an additional separately normalized distribution with relaxation strength $\Delta_P$ can be incorporated (e.g. Jackson & Faul 2010; Faul & Jackson 2015):

$$D_P(\ln\tau) = \frac{1}{\sigma\sqrt{2\pi}} \exp \left\{ \frac{\left[ \ln \left( \frac{\tau}{\tau_P} \right) / \sigma \right]^2}{2} \right\}. \quad (6)$$

Here, $\tau_P$ and $\sigma$, which are the location in relaxation time of the peak centre and its width, and the relaxation strength $\Delta_P$ can be constrained by the experimental data.

# 3 RESULTS

## 3.1 Permeability

Using the transient-flow technique (Section 2.2.1), we have measured the permeability for each of the fluid-saturated conditions of the Wilkeson specimen, as a prelude to testing the seismic properties. Fig. 4 illustrates the typical time dependence of the pore-pressure differential, and the exponential fit, from which we obtained permeability as a function of differential pressure $P_d$. The results obtained using such approach for all the fluid-saturated conditions are presented as a function of differential pressure $P_d$ (confining pressure – pore pressure) in Fig. 5.

The measured permeability decreases by more than an order of magnitude with increase of differential pressure from 5 to 10 MPa for the fluid-saturated Wilkeson sandstone (Fig. 4), signifying how the pressure sensitivity of the specimen's compliant microstructure affects its fluid flow behaviour. The permeability measured with argon as pore-fluid water (~$10^{-17}$ m² at $P_d = 5$ MPa) is lower by about half an order of magnitude than that measured at the same value of differential pressure either water- or glycerine-saturated.

## 3.2 Torsional oscillation: shear modulus and associated dissipation

In Fig. 6, we present a summary of the measurements of shear modulus and dissipation obtained from torsional forced oscillation

for the dry and fluid-saturated Wilkeson specimen. We note that dry and argon-saturated conditions were investigated twice: both before the abortive initial attempt at glycerine saturation and following the ultimately successful successive saturation with glycerine and water (Section 2.2.1). Markedly lower shear moduli measured dry and argon-saturated following glycerine and water saturation (Figs 6c and d) suggest that the crack density was increased by the transient glycerine overpressure during the first attempt at glycerine saturation. We will focus our attention upon the results obtained following successful glycerine saturation.

Glycerine saturation (Fig. 6a) results in a measured shear modulus that is both markedly higher and much more strongly dispersed than that subsequently observed under conditions of water and argon saturation, and dry. The average value for the wider frequency range at 5 MPa confining pressure is 39 per cent higher than for dry conditions (Fig. 6d) with 35 per cent dispersion. Correspondingly, the dissipation is also much more intense at an average value of $Q_G^{-1}$ of 0.062 compared with 0.028 for dry conditions (Fig. 6d).

During the substitution of water for glycerine as pore fluid, the measured modulus and dissipation define trends (labelled ‘10a’, ‘10b’ and ‘10c’ in Fig. 6b) that are transitional between those of glycerine saturation and those characteristic of dry or argon-saturated conditions. Relative to the dry conditions, the shear moduli measured at $P_d = 5$ MPa under argon and water-saturated conditions are actually systematically lower by about 10 and 6 per cent, respectively, than for dry conditions. At 5 MPa differential pressure, both argon- and water-saturated conditions yield more modest modulus dispersion of 7–11 per cent across the wider range of measurement frequencies, broadly comparable with 8 per cent for the more limited frequency range of dry conditions.

Very similar behaviour is observed under argon-saturated and dry conditions (Figs 6c and d). The shear modulus increases systematically with increasing confining (or differential) pressure. For dry conditions, the frequency-averaged shear modulus increases by 31 per cent with increasing confining pressure from 5 to 15 MPa. Secondly, the measured shear modulus shows a mild positive dispersion across the frequency range of the test, which decreases from ~8 per cent at 5 MPa to ~2 per cent at 15 MPa. Thirdly, the dry specimen shows a non-negligible, yet systematically pressure-dependent shear dissipation, whereby the frequency-averaged value of $Q_G^{-1}$ decreases from 0.028 to 0.021 across the pressure range 5–15 MPa.

Results of the shear modulus and dissipation for the dry, argon- and water-saturated conditions, therefore, can be summarized as broadly similar, while that of the glycerine-saturated specimen displays the most remarkable of the mechanical behaviours.

## 3.3 Flexural oscillation: Young's modulus and dissipation

Next, we present the results concerning Young's modulus and associated dissipation obtained using the forced flexural oscillation technique (Fig. 7). As with the shear modulus, we group the results under dry and fluid-saturated conditions for highlighting the key observations concerning the variations with pressure, frequency and fluid saturation. As explained in Section 2.2.2, the modelling of flexural oscillation is constrained to yield a period-averaged Young's modulus for dry Wilkeson sandstone of 20 GPa at 15 MPa pressure.

