Twin Shear Unified Strength Solution of Shale Gas Reservoir Collapse Deformation in the Process of Shale Gas Exploitation

Abstract

The collapse deformation of shale has a significant influence on the exploitation process. Experimental analysis has indicated a correlation coefficient range from 0.9814 to 0.9981 and the established sample regression formula could be used to express the relationship between the dynamic elastic modulus and static elastic modulus of shale specimens. Based on the twin shear unified-strength theory, where coefficient b was considered to express the effect of intermediate principal stress, with the deduced regression formula, the unified solution of major principal strains describing a critical collapse of the shale shaft wall was derived. The results showed that the intermediate principal stress had a significant influence on the major principal strain, describing the critical collapse of the shale shaft wall. At the same depth, the critical collapse major principal strain increased with the increase in the b values. With the change in b value from 0 to 1, the calculated difference in critical collapse major principal strain with the same wellbore depth would change from 22.1% to 45.5%. With the change in b value from 0 to 1, the calculated difference in critical collapse major principal strain with the same wellbore temperature would change from 22.1% to 45.6%. The unified solution formula of the major principal strain, describing the critical collapse of the shale shaft wall expressed by the dynamic elastic modulus, could adjust the contribution of intermediate principal stress by changing the values of b, while considering the influence of temperature and confining pressure. The twin shear unified-strength solution of the shale gas reservoir collapse deformation could be used to effectively evaluate the shale gas reservoir stability during shale gas exploitation.

1. Introduction

Shale gas is an important and unconventional form of energy. However, shale gas reservoirs have strong heterogeneity and anisotropy due to their argillaceous development and layered structure, which results in mechanical instability in the shale shaft wall during shale gas exploitation. The fractures and tracks in the shale reservoir will significantly affect the strength of the shale reservoir [1,2]. Determination of the shale reservoir’s strength parameters and the adoption of strength criteria are important factors when evaluating the strength of shale and calculating the stability of the shale shaft wall during the drilling process [3,4]. In the field of oil and gas engineering, the static elastic modulus and the dynamic elastic modulus of reservoir rock are also important for the evaluation of shaft wall stability. The rock’s elastic parameters are mainly obtained using the static method and dynamic method [5,6,7]. To obtain static elastic parameters of the rock under real reservoir conditions, corresponding temperature and pressure conditions all need to be simulated, which requires huge experimental work and data analysis [8,9]. Therefore, in practical engineering, the acoustic logging dynamic method is normally considered in advance, to obtain the dynamic elastic properties of rock, and then the static elastic parameters of rock are obtained by conversion [10].

Much of the research has been focused on the investigation of the relationship between the dynamic and static elastic parameters of various rocks [11,12,13,14,15,16,17], as well as the dynamic and static mechanical behavior and failure modes of shale [18,19,20]. The current calculation method of shaft wall rock strength is mostly based on the Mohr–Coulomb strength theory, a single shear strength theory suitable for rock and soil. However, the Mohr–Coulomb strength theory was not able to cover the influence of intermediate principal stress on rock strength, which may lead to the calculation result differing from the real situation [21,22,23,24]. Much of the research has shown that the intermediate principal stress affects the strength calculation of rock and soil [25,26,27,28]. The twin shear unified-strength theory, as proposed by Yu [29], can reasonably illustrate the intermediate principal stress effect of materials under triaxial stress conditions and has been widely considered [30,31,32,33,34,35,36]. Therefore, the investigation of the relationship between the dynamic and static elastic mechanical properties of rock, and the calculation of shale shaft wall strength based on the dynamic elastic mechanical properties rather than the static ones, are of great importance. Furthermore, when considering the effect of intermediate principal stress on of shale shaft wall rock strength calculation, a more reasonable critical collapse deformation of the wellbore is determined, obtaining an effective and reasonable scientific evaluation method of the shale shaft wall stability in the drilling process.

