A Thermal-Fluid-Solid Coupling Computation Model of Initiation Pressure Using Supercritical Carbon Dioxide Fracturing

: With the characteristics of low fracturing pressure, little damage to the reservoirs, and assuming the role of carbon storage, supercritical carbon dioxide (SC-CO 2 ) fracturing is suitable for the development of unconventional oil and gas resources. Based on the tensile failure mechanism of rocks, this paper establishes a thermal-ﬂuid-solid coupling initiation pressure model for SC-CO 2 fracturing. Using this model, the changes in formation temperature and pore pressure near a wellbore caused by invasion of CO 2 into the formation are analyzed, as well as the impact of these changes on the tangential stress of reservoir rocks. The ﬁeld data of SC-CO 2 fracturing in a sandstone gas well are used to validate the reliability of the model. The results show that SC-CO 2 fracturing can signiﬁcantly reduce the initiation pressure, which decreases with the increase in fracturing ﬂuid injection rate. The minimum value of tangential stress is located at the well wall, and the direction of tangential stress caused by formation temperature and pore pressure is opposite, with the former greater than the latter. The increase in Poisson’s ratio, the increase in elastic modulus and the decrease in bottom hole temperature can reduce the initial fracturing pressure of the reservoir. The computation model established in this paper provides an effective method for understanding the reservoir fracturing mechanism under the condition of SC-CO 2 invasion.


Introduction
Unconventional oil and gas resources in China are characterized by tight reservoir, complex geological structure, and high clay mineral content, which increases the difficulty of unconventional oil and gas field development [1][2][3][4]. Practice shows that large-scale hydraulic fracturing is one of the most effective technical means in unconventional oil and gas development fields [5,6]. However, the traditional hydraulic fracturing poses problems such as high water consumption, reservoir damage, and flowback fluid pollution [7], which restrict the scale and effectiveness of reservoir fracturing, and hinder the process of commercial development of unconventional oil and gas resources in China. Therefore, it is urgent to explore a new environmentally friendly water free fracturing technology.
SC-CO 2 fracturing is a new technology that uses SC-CO 2 (critical point: temperature 304.1 K, pressure 7.38 MPa) as the fracturing fluid to fracture the reservoir [8,9]. During

Tangential Stress Model around Well
The fracture is related to tensile failure during fracturing. Taking the compressive stress as positive, the condition for formation fracturing is as follows: where σ θ is effective tangential stress, MPa; σ t is the tensile strength of reservoir rock, MPa. CO 2 can invade the formation after reaching the bottom of the well for its strong permeability, and then change the pore pressure around the well and affect the physical properties of the rock, thus change the tangential stress around the well. At the same time, the temperature difference between the fluid and the reservoir also causes the change in the tangential stress around the well.
According to the analysis above, the tangential stress around the well during SC-CO 2 fracturing is mainly caused by four basic loads: external radial stress, wellbore pressure, pore pressure, and thermal effect. According to the superposition principle [24], the expression of tangential stress around the well is: where σ θ1 is the tangential stress caused by external radial stress, MPa; σ θ2 is the tangential stress caused by the wellbore pressure, MPa; σ θ3 is the tangential stress caused by fluid seepage effect in the wellbore, MPa; σ θ4 is the tangential stress caused by thermal effect, MPa; σ H , σ h is the maximum and minimum horizontal principal stress, respectively, MPa; R w is the borehole radius, m; r is the distance from the borehole in the horizontal direction, m; p w is the bottom hole pressure, MPa; and θ is the well circumference angle, ( • ). The mud cake cannot be formed inside the well wall for CO 2 has little wall building property [31]. At the same time, the bottom hole pressure increases during injection, and the pore pressure increases caused by invasion of SC-CO 2 into the formation, thus the tangential stress changes. The pore pressure change ∆p p is defined as where p p (r,t) is the formation pore pressure around the well, MPa; p po is the original formation pressure, MPa; and t is the time, s. The tangential stress distribution formula caused by the change in pore pressure is where η is the stress coefficient of porous elastic medium; G and γ are the Lame coefficient of rock; ν is Poisson's ratio; α is the Biot coefficient; and r is the variable in the integral. The thermal stress is generated when there is a temperature difference between the fluid and the reservoir, which causes rock damage [32,33]. During SC-CO 2 fracturing, there is a temperature difference between the injected low temperature CO 2 and the reservoir when it reaches the bottom of the well, and the resulting thermal stress affects the tangential stress around the well. Define the temperature change field ∆T of rock around the well as where T(r,t) is the formation temperature around the well, K; and T 0 is the original formation temperature, K. Then the tangential stress caused by temperature difference in the wellbore is where E is the elastic modulus of reservoir rock, MPa; and α m is the volume thermal expansion coefficient of rock, K −1 . If r = R w , the effective tangential stress at the wellbore under the influence of multiple factors is where Tw is the temperature when the fluid reaches the bottom hole, K. Fracturing occurs when the effective tangential stress in the borehole wall reaches the tensile strength of the reservoir rock. Therefore, it can be calculated that the fracture initiation pressure pf is