The most striking feature of the flexural oscillation data is the pronounced dispersion of Young's modulus only for glycerine saturation (Fig. 7a) and in the tests conducted during the replacement of glycerine with water as pore fluid (labelled ‘10a’ and ‘10b’ in Fig. 7b). Close consistency between the final set of measurements
<figure><img src="image_6.png" /><figcaption>**Figure 4.** Representative example of the pore pressure re-equilibration, and the exponential decay of the upper-reservoir pore pressure, following application of a small pressure perturbation in the lower reservoir, for argon-saturated conditions at 7.5 MPa differential pressure. Lower panel shows the fit to the exponential decay, providing the rate constant that is used in the calculation for the permeability (see details in text).</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>**Figure 5.** Pressure-dependent permeability of Wilkeson sandstone specimen, tested with water saturation by Pimienta *et al.* (2017) and with argon, water and glycerine as pore fluid in this study. Measurements with water and argon as pore fluids were performed after the successful saturation with glycerine pore fluid. Estimated uncertainty ~10 per cent.</figcaption></figure>

for water saturation at 10 MPa (labelled ‘10c’) and those subsequently obtained at differential pressures of 7 and 5 MPa suggests that the exchange of water for glycerine was then complete. Following completion of the substitution of water for glycerine, there is in fact broad consistency among the values of Young’s modulus measured water- and argon-saturated, and dry, with a significant

positive pressure dependence. Also noteworthy is the generally unusually high level of dissipation $Q_E^{-1}$ approaching and exceeding 0.2, even for conditions where there is negligible modulus dispersion.

# 4 DISCUSSION

## 4.1 Mechanical behaviour under dry conditions

### 4.1.1 Pressure sensitivity

For the composite Wilkeson sandstone specimen that we have tested, we observed the pressure dependence of both Young’s and shear moduli in dry conditions. We ascribe such pressure sensitivity to the progressive pressure-induced closure of the compliant microstructure (presumably the intergranular contacts and cracks). Walsh’s (1965) expression $P = Ea$ for the closure pressure (with Young’s modulus $E$ of ~15 GPa, measured at 5 MPa) implies aspect ratios $\alpha < 3 \times 10^{-4}$ for the cracks closed by 5 MPa confining pressure. Such indication is broadly consistent with the value of $\alpha \sim 10^{-4}$ that was suggested in Pimienta *et al.* (2017) for the same specimen, and published estimates of aspect ratios in sandstones (e.g. as obtained by the inversion of velocity data for Navajo sandstone and Berea sandstone by Cheng & Toksoz 1979).

Comparison of the shear moduli measured under dry and argon-saturated conditions prior to the abortive first attempt at glycerine saturation with those made under the same conditions following glycerine and water saturation (Figs 6a and b) indicates a marked
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Frequency-dependent shear modulus and dissipation of the Wilkeson sandstone specimen, obtained from forced-oscillation experiments. The test was performed at a range of confining pressures ($P_c$) under dry conditions and at 5–10 MPa differential pressures, $P_d$ ($P_c - P_f$) for the argon-, glycerine- and water-saturated conditions. Labels 10a, 10b, 10c in panel (b) indicate successive measurements obtained at 10 MPa $P_d$ during the pore fluid replacement of glycerine with water. Open symbols labelled ‘initial’ in panels (c) and (d) denote measurements conducted prior to the initial abortive attempt at glycerine saturation, whereas solid symbols in all panels represent data subsequently obtained successively with glycerine, water and argon saturation, and dry. Representative error bars indicated at $log_{10}(freq, Hz) = -2$.</figcaption></figure>

reduction averaging 22 per cent for dry conditions—presumably the result of microcracking caused by locally transient fluid overpressure during the first attempt at glycerine saturation.

### 4.1.2 Modulus dispersion and dissipation

We observed mild dispersion, associated with a non-negligible dissipation ($Q_G^{-1}$) for dry conditions of the Wilkeson sandstone specimen. (Fig. 6). Such dispersion of shear modulus $G$ and dissipation associated with the ‘dry’ condition might be related to the presence of adsorbed moisture in the specimen especially at frictional contacts within the compliant microstructure. Any compliance of the interfaces within the experimental assembly, each loaded by the differential pressure, may also be a contributing factor.

That such mechanical properties measured under our ‘dry’ conditions are related to the effect of moisture in the pore space (e.g. Clark *et al.* 1980; Vigil *et al.* 1994; Pimienta *et al.* 2014; Yurikov *et al.* 2018) is in fact highly likely given the difficulty of achieving complete dehydration of this clay-bearing sandstone specimen. The effect of such moisture adsorbed on the surfaces of quartz grains is to reduce the surface energy, resulting in reduced moduli and associated dissipation when stressed at appropriate frequencies

(Pimienta *et al.* 2014; Yin *et al.* 2019). The water-sensitive clay layers might also play a role. For instance, the shear modulus under water-saturated conditions at 5 MPa tends to be slightly lower than those of the dry conditions, suggesting some possible clay-related shear weakening effect, which is consistent with several other published observations on the mechanical properties of clay-bearing sandstones (Han *et al.* 1986; Murphy *et al.* 1986; Yin *et al.* 2019).

The much higher levels of dissipation evident in the results for flexural oscillation, even dry and argon-saturated, require additional consideration. It proved impossible to simultaneously fit both the Young’s modulus dispersion and associated dissipation to a model of the extended standard anelastic solid type (Section 2.4). Instead, the background-only model that best fits the modulus dispersion (Fig. 8a) grossly underestimates the observed dissipation (Fig. 8b). The difference between the observed and modelled dissipation, termed ‘excess dissipation’, varies systematically with frequency $f$ approximately as $a + b \log f$ (Fig. 8c).