In this research, acoustic and mechanical experiments with the three shale specimens were considered. The relationship between the dynamic and static elastic moduli of shale under the conditions of different temperature and confining pressure was studied and established. Assuming that shaft wall rock was in the plane strain state, with the twin shear unified-strength theory and general Hooke’s law, the unified solution of the major principal strain describing the critical collapse of shale shaft wall, which considered the relationship between the dynamic and static elastic modulus, was derived through theoretical analysis and practical verification. The intermediate principal stress effect on the critical collapse deformation of shale shaft wall under different depths and temperatures was also discussed. The results from this research have practical and theoretical significance for improving the application of dynamic elastic shale parameters and determining the stability of the shale shaft wall during the process of shale gas exploitation.

2. Materials and Methods

2.1. Materials

Three shale rock specimens from Changqing oil field (China), with a burial depth of 2800 m~3200 m, were considered in this experiment. The rock specimens with a full diameter were cored by drilling and were of gray–black shale. The specimens were processed as follows. A cylinder rock sample with a diameter of 1 inch was drilled along the horizontal direction. Then, the end face of the cylinder rock sample was cut and ground, and finally the cylinder rock sample was processed into a standard cylindrical sample with a length of 4~5 cm. Water was avoided for all specimen processing due to the natural fractures that exist in the shale rock specimens. The specimens are shown in Figure 1. The main physical characteristics of rock specimens are shown in

Table 1. The main physical characteristics of rock specimens.

The Serial Number	The Lithology	The Specimen Number	Grain Density (g/cm3)	Porosity (%)	Permeability (10−3 μm2)	Rock Density (g/cm3)
1	Gray-black calcareous shale	#5	2.75	2.8	0.149	2.67
2		#9	2.75	2.9	0.022	2.67
3		#76	2.71	0.6	0.010	2.69

.

The experiment was conducted according to the Chinese national standard “Method for Determination of Physical and Mechanical Properties of Coal and Rock” (GB/T 23561.8-2009) [37] and “Test Rules for Physical and Mechanical Properties of Rock” (DZ/T 0276.24-2015) [38]. The acoustic and mechanical parameters of rock specimens at high temperature and high pressure were measured on AutoLab 1500 system (AutoLab Software Version 1.0, New England Research Inc, Houston, TX, USA). The longitudinal and shear wave velocities of rock specimens #5, #9, and #76 were measured at different temperatures (changed from 25 to 110 °C) and confining pressures (changed from 5 to 50 MPa). The dynamic elastic modulus was calculated according to the experiment results. The effects of different temperatures and confining pressure conditions on the acoustic propagation velocity and dynamic elastic modulus of the shale were investigated. At the same time, the static elastic modulus of rock specimens was measured. The variation rule of the static elastic modulus of shale under different temperature and confining pressure conditions was also studied. Finally, the relationship between the dynamic and static elastic modulus was analyzed.

2.2. Experimental Study on Dynamic Elastic Modulus

The variations in the longitudinal and shear wave velocities and dynamic elastic modulus with temperature and confining pressure in the three shale specimens were similar. By evaluating the prominence of experimental results in the three specimens, specimen #9 was the most reliable, so the results were analyzed with specimen #9. The variation in the dynamic elastic modulus of specimen #9 under the conditions of different confining pressures and temperatures is presented in Figure 2.

From Figure 2, it can be seen that dynamic elastic modulus of the specimen increased with the increase in confining pressure at different temperatures and decreased with the increase in temperature at different confining pressures.

2.3. Experimental Study on Static Elastic Modulus

The experimental process showed that the static elastic modulus of the three rock specimens varied similarly with temperature and confining pressure. By evaluating the prominence of experimental results among the three specimens, specimen #5 was the most reliable, so specimen #5 was taken as an example for analysis. The variation in the static elastic modulus of the rock specimen under the condition of different confining pressures and temperatures is shown in Figure 3. The static elastic modulus increased with the increase in confining pressure and the increased range was larger at high temperatures. At the same time, the static elastic modulus increased with the increase in temperature, especially under higher confining pressure conditions. These results showed that the increase in temperature and confining pressure would lead to the increase in static elastic modulus of the shale specimen. The analysis of this phenomenon could be illustrated as follows: the increase in confining pressure would lead to the decrease in the axial strain of the specimen under the same axial pressure condition, which leads to the increase in the static elastic modulus determined by the ratio of axial stress to strain. On the other hand, the temperature increase would lead to the expansion of some minerals in the shale specimen, which results in the decrease in axial strain and the increase in the static elastic modulus of the specimen under the same axial pressure condition. Furthermore, as both temperature and effective pressure increased with the increase in the shaft wall rock burial depth, it can be inferred that the static elastic modulus of the rock specimens would also increase.