Reservoir Temperature and Pore Pressure Model
It can be seen from Equations (5) and (7) that the tangential stress of the near-wellbore formation is related to formation temperature and pore pressure. Therefore, a reservoir temperature and pressure model with mechanical and thermal coupling is established to analyze the initiation conditions of fractures after CO 2 invasion.
Based on the mass conservation equation and energy conservation equation of fluid in pores, it can be concluded: where ρ f is the density of carbon dioxide, kg/m 3 ; φ is porosity; ρ s is the density of reservoir rock, kg/m 3 ; c ps is the specific heat capacity of reservoir rock, J/(kg·K); k T is the thermal conductivity coefficient, W/(m·K); Qs is the heat source intensity, J/(m 3 ·s); and J v is the fluid velocity, m/s. After CO 2 invades the reservoir, the fluid velocity is affected by temperature and pressure. The velocity divergence is where K is permeability, µm 2 ; µ is viscosity, µPa·s; p is pore pressure, MPa; and K T is the thermal permeability coefficient, m 2 /(s·K). For the same formation, assuming that the thermal conductivity coefficient k T is constant [34], and there is no internal heat source in the reservoir, then: In this paper, only the reservoir temperature at the distance r from the borehole is required. Therefore, Equation (13) degenerates into a one-dimensional heat conduction equation, that is, a parabolic equation. The analytical solution form can be obtained by Substitute Equation (12) into continuity equation, then: where kρ f µ ∇ 2 p, k µ ∇p∇ρ f and k T ∇T∇ρ f are small quantities which can be ignored, and set the coupling coefficient c cou = ρ 0 c p K T k T , then: Equation (13), Equation (16), and CO 2 physical property equation are comprehensively solved to obtain the change relationship of reservoir temperature and pore pressure with time and location.
The boundary conditions are

Model Solving Method
Due to the compressibility of CO 2 , its physical parameters are sensitive to the changes in temperature and pressure. At the same time, the physical parameters of CO 2 also change the distribution of reservoir temperature and pore pressure near the well, thus affecting the local stress state of the reservoir. CO 2 interacts with rock after invasion into reservoir, and affects the physical and mechanical properties of rock [35,36]. Therefore, when calculating the fracture initiation pressure, it is necessary to comprehensively consider the following parameters such as influence of CO 2 on the physical and mechanical properties of rocks, fluid seepage and heat transfer, CO 2 physical property changes, and other factors before conducting a coupling solution. The calculation process is shown in Figure 1. In this paper, the Span Wagner equation [37] and Vesovic equation [38,39] with internationally recognized high accuracy are used to calculate the thermophysical properties (density, specific heat capacity, etc.) and migration properties (viscosity, thermal conductivity, etc.) of CO 2 to ensure the accuracy of the model.

