Characteristics of elastic wave dispersion and attenuation induced by microcracks in complex anisotropic media


 The mechanism of dispersion and attenuation induced by fluid flow among pores and microcracks in rocks is an important research topic in geophysical domain. A generalised frequency-dependent fourth-rank tensor is proposed and derived herein by combining Sayers's discontinuity tensor formula and Gurevich's squirt flow model. Furthermore, a proposed method for establishing a cracked model with cracks embedded in a transversely isotropic (TI) background medium is developed. Based on the new formulation, we investigate the characteristics of dispersion, attenuation and azimuthal anisotropy of three commonly encountered vertical crack distributions, including aligned cracks, monoclinic cracks and cracks with partial random orientations. We validate the developed model by comparing its predictions with those of the classic anisotropic squirt flow model for an aligned crack. The numerical analyses indicate that the azimuth is independent of frequency when the maximum attenuation is observed for all three crack distributions. In a low-frequency range in the case of an anisotropic background, the attenuation of the qP-wave is inversely proportional to velocity, whereas the attenuation of the qSV-wave is proportional to velocity. In addition, the inherent anisotropy of the rock does not significantly affect the dispersion and attenuation owing to squirt flow. Finally, to investigate the applicability of the theory, we model laboratory data of a synthetic porous sandstone with aligned cracks. Overall, the models agree well with laboratory data. The complex characteristics determined through this study may be useful for the seismic characterisation of fractured reservoirs.


