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Compared with the core‐parallel fracture case, the modeling results of the core‐perpendicular case are qualitatively different from the experiment (Fig. 5.8b). Although the impact of the fluid substitution on the Young's modulus (through changes in the drained normal fracture compliance and the storage fracture compliance) is clearly seen, the model predicts much smoother and more monotonic changes than the experiment. (Note that, in Fig. 5.8b, similar to the core‐parallel case, assuming a larger shear modulus by 4.5% results in better predictions for the intact core case.) This is because equation 5.10 assumes that reductions in the fluid bulk modulus by scCO₂ injection occur throughout the sample (i.e., quasi‐static assumption). In reality, because of the finite frequency of the experiment, the effect of bulk fluid modulus reduction is limited by the fluid pressure diffusion length (the relative amplitude decays to e ^–1 over the distance) in the water‐saturated part of the rock. This is given approximately by (e.g., Pride, 2003)

(5.12)

where k ₀ is the static permeability of the rock (34.5 md) and η _f is the fluid viscosity (0.47 cP). For a frequency of 1.5 kHz and the material properties in Table 5.2, δ _D is computed as 12.5 mm (Carbon Tan #1 core) and 13.6 mm (Carbon Tan #2 core).

Attenuation of the Young's modulus appears to be more strongly affected by the finite frequency effect than the stiffness (the real part of the moduli), as seen in the increasing attenuation during the experiment (Fig. 5.5 a and c). Particularly, for the core‐perpendicular fracture case (Fig. 5.5 b and d), the frequency effect is quite dramatic. Figure 5.9 shows the observed changes during scCO₂ injection into Frac IIa and Frac IIb samples, extracted from Figure 5.5b and d and correlated to the X‐ray CT images of scCO₂. In contrast to the behavior predicted by the Gassmann model, the largest changes in the Young's modulus occur once scCO₂ reaches the fracture (pore space CO₂ saturation of the core ~2% for Frac IIa and ~5% for Frac IIb), corresponding to a peak and a subsequent rapid drop in the attenuation. Attenuation exhibits secondary increases as scCO₂ invades more of the fracture and the rock matrix.

Table 5.2 Modeling parameters

Parameter	Symbol	Value
Undrained frame Young's modulus	E U	20.5 GPa
Matrix frame shear modulus	G	7.60 GPa
Matrix Biot‐Willis effective stress coefficient	α	0.769
Matrix porosity	φ	16.7%
Mineral bulk modulus	K s	38 GPa
Mineral density	ρ s	2,650 kg/m3
Water density		989 kg/m3
scCO2 density		535 kg/m3
Water bulk modulus		2.46 GPa
scCO2 bulk modulus		0.04 GPa
Sample length	H	10.0 cm
Sample diameter	a	3.81 cm
Fracture aperture (core parallel)	h	0.54 mm
Fracture aperture (core perpendicular)	h	0.26 mm

Note: Unlisted poroelastic parameters such as α , B, K _U , K _D , E _D , M are derived from these parameters using Gassmann relationships for isotropic poroelastic media. The fluid bulk modulus K _f is given as a function of the scCO₂ pore saturation via

The rock’s effective stress coefficient α is computed from undrained bulk modulus K _U = E _U G / 3(3G – E _U ), K _f , and the porosity φ via

The abrupt changes in the attenuation can be explained by coexisting two attenuation mechanisms. The first mechanism is the effect of the heterogeneous and patchy scCO₂ distribution in the rock matrix, as seen for the intact cores. As assumed by the patchy‐saturation model (e.g., Azuma et al., 2013), pressure in these patches does not equilibrate with the water in the surrounding rock if ultrasonic waves are used for the measurements (i.e., there is no fluid flow across the boundaries). However, with the current, sonic‐frequency measurements, seismic waves cause higher pressure within the stiff, water‐saturated part of the rock, which drives the water toward the softer, scCO₂‐saturated part. With increasing volume of the rock where the two fluids coexist, the overall attenuation of the sample increases as scCO₂ is injected into the sample.

The second mechanism is the attenuation caused by the interaction between a high‐porosity, high‐compliance fracture and a lower‐porosity, low‐compliance rock matrix. At the initial, water‐saturated state, attenuation in the sample is large. This is because seismic waves induce enhanced pressure changes within the compliant fracture, which drives dynamic fluid exchange with the matrix and dissipates a large amount of energy. This attenuation becomes small once the compliance of the fluid in the fracture increases, and the fracture‐driven motions of the water in the rock matrix diminish.