In order to understand the origin of this substantial excess dissipation, we need to review the way in which the specimen assembly is located within the pressure vessel. For unhindered flexural oscillation, the multicomponent experimental assembly with a total length >1 m (Figs 2b and c) must be aligned within ±1.5 mm to
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Young's modulus *E* and associated dissipation *Q*<sub>E</sub><sup>-1</sup> for Wilkeson sandstone tested in flexural oscillation, successively glycerine-, water-, and argon-saturated, and finally 'dry', labelled with the differential or confining pressure as appropriate. The values of *E* and *Q*<sub>E</sub><sup>-1</sup> were obtained by finite-difference modelling of the flexural oscillation of the specimen assembly with the finite-difference filament elongation model (Section 2.2.2).</figcaption></figure>

avoid contact with the close-fitting compound pressure vessel. The spiral steel vent tube, fed through the lower closure of the pressure vessel to connect with the lower pore-fluid reservoir, has an important influence on the overall alignment of the assembly because of its potential to impose a near-axial location for the lower end of the assembly. As noted in Section 2.2.2, it was necessary during the course of this study to replace the vent tube. The significantly less stiff replacement vent tube was repeatedly reshaped in order to achieve alignment compatible with values of the flexural compliance within the normal range. However, residual grazing contact between the assembly and the pressure vessel at the lower end provides a plausible explanation for the excess strain-energy dissipation of similar magnitude under both dry and fluid-saturated conditions, as follows.

The finite-difference filament-elongation model, used to relate the observed complex flexural compliance $S_{NF}$ of the specimen assembly to the Young's modulus and dissipation of the specimen, has been modified to include the effect of a secondary force of amplitude $F_G$ applied at the top of the model vent tube. When assigned a phase angle of $\pi/2$ relative to the applied bending force, the grazing force results in maximal increase in the assembly phase lag, along with minimal impact upon the compliance. With such optimal phase, a grazing force $F_G$ with an amplitude $\sim 10$ per cent of the applied bending force is sufficient to explain the excess dissipation. On the assumption that the excess dissipation attributable to the grazing force is common to dry and fluid-saturated conditions, we subtract the excess dissipation observed under dry conditions from

that measured with fluid saturation. By so doing, we are emphasizing the difference in flexural behaviour between fluid-saturated and dry conditions, rather than the absolute values of dissipation. The fact that the excess dissipation measured dry in flexure decreases markedly with increasing pressure (Fig. 8c) suggests a systematic reduction in the amplitude of the grazing force with increasing pressure, perhaps reflecting diminishing normal force at the grazing contact between the assembly and the pressure vessel.

## 4.2 Permeability

The Wilkeson sandstone specimen that we have tested is the smaller-diameter version of the one tested by Pimienta *et al.* (2017). The permeabilities, newly measured with water as pore fluid at $P_d$ = 5 MPa, are higher by almost an order of magnitude than those obtained by Pimienta *et al.* (2017) on the same specimen but of larger diameter, but are comparable at 10 MPa on account of the stronger pressure dependence of the permeability measured in this study. Permeabilities ranging between $5 \times 10^{-18}$ and $1 \times 10^{-17} \text{ m}^2$ for pressures of 10–100 MPa, broadly consistent with those of Pimienta *et al.* (2017), have been reported for multiple specimens of Wilkeson sandstone by Ahrens *et al.* (2018). Some of this variation may reflect the differences in the measurement methodology. We used the transient-flow approach whereas the Darcy (steady-flow) method was employed by Pimienta *et al.* (2017) and an oscillating pore-pressure technique by Ahrens *et al.* (2018). However, the reason for the substantial difference between the permeabilities measured
<figure><img src="image_10.png" /><figcaption>**Figure 8.** The implications of fitting a background-only model of the extended anelastic solid type to the observed dispersion of the Young's modulus for the Wilkeson sandstone specimen tested under dry conditions following successive saturation with glycerine, water and argon. (a) Data indicated by plotting symbols and ESAS model by curves, both colour-coded for pressure. (b) Comparison of the observed dissipation (plotting symbols) with that associated with the ESAS model fitted to the modulus dispersion only (curves). (c) The 'excess' dissipation $d(1/Q)$, fitted to a function of the form $dQ^{-1}$, $= a + b \log f$, is used in the following analysis to apply a correction to the dissipation measured under conditions of fluid saturation.</figcaption></figure>

under conditions of argon/water and glycerine saturation remains unclear. Overall, the permeability is significantly higher than anticipated at the lowest pressure and more strongly pressure sensitive. The higher permeability, along with the reduction in shear modulus discussed in Section 4.1.1, is consistent with the suggestion that the crack density and permeability were significantly increased by the transiently high pore pressure realized during the initial aborted attempt at glycerine saturation.

