The full diameter carbonate core samples in experimentation were collected form the two adjacent ultra deep wells of Gucheng area of the Tarim Basin, Xinjiang Oil Field (as shown in Figure 2), where gathered 70 percent of ultradeep wells and 90 percent of special ultradeep wells of China. Samples with basically the same composition were divided into five groups based on their burial depth, respectively, 6077, 6234, 6488, 6652 and 6738 m underground, and because the high coring failure rate, there were merely four to six samples per group. According to the standards specified by International Society for Rock Mechanics, the rock cores were processed into cylinders with a 50 mm dimensions of in diameter and 100 mm in height, the roughness of two end surface of the rock was confined into 0.05 mm, and the deviation of diameter was not exceed 0.5 mm. Then, to avoid the influence of oil and inorganic salt in the primary water on parameters, the samples were washed in the core automatic oil washing instrument with the organic solvent extraction method.

As shown in Figure 3, the experimental section was mainly composed of three parts: a triaxial compression experiment, acoustic emission velocity test and X-ray computed tomography (CT) scanning. An RTR-2000 high pressure rock triaxial dynamic testing system produced by GCTS was used for the triaxial load part to obtain the peak strength, long term strength, static elasticity modulus, static Poisson’s ratio and Biot coefficient. This system can measure rock physical parameters under high temperature and pressure, and its maximum capacity of confining pressure and temperature can reach 210 MPa and 200°C, respectively, which generally conforms to the conditions tested in situ at a depth of 6,000 m. At the same time that the triaxial test was carried out, the dynamic elasticity modulus, dynamic Poisson’s ratio, and velocity of transverse and longitudinal waves were detected by a ULT-100 ultrasonic testing instrument produced by GCTS. The instrument was equipped with a corresponding 1 MHZ ultrasonic transmission and digital acquisition function to achieve the whole acoustic test process. Finally, the grayscale images of rock samples used to extract the microscopic parameters and numerical simulation were obtained from a micronanometer three-dimensional topological imaging microscopy system (MicroXCT-400, Xradia, United States). In addition, the normal triaxial confining pressure (σ₁>σ₂ = σ₃) was adopted in this experiment.

Before carrying out the triaxial test, all of the samples were measured for basic physical properties, including the mass and density of the rock specimen, porosity, and permeability, with a vernier caliper, electronic balance and nitrogen porosity meter. Among them, five samples were selected from different burial depths H _dep for full-field CT scanning at a 12 micron resolution. The CT scanning images of the samples are shown in Figure 4.

To further elucidate the mechanical properties of carbonate rock in ultradeep reservoirs, the experiment was carried out under quasi in situ conditions. In situ geology generally includes the in situ temperature, humidity, composition, stress, osmotic pressure, and other geological parameters (Xie et al., 2021). In this paper, we mainly consider the stress and temperature that most obviously affect the mechanical response. Temperature gradients typically range from 20°C/km to 30°C/km as the crust deepens, and this value in western and central China is approximately 25°C/km (Pang et al., 2015). According to the in situ stress measurement of the target area, the overburden pressure gradient is approximately 23.8 MPa/km. Thus, the ranges of the in situ temperature and stress of the formation in which the samples were located are approximately 150–168°C and 143–160 MPa. Different from previous studies focusing on the effect of confining pressure and temperature, this experiment aims to investigate the influence of the intrinsic properties, especially the microstructure of rocks at different burial depths, on mechanical parameters. Therefore, samples of different burial depths were tested under the same conditions of confining pressure and temperature in this work, and the specific parameter values take the value around the middle section of the target formation, i.e., the experimental temperature was 160°C, and the confining pressure was 150 MPa. Meanwhile, to study the influence of the saturation state on the transition relationship between dynamic and static mechanical parameters, samples were subjected to immersion treatment and drying treatment in sequence to obtain the mechanical parameters in the saturated state and the dry state.