Model Validation
This paper uses the SC-CO 2 fracturing field data from a tight sandstone gas well to verify the model ( Table 1). The field data show that the fracturing occurred at the initial stage of injection, the initial pressure decreases from 66.02 MPa to 48.62 MPa, the bottom hole temperature was 368.97 K, and the original formation temperature was 382.15 K. The fracture initiation pressure calculated by the model in this paper is 48.94 MPa, and the absolute error is about 0.32 MPa, which is within 1%, indicating that the model in this paper has high calculation accuracy.
properties of rocks, fluid seepage and heat transfer, CO2 physical property changes, and other factors before conducting a coupling solution. The calculation process is shown in Figure 1. In this paper, the Span Wagner equation [37] and Vesovic equation [38,39] with internationally recognized high accuracy are used to calculate the thermophysical prop erties (density, specific heat capacity, etc.) and migration properties (viscosity, therma conductivity, etc.) of CO2 to ensure the accuracy of the model.

Model Validation
This paper uses the SC-CO2 fracturing field data from a tight sandstone gas well to verify the model ( Table 1). The field data show that the fracturing occurred at the initia stage of injection, the initial pressure decreases from 66.02 MPa to 48.62 MPa, the bottom hole temperature was 368.97 K, and the original formation temperature was 382.15 K. Th fracture initiation pressure calculated by the model in this paper is 48.94 MPa, and th absolute error is about 0.32 MPa, which is within 1%, indicating that the model in thi paper has high calculation accuracy.

Distribution of Formation Temperature and Pressure
During the fracturing and injection of SC-CO 2 , the bottom hole temperature gradually decreases and is lower than the formation temperature, while the bottom hole pressure is higher than the pore pressure, so CO 2 is easily invades the formation, affecting the formation temperature and pore pressure near the well [40,41]. Previous research shows that the fracturing fluid injection rate is the main control factor affecting the temperature and pressure of CO 2 in the wellbore [40]. The bottom hole temperature decreases with the increase in injection rate. This paper refers to the calculation of bottom hole temperature and pressure in reference [29] and analyzes the distribution rule of formation temperature and pore pressure near the well under different injection rate (0.5 m 3 /min, 2.0 m 3 /min) (Figures 2 and 3).
It can be seen from Figure 2 that when the injection rate increases from 0.5 m 3 /min to 2.0 m 3 /min, the bottom hole temperature decreases from 368.97 K to 328.83 K, a decrease of 40.14 K. Under a different injection rate, the formation near the bottom of the well has an evident temperature drop with the increase in time, and the influence range gradually expands. In addition, it can be seen from Figure 2 that the injection rate has little influence on the range of formation temperature drop.    MPa. This is because the pressure increment caused by the increase in gravity is greater than the frictional pressure drop increment caused by the increase in injection rate. Under a different injection rate, the formation pore pressure near the bottom hole increases with the increase in time, and the influence range increases. In addition, it can be found that the sweep velocity of bottom hole pressure is significantly higher than that of bottom hole temperature by comparing Figures 2 and 3. It can be seen from Figure 2 that when the injection rate increases from 0.5 m 3 /min to 2.0 m 3 /min, the bottom hole temperature decreases from 368.97 K to 328.83 K, a decrease of 40.14 K. Under a different injection rate, the formation near the bottom of the well has an evident temperature drop with the increase in time, and the influence range gradually expands. In addition, it can be seen from Figure 2 that the injection rate has little influence on the range of formation temperature drop. Figure 3 shows the distribution of pore pressure near the wellbore formation with time and radial distance under different injection rate (0.5 m 3 /min, 2.0 m 3 /min). It can be seen from Figure 3 that under the same wellhead pressure, when the injection rate increases from 0.5 m 3 /min to 2.0 m 3 /min, the bottom hole pressure increases from 32.35 MPa to 34.17 MPa, an increase of 1.82 MPa. This is because the pressure increment caused by the increase in gravity is greater than the frictional pressure drop increment caused by the increase in injection rate. Under a different injection rate, the formation pore pressure near the bottom hole increases with the increase in time, and the influence range increases. In addition, it can be found that the sweep velocity of bottom hole pressure is significantly higher than that of bottom hole temperature by comparing Figures 2 and 3.