2.4. Comparison of Dynamic and Static Elastic Moduli of the Shale Specimens

The variations in the dynamic and static elastic moduli of shale specimens with confining pressure and temperature are summarized in

Table 2. Variation in dynamic and static elastic modulus under different confining pressures and temperatures.

	Parameter	Confining Pressure Increase	Temperature Increase	Buried Depth Increase
Category
Dynamic elastic modulus		Increase	Decrease	Not clear
Static elastic modulus		Increase, especially faster at high temperature	Increase, especially faster at high pressure	Increase

. The static elastic modulus of shale specimens increased with the increase in buried depth. However, with the increase in burial depth, the variation tendency of the dynamic elastic modulus of shale specimens could not be inferred from the experiment results. In addition, as shown in Table 2, the static elastic modulus increased while the dynamic elastic modulus decreased under increasing temperature. Therefore, if the dynamic elastic mechanical parameters obtained by the acoustic waves, were directly adopted to describe the static elastic mechanical properties of shale gas reservoir, some negative uncertainties would be present in the real engineering project. Some research shows that extraneous variables may interfere with this information and, thus, the outcomes can be adversely impacted by the quality of the work [39,40,41]. However, by evaluating the prominence of the experimental results, #9 and #5 were the least affected by extraneous variables. Specimens #9 and #5 could be taken as representative for analysis.

3. Results

Rock is a multi-phase composite medium and its heterogeneity will lead to discrete and abnormal corresponding relationships between the dynamic and static elastic parameters of rock [42,43]. In order to avoid the discrete effects caused by the heterogeneity of the three shale specimens, specimen #5, with the fewest abnormal measure points, was taken as an example to study the relationship between the dynamic and static elastic moduli.

Figure 4 shows the cross-plot of dynamic and static elastic moduli of specimen #5, which expresses vertical and abscissa coordinates at the same scale. It was obvious that the dynamic and static elastic moduli of rock specimens presented a clear correlation at the same temperature, and the dynamic elastic modulus was higher than the static elastic modulus.

Based on the data in Figure 4 and using linear regression analysis method, the transformation relationship between the dynamic and static elastic moduli of specimen #5 could be expressed as follows:

$$E_S=a{E}_D+c$$

(1)

where $E_S$ is the static elastic modulus, MPa; $E_D$ is the dynamic elastic modulus, MPa; and a and c are linear coefficients without units. According to Formula (1), the relationship between the dynamic and static elastic moduli of specimen #5 was found under different temperature conditions and confining pressures. Uncertainty analysis is important in data analysis and the measurement of measurand [44,45,46,47,48]. Here, correlation coefficient R, without units, was adopted to evaluate the accuracy of linear regression. The fitting results are shown in

Table 3. Linear regression result of the relationship between the dynamic elastic modulus and static elastic modulus of #5 rock specimen.

	Coefficients	a	c	R
Temperature
25 °C		1.010	−26.163	0.9814
50 °		2.174	−87.452	0.9940
70 °C		2.474	−100.75	0.9928
90 °C		2.721	−111.12	0.9961
110 °C		3.029	−124.09	0.9981

and correlation coefficients R were all greater than 0.9. The data and analysis results indicated that Formula (1) could represent the dynamic and static elastic modulus conversion relationship of shale specimen #5.