Introduction
When rocks are subjected to external forces, a large number of cracks and microcracks are formed. Generally, these cracks and microcracks are not randomly oriented; therefore, the rocks exhibit elastic anisotropy (Sayers & Kachanov 1995). Directional regional stress is exerted on the rock such that existing cracks, including interparticle microcracks, further develop or close in a certain direction, which also causes anisotropy. In logging and seismic exploration, the preferential alignment of clays and organic materials in rocks, such as mudstone and shale, and alternating non-fractured thin layers with different velocities causes inherent anisotropy, which can generally be attributed to transversely isotropic (TI) media (Backus 1962;Wang 2002).
Fractured rocks typically contain two types of pore: hard and soft pores (or compliant pores). Microcracks are generally known as compliant pores, and their porosity is low (generally less than 0.1%); therefore, they are sensitive to external pressure and primarily affect the permeability of rocks. The background equant pores of rocks are generally known as hard pores, which are insensitive to pressure and have a higher porosity (Liu et al. 2016). In general, compliant and hard pores in rocks are likely to be filled with fluids. Therefore, understanding the interaction mechanism among pores, cracks and fluids as well as the characteristics of velocity dispersion and attenuation caused by them is crucial for identifying subsurface crack distributions and characterising oil and gas reservoirs (Galvin & Gurevich 2015).
The Gassmann equation is suitable for describing the rock physics of fluid-saturated rocks at the low-frequency limit (Gassmann 1951); Brown & Korringa (1975) extended the Gassmann equation to an anisotropy case. It is widely acknowledged that the mechanism of dispersion and attenuation caused by the presence of fluid is wave-induced fluid flow (WIFF) (Müller et al. 2010). According to the scale of fluid flow, WIFF can be divided into three categories. The wavelength-scale pressure gradient is called macroscopic or global flow. When the pressure gradient occurs on a scale far less than the wavelength and much larger than the pore scale, the flow in the rock is called mesoscopic flow. This paper, however, focuses on the WIFF resulting pore-scale pressure gradients. In addition, the fluids in this article mainly refer to liquid fluids, such as oil and water. At the microscale, fluids flow between pores of different sizes, directions and shapes. Such fluid flows are known as local or squirt flows, which primarily exist in the ultrasonic frequency band but may also be at play in seismic and logging frequency bands (Pride et al. 2004;Müller et al. 2010;Zhang et al. 2017). At a high-frequency limit, fluids do not have sufficient time to flow and reach a pressure gradient equilibrium during half-wave cycle, which makes the rock harder. For this squirt effect, Mavko & Jizba (1991) established isotropic squirt relations, and Mukerji & Mavko (1994) derived simple anisotropic squirt formulas. Chapman et al. (2002) established a microscope squirt flow model and the prediction made based on the model is in accordance with the Gassmann equation at low frequencies. Gurevich et al. (2009Gurevich et al. ( , 2010) extended Mavko's isotropic squirt formulas and established a frequency-dependent isotropic squirt model based on Murphy's microscale pore structure (Murphy et al. 1986).
Recently, many scholars have combined the squirt model and crack media to investigate crack-induced squirts. Guéguen & Sarout (2011) established an anisotropic squirt model considering the fluid flow among cracks and analysed the characteristics of aligned, randomly distributed, and vertical cracks. However, the model established by Guéguen & Sarout provides saturated elastic modulus only under the low-and high-frequency limit conditions. Combining linear sliding theory (Schoenberg & Sayers 1995) and the Gurevich squirt formulas, Collet & Gurevich (2016) established a frequencydependent aligned crack squirt model that crossed the intermediate frequency. Based on the scattering relations of a single crack, Galvin & Gurevich (2009 used multiple scattering theory to obtain the normal frequency-varying modulus of an aligned crack medium and determined the global frequency-varying modulus using the Johnson branch function approximation method ( Johnson 2001). Wu et al. (2015) established a squirt flow mechanism involving both coin-shaped and pinched-out cracks and investigated the effects of different scales and cracks on velocity dispersion and attenuation. Zhang et al. (2019) regarded fractures as a periodic structure of thin layers with high porosity and derived the relationship between the stiffness matrix and frequency by considering the effect of squirt flow and combining the improved Biot equation (Tang 2011). Alkhimenkov et al. 2020, 2020a, 2020b) performed a three-dimensional finite element numerical simulation to investigate the dispersion and attenuation of squirts at the pore scale, verified the previously proposed analytical model (Collet & Gurevich 2016) and analysed the dependence of velocity on the azimuth, incident angle and frequency.
However, most current investigations on squirt dispersion caused by microcracks have primarily focused on aligned cracks and an isotropic background. There are few studies pertaining to microcrack squirts in complex crack media and models with an inherent anisotropic background medium. Based on the discontinuous tensor formulas of Sayers andKachanov 1991, 1995) and Gurevich's squirt model ), a frequency-dependent fourth-rank elastic tensor is proposed and derived herein, through which various complex anisotropic fractured media can be easily constructed for modeling. In addition, a model for calculating the elastic modulus of rocks with cracks embedded into an inherent anisotropic background is also developed based on the Backus average, the results of which are compared with experimental data (Ding et al. 2017).
We investigated several complex medium models, including vertically aligned, vertical monoclinic and vertical cracks with partial random orientations (cracks randomly normal to the horizontal plane). This paper mainly describes the numerical simulation results of elastic wave dispersion and attenuation characteristics under monoclinic crack media conditions with different backgrounds, including both isotropic and inherent anisotropy, which is helpful for understanding the characteristics of velocity dispersion and attenuation induced by microcracks. 789

