GEOPHYSICS, VOL. 80, NO. 4 (JULY-AUGUST 2015); P. D403–D415, 15 FIGS., 2 TABLES. 10.1190/GEO2014-0289.1
A
Saturation-path dependency of P-wave velocity and attenuation in sandstone saturated with CO2 and brine revealed by simultaneous measurements of waveforms and X-ray computed tomography images
Yi Zhang1, Osamu Nishizawa1, Tamotsu Kiyama1, and Ziqiu Xue1
saturation model by using exponent-type autocorrelation. The nonunique relationship of velocity to the CO2 fraction was explained by the model with a correlation length from an order of tens of millimeters in drainage to an order of hundreds of microns in imbibition. We also examined the potential influence of the fractal dimension of CO2 distribution by using the fractal patchy saturation model and a von-Kármán-type autocorrelation function. The fractal dimension of pore-space modulus as related to CO2 distribution was estimated on the basis of the CT images. The fractal dimension during imbibition found a larger value than that during drainage. The change in fractal dimension had a limited effect on velocity and attenuation changes. These nonunique relationships might have a significant impact on the use of seismic methods for estimation of CO2 volume in an inhomogeneous storage reservoir.
ABSTRACT We performed a laboratory experiment to measure compressional (P)-wave velocity and its attenuation in low-permeability striped sandstone at ultrasonic frequency during drainage (CO2 injection) and imbibition (brine injection). X-ray computed tomography (CT) images of the rock sample were simultaneously recorded during the injections. On the basis of the CT images, we calculated the CO2 saturation inside a prolatespheroid-shaped volume (the first Fresnel zone) that enclosed the wave-propagation path. The relationship of P-wave velocity and that of attenuation to the CO2 fraction were then obtained. Both relationships showed nonuniqueness with remarkable hysteresis in a drainage-imbibition cycle. To explain the nonunique relationship of P-wave velocity to the CO2 fraction, we investigated the continuous random patchy
tion and seismic velocity/attenuation is unique or nonunique during drainage and imbibition. Several experimental studies have been performed in an attempt to understand the seismic wave characteristics in partially saturated porous media (Murphy, 1982; Knight and Nolen-Hoeksema, 1990; Gist, 1994; Cadoret et al., 1995; Geller et al., 2000; Monsen and Johnstad, 2005; Xue and Lei, 2006; Shi et al., 2007; Lebedev et al., 2009; Xue et al., 2009; Lebedev et al., 2013). In conventional displacement experiments between gas (such as air or nitrogen) and water, some experimental studies have evidenced clear hysteresis in the relationship between P-wave velocity/attenuation and fluid saturation in a drainage-imbibition cycle (Domenico, 1976; Knight and Nolen-Hoeksema, 1990; Yin et al., 1992; Cadoret et al., 1995,
INTRODUCTION Recently, there has been increasing interest in using time-lapse seismic methods to monitor and estimate the amount of injected supercritical CO2 (scCO2 ) in deep subsurface reservoirs for CO2 geologic storage purposes (Arts et al., 2008; Daley et al., 2008; Spetzler et al., 2008; Cairns et al., 2012; Li et al., 2013). To convert observed seismic attributes to a degree of scCO2 saturation and to accurately map scCO2 spatial distribution in a reservoir, an understanding of the relationship between seismic wave velocity and its attenuation (hereafter velocity/attenuation) and scCO2 saturation is essential (Caspari et al., 2011; Ajo-franklin et al., 2013). One of the important issues is whether the relationship between scCO2 satura-
Manuscript received by the Editor 17 June 2014; revised manuscript received 30 December 2014; published online 10 June 2015. 1 Research Institute of Innovative Technology for the Earth (RITE), Kizugawa, Kyoto, Japan. E-mail:
[email protected];
[email protected]; kiyama@ rite.or.jp;
[email protected]. © 2015 Society of Exploration Geophysicists. All rights reserved. D403
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1998). Cadoret et al. (1995, 1998) study the relationships of seismic wave velocity and attenuation to the air/water saturation during drainage (by drying) and imbibition (by depressurization). The extensional wave (similar to P-wave) velocity and attenuation were higher during drainage than imbibition at the same fluid saturation level. They also observe the distribution of two fluids by X-ray computed tomography (CT), and they find that imbibition by depressurization produces a very uniform distribution of fluids, whereas drainage by drying produces rather concentrated distribution with mesoscopic patches. However, their approach of degassing for drying and depressurization for wetting does not mimic the fluid flow of natural flooding experiments. Moreover, the experimental results of Yin et al. (1992) also show a strong dependence of the extensional-wave attenuation on the saturation history, with the attenuation peak located at 90% water saturation in drainage and 98% water saturation in imbibition. The conventional drainage and imbibition experiments thus suggest the nonuniqueness in the relationship between seismic velocity (or attenuation) and the ratio of wetting fluid to nonwetting fluid. Nakagawa et al. (2013) also show faster velocities and smaller attenuation of extensional waves during CO2 drainage than during water imbibition. However, the large wavelength compared with the fluid distribution scale in their experiment could not reveal the effect of the small-scale inhomogeneity of fluid saturation on velocity/attenuation. On the other hand, the Alemu et al. (2013) show no evident differences in velocity and attenuation in drainage and imbibition of CO2 /brine displacements. In all the experiments so far, CO2 saturation is estimated for the entire core or the entire slicing image, not for the specific wave-propagation path between the source and receiver. However, the Fresnel-zone concept of wave propagation proposes that the wave velocity and attenuation are mostly affected by the materials within a prolate-spheroid-shaped volume, the major axis of which is located on the propagating path; CO2 saturation outside this volume has little effect on wave velocity and attenuation. Although the question of whether the relationship between Pwave velocity/attenuation and scCO2 -brine saturation is unique remains unclear, most of the theoretical models suggest nonunique relationships between P-wave velocity/attenuation and scCO2 -brine saturation. On the basis of the concepts of mesoscale patchy saturation and wave-induced flow, White (1975) and Dutta and Odé (1979) consider the simple spherical geometry of patches and present a method to predict velocity and attenuation for various saturation degrees and patch sizes. The model has been numerically verified (Carcione et al., 2003), and it is widely used in practice (Lei and Xue, 2009; Rubino and Velis, 2011; Ajo-franklin et al., 2013). Le Ravalec et al. (1996) present a two-scale model, using pore/crack and patchy scales to explain the hysteresis of P-wave velocity between drainage and imbibition. Toms et al. (2007) develop a 3D continuous random model (CRM) by considering that the size and spatial distribution of patches are characterized by a continuous random function that is expressed by the Gaussian or exponential autocorrelation function. Müller et al. (2008) later suggest that the distributions include not only the correlation scale lengths but also the fractal dimensions. Using the von Kármán power spectral density function, they derive equations for velocity and attenuation with various fractal dimensions. Caspari et al. (2011) apply the CRM to estimate representative patch size and CO2 -brine saturation on the basis of well-log data at the Nagaoka
CO2 storage site in Japan. More recently, Masson and Pride (2011) analyze the effects of patchy saturation distribution using forward numerical simulations based on artificial models generated by an invasion percolation algorithm. Lo and Sposito (2013) discuss the impact of the effect of hysteresis in hydraulic properties on the behavior of acoustic waves using numerical simulations. The actual patch in porous media is imaged by Iglauer et al. (2011) by means of micro-CT during CO2 -brine drainage and imbibition. They find a power law distribution between the cluster size and its population, suggesting fractal distribution. To overcome the drawbacks of the previous experimental studies of the velocity/attenuation-saturation relationship(s), we conducted a unique experiment that determined exact scCO2 -brine saturation within the first Fresnel zone of the P-wave paths. The P-waveform was obtained during scCO2 -brine drainage and imbibition at the same time that scCO2 -brine saturation was imaged using a highspeed medical CT scanner. In this paper, we first introduce the details of our experiment, including sample characterization, experimental equipment, and procedure, and the methods used to obtain saturation from CT images and relative P-wave attenuation from wave amplitudes. Next, we present the measured waveform, saturation maps, and changes in P-wave velocity/attenuation during CO2 drainage and brine imbibition. Then, we describe the relationships between P-wave velocity/attenuation and CO2 -brine saturation as determined by the local saturation images within the first Fresnel volumes. Finally, we analyze CO2 distribution by measuring the fractal dimension of the distribution of the pore-space modulus associated with the saturation value distribution. On the basis of experimental results and model calculations, we discuss the effects of fractal dimension and patch size on P-wave velocity/attenuation.
