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Cryogenic liquid nitrogen fracturing is expected to provide an effective stimulation method for hot dry rock reservoirs to increase heat production. This paper establishes a three-dimensional model to calculate the distributions of temperature and stress of the reservoir rock when liquid nitrogen is injected into the wellbore. The sensitivity of different parameters and water fracturing to the stress state is studied. The results indicate that when liquid nitrogen is injected into the bottom of well, a huge heat exchange occurs on the rock surface, which generates great thermal stress on the fluid-solid interface, and the value of thermal stress exceeds the tensile strength of rock. For the effect of parameters, the primitive temperature of the rock has a significant impact on the value of maximum principal stress. The pressure drop and ambient pressure affect the thermal stress slightly. At the same time, a series of experiments are conducted to validate the effect of thermal stress induced by liquid nitrogen injection on the rock fracture. As the temperature rises, the shale samples are broken more severely at the action of thermal stress. Thus, the study of liquid nitrogen fracturing provides a scientific and effective method for geothermal exploitation.

The hot dry rock (HDR) is green, low-carbon renewable energy in the reservoirs, which is regarded as an important alternative for conventional energy [

At present, the main stimulation method of HDR resources is hydraulic fracturing, which has been widely used in increasing production of unconventional oil and gas [

As a method of waterless fracturing, cryogenic liquid nitrogen (LN) fracturing has its unique advantages in the process of fracturing HDR because of its extremely cryogenic property [

Many previous investigations have been performed to study the rock damage due to LN cooling. Ren et al. [

According to the literature review, most of studies focused on the mechanical properties deterioration of granite caused by thermal stress after LN treatment. The fracture characteristic of granite attributed to LN was discussed in laboratory investigation. However, the effect after injecting LN into the bottom of reservoir is still not clear. Hence, before the LN can be extensively implemented as a fracturing fluid instead of water-based fracturing fluid, many fundamental problems should be addressed to provide guidance on HDR stimulation. These problems included the heat transfer and the distribution of stress at the bottom of well during injecting LN into wellbore. The heat transfer contributes to the change of fracturing conditions, and the distribution of stress is directly related to the effect of fracturing.

This paper conducted a numerical simulation study on the heat transfer and the stress distributions of reservoir when LN was injected into the HDR reservoir. During the process of LN flowing into the wellbore, the tensile stress was generated due to the thermal stress and fluid pressure. The maximum principal stress was selected for analyzing the effect of thermal stress on rock failure, because the maximum principal stress can effectively determine the rock fracture when the tensile failure was generated in rock. The computational fluid dynamics (CFD) was used in the fluid region with standard k-ɛ turbulence model. The fluid-solid interface was adapted to conjugate heat transfer method. In the simulation, granite was adopted in solid region, which is common in HDR. Due to the large elastic modulus and small deformation in granite, the thermoelastic mechanical model was adapted to calculate the stress and strain in solid region.

A three-dimensional model is established to study the heat transfer between LN and high-temperature rock and the stress distributions of rock during injecting LN into the reservoirs. The schematic of model is shown in Figure

The geometry model of computation zones.

Geometric parameters of the model shown in Figure

Pipe inner diameter (mm) | Pipe outer diameter (mm) | Pipe length (mm) | Wellbore diameter (mm) | Well depth (mm) | Model |
---|---|---|---|---|---|

25.4 | 31.8 | 270 | 50.8 | 300 | 400 × 400 × 400 mm^{3} |

The model mainly includes two regions, the fluid region and the solid region. In the fluid region, the high-pressure LN is injected into a pipe and then flows through the annulus between the drill hole and pipe. The inlet and outlet boundary conditions of fluid region are set as pressure inlet and pressure outlet boundary condition, respectively. During LN flowing through the hole of rock, the heat of the rock will be transferred to the cryogenic fluid quickly, which results in rock temperature decreasing. Consequently, the thermal stress is generated in reservoir rock, leading to the variation of original stress distribution and even rock damage.

Based on the present analysis, the following assumptions are proposed: (1) the influence of seepage flow in rock is ignored in the calculation. (2) No deformation of rock is assumed in flow field computation. (3) The rock is assumed as a homogeneous, isotropic, and linearly elastic material. (4) The calculating process does not involve rock failure.

For the progress of injecting fluid into the wellbore, the standard k-ɛ model is adapted for the simulation of turbulent flow because of the larger Reynolds number. Besides, the intense heat transfer will also occur between the cryogenic fluid and warm rock. Thus, the equations of mass conservation, momentum, and energy should be solved in the flow field.

Continuity equation:

where _{l} is fluid density, kg/m^{3};

Momentum equation:

where ^{2}; and g is gravitational acceleration, m^{2}/s.

Energy equation of fluid:

where _{l} is fluid temperature, °C; _{l} is thermal conductivity of fluid, W/m·K; and _{p} is specific heat at constant pressure, J/kg·K.

Turbulent kinetic energy equation:

where ^{2}/s^{2}; _{t} is turbulent viscosity, N·s/m^{2}; _{k} is the turbulent Prandtl number for _{k} is generation of turbulence kinetic energy; _{M} is the contribution of the fluctuating dilatation.

