Seismic and ultrasonic frequency measurements on partially saturated rocks under X-ray

SUMMARY

Low-frequency laboratory measurements provide direct access to the elastic properties of samples within the seismic frequency band, offering calibration data for seismic survey analysis. Additionally, µCT imaging can quantify actual saturations and provides insights into phase distributions at the pore scale. To conduct laboratory triaxial measurements at seismic frequencies while simultaneously imaging the rock interior, we developed an X-ray transparent low-frequency apparatus. Our apparatus determines rock mechanical properties at seismic frequencies (0.5–150 Hz) and strain amplitudes (10|$\mathrm{^{-7}}$|–10|$\mathrm{^{-5}}$|⁠), measuring Young’s modulus, Poisson’s ratio and attenuation. In addition P- and S-wave velocities at ultrasonic frequencies are measured. We conducted imbibition-drainage experiments to assess the effect of saturation and patch size on seismic and ultrasonic elastic properties in sandstone. Additional tests with liquid and gaseous |$\mathrm{CO_2}$| reveal the impact of partial |$\mathrm{CO_2}$|-gas saturation. The imbibition-drainage experiment demonstrated that P-wave velocity at ultrasonic frequencies was elevated during drainage and reduced during imbibition. Drainage caused patchy saturation, while imbibition resulted in uniform saturation. This implies that ultrasonic measurements, with wavelengths comparable to the pore fluid patch size, are likely influenced by scattering. In contrast, low-frequency measurements, where the wavelength surpasses the patch size, capture effective medium properties and therefore are not affected by scattering effects. The results of the |$\mathrm{CO_2}$| test suggest that low-frequency measurements can detect even low gas saturations (4 per cent gaseous |$\mathrm{CO_2}$|⁠). In contrast, ultrasonic velocity measurements primarily reflect the response of the fully saturated sample at low gas saturations and do not indicate a reduction in velocity. Identifying fluid–solid interactions and estimating saturation via µCT imaging is crucial, especially with minimal gas presence. Our combined approach allows precise determination of elastic properties at seismic frequencies and shows the importance of low-frequency over ultrasonic measurements.

1 INTRODUCTION

One of the critical challenges in carbon capture and storage (CCS) is to ensure that the |$\mathrm{CO_2}$| is stored safely in the reservoir over a long time. The approach is to monitor the subsurface with seismic surveys that measure the change in seismic wave propagation as the amount of stored |$\mathrm{CO_2}$| increases. This is a well-established tool to monitor high |$\mathrm{CO_2}$| concentrations. To effectively bridge the gap between field-scale seismic monitoring and geomechanical modelling, the development of comprehensive rock physics models is essential. However, the development of a reliable rock physics model for CCS remains an ongoing challenge (Furre et al. 2017). This challenge is particularly critical when accessing the regime of low |$\mathrm{CO_2}$| concentrations. Addressing this issue requires a thorough investigation of mechanical behaviours at the laboratory scale and an understanding of the physical mechanisms governing |$\mathrm{CO_2}$|-water replacement. Integrating these insights into rock physics models is pivotal and enhances the precision of |$\mathrm{CO_2}$| monitoring.

Over several decades, rock physics models have evolved to incorporate diverse mechanisms. White (1975), Dutta & Ode (1979b), Jones (1986) and Pride et al. (2004) introduce a parameter for length scale, linked with patch size, to describe wave-induced local flow and its impact on velocities, seismic dispersion (meaning the change in propagation velocity with frequency) and attenuation. In White (1975), the focus is on porous rocks predominantly filled with water but interspersed with gas-filled pockets. The approach of Dutta & Ode (1979b, a) builds upon White’s model by implementing Biot (1962)’s systematic equations for poroelasticity, correlating velocity and attenuation with wave frequency, gas saturation and the spacing between gas pockets. Dutta & Seriff (1979) highlight the likelihood of high attenuation at low gas saturations. Other models, such as those by Mavko & Nur (1979), O’Connell & Budiansky (1977) and Dvorkin et al. (1995), integrate squirt flow between compliant cracks and less compliant pores. Mavko et al. (2009) reconcile the effective bulk modulus at low and high frequencies using Reuss and Voigt averages, respectively. Onuki (1991), Tisato et al. (2015) and Chapman et al. (2017) discuss wave-induced dissolution–exsolution effects. The introduction of gas, as noted in Krevor et al. (2011), Tanino & Blunt (2012) and Pak et al. (2015), results in patchiness. More contemporary models, like those cited in Brunner & Spetzler (2001), Papageorgiou et al. (2016) and Rozhko & Bauer (2019), incorporate capillary effects. Papageorgiou & Chapman (2017) demonstrate that capillary effects explain deviations from the Gassmann model at low frequencies, addressing both capillary and squirt flow effects. In a novel analytical model, attenuation and dispersion are caused by squirt flow in isotropic porous rocks (Alkhimenkov & Quintal 2023). Brie et al. (1995) formulate an empirical model for wave velocities in rocks saturated with two-phase fluids. This model has seen application in |$\mathrm{CO_2}$| sequestration projects like Sleipner, though uncertainties persist due to the unconstrained exponent in Brie’s law (Queißer & Singh 2013; Dupuy et al. 2017). Calibration of these models to laboratory measurements is vital for accurately quantifying the effects related to |$\mathrm{CO_2}$|⁠.

Wang & Nur (1989a) were pioneers in |$\mathrm{CO_2}$| testing, concentrating the relationship between elastic waves and fluid saturation. Electrical resistivity can also be an indicator for the degree of saturation in |$\mathrm{CO_2}$|-brine multiphase systems, although this approach might underestimate |$\mathrm{CO_2}$| saturation (Falcon-Suarez et al. 2017). Since then, numerous studies have focused on tests involving |$\mathrm{CO_2}$| (Xue & Ohsumi 2004; Xue & Lei 2006; Shi et al. 2007; Lei & Xue 2009; Perrin & Benson 2010; Alemu et al. 2011, 2013; Iglauer et al. 2011; Kim et al. 2011; Shi et al. 2011; Pini et al. 2012; Lebedev et al. 2013; Kitamura et al. 2014; Lebedev et al. 2014; Zhang et al. 2015; Falcon-Suarez et al. 2016, 2017, 2018; Agofack et al. 2018; Papageorgiou et al. 2018). A notable observation from these studies is a decrease in P-wave velocity, with S-wave velocity being less impacted (Xue & Ohsumi 2004; Xue & Lei 2006; Lei & Xue 2009; Kim et al. 2011; Lebedev et al. 2013, 2014; Zhang et al. 2015; Falcon-Suarez et al. 2017, 2018). The highest drop in P-wave velocity occurs immediately after the first fluid change, followed by a gradual decrease (Xue & Ohsumi 2004; Shi et al. 2007; Lei & Xue 2009; Alemu et al. 2013; Kitamura et al. 2014; Falcon-Suarez et al. 2016). A detailed overview of performed tests from the literature is shown in Table 1. Initial studies primarily focused on enhanced oil recovery (EOR). At the same time recent research has shifted towards examining the effects of supercritical |$\mathrm{CO_2}$| for |$\mathrm{CO_2}$|-sequestration, as P- and S-wave velocities are good indicators distinguish between pore fluid distribution and mechanical deformation during |$\mathrm{CO_2}$|-geosequestration (Falcon-Suarez et al. 2017). However, it’s important to note that velocity measurements at ultrasonic frequencies in the laboratory (Mavko & Nolen-Hoeksema 1994; Lebedev et al. 2009) may not be fully representative of wave propagation at seismic frequencies as observed in field applications.

Ultrasonic P- and S-wave velocities are measured using pulse-transmission technique at a central frequency between 100 kHz and 1 MHz. It is a precise and rapid method for characterizing the mechanical properties of materials. However, this technique can come along with unwanted wave scattering and is influenced by the ratio of wavelength to layer spacing (Marion & Coudin 1992). Additionally, there are scale-related considerations; imbibition typically results in homogeneous saturation, while drainage can create heterogeneous gas patches at the mesoscopic scale (Cadoret et al. 1995; Müller et al. 2008; Zhang et al. 2015).