Derivation and establishment of frequency-dependent elastic tensor
Microcrack in rocks can be regarded as displacement discontinuities embedded in both isotropic and anisotropic background media. The effects of these displacement discontinuities can be quantitatively described by the Sayers discontinuity tensor equation (Sayers & Kachanov 1995;Sayers 2002), where S ijkl is the elastic compliance tensor of the entire medium; S 0 ijkl is the compliance tensor of the background medium, which is inversed by the stiffness tensor c 0 ijkl ; ΔS ijkl is the excess compliance tensor caused by the displacement discontinuities.
where ij is the second-rank tensor; ijkl is the fourth-rank tensor; B T are the normal and tangential compliances of the rth microcrack, respectively; n (r) i is the ith component of the normal vector of the rth microcrack; V is the volume of rock and A (r) is the area of the rth microcrack. It is assumed that B (r) T satisfies the rotation invariance and is independent of the tangential direction of the microcrack.
When the microcracks in the medium have the same size and parameters as in the case of dry rock, equations (3) and (4) can be simplified to equations (5) and (6), respectively.
= s where s = A∕V; A is the total area of cracks and s is the total surface of cracks in the unit volume of the rock; N is the number of cracks in the unit volume and Z dry N and Z dry T are the normal and tangential compliances of dry cracks, respectively. When the background medium is an isotropic medium, the expressions for the coin-shaped dry crack parameters with radius a are as shown in equations (9) and (10) (Hudson 1981;Schoenberg & Douma 1988): where the crack density e = Na 3 ∕V (Kachanov 1992). 790 In the low-frequency limit, the fluid pressure equilibrates throughout the pore space. Therefore, the anisotropic Gassmann equation (Brown & Korringa 1975), as shown in equation (11), can be used for saturated anisotropic rocks: where S s ijkl , S d ijkl and S g ijkl are the compliance tensors of saturated rocks, dry rocks and mineral grains, respectively, and the repeated parts of the subscripts are defined as summations. t is the total porosity, whereas g and f are the compression coefficients of mineral grains and fluids, respectively. Rewriting equation (11) using the Voigt 6 × 6 matrix notation, in which 11 → 1, 22 → 2, 33 → 3, 23 → 4, 13 → 5, and 12 → 6, results in where d and k are defined as follows: At a high frequency, the fluid does not have sufficient time to flow such that the fluid pressure reaches equilibrium among the pores; consequently, the rock becomes harder. The elastic coefficient of anisotropic rocks at high frequencies can be calculated using the anisotropic squirt equation (Mukerji & Mavko 1994). The frame matrix of the background medium is not affected by the compliance space filled by the fluid; therefore, the unrelaxed frame at high frequency is only related to the microcracks (Collet & Gurevich 2016). Hence, the unrelaxed elastic modulus S f ijkl of a rock at high frequencies can be calculated as follows: where Z uf N and Z uf T are the unrelaxed normal and tangential compliance of cracks, respectively; uf ij and uf ijkl are the secondand fourth-rank tensors of the unrelaxed cracks, respectively.
If Mukerji & Mavko's (1994) high-frequency squirt formula (equation (A.1)) is used directly, then it will be trivial to obtain the formulas of the high-frequency unrelaxed compliance tensor for complex anisotropic media. Equations (19) and (20), derived by Gurevich (2003) and Gurevich et al. (2009), are often used to calculate the parameters of the unrelaxed high-frequency state of crack media.
where c is the porosity of the microcracks; K f and K g are the bulk moduli of the fluids and mineral particles, respectively, and are the reciprocal of the compression coefficients.
In Appendix A, we prove that equation (19) can be used to correctly express the high-frequency limit state of complex media for any distribution of cracks.
The high-frequency unrelaxed elastic parameter S uf ijkl calculated using equation (15) is introduced into equation (11) to replace the dry rock elastic parameter S d ijkl such the saturated rock elastic parameters for high frequencies can be obtained. To calculate the elastic coefficient at an intermediate frequency, Gurevich et al. (2010) derived a modified fluid modulus, as shown in equation (21), by assuming fluid-filled compliant pores and dry hard pores: where is the aspect ratio of microcracks; is the viscosity coefficient of fluids; K f is the bulk modulus of fluids and J 0 and J 1 are the zeroth-and first-order Bessel functions, respectively. Substituting equations (19-21) into equations (17) and (18), the tensors of unrelaxed cracks as a function of frequency, which are suitable for the entire frequency range, are obtained as follows: As shown in equations (25) and (26), the fourth-rank tensor is required to characterise the dispersion characteristics of saturated rocks, even when B N = B T .
Equations (23) and (24) are introduced into equation (15) to calculate the frequency-dependent elastic parameters of unrelaxed rock. Subsequently, the parameters are introduced into the anisotropic Gassmann equation (11) to obtain the frequency-dependent elastic coefficient S s ijkl ( ) of a saturated medium with microcracks. Therefore, we can use S s ijkl ( ) to calculate the stiffness tensor c s ijkl ( ). In Appendix B, we present the stiffness matrix expression derived using the compliance matrix in a monoclinic medium.
Overall, the calculation processes are divided into three steps. First, combining Gurevich's high-frequency unrelaxed cracks (equations (19) and (20)) and modified fluid modulus (equations (21) and (22)), the frequency-dependent normal compliances of cracks (equations (25) and (26)) are calculated. Then the frequency-dependent crack parameters (equations (25) and (26)) are substituted into the ) to obtain the excess compliance matrix (equations (23) and (24)). Finally, the elastic modulus of the entire rock, calculated by the sum of additional compliance and background compliance, is substituted into the anisotropic ) to get the final frequency-dependent elastic modulus. Next, we give the corresponding specific excess compliances tensors for several different arrangements of cracks. 792 x 1