## 4.3 Mechanical behaviour under fluid-saturated conditions

### 4.3.1 Fluid-flow regimes

The frequency-dependent mechanical behaviour of a fluid-saturated porous medium is expected to be comprised of a series of regimes, each defined by the spatial scale on which stress-induced gradients in pore pressure can be relaxed by fluid flow. With decreasing frequency, the saturated-isolated (unrelaxed), saturated-isobaric (undrained) and drained regimes are successively expected (e.g. Gassmann 1951; Mavko & Nur 1975; O'Connell & Budiansky

1977; Cleary 1978). In the following discussion, we use the theoretical framework of O'Connell & Budiansky (1977) for interpretation of the results, with one important modification. During flexural oscillation of a fluid-saturated specimen, a gradient in the local average pore pressure will develop between that side of the specimen that is instantaneously under compression and the other side under tension. Such gradients in pore pressure can be relaxed on appropriate timescales by lateral fluid flow.

At sufficiently high frequency of stress application, there is insufficient time for fluid flow between differentially compliant/pressurized parts of the pore space, precluding even grain-scale pore pressure re-equilibration in a regime appropriately termed saturated isolated (O'Connell & Budiansky 1977) or unrelaxed (Guéguen & Kachanov 2011). Within the saturated-isolated regime, the incompressibility of the fluid results in a higher shear modulus than if pore-pressure gradients were relaxed by grain-scale fluid flow (Endres & Knight 1997; Adelinet *et al.* 2011), and for a distribution of crack/pore aspect ratios, also a higher bulk modulus.

With decreasing frequency from the saturated-isolated regime, pore-pressure equilibration between cracks and pores, and between cracks of different orientation, first occurs locally by grain-scale
squirt flow. This establishes a locally uniform pore pressure, but in flexural oscillation, the locally uniform pore pressure established by such fluid flow potentially varies spatially across the diameter of the specimen. With further decrease in frequency, lateral fluid flow begins to eliminate pore pressure differentials initially on scales only somewhat greater than the grain size, but eventually across the entire diameter. Only then do conditions become genuinely, that is, globally, isobaric. Thus, the transition from saturated-isolated to saturated-isobaric conditions with decreasing frequency of flexural oscillation occurs in two conceptually distinct stages. However, for the Wilkeson sandstone specimen with its substantial equant porosity, local squirt flow between cracks of low aspect ratio and adjacent pores will be sufficient to achieve effectively isobaric conditions. Accordingly, for this specimen, no significant modulus dispersion and dissipation is expected from lateral fluid flow.

Under saturated-isobaric conditions, fluid flow enables the perturbed pore pressure to return to equilibrium conditions between differentially pressurized parts of the pore space (of different orientation relative to the applied stress field and/or different aspect ratio) but without an exchange of fluid with an external reservoir. The Biot-Gassmann equations of poroelasticity describe the mechanical properties (bulk and shear moduli) of fluid-saturated porous media under such saturated-isobaric conditions.

Finally, for drained conditions, the perturbation in uniform pore pressure within the pore space of the specimen, induced by any change in volume of the bulk specimen, is fully relaxed by the fluid exchange between the specimen's interior and an external reservoir of sufficient volume. So the bulk and shear moduli are unaffected by the presence of the fluid, and the specimen behaves as if it were dry.

These conceptually distinct fluid-flow regimes (O'Connell & Budiansky 1977) are separated by characteristic frequencies: the squirt flow frequency $f_{sq}$ for grain-scale fluid flow between adjacent differentially pressurized parts of the pore space, which is also the upper bound for lateral fluid flow in flexure, the lower bound $f_{lt}$ for lateral fluid flow, that is, for fluid flow across the specimen diameter during flexural oscillation, and the drainage frequency $f_{dr}$ for fluid exchange between the stressed specimen and an external reservoir. As explained in Section 1, draining is not relevant for our torsional and flexural mode tests as they involve no net change in volume of the specimen. However, the lower bound $f_{lt}$ on the characteristic frequency for lateral fluid flow is estimated from that for draining (Cleary 1978) by substitution of the radius $r$ of the specimen cross-section for its half-length, that is,

$$f_{lt} = \frac{k K_f}{\phi \eta r^2}, \qquad (7)$$

where $k$, $\phi$, $K_f$ and $\eta$ are the permeability and porosity, and fluid bulk modulus and viscosity, respectively. For squirt flow, the characteristic frequency can be given as (O'Connell & Budiansky 1977)

$$f_{sq} = \frac{K \alpha^3}{2\pi \eta}, \qquad (8)$$

where $K$ is the bulk modulus of the medium, $\alpha$ is crack aspect ratio and $\eta$ is the dynamic viscosity of the pore fluid. The frequency $f_{sq}$ is also the upper bound for the lateral fluid flow in flexure.

For the water-saturated specimen of Pimienta et al. (2017) of length $L = 80$ mm and diameter 40 mm, draining was observed at 0.3 Hz. For fixed porosity and permeability, the draining and lateral fluid-flow frequencies scale as $K_f/(\eta x^2)$ with $x = L/2$ and $x = r = 7.5$ mm, respectively. Thus, the lower bounds $f_{lt}$ for lateral

