Surrounding rock burst monitoring and early warning method after concentrated stress is transferred from kerf to deep part

By designing targeted cut-off methods in rock burst monitoring, stress is transferred to deep surrounding rocks, and using accurate inversion algorithms and calculation models, the problem of insufficient microseismic positioning accuracy and energy level in the existing technology is solved, and more accurate rock burst risk assessment and early warning is achieved, ensuring the construction safety of high-stress hard rock tunnels.

CN120159529APending Publication Date: 2025-06-17WUHAN UNIV
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Patent Information

Application Number
CN202510433340.1
Authority / Receiving Office
CN · China
Patent Type
Applications(China)
Current Assignee / Owner
Filing Date
2025-04-08
Publication Date
2025-06-17

AI Technical Summary

Technical Problem

In the existing rock burst monitoring technology, there are significant deviations in the three-dimensional positioning accuracy of microseismic events and the inversion of rupture energy levels, resulting in limited rock burst disaster warning efficiency.

Method used

By designing a targeted slit method in the surrounding rock after transferring concentrated stress to the deep slit, the tangential concentrated stress of the excavated palm surface and hole wall is transferred to the deep surrounding rock, and the stress is redistributed to concentrate at the end of the slit. Sensors are used to monitor the acoustic emission or microseismic signals of rock rupture in the stress concentration area near the cutting slit, and position and energy level inversion are performed through accurate inversion algorithms, and a calculation model is established for theoretical solution or numerical simulation, and a comprehensive warning of rock burst risk is comprehensively warned.

Benefits of technology

The three-dimensional positioning accuracy and rupture level inversion accuracy of microseismic events are significantly improved, and the effect of concentrated stress transfer and the risk of rock bursts are more accurately evaluated, ensuring the construction safety of high-stress hard rock tunnels.

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Abstract

The invention discloses a surrounding rock burst monitoring and early warning method after concentrated stress is transferred to a deep part from a kerf, which comprises the following steps: cutting the surrounding rock of a tunnel face and a tunnel wall along the minimum principal stress direction, arranging sensors along the two sides of the kerf, and monitoring rock fracture acoustic emission or microseismic signals in a stress concentration area near the kerf; positioning rock fracture acoustic emission or micro-seismic events and determining energy level inversion through a precise inversion algorithm for controlling an inversion area and a retrieval range; according to the positioning and energy level inversion result of the rock fracture acoustic emission or micro-seismic event, the surrounding rock stress concentration and rock fracture conditions obtained through analysis of the calculation model are combined, and the rock burst risk level is comprehensively pre-warned. According to the surrounding rock rockburst monitoring and early warning method, targeted monitoring is implemented by narrowing the monitoring range, the three-dimensional positioning precision and the fracture energy level inversion precision of a micro-seismic event are greatly improved, the effect of transferring concentrated stress by a kerf is evaluated more accurately, the risk of rockburst is evaluated, and construction safety is guaranteed.
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Description

Technical Field

[0001] The present invention relates to the technical field of rock mass construction, and particularly relates to a method for monitoring and warning rockburst of surrounding rock after concentrated stress is transferred to the deep part in a cut seam. Background Art

[0002] When a tunnel (cavity) is excavated in a relatively intact rock mass under a high in-situ stress environment, unloading occurs in the direction perpendicular to the excavation face and stress concentration occurs in the direction parallel to the excavation face. Under the conditions of radial unloading and tangential stress concentration on the tunnel wall, slab cracking failure and rockburst dynamic disasters will occur, significantly reducing the construction efficiency. In severe cases, it will lead to equipment damage and casualties. Rockburst monitoring, as an important technical means for rockburst dynamic disaster warning, can effectively reveal the catastrophic evolution process of rock mass from energy accumulation to sudden instability by capturing the acoustic emission signals, microseismic activities and deformation characteristics of rock fractures in a high-stress geological environment in real time, providing key data support for disaster precursor identification and warning decision-making. However, the existing technology still faces bottleneck problems such as insufficient three-dimensional positioning accuracy of microseismic events and significant deviations in the inversion of fracture energy levels, which restricts the further improvement of the rockburst disaster warning efficiency. Summary of the Invention

[0003] The purpose of the present invention is to solve the problems existing in the prior art. In order to improve the accuracy of rockburst monitoring and warning, a method for monitoring and warning rockburst of surrounding rock after concentrated stress is transferred to the deep part in a cut seam is provided. By implementing targeted monitoring, the effect of the cut seam transferring concentrated stress is accurately evaluated, the risk of rockburst occurrence is evaluated, and the construction safety of high-stress hard rock tunnels is ensured.