Laboratory measurements at seismic frequencies, often referred to as low-frequency measurements using forced-oscillation technique, involve axial oscillations generated by piezoelectric actuators or dynamic shakers to determine Young’s modulus and Poisson’s ratio (Spencer 1981; Batzle et al. 2006; Mikhaltsevitch et al. 2011; Tisato & Madonna 2012; Szewczyk et al. 2016; Sun et al. 2018; Borgomano et al. 2020). These low-frequency measurements (⁠|$\approx$| 0.5 to 150 Hz) align closely with seismic survey frequencies, offering direct insights into elastic properties (Szewczyk et al. 2016). Agofack et al. (2018) extended this methodology to measure the elastic properties of two-phase fluid-saturated sandstone samples with theoretical computed expected saturations under |$\mathrm{CO_2}$| exposure across a broad frequency band. Additional µCT imaging can further validate saturations and give insight into dispersion mechanisms at the pore scale. It allows for distinguishing between variations due to fluid substitution and changes in fluid distribution patterns (Alemu et al. 2013). While studies integrating rock mechanical testing with µCT scanning in the context of |$\mathrm{CO_2}$| have been conducted (Lebedev et al. 2009; Alemu et al. 2011; Pini et al. 2012; Alemu et al. 2013; Lebedev et al. 2014), few have simultaneously determined elastic properties at seismic frequencies and performed µCT imaging (Tisato et al. 2014; Tisato & Zhao 2015). Notably, there is a distinct gap in laboratory measurements that assess the effects of |$\mathrm{CO_2}$| across both seismic and ultrasonic frequencies while incorporating µCT imaging. This represents a significant opportunity for advancing our understanding of |$\mathrm{CO_2}$|’s impact on rock mechanical properties, underscoring the need for comprehensive, multifrequency studies integrated with advanced imaging techniques.

Dispersion between laboratory data acquired at seismic and ultrasonic frequencies is well-documented (Hofmann 2006; Mikhaltsevitch & Lebedev 2014; Szewczyk et al. 2018; Borgomano et al. 2020; Sun et al. 2022). It’s observed that |$\mathrm{CO_2}$| significantly impacts low-frequency measurements, whereas the velocity reduction at higher frequencies is less pronounced (Cadoret et al. 1998; Batzle et al. 2006). These laboratory insights are crucial despite the differences between ultrasonic laboratory measurements and field surveys at seismic frequencies, but highlights the need for laboratory measurements at a similar wavelength to seismic surveys.

To perform laboratory measurements at both seismic and ultrasonic frequencies while simultaneously capturing processes at the pore-scale, we have built an X-ray transparent triaxial stress apparatus. This apparatus is designed to measure seismic frequencies in the range of 0.5–143 Hz and ultrasonic frequencies at 500 kHz for P waves and at 250 kHz for S waves. Its compact and lightweight design allows for portability and compatibility with µCT scanners. One of the key features of our apparatus is its capability to enable high-resolution scans with a resolution of down to 15 μm allowing for studying dispersion mechanisms, particularly patchiness. By employing ultrasonic velocity measurements, the apparatus effectively measures the impact of small patches through averaged velocities throughout the sample length, while larger patches are quantified through low-frequency measurements. Additionally, the patch size is estimated from 3-D µCT imaging. This multifaceted approach not only provides a comprehensive understanding of the dispersion mechanisms but also plays a pivotal role in the validation and calibration of rock physics models for partially saturated media for |$\mathrm{CO_2}$|-sequestration.

In this study, we introduce the new X-ray transparent triaxial stress apparatus, showcasing its application through two tests on Bentheimer sandstone: an imbibition-drainage test with brine and air, and a |$\mathrm{CO_2}$|-injection test involving a |$\mathrm{CO_2}$|–water mixture. The latter test transitions from liquid to gaseous phase by varying pressure. The paper is organized as follows: We commence with a detailed introduction of the low-frequency apparatus, and the nuances of CT scanner measurements. This is followed by a comprehensive explanation of the experimental procedures, encompassing the description of the sample material, sample mounting techniques and the specific equipment employed for sample saturation and methodology in the respective tests. We further present the calibration and address potential experimental errors. Subsequent sections present the methodology for CT scan evaluation and discuss the results from both the imbibition-drainage and |$\mathrm{CO_2}$| tests. The paper concludes with a discussion of these results, highlighting their implications and contributions to the understanding of rock mechanical and rock physical processes, particularly in the context of |$\mathrm{CO_2}$|-sequestration.

2 EXPERIMENTAL SETUP

In this section, we present a new triaxial laboratory apparatus to acquire frequency-dependent rock mechanical properties. Laboratory measurements were conducted at both seismic and ultrasonic frequencies, alongside simultaneous capture processes at the pore scale, which were monitored using Computed Tomography (CT) imaging techniques. Tests were carried out in the Formation Physics laboratory at SINTEF, employing insights gained from the previously developed low-frequency apparatus at the same institution.

The low-frequency apparatus is comprised of multiple units, with the components and the modelling of the X-ray transparent low-frequency apparatus outlined in subsequent sections.

2.1 Mechanical design

We began by conducting numerical modelling of the low-frequency apparatus. The design challenge was to minimize attenuation during CT measurements by engineering the frame’s walls where the sample resides to be slim. The optimized wall thickness was achieved at 5 mm. Finite element method (FEM) simulations were conducted utilizing the SolidWorks software to analyse the mechanical integrity of the experimental apparatus. The materials selected for the simulations included Aluminium 7075, characterized by an elastic modulus of 72 GPa, a density of 2.8 g cm⁻³ and a yield strength of 505 MPa; and AISI 316 stainless steel, with an elastic modulus of 193 GPa, a density of 8 g cm⁻³ and a yield strength of 172 MPa. The simulated components comprised a steel bottom part weighing 4.0 kg, an aluminium pressure vessel at 6.1 kg, a steel clamp at 1.7 kg and a steel top nut at 5.1 kg, culminating in a total mass of 17 kg for the primary components.

Fig. 1 presents the outcomes from the FEM analysis performed for the testing apparatus. The solid mesh used in the model consists of approximately 105 000 nodes and 65 000 elements with an element size of 7.6 mm and 16 Jacobian points, solved using the FFEPlus solver type. The factor of safety (FOS), shown in Fig. 1(a), indicates a minimum FOS of 1.6 for the steel bottom part, while the aluminium wall around the sample location has an FOS of 2. The stress distribution (Fig. 1b), calculated according to the von Mises criterion, reveals that stresses range from 21 MPa in areas surrounding the sample to a lower 5 MPa at the top and bottom regions. Strain calculations, following the ESTRN model (Fig. 1c), anticipate maximum strain values to be around 2.5|$\mathrm{\times 10^{-7}}$| m m⁻¹.

The low-frequency setup described below is fundamentally similar to the setup described by Szewczyk et al. (2016). Modifications include the construction of a smaller, more portable apparatus, weighing approximately 30 kg, with a height of 39 cm (16.3 cm shorter than the previous LF apparatus) to minimize nonlinearities. The pressure vessel can be rapidly mounted due to its screw-on design, rather than requiring mounting with screws. This vessel is crafted from X-ray transparent aluminium to allow for CT imaging. Vertical stress is applied simply by screwing a nut on top that presses down a piston, obviating the need for alignment during test setup and/or a mechanical load frame.

The vertical stress is measured using an aluminium piece equipped with strain gauges, which also serves a dual purpose in Young’s modulus and attenuation measurements. This method replaces the need for internal load cells. Additionally, the absence of LVDTs means that static deformation cannot be measured globally rather only locally using strain gauges. The new design includes three pore fluid outlets towards the sample, facilitating faster and more even sample saturation process.