<table><caption>**Table 1.** Expected characteristic frequencies for lateral flow (lower bound) and squirt flow for each of the fluid-saturated conditions (at 5 MPa pore pressure) based on characteristic frequencies ($f_{dr} = 0.3$ Hz and $f_{sq} = 30$ Hz) for the water-saturated conditions tested by Pimienta et al. (2017). For this purpose, it is assumed that the permeability (and porosity) is unchanged from the study by Pimienta et al. (2017). The effect of higher crack density and permeability resulting from transient pore-fluid overpressure during the first aborted attempt at glycerine saturation is discussed in Section 4.4. Values of the fluid bulk modulus $K_f$ (GPa) and viscosity $\eta$ (mPa s) are (0.005, 0.03), (2.25, 1) and (4.46, 1300), for argon, water and glycerine, respectively. The radius $r$ of the specimen of this study is 7.5 mm.</caption><thead><tr><th rowspan="2">Pore fluid</th><th colspan="2">Characteristic frequency (Hz)</th></tr><tr><th>Lateral flow ($f_{lt}$)</th><th>Squirt flow ($f_{sq}$)</th></tr></thead><tbody><tr><td>Argon</td><td>0.63</td><td>1000</td></tr><tr><td>Water</td><td>8.53</td><td>30</td></tr><tr><td>Glycerine</td><td>0.013</td><td>0.023</td></tr></tbody></table>

flow in the variously saturated specimen of this study are estimated from the draining frequency observed for water saturation by Pimienta et al. (2017) (Table 1). For squirt flow, the characteristic frequency $f_{sq}$ of a given medium (i.e. of fixed $K$ and $\alpha$) scales simply as $1/\eta$ (Table 1). Note that for our glycerine saturation, $f_{lt}$ is only marginally lower than $f_{sq}$, and that any need for lateral fluid flow to achieve specimen-wide isobaric conditions applies only to the flexural mode of forced oscillation.

### 4.3.2 Fluid sensitivity of seismic properties

The low-frequency mechanical properties for the Wilkeson sandstone specimen under various conditions of fluid saturation are summarized in Figs 6 and 7. For water- and argon-saturated conditions, the measured moduli are broadly comparable with those for dry conditions, with one important exception. During the replacement of glycerine with water as pore fluid, the behaviour of the nominally water-saturated specimen is transitional between that for glycerine saturation and the behaviour common to fully water-saturated, argon-saturated and dry conditions (Figs 6c and 7b).

For the glycerine-saturated conditions, both shear and Young's moduli are systematically higher than for dry conditions (Figs 6d and 7a) and increase markedly with increasing frequency. Such dispersion is potentially attributable to the presence of glycerine as a relatively incompressible and viscous pore fluid. As noted previously, the torsional and flexural oscillation tests involve no change in volume of the whole specimen, and therefore there is no driving force for drainage between the specimen and the external reservoirs. Also, as noted in Section 4.3.1, grain-scale fluid flow between cracks of low aspect ratio and adjacent pores in the specimen will be sufficient to achieve effectively isobaric conditions, obviating the need for significant lateral fluid flow. Accordingly, the following discussion seeks to relate the observed dispersion and dissipation to the characteristic frequency for squirt flow with emphasis on the results obtained at the lowest differential pressure of 5 MPa.

Optimal models of the background-only and peak-only types were fitted to the constraining torsional oscillation data for glycerine-saturated conditions, but only the ESAS peak-only fits are displayed in Fig. 9(a). The alternative background-only and peak-only viscoelastic models each provide a reasonable representation of the torsional oscillation data within the observational window but diverge significantly at lower and higher frequencies. However, the peak-only model is preferred for consistency with the fitting of the flexural oscillation described below. The fact that the models fit quite well both the shear modulus and associated dissipation
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Shear modulus (a) and Young's modulus (b) and associated dissipations from the torsional and flexural oscillation tests, for the glycerine-saturated Wilkeson sandstone specimen. (c) The bulk modulus and associated dissipation derived by combining the complex shear and Young's moduli. The filled circles, open triangles and squares represent the data at differential pressures $P_d$ of 5, 7 and 10 MPa, respectively. The curves represent the peak-only ESAS model fitted simultaneously to the observed dispersion and the dissipation (for flexural oscillation, adjusted for excess dissipation as described in Section 4.1.2). The dashed lines labelled '$\omega\tau_p = 1$' denote the centre frequencies of the dissipation peaks as modelled for $P_d = 5$ MPa.</figcaption></figure>

data is an illustration of the quantitative consistency between the substantial dispersion across the measurement frequency interval of 0.001–0.3 Hz and the measured level of dissipation (+22 per cent dispersion and average $log(Q^{-1}) = -1.22(\pm0.01)$ for $P_d = 5$ MPa). Such consistency is expected from the Kramer–Kronig integral relations of linear viscoelasticity. Although the dissipation data reveal no distinct peak, a reasonable fit can be obtained with a viscoelastic model having a broad dissipation peak with logarithmic width parameter $\sigma = 6$ centred at $log \tau_p = 1.3$ (Table 2). However, alternative values of $\sigma$ as low as 4 involve only a modest increase in misfit.

For the Young's modulus and associated dissipation, we previously noted that unusually strong dissipation is observed in flexural oscillation for both dry and the variously fluid-saturated conditions, and that the dissipation measured under dry conditions is much greater than that expected from the observed modulus dispersion. Accordingly, on the assumption that the excess dissipation is associated with residual grazing contact between the assembly and pressure vessel, all flexural dissipation data were corrected downwards by $dQ_E^{-1} = a + b\log f(\text{Hz})$ as explained in Section 4.1.2. It is demonstrated in Fig. 9(b) that the flexural dissipation data for the glycerine-saturated specimen thus adjusted are quantitatively compatible through the peak-only ESAS model with the observed strong dispersion of the Young's modulus for glycerine-saturated conditions.