[0004] To achieve the above purpose, the technical solution adopted by the present invention is: A method for monitoring and warning rockburst of surrounding rock after concentrated stress is transferred to the deep part in a cut seam, comprising the following steps: Determine the principal stress directions in the cross-sectional plane perpendicular to the tunnel, and the principal stress directions in the cross-sectional plane include the maximum principal stress direction and the minimum principal stress direction; Cut the surrounding rock of the excavation face and the tunnel wall along the minimum principal stress direction; Arrange sensors on both sides of the cut seam to monitor the acoustic emission or microseismic signals of rock fractures in the stress concentration area near the cut seam; According to the monitored acoustic emission or microseismic signals of rock fractures, through an accurate inversion algorithm that controls the inversion area and retrieval range, determine the positioning and energy level inversion of the acoustic emission or microseismic events of rock fractures; Establish a calculation model for the monitoring area, and perform theoretical calculation or numerical simulation of the stress concentration and rock fracture conditions of the surrounding rock after the cut seam; According to the positioning and energy level inversion results of the acoustic emission or microseismic events of rock fractures, combined with the stress concentration and rock fracture conditions of the surrounding rock obtained by analyzing the calculation model, comprehensively warn the rockburst risk level.

[0005] This surrounding rock rockburst monitoring and early warning method is carried out after the concentrated stress is transferred to the deep part in the direction of the cut seam. By specifically designing a simple cut seam method, it effectively realizes the transfer of the tangential concentrated stress of the excavation face and the tunnel wall to the deep surrounding rock, so that the redistributed stress is concentrated at the end of the cut seam, and the induced stress is concentrated at the end of the cut seam. By narrowing the monitoring range to implement targeted monitoring, the three-dimensional positioning accuracy of microseismic events and the inversion accuracy of rupture energy levels are greatly improved, the effect of the cut seam transferring concentrated stress is more accurately evaluated, the risk of rockburst occurrence is evaluated, and the construction safety of high-stress hard rock tunnels (caverns) is guaranteed.

[0006] Furthermore, the position of the sensor relative to the cut seam is arranged within the range of 0.5H to 1.5H with reference to the depth H of the cut seam.

[0007] Furthermore, assuming the depth of the cut seam is H and the tunnel diameter or equivalent tunnel diameter of the surrounding rock is D, the relationship between H and D is as follows: When preventing medium rockburst, H ≥ D / 20; When preventing strong rockburst, H ≥ 2D / 20; When preventing extremely strong rockburst: H ≥ 3D / 20; When preventing damage and collapse of the surrounding rock of the high-stress tunnel wall, H ≥ 0.5m.

[0008] Furthermore, if there are rock mass structural planes developed in the surrounding rock near the cut seam, the cut seam should avoid intersecting with the rock mass structural plane at the upper part, and the sensor is arranged at the middle part between the cut seam and the rock mass structural plane.

[0009] Furthermore, the spacing L of the sensors arranged along the direction of the cut seam is determined as follows: For the cut seam on the tunnel wall, the spacing L is taken as 10 - 30m; for the cut seam on the excavation face, the spacing L is taken as 5 - 10m.

[0010] Furthermore, in the precise inversion algorithm, the inversion area and the retrieval range are concentrated in the stress concentration area at the end of the cut seam.

[0011] Furthermore, the precise inversion algorithm includes the following steps: Through the established calculation model, the stress concentration situation of the surrounding rock after cutting the seam is theoretically calculated or numerically simulated, and the inversion area is demarcated in the area of the stress concentration of the surrounding rock; The wave velocity iteration algorithm is used to calculate the rock wave velocity in the monitoring area to make it approach the true rock wave velocity in the monitoring area; Then, using the rock wave velocity obtained by iterative calculation, the rupture events within a certain range from the stress concentration area of the surrounding rock in the retrieval range are calculated for precise positioning; Based on the relationship between the wave velocity of the rock and the energy attenuation, the energy level of the rupture event is inverted.

