Study on the Distribution Law of Crustal Stress in Fault Fracture Area

Abstract

Determining crustal stress is crucial in the design and construction of underground engineering projects, particularly in fault fracture zones. This paper aims to identify the characteristic factors that influence the crustal stress of faults, including the difference in deformation characteristics between faults and ordinary formations (expressed by the elastic modulus ratio E_r/E_f), the degree of fault fragmentation P, and the angle α between fault tendency and the principal stress direction. Numerical simulations are conducted to investigate the impact of these three factors on the magnitude and direction of the principal stress. Additionally, measured stress values from a specific tunnel are analyzed to validate the findings. The results demonstrate the crucial role of α in determining the increase or decrease of principal stress and the direction of the offset. When α is less than 45°, σ₁ and σ₃ within the fault experience an increase, while σ₁ and σ₃ in the ordinary surrounding rock area near the fault exhibit a slight decrease. Moreover, the σ₁ direction inside the fault deviates towards the parallel fault, whereas σ₁ near the outside slightly deviates towards the vertical fault. Conversely, when α is greater than 45°, the trend of principal stress is reversed. E_r/E_f affects the magnitude of the principal stress change, with larger values resulting in more pronounced changes. P only affects the stress distribution inside the fault.

1. Introduction

High geostress is a critical issue that poses a significant threat to the safety of tunnel engineering and has garnered considerable attention from rock mechanics and engineering experts worldwide. According to Ask et al. [1], crustal stress in underground excavation engineering can be classified into two types: absolute crustal stress and relative crustal stress. Among these, the initial absolute crustal stress of the original rock plays a more crucial role. Therefore, it is imperative to determine the distribution of crustal stress in tunnel engineering to facilitate the project’s design and construction.

In order to determine the distribution of crustal stress, scientists from various countries have proposed many exploration methods, which are mainly divided into two categories. The first type is the direct determination method, including hydraulic fracturing [2,3,4,5] and stress relief [6,7,8], which is also the most widely used method in engineering; however, these two types of practical methods have a certain simplification at the theoretical level, for example, the hydraulic fracturing method assumes that the direction of the principal stress is consistent with the direction of the drilling axis, and the principal stress value of this direction is estimated; therefore, the measurement data in the area with complex terrain and geological conditions may be quite different from the actual crustal stress; The measurement accuracy of the stress relief method depends on the choice of constitutive relationship of the rock mass, and the classical theoretical basis is to assume that the rock is continuous, homogeneous, and isotropic, while the general rock mass has different degrees of discontinuity, heterogeneity, and anisotropy [9], which will lead to a certain gap between the calculated crustal stress and the actual crustal stress value on site. In addition, with the deepening of human understanding of materials, an indoor acoustic emission test has gradually been favored by many researchers to explore crustal stress. Yoshikawa [10] carried out cyclic loading experiments on andesite, compared and analyzed the differences in acoustic emission phenomena of each cyclic rock, and proposed that when the single-loading Kaiser effect is not obvious, the cyclic loading method can be used to measure the crustal stress. Shilin [11] used the anisotropy of the acoustic wave velocity in different principal stress directions to determine the orientation of the crustal stress after the core was unloaded and measured the magnitude of the stress component in various places through the acoustic emission method. Funato and Ito [12] proposed a new radial core deformation analysis (DCDA) method for assessing rock ground stresses in elliptically deformed borehole cores after stress relief. The results of the study showed that the magnitude and direction of stresses estimated by the proposed DCDA method from measured core diameters closely matched the actual applied stresses. Chen et al. [8] investigated the possibility that bedrock temperatures could be used to explore the state of ground stress. Yamamoto [13] determined the magnitude of ground stress from core samples of the borehole by Deformation Rate Analysis (DRA). This method of determining crustal stress through field tests or indoor tests is accurate, but it can only reflect local stresses, which is of limited significance for large-scale projects. Therefore, scholars have proposed a second method to try to solve this problem: the crustal stress inverse analysis method. This method mainly includes stress inverse analysis [14,15], displacement inverse analysis [16], and mixed inverse analysis methods, regardless of which type of analysis relies on numerical simulation to establish tunnel engineering geological models and inversion models, such as linear regression [17], neural networks [18], particle swarm optimization algorithms [19], support vector machines [20] and so on. The use of a computer to invert and fit the initial crustal stress field has advantages for large-scale and large-buried engineering areas, but there are disadvantages with complex geological conditions, such as fault fracture zones. The obvious difference between the stress of the fault area and the main stress of the adjacent formation interferes with the boundary conditions of the model, which causes difficulties in the establishment of the model, resulting in inversion results and the actual measurement cannot be consistent. Therefore, the disturbance law of fault to crustal stress has an important effect on the establishment of the inversion model. The existing research has paid less attention to this.