The measured values of the complex shear modulus $G^*$ and Young's modulus $E^*$ have been combined in the usual way to estimate the corresponding value of the bulk modulus $K^*$ from

$$1/E^* = 1/(3G^*) + 1/(9K^*). \qquad (9)$$

Both the bulk modulus $K$ and associated dissipation $Q_K^{-1}$ thus derived for $\log f < -1.3$ are presented in Fig. 9(c). At higher frequencies, the values of $Q_K^{-1}$ thus estimated are (unphysically) negative—plausibly the result of uncertainties in adjusting the flexural oscillation data for the excess dissipation. Compared with the other moduli, the bulk modulus is even more strongly frequency-dependent and such dispersion is associated with a pronounced dissipation peak centred near $\log f (\text{Hz}) = -2.5$.

### 4.3.3 Effective frequency for squirt flow

Because the characteristic frequency (eq. 5) for squirt flow varies inversely with fluid viscosity, we have further analysed the variations of modulus and dissipation for saturation with a relatively incompressible fluid (i.e. water or glycerine) by using an effective frequency $f^* = (\eta/\eta_w)f$, where $\eta$ and $\eta_w$ are, respectively, the viscosities of the pore fluid and water, and $f$ is the experimental oscillation frequency (cf. Batzle et al. 2006; Pimienta et al. 2015). The optimal ESAS background-plus-peak models, fitted simultaneously to the
<table><caption>**Table 2.** Optimal models of the extended standard anelastic solid type (Section 2.4) fitted simultaneously to the variations of modulus and dissipation with oscillation period for the lowest differential pressure $P_d$ of 5 MPa. Values of selected parameters were refined by the Levenberg-Marquardt scheme for iterative non-linear least-squares fitting to minimize the objective function $\chi^2 = \chi_M^2 + \chi_Q^2$. Values in parentheses indicate the uncertainties in the last digit of the refined value, whereas parameter values in square brackets were held constant. ‘Background + peak’ models are denoted ‘bg + pk’, whereas peak-only models are denoted ‘pk’.</caption><thead><tr><th rowspan="3">Parameter, unit</th><th colspan="2">Glycerine saturation</th><th colspan="3">Glycerine and water saturation</th></tr><tr><th>G & Q<sub>G</sub><sup>-1</sup></th><th>E & Q<sub>E</sub><sup>-1</sup></th><th>G & Q<sub>G</sub><sup>-1</sup></th><th>E & Q<sub>E</sub><sup>-1</sup>(adj)</th><th>K & Q<sub>K</sub><sup>-1</sup></th></tr><tr><th>pk</th><th>pk</th><th>bg + pk</th><th>bg + pk</th><th>bg + pk</th></tr></thead><tbody><tr><td>N, (M, logQ<sub>M</sub><sup>-1</sup>) pairs</td><td>9</td><td>9</td><td>18</td><td>18</td><td>11</td></tr><tr><td>P<sub>d</sub>, MPa</td><td>5</td><td>5</td><td>5</td><td>5</td><td>5</td></tr><tr><td>G<sub>U</sub>/E<sub>U</sub>/K<sub>U</sub>, GPa</td><td>14.7(3)</td><td>28.8(4)</td><td>11.9(2)</td><td>26.0(3)</td><td>16.6(3)</td></tr><tr><td>Δ<sub>B</sub></td><td>(0)</td><td>(0)</td><td>(0.6)</td><td>1.1(3)</td><td>2.1(6)</td></tr><tr><td>α</td><td></td><td></td><td>(0.25)</td><td>0.39(6)</td><td>0.36(8)</td></tr><tr><td>log(τ<sub>LR</sub>, s)</td><td></td><td></td><td>(-5.0)</td><td>(-5.0)</td><td>(-5.0)</td></tr><tr><td>log(τ<sub>HR</sub>, s)</td><td></td><td></td><td>(5.0)</td><td>(5.0)</td><td>(5.0)</td></tr><tr><td>Δ<sub>P</sub></td><td>0.86(5)</td><td>0.35(3)</td><td>0.48(2)</td><td>0.45(2)</td><td>1.11(5)</td></tr><tr><td>log(τ<sub>PR</sub>, s)</td><td>1.2(4)</td><td>1.8(1)</td><td>-2.20(6)</td><td>-1.26(3)</td><td>-0.89(2)</td></tr><tr><td>σ</td><td>(6)</td><td>1.7(2)</td><td>(3)</td><td>1.42(7)</td><td>0.55(10)</td></tr><tr><td>χ<sub>M</sub><sup>2</sup></td><td>0.29</td><td>0.9</td><td>156.2</td><td>212.7</td><td>325.0</td></tr><tr><td>χ<sub>Q<sup>-1</sup></sub><sup>2</sup></td><td>2.95</td><td>19.0</td><td>32.5</td><td>92.1</td><td>215.8</td></tr><tr><td>χ<sub>T</sub><sup>2</sup></td><td>3.24</td><td>19.9</td><td>188.7</td><td>304.8</td><td>540.8</td></tr><tr><td>(χ<sub>T</sub><sup>2</sup>/2 N)<sup>1/2</sup></td><td>0.42</td><td>1.05</td><td>2.29</td><td>2.91</td><td>4.96</td></tr></tbody></table>