[0012] Furthermore, the calculation model is a three-dimensional model or a two-dimensional model; a three-dimensional calculation model is established at the cut seam of the heading face or at the cut seam of the surrounding rock of the tunnel wall near the heading face; a two-dimensional calculation model is established at the cut seam of the surrounding rock of the tunnel wall farther away from the heading face.

[0013] Furthermore, the position of the cut seam is set at the center and the lower part of the heading face, and at the part below the arch waist of the tunnel wall; the width of the cut seam ranges from a few millimeters to a few centimeters and varies with the level of the maximum principal stress.

[0014] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The present surrounding rock rockburst monitoring and early warning method is carried out after the cut seam transfers the concentrated stress to the deep part. By designing a simple cut seam method specifically, it effectively realizes the transfer of the tangential concentrated stress of the excavation heading face and the tunnel wall to the deep surrounding rock, so that the redistributed stress is concentrated at the end of the cut seam, and the induced stress is concentrated at the end of the cut seam. By narrowing the monitoring range and implementing targeted monitoring, the three-dimensional positioning accuracy of microseismic events and the inversion accuracy of the rupture energy level are greatly improved, and the effect of the cut seam transferring the concentrated stress is evaluated more accurately, and the risk of rockburst occurrence is evaluated to ensure the construction safety of high-stress hard rock tunnels (caverns); 2. The direction of the cut seam is cut along the direction of the minimum principal stress, that is, perpendicular to the direction of the maximum principal stress in the cross-sectional plane, which can unload the concentrated stress from the surface of the heading face and / or the tunnel wall to the bottom of the crack, and transfer the concentrated stress to the deep surrounding rock at a certain distance from the heading face and the tunnel wall at the bottom of the crack. In this way, the stress concentration situation in the rock mass can be guided to the end of the cut seam, which is beneficial to subsequent targeted monitoring and improves the accuracy of subsequent monitoring; 3. By arranging a number of monitoring sensors in the area on both sides of the cut seam, the rock fracture acoustic emission or microseismic signals in the stress concentration area near the cut seam can be monitored in real time, and the precise inversion algorithm is used for the positioning and energy level inversion of the rock fracture acoustic emission or microseismic signals. By concentrating the inversion range in the stress concentration area at the end of the cut seam, the inversion efficiency and inversion accuracy can be significantly improved, and efficient and accurate monitoring and early warning of rockburst can be realized. Description of the Drawings

[0015] Figure 1 It is a schematic diagram of the cut seam and sensor arrangement on the tunnel wall in the surrounding rock rockburst monitoring and early warning method of the present invention; Figure 2 is Figure 1 Schematic diagram of the A-A cross-section in Figure 3 It is a cross-sectional schematic diagram of the cut seam at the tunnel heading face in the surrounding rock rockburst monitoring and early warning method of the present invention; Figure 4 is Figure 3Schematic diagram of the cut seam in the middle heading face and the arrangement of sensors; In the figure: 1. Cut seam; 2. Heading face; 3. Tunnel wall; 4. Sensor. Specific implementation method

[0016] The following will clearly and completely describe the technical solutions of the present invention in conjunction with the accompanying drawings in the present invention. Obviously, the described embodiments are only a part of the embodiments of the present invention, rather than all the embodiments. Based on the embodiments in the present invention, all other embodiments obtained by those of ordinary skill in the art without creative work conditions belong to the scope of protection of the present invention.

[0017] In the description of the present invention, it should be noted that the orientation or positional relationship indicated by the terms "middle", "upper", "lower", "left", "right", "inner", "outer", etc. is based on the orientation or positional relationship shown in the accompanying drawings. It is only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as a limitation of the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance.