MATERIALS AND METHODS Sample and equipment We use a reservoir sandstone collected from a confidential location as a sample. Its mineral composition and other petrological properties are as follows: The sample’s constituent minerals are quartz (64%), plagioclase (12%), Fe-rich cement material (16%), and clay (3%). Open- and cross-Nicol images of the sample are shown in Figure 1a and 1b, respectively. The subcore scale striped texture of the sample is remarkable. It includes layers of high and low porosity that, respectively, correspond to the dry-state low- and high-CT number (a Hounsfield unit number representing the relative density) zones (Figures 1c and 2). The porosity of the sample is 13.5%, and its permeability is approximately 0.72 × 10−15 m2 (0.72 mD). A column of 35 mm in diameter and 70 mm in length was prepared for this experiment. A schematic description of the experimental setup is shown in Figure 3. A medical X-ray CT scanner (Toshiba Medical Systems Corp. Aquilion ONE TSX 301A) with a high-speed helical scan system was used for imaging. The shortest scan interval was 2 s, allowing us to obtain quick snapshots of the CO2 flow inside the sandstone. In each scan, slices were reconstructed at 512 × 512 pixels and with a 0.5-mm interval between slices. The voxel size was 0.159 mm (width) × 0.159 mm (height) × 0.50 mm (depth), viewed from the wave-propagation direction. An upstanding, cylindrical, high-pressure vessel made of carbonfiber-immersed polyetheretherketone (PEEK) was inserted into the CT scanner’s gantry. Both ends of the vessel were sealed with
V P & Q−1 versus scCO2 -brine saturation
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striped texture in the rock sample causes velocity anisotropy. HowPEEK closures, using a conventional O-ring seal. To balance the force exerted from the confining pressure, carbon-fiber-reinforced ever, we are not engaged in the velocity anisotropy and are only plastic (CFRP) bars were used as joists to support the PEEK vessel concerned about relative velocity changes for the raypath channels. closures. The joists were aligned parallel to the scanner’s rotation Injection was controlled by high-precision syringe pumps at both axis. Both ends of the CFRP joists were fixed to steel frames located ends of the sample; one end was used for drainage, and the other outside the scanning area. The wavepath between the source and was used for imbibition (Figure 3). The injection direction was set receiver transducers (lead zirconate titanate [PZT]) was aligned parnear perpendicular to the striped layers, with a small intersecting allel to the scanner axis, avoiding the X-ray shadows of PZTs on the angle. The confining pressure was set at 12 MPa. Inlet and outlet rock image. The lead wires of the PZTs were connected to very fine pressures were approximately 10 MPa. Throughout the experiment, coaxial cables (0.35 mm in diameter) and fed out from the vessel the temperature was set at 50°C. through 1/16-in. PEEK tubes that were attached to the vessel cloThe temperature and pressure were above the critical point sures by ordinary high-performance liquid chromatography (HPLC) (30.95°C and 7.38 Mpa) of CO2 . These conditions were maintained fittings. The tubes were connected to other small steel vessels, and throughout the experiment, keeping CO2 in a supercritical state. To the fine coaxial cables were connected to conventional coaxial-type feedthroughs (Nishizawa, 1997; Yoshida, 2001). Both ends of the sample were attached to the PEEK end pieces, where the 1/16-in. PEEK tubes were connected using HPLC fittings for charge and recharge of injection fluids. For temperature control, a carbon-coated sheet heater was mounted on the lateral face of the vessel. It was coated with aramid-fiber thermal insulation cloth. Because of the X-ray transparency of the CFRP, PEEK, carbon sheet, and aramid-fiber, the images show the inside of the rock sample without strong shadows. Although an “X”shaped artifact appears in the section image perpendicular to the scanner axis, it does not impede calculation of the CO2 -brine saturation values within the first Fresnel volume of the wavepropagation paths. Therefore, we can simultaneously observe elastic waves and CT images under controlled temperature and pressure conditions. Figure 3 outlines the fluid line control system and the waveform measurement system. Three Figure 1. (a) Open and (b) cross Nicol images of the sample under an optical micropairs of PZTs were placed on the cylindrical samscope. (c) Pore-radius distribution of the sample. High- and low-CT number layers show ple’s lateral surface, allowing nine wave-propadifferent pore-size distributions. gation paths (Figure 2). A one-cycle 500-kHz sine wave burst was generated and amplified for driving PZTs. The transmitted wave was amplified by each low-noise amplifier and then recorded at a 10-ns sampling rate and 16-bit resolution. The wavelength of P-waves at the source frequency was 6–7 mm. It is worth mentioning that in most of the previous experiments for simultaneous measurements of elastic wave and CT images, the propagation direction was located along the core axis, where a larger gradient of CO2 saturation could exist. This gradient of saturation is undesirable and gives uncertainty in the relationship between CO2 saturation and seismic velocity/attenuation. Our setting avoided this problem. Shear (S)-wave velocity is necessary to calculate shear modulus and bulk modulus. We measFigure 2. (a) A photo of a vertical section of the sample. The transducer installation is urer S-wave velocity for three states, dry, brinedepicted. Three pairs of PZTs (1-1′, 2-2′, and 3-3′) were installed at both sides of the saturated, and CO2 -saturated rock, using another sample, configuring nine channels of wave-transmitting paths. (b) X-ray CT image of sample from the same rock. We used the S-wave the dry sample. (c) The section image of rock porosities. The dashed ellipsoid is the first Fresnel zone of the 1-1′ channel. PZTs for the source and receiver. Generally, the
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increase the X-ray CT image contrast between brine and CO2 , the selected potassium iodide (KI) concentration was 12.5 wt.%.
end at a flow rate of 0.20 mL∕ min (0.0038 kg∕ðs · m2 Þ), and then it was increased to 0.5 mL∕ min (0.0094 kg∕ðs · m2 Þ).