The thin walls of the vessel may restrict its safe use to a confining pressure of up to 20 MPa due to possible material flaws, although the design has been tested and modelled to withstand up to 30 MPa. While tests were carried out at room temperature, the system can be modified to include a temperature control component. The equipment is specifically tailored for cylindrical samples measuring 50 mm in length and 25 mm in diameter. Table 2 gives an overview of measured parameters and computed elastic properties.

2.2 Static measurements and pore pressure system

Fig. 2 illustrates the testing setup including the low-frequency apparatus in the centre. This apparatus features a piezoelectric actuator that generates axial strain amplitudes ranging from 1 to 50 µStrain. A piezoelectric force sensor monitors dynamic forces, while an aluminium component records force and attenuation (1/Q). Endcaps at the top and bottom house P- and S-wave transducers operating at 500 and 250 kHz, respectively, and integrate pore fluid lines for bottom-up sample saturation. A detailed specification of the components in Fig. 2 is given in Appendix  A, Table A1. Photographs of the apparatus are shown in Figs A1.

The static confining pressure unit consist of a pressure pump P1 which is supplied with Paratherm oil. The pore pressure unit setup depends on the performed test and is in greater detail explained in Fig. 7 for the imbibition-drainage test and in Fig. 9 for the |$\mathrm{CO_2}$| test.

2.3 Dynamic measurements

The low-frequency unit features a digital lock-in amplifier that produces sinusoidal signals at frequencies ranging from 0.5 to 143 Hz. These signals are then amplified by a voltage amplifier before being sent to the piezoelectric actuator. The dynamic force and axial and radial strains from both the aluminium piece and the sample are recorded by an HBM system. Strains from the sample are measured and averaged using four Wheatstone bridges. All data are subsequently stored on a computer system, which also facilitates real-time visualization. Low-frequency measurements were performed at about 500 N (here |$\approx$| 1 MPa) axial deviatoric stress (⁠|$\mathrm{\sigma _{dev}=\sigma _{\rm ax}-P_{conf}}$|⁠).

The dynamic stress magnitude is derived by dividing the force amplitude F by the sample’s cross-sectional area A:

The dynamic strain amplitudes |$\varepsilon _{\rm ax}$| and |$\varepsilon _r$| are calculated as follows:

|$B_{\rm ax}$| and |$B_{\rm r}$| represent the axial and radial voltage amplitudes from Wheatstone Bridges, while |$V_{\rm ax}$| is the bridge excitation voltage and |$G_{\rm F}$| is the gauge factor. Fig. 3 provides a demo of the strain measurement as time-series showing the axial strains, radial strains and the dynamic force.

Attenuation |$1/Q$| is calculated from the phase angle |$\delta$| between the applied stress and the resulting strain Jackson et al. (1984):

The ultrasonic velocity unit is equipped with a signal generator and an amplifier that enhances the signal prior to transmission through the sample. The P-wave signal undergoes further amplification before it is relayed to the oscilloscope, while the S-wave signal is directly fed into the oscilloscope and then to a computer for automated data acquisition. During critical phases of the experiment, signals are collected at 60-s intervals.

P- and S-wave velocities in the axial direction were measured using the pulse-transmission technique as described by Hughes et al. (1949). The ultrasonic P-wave and S-wave transducers and receivers, with central frequencies of 500 and 250 kHz respectively, are integrated within titanium endcaps. The velocities |$V_{P,S}$| are calculated using the following formula:

where L is the length of the sample at the start of the test, |$\Delta L$| is the change in sample length due to applied stress, |$T_{P,S}$| is the total traveltime from send to receive and |$T_0$| is the traveltime of the system. |$T_0$| is derived from measurements conducted on a standard aluminium sample, which possesses well-characterized elastic properties. Fig. 4 presents exemplary waveforms for both aluminium and Bentheimer sandstone. Even though S-wave signals are difficult to determine, we were able to get through the design of the end-caps a steady S-wave signal at the second zero-crossing. This was most reliable as it was pronounced in all saturations. By following the arrival from low to high pressure and from dry to saturated stages we ensure that the signals are picked correctly.

A comparison between derived parameters from low-frequency and ultrasonic velocity measurements is given in Table 3.

2.4 CT scanner measurements

In this study, we utilized the Nikon 225 kV Ultrafocus walk-in CT scanner housed at the Pore Imaging Laboratory at SINTEF for our measurements. Data acquisition was managed using the Inspect-X XT 6.11 software suite. The system’s X-ray apparatus features a 225 kV reflection rotating target, operating at 450 W. We procured helical scan data across 4892 projections, with a single frame captured per projection. The system’s detector was set to a gain of 24 dB, with a pixel size tuned to 200 |$\mathrm{\mu }$|m. Notably, our methodology did not incorporate the use of a filter.

The shading correction was executed using 120 frames. For this purpose, an aluminium reference piece was securely fastened atop the experimental setup. This reference piece was selected for its material consistency with the frame and was machined to match the thickness of the pressure vessel’s sample-adjacent region, featuring a 5 mm thick wall. We then filled the reference piece with Paratherm oil, mirroring the confining fluid used within the cell.

The acquisition of a single scan required approximately 80 min, with an additional 5 min allocated for shading correction. This process yielded a high-resolution 3-D image with a detail level of 15 μm. For a comparison, the X-ray transparent low-frequency apparatus by Tisato & Zhao (2015) and Zhao et al. (2017) had a lower resolution of 25 to 37 μm. Fig. 5 provides a detailed view from the CT scanner’s measurements, illustrating the precise dimensions that the X-rays traverse during the scanning process.

The CT scans of imbibition-drainage test were evaluated using SINTEF’s in-house Python script ‘Basilisk Gaze’ based on statistical method of image segmentation. Since the sample was saturated with potassium iodide (KI) brine, brine’s intensity was matching intensity of rock matrix. Therefore, all tested brine saturations spanning from dry to fully saturated resulted in two distinct populations of pixel intensities. Lower intensities represented free porosity and higher intensity represented rock matrix and brine. These two pixel populations were quantified by approximation with normal distribution functions. |$\mathrm{CO_2}$| test scans contained three pixel populations, and therefore were evaluated using AVIZO software (see Table A4 for estimated errors). First, the sample positions of all scans were aligned through rotation, height and vertical adjustment. Then the coordinate system was resampled to align all scans to the dry one. Afterwards, we applied smoothing and denoising filter to reduce noise and artefacts. Subvolumes (600 × 600 × 600 voxels) below and above the strain gauges were cropped to extract saturations. We used different intensity contrasts to separate the mineral matrix from the pore space in the dry and fully saturated sample. To differentiate the third phase, free gas, we subtracted the dry scan, isolating two phases: liquid and gaseous. The difference was determined by adjusting the intensity contrast to match the free gas saturation.

3 EXPERIMENTAL PROCEDURES

In this section, we present the tested sample material, the sample mounting process and the detailed workflow for both the imbibition-drainage test and the |$\mathrm{CO_2}$| test. The calibration of the apparatus is presented in the Appendix A2.

3.1 Sample material

Bentheimer sandstone, selected for this study, originates from an outcrop in Germany and is of Lower Cretaceous age. The sandstone is composed predominantly of quartz, constituting 92 wt. per cent, accompanied by feldspar at 5 wt. per cent and clay minerals making up the remaining 3 wt. per cent (Peksa et al. 2015).

Samples were carefully drilled and trimmed to achieve a uniform size, resulting in dimensions of approximately 50 mm in length and 25 mm in diameter (2 arcsec length, 1 arcsec diameter). Additionally, the top and bottom surfaces were grinded to ensure parallelism. Fig. 6 illustrates the drilled sample prior to the |$\mathrm{CO_2}$| test.

Attaching resistive strain gauges directly onto the sandstone sample can lead to irreversible damage to the gauges due to the possible presence of large voids in the sample. To circumvent this issue, a layer of epoxy is applied to the designated area on the sample surface where the active part of the strain gauges will be positioned. This process involves applying a thin epoxy layer followed by careful grinding with sandpaper, a method previously employed and documented in Lozovyi et al. (2017). This procedure does not affect the mechanical properties of the sample (Lozovyi et al. 2017). The dimensions and elastic properties are shown in Table 4.