The specific experimental steps were as follows: First, the samples were put into a core vacuum saturation apparatus for 24 h, and then the saturated samples were fixed with a thermplastic tube on the platform and sealed with a slightly heated hot fan. A pair of deformation sensors were attached to the radial and axial directions of the sample. Second, an acoustic sensor was adhered to the sample, and a coupling agent was evenly applied between the sample and the end cap of the acoustic probe to ensure sufficient coupling. Then, the experimental setup began to load the confining pressure and heat after debugging the initial sensor value. Finally, the displacement loading method was used with a loading rate of 0.002 mm/min until the axial loading pressure reached approximately 100 MPa. Considering that the peak strength of the sample is generally large, and the sample has a long elastic stage during the axial compression loading process, in order not to damage the sample as much as possible, and to ensure the accuracy of the subsequent experiments, it is not necessary to load the axial compression too much when measuring the chord modulus. Therefore, the axial pressure is added to 100 MPa when the saturated elastic modulus is measured. During the experiment, the acoustic emission signal not received by the sensor increased significantly, which proved that the microstructure of brittle carbonate samples was not damaged.

Similarly, the above process was conducted to measure the same parameters of the samples in the dry state after drying the saturated sample in a constant temperature chamber at 40°C for 24 h. However, the loading process continued until the sample broke to obtain the compressive strength and postpeak characteristics of the samples. In addition, it is worth noting that a small initial mechanical load added in this triaxial test has the ability to avoid the initial friction effect on the experimental results.

Pore structure is an important factor affecting the mechanical properties and dynamic static parameter relationship of rock, especially for carbonate rocks with strong anisotropy. To describe the inner pore and fracture structures of samples from different burial depths, the actual digital rocks were constructed using a physical experimental method (Arns et al., 2004). Based on grayscale CT scanning images, we adopted the maximum classification method to generate binary images of samples after nonlocal filtering (Yu et al., 2008). Then, the rock matrix and pore–fracture space were extracted, as shown in Figure 5. Finally, the 3D actual digital rock was constructed by superimposing the segmented images.

According to the segmentation of pore–fracture space, the characteristics of pore structure could be quantitatively described by extracting the corresponding pore network model of samples. The model proposed that the pore network comprises pores and throats. Using the maximal ball method, the irregular cross section of pores and throats can be interpreted as different regular shapes (Pelayo and Schmidt, 2008).

The specific procedure is as follows: The isolated rock skeleton particles of the rock are removed, and then the central axis of the pore space is established by the Lee–Kashyap–Chu algorithm (Wang C.-c. et al., 2013). Then, the center position of each pore in the central axis system is determined, and the optimized pore space is optimized into pores and throats. Finally, the model of the pore network was established to reflect real pore space topology and geometric features (Wang C.-c. et al., 2013).

According to the statistical information of the pore network model extracted by using the maximal ball method, the structural parameters can be obtained, including the pore throat ratio, pore and throat radius, coordination number, and throat and pore shape factor. Among them, the nondimensional shape factor G characterizes the degrees of irregularity of pores or throats defined by: G = V L A s 2 (1) where V is the volume of the pore or throat space, L is the length of the pore or throat space, and As2 is the surface area of the pore or throat space. Therefore, the positive correlation of the value of G indicates the regularity of the pore or throat space; for instance, G reaches the maximum when the space is round, and a triangular G is in the range of 0–0.0481. In addition, the process of simplification follows the principle of shape factor conservation, i.e., the geometry used to characterize a pore or throat has a shape factor equal to the shape factor of the pore or throat.

The stress-strain curves of the samples chosen for CT scanning obtained under triaxial compression are displayed in Figures 9, 10. Clearly, there is a large span between the peak strength of different burial depths; the minimum peak strength of the five sets of rock samples is 262.5 MPa, and the maximum peak reaches 612.8 MPa. The results reflect the strong heterogeneity of the carbonate reservoir, which does not show a significant correlation with burial depth.

Remarkably, samples of different burial depths present discrepant postpeak characteristics, all of which exhibit plastic flow, especially for a sample burial depth of 6738 m, but the samples buried at 6488, 6652 and 6738 m alternately exhibit strain hardening and strain softening; this seems to be more pronounced in samples of low strength. In Figure 7C–E, this phenomenon could be construed as samples with weaker structures (fractures and caves, Figure 7C–E) becoming more prone to local instability after compaction and developing microfractures; after the strain softening process during the recompaction of expanded microfractures, the sample has increased strength manifested as strain hardening, and the effect of fracture on rock strength is more obvious.