Experimental procedure
Estimation of fluid saturation by X-ray computed tomography
Measurements were conducted for a dry stage, a full-brine saturation stage, and multiple stages of drainage and imbibition with stepped increases in the flow rate, as shown in Table 1. In drainage (CO2 injection into core saturated with KI brine), the initial flow rate was set at 0.1 mL∕ min (0.0011 kg∕ðs · m2 Þ). However, no apparent flow of CO2 was observed in the X-ray CT images after 24 min. The injection rate was then set to 0.2 mL∕ min with a 50-min duration, and then to 0.5 mL∕ min with a 20-min duration. After CO2 drainage, there was a 16-h interruption during which time the fluids could redistribute in the sample. Then, the imbibition was commenced by injecting brine into the sample from the other
The time-lapse variations in CT number linearly correspond to the change in the local CO2 fraction in the rock sample. Through simple computation of CT numbers during a CO2 -flooding experiment, the in situ CO2 saturation can be continuously monitored in 3D voxels. The following equation yields the degree of CO2 saturation (SCO2 ) from voxel CT numbers (Akin and Kovscek, 2003):
SCO2 ¼ cðCTobs − CTsat brine Þ ¼
CTobs − CTsat brine sat ; CTsat CO2 − CTbrine
(1)
sat where CTsat brine , CTCO2 , and CTobs are the CT numbers of each voxel for the brine-saturated core, CO2 -saturated core, and each examining scan, respectively. In this equation, c is the coefficient that relates the CO2 saturation to the difference in the CT number from each observation scan and the brine-saturated state. When an elastic wave transmits through a saturated sample, the wave is affected by rock properties in a specific area, and not the entire sample area. Therefore, the wave velocity/attenuation should only be relevant to CO2 saturation, where most of the wave energy concentrates. Using X-ray CT imaging, the CO2 saturation of each voxel can be acquired and the local average CO2 saturation in a Fresnel volume (Appendix A) along the wave raypath can be estimated (see Figure 2). This kind of saturation value describes an effective CO2 fraction relevant to wave velocity/attenuation.
Estimation of wave velocity and relative attenuation We use the first-arrival time for velocity measurements and the amplitude of the first wave cycle for attenuation measurements. The firstarrival time was picked using a waveform processing tool called the TSpro (Lei and Xue, Figure 3. A schematic of the equipment, including injection components, velocity measurement components, and the imaging area of the X-ray CT. 2009). The crosscorrelation technique used in this software ensures the accuracy of the first arrival time. The P-wave velocity was calculated using the ratio of transmitting distance to the first arrival time. Table 1. Experimental procedure and parameters. The attenuation of the P-wave, which is represented by the inverse of the quality factor Q−1, can be related to the variation of Flow rate Duration Flow the wave amplitude through the following equation (Aki and RiProcess Code (mL∕ min) (min) volume (PV) chards, 2002): Drainage
Imbibition
D-1 D-2 D-3 I-1 I-2
0.1 0.2 0.5 0.2 0.5
24 50 20 180 60
0 1 1 3 3
26 08 08 87 23
AðxÞ ¼ A0 e−πfx∕cQ ;
(2)
where A0 , f, c, and AðxÞ correspond to the source amplitude, the wave frequency, the wave velocity, and the wave amplitude at x, respectively. The transmitting distance x equals the product of the wave velocity (c) and traveltime (t).
V P & Q−1 versus scCO2 -brine saturation Equation 2 can be reformulated as
ln
AðxÞ −πft : ¼ A0 Q
(3)
Because there was only one wave receiver along each wave raypath and the source amplitude is unknown, the absolute attenuation could not be directly obtained from recorded waveforms. However, the absolute attenuation is not necessary if one is only interested in the relative changes in attenuation due to fluid displacement (Lei and Xue, 2009). If we take the amplitude of the dry state (AR ) as a reference and assume its Q value as QR , we can formulate the ratio of the amplitudes of each state (Ai ) to that of the dry state as
ln
Ai −1 ¼ πfðtR Q−1 R − ti Qi Þ; AR
(4)
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images corresponds to the measured waveforms (Figure 4). CO2 was injected from the bottom, whereas brine was injected from the top. An inhomogeneous distribution of CO2 is shown in each image. During drainage, the high-CO2 saturation zones correspond to the high-porosity zone shown in Figure 2b. The CO2 plume shape and distribution during brine imbibition are different from those during drainage. Besides the differences in plume shapes, the sizes of high-CO2 areas seem smaller in imbibition and the spatial pattern is more uniform. These results can be seen in images 4 and 8 of Figure 5. Figure 6, which displays the selected cross sections near wave channel 1-1′ at the same CO2 saturation, further demonstrates this change in spatial distribution of CO2 . High- and low-CO2 -saturation areas are concentrated during drainage, whereas CO2 is more
a) SCO 2 =0.105
where tR and ti are the traveltime of the dry state and each state, respectively, and Qi is the quality factor of each state. Here, we know tR and ti from measurements and we give the QR a proper value (such as 50, for sandstone). Therefore, by matching the amplitude spectra of Ai ðfÞ and AR ðfÞ, we can obtain the best Q−1 i for the waveform. The best match is obtained by minimizing the following objective function:
X
ti tR Q−1 − ln R
Ai AR
∕πf
− Qi
0.209 0.327 0.401 0.470 0.390
2 :
0.026
(5)
Subsequently, we can obtain the relative attenuation ΔQ−1 by subtracting the Q−1 i value in a fully brine-saturated state. Here, we only care about the relative attenuation, and the choice of absolute Q−1 R (0–0.03) results in small uncertainty in the final result (this fact was verified by a trial-and-error evaluation).
Drainage Imbibition 13
b)
RESULTS
Waveforms An example of time-lapse waveforms corresponding to raypath channel from 1 to 1′ (1-1′) is shown in Figure 4a. During drainage, the first-arrival delays and waveform amplitude show a significant reduction as the CO2 saturation in the Fresnel volume (SCO2 ) increases. Then, they recover during imbibition. It is interesting to note that the first arrival continues to delay after imbibition starts, although the CO2 saturation is decreasing. Changes in amplitude during drainage and imbibition also lead to remarkable changes in power spectral density. Figure 4b shows the power spectral density calculated from the autoregressive model algorithm. The initial spectral peak, approximately 400 kHz in the dry state, shifted to approximately 330 kHz during CO2 -brine injection. Furthermore, the low-frequency (10–200 kHz) area shows correlation between power spectra and CO2 saturation: usually, the higher the CO2 saturation, the lower the spectral power. CO2 distribution Figure 5 shows the time-lapse variation of the CO2 saturation map for the section parallel to the scanner axis. The sequence of
14
15
16 Time (ms)
17
18
19
10 11
10 10
Power Spectral Density (relative scale)
Experimental results
0.300 0.208 0.103
10 9
10 8
0.026 0.327
0.105
0.209
0.401 0.103
0.208
0.300 0.470 0.390
10 7
10 6
10 5
10 4
Drainage Imbibition
100
1000 Frequency (kHz)
10000
Figure 4. (a) Time-lapse waveforms of path 1-1′ during drainage and imbibition. The colored part denotes the first period beginning from the first-arrival time. The local CO2 -saturation values inside the first Fresnel zone are marked. (b) The power spectrum of each waveform. The corresponding CO2 saturations are denoted. The color and line style correspond to the waveforms in panel (a).
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uniformly distributed during imbibition. The difference in distribution is mainly led by the capillary force field, which, in turn is related to the pore-size distribution (Zhang et al., 2014). The effect of the capillary force field is enhanced by the invasion-percolation mechanism. Velocity, attenuation, and saturation Figure 7 shows time-lapse changes in CO2 saturation within the first Fresnel volume along each raypath. It shows the relative velocity reduction and the relative change in wave amplitude A with reference to those in the fully brine-saturated state, from the upper to the bottom panel. The fluctuation in the plotted changes shows stability of the measurement. Generally, the short-path channels have small fluctuations of velocity and wave amplitude, whereas the long-path channels have large fluctuations due to the lower signal-to-noise ratio. The relatively smooth curves of saturation suggest the high stability of saturation estimated from X-ray CT images. The drainage and imbibition can be distinguished from the locations of the saturation peaks. However, for several channels (e.g., 1-1′, 2-2′, and 21′), the increase in CO2 saturation continued after the start of imbibition. This may be caused by slight but unavoidable pore-pressure fluctuations caused by the shift from the drainage to imbibition, leading to CO2 volume expansion and exsolution.