variation with effective frequency of the modulus and dissipation for both water- and glycerine-saturated conditions, evidently provide a broadly satisfactory description of the data (Fig. 10). The comparisons between model and data presented in Fig. 10 emphasize our key observation concerning the frequency-dependent mechanical behaviour of the Wilkeson sandstone: the marked contrast between strongly dispersive and dissipative behaviour for glycerine saturation and the generally much milder dispersion and dissipation at sub-Hz frequencies for water-saturated conditions (and also dry and argon-saturated conditions). We interpret the frequency-dependent mechanical behaviour to reflect squirt flow across several orders of magnitude of effective frequency. Through eq. (5) for the characteristic frequency for squirt flow, we can associate squirt flow across this range of effective frequency with aspect ratios ranging over an order of magnitude from ~0.5 × 10⁻⁴ on the low-frequency side of the inferred dissipation peak ($f^* = 10^{-0.5}$ Hz) to ~5 × 10⁻⁴ on the high-frequency side of the peak ($f^* = 10^{2.5}$ Hz).

The broadly similar ranges of frequency/effective frequency for poroelastic relaxation in shear and flexure are consistent with the notion that the effect of squirt flow dominates over lateral fluid flow. As noted in Section 4.3.1, poroelastic relaxation associated with lateral fluid flow is not expected to play a significant role in this specimen, because local squirt flow between cracks and pores will realize effectively isobaric conditions.

Forced oscillation data obtained under conditions of water and glycerine saturation are thus reconciled through the use of an effective frequency that scales with fluid viscosity. The pattern that is revealed by both torsional oscillation and flexural oscillation tests involves superposition of a squirt-related peak upon a poorly resolved background that increases with decreasing frequency. When the complex shear and Young’s moduli are combined to estimate the bulk modulus, it too shows strong dispersion and an associated dissipation peak. The observation of strong dispersion and dissipation in both shear and hydrostatic compression is consistent with a dominant role for stress-induced fluid flow between cracks and pores. The background dissipation, increasing with decreasing frequency, might reflect sliding on grain boundaries under the prevailing conditions of relatively low normal stress.

## 4.4 Comparison with previous studies

In their study of water-saturated Wilkeson sandstone, Pimienta et al. (2017) reported the findings of their axial stress oscillation experiments as Young’s modulus (Fig. 11b) and Poisson’s ratio and the associated reciprocal quality factors (1/Q). However, it is instructive to separate the contributions of the hydrostatic and shear components of the deformation. For this purpose, the complex bulk $K^*$ and shear moduli $G^*$ are calculated from the complex Young’s modulus $E^*$ and Poisson’s ratio $\nu^*$ through the standard relationships for isotropic elasticity:

$$K^* = E^*/3(1 - 2\nu^*) \text{ and } G^* = \frac{\nu^*}{2(1 + \nu^*)}. \qquad (10)$$

The advantage of this alternative representation is that the dissipation and dispersion associated with the draining and squirt transitions are more clearly separated (Fig. 11). In Fig. 11(c), two dissipation peaks, each with $\log Q_K^{-1} > -1$ and associated with strong bulk modulus dispersion, are observed at frequencies near 0.3 and 30 Hz ($\log f \sim -0.5$ and $+1.5$), the frequencies associated by Pimienta et al. (2017) with the draining and squirt relaxations. That the former is associated with draining is clear from the associated substantially positive values of the pseudo-Skempton coefficient $\gamma$, confirming pore-pressure oscillations within the finite dead volume of the pore-fluid system associated with fluid flow between the specimen and the external reservoir (Fig. 11d). In fact, the data presented in Fig. 11(d) suggest that draining extends to frequencies as high as 3 Hz, consistent with a location nearer $\log f = 0$ (i.e. Hz) for the centre of the dissipation peaks ($Q_E^{-1}$ and $Q_K^{-1}$) ascribed to draining. In marked contrast, the behaviour in shear involves only a single major relaxation (clearly evident in both dissipation and dispersion) centred near $\log f = 1.5$. The absence of any significant impact on the shear modulus of the transition between saturated-isobaric and drained conditions is expected (Gassmann 1951), and the observed relaxation was plausibly attributed by Pimienta et al. (2017) to stress-induced squirt flow.

Because the characteristic frequency for squirt flow varies inversely with viscosity, the observations of Pimienta et al. (2017) suggest a frequency of 0.023 Hz (i.e. ~40 s period) (Table 1) for the
<figure><img src="image_12.png" /><figcaption>**Figure 10:** Frequency-dependent moduli and corresponding dissipations for water (i.e. triangles)—and glycerine (i.e. dots)—saturated conditions of the Wilkeson specimen. The symbols indicate the laboratory data points, whereas the curves represent the optimal background + peak ESAS model fitted to the observed dispersion and dissipation behaviour. (a) Shear modulus and associated dissipation. (b) Young's modulus and associated dissipation. (c) Bulk modulus and associated dissipation inferred by combining the measured complex shear and Young's moduli. For panels (b) and (c), the dissipation inferred from the flexural oscillation data has been adjusted as explained in the text. The dashed lines labelled '$\omega*\tau_p = 1$' denote the effective frequencies of the dissipation peak centres as modelled for $P_d = 5$ MPa.</figcaption></figure>

squirt transition for glycerine-saturated conditions, providing the motivation for this study. Moreover, because the torsional and flexural oscillation tests employed in this study involve no net change in volume of the specimen, there are no fluid pressure gradients established between the specimen and external pore-fluid reservoir, and accordingly, there is no driving force for draining.