In this article, a numerical simulation is used to establish a stratum stress model that incorporates faults. The aim is to investigate the impact of three key factors on the magnitude and direction of crustal stress, namely: (1) the distinct deformation characteristics of faults and ordinary strata, (2) the degree of fault fragmentation, and (3) the angle between the fault trend and the horizontal principal stress direction of the formation. By analyzing these factors and their influence on the distribution of crustal stress, this article aims to enhance our understanding of crustal stress distribution in underground engineering.

2. Model Establishment

To consider the influence of crushed rocks on stress, it is crucial to take into account the varying degrees of fault fracturing commonly observed in fault zones as a result of strata extrusion [21,22,23]. Within the fault zone, crushed rocks can manifest in different forms, including fragmented structures and block structures. In this study, different crushed rock conditions are incorporated to represent distinct fracture degrees of faults, with content ranging from 0% to 15%, as depicted in Figure 1. The model employed in this research has dimensions of 90 m in length and 30 m in width, with the fault situated at the center spanning 10 m in width and 30 m in length, aligned vertically. The fragmented rock mass within the fault is generated randomly, adhering to specific requirements using the macro language in ABAQUS. The model is simulated using planar strain elements with a mesh size of 1 m.

To focus solely on the effects of material property variations on stress distribution, a Mohr-Coulomb elastoplastic model with mechanical parameters (shown in

Table 1. Mechanical Parameters of Model Materials.

Material	Density ρ (g/cm3)	Elastic Modulus E (GPa)	Poisson’s Ratio μ	Cohesion c (MPa)	Angle of Internal Friction (°)
Ordinary stratigraphic area	2.7	30, 60, 90, 120	0.27	2	56
fault	2.63	30	0.35	1.5	45
Crushed rock	2.8	100	0.26	1.8	50

) is employed for each material in this article [24,25]. The elastic modulus of the ordinary formation model is set to four different working conditions to investigate the impact of different E_r/E_f ratios on stress changes.

To simplify the calculation, it is assumed that the maximum principal stress acting on the model area σ₁ is 30 MPa, and the minimum principal stress σ₃ is 10 MPa. Under different working conditions, σ₁ acts on the model boundary from different angles. The model constrains normal displacement on all sides, with a free interface at the top and all degrees of freedom constrained at the bottom boundary [26,27], as shown in Figure 2.

Several different principal stress directions are designed, including α = 0°, 30°, 60°, and 90°, where α represents the angle between the maximum principal stress direction and the x-axis direction. To apply the model boundary conditions of σ_x, σ_y, and τ_xy, it is required to convert the coordinates using Equations (1)–(3).

$${\sigma }_x=\frac{{\sigma }_1+{\sigma }_3}{2}+\frac{{\sigma }_1-{\sigma }_3}{2}\cdot \cos 2\alpha$$

(1)

$${\sigma }_y=\frac{{\sigma }_1+{\sigma }_3}{2}-\frac{{\sigma }_1-{\sigma }_3}{2}\cdot \cos 2\alpha$$

(2)

$${\tau }_{xy}=\frac{{\sigma }_1-{\sigma }_3}{2}\cdot \sin 2\alpha$$

(3)

To summarize, this article’s model considers three factors: the degree of fault fragmentation P, the ratio of elastic modulus between strata and faults E_r/E_f, and the angle α between the fault inclination and the direction of horizontal principal stress in the strata. Each factor has four situations, resulting in 64 operating conditions, as shown in

Table 2. Variable Values of the Model under Different Operating Conditions.

Working Condition	P (%)	Er/Ef	α (°)
No. 1~No. 16	0	1, 2, 3, 4	0, 30, 60, 90
No. 17~No. 32	5	1, 2, 3, 4	0, 30, 60, 90
No. 33~No. 48	10	1, 2, 3, 4	0, 30, 60, 90
No. 48~No. 64	15	1, 2, 3, 4	0, 30, 60, 90

.

3. Stress Change Results

The stress distribution was evaluated under various operating conditions, and the unit stress scenario was extracted, as illustrated in Figure 3. In this figure, σ₁ denotes the maximum principal stress traversing the fault zone horizontally at different locations, while σ₃ represents the minimum principal stress. Additionally, the variation in the direction of σ₁ is also considered. The outcomes of the analysis are presented in Figure 4, Figure 5, Figure 6 and Figure 7.