Figure 5. Time-lapse CO2 saturation images corresponding to waveforms (Figure 4). The Fresnel volume for channel 1-1′ is denoted by the red dashed line.
Changes in the velocity and amplitude appear in a sequential order in association with the locations of PZTs and flow fronts of CO2 and brine. These changes suggest that the CO2 injection front can be effectively monitored. The measured absolute values of V P are different for each raypath in the fully brine-saturated state, which range from 3.3 to 3.5 km∕s. Overall, within the saturation scope of this study (SCO2 ¼ 0%– 45%), the velocity and amplitude decrease when CO2 saturation increases during drainage, and a reverse process appears when CO2 saturation decreases during imbibition. At the end of drainage, the velocity reduction is approximately 12%, and the CO2 saturation is approximately 0.45. The curves show retardant phenomena such that the velocity and amplitude continue decreasing when drainage turns to imbibition, where CO2 saturation starts decreasing. The largest velocity reduction during imbibition is approximately 14%, whereas the largest reduction in the amplitude ratio is approximately 90%. Different channels show large differences in velocity and amplitude, reflecting the rock heterogeneities. Usually, when differences in saturation in different channels are large, the differences in velocity and amplitude are also large. Figure 8 shows the relationship between CO2 saturation and velocity, and Figure 9 shows the relationship between CO2 saturation and relative attenuation for channels 1-1′, 2-2′, and 3-3′. It is
Figure 6. (a) Porosity map across wavepath 1-1′. (b) Comparisons of CO2 distribution maps at the same average CO2 saturations in the Fresnel volume (wavepath 1-1′) in drainage and imbibition.
V P & Q−1 versus scCO2 -brine saturation
has larger values than those in drainage. This may suggest that the CO2 -brine distribution affects the wave energy transmission and wave velocity. Compared with velocity, the attenuation shows more complex changes. Figure 10 shows a crossplot of the velocity and attenuation of channel 1-1′. There is a negative correlation between velocity and attenuation. As drainage proceeds, velocity reduces and attenuation increases. As the imbibition proceeds, velocity and attenuation recover; however, the velocity-attenuation relationship is not the same during drainage and imbibition. Imbibition shows larger attenuation than drainage at the same velocity.
interesting to note that the velocity reduction and the change in relative attenuation are not single-valued functions with respect to CO2 saturation, but they depend on the saturation history. Here, we take channel 1-1′ as an example. Immediately after the start of imbibition, the velocity continued to decrease with decreasing CO2 saturation. The largest velocity reduction appeared at the early stage of imbibition, which was when the CO2 saturation had already been recovered to approximately 30%. The CO2 saturation patterns that were evidently different from those during drainage (Figure 6) appeared at this stage. At this saturation, the velocity reduced by 6% during drainage and approximately 13% during imbibition. A similar phenomenon was observed in the amplitude attenuation. Generally, the attenuation values during imbibition were significantly higher than those during drainage. Furthermore, the peak of attenuation in imbibition appears at smaller CO2 saturation and
Imbibition
1-1' 1-2' 1-3'
Amplitude ratio
VP ratio
SCO2
0.5 Drainage 0.4 0.3 0.2 0.1 0 1 0.975 0.95 0.925 0.9 0.875 0.85 1 0.8
DISCUSSION
2-1' 2-2' 2-3'
0.6 0.4 0.2 0 0
50 100 150 200 250 300 0 Time (min)
50 100 150 200 250 300 Time (min)
0
50 100
We have shown a clear relationship between P-wave velocity and CO2 saturation. Now, we will try to interpret these results by using the continuous random patchy saturation model developed by Toms et al. (2007) and Müller et al. 3-1' (2008). The patchy size and the fractal dimension 3-2' 3-3' of the fine structures in velocity fluctuation play a major role in this model. We first describe some basic equations of the random patchy saturation model. We then try to interpret our experimental results by applying the exponential autocorrelation function and a von-Kármán-type autocorrelation function to observe changes in the correlation length of the velocity fluctuation caused by patchy saturation and the fractal dimension of the fine structures in velocity fluctuation. Finally, we discuss the implications of the present study for seismic monitoring of 150 200 250 300 350 CO2 storage sites.
Time (min)
Figure 7. Time-lapse changes of CO2 saturation (SCO2 ), normalized P-wave velocity (V P ), and amplitude data (presented as measured velocity or amplitude normalized in the brine-saturated state) for nine channels.
1 VP ratio
0.975 0.95
1-3'
0.13
Drainage Imbibition Wood Hill
Gassmann model
K sat ¼ K dry þ B2 M; 0.3
(6)
2-1'
2-2'
2-3'
where K sat and K dry denote the bulk modulus of the rock under fully saturated and dry states, respectively (Müller et al., 2008; Mavko et al., 2009). Here, B is the Biot-Willis coefficient given by
3-1'
3-2'
3-3'
B ¼ 1 − K dry ∕K 0 ;
0.95 0.925 0.9
0.95 0.925
(7)
where K 0 is the bulk modulus of the grains, and M is the pore space modulus given by
0.9 0.875 0.85 0
Effective modulus theory
The well-known Gassmann model is commonly used to explain the relationship between P-wave velocity and fluid saturation in partially saturated rocks. It can be formulated as
0.9
0.875 0.85 1 0.975 VP ratio
1-2'
0.925 0.875 0.85 1 0.975
VP ratio
1-1'
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0.2
0.4 0.6 SCO2
0.8
10
0.2
0.4 0.6 SCO2
0.8
10
0.2
0.4 0.6 SCO2
0.8
1
Figure 8. The V P ratio of nine channels with respect to CO2 saturation. The GH and GW bounds are shown for comparison.
M ¼ ½ðB − ϕÞ∕K 0 þ ϕ∕K f −1 ;
(8)
where K 0 , K f , and ϕ are the grain bulk modulus, fluid bulk modulus, and porosity, respectively.