[0018] A method for monitoring and warning rock bursts in surrounding rock after concentrating stress deep into the cut seam includes the following steps: Step 1: Determine the principal stress direction of the cross-sectional plane perpendicular to the tunnel, and the principal stress direction of the cross-sectional plane includes the maximum principal stress direction and the minimum principal stress direction; Step 2: As shown in Figures 1 to 4 , cut the surrounding rock of the heading face 2 and the tunnel wall 3 along the minimum principal stress direction; Step 3: Arrange sensors 4 on both sides of the cut seam 1 to monitor the rock fracture acoustic emission or microseismic signals in the stress concentration area near the cut seam 1; Step 4: According to the monitored rock fracture acoustic emission or microseismic signals, through an accurate inversion algorithm that controls the inversion area and retrieval range, determine the location and energy level inversion of the rock fracture acoustic emission or microseismic events; Step 5: Establish a calculation model for the monitoring area, and perform theoretical calculation or numerical simulation of the stress concentration and rock fracture conditions of the surrounding rock after cutting the seam; Step 6: According to the location and energy level inversion results of the rock fracture acoustic emission or microseismic events, combined with the stress concentration and rock fracture conditions of the surrounding rock obtained by analyzing the calculation model, comprehensively warn the rock burst risk level.

[0019] This surrounding rock rockburst monitoring and early warning method is carried out after the concentrated stress is transferred to the deep part in the direction of the cut seam. By specifically designing a simple cut seam method, the tangential concentrated stress of the excavation face and the tunnel wall is effectively transferred to the deep surrounding rock, so that the redistributed stress is concentrated at the end of the cut seam, and the induced stress is concentrated at the end of the cut seam. By narrowing the monitoring range and implementing targeted monitoring, the three-dimensional positioning accuracy of microseismic events and the inversion accuracy of rupture energy levels are greatly improved, the effect of the cut seam transferring the concentrated stress is more accurately evaluated, the risk of rockburst occurrence is evaluated, and the construction safety of high-stress hard rock tunnels (caverns) is guaranteed.

[0020] In the step 1, the direction of the principal stress in the cross-sectional plane perpendicular to the tunnel is inferred and determined by in-situ stress testing, in-situ stress inversion or observation of the fracture characteristics of the surrounding rock of the tunnel wall. The maximum principal stress is σ1 and the minimum principal stress is σ2.

[0021] In the step 2, the direction of the cut seam 1 is cut along the direction of the minimum principal stress, that is, perpendicular to the direction of the maximum principal stress in the cross-sectional plane, which can unload the concentrated stress from the surface of the heading face and / or the tunnel wall to the bottom of the crack, and transfer the concentrated stress to the deep surrounding rock at a certain distance from the heading face and the tunnel wall at the bottom of the crack. In this way, the stress concentration in the rock mass can be guided to the end of the cut seam, which is beneficial to subsequent targeted monitoring and improves the accuracy of subsequent monitoring.

[0022] In steps 3-6, by arranging a number of monitoring sensors (such as vibration sensors, mechanical wave sensors, etc.) in the area on both sides of the cut seam, the rock fracture acoustic emission or microseismic signals in the stress concentration area near the cut seam can be monitored in real time, and the precise inversion algorithm is used for positioning and energy level inversion of the rock fracture acoustic emission or microseismic signals. By concentrating the inversion range in the stress concentration area at the end of the cut seam, the inversion efficiency and inversion accuracy can be significantly improved, and efficient and accurate monitoring and early warning of rockburst can be realized.

[0023] Further, in step 3, the position of the sensor 4 from the cut seam 1 should be appropriate, neither too close nor too far. With reference to the depth H of the cut seam 1, the sensor 4 is arranged in the range of 0.5H to 1.5H.

[0024] Further, during the process of arranging the sensor 4, its setting position needs to be adjusted in real time. If there are rock mass structural planes developed in the surrounding rock near the cut seam 1, the cut seam should avoid intersecting with the rock mass structural plane at the upper part, and the sensor 4 is arranged in the middle part between the cut seam 1 and the rock mass structural plane, so as to avoid arranging the sensor 4 at a position far from the cut seam on the structural plane, which may cause deviations in the monitoring position and energy level due to the propagation of rock fracture signals passing through the structural plane.

[0025] Further, generally, the position of the sensor 4 is set according to the surrounding rock conditions, monitoring purposes and specific requirements. The spacing L of the sensors 4 arranged along the direction of the cut seam 1 is determined as follows: For the cut seam on the tunnel wall, the spacing L is taken as 10 - 30 m; for the cut seam on the heading face, the spacing L is taken as 5 - 10 m. If the integrity of the surrounding rock is better, a larger value can be taken for the spacing L.