Models of cracks
We propose the use of the second-rank tensor and frequency-dependent fourth-rank tensor to calculate the dynamic elastic coefficient for crack media to investigate the elastic wave dispersion and attenuation induced by microcracks. This approach allows us to easily construct various complex crack media for modeling. We investigated various complicated models with cracks, as illustrated in figure 1, primarily including vertically aligned cracks, vertical monoclinic cracks (two groups of aligned cracks in different directions) and vertical cracks with partial random orientation (cracks that are randomly normal to the horizontal plane). Considering previous studies, we compared an aligned crack model with the model by Collet & Gurevich (2016) to validate our proposed method, whereas the monoclinic crack model and other models were primarily analysed in this study.

Vertically aligned cracks.
As shown in figure 1a, the normal direction of the microcracks is parallel to the Ox 3 axis, i.e. n = (n 1 , n 2 , n 3 ) = (1, 0, 0). Furthermore, only components uf 11 and uf 1111 exist, and the remainder are zero. As such, equations (23) and (24) are converted to equations (27) and (28), respectively: Next, we substituted equations (27) and (28) into equation (16) and changed the subscript to a 6 × 6 tensor to obtain the excess compliance matrix of vertically aligned microcracks, Collet & Gurevich's (2016) models are consistent with equation (29). In this study, second-rank and frequency-dependent fourth-rank tensors were used to construct the excess compliance matrix; this method may be more convenient for expanding the cracked state. For example, to express the additional compliance of a set of aligned cracks distributed in any direction, we are only required to set the normal unit vector of cracks to be n = (sin cos , sin sin , cos ), and the excess 793 compliance matrix is as shown in equation (30).

Vertical monoclinic cracks.
As shown in figure 1b, when the rock is subjected to stress in different directions and different periods, multiple sets of aligned cracks in different directions may be generated, which often occur in nature and result in azimuthal anisotropy. A nature example of related rocks can be referred to the actual rocks containing two or several sets of fractures as illustrated in Far et al. (2013). Without considering the interaction among the different groups of cracks, the normal direction of the first group of cracks is parallel to the Ox 1 axis, and the angle between the normal direction of the second group and Ox 1 is . The normal unit vector of the second group of microcracks is n = (cos , sin , 0), and the excess compliance matrix of the monoclinic cracks can be calculated using equation (30). Similarly, multiple groups of aligned cracks with arbitrary directions can be constructed.

Vertical cracks with partial random orientation.
The second-and fourth-rank tensors can represent the excess compliance matrix of cracks with a partial random distribution, as shown in figure 1c. In vertical cracks with partial random orientation, it is assumed that the normal directions of the cracks are randomly distributed in the horizontal plane. This model is suitable when the rock is subjected to high pressure in a certain direction (e.g. uniaxial high-pressure measurements in the laboratory), or when the subsurface rock is subjected to directional regional stresses for a significant amount of time (Sayers & Kachanov 1995). Referring to Gurevich et al. (2011) to calculate the crack density tensor in the spherical coordinate system, when the cracks are randomly normal to the horizontal plane, it is relatively easy to calculate the crack density tensor under the polar coordinates with x i as the polar axis (the polar axis is not a fixed Ox 1 or Ox 2 axis, but changes with the calculated targets).
Then the tensor In equations (31)