Distribution of In Situ Stress
The near-wellbore temperature and pore pressure are substituted into Equations (5) and (7) to obtain the stress changes caused by formation temperature and pore pressure (Figure 4). It can be seen from Figure 4a that the thermal stress near the well wall (σ θ4 ) is negative, which can effectively reduce the tangential stress at the well wall, thus reducing the fracture initiation pressure. Maximum temperature stress (|σ θ4 |) is 5.24 MPa, which is located at the well wall. The absolute value of temperature stress gradually decreases and changes to a positive value with the increase in radial distance; the negative range of temperature stress increases with the increase in time.  In general, the stress caused by the change in formation temperature during SC-CO2 fracturing is greater than that caused by the change in pore pressure. This is because the injected low temperature CO2 can reach the supercritical state when reaching the bottom of the well, but the temperature is lower than the reservoir temperature, resulting in greater thermal stress. Figure 5 shows the distribution of tangential stress in the near-wellbore formation. It can be seen from the figure that the minimum tangential stress is 41.16 MPa, which is located at the well wall. With the increase in the radial distance, the tangential stress first increases, then decreases and gradually plateaus. There is a stress concentration area within the range of 2~4 times the borehole diameter. With the increase in time, the stress concentration area is gradually released, which is the result of the comprehensive effect of formation temperature and pore pressure changes near the well. It can be seen from Figure 4b that the tangential stress caused by pore pressure near the wellbore (σ θ3 ) is a positive value, indicating that the increase in pore pressure can cause the increase in tangential stress and the increase in fracture initiation pressure. The maximum value of σ θ3 also appears at the well wall, which is 0.92 MPa. In the process of CO 2 fracturing, the effect of pore pressure change on the tangential stress at the well wall is less than that of the reservoir temperature; as the radial distance increases, σ θ3 gradually decreases and turns into a negative value. The range of positive values of σ θ3 increases gradually with the increase in time. In addition, it can be found from Figure 4b that the tangential stress concentration area caused by pore pressure is concentrated near the well wall.
In general, the stress caused by the change in formation temperature during SC-CO 2 fracturing is greater than that caused by the change in pore pressure. This is because the injected low temperature CO 2 can reach the supercritical state when reaching the bottom of Processes 2023, 11, 437 9 of 14 the well, but the temperature is lower than the reservoir temperature, resulting in greater thermal stress. Figure 5 shows the distribution of tangential stress in the near-wellbore formation. It can be seen from the figure that the minimum tangential stress is 41.16 MPa, which is located at the well wall. With the increase in the radial distance, the tangential stress first increases, then decreases and gradually plateaus. There is a stress concentration area within the range of 2~4 times the borehole diameter. With the increase in time, the stress concentration area is gradually released, which is the result of the comprehensive effect of formation temperature and pore pressure changes near the well.  In general, the stress caused by the change in formation temperature during SC-CO2 fracturing is greater than that caused by the change in pore pressure. This is because the injected low temperature CO2 can reach the supercritical state when reaching the bottom of the well, but the temperature is lower than the reservoir temperature, resulting in greater thermal stress. Figure 5 shows the distribution of tangential stress in the near-wellbore formation. It can be seen from the figure that the minimum tangential stress is 41.16 MPa, which is located at the well wall. With the increase in the radial distance, the tangential stress first increases, then decreases and gradually plateaus. There is a stress concentration area within the range of 2~4 times the borehole diameter. With the increase in time, the stress concentration area is gradually released, which is the result of the comprehensive effect of formation temperature and pore pressure changes near the well.

Influencing Factors of Fracture Initiation Pressure
It can be seen from Equation (10) that there are many factors that affect the fracture initiation pressure of SC-CO 2 fracturing. In this paper, the effects of rock mechanical properties (Poisson's ratio, elastic modulus) and bottom hole temperature are analyzed.