In Figure 9, we can see that the samples have both high levels of peak strength and residual strength. After the process of compression achieves the ultimate failure stress, the nonlinear rebound of the stress–strain curves is more evident as the burial depth increases. It is worth noting that the residual strength is closer to its corresponding peak strength as the burial depth increases, and the ratios of residual strength to peak strength in sequence are 76.3%, 76.6%, 78.4%, 84.4%, 91.5% as the burial depth increases. This may be due to the pore structure, the method and degree of cementation and other factors of the more deeply buried samples under the long-term “three high” geological environment that make the sample exhibit more plasticity under high stress. Although the influence of temperature on the dislocation motion of the carbonate is not significant (Araújo et al., 1997), the influence of pressure on mechanical twin crystals, such as calcite in carbonate, is obvious.

As shown in Figure 10A, the radial strain obviously lags behind the axial strain, and the peak of the radial stress–strain curves are more rounded compared with the axial stress–strain curves, which may indicate that the radial samples have a stronger resistance to damage. As shown in Figure 10B, the samples buried at 6234 m, 6488 m and 6738 m exhibit rock dilatancy (Cook, 1970). Alkan et al. (2007) defined the turning point of the volume from compression to expansion as the compression–expansion boundary (C/D boundary), and Martin (Martin and Read, 1992) considered that the axials stress corresponding to the C/D boundary is the crack damage stress, namely, the long-term strength σ _B . In general, the long-term strength is between 60% and 80% of the peak strength (Szczepanik et al., 2003); however, the long-term strength reaches 94%–99% of the peak strength in this experiment (as shown in Table 1). This is also the result of the rock being under high in situ stress for a long time.

The dynamic and static mechanical parameters and the conversion relationship between them play an important guiding role in underground engineering and the process of oil and gas development. Compared with sandstone, limestone or mudstone, carbonate rocks have a poorer dynamic and static correlation, and the accuracy of the dynamic–static parameter conversion model often decreases with increasing burial depth. The main factors causing the difference of dynamic and static elastic moduli are the effect of pore structure on wave propagation, the occurrence state of fluid and the velocity dispersion (Warpinski et al., 1998), Jing et al.(2016; 2017) has done meaningful work on the analysis and understanding of strong compressional wave velocity dispersion and the influence of fabric and saturation inhomogeneity on wave propagation by establishing theoretical models. And the relationship between the static and dynamic mechanical parameters of all samples in the dry or saturated state is displayed in Figure 11.

As shown in Figures 11A,B, there is a certain linear relationship between E ^d _D and E ^s _D ; however, E ^d _S and E ^s _S are not the case, and compared to the dynamic modulus, the presence of pore water has a more obvious impact on the static mechanical parameters (Figure 12A, C). In general, for rocks that are not prone to physicochemical reactions with water, the presence of incompressible pore water has an effect of resistance on longitudinal deformation of the sample during loading, which is reflected in the macromechanical parameter being the increase in elastic modulus. However, in this experiment, the influence of pore water on the elastic modulus may be increased or decreased. By analyzing the relationship between the E ^s - E ^d of the samples and their burial depth, Figure 13 illustrates that the effect of pore water on the elastic modulus is related to the burial depth of the sample. At a relatively shallow position, pore water increases the elastic modulus, but with increasing depth, the influence of pore water on the elastic modulus tends to decrease. Combining the conclusions from the previous section that pore structure, the method and degree of cementation and other factors of the buried deeper samples under the long-term “three high” geological environment make the sample exhibit more plasticity with increased burial depth, the results in Figure 13 can be interpreted as the deeper-buried samples under high stress for a long time exhibit more plasticity, and their dynamic and static elastic modulus is more likely to be weakened by pore water. This phenomenon seems to be more pronounced for the dynamic modulus than for the static modulus. In addition, the mean values of E ^s - E ^d of all samples show a similar trend with the E ^s - E ^d of five samples for the CT scan (Figure 13A,C, Figure 13B,D), which reflects the representativeness of the sample selected for scanning to some extent.