TI background medium
In addition to the crack models, the background medium should be considered in this study. Most previous studies were based on an isotropic background medium; however, the inherent anisotropic background was considered as well. The inherent anisotropy of unfractured rocks is caused by the preferential alignment of clays and organic materials, such as shale. In addition, in logging and seismic exploration, thin interbedded reservoirs with different velocities typically exhibit inherent anisotropy, particularly when the thickness of the thin layers is much smaller than the wavelength (Backus 1962). Moreover, most inherent anisotropy can be regarded as transverse isotropy (Wang 2002). For anisotropy caused by the interbedded layers with different velocities, we assume that the thickness of the layer is much larger than the scale of microcracks and significantly less than the wavelength. Hence, we calculated the dynamic stiffness matrix c s ij ( ) of each layer separately, and then used the Backus average (Backus 1962) to calculate the dynamic elastic coefficient of the equivalent medium. For thin layers with arbitrary anisotropy, when the axis is perpendicular to the formations, the stiffness matrix of the equivalent medium is as showed in Appendix C.
For inherent anisotropy apart from non-interbedded anisotropy, Guo et al. (2019) developed a theoretical method for calculating the compliance of cracks based on penny-shaped cracks embedded in a TI background medium. However, the calculation error increased with the angle between the fracture and isotropic planes, and TI background media must exhibit the characteristics of elliptical TI media. Hence, we propose a method for calculating the stiffness matrix of rocks with cracks embedded in the TI background medium. We assume that the rocks are composed of thin interbedded layers, and the inherent anisotropic media are equivalent to the interbedded layers, although the values from the solutions might not reflect the real situation. Then, we calculate using the previous steps. Next, we validate this method by comparing the results obtained with experimental data.
The elastic moduli of the inherent TI background medium are represented by five elastic parameters (c 11 , c 33 , c 44 , c 66 and c 13 ); therefore, we assume that the background medium comprises two types of isotropic thin layers with five parameters (the Lamé constants of each layer and the ratio of the layer thickness). Therefore, the parameters of the thin interbedded cracks should be accompanied by a unique set of solutions, as follows: where c ij is the parameter of the inherent TI background medium; 1 , 2 , 1 and 2 are the Lamé constants of the equivalent medium composed of two kinds of thin layers; h is the ratio of the thickness of the first layer to that of the entire medium. The parameters m, a and b are expressed as follows:

Numerical simulation
We performed a numerical simulation based on the previously mentioned models. We used the formulas derived by Carcione et al. (2012) to calculate the velocities and attenuation of monoclinic media, as shown in Appendix D The parameters of the background rock, including the porosity, can be calculated using the parameters of the mineral particles and the background porosity, as follows (Krief et al. 1990): where K and are the bulk and shear moduli of the rock containing pores, respectively; K g and g are the bulk and shear moduli of the mineral particles, respectively, and p is the porosity of the background. The parameters for the microcracks are listed in Table 1. The numerical simulation was performed in a plane perpendicular to the cracks based on the parameters listed. Figure 2 shows the relationship between the attenuation and azimuth of the qP-and qSV-waves in the medium with aligned cracks. It was observed that the attenuation pattern with the azimuth was consistent for different frequencies, although the magnitude of the attenuations differed; this indicates that the attenuation patterns with direction will not be affected by frequency. Additionally, figure 2 shows that the attenuation of the qP-wave was the strongest in the normal directions of the cracks and weakest in the tangential directions of the cracks. The attenuation of the qSV-wave was the strongest when the azimuth was 45°to the normal direction of the cracks and the weakest in the normal and tangential directions of the cracks.  Collet & Gurevich's (2016) model. The results from these two methods were similar, thereby validating our proposed method. However, the proposed method can be applied to more general models, including arbitrary anisotropic media induced by microcracks.

Model of aligned cracks
In figures 3 and 4, angles of 0°and 90°indicate incidence perpendicular and parallel to the crack surface, respectively. In the normal direction of the cracks, the velocity dispersion of the qP-wave was the maximum, whereas it was the weakest in the direction parallel to the cracks. For the qSV-wave, the dispersion at approximately 45°was the strongest; however, its magnitude can be ignored, unlike the velocity dispersion of the qP-wave, which is not shown in figure 4. The qSH-wave did not indicate dispersion, which is independent of frequency. The vibration direction of qP-wave particle-motion is similar to the phase velocity direction, so the qP-wave particle-motion is perpendicular to the crack at 0°incidence, resulting in the maximum dispersion and attenuation in this direction. As for qSH-wave, the pure S-wave oscillated horizontally on the cracked surface and did not squeeze the cracks; consequently, no dispersion or attenuation occurred (Liu et al. 2014). When the incident angle of qSV-wave is 0°, the particle vibration is parallel to the crack. Therefore, like qSH, there is no dispersion effect in this direction. With the increase of incident angle, the dispersion effect gradually increases. The qSV-wave variation repeats every ∕2 radians (Crampin 1978), so the dispersion and attenuation reach the maximum at 45°.