3.1. Twin Shear Unified-Strength Theory

According to the twin shear unified-strength theory established by Yu [29], when the influence function of the two larger shear stresses on the twin shear element and the normal stress on the surface reach a certain limit value, the material begins to fail. By introducing the shear strength parameters $C_0$ and rock internal friction angle $\phi$, the formula of the twin shear unified-strength theory, in the form of principal stress, with positive compressive stress in rock mechanics, is derived as follows:

$$F={\sigma }_1-\frac{1+\sin \phi }{(1+b)(1-\sin \phi )}(b{\sigma }_2+{\sigma }_3)=\frac{2C_0\cos \phi }{1-\sin \phi }$$

(2)

$$\left({\sigma }_2\le \frac{1}{2}({\sigma }_1+{\sigma }_3)-\frac{\sin \phi }{2}({\sigma }_1-{\sigma }_3)\right)$$

$$F^{\prime }=\frac{1}{1+b}({\sigma }_1+b{\sigma }_2)-\frac{1+\sin \phi }{1-\sin \phi }{\sigma }_3=\frac{2C_0\cos \phi }{1-\sin \phi }$$

(3)

$$\left({\sigma }_2\ge \frac{1}{2}({\sigma }_1+{\sigma }_3)-\frac{\sin \phi }{2}({\sigma }_1-{\sigma }_3)\right)$$

$$\alpha =\frac{f_t}{f_c}; b=\frac{(1+\alpha ){\tau }_0-f_t}{f_t-{\tau }_0}$$

(4)

where ${\sigma }_1$ is the major principal stress, ${\sigma }_2$ is the intermediate principal stress and ${\sigma }_3$ is the minor principal stress; $\alpha$ is the tension–compression strength ratio coefficient of material; ${\tau }_0$ is the shear yield limit of material; $f_t$ is the tensile yield limit of material; $f_c$ is the compressive yield limit of material; b is the coefficient, which can express the influence of the principal shear stress and the normal stress on the corresponding surface of the material, different b values (from 0 to 1) can be used to express the different effects of intermediate principal stress.

3.2. Determination of the Principal Stress at the Collapse of Shale Shaft Wall

The surrounding rock of a vertical well is usually subjected to a combination of overburden stratum stress ${\sigma }_v$, drilling fluid pressure $P_i$, two horizontal stratum stresses ${\sigma }_H$ and ${\sigma }_h$ and stratum pore pressure $P_p$. The vertical well radius is set as $r_0$, the center of the vertical well is taken as the origin, the radial coordinate is $r$ and the angle between the radius vector on the shaft wall and ${\sigma }_H$ is the circumferential coordinate $\theta$ and the polar coordinate system is established. The three-dimensional mechanical model is shown in Figure 5.

Assuming that the linear elastic porous material for the stratum and shaft wall rock are in the plane strain state, based on the theory of elastic mechanics, without considering the formation permeability ( $P_p$ and $P_i$ not affecting each other) and stress nonlinear correction, the stress distribution of the shaft wall rock can be expressed by $P_i$, ${\sigma }_H$, ${\sigma }_h$ and ${\sigma }_v$, shown in Figure 5, based on the linear superposition principle. The simplified plane mechanical model of the shaft wall is shown in Figure 6.

In addition, according to the principle of linear superposition, Figure 6 can be further expressed by Figure 7.

According to Figure 7, under the combined action of drilling fluid pressure and stratum stress, the stress distribution of shaft wall rock can be superimposed as follows:

$$\{\begin{array}{c}{\sigma }_r=\frac{r_0{}^2}{r^2}{P}_i+\frac{({\sigma }_H+{\sigma }_h)}{2}(1-\frac{r_0{}^2}{r^2})+\frac{({\sigma }_H-{\sigma }_h)}{2}(1-\frac{r_0{}^2}{r^2})(1-\frac{3r_0{}^2}{r^2})\cos 2\theta \\ {\sigma }_{\theta }=-\frac{r_0{}^2}{r^2}{P}_i+\frac{({\sigma }_H+{\sigma }_h)}{2}(1+\frac{r_0{}^2}{r^2})-\frac{({\sigma }_H-{\sigma }_h)}{2}(1+\frac{3r_0{}^4}{r^4})\cos 2\theta \\ {\sigma }_z={\sigma }_v-2\mu ({\sigma }_H-{\sigma }_h){(\frac{r_0}{r})}^2\cos 2\theta \\ {\tau }_{r\theta }=\frac{({\sigma }_h-{\sigma }_H)}{2}(1-\frac{r_0{}^2}{r^2})(1+\frac{3r_0{}^2}{r^2})\sin 2\theta \end{array}$$

(5)

where $P_i$ is the drilling fluid pressure, ${\sigma }_H$ is the maximum horizontal stratum stress, ${\sigma }_h$ is the minimum horizontal stratum stress, ${\sigma }_v$ is the overburden stratum stress, $\mu$ is the Poisson’s ratio of the shaft wall rock, and $\theta$ is the angle between the radius vector on the shaft wall and ${\sigma }_H$.