The main internal factors that cause the difference in dynamic and static mechanical parameters are pore structure, fracture and pore fluid. As shown in Figure 11C and Figure 12D, the correlation between E _D and E _S is poor regardless of the presence of pore fluid for different burial depths. On the other hand, the difference in dynamic parameters between samples is smaller than that in static parameters, the maximum values of E ^d _D and E ^s _D are 104.91 and 104.70 GPa, and the minimum values of E ^d _D and E ^s _D are 93.65 and 89.94 GPa. Correspondingly, the maximum values of E ^d _S and E ^s _S are 74.56 and 78.43 GPa, and the minimum values of E ^d _S and E ^s _S are 32.83 and 32.94 GPa. The ratio of E _S to E _D is approximately 0.4–0.7, and the average values of E ^d _S /E ^d _D and E ^s _S /E ^s _D are 0.592 and 0.578, respectively. This is approximate to the Evans. (1973) and Warpinski et al. (1998) view that the static elastic modulus is one half of its dynamic elastic modulus.

A schematic of the parameters of the pore structure probability distribution from different burial depths is shown in Figures 14, 15, and the pore size variation with burial depth is presented in Figure 12D. The probability distribution of these structural parameters reflects the geometric properties of the samples and attempts to display the effect of depth change on the geometric size and shape of the pore and throat in the pore network model. Figure 13D shows that the pore–throat size of the sample decreases first and then increases with increasing depth. As shown in Figure 14A and Figure 14B, the pore radius and throat radius of the samples are situated on the same order of magnitude, reflecting that the samples have small pores and thin throats in porous media. The distributions of pore and throat radii differ widely among the samples and have a complex pore size structure. The pore and throat degree of sorting, which represents the concentration of the size distribution, first decreases and then significantly increases with burial depth, and it is positively related to pore throat size. The pore throat degrees of sorting of samples at 6652 m and 6738 m are smaller than those of the shallower rocks; moreover, the pore throat degree of sorting of samples at 6488 m is maximal, and the peaks are focused at approximately 20 and 12 μm.

In Figure 14C, we can see that, except for the 6738 m sample, the pore shape factors of other samples have normal distribution characteristics, of which the probability distributions of the burial depths at 6077, 6234 and 6652 m are similar and their pore shape factors are mainly approximately 0.03. In contrast, the probability distributions of the 6738 m sample have three peaks near 0.022, 0.034 and 0.054, which indicates that the samples have more abundant cave and fracture structures. The probability distribution of throat lengths displayed in Figure 14D presents a contradistinctive result to the probability distribution of pore and throat radius, the samples of 6077 and 6738 m have a longer throat compared to the other samples, and even those above 300 μm are still distributed.

The pore–throat ratio, which refers to the ratio of the pores in the local range to the average of the radius of all the throat connections, is regarded as a commonly used parameter to evaluate the homogeneity of the pore network model. Specifically, the decrease in the pore throat ratio is embodied in the decrease between the pore radius and throat radius, namely, the more uniform the development of the pore space on the microscopic scale. The probability distribution of the pore–throat ratio for different burial depths is displayed in Figure 15A.

As shown in the plot of E ^s _D - E ^s _S and the mean pore–throat ratio of samples as a function of burial depth in Figure 15B, it is obvious that limited by the strong heterogeneity of the reservoir and the number of samples, although E ^s _D - E ^s _S does not present an apparent regularity of variation with burial depth, the values of E ^s _D - E ^s _S are significantly connected with the mean pore–throat ratios. E ^s _D - E ^s _S , as a function of the mean pore–throat ratio R _p-t , can be expressed as: E D s − E S s = 11.396 R p − t + 4.0726 ( R 2 = 0.7966 ) (5)

The pore–throat ratio has always been regarded as an important microstructural parameter that affects the permeability and electrical properties of rocks; however, this result indicates that the pore–throat structure form has an influence on the difference in dynamic and static elastic modulus in the presence of pore water.