Poisson's Ratio
After the invasion of CO 2 , the axial strain of reservoir rock increases, leading to a decrease in Poisson's ratio. Ding et al. [30] found through shale core experiments that the Poisson's ratio of reservoir rock decreases from 0.280 to 0.257, a decrease of 8.2 % after the invasion of CO 2 .
Porous elastic stress coefficient η and thermal stress σ θT are affected by Poisson's ratio. It can be seen from Equations (5) and (7) that the former decreases, while the latter increases with the increase in Poisson's ratio. Figure 6 shows the variation in the porous elastic stress coefficient and the fracture initiation pressure with Poisson's ratio under different Biot coefficients.
It can be seen from Figure 6 that η and pf increase with the increase in Biot coefficient α. Under different Biot coefficients, η and pf reduce with the increase in Poisson's ratio of rock. Combined with the research results of Ding et al. [34], the Poisson's ratio of rocks after CO 2 invasion is reduced, so the variation in CO 2 into the formation may increase the initial fracturing pressure of the reservoir.
Porous elastic stress coefficient η and thermal stress σθT are affected by Poisson's ratio. It can be seen from Equations (5) and (7) that the former decreases, while the latter increases with the increase in Poisson's ratio. Figure 6 shows the variation in the porous elastic stress coefficient and the fracture initiation pressure with Poisson's ratio under different Biot coefficients. It can be seen from Figure 6 that η and pf increase with the increase in Biot coefficient α. Under different Biot coefficients, η and pf reduce with the increase in Poisson's ratio of rock. Combined with the research results of Ding et al. [34], the Poisson's ratio of rocks after CO2 invasion is reduced, so the variation in CO2 into the formation may increase the initial fracturing pressure of the reservoir.

Elastic Modulus
Ding et al. found that the strain difference that the core can withstand decreases and the elastic modulus increases after intrusion of CO2. According to Equation (10), when the bottom hole fluid temperature is low, the initiation pressure decreases with the increase in elastic modulus. Figure 7 shows the change in the fracture initiation pressure with the elastic modulus. It can be seen from the figure that the fracture initiation pressure decreases linearly with the increase in elastic modulus.

Elastic Modulus
Ding et al. found that the strain difference that the core can withstand decreases and the elastic modulus increases after intrusion of CO 2 . According to Equation (10), when the bottom hole fluid temperature is low, the initiation pressure decreases with the increase in elastic modulus. Figure 7 shows the change in the fracture initiation pressure with the elastic modulus. It can be seen from the figure that the fracture initiation pressure decreases linearly with the increase in elastic modulus.

Bottom Hole Temperature
Bottom hole temperature is an important factor affecting the fracture initiation pressure of SC-CO2 fracturing reservoirs. The change in initiation pressure with Poisson's ratio at different bottom hole temperatures is shown in Figure 8. In the figure, the injection rate is 0.5 m 3 /min and 2.0 m 3 /min, respectively, and the corresponding bottom hole temperature difference and formation temperature difference are 13.18 K (bottom hole temperature 368.97 K) and 53.32 K (bottom hole temperature 328.83 K), respectively.
It can be seen from Figure 8 that under different temperature differences, the reservoir fracture initiation pressure decreases linearly with the increase in Poisson's ratio. The reservoir pressure decreases with the increase in temperature difference between the bottom hole and the reservoir.