By substituting r = r₀ into Equation (5), the stress distribution on the surface of shaft wall rock can be expressed as:

$$\{\begin{array}{c}{\sigma }_r=P\\ {\sigma }_{\theta }=(1-2\cos 2\theta ){\sigma }_H+(1+2\cos 2\theta ){\sigma }_h-P\\ {\sigma }_z={\sigma }_v-2\mu ({\sigma }_H-{\sigma }_h)\cos 2\theta \\ {\tau }_{r\theta }=0 \end{array}$$

(6)

Considering the stratum pore pressure $P_p$, based on Equation (6), the effective stress distribution of the shaft wall rock can be expressed as follows:

$$\{\begin{array}{c}{{\sigma }^{\prime }}_r=P_i-\xi P_p\\ {{\sigma }^{\prime }}_{\theta }=(1-2\cos 2\theta ){\sigma }_H+(1+2\cos 2\theta ){\sigma }_h-P_i-\xi P_p\\ {{\sigma }^{\prime }}_z={\sigma }_v-2\mu ({\sigma }_H-{\sigma }_h)\cos 2\theta -\xi P_p \end{array}$$

(7)

where $\xi$ is effective stress coefficient and $P_p$ is stratum pore pressure.

The main mechanical reason for shaft wall rock collapse is that the pressure of drilling fluid in the wellbore is low and the stress of the shaft wall rock exceeds the strength of the rock itself, which results in shear failure. Therefore, considering the drilling fluid pressure in the wellbore, the expression of radial, tangential and vertical effective stress of the shaft wall at collapse ( $\theta ={90}^{\circ }$ or $\theta ={270}^{\circ }$) is:

$$\{\begin{array}{c}{{\sigma }^{\prime }}_{rc}=P_i-\xi P_p\\ {{\sigma }^{\prime }}_{\theta c}=3{\sigma }_H-{\sigma }_h-P_i-\xi P_p\\ {{\sigma }^{\prime }}_{zc}={\sigma }_v+2\mu ({\sigma }_H-{\sigma }_h)-\xi P_p \end{array}$$

(8)

where ${{\sigma }^{\prime }}_{rc}$ is the radial effective stress of the shaft wall at collapse, ${{\sigma }^{\prime }}_{\theta c}$ is the tangential effective stress of the shaft wall at collapse, and ${{\sigma }^{\prime }}_{zc}$ is the vertical effective stress of the shaft wall at collapse.

Considering the shear failure mode and Equation (8), the principal stress can be expressed as follows:

$${\sigma }_1={{\sigma }^{\prime }}_{\theta c},{\sigma }_2={{\sigma }^{\prime }}_{zc},{\sigma }_3={{\sigma }^{\prime }}_{rc}$$

(9)

3.3. Twin Shear Unified-Strength Solution of Critical Collapse of the Shale Shaft Wall

By substituting Equation (9) into the discriminant formula of the twin shear unified-strength theory for comparison, ${\sigma }_2\le \frac{1}{2}({\sigma }_1+{\sigma }_3)-\frac{\sin \phi }{2}({\sigma }_1-{\sigma }_3)$ could be obtained. Therefore, by substituting Equation (9) into Formula (2) and considering general Hooke’s law, the calculation formula for the twin shear unified-strength solution of the critical collapse of shale shaft wall was obtained:

$${\epsilon }_1=\left\{\left[b\frac{1+\sin \phi }{(1+b)(1-\sin \phi )}-\mu \right]{\sigma }_2+\left[\frac{1+\sin \phi }{(1+b)(1-\sin \phi )}-\mu \right]{\sigma }_3+\frac{2C_0\cos \phi }{1-\sin \phi }\right\}{E}^{-1}$$

(10)

By substituting Equation (1) of the transformation between dynamic and static elastic modulus, Equation (10) is transformed as follows:

$${\epsilon }_1=\left\{\left[b\frac{1+\sin \phi }{(1+b)(1-\sin \phi )}-\mu \right]{\sigma }_2+\left[\frac{1+\sin \phi }{(1+b)(1-\sin \phi )}-\mu \right]{\sigma }_3+\frac{2C_0\cos \phi }{1-\sin \phi }\right\}{(a{E}_D+c)}^{-1}$$

(11)

Formula (11) was the unified solution expression of the major principal strain describing the critical collapse of shale shaft wall with dynamic modulus based on twin shear unified-strength theory.