Models of monoclinic cracks
3.2.1. Isotropic background. Studies of multiscale crack media by Wu et al. (2015) showed that if a medium includes two groups of cracks with different parameters, such as the viscosity of fluids and aspect ratio of cracks, then the curves of velocity dispersion and attenuation as a function of frequency will contain multiple peaks and frequency bands. Hence, it can be concluded that when a rock contains two sets of cracks with different scales and directions, such as the monoclinic crack model, the velocity and attenuation curves will exhibit multiple peaks and multi-frequency bands. To reduce the interference of multiple peaks and emphasize the velocity dispersion and attenuation as a function of the angle of cracks, we considered two sets of cracks with the same parameters in the model, which differed only in terms of the cracked density (the number of cracks differed in different directions). The other parameters are listed in Table 1. The densities of the first and second sets of cracks was 0.02 and 0.04, respectively, and the normal azimuths of the second set of cracks were 30°, 60°and 90°(the angles were in the normal directions of the cracks, as shown in figure 5). Conclusions regarding the attenuation pattern for the model are generally similar, i.e. the attenuation of the qP-wave in the normal direction of the cracks is the strongest, whereas that in the tangential direction is the weakest; furthermore, the qSV-wave has the strongest attenuation at an angle of 45°to the normal direction of the crack, and the tangential and normal attenuations are almost zero. However, it was observed that the two sets of cracks exerted a superimposed effect on each other for the abovementioned patterns. As shown in figure 5, the attenuation pattern of the qSV-wave was a quatrefoil with strong symmetry, and the orientation and value of the attenuation changed with the angle and cracked density. When the angle of the two sets of cracks was small, the maximum attenuation of the qP-wave was between the normal directions of the two sets of cracks, near the side of the larger density of cracks. Meanwhile, when the angle was large, the qP-wave attenuation appeared at four maximum points, which were the normal directions of the two sets of cracks (figure 5c). Figures 6 and 7 show the variations in the qP-and qSV-wave velocity dispersions with frequency in the monoclinic medium. Based on figures 6 and 7, we can conclude that the qSV velocities for different directions at low and high frequencies  remained unchanged, unlike the qP velocities, particularly in the directions with lower velocities at the low-frequency limit. Although this conclusion is not absolute, it is applicable in most cases, as shown in figures 6 and 7. For low frequencies, the maximum attenuation of the qP-wave in figure 5a appeared at approximately 20°, whereas the minimum appeared at approximately 110°. In figure 6a, the 30°curve indicated the lowest velocity and was the closest to that of the 20°curve; the 120°curve indicated the highest velocity and was the closest to that of the 110°curve. For the qSVwave at low frequencies, the direction with the least attenuation shown in figure 5a indicated a lower velocity in the direction corresponding to figure 7a, wherein a larger attenuation corresponded to a higher velocity. Combining the observations with the previous aligned crack model, the following general conclusion can be made: for the low frequencies, in the direction where the qP-wave attenuation is significant, the velocity is low, whereas in the direction where the attenuation is small, the velocity is high, but the conclusion for the qSV-wave is contrary to that for the qP-wave. This conclusion is valid for a medium with an angle less than 60°between the two sets of cracks; however, a deviation occurs for the medium with a 90°angle. In general, the conclusion is valid for most situations, particularly for models with low-frequency bands and small angles among the cracks.