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The Gassmann model basically deals with the relaxation states in which the pore pressure equilibrium has been reached. This equilibrium is achieved where the fluid flow has been stopped after deformation of the porous medium. In elastic-wave propagation, the equilibrium state can be achieved when the fluid-flow diffusion lengths are sufficiently smaller than the wavelength of the elastic wave. For two-fluid saturated rock, the bulk modulus can be obtained by the Gassmann model with one of two mixing rules: those of Wood and Hill (Müller et al., 2008). Specifically, if CO2 and brine coexist in the same pore or the same small pore network, the Wood average is used to describe the bulk modulus of the fluid mixture. In other words, the Wood average is used in cases in which CO2 and brine are well mixed with a characteristic patch size that is much smaller than the fluid diffusion distance during one period of the propagating wave. The Wood average is a saturation-weighted harmonic average (also called the Reuss or isostress average; Mavko et al., 2009) of the bulk modulus of two fluids: K f1 and K f2 . It is expressed as
K f ¼ ðS1 ∕K f1 þ S2 ∕K f2 Þ−1 ;
(9)
where S1 and S2 are the saturations of fluids 1 and 2, respectively. This description represents the fact that the wave-induced pore-
a)
K sat ¼ ðS1 ∕K 1 þ S2 ∕K 2 Þ−1 ;
VP ¼
sffiffiffiffiffiffi κN λd ¼ ; ωη
0.03
0.01
(13)
where κ, ω, and η are the local permeability, the angular frequency of the wave, and the viscosity of the fluid, respectively. Here, N is the combination of poroelastic moduli of dry (H dry ) and saturated (H sat ) rocks, as shown in the following equation:
0 0.06
2-2'
0.04
ΔQ –1
(12)
The fluid diffusion distance is given by (Toms et al., 2007)
Drainage Imbibition
0.05
(11)
4 H ¼ K þ μ ¼ ρV P 2 : 3
0.02
b)
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðK sat þ 4μ∕3Þ∕ρ;
where μ is the shear modulus and ρ is the density of the saturated rock. When the S-wave velocity is not available for the bulk modulus calculation, the bulk modulus K in the above equations (equation 6, 9, and 10) can be approximately replaced by P-wave modulus H (Müller et al., 2008; Mavko et al., 2009):
1-1'
0.04
(10)
where K 1 and K 2 represent the bulk modulus when the rock is fully saturated by fluids 1 and 2, respectively. The P-wave velocity then can be calculated by
0.06 0.05
ΔQ –1
pressure disturbance at the two-fluid interface can be relaxed during the wave period. The Hill mixing rule is used when the wave-induced pore pressure disturbance is in an unrelaxed regime, in which the characteristic patch size is significantly larger than the fluid diffusion distance and a stiffening effect appears due to the more incompressible fluid. In this situation, the overall elastic properties of the porous medium will be considered as the mixture of the two porous media having different pore fluids, resulting in
N¼M
0.03
H dry : H sat
(14)
0.02 0.01
c)
0.06
0 0.06
0.04
ΔQ –1
ΔQ –1
0.04 0.03
0.02
0.01
0.01
0
0.2
0.4
0.6
0.8
1
SCO
2
Figure 9. Relative attenuation (ΔQ−1 ) with respect to CO2 saturation.
Drainage Imbibition
0.03
0.02
0
1-1'
0.05
3-3'
0.05
0
3
3.1
3.2
3.3
3.4
VP (km/s)
Figure 10. Relationship of relative attenuation (ΔQ−1 ) to wave velocity (V P ).
V P & Q−1 versus scCO2 -brine saturation We conducted an independent experiment of measuring the S-wave velocity to estimate μ. Table 2 lists the necessary parameters for the modeling. Figure 8 shows that most of our measurements are within the two bounds of the Gassmann model. The data points during drainage are near the Hill bound, whereas the points during imbibition are located near the intermediate range of the two bounds. The results for drainage and imbibition cannot be simply predicted by the Gassmann model. The estimated diffusion distance is 0.045– 0.16 mm, which is much smaller than the wavelength in ultrasonic waves (6–7 mm). Random patchy saturation model Considering a homogeneous porous frame saturated with a random patchy distribution of fluids, Toms et al. (2007) and Müller et al. (2008) formulate an approach to calculate the effective complex P-wave modulus. In their model, the spatial fluctuation of the pore space modulus M may follow the exponential, the Gaussian, or the von Kármán autocorrelation function. Using the von Kármán autocorrelation function in random patchy saturation models allows us to describe the effects of the characteristic patch size and the fractal nature of the fine-scale structure on the effective complex P-wave modulus, consequently, the wave velocity and attenuation. In the following, we briefly show some key concepts of the fractal patch model. The spectral density of the 3D von Kármán autocorrelation function ΦðkÞ for unit variance is given by
ΦðkÞ ¼
a Γðυ þ 32Þ π 3∕2 ΓðυÞð1 þ k2 a2 Þυþ3∕2
The von Kármán spectral density function ΦðkÞ is related to the effective P-wave modulus H eff by the following equations:
H eff ¼ H W ð1 þ δ½τξ2 þ ðτ − 1ÞξÞ;
(19) wherepkffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L is theffi wavenumber of the slow P-wave given by kL ¼ iω¯η∕ðκNÞ, a is the correlation length; κ is the permeability; η¯ is the viscosity of the mixing fluids calculated by the saturationweighted arithmetic average; F2f1 is the generalized hypergeometric function; H W and H H are the Gassmann-Wood (GW) and the Gassmann-Hill (GH) P-wave moduli, respectively; and δ is a scaling coefficient between the two moduli:
δ ¼ ðH H − H W Þ∕H W ;
υ ¼ E þ 1 − D;
(16)
where E is the Euclidean dimension. When the Hurst exponent equals 0.5, equation 15 is identical to an exponential spectral density function:
ΦðkÞ ¼
γa3 ; ð1 þ k2 a2 Þ2
(17)
where γ represents a constant. In this form, only the correlation length has a role in spectral density.
(20)
where H H and H W are calculated from equations 9 and 10. The P-wave modulus gives the effective phase velocity of the Pwave:
υeff ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RefH eff g∕ρ;
(21)
and its attenuation (the inverse of the quality factor)
Q−1 eff ¼ ImfH eff g∕RefH eff g.
(15)
where a is the correlation length, Γ is the Gamma function, k is the wavenumber, and υ is the Hurst exponent (Müller et al., 2008; Sato et al., 2012). The Hurst exponent υ is related to fractal dimension D as
(18)
pffiffiffi ξ ¼ 2υðkL aÞ2 F2f1 ð½1; υ þ 1; 1∕2; −k2L a2 Þ þ i 2π 2 k3L ΦðkÞ;
3
;
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(22)
The parameters a, kL , and υ in the von Kármán spectral density function affect the P-wave phase velocity and its attenuation. For example, when assuming a constant Hurst exponent of 0.5, the velocity-saturation relationship depends on the product of kL and a or the ratio of patch size a to diffusion distance kL (Figure 11). The wavenumber kL falls in the range 0.6 × 104 to 2.4 × 104 m−1 . Depending on kL , the characteristic patch size can significantly vary. For instance, when the CO2 saturation is 0.2, the estimated characteristic patch size varies from 0.01 to 10 mm. In the following section, on the basis of the von-Kármán-type random distribution of patches (or exponential, when υ is 0.5), we try to understand the hysteresis of P-wave velocity and attenuation. We first address the role of characteristic patch size on velocity- and attenuation-saturation relationships when the Hurst exponent equals 0.5. We then discuss the effects of fractal dimension of random patchy distribution using the fractal dimension estimated from the CO2 -brine saturation images.
Table 2. Parameters for modulus calculation. State Dry Brine-saturated CO2 -saturated
V S (km∕s)
V P (km∕s)
Density ρ (kg∕m3 )
Shear modulus μ (Pa)
Bulk modulus K (Pa)
P-wave modulus H (Pa)
2.112 2.037 2.042
3.024 3.458 3.122
2226 2407 2290
9.933 × 109 9.986 × 109 9.550 × 109
7.120 × 109 1.546 × 1010 9.590 × 109
2.036 × 1010 2.878 × 1010 2.232 × 1010
Zhang et al.