4. Discussion

In order to confirm the effect of intermediate principal stress using Formula (11), one real shale reservoir was taken as an example for analysis. With the calculation results, the variation in major principal strain describing the critical collapse of shale shaft wall under different conditions (depth, b values and temperature) was also discussed.

The basic geological parameters of Well Hu 2 of Karamay oil field are as follows [49]:

Interval range: $H=2900-2940$ m, in situ stress gradient: ${\sigma }_H=2.73$ MPa/100 m, ${\sigma }_h=1.82$ MPa/100 m, ${\sigma }_v=2.34$ MPa/100 m, stratum pore pressure coefficient: $p_p=1.2$, effective stress coefficient: $\xi =0.4$, Poisson’s ratio: $\mu =0.2$, drilling fluid density: 1.80 g/cm³, shale strength parameters: $C_0=11.08$ MPa, $\phi ={19.8}^{\circ }$, dynamic elastic modulus of shale: E_D = 21.5 GPa.

The relationship between the dynamic elastic modulus and static elastic modulus at 25 °C, as well as the above parameters, was substituted into Formula (10) to explore the variation in the critical collapse major principal strain of the shale shaft wall at different depths under the condition of a change in intermediate principal stress coefficient b. When coefficient b was zero, Formula (11) was expressed by Mohr–Coulomb strength theory. To research the effect of intermediate principal stress, taking the calculation results of b being zero as the standard, different b values (0.25, 0.5, 0.75 and 1) were selected in Formula (11) along the well depth. The results are shown in Figure 8.

Figure 8 shows that, with the increase in the well depth, the critical collapse major principal strain gradually increased under different b values. This variation indicated that, with an increase in the well depth and the constant drilling fluid density, the shaft wall was more prone to borehole diameter enlargement and collapse. At the same depth, the critical collapse major principal strain increased with the increase in the b value. The critical collapse major principal strain at the same depth with b = 0 differed significantly, compared to other different b value conditions. This showed that the increase in the shale deformation could be further aggravated and resulted in the collapse of shale shaft wall when the effect of intermediate principal stress was considered.

The critical major principal strain values under different b values were selected at three depths (600 m, 1800 m and 3000 m), as shown in

Table 4. The critical collapse major principal strain with the same wellbore depth under different b values.

	Well Depth/m	600	1800	3000	Intermediate Principal Stress Effect Comparison
b Value					600 m	1800 m	3000 m
0		0.00176	0.00237	0.00298	22.1%	38.8%	45.5%
0.25		0.00209	0.00337	0.00464
0.5		0.00218	0.00361	0.00505
0.75		0.00220	0.00370	0.00519
1		0.00226	0.00387	0.00547

. Taking the calculation results of b being zero as the standard, the influence of intermediate principal stress at 600 m on the calculation results was 22.1%, and that at 3000 m was 45.5%. The difference between the calculation result at two positions was nearly doubled. The results indicated that the influence of intermediate principal stress effect on the critical collapse major principal strain increased with the increase in well depth.

Considering the confining pressure conditions in the example and the experimental results, the variation in the critical major principal strain of shale shaft wall with the well depth at different temperatures and b values was further investigated, taking the calculation results of b being zero as standard. Different b values (0.5 and 1) were selected in Formula (11) along the well depth. The results are shown in Figure 9.

Figure 9 shows that, under the same b value, the critical collapse major principal strain increased with an increase in well depth. The increase in the critical collapse major principal strain at a high temperature was lower than that at low temperatures. The critical collapse major principal strain at the same depth decreased with the increase in temperature. Under the condition of different b values, the critical collapse major principal strain with the same temperature and the same depth increased with the increase in b values. The above analysis results showed that the static elastic modulus increased with the increase in temperature and the critical collapse major principal strain decreased. However, the influence of temperature on the critical collapse major principal strain gradually weakened with the increase in temperature.