Specific dissipation rate of turbulent kinetic energy:

where _{ɛ} is the turbulent Prandtl number for _{1ɛ} and _{2ɛ} are turbulent constant.

Besides, _{t} is defined as_{μ} is turbulent constant. The empirical constants appearing in the above equations are given by the following values: _{1ɛ} = 1.44, _{2ɛ} = 1.92, _{μ} = 0.09, _{k} = 1.0, _{ɛ} = 1.3.

In the solid region, the heat conduction equation will be solved to calculate the temperature distribution. Furthermore, the stress state of rock owing to thermal stress is also considered in our model. The governing equations of the thermoelastic model are considered in the calculation.

The heat conduction of equation for rock:_{s} is thermal conductivity of rock, W/m·K; _{s} is rock density, kg/m^{3}; and _{s} is rock temperature, °C.

Physical equation:

where _{s} is rock temperature difference, °C.

Equilibrium equation:

where _{s} is rock density.

The strain-displacement relation is

To analyze the temperature and stress distributions of the rock during the process of injecting LN into the wellbore, the simulation parameters are set as in Table

Simulation parameters of fluid region.

Inlet pressure (MPa) | Outlet pressure (MPa) | Initial temperature (°C) | Injection temperature |
---|---|---|---|

25 | 20 | 150 | −173°C |

In addition, the surface contacted the fluid region, and the solid region is set as the fluid-solid interface, which causes a large amount of heat exchange. The conjugate heat transfer method is used to simulate the heat transfer in the fluid-solid interface. Under conjugated boundary conditions, the temperature and the heat flux of the fluid and solid are equal, respectively. The corresponding mathematical expressions are as follows:_{l} and _{s} are the fluid temperature and the solid temperature at the solid-fluid interface, respectively; _{l} is the thermal conductivity of fluid; _{s} is the solid’s thermal conductivity; and

In the progress of the simulation, the detailed modeling parameters used are shown in Table

Material parameters.

LN | Granite | ||
---|---|---|---|

Density | 806.08 kg m^{3} | Density | 2700 kg/m^{3} |

Specific heat | 2041.50 J (kg·K) | Elastic model | 43.20 GPa |

Thermal conductivity | 0.15 W (m·K) | Poisson’s ratio | 0.22 |

Viscosity | 1.61 × 10^{−4} kg (m·s) |

The specific heat of granite:_{s} is rock specific heat, J/kg·K.

The thermal conductivity coefficient of granite:_{s} is thermal conductivity coefficient of rock, W/m·K.

The thermal expansion coefficient of granite:_{s} is thermal expansion coefficient of rock.

To ensure the accuracy of the simulation, mesh independence is validated with four different cell number cases (see Table

Mesh scheme.

Mesh study case | Case 1 | Case 2 | Case 3 | Case 4 |
---|---|---|---|---|

Grid number | 1452566 | 1757972 | 2071302 | 2250028 |

Sensitivity of mesh and time step: (a) the monitoring point temperature at 1 s with different grid number; (b) the monitoring point temperature at 1 s with different time step.

In the meantime, time step is also an important influencing parameter in transient calculation. Different time steps, including 0.05 s, 0.1 s, 0.2 s, 0.5 s, and 1 s, are computed to verify that the time step is independent of simulation results. As shown in Figure

The fluid flow numerical solution is performed based on the CFD code of Fluent with coupled algorithm. It can simulate flow field and heat transfer by coupling mass conservation, momentum, energy, and heat conductivity equations. The computation results indicate that the parameters of pressure and velocity change slightly with time during the process of computation. Thus, the steady state flow is analyzed in this section. Figure

The contours of velocity and pressure in flow field of LN under steady state.

As shown in Figure

The temperature contours in the symmetry plane of model at different times are shown in Figure

The temperature contours in the symmetry plane of model at different times: (a) 0.1 s; (b) 20 s; (c) 40 s; (d) 60 s; (e) 80 s; (f) 100 s.

Temperature distribution at different times along path 1 and path 2, respectively: (a) path 1; (b) path 2.

For the inner region of rock, its cooled-region is constantly increasing, but the growth trend in cooled-region becomes slower and slower. Based on (

The thermal stress will be generated due to the rock temperature decreasing. Therefore, the stress state in rock would be changed with the effects of thermal stress and fluid pressure. A vertical plane 1 which is 100 mm above the downhole is chosen to show the maximum principal stress contour of rock at different times (see Figure

The maximum principal stress contours of rock plane 1 at different times: (a) 0.1 s; (b) 20 s; (c) 40 s; (d) 60 s; (e) 80 s; (f) 100 s.

Figure

The maximum stress in hot rock along path 2 at different times.

Figure

The maximum principal stress in rock along path 2 with different primitive temperature of rock at 20 s.

The difference between the inlet and ambient pressure is defined as the pressure drop. Figure

The maximum principal stress along path 2 with different pressure drop at 20 s.

The ambient pressure is also a significant factor on rock fracturing. The maximum principal stress along path 2 with different ambient pressure is shown in Figure

The maximum principal stress in rock along path 2 with different inlet pressure at 20 s.