3.2 Sample mounting

Sample mounting is completed within a 15-min time frame. This rapid mounting procedure is particularly beneficial for testing samples that are critically saturated at the time of mounting, as it minimizes water loss. The following sections provide a detailed description of the steps:

Strain gauges are affixed to the sample’s surface, with their active parts positioned on the epoxy. The coupling between the strain gauges and the sample is achieved by increasing the confining pressure during the test. The sample is then placed on the bottom end-cap, and an axial air pressure of about 2 bar is applied to maintain the sample’s position during the mounting process. Sixteen wires, originating from the four tee rosette strain gauges, are held in place to the bottom end-cap using vulcanizing sealing tape. The strain gauge wires are embedded between two layers of tape wrapped around the end-cap. Following this, the sleeve is mounted; a Viton sleeve is used for the imbibition-drainage test due to its elasticity, while a Teflon sleeve is employed for the |$\mathrm{CO_2}$| test to prevent gaseous |$\mathrm{CO_2}$| migration from the sample through the sleeve. Both the top and bottom end-caps are equipped with O-rings to ensure a tight seal.

3.3 Workflow imbibition-drainage

3.3.1 Pore pressure unit imbibition-drainage

The pore pressure unit for the imbibition-drainage test is shown in Fig. 7. The unit comprises a hydraulic pump (P2), an air bottle, a nitrogen bottle and a vacuum pump.

The confining pressure was increased to 5 MPa through P1 at a consistent rate of 4 MPa hr⁻¹. Subsequent procedures for attaining various saturation levels during the imbibition and drainage processes are outlined in the following section.

3.3.2 Flowing procedure

In the described procedure, specific volumes of 12.5 wt. per cent potassium iodide (KI) brine are injected. The rationale behind selecting this particular brine concentration is elaborated in the subsequent section.

Fig. 8 shows the static data measured throughout the test and indicates when CT scans were performed. Low-frequency measurements were performed before or after the CT scan at 2 and 20 µStr strain amplitude.

During the imbibition phase, all valves remain open except for valves 6 and 4. We inject 3.57 ml of brine using Pump P2 at a flow rate of 0.1 ml min⁻¹. Out of this, 0.38 ml represents the dead volume from valve c1 to the sample, leading to a sample saturation level of 60 per cent. To attain a higher saturation, an additional 1.92 ml of brine is injected at the same rate, marginally increasing the saturation to 63 per cent.

For achieving a higher saturation level (86 per cent), valves c1 and 3 were closed while valves c2 and 4 were opened. Subsequently, the vacuum pump was activated to maintain a pressure of approximately 300 mbar. Following this step, valve c2 was closed. Valve c1 was then opened, allowing for the injection of brine via pressure pump P2. This injection process continued with the subsequent opening of valve c2 to ensure thorough fluid permeation.

To attain complete saturation, the procedure for reaching 86 per cent saturation is replicated, with the vacuum pump set to a more rigorous 10 mbar. Subsequent to establishing the vacuum, a pressure of 2 MPa is applied via the nitrogen gas bottle to the back pressure valve, at which point valve 4 is closed while valves c2 and 3 are opened. The system’s confining pressure is then escalated from 5 to 7 MPa and the pore fluid pressure from 0 to 2 MPa, both at a consistent loading rate of 4 MPa per hr. At this point, 30 ml of brine is injected utilizing the P2 pump. This resulted in 100 per cent saturation. Upon completion of the injection, the confining pressure and the pore fluid pressure are both gradually reduced to their initial states of 5 and 0 MPa, respectively, at the original loading rate.

To initiate drainage, valves 3 and 5 were closed, and valves c1, c2, 4 and 6 were opened. Air injection commenced at a pressure of 1 bar, achieving an approximate saturation of 83 per cent. Subsequently, to reduce the saturation level further, air pressure was intensified to 14 bars, culminating in a 79 per cent saturation state.

3.3.3 Saturation fluid

The test’s saturation fluid is selected to have a higher density, which improves the contrast distinction between air and fluid. For our purposes, we selected a 12.5 wt. per cent KI brine, a decision based on existing literature (see Appendix  B, Table A3). Also, the solubility of KI in water is notably high, at 148 g/100 g at 25 |$^\circ$|C (Lide 2005). To minimize risks such as crystallization and pore throat clogging, as well as to reduce the potential for system corrosion, we aim to keep the testing duration as brief as feasible.

3.4 Workflow |${\rm CO}_{2}$| test

3.4.1 Pore pressure unit |$\mathrm{CO_2}$|

A |$\mathrm{CO_2}$|-water mixing unit was constructed, as depicted in Fig. 9. This unit comprises two hydraulic pumps, two accumulators, a nitrogen bottle, a |$\mathrm{CO_2}$| bottle and a vacuum pump. The |$\mathrm{CO_2}$| bottle possesses a capacity of 13.4 L and is pressurized to 50 bar at a room temperature of 23 |$^{\circ }$|C. The bottle was warmed to 33 |$^{\circ }$|C using a heating blanket, and the pressure was monitored and controlled until it reached 80 bar. Initially, one side of Accumulator #1 contains 600 mL of paraffin oil, which is pressurized and regulated by pressure pump P3. The opposite side is filled with 400 mL of water, pressurized and governed by pressure pump P2. Accumulator #1 functions as a mixer for water and |$\mathrm{CO_2}$|⁠, and it can be manually rotated by 180|$^{\circ }$|⁠. Approximately 50 per cent of Accumulator #2 is filled with water from both sides. This accumulator manages the outlet pressure and, later in the test, receives a mixture of |$\mathrm{CO_2}$| and water.

In the initial setup, both accumulators were completely filled with fluid and the valves at the ends were closed. Subsequently, the system was evacuated by the vacuum pump. All valves, except for valves 5, 8 and 12 were then opened and finally, the entire system, including the sample, was filled with water.

The confining pressure P|$\mathrm{_{c}}$| was increased through P1 and the pore fluid pressure P|$\mathrm{_{f}}$| through P2 (with valve 10 open) at a consistent rate of 4 MPa hr⁻¹. P|$\mathrm{_{c}}$| was increased from 5 to 7 MPa and P|$\mathrm{_{f}}$| from 0 to 2 MPa to maintain a constant effective stress. Following this, 50 ml of fluid was flushed against the back pressure valve. Subsequently, valve c2 was closed, and P|$\mathrm{_{conf}}$| was further increased to 12.5 MPa and P|$\mathrm{_{f}}$| to 7.5 MPa. To execute an undrained low-frequency measurement at 100 per cent water saturation, valve c1 was closed.

To build up pressure in accumulator #2 towards the pump, both valves 2 and 7 were closed and remained closed thereafter. A pressure of 7.5 MPa was established using pressure pump P2. Valve 12 was opened to build up pressure from both sides in accumulator #2. It should be noted that accumulator #2 is not essential if the pressure pump (in this case, P2) operates accurately when |$\mathrm{CO_2}$| is present in the system.

Subsequently, pressure in accumulator #1 was built up on the side facing pressure pump P3. The pressure in P3 was raised to 7.5 MPa, after which valve 8 was opened. At this point, the entire system was filled with fluid and was pressurized to 7.5 MPa pore pressure.