[0026] Since the position of the sensor is closely related to the position and depth of the cut seam, the determination of the cut seam depth is also crucial. In this embodiment, let the depth of the cut seam be H, and the diameter or equivalent diameter (if it is a non - circular cross - section, calculate the equivalent diameter) of the tunnel where the surrounding rock is located be D. Then the relationship between H and D is as follows: When preventing medium rockburst, H≥D / 20; When preventing strong rockburst, H≥2D / 20; When preventing extremely strong rockburst: H≥3D / 20; When preventing damage and collapse of the surrounding rock of the high - stress tunnel wall, H≥0.5 m.

[0027] By the rockburst grade to be prevented and the diameter of the tunnel to be excavated, the depth of the cut seam is determined. Then, setting the arrangement position of the sensor with this depth value can well improve the accuracy of the sensor monitoring and reduce the error caused by inaccurate monitoring positions.

[0028] Further, in the precise inversion algorithm, the inversion area and the retrieval range are concentrated in the stress - concentration area at the end of the cut seam.

[0029] The precise inversion algorithm includes the following steps: Through the established calculation model, theoretical solution or numerical simulation is carried out on the stress concentration situation of the surrounding rock after cutting the seam, and the inversion area is delineated in the area of stress concentration of the surrounding rock; The wave velocity iteration algorithm is used to calculate the rock wave velocity in the monitoring area to make it approach the true rock wave velocity in the monitoring area; Then, using the rock wave velocity obtained by iterative calculation, the fracture events within a certain range from the stress - concentration area of the surrounding rock in the retrieval range are calculated for precise positioning; Based on the relationship between the rock wave velocity and energy attenuation, the energy level of the fracture event is inverted.

[0030] Further, the calculation model is a three - dimensional model or a two - dimensional model; for the cut seam on the heading face or the cut seam on the surrounding rock of the tunnel wall near the heading face, a three - dimensional calculation model is established; for the cut seam on the surrounding rock of the tunnel wall far from the heading face, a two - dimensional calculation model is established.

[0031] During the above-mentioned slot cutting process, the slot cutting will also be optimized in real time. The methods and principles for determining technical parameters such as the position, width, and depth of the slot cutting also include the following points: The position of the slot cutting is set at the center and lower part of the heading face, and the part below the arch waist of the tunnel wall; the width of the slot cutting ranges from a few millimeters to a few centimeters and varies with the level of the maximum principal stress.

[0032] To reduce the risks of high-stress slab cracking failure and rockburst dynamic disasters, the width of the slot cutting generally ranges from a few millimeters to a few centimeters. The higher the level of the maximum principal stress and the lower the modulus of the surrounding rock, the wider the required slot cutting. The specific width is based on the principle that the slot cutting saw blade will not be stuck due to the deformation of the surrounding rock during the slot cutting process.

[0033] To reduce the large deformation of high-stress surrounding rock, the width of the slot cutting is generally above the centimeter level, or multiple parallel narrow cracks can also be set to achieve the purpose of reducing the large deformation of high-stress surrounding rock.

[0034] The slot cutting on the heading face needs to ensure a certain overlap length h between two consecutive slot cuttings. For the heading face sections with frequent rockbursts: h ≥ 0.3 m.

[0035] During the actual operation process, rock cutting machines can be respectively set in the cutter head area, the cutter head edge and shield connection area, and the shield tail area of the roadheader. Using rock cutting machines (such as different types of cutting machines like disc saw cutting machines, high-pressure water jet cutting machines or diamond wire saw cutting machines), slot cutting is carried out on the rock. The slot cutting on the tunnel wall needs to advance along with the excavation of the tunnel (cavity). In the heading face sections with frequent rockbursts, the front end of the slot cutting should closely follow the heading face.

[0036] Although the embodiments of the present invention have been shown and described, for those of ordinary skill in the art, it can be understood that various changes, modifications, substitutions, and variations can be made to these embodiments without departing from the principles and spirit of the present invention. The scope of the present invention is defined by the appended claims and their equivalents.