To study the influence of different stress boundary on stress distribution, we calculate the maximum principal stress with different stress boundary as shown in Figure _{H}, the maximum principal stress area in the axial direction will decrease gradually, and the maximum principal stress near wellbore along the radial direction also decreases at the same time. The greater the difference is, the more obvious the trend will be. Figure _{H} improving. Thus, it could be reasonably predicted that the rock will be fractured toward the direction of larger in situ stress when LN flows into wellbore.

Maximum principal stress contours of hot rock plane 1 with different stress boundary at 100 s: (a) _{H} = 22.35 MPa, _{h} = 22.35 MPa; (b) _{H} = 27.35 MPa, _{h} = 22.35 MPa; (c) _{H} = 22.35 MPa, _{h} = 32.35 MPa.

The maximum principal stress in rock along path 3 with different stress boundary at 100 s.

To analyze the effect of thermal stress induced by LN cooling, the water cooling is also considered under the same conditions. The physical properties of water are set as follows: density is 998.2 kg/m^{3}, specific heat is 4182 J/(kg·K), thermal conductivity is 0.6 W/(m·K), and viscosity is 0.001003 kg/(m·s). When the stress caused by injecting water into the reservoirs is calculated, the inlet temperature is set as 25°C. Figure

The maximum principal stress in rock along path 2 with different working fluid at 100 s.

In order to study the effect of inlet pressure on stress distribution around wellbore as water flows into the bottom of the wellbore, the maximum principal stress of wellbore surface at 100 s under the same pressure drop is calculated. Figure

The maximum principal stress in rock with different water inlet pressure at 100 s.

In addition, the value of maximum principal stress is about 50.65 MPa with the inlet pressure of 55 MPa, which is similar to the value caused by LN flowing at 25 MPa under the same conditions. This shows that LN injection into the wellbore for fracturing operations can effectively reduce the inlet pressure due to its cryogenic properties compared with fracturing by water.

The variation of stress values due to water injection under different in situ stress conditions was analyzed. Figure _{H}. It indicated that as _{H} increases, the value of maximum principal stress decreases at the surface of wellbore in the direction of minimum horizontal in situ stress, which is the same trend as the effect of LN injection on the values of maximum principal stress. However, when _{H} is 32.35 MPa, the maximum principal stress in the direction of maximum horizontal in situ stress is about 31.04 MPa generated by the water injection, which is lower than the stress value of 35.04 MPa at the direction of minimum horizontal in situ stress by the injected LN. It can be concluded that the tensile damage could be generated in the direction of minimum horizontal in situ stress by the LN injection during the fracture process by injecting water and LN at the same inlet pressure. This is mainly due to the fact that the LN injection process generates a huge thermal stress in the surface of wellbore. The value of thermal stress is not affected by the in situ stress distribution, so the thermal cracks are formed under the thermal stress, which is beneficial to the cracks expansion. Based on this, a more complex fracture network is formed in the reservoir. It has been demonstrated that the main fracture direction of rock samples after LN fracturing does not extend exactly in the direction of the maximum principal stress [

Maximum principal stress at the surface of the wellbore under different in situ stress conditions after LN and water injection for 100 s.

In this paper, the heat transfer and stress state during injecting LN into HDR reservoirs are analyzed based on a 3D thermal-hydraulic-mechanical coupling numerical model. The parameters’ sensitivity is analyzed in detail, and the injection of water under the same conditions is also considered. Finally, a set of experiments is conducted to validate the effect of thermal stress on the rock. The main conclusions of this paper are as follows:

The LN injected into the wellbore causes the rocks around the wellbore to cool down. As time passes by, the increasing amplitude of cooled-region is weakened. During this progress, the great tensile stress that exceeds the tensile strength of granite is generated around the wellbore due to the action of thermal stress and fluid pressure.

The primitive temperature of the reservoir has a significant impact on the stress distributions around the wellbore during LN injecting. With the growth of the primitive temperature, the thermal stress value around wellbore becomes larger. The inlet pressure and the ambient pressure have little effect on the thermal stress generated by LN cooling.

The in situ stresses affect the stress distribution during LN fracturing. Under the unequal in situ stress in different directions, the value of maximum principal stress is larger in the direction with larger in situ stress; i.e., the rock has a tendency to break in the direction of the larger in situ stress value.

Compared with the injection of LN, the value of thermal stress around the wellbore caused by water injection is reduced under the same conditions. The thermal stress value around the wellbore during LN injecting with injection pressure of 25 MPa is similar to that during water injecting with injection pressure of 55 MPa. This means that LN fracturing can reduce the injection pressure effectively and achieve a better fracturing effect.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Chengzheng Cai contributed to conceptualization. Keda Ren participated in data curation and reviewed and edited the manuscript. Both authors were responsible for methodology, software, supervision, and original draft preparation.

This research was funded by the National Key R&D Program of China (2020YFA0711800), the Assistance Program for Future Outstanding Talents of China University of Mining and Technology (2020WLJCRCZL010), the Postgraduate Research and Practice Innovation Program of Jiangsu Province (KYCX20_2042), and the National Natural Science Foundation of China (51604263).