3.4.2 Mixing procedure

To fill accumulator #1 with |$\mathrm{CO_2}$|⁠, valve 3 was also closed to prevent fluid flow into the sample. The |$\mathrm{CO_2}$| bottle had a pressure of 8 MPa, while the system’s pressure stood at 7.5 MPa. This pressure disparity facilitated the flow from the bottle to the system. Valves 1 and 4 were opened in sequence. Pump P3 was then set to intake 90 mL (comprising 80 mL and an additional 10 mL from the dead volume of the tubings) at a rate of 2 mL min⁻¹, allowing for 20 per cent liquid |$\mathrm{CO_2}$| to flow into accumulator #1. After the filling of accumulator #1, valve 4 was closed, and the accumulator was rotated to facilitate the mixing of liquid |$\mathrm{CO_2}$| with water. This mixing resulted in a decline in pressure. It was imperative to ensure that the pressure did not fall below 6.5 MPa to keep the |$\mathrm{CO_2}$| in its liquid state. Valve 4 was then reopened to infuse more liquid |$\mathrm{CO_2}$| into accumulator #1, after which it was closed, and the mixture was agitated once more. This process was repeated until no further pressure decrease was observed upon mixing, a procedure that lasted approximately 30 min. Following this, valve 1 was closed, and the |$\mathrm{CO_2}$| bottle was disconnected from the system. Pressure pump P3 also reached a pressure of 8 MPa. At this stage, the system was filled with both liquid |$\mathrm{CO_2}$| and water.

3.4.3 Flowing procedure

Following this, the sample was flushed with the mixture of liquid |$\mathrm{CO_2}$| and water. To achieve this, valves c1, c2 and 3 were opened in the direction of the apparatus, while valve 9 was closed. Given that P3 maintained a pressure of 8 MPa and P2 was at 7.5 MPa, flow was ensured through the sample, moving from the inlet (c1) to the outlet (c2). Pump P2 then received 100 mL of the fluid. Subsequently, valves c1 and c2 were closed to maintain undrained conditions.

Next, the pressure in accumulator #1 is stepwise reduced to 10.5 MPa confining pressure through P1 and 5.5 MPa pore fluid pressure through P3 to release free gas. Therefore, valve c1 is opened.

Throughout the CT scans, pressures were maintained constant. We maintained the confining pressure using an additional accumulator, half-filled with Paratherm oil, connected to the sample on one side and to a nitrogen bottle on the other. To minimize fluctuations in pore pressure, we closed valves 8 and 12 and disconnected the pumps from the system. With valves c1 and c2 leading to the sample left open, minor temperature variations did not induce pressure changes, as the system could compensate.

Fig. 10 shows the static data measured throughout the test and indicates when CT scans were performed. Low-frequency measurements were performed before or after the CT scan at 2 and 20 µStr strain amplitude. Table 5 gives an overview of the notation for these tests, the pore fluid and the pressures and stresses at the respective saturation stage.

4 RESULTS

4.1 Overview

We present findings from two studies on Bentheimer sandstone. The first involves an imbibition-drainage test, while the second examines the effect of liquid and free |$\mathrm{CO_2}$|⁠. Table 6 provides a summary of saturation levels observed in these experiments. We had a systematic error between seismic and ultrasonic measurements. Therefore, we only present normalized values when results at seismic frequencies are directly compared to results at ultrasonic frequencies. Not normalized results are summarized in the Tables A5 and A6. Attenuation measurements are included in Appendix  E.

4.2 Imbibition-drainage

Fig. 11 displays the variations in P- and S-wave velocities at different saturation levels. The P-wave velocity is highest at 100 per cent saturation, followed closely by velocities in the lowest saturation during drainage and under dry conditions. The P-wave velocity is reduced at partial saturation due to high attenuation. For S waves, the highest velocities are observed in dry conditions, with fairly consistent velocities across saturated conditions.

Fig. 12 presents the measurements of elastic properties across seismic frequencies. Young’s modulus is observed to be highest under dry conditions, followed by 100 per cent saturation state, and is lowest during partial saturation. Young’s modulus measured during imbibition exceeds that during drainage. Poisson’s ratio is highest at full saturation and lowest in dry conditions, with intermediate values observed at partial saturation during drainage and imbibition. The bulk modulus peaks at 100 per cent saturation. During imbibition, as saturation increases, the bulk modulus modestly decreases, then slightly rises during drainage as water saturation declines. The shear modulus is highest under dry conditions and exhibits a slight decrease upon saturation. During drainage, shear modulus values are marginally lower than during imbibition and at full saturation. Additionally, for both tests, frequencies above 100 Hz show a slight reduction in elastic properties in saturated samples, likely due to resonances.

In Fig. 13, we look at normalized elastic properties as a function of saturation, both at seismic and ultrasonic frequencies. At ultrasonic frequencies, Young’s modulus exhibits an increase with saturation during imbibition, followed by a decrease during drainage, with higher velocities noted during the drainage phase than during imbibition. Conversely, at seismic frequencies, a different pattern emerges: Young’s modulus is maximal under dry conditions, decreases during imbibition, rises again at full saturation, and decreases during drainage. Young’s modulus during the imbibition phase is slightly higher than during drainage. Poisson’s ratio maintains consistency across both frequency ranges, but with higher values observed during drainage compared to imbibition at ultrasonic frequencies, as previously indicated. Ultrasonic bulk modulus increases with increasing saturation and is higher during drainage than during imbibition. Bulk modulus at seismic frequencies shows a weak dependence of stiffness upon saturation during imbibition and drainage stage and increases at full saturation. At ultrasonic frequencies, shear modulus remains relatively constant with rising saturation indicating no difference between drainage and imbibition. Conversely, shear modulus at seismic frequencies decreases with partial saturation, but remains relatively consistent across different saturation levels. Shear modulus during drainage is slightly lower than during imbibition.

Ultrasonic P-wave velocities in Fig. 13(e) are higher during drainage compared to imbibition, a trend in elastic properties previously described for Young’s modulus, Poisson’s ratio and bulk modulus. At seismic frequencies, P-wave velocities decrease with increasing saturation during the imbibition stage and increase again towards 100 per cent saturation. Seismic frequencies show a stronger effect towards saturation level, whereas ultrasonic velocities change only minimally and increase significantly only at full saturation. For ultrasonic S wave (Fig. 13f), the dependence is less pronounced with only a minor decrease from dry to saturated state. At seismic frequencies, S-wave velocities decrease from dry to saturated and show a weak dependence thereafter, similar to shear modulus.

Fig. 14 displays the sample’s saturation as a function of vertical position. Saturation profiles were obtained from processing over 4000 slices for each scan using method described in Section 2.4. Total saturations were obtained by averaging and interpolating (though areas affected by strain gauges) all slices. Complementing this, Fig. 15 presents cross-sectional views from CT-scan image analysis at various positions throughout the sample. It indicates that imbibition achieves a homogeneous saturation across the sample with air patches at the pore size scale. At the first drainage stage (83 per cent saturation), the injection of air into the 100 per cent saturated sample formed mesoscopic scale patches. At the second drainage stage (79 per cent saturation), a significant increase in air flow rate led to a more homogeneous fluid distribution and a minor reduction in saturation.

4.3 |${\rm CO}_{2}$|

This section shows the results for the |$\mathrm{CO_2}$|-test (also see Section 3.4). Ultrasonic data at 12 per cent free |$\mathrm{CO_2}$| is missing.

Fig. 16 presents P- and S-wave velocity waveforms at various saturation stages. The ultrasonic P wave for 100 per cent water and 4 per cent free |$\mathrm{CO_2}$| arrives first, followed by the 100 per cent water + |$\mathrm{CO_2}$| stage with 80 per cent water and 20 per cent liquid |$\mathrm{CO_2}$|⁠, as P-wave velocity reductions due to |$\mathrm{CO_2}$| are less sensitive when |$\mathrm{CO_2}$| saturation exceeds 20 per cent (Kim et al. 2011). The 31 per cent free |$\mathrm{CO_2}$| stage shows a significant delay, with the dry stage being the slowest. In contrast, S-wave velocities exhibit minimal variation in arrival times compared to P-wave velocities. The highest S-wave velocity is observed at 0 per cent saturation, followed by 100 per cent water and 4 per cent free |$\mathrm{CO_2}$|⁠. The S-wave velocity decreases for 100 per cent water + |$\mathrm{CO_2}$| and is the lowest for 31 per cent free |$\mathrm{CO_2}$|⁠.