This reduction is expected to be large if scCO₂ fills the fracture faster than the surrounding matrix, resulting in similar wave‐induced pressures within the fracture and the matrix. As the CT images in Figure 5.9 indicate, this situation appears to be the case in our experiments: fast passing of the fluid is seen along high‐permeability sedimentary layers (Frac IIb test) and the sample‐Mylar interface (Frac IIa test). Figure 5.10 compares the scCO₂ saturations for only the fracture and for the entire core for Frac IIb test. This plot indicates that when rapid changes in the Young's modulus and the attenuation occur (core scCO₂ saturation ~5%–7%), nearly 50% of the fracture is already saturated with scCO₂, although from the CT images, most of the matrix near the fracture is still fully saturated with water. The final saturation of the fracture is also much higher (~83%) compared with the average over the entire core (~17.5%). As more pores near the fracture are invaded by scCO₂, because wave‐induced pore pressure in the matrix also decreases, the attenuation caused by the fluid movement between the fracture and the matrix increases again. Additionally, introduction of scCO₂ in the second half of the core increases attenuation further.

The above effect can be examined semiquantitatively by comparing the changes in the Skempton coefficients of the fracture and the rock matrix during fluid substitution. Under uniaxial stress, assuming an undrained state (i.e., only the “driving force” of the fluid flow is examined here), pore pressure and axial stress in the rock matrix are related by

Figure 5.9 Young's modulus E and its related attenuation a_E compared with the scCO₂ distribution in Frac IIa and Frac IIb cores. The maximum decreases in E occur when scCO₂ reaches the fracture but before the saturation in the adjacent rock matrix becomes high. Attenuation also drops abruptly at the same time. Green boxes indicate the responses due to possible relaxation of the fractures resulting from the invasion of scCO₂. However, the attenuation starts to recover as more scCO₂ saturates the fracture and the rock matrix. Red oval indicates artifacts due to CT image registration errors. Blue oval indicates fast passing of scCO₂ along a Mylar sheet, mimicking the behavior of a highly permeable sedimentary layer or a core‐parallel fracture in the sample: (a) Young's modulus and (b) Young's modulus attenuation.

(5.13)

Figure 5.10 Comparison between the scCO₂ saturations in Frac IIb sample within only the fracture versus the entire core. The saturation within the fracture was determined from CT images. When rapid changes in the Young's modulus and the attenuation occur in Figure 5.8 (core scCO₂ saturation ~5%–7%), nearly 50% of the fracture is already saturated with scCO₂, although from the CT images, most of the matrix near the fracture is still fully saturated with water. The final saturation of the fracture is also much higher (~83%) compared with the entire core (~17.5%). Note that the second data point in the plot was affected by CT image registration errors near the bottom of the sample.

where the fractor 1/3 is due to the fact the radial total stresses τ ₁₁ and τ ₂₂ are zero. B can be expressed as (e.g., Mavko et al., 1998)

(5.14)

where K _f is the bulk fluid modulus for mixed water and scCO₂. Also, from equations (5.5) and (5.6), by assuming no fluid motion in the fracture (i.e., ),

(5.15)

where we used α ^F ~ 1 and η _M ~ φ ^F /K _f , and φ ^F ~ 1.

The resulting uniaxial Skempton coefficients B/3 and B ^F /3 are plotted in Figure 5.11 for a range of scCO₂ saturation within the matrix rock and the frac ture. At the initial stage (the fracture and matrix pores are fully water saturated), there is a large difference in the uniaxial Skempton coefficient values, which would result in rapid fluid exchange and energy dissipation. At the intermediate stage, fast passing of scCO₂ results in large increases in the scCO₂ saturation in the fracture (50% is assumed here) compared with the matrix (0% is assumed). This results in much smaller differences in the Skempton parameters, which indicates much less energy loss from the fluid movement. At the final stage (scCO₂ saturations of the fracture 83%, matrix 16.5% [calculated from the saturation of the entire core, 17.5%]), the difference becomes somewhat larger, although much less than the initial stage. Although admittedly this analysis ignores the dynamic effect of fluid flow between a fracture and a matrix, it indicates that the fluid‐substitution‐induced changes in the dynamic pore pressure differences may be responsible for the observed unusual behavior of the dynamic Young's modulus and the attenuation.

Figure 5.11 Changes in the uniaxial Skempton coefficients B/3 and B^F/3 during Frac IIb scCO₂ injection test. The difference between the fracture and the matrix is proportional to the wave‐induced pressures between them if the fluid within the pores and the fracture is not allowed to move. A large difference at the initial water‐saturated state reduces when the scCO₂ saturation of the fracture increases by fast passing of the fluid to the fracture (here, the matrix scCO₂ saturation is assumed to be unaffected). In the final stage, the difference between the coefficients recovers to some degree.

When the two mechanisms (near‐monotonic decreases in the modulus and increases in attenuation from the heterogeneous invasion of scCO₂ in the matrix, and nonmonotonic changes in the modulus and attenuation from interactions between the matrix and the fracture) are combined, the observed complex behavior of the sample's dynamic properties can be explained. However, dynamic poroelastic modeling of the scCO₂‐invasion‐induced dynamic rock property changes would be needed for their quantitative validation.