Bottom Hole Temperature
Bottom hole temperature is an important factor affecting the fracture initiation pressure of SC-CO 2 fracturing reservoirs. The change in initiation pressure with Poisson's ratio at different bottom hole temperatures is shown in Figure 8. In the figure, the injection rate is 0.5 m 3 /min and 2.0 m 3 /min, respectively, and the corresponding bottom hole temperature difference and formation temperature difference are 13.18 K (bottom hole temperature 368.97 K) and 53.32 K (bottom hole temperature 328.83 K), respectively.
It can be seen from Figure 8 that under different temperature differences, the reservoir fracture initiation pressure decreases linearly with the increase in Poisson's ratio. The reservoir pressure decreases with the increase in temperature difference between the bottom hole and the reservoir.
is 0.5 m /min and 2.0 m /min, respectively, and the corresponding bottom hole temperature difference and formation temperature difference are 13.18 K (bottom hole temperature 368.97 K) and 53.32 K (bottom hole temperature 328.83 K), respectively.
It can be seen from Figure 8 that under different temperature differences, the reservoir fracture initiation pressure decreases linearly with the increase in Poisson's ratio. The reservoir pressure decreases with the increase in temperature difference between the bottom hole and the reservoir. According to the analysis above, the Poisson's ratio and elastic modulus of reservoir rock change distinctly after contacting with CO2. In order to comprehensively analyze the influence of changes in Poisson's ratio and elastic modulus of rock on the initiation pressure, the contour of the initiation pressure changes with Poisson's ratio and elastic modulus when the bottom hole temperature is 368.97 K as provided in Figure 9. It can be seen from the figure that the increase in Poisson's ratio and elastic modulus can lead to the decrease in reservoir initiation pressure. In practical application, the change in initiation pressure can be judged according to the increment of Poisson's ratio and elastic modulus after CO2 invasion. According to the analysis above, the Poisson's ratio and elastic modulus of reservoir rock change distinctly after contacting with CO 2 . In order to comprehensively analyze the influence of changes in Poisson's ratio and elastic modulus of rock on the initiation pressure, the contour of the initiation pressure changes with Poisson's ratio and elastic modulus when the bottom hole temperature is 368.97 K as provided in Figure 9. It can be seen from the figure that the increase in Poisson's ratio and elastic modulus can lead to the decrease in reservoir initiation pressure. In practical application, the change in initiation pressure can be judged according to the increment of Poisson's ratio and elastic modulus after CO 2 invasion.

Application Example
Based on the analysis above, the initiation pressure at the injection rate of 0.5 m 3 /min and 2.0 m 3 /min is calculated, respectively (Table 2). It can be seen from Table 2 that SC-CO2 fracturing can effectively reduce the initial pressure of the reservoir. When the injection rate increases from 0.5 m 3 /min to 2.0 m 3 /min, the initial pressure decreases from 48.94 MPa to 40.99 MPa, reducing by nearly 8 MPa. It should be noted that although increasing the injection rate can reduce the fracture initiation pressure, it also brings large energy consumption. In the fracture initiation stage, it is not appropriate to excessively pursue the low fracture initiation pressure under high injection rate.

Application Example
Based on the analysis above, the initiation pressure at the injection rate of 0.5 m 3 /min and 2.0 m 3 /min is calculated, respectively ( Table 2). It can be seen from Table 2 that SC-CO 2 fracturing can effectively reduce the initial pressure of the reservoir. When the injection rate increases from 0.5 m 3 /min to 2.0 m 3 /min, the initial pressure decreases from 48.94 MPa to 40.99 MPa, reducing by nearly 8 MPa. It should be noted that although increasing the injection rate can reduce the fracture initiation pressure, it also brings large energy consumption. In the fracture initiation stage, it is not appropriate to excessively pursue the low fracture initiation pressure under high injection rate. From the perspective of dynamic fracturing process, low initial pressure is conducive to the formation of fractures, which expands rapidly under the bottom hole pressure. Due to the slow propagation speed of the temperature field, the thermal stress generated by the temperature field can induce the generation and expansion of secondary fractures, and thus create a fracture network system with high conductivity in the reservoir.

Conclusions
SC-CO 2 fracturing can effectively reduce the fracture initiation pressure of reservoir. The fracture initiation pressure decreases with the increase in injection rate.
With the injection of CO 2 , the influence of bottom hole temperature and pressure on the near-wellbore formation expands radially. The sweep velocity of bottom hole pressure is higher than that of bottom hole temperature. The injection rate mainly affects the near-wellbore formation temperature and pore pressure, but has little influence on the sweep velocity.
The maximum tangential stress caused by formation temperature and pore pressure are both located at the well wall. The former reduces the tangential stress at the well wall, while the latter increases the tangential stress at the well wall. The stress caused by formation temperature change is greater than that caused by pore pressure change. On the whole, the minimum tangential stress is also located at the well wall.
The increase in Poisson's ratio, the increase in elastic modulus and the decrease in bottom hole temperature can all contribute to reducing the initial pressure of the reservoir.