Fig. 17 presents the measurements of elastic properties across seismic frequencies. Young’s modulus is highest at 100 per cent water saturation and slightly lower under dry conditions. When water is mixed with liquid |$\mathrm{CO_2}$|⁠, Young’s modulus decreases. It is lowest when free |$\mathrm{CO_2}$| is present in the system, with no significant difference between 12 per cent and 31 per cent free |$\mathrm{CO_2}$|⁠. Poisson’s ratio is highest at 100 per cent water saturation, slightly lower when water is mixed with liquid |$\mathrm{CO_2}$|⁠, and under dry conditions. It is lowest when free |$\mathrm{CO_2}$| is present. The bulk modulus at 100 per cent water saturation is the highest. Similar to Young’s modulus, the bulk modulus decreases when liquid |$\mathrm{CO_2}$| is present and is lowest when free |$\mathrm{CO_2}$| is in the system. The shear modulus shows only minor differences, being highest under dry conditions, followed by full water saturation, the water and |$\mathrm{CO_2}$| mixture, and is lowest when free |$\mathrm{CO_2}$| is present. Measurements for 4 per cent free |$\mathrm{CO_2}$| exhibit notable increases at frequencies above 56 Hz. The reasons behind these observations remain unclear, but similar trends are observed at 2 µStrain amplitude measurements.

Fig. 18 shows normalized elastic properties as a function of saturation, both at seismic and ultrasonic frequencies. At ultrasonic frequencies, Young’s modulus increases with saturation, with the highest values observed at 100 per cent water saturation. Conversely, at seismic frequencies, Young’s modulus is high under dry conditions, decreases when free |$\mathrm{CO_2}$| is present and rises again at full saturation, with the highest values again at 100 per cent water saturation. At ultrasonic frequencies, Poisson’s ratio increases with increasing saturation. Conversely, Poisson’s ratio remains consistent across seismic frequencies and increases only at 100 per cent saturation. At ultrasonic frequencies, bulk modulus shows a dependence on saturation level. At seismic frequencies, bulk modulus shows only a weak dependence on saturation. The shear modulus at ultrasonic frequencies remains relatively consistent as saturation increases, except at 96 per cent saturation and higher where the shear modulus is slightly elevated. At seismic frequencies, shear modulus shows decreases slightly with increasing saturation, but increases again at higher saturations to values similar as the dry measurement.

Ultrasonic P-wave velocities in Fig. 18(e) increase as saturation increases. At seismic frequencies, P-wave velocities decrease with increasing saturation and increase again towards 100 per cent saturation. Shear wave velocity shows a decrease from dry to saturated at ultrasonic and seismic frequencies and a weak dependence thereafter.

5 DISCUSSION

5.1 Effect of patch size during imbibition-drainage

During imbibition, brine displaces the non-wetting fluid (gas) within the pore spaces. Conversely, during drainage, the wetting brine is displaced by gas. These processes are primarily driven by capillary forces, influenced by factors such as rock wettability, pore size distribution, fluid properties and the interfacial tension between the fluids. The effect on elastic properties at high frequencies (e.g. ultrasonic velocity measurements) is described by the Hill limit, where fluid pressure does not equilibrate during a wave cycle, leading to the isolation of patches from each other (Müller et al. 2008). In contrast, the low-frequency limit is described by the Wood limit, where pore pressure is uniform during a wave cycle, providing a comprehensive description of high- and low-frequency limits as patchy and homogeneous, respectively (Müller et al. 2008).

Young’s modulus is observed to be highest under dry conditions, followed by the fully saturated state, and is lowest during partial saturation (Fig. 12), consistent with findings from the literature (Chapman et al. 2016). The bulk modulus peaks at 100 per cent saturation, in alignment with Gassmann’s model predictions (Gassmann 1951), though it is slightly underestimated by the model under dry conditions. According to Wood’s model, the bulk modulus shows a weak dependence on stiffness during the imbibition stage and increases at full saturation, mirroring our seismic frequency measurements (Fig. 13). During drainage, the seismic response continues to follow Gassmann–Wood predictions, while the ultrasonic bulk modulus aligns more closely with Hill’s model. Overall, the bulk modulus at seismic frequencies is similar during imbibition and drainage. However, at ultrasonic velocities, the bulk modulus is lower during imbibition and higher during drainage.

That is exactly what has been observed for measurements at ultrasonic velocities: Falcon-Suarez et al. (2020) observed in partially saturated sandstone samples that the P-wave velocity was higher during drainage and lower during imbibition. A similar trend was noted in ultrasonic P-wave velocities by Knight & Nolen-Hoeksema (1990), where P-wave velocity remained relatively constant during imbibition, decreasing only slightly with increasing saturation. During drainage, P-wave velocity increases as saturation increases, resulting in overall higher P-wave velocities for drainage compared to imbibition, with significantly higher velocities at full saturation (Fig. 19). This behaviour is explained, similar to Müller et al. (2008), by differences in pore-fluid distribution, with drainage resulting in patchy saturation and imbibition leading to uniform saturation (Knight & Nolen-Hoeksema 1990).

For low-frequency measurements of partially saturated limestone, Sun et al. (2022) found that velocity is related to WIFF (wave-induced fluid flow) at a mesoscopic scale, controlled by the geometry and distribution of gas patches. During drainage, this leads to a more heterogeneous fluid distribution with larger air patches. Similarly, Chapman et al. (2017) observed a shift from a homogeneous distribution of pore-scale gas bubbles to a more heterogeneous gas patch distribution.

When discussing the reverse trend between elastic properties measured at seismic and ultrasonic frequencies, it is essential to consider the wavelength (⁠|$\lambda$|⁠):

whereas v is the wave velocity, and f denotes the frequency of the wave. For instance, a P-wave velocity of 2500 m s⁻¹ results in wavelengths of 5 mm at ultrasonic frequencies of 500 kHz, making a 50 mm sample length feasible. Estimates of patch sizes from CT scans range from 5 to 10 mm at 83 per cent saturation during drainage, indicating that ultrasonic measurements have wavelengths on the same scale as the pore fluid patch size, and are therefore likely affected by scattering. During the second drainage step at 79 per cent saturation, a significant increase in air flow rate resulted in a more homogeneous fluid distribution with fewer patches, aligning with Hill’s model.

However, at a seismic frequency of 1 Hz, the wavelength extends to an impractical 2.5 km, and even at 100 Hz, it reaches 25 m, far exceeding the patch size. Thus, low-frequency measurements do not capture patches on the pore scale due to the much higher wavelength, and therefore do not exhibit scattering effects. This results in similar measurements between drainage and imbibition. That is exactly why low-frequency measurements are important as they are directly comparable to the response from seismic surveys, whereas ultrasonic measurements during drainage are dominated by the patch sizes.

Thus, we conclude that Hill’s model adequately captures the response during imbibition, showing a weak dependence on saturation level. Measurements at low frequencies are also captured by Hill’s model during drainage, as the patch size is much smaller than the wavelength. At ultrasonic frequencies during drainage, however, Wood’s model adequately captures the data since the patch size is on the same scale as the wavelength. This underscores the importance of considering scale effects in interpreting the laboratory measurements.

In these experiments we did not observe significant dispersion. Pimienta et al. (2017) tested Bentheimer sandstone and did observe significant dispersion in Young’s modulus attributed to squirt flow in the low frequency range. However, the tests were performed with a water–glycerine mixture encountering a higher viscosity than the fluids used in this study. Higher viscosity shifts squirt flow towards lower frequencies, explaining that the dispersion in our test would likely fall between low-frequency and ultrasonic velocity measurements. Moreover, the tested sample was intact and has a higher permeability and high porosity. Under these conditions we would not necessarily expect squirt flow.

5.2 Effect of |${\rm CO}_{2}$| on elastic properties

The injection of |$\mathrm{CO_2}$| in the pore space can change the elastic properties quite drastically, (see Table 1), whereas P-wave velocities are decreased of up to 16 per cent for supercritical |$\mathrm{CO_2}$|⁠.