Considering the temperature conditions, the critical major principal strain values were selected under the condition of different b values at three depths (600 m, 1800 m and 3000 m), as shown in

Table 5. The critical collapse major principal strain with the same temperature under different wellbore depths and different b values.

	Conditions	25 °C			70°C			110°C
bValue		600 m	1800 m	3000 m	600 m	1800 m	3000 m	600 m	1800 m	3000 m
0		0.00176	0.00237	0.00298	0.00158	0.00212	0.00267	0.00150	0.00202	0.00254
0.5		0.00218	0.00361	0.00505	0.00195	0.00324	0.00453	0.00186	0.00309	0.00432
1		0.00226	0.00386	0.00547	0.00203	0.00346	0.00490	0.00193	0.00330	0.00467
	Conditions	Intermediate principal stress effect comparison
bValue		25°C			70°C			110°C
0–1		22.1–45.5%			22.2–45.5%			22.3–45.6%

. The data in Table 5 showed that the critical collapse major principal strain of the shaft wall increased, considering the effect of intermediate principal stress. However, the influence of intermediate principal stress on the critical collapse major principal strain gradually decreased with the increase in b value. At the same time, the influence of different b values on the critical collapse major principal strain was basically the same under different temperatures. Taking the calculation results of b being zero as standard (using Mohr–Coulomb strength theory), the influence of intermediate principal stress at 25 °C on the calculation results was 22.1%, the influence at 70 °C was 45.5% and that at 110 °C was 45.6%, which indicated that the influence on the deep shaft wall rock was greater than that on the shallow shaft wall rock.

5. Conclusions

In this study, the transformation relationship between dynamic elastic modulus and static elastic modulus was derived by experimental and linear regression analysis. In the real projects, the dynamic elastic modulus could be used to calculate the critical collapse major principal strain based on the derived transformation relationship, which simplified the static elastic modulus measurement procedures. Furthermore, considering the intermediate principal stress effect is meaningful to evaluate the shale gas reservoir stability.

The research results suggested that both temperature and confining pressure had an impact on the dynamic and static elastic moduli of shale and the impact from dynamic elastic modulus was more significant than the static one. The increase in confining pressure led to the gradual closure of micro-cracks in rock specimens during the dynamic test. However, due to the “undrained” state during the dynamic test, the fluid in the cracks could not be discharged and resulted in an increase in pore pressure, which was macroscopically shown as the increase in dynamic elastic modulus. On the other hand, with the increase in temperature, the micro-cracks in the rock specimen gradually expanded and the dynamic elastic modulus decreased macroscopically.

Furthermore, the intermediate principal stress effect had a direct influence on the calculation of the critical collapse major principal strain of shale shaft wall. With the same wellbore depth, the calculation difference of the critical collapse major principal strain would change from 22.1% to 45.5% when the b value changed from 0 to 1. With the same wellbore temperature, the calculation difference in critical collapse major principal strain would change from 22.1% to 45.6% when the b value changed from 0 to 1. That influence increased with increase in the wellbore depth, which indicated that the deep shaft wall was more prone to diameter enlargement and collapse. Therefore, it is necessary to reasonably adjust the drilling fluid density to ensure the shaft wall stability during the drilling. Lastly, the critical collapse major principal strain decreased with increase in temperature under the same intermediate principal stress conditions. The critical collapse strain increased with the increase in the b value under the same temperature and the same depth. Therefore, the temperature and intermediate principal stress effects should be considered carefully when determining the deep shaft wall collapse, to determine the relatively accurate critical collapse conditions of the shale shaft walls.

All of the results showed that the established twin shear unified-strength solution of shale gas reservoir collapse deformation, considering the dynamic elastic modulus, could be used to effectively determine the shale gas reservoir stability during the shale gas exploitation. In addition, the twin shear unified-strength solution of shale gas reservoir collapse deformation could adapt to different geological shale conditions by adjusting the intermediate principal stress coefficient b according to practical engineering.