Role of characteristic patch size of patchy distribution To explain the observed P-wave velocity and attenuation, we use the exponential autocorrelation function (see equation 17), which is identical to a Hurst exponent of 0.5 in the von Kármán correlation function. The model includes the change in the characteristic correlation length in random patchy saturation. Figure 12a and 12b shows the calculated relationships of P-wave velocity and attenuation to the CO2 fraction for different correlation lengths, together with the measured P-wave velocity and attenuation in the channels 1-1′, 2-2′, and 3-3′. The calculated curves indicate that the velocity and attenuation are significantly affected by the characteristic patch size; even at the same CO2 -brine saturation, different values appear depending on the patch size. Most of the velocity values during drainage are in the region of large correlation lengths (0.5–5 mm) of calculated curves. In the velocity plots, the correlation length decreases with increasing CO2 saturation. The velocity values during imbibition indicate smaller correlation lengths compared with those in drainage. The velocity values of the initial stage of imbibition (the transition part) indicate a reduction of the correlation length, changing roughly from 0.5 to 0.1 mm. As the imbibition continues, the correlation length gradually recovers from 0.1 mm to larger sizes. The changes in attenuation indicate almost similar trends to the changes of correlation length in velocity. The attenuation values during initial imbibition change along the curves with specific correlation lengths. For example, the attenuation in channel 3-3′ changes along the calculated curve of correlation length 0.5 mm at low CO2 saturation. However, the values of characteristic patch sizes for the attenuation do not correlate well to those for the velocity data. Furthermore, the peaks in the measured attenuation during drainage cannot be predicted by the model. Figure 12b indicates that increase of the measured attenuation values correspond to the decrease of patch size. However, the patch size estimated from attenuation varies widely, and it is not consistent with the patch size estimated from velocity for the same CO2 saturation. So far, the reason of inconsistency is not clear, but the patch-size changes in velocity are more stable, and thus they will be more reliable. Moreover, the estimated correlation lengths can be as small as 0.1 mm, which is below the image voxel length
(0.5 mm). We cannot reveal the full range of patch sizes by means of the present medical X-ray CT technique.
Role of fractal dimension of patchy distribution In the previous section, we saw that although the observed velocity hysteresis can generally be explained by variations in patch size, inconsistencies still exist. In addition to patch size, the theoretical work of Müller et al. (2008) addressed the potential role of fractal dimension in velocity and attenuation. Therefore, an attempt to evaluate changes in the fractal dimension during drainage and imbibition may be useful. Müller et al. (2008) use P-wave modulus to reflect the velocity fluctuation. We use the pore space modulus M for calculating velocity inhomogeneity because M is the major factor that controls velocity fluctuation due to inhomogeneous CO2 saturation (see equations 6–9). Fluctuation of M is practically useful because the value is directly calculated from the CO2 saturation. We first try to obtain the fractal dimension of the M-value fluctuation due to CO2 and brine distribution in a porous sandstone using CT images. On the basis of the fractal patchy saturation model, we then discuss whether the change in fractal dimension could cause the differences in P-wave velocity in drainage and imbibition. Our data, including the CO2 -brine saturation and porosity of each pixel, allow us to calculate M at each pixel (see equation 8) and obtain the spatial distribution of M. Applying the 2D fast Fourier transform (FFT) to the 2D fluctuation of M, we can obtain the power spectral density of the fluctuation in M distribution. Then, by measuring the power-law decay between the power spectral density and wavenumber in the large wavenumber region, we can
a)
1 0.975
VP ratio
D412
Velocity
a = 5 mm
1
Wood Hill 1-1' 2-2' 3-3'
0.95 0.5 0.3
0.925
0.2
0.9
0.1
0.875 1 0.975
Velocity
11 8
Wood Hill
6
0.925
b)
> 100
0.07 0.06
0.9 1
Attenuation
0.04 0.03
0.875 0.02 0.85 0
0.1 0.2
0.05 0.3
3 2
ΔQ –1
VP ratio
0.95
0.85
a kLa = λd
0.5 0.7 1
< 0.1
0.2
0.4
0.6
0.8
1
SCO
2
Figure 11. Velocity-saturation relationships as a function of the product of the wavenumber of the flow diffusion distance kL and the characteristic patch size a. The product can also be theoretically written as a ratio of patch size a to the diffusion distance kL .
2
0.01
a = 5 mm
0 0
0.2
0.4
0.6
SCO
0.8
1
2
Figure 12. Effects of characteristic correlation length a on (a) velocity and (b) attenuation. The measured velocity and attenuation data of the channels 1-1′, 2-2′, and 3-3′ are plotted.
V P & Q−1 versus scCO2 -brine saturation obtain the fractal dimension for the M fluctuation in the rock sample (Russ, 1994). The normalized fluctuation intensity is computed as
M 0 ðrÞ ¼
¯ MðrÞ − M ; ¯ M
Taking the average slope β in the large wavenumber region of ε2 ðkÞ map, we obtain the fractal dimension D by the following equation:
D ¼ ð6 þ βÞ∕2.
(25)
(23)
¯ is the mean of pore space modulus M. The power specwhere M trum of 2D fluctuation in M is given as a function of the wavenumber vector k:
ε2 ðkÞ ¼ FðM 0 Þ FðM 0 Þ ;
D413
(24)
where FðM 0 Þ and FðM 0 Þ are the Fourier transform of M 0 and its conjugate, respectively.
Figure 13. (a) Two images of pore space modulus M distribution from the drainage and imbibition processes with the same saturation state (SCO2 0.29; see the saturation images in Figure 6). (b) Estimation of fractal dimension of M distribution (a) by the slope of log spectral density against the log wavenumber of M fluctuation. (c) Changes in fractal dimension with respect to CO2 saturation during drainage and imbibition stages.
Figure 13 shows the estimated fractal dimension based on the FFT spectral slope fitting. Figure 13a shows examples of M value distribution in drainage and imbibition processes with the same saturation (0.29, corresponding to Figure 6b). The average slope of the power spectral during drainage is approximately −2.808, and the fractal dimension D is 1.599. The slope during imbibition is approximately −2.745, and the fractal dimension is 1.628. Figure 13c shows a crossplot of the fractal dimension with CO2 saturation. The overall fractal dimensions range between 1.55 and 1.7. Although the estimated fractal dimensions show large fluctuations due to the reconstruction noise on the CT images, the images during imbibition generally have larger fractal dimensions than those during drainage. Figure 14 shows the calculated velocity and attenuation with respect to the CO2 fraction for the Hurst exponent υ ¼ 0.45 and 0.3 (2D fractal dimensions of 1.55 and 1.7, respectively) with the correlation length a ¼ 0.1 and 1 mm. The measured velocity and attenuation of channel 1-1′, 2-2′, and 3-3′ are also plotted in the same figure. The P-wave velocity slightly reduces with the increasing fractal dimension, irrespective of the correlation length. Attenuation, however, either increases or decreases with the fractal dimension, depending on the correlation length. It should be noted that the apparent dimension we estimated is based on the voxel-scale resolution, and it cannot cover the subvoxel scale. In the subvoxel scale, the actual fractal dimension may be different from the values
Figure 14. Effects of fractal dimension D of P-wave modulus distribution on (a) velocity and (b) attenuation. The Hurst exponent υ ¼ 2 − D is used for the von Kármán function. The measured data of channels 1-1′, 2-2′, and 3-3′ are plotted.
Zhang et al.
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estimated from the voxel scale, which may involve changes of the fractal dimension during drainage and imbibition. Therefore, the actual changes in fractal dimension may be different than what the present results show. Thus, increasing the fractal dimension brings the velocity-saturation relationship closer to the GW bound (Müller et al., 2008). It is, therefore, reasonable to consider the large-sized patches during drainage and the smaller-sized patches during imbibition, along with the slight increase of the fractal dimension in the patch distributions during imbibition for the same saturation states.