Falcon-Suarez et al. (2017) found that S-wave velocity is an excellent indicator of the mechanical deformation during |$\mathrm{CO_2}$| injection, whereas P-wave gives the mechanical and pore fluid distribution information. They found that even long-term exposure to |$\mathrm{CO_2}$|-rich solutions leads to minor variations of geomechanical properties from synthetic sandstone tests (Falcon-Suarez et al. 2017). The highest drop in P-wave velocity occurs immediately after the initial fluid change to |$\mathrm{CO_2}$|⁠, followed by a gradual decrease, as reported in other experimental works (Xue & Ohsumi 2004; Shi et al. 2007; Lei & Xue 2009; Alemu et al. 2013; Kitamura et al. 2014; Falcon-Suarez et al. 2016).

In this study, the presence of |$\mathrm{CO_2}$| resulted in a 3 per cent reduction in P-wave velocity for a sample saturated with 100 per cent saturation water and a mixture of 80 per cent water and 20 per cent liquid |$\mathrm{CO_2}$| (Table 7). The largest P-wave reduction, 7 per cent, occurred when the sample contained 12 per cent free |$\mathrm{CO_2}$| gas. Increasing the free |$\mathrm{CO_2}$| gas content to 31 per cent did not further reduce the velocity. Overall the P-wave velocities are fairly consistent between measurements at seismic and ultrasonic frequencies with one exception:

For the test with 4 per cent free |$\mathrm{CO_2}$|⁠, low-frequency measurements show a decrease in Young’s modulus resulting in values between the 100 per cent water + |$\mathrm{CO_2}$| and the 12 per cent and 31 per cent free |$\mathrm{CO_2}$| measurement, but at ultrasonic frequencies, it showed no reduction. This indicates that small gas concentrations are only captured in low-frequency measurements, but are not visible to tests performed at ultrasonic frequencies showing the urgent need for low-frequency measurements. These finding for low gas saturation (below 10 per cent) are consistent with the measurements of Agofack et al. (2018). We observed minimal effects on ultrasonic frequencies but significant effects in the low-frequency regime. Agofack et al. (2018) argue that ultrasonic velocities are nearly unaffected by free |$\mathrm{CO_2}$| because the rock is near the high-frequency limit. In contrast, elastic properties at seismic frequencies lead to a significant velocity reduction. It is possible that small |$\mathrm{CO_2}$| saturations are not detected by ultrasonic measurements. The ultrasonic wave travels the fastest way possible between transducer and receiver, and thereby saturations, which would lower the velocity, could be outside the wave propagation path. The fastest ultrasonic path traverses the centre of the sample, which may not intersect the locations small amounts of free |$\mathrm{CO_2}$|⁠. Additionally, the ultrasonic response is dominated by the fully saturated state, as small amounts of free |$\mathrm{CO_2}$| have a negligible impact over the entire travel distance of the ultrasonic wave. When higher amounts of free gas are present, the velocity reduction becomes evident as the fully saturated response no longer dominates. Low-frequency measurements, however, capture small amounts of free gas because the entire sample is oscillated, allowing detection even if saturations are localized.

S-wave velocities showed a maximum reduction of 4 per cent at 31 per cent free |$\mathrm{CO_2}$| in ultrasonic tests, while at seismic frequencies, S-wave velocities exhibited a marginal change of up to 2 per cent with 12 per cent free |$\mathrm{CO_2}$|⁠. Overall the changes in ultrasonic S-wave velocities seem to be slightly higher compared to S-wave velocity changes measured at seismic frequencies.

Fig. 20(a) shows the ratio of P- and S-wave velocity as a function of saturation for the imbibition-drainage test. At ultrasonic velocities, the ratio is higher during drainage than during imbibition. At seismic frequencies the ratio maintains consistent throughout imbibition and drainage and increases only towards full saturation. Fig. 20(b) shows an increase in the ratio of ultrasonic P-wave over S-wave velocity from 1.51 (dry) to 1.64 for partial saturations at 12 and 31 per cent free |$\mathrm{CO_2}$| to 1.67 at 100 per cent saturation (independent of presence of |$\mathrm{CO_2}$|⁠), exactly what Falcon-Suarez et al. (2017) observed and argued that is due to a reduction in pore fluid compressibility (Wang et al. 2012).

We aimed for low free |$\mathrm{CO_2}$| saturations of 2 per cent and 5 per cent, but observed saturations approximately six times higher. We hypothesize that incomplete mixing of |$\mathrm{CO_2}$| in water contributed to this discrepancy. Additionally, the solubility of |$\mathrm{CO_2}$| is highly temperature-dependent. Without temperature control, room temperature fluctuations likely caused further deviations. According to Agofack et al. (2018), pure |$\mathrm{CO_2}$| undergoes significant expansion with pressure around 6 MPa, which coincides with our target range of 2 per cent to 5 per cent. This phenomenon likely contributed to the higher-than-expected saturations.

Overall, Gassmann fluid substitution worked fine for interpretation of results with |$\mathrm{CO_2}$|⁠, similar as observed in Mikhaltsevitch et al. (2014). The X-ray transparent low-frequency apparatus allows for capturing multiphase fluid saturations, even at low concentrations concentrated at one place.

The X-ray transparent low-frequency apparatus has potential applications in testing rocks with complex geometries, fractured rocks and carbonates. Specifically, it enables quantification of dissolution effects on elastic properties in carbonates. Modification of the setup can facilitate testing with methane, thereby assessing the impacts of depletion and injection in oil and gas reservoirs.

These findings are particularly relevant for CCS initiatives, bridging the gap between seismic surveys and rock physics modelling. A temperature component could be integrated into the system to extend testing capabilities to supercritical |$\mathrm{CO_2}$| conditions. During |$\mathrm{CO_2}$| injection, patches are created where ultrasonic velocity measurements may deviate from seismic surveys. Specifically, during drainage, ultrasonic velocity measurements are dominated by these patches, leading to an overestimation of elastic properties. At low gas saturations, measurements are influenced predominantly by fluid saturation, similarly resulting in an overestimation.

6 CONCLUSIONS

The developed apparatus allows for 3-D pore-scale imaging of |$\mathrm{CO_2}$| and other gases in a fluid and gaseous phase with a resolution of up to 15 μm while measuring frequency-dependent elastic properties. Identifying fluid–solid interactions and estimating saturation via µCT-imaging is particularly critical when only a small amount of gas is present. Therefore, our combined approach allows precise determination of elastic properties at seismic and ultrasonic frequencies.

The imbibition-drainage experiment indicated that P-wave velocity at ultrasonic frequencies was higher during drainage and lower during imbibition. Drainage resulted in patchy saturation, whereas imbibition led to uniform saturation. Estimates of patch sizes from CT scans ranged from 5 to 10 mm during drainage. This suggests that ultrasonic measurements, with wavelengths on the same scale as the pore fluid patch size, are likely affected by scattering. In contrast, low-frequency measurements, where the wavelength is much greater than the patch size, capture effective medium properties and therefore are not affected by the scattering effects.

Gassmann–Wood’s model effectively captures the response during imbibition, demonstrating a weak dependence on saturation when gas is present. During drainage, Gassmann–Wood’s model aligns well with low-frequency measurements, whereas Gassmann–Hill’s model may better describe ultrasonic frequencies due to the alignment of patch size with the wavelength.

The results suggest that low-frequency measurements capture even low gas saturations (4 per cent free |$\mathrm{CO_2}$|⁠), whereas ultrasonic velocity measurements are dominated by the response of the fully saturated sample and show no velocity reduction. |$\mathrm{CO_2}$| presence led to a 3 per cent P-wave velocity reduction at 20 per cent liquid |$\mathrm{CO_2}$| saturation and a 7 per cent reduction at 12 per cent free |$\mathrm{CO_2}$| gas saturation. Increasing free gas to 31 per cent did not further reduce velocity. S-wave velocities decreased by 4 per cent at 31 per cent free gas in ultrasonic tests and by up to 2 per cent at seismic frequencies with 12 per cent free gas. Overall, ultrasonic S-wave changes were slightly higher than those at seismic frequencies.