Claims

1. A method for monitoring and early warning of rock burst in surrounding rock after the concentrated stress is transferred from the cutting seam to the deep part, characterized in that: The steps include: Determining the principal stress direction of the cross-sectional plane perpendicular to the tunnel, wherein the principal stress direction of the cross-sectional plane includes the maximum principal stress direction and the minimum principal stress direction; Cutting the surrounding rock of the tunnel face and the tunnel wall along the direction of the minimum principal stress; Arrange sensors along both sides of the cut to monitor acoustic emission or microseismic signals of rock fracture in stress concentration areas near the cut; According to the monitored rock fracture acoustic emission or microseismic signal, the rock fracture acoustic emission or microseismic event is located and the energy level inversion is determined by controlling the accurate inversion algorithm of the inversion area and the search range; Establish a calculation model for the monitoring area, and perform theoretical calculations or numerical simulations on surrounding rock stress concentration and rock fracture after cutting; According to the positioning and energy level inversion results of rock fracture acoustic emission or microseismic events, combined with the surrounding rock stress concentration and rock fracture conditions obtained by the calculation model analysis, a comprehensive warning of the rock burst risk level is given.

2. The method for monitoring and early warning of surrounding rock burst after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1, characterized in that: The distance between the sensor and the slit is within the range of 0.5H to 1.5H, with the depth H of the slit as a reference.

3. The method for monitoring and early warning of surrounding rock burst after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1 or 2, characterized in that: Assuming that the depth of the cut is H, and the diameter of the tunnel where the surrounding rock is located or the equivalent diameter is D, the relationship between H and D is as follows: To prevent moderate rock burst, H≥D / 20; To prevent severe rock burst, H≥2D / 20; To prevent extremely strong rock burst: H≥3D / 20; To prevent damage and collapse of high-stress cave wall surrounding rock, H≥0.5m.

4. The method for monitoring and early warning of surrounding rock burst after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1, characterized in that: If a rock mass structural surface is developed in the surrounding rock near the cutting seam, the cutting seam is prevented from intersecting with the rock mass structural surface at the upper part, and the sensor is arranged in the middle part between the cutting seam and the rock mass structural surface.

5. The method for monitoring and early warning of surrounding rock burst after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1, characterized in that: The spacing L of the sensors arranged along the slit direction is determined as follows: For the cuts on the tunnel wall, the spacing L is 10-30 m; for the cuts on the tunnel face, the spacing L is 5-10 m.

6. The method for monitoring and early warning of rock burst in surrounding rock after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1, characterized in that: In the precise inversion algorithm, the inversion area and the search range are concentrated on the stress concentration area at the end of the slit.

7. The method for monitoring and early warning of rock burst in surrounding rock after the concentrated stress is transferred to the deep part by the cutting seam according to claim 1, characterized in that: The precise inversion algorithm comprises the following steps: Through the established calculation model, theoretical calculation or numerical simulation is performed on the stress concentration of the surrounding rock after the cutting, and the inversion area is delineated in the area where the surrounding rock stress is concentrated; The wave velocity iteration algorithm is used to calculate the rock wave velocity in the monitoring area, making it close to the real wave velocity of the rock in the monitoring area; Then, the rock wave velocity obtained by iterative calculation is used to calculate the fracture events within a certain range of the surrounding rock stress concentration area for accurate positioning; Based on the relationship between rock wave velocity and energy attenuation, the energy level of the rupture event is inverted.

8. The method for monitoring and early warning of rock burst in surrounding rock after the concentrated stress is transferred to the deep part by the cutting seam according to claim 1, characterized in that: The calculation model is a three-dimensional model or a two-dimensional model; a three-dimensional calculation model is established at the cut of the tunnel face or at the cut of the tunnel wall surrounding rock close to the tunnel face; a two-dimensional calculation model is established at the cut of the tunnel wall surrounding rock far away from the tunnel face.

9. The method for monitoring and early warning of surrounding rock burst after the concentrated stress is transferred from the cutting seam to the deep part according to claim 1, characterized in that: The positions of the cuts are set at the center and lower part of the tunnel face, and the part below the waist of the tunnel wall; the width of the cuts is in the range of several millimeters to several centimeters, and varies with the level of the maximum principal stress.