Application to reservoir monitoring Our experimental results indicate large hysteresis of velocity and attenuation during drainage and imbibition of CO2 -brine saturation. The nonuniqueness in P-wave change may have a significant meaning for reservoir monitoring via time-lapse seismic methods because the saturation state of the reservoir cannot be uniquely determined by velocity or attenuation. This nonuniqueness is caused by changes in patchy distribution of CO2 and brine. For example, even after CO2 injection, the velocity may continue to decrease and the attenuation may increase by the patch size reduction by brine imbibition. A retardant recovery of P-wave velocity may occur sometime after CO2 injection. The recovery is abrupt with respect to CO2 saturation. This can happen when there is an operation of forced imbibition of brine at a relatively high flow rate or a long-term imbibition by groundwater, leading to the redistribution of injected CO2 in a meso- to microscale. This is the case for highfrequency seismic waves such as those observed in well logging. On the other hand, for low-frequency seismic waves, such as those observed in seismic reflection profiling, the pore-pressure equilibrium is always fulfilled in whole patches and the seismic velocity changes along the GW bound. The relationship between seismic velocity and CO2 saturation becomes unique along the GW bound. However, geologic inhomogeneity in a reservoir formation (e.g., porosity and permeability fluctuation) brings large-scale variations in CO2 saturation and seismic velocity. In such cases, the reservoir will be considered as an aggregate of the large-scale materials, each having different velocities. The overall velocity of the reservoir will be (or will be close to) the GH average of the velocities of the materials. This is because the distances of the wave-induced fluid flow during one cycle of wave are much shorter than the inhomogeneity. Thus, it is important to consider the scaling factor between the seismic wave and the inhomogeneity produced by fluctuations in CO2 saturation.
CONCLUSION We demonstrate that the relationship between P-wave velocity and scCO2 saturation is nonunique during a drainage and imbibition cycle by conducting a robust experiment. In this experiment, we eliminate most of the uncertain factors in saturation value estimation by eliminating saturation gradients and using the first Fresnel volume for saturation calculations. Applying the continuous random patchy saturation model, the nonuniqueness is closely related to the change of patch size of brine and scCO2 . The experimental results and interpretation are consistent with most of the previous experimental results, and their interpretations provided for air (gas) and water (brine). The fundamental mechanism controlling the relationship of the velocity to scCO2 saturation is the difference in the velocity fluctuation, which is mainly controlled by the distribution of the pore-space modulus M.
The X-ray image analysis suggested changes in the space fluctuation of M between scCO2 drainage and brine imbibition. Therefore, simultaneous observation of the fluid distribution and seismic velocity gives clear-cut results not only for the velocity-saturation relationship but also for the mechanism responsible for creating the nonunique relationship. The nonuniqueness of the velocity-scCO2 saturation relationship will affect seismic monitoring methods, especially for high-frequency seismic methods. To understand the effect of scCO2 saturation on lower-frequency seismic waves, the effect of geologic-scale inhomogeneity on the M value distribution will be important. Note that the sandstone sample used in this study develops anisotropy (e.g., stripes of high and low porosity). This anisotropy characteristic itself, and the leading anisotropy in fluid distribution, may result in different velocity changes along directions. The anisotropy effects should be investigated in the future.
ACKNOWLEDGMENTS We thank Y. Guéguen, two anonymous reviewers, and the editors for their constructive comments and suggestions. This work was supported by the Ministry of Economy, Trade and Industry of Japan under the research contract Development of Safety Assessment Technology for Carbon Dioxide Capture and Storage.
APPENDIX A FRESNEL VOLUME When seismic signals transmit through media, previous studies suggest that the area around the wave raypath affects the wave propagation. The area is limited in a volume; namely, the Fresnel volume depends on the wave frequency (Spetzler and Snieder, 2004). The geometry of the Fresnel volume is similar to an ellipsoid (Figure A-1). For a point P within the Fresnel volume, the traveltime from the source S to P (τSP ), the traveltime from P to the receiver R (τPR ), and the traveltime from S to R (τSR ) satisfy the following equation (Cervený and Soares, 1992):
τSP þ τPR − τSR ≤
T : 2
(A-1)
Here, T is the wave period. If the difference between the traveltime of the ray that passes point P and the shortest traveltime is less than half the period, point P is within the Fresnel volume. If we denote the velocity by v (= const.), we can rewrite the boundary of the Fresnel volume as follows:
λ ιSP þ ιPR − ιSR ¼ : 2
(A-2)
Here, λ ¼ vT is the wavelength and ι denotes the distance between points. For our case, the wavelength is 6–7 mm, because the major
Figure A-1. Schematic illustration of a first Fresnel volume between a point source S and a receiver R.
V P & Q−1 versus scCO2 -brine saturation axis of the ellipsoid (equals to the distance between the source and receiver) is approximately 35 mm, and the length of the minor axis is approximately 14–16 mm.
REFERENCES Ajo-Franklin, J. B., J. Peterson, J. Doetsch, and T. M. Daley, 2013, Highresolution characterization of a CO2 plume using crosswell seismic tomography: Cranfield, MS, USA: International Journal of Greenhouse Gas Control, 18, 497–509, doi: 10.1016/j.ijggc.2012.12.018. Aki, K., and P. G. Richards, 2002, Quantitative seismology: Theory and methods: University Science Books. Akin, S., and A. Kovscek, 2003, Computed tomography in petroleum engineering research: Geological Society, London, Special Publications, 215, 23–38, doi: 10.1144/GSL.SP.2003.215.01.03. Alemu, B. L., E. Aker, M. Soldal, Ø. Johnsen, and P. Aagaard, 2013, Effect of sub-core scale heterogeneities on acoustic and electrical properties of a reservoir rock: A CO2 flooding experiment of brine saturated sandstone in a computed tomography scanner: Geophysical Prospecting, 61, 235–250, doi: 10.1111/j.1365-2478.2012.01061.x. Arts, R., A. Chadwick, O. Eiken, S. Thibeau, and S. Nooner, 2008, Ten years’ experience of monitoring CO2 injection in the Utsira sand at Sleipner, offshore Norway: First Break, 26, 65–72 Cadoret, T., D. Marion, and B. Zinszner, 1995, Influence of frequency and fluid distribution on elastic wave velocities in partially saturated limestones: Journal of Geophysical Research, 100, 9789–9803, doi: 10 .1029/95JB00757. Cadoret, T., G. Mavko, and B. Zinszner, 1998, Fluid distribution effect on sonic attenuation in partially saturated limestones: Geophysics, 63, 154– 160, doi: 10.1190/1.1444308. Cairns, G., H. Jakubowicz, L. Lonergan, and A. Muggeridge, 2012, Using time-lapse seismic