These findings underscore the importance of low-frequency measurements over ultrasonic velocity measurements due to scaling effects. They also provide a basis for selecting and calibrating relevant rock physics models to predict elastic properties of partially saturated reservoirs, ensuring safe oil and gas extraction and CCS operations through improved monitoring.

ACKNOWLEDGMENTS

The authors wish to express their gratitude for the financial support received from the Research Council of Norway, facilitated through the PETROMAKS 2 program researcher project (grant number 301910) at SINTEF: Calibrated rock physics model for quantitative seismic analysis of two-phase fluid saturation. Special thanks are extended to Rune M. Holt for valuable discussions. We also appreciate the insightful dialogues with Jørn Stenebråten and Nicola Tisato regarding the experimental equipment. Our acknowledgment extends to Blandine Feneuil, Amir Ghaderi and Albert Barabino for their efforts in the acquisition of CT images, and to Marcin Duda and Hailil Long for the evaluation of these images. During project meetings we had valuable discussions with Anne-Kari Furre, Andreas Bauer, Giorgos Papageorgiou, Mark Chapman and Alexander Rozhko.

DATA AVAILABILITY

All experimental measurement data are available through doi:10.5281/zenodo.12506772. CT scans are made available upon request due to the limited data repository storage space.

References

APPENDIX A: LOW-FREQUENCY APPARATUS DETAILS

In the following we show an overview of low-frequency components built inside the apparatus and the calibration protocol.

A1 LF components

Table A1 provides a detailed overview of the specific components of the X-ray transparent low-frequency apparatus and their manufacturers. Photographs of the apparatus are shown in Fig. A1.

A2 Calibration

The apparatus was calibrated with Aluminium (type 6061) and PEEK (Polyetheretherketone). Aluminium has distinct mechanical properties. PEEK is not a certified material, but the results shown are compared to PEEK samples made from the same bolt and tested in a different triaxial apparatus and are, therefore, expected to have similar properties. Overall, PEEK is significantly softer compared to aluminium. The aluminium standard properties are compared against calibration measurements in this apparatus. PEEK measurements are compared against LF measurements in the old LF apparatus described in Szewczyk et al. (2016).

For Aluminium, the calibration procedure entailed conducting three cyclic loadings ranging from 1 to 20 MPa at a controlled rate of 20 MPa per hr. Upon reaching 10 MPa, the material was maintained at this stress level to facilitate low-frequency measurements. Concurrently, ultrasonic velocity was recorded at 60-s intervals throughout the process. To ensure accuracy in the force measurements, corrections were made to account for the confining pressure. A linear correction method was employed to ensure that the recorded force remained unaffected by variations in external pressure. Similarly, ultrasonic measurements were adjusted for the sample’s length based on data from strain gauges, providing a direct measure of deformation. Additionally, a linear correlation was applied to mitigate the effects of confining pressure on the ultrasonic system’s traveltime, thereby preventing any external pressure changes from influencing the results. These methodical corrections were critical for the integrity of the calibration process.

The calibration results are summarized in Table A2 and visualized in Fig. A2. The Young’s modulus value for aluminium is, on average, marginally higher than anticipated, while Poisson’s ratio is underestimated. Young’s modulus in PEEK is slightly lower, and Poisson’s ratio is almost the same. Attenuation measurements are, on average, zero in aluminium as one would expect. P-wave velocities are, on average, 0.2 per cent higher than the reference material and S-wave velocities 0.6 per cent. The underlying reasons for these observations are thoroughly discussed, and error estimates are computed in section  Experimental Error analysis.

Fig. A2 presents a detailed analysis of the averaged data derived from Table A2. Aluminium exhibits a slight reduction in Young’s modulus at a frequency of 0.5 Hz, with a general trend of decreasing modulus values as the frequency increases. In the case of Poisson’s ratio, the data indicate a relative stability for frequencies exceeding 1 Hz. Conversely, the Young’s modulus for PEEK-11 displays a consistent pattern across various frequencies, with only minor deviations observed. The Poisson’s ratio for PEEK-11, however, exhibits more significant fluctuations when compared to baseline measurements obtained from the older LF apparatus. Regarding attenuation, which ideally should approach zero in standard samples, our investigation reveals that aluminium shows minor fluctuations around this ideal value. In contrast, the attenuation for PEEK-11 demonstrates a decreasing trend as frequency increases.

APPENDIX B: BRINE COMPOSITION

Table A3 compiles various brine solutions used in previous studies involving µCT scans. This was the basis for our decision to choose a 12.5 wt. per cent brine.

APPENDIX C: EXPERIMENTAL ERROR ANALYSIS

Low-frequency measurements were conducted using factory-calibrated sensors and equipment, with adjustments made only in specified instances. The primary errors identified include inaccuracies in calculating Young’s modulus and Poisson’s ratio from measurements on the aluminium standard. Furthermore, strain gauge measurements, as detailed in Lozovyi & Bauer (2019), were subject to errors such as angular misalignment, temperature variations, confining pressure effects, Wheatstone bridge input voltage inaccuracies and transverse sensitivity. However, these errors were considered negligibly small. Additionally, differences between glued and taped strain gauges were minimal, less than 1 per cent, as reported in Szewczyk et al. (2016) and Lozovyi et al. (2017), affirming the reliability of the measurement techniques used. The error of the ultrasonic velocity measurements is estimated to be less than 1 per cent (Lozovyi & Bauer 2019). The margin of error might be elevated for S-wave velocity because it frequently overlaps with the pronounced converted P-wave signal. The systematic error of low-frequency measurements is estimated to be 1.1 for Young’s modulus and −0.03 for Poisson’s ratio derived from calibration measurements.

During the imbibition-drainage test (see section 4.2), the ultrasonic measurements yielded lower properties compared to the low-frequency measurements. We believe this discrepancy is due to a systematic error in the measured low-frequency Young’s moduli and Poisson’s ratios, likely attributed to strain gauges. Additionally, there was a drift in the measured strains (see Fig. 8) over a longer period, although not within the duration of a single low-frequency measurement. We assumed the same length for the ultrasonic data because it could not be accurately determined due to the drifting strain gauges. When comparing the V_p/V_s ratios, the results showed expected trends, indicating that the relative trends are valid.

C1 Error from CT scanner

Uncertainties from CT scan evaluations in AVIZO are summarized in Table A4. We converted the properties using the fluid and sample densities. Although the uncertainties appear high, their impact on velocities and moduli is negligible.

The error from the CT scanner is significant due to variations in grey values between scans. The complexity of the test, which takes a week to complete, introduces instrumental deviations between the first and last scans. Alemu et al. (2013) report an average error of |$\pm$| 7–10 per cent in calculated saturations for CT-image analysis, which aligns with our measurements.

APPENDIX D: CT SCAN IMAGES |$\mathrm{CO_2}$|-TEST

Fig. A3 and A4 show CT scan images for the |$\mathrm{CO_2}$|-test.

APPENDIX E: ATTENUATION MEASUREMENTS

Fig. A5 illustrates attenuation measurements conducted at seismic frequencies. In the imbibition-drainage experiments, attenuation is highest during the dry measurement and exhibits variability at full saturation. Both imbibition and drainage stages display comparable attenuation levels. Frequency-dependent trends are evident, except for a notable anomaly at 146 Hz, where attenuation increases during full saturation but decreases at other saturation stages.

For the |$\mathrm{CO_2}$| (b) experiments, the greatest attenuation is observed at 4 per cent free |$\mathrm{CO_2}$|⁠, followed by measurements at 100 per cent water + |$\mathrm{CO_2}$|⁠. Attenuation responses at other saturation stages are similar, showing no discernible frequency-dependent trend.

APPENDIX F: RESULT TABLES

Tables A5 and A6 show the results of measured parameters.