The results of our interlaboratory study of the same fluid-saturated Wilkeson specimen are broadly consistent with such expectations. Under glycerine-saturated conditions, we observe a single dissipation peak superimposed upon a dissipation background along with pronounced modulus dispersion. Figs 10(a) and (b) reveal strong dispersion and dissipation for glycerine-saturated conditions at an effective frequency of 30 Hz, as expected from the study of Pimienta *et al.* (2017) under conditions of water saturation. However, it is also clear from comparison of Figs 10 and 11 that the dissipation peak observed in our study is substantially broader than that of Pimienta *et al.* (2017), with strong dissipation and dispersion extending to markedly lower effective frequencies. Relaxation by squirt flow extending across the broader range ~0.3–300 Hz of effective frequency requires a broader distribution ~($0.5–5) \times 10^{-4}$ of crack aspect ratios than inferred in the previous study on the same specimen by Pimienta *et al.* (2017).

Some modification of the controlling microstructure during the multiple stages of fluid saturation is plausible given the presence in the rock specimen of a significant clay fraction and the potential microcracking caused by transiently high pore pressure during the first aborted attempt at charging the specimen with glycerine. Concerning the latter issue, our measurements indicate a systematically higher permeability than that of Pimienta *et al.* (2017) but converging towards similar values for $P_d$ > 10 MPa (Fig. 5). If it is therefore concluded that the additional cracks are closed by 10 MPa differential pressure, their aspect ratios must be less than $P_c/E < 10$ MPa/30 GPa ~ $3 \times 10^{-4}$) which is comparable with an aspect ratio of $2 \times 10^{-4}$ for the 30 Hz squirt frequency of Pimienta *et al.* (2017) for water saturation. The range of generally lower, but also somewhat higher, aspect ratios inferred for the newly introduced microcracks through eq. (6) might thus explain the broadening of the squirt peak mainly towards lower frequencies.

## 5 CONCLUDING SUMMARY

Motivated by the results of Pimienta *et al.* (2017) concerning dispersion and dissipation in a water-saturated specimen of the relatively impermeable Wilkeson sandstone, we have measured its
<figure><img src="image_13.png" /><figcaption>**Figure 11.** The results and implications of the experiment by Pimienta *et al.* (2017) involving oscillation of the axial stress imposed upon the specimen of water-saturated Wilkeson sandstone from which the specimen of this study was cored (Section 2.1). The Young's modulus and associated dissipation, and pseudo-Skempton coefficient *γ*, reported by Pimienta *et al.* (2017) for a differential pressure of 5 MPa are displayed in panels (b) and (d), respectively. The corresponding values of the shear and bulk moduli and associated dissipations, derived as explained in the text from *E*\* and *ν*\*, are displayed in panels (a) and (c), respectively. The frequencies of 0.3 and 30 Hz associated by Pimienta *et al.* (2017) with the draining and squirt relaxations, respectively, are indicated by the dashed vertical lines. To facilitate comparison with the results of this study, the same scales are used for the *x*- and *y*-axes as in Fig. 10.</figcaption></figure>

low-frequency seismic properties under dry and argon-, water- and glycerine-saturated conditions. We used the ANU attenuation apparatus to investigate the same specimen in both torsional and flexural oscillation at frequencies between 0.001 and 1 Hz. Shear modulus and dissipation were directly obtained from the torsional oscillation test while the flexural oscillation test provides Young's modulus and the associated dissipation.

The results show a marked increase in both shear and Young's moduli for the dry specimen with increasing pressure, indicative of progressive closure of the compliant intergranular contacts. The permeability, measured under argon-, water- and glycerine-saturated conditions with the transient-flow technique, decreases markedly with increasing differential (confining – pore) pressure to 10 MPa, consistent with pressure-induced closure of cracks of low aspect ratio. Shear and Young's moduli measured under argon- and water-saturated conditions show no significant stiffening or dispersion and are broadly similar to those of the dry conditions, suggesting that such measurements probe saturated isobaric conditions. In marked contrast, under glycerine-saturated conditions, the measured moduli

decrease systematically with decreasing frequency. This behaviour is interpreted as the transition from saturated-isolated to saturated-isobaric conditions. We modelled this transition with appropriate log-normal distributions of relaxation times. These findings provide additional insight into the squirt flow mechanism for poroelastic relaxation and its role in linking conventional ultrasonic data with the lower wave speeds of low-permeability media at seismic frequencies.

## ACKNOWLEDGMENTS

Technical support by Hayden Miller is gratefully acknowledged.

## DATA AVAILABILITY

The laboratory data supporting this paper are available through the ANU Open Research Repository in the PhD thesis of the first author (https://doi.org/10.25911/T3C3-M673) and supplementary information (https://hdl.handle.net/1885/733731467).
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