monitoring to identify trapping mechanisms during CO2 sequestration: International Journal of Greenhouse Gas Control, 11, 316–325, doi: 10.1016/j.ijggc.2012.08.014. Carcione, M., H. B. Helle, and N. H. Pham, 2003, White’s model for wave propagation in partially saturated rocks : Comparison with poroelastic numerical experiments: Geophysics, 68, 1389–1398, doi: 10.1190/1 .1598132. Caspari, E., T. M. Müller, and B. Gurevich, 2011, Time-lapse sonic logs reveal patchy CO2 saturation in-situ: Geophysical Research Letters, 38, L13301, doi: 10.1029/2011GL046959. Cervený, V., and J. E. P. Soares, 1992, Fresnel volume ray tracing: Geophysics, 57, 902–915, doi: 10.1190/1.1443303. Daley, T., L. Myer, J. Peterson, E. Majer, and G. Hoversten, 2008, Timelapse crosswell seismic and VSP monitoring of injected CO2 in a brine aquifer: Environmental Geology, 54, 1657–1665, doi: 10.1007/s00254007-0943-z. Domenico, S. N., 1976, Effect of brine-gas mixture on velocity in an unconsolidated sand reservoir: Geophysics, 41, 882–894, doi: 10.1190/1 .1440670. Dutta, N., and H. Odé, 1979, Attenuation and dispersion of compressional waves in fluid-filled porous rocks with partial gas saturation (white model). Part I: Biot theory: Geophysics, 44, 1777–1788, doi: 10.1190/ 1.1440938. Geller, J. T., M. B. Kowalsky, P. K. Seifert, and K. T. Nihei, 2000, Acoustic detection of immiscible liquids in sand: Geophysical Research Letters, 27, 417–420, doi: 10.1029/1999GL010483. Gist, G. A., 1994, Interpreting laboratory velocity measurements in partially gas-saturated rocks: Geophysics, 59, 1100–1109, doi: 10.1190/1 .1443666. Iglauer, S., A. Paluszny, C. H. Pentland, and M. J. Blunt, 2011, Residual CO2 imaged with X-ray micro-tomography: Geophysical Research Letters, 38, L21403, doi: 10.1029/2011GL049680. Knight, R., and R. Nolen-Hoeksema, 1990, A laboratory study of the dependence of elastic wave velocities on pore scale fluid distribution: Geophysical Research Letters, 17, 1529–1532, doi: 10.1029/ GL017i010p01529. Lebedev, M., M. Pervukhina, V. Mikhaltsevitch, T. Dance, O. Bilenko, and B. Gurevich, 2013, An experimental study of acoustic responses on the injection of supercritical CO2 into sandstones from the Otway Basin: Geophysics, 78, no. 4, D293–D306, doi: 10.1190/geo20120528.1. Lebedev, M., J. Toms-Stewart, B. Clennell, M. Pervukhina, V. Shulakova, L. Paterson, T. Müller, B. Gurevich, and F. Wenzlau, 2009, Direct laboratory
D415
observation of patchy saturation and its effects on ultrasonic velocities: The Leading Edge, 28, 24–27, doi: 10.1190/1.3064142. Lei, X., and Z. Xue, 2009, Ultrasonic velocity and attenuation during CO2 injection into water-saturated porous sandstone: Measurements using difference seismic tomography: Physics of the Earth and Planetary Interiors, 176, 224–234, doi: 10.1016/j.pepi.2009.06.001. Le Ravalec, M., Y. Guéguen, and T. Chelidze, 1996, Elastic wave velocities in partially saturated rocks: Saturation hysteresis: Journal of Geophysical Research: Solid Earth, 101, 837–844, doi: 10.1029/95JB02879. Li, J., C. Liner, P. Geng, and J. Zeng, 2013, Convolutional time-lapse seismic modeling for CO2 sequestration at the Dickman oilfield, Ness County, Kansas: Geophysics, 78, no. 3, B147–B158, doi: 10.1190/ geo2012-0159.1. Lo, W.-C., and G. Sposito, 2013, Acoustic waves in unsaturated soils: Water Resources Research, 49, 5674–5684, doi: 10.1002/wrcr.20423. Masson, Y. J., and S. R. Pride, 2011, Seismic attenuation due to patchy saturation: Journal of Geophysical Research, 116, B03206, doi: 10.1029/ 2010JB007983. Mavko, G., T. Mukerji, and J. Dvorkin, 2009, The rock physics handbook: Tools for seismic analysis of porous media: Cambridge University Press. Monsen, K., and S. Johnstad, 2005, Improved understanding of velocity-saturation relationships using 4D computer-tomography acoustic measurements: Geophysical Prospecting, 53, 173–181, doi: 10.1111/j.13652478.2004.00464.x. Müller, T. M., J. Toms-Stewart, and F. Wenzlau, 2008, Velocity-saturation relation for partially saturated rocks with fractal pore fluid distribution: Geophysical Research Letters, 35, L09306, doi: 10.1029/2007GL033074. Murphy, W. F., 1982, Effects of partial water saturation on attenuation in Massilon sandstone and Vycor porous glass: Journal of the Acoustical Society of America, 71, 1458, doi: 10.1121/1.387843. Nakagawa, S., T. J. Kneafsey, T. M. Daley, B. M. Freifeld, and E. V. Rees, 2013, Laboratory seismic monitoring of supercritical CO2 flooding in sandstone cores using the split Hopkinson resonant bar technique with concurrent X-ray computed tomography imaging: Geophysical Prospecting, 61, 254–269, doi: 10.1111/1365-2478.12027. Nishizawa, O., 1997, New multi-wire type and co-axial type feedthroughs for an oil pressure-medium vessel: Bulletin of Geological Survey Japan, 48, 431–438. Rubino, J., and D. R. Velis, 2011, Seismic characterization of thin beds containing patchy carbon dioxide-brine distributions: A study based on numerical simulations: Geophysics, 76, no. 3, R57–R67, doi: 10.1190/ 1.3556120. Russ, J. C., 1994, Fractal surfaces: Plenum Press. Sato, H., M. Fehler, and T. Maeda, 2012, Seismic wave propagation and scattering in the heterogeneous earth: Springer. Shi, J., Z. Xue, and S. Durucan, 2007, Seismic monitoring and modelling of supercritical CO2 injection into a water-saturated sandstone: Interpretation of P-wave velocity data: International Journal of Greenhouse Gas Control, 1, 473–480, doi: 10.1016/S1750-5836(07)00013-8. Spetzler, J., and R. Snieder, 2004, The Fresnel volume and transmitted waves: Geophysics, 69, 653–663, doi: 10.1190/1.1759451. Spetzler, J., Z. Xue, H. Saito, and O. Nishizawa, 2008, Case story: Timelapse seismic crosswell monitoring of CO2 injected in an onshore sandstone aquifer: Geophysical Journal International, 172, 214–225, doi: 10 .1111/j.1365-246X.2007.03614.x. Toms, J., T. M. Müller, and B. Gurevich, 2007, Seismic attenuation in porous rocks with random patchy saturation: Geophysical Prospecting, 55, 671– 678, doi: 10.1111/j.1365-2478.2007.00644.x. White, J., 1975, Computed seismic speeds and attenuation in rocks with partial gas saturation: Geophysics, 40, 224–232, doi: 10.1190/1.1440520. Xue, Z., J. W. Kim, S. Mito, K. Kitamura, and T. Matsuoka, 2009, Detecting and monitoring CO2 with P-wave velocity and resistivity from both laboratory and field scales: Presented at SPE International Conference on CO2 Capture, Storage, and Utilization. Xue, Z., and X. Lei, 2006, Laboratory study of CO2 migration in water-saturated anisotropic sandstone, based on P-wave velocity imaging: Exploration Geophysics, 37, 10–18, doi: 10.1071/EG06010. Yin, C.-S., M. Batzle, and B. Smith, 1992, Effects of partial liquid/gas saturation on extensional wave attenuation in Berea sandstone: Geophysical Research Letters, 19, 1399–1402, doi: 10.1029/92GL01159. Yoshida, S., 2001, Convection current generated prior to rupture in saturated rocks: Journal Geophysical Research, 106, 2103–2120, doi: 10.1029/ 2000JB900346. Zhang, Y., O. Nishizawa, T. Kiyama, S. Chiyonobu, and Z. Xue, 2014, Flow behavior of supercritical CO2 and brine in Berea sandstone during drainage and imbibition revealed by medical X-ray CT images: Geophysical Journal International, 197, 1789–1807, doi: 10.1093/gji/ggu089.