TI background medium.
When investigating the case involving a TI background medium with isotropic thin layers interbedded, as described in section 2, it is assumed that monoclinic cracks are embedded; therefore, the entire medium exhibits monoclinic anisotropy. The parameters of the thin layers are listed in Table 2, and the other parameters are listed in Table 1. Similar to the case involving an isotropic medium for the background, we set two sets of cracks, in which the densities of the first and second sets of cracks were 0.02 and 0.04, respectively, and the angles between the cracks were 30°, 60°and 90°. Figure 8 shows the attenuation patterns of qP and qSV with the direction (azimuth). The results are similar to figure 5. After defining the difference between the velocities of different frequencies and the low-frequency limit as the magnitude of dispersion, we obtained the dispersion patterns of the qP-and qSV-waves, as shown in figure 9. Figures 8 and 9 are similar, indicating that the dispersion of velocities was greater in the direction where the attenuations were greater. Moreover, this observation is similar to the case involving an isotropic background.
These results show that the inherent anisotropy of the rock may exert less effects on the squirt dispersion and attenuation caused by the microcracks. This property may enable one to understand and distinguish the rock anisotropy mechanism when it appears, e.g. when performing cross-frequency and multi-azimuth measurements for rocks.

Comparison with experimental data
To further verify the proposed method, we compared the theoretical predictions from this study with the experimental data of a saturated synthetic fractured rock sample by Ding et al. (2017). The synthetic rock comprised a powder of silica and clays, and the penny-shaped cracks pre-fabricated using polymeric material disks were randomly added onto the surface of each layer. The details of rock production are available in Ding et al. (2017). The manufacturing process of the rock yielded a thin interbedded structure, which is consistent with our proposed model. This rock was vacuumed and then immersed in water to ensure that the rock was saturated with water. Subsequently, the saturated rock was measured using an ultrasonic measurement system at 0.5 MHz, and some of the measurement results are shown in Table 3. According to the original 800   Table 3, the elastic coefficient of the blank rock and crack parameters can be calculated, and the results are shown in Table 4.
In our study, we calculated the compliance of cracks for an isotropic background, whose average P-and S-wave velocities were 3.417 and 1.975 km s −1 , respectively. Moreover, the porosity of the background medium was not measured. All rocks, including the blank rock without cracks, were measured in the water-saturated state. Therefore, we regarded a blank rock (including fluid and skeleton) as a whole, with zero porosity. However, in the fractured rock, owing to the destruction of the original structure by the addition of cracks, we assumed that the background porosity of the rock with cracks was small (0.02). Figure 10 shows the results of the comparison between the theoretical predictions and experimental data as a function of the incidence angle. To quantitatively analyze the comparison between the experimental results and the theoretical predictions, we listed the velocity values of each incident direction in detail and calculated the Euclidean distance between the theoretical and experimental results. The results are shown in Table 5. For the qP-and qSH-waves, the change trend of the isotropic background was inconsistent with the laboratory data. However, the modeling of the thin interbedded background was consistent with the laboratory results. Regarding the qSV-wave, our model was better than the model with an isotropic background, although they had a consistent trend. The theoretical prediction accuracy is improved significantly when an anisotropic instead of an isotropic background was considered. These phenomena indicate that the anisotropic background cannot be ignored when predicting the velocity of fractured rocks.
where soft is the soft porosity, which is the porosity of the microcracks c in this study. S dry−hp ijkl represents the compliance of the rock frame under high pressure. At this time, the soft pores are closed, and the frame compliance is equal to the background compliance without cracks, i.e. S dry−hp ijkl 4) The tensor G ijkl represents a fraction of the total compliance caused by the volumetric deformation of the microcracks.
G ijkl = ∫ f (Ω)n i n j n k n l dΩ, (A.5) where f (Ω) is the normalised crack-orientation distribution function, and its integral over all angles is equivalent to the unit quantity. Using the distribution of each crack to represent the tensor G ijkl , the following expression is derived: Based on equation (2), we express the excess compliance matrix of the cracks in a simple form, as follows: , and tensor ijkl represents the coefficient of the second-rank crack density tensor. We consider arbitrarily distributed cracks, assuming that the normal direction of cracks is n = (n 1 , n 2 .n 3 ) = (sin cos , sin sin , cos ), and the normalised distribution function of cracks is f (Ω). Therefore, the parameter ΔS * can be calculated using equations (5) and (6), as follows:  (19), thereby proving that the Mukerji & Mavko squirt formulas are consistent with Gurevich's high-frequency unrelaxed formulas in media with randomly distributed cracks.