To begin with, a relatively straightforward scenario is examined where the effect of the elastic modulus ratio and stress direction on the principal stress disturbance is investigated under the condition of P = 0. As shown in Figure 4, it is evident that there are notable changes in stress magnitude and direction within the fault zone:

(1) When the direction of σ₁ is perpendicular to the direction of the fault (α = 0°), the σ₁ inside the fault increases, while the σ₁ in the surrounding ordinary rock area decreases slightly. Similarly, σ₃ also increases inside the fault and decreases outside. In this scenario, the σ₃ in the model region is 0, so σ₃ remains 0 in the strata far from the fault. Inside the fault, σ₃ > 0, indicating compression, while outside the fault near the fault zone, σ₃ < 0, indicating tension. This phenomenon becomes more pronounced as the difference in deformation modulus between the two increases, as the deformation capacity of the ordinary strata and the fault are inconsistent. The E in the fault zone is smaller and the μ is larger, so the deformation is larger than that of the ordinary strata. At the junction of the fault and the strata, in order to ensure displacement continuity, the fault element tends to deform the strata element upward and downward. Therefore, some strata elements will be under tension in the y direction. Similarly, for the fault element, adjacent strata elements will limit its deformation, so some fault elements will be under compression in the y direction. This is also the reason there is a sudden change in the curve in Figure 4b. In addition, from Figure 4c, it can be seen that when the direction of σ₁ is perpendicular to the direction of the fault, the disturbance of the fault to its direction change is small.

(2) When the direction of σ₁ is parallel to the direction of the fault (α = 90°), the change in stress magnitude presents a completely opposite situation to the discussion in (1). That is, σ₁ and σ₃ will both decrease inside the fault while they will increase in the surrounding ordinary rock area near the fault. The greater the E_r/E_f, the more this change will be amplified. Similarly, this phenomenon is also caused by the difference in deformation capacity between the two. Compared with the ordinary strata, the deformation in the fault zone is greater. This may lead to stress release and stress transfer phenomena similar to those in tunnel engineering caused by rock convergence. Therefore, in relatively hard strata, σ₁ will increase, while it will decrease in the relatively “soft” fault zone. As for σ₃, due to the large lateral deformation in the fault zone, there may be a tensile crack area, while the ordinary strata on both sides will be subjected to lateral compression from the fault. In addition, from Figure 4c, it can be seen that when the direction of σ₁ is parallel to the direction of the fault, the disturbance of the fault to its direction change will also be relatively small.

(3) When the angle between the direction of σ₁ and the vertical fault direction is less than 45° (α = 30°), the variation of stress in the formation is similar to that of the α = 0° case. The difference is that the amplitude of the changes in σ₁ and σ₃ is reduced. In fact, the inclined stress condition can be considered as a combination of horizontal stress and vertical stress, as shown in Figure 8. When the inclination angle α is less than 45°, the horizontal stress dominates, and the variation pattern of stress in the formation is the same as that of α = 0°, but the amplitude of stress change is reduced due to the existence of vertical stress. Additionally, in the case of an angle, the direction of σ₁ in the fault zone will also change. When α is small, the direction of the principal stress in the fault zone will deviate slightly toward the direction parallel to the fault, while the direction of the principal stress in the formations near the fault will deviate slightly toward the direction perpendicular to the fault.

(4) When the angle between the direction of σ₁ and the vertical fault direction is greater than 45° (α = 60°), the variation pattern of the earth’s stress is similar to that of the α = 90° condition, which can also be explained by Figure 8. When α is greater than 45°, the vertical stress dominates, but due to the existence of horizontal stress, its amplitude of stress change will also be weakened. Under this condition, the direction of the main stress inside the fault will deviate towards the direction perpendicular to the fault trend, while the direction of the main stress in the layers near the outside of the fault will deviate in the opposite direction.

Figure 5, Figure 6 and Figure 7 illustrate that the stress distribution outside the fault remains consistent with the P = 0 condition when crushed rock is present within the fault, while stress fluctuations occur inside the fault. To gain a better understanding of the impact of stress on crushed rock, we extracted and plotted the changes in the σ₁ curve under different working conditions with α = 0° in Figure 9.

Based on the analysis of the aforementioned σ₁ change law, it can be observed that under the conditions of P = 0 and α = 0°, the stress inside the fault increases, but the presence of crushed rock causes significant fluctuations in internal stress. Figure 9 highlights the stress point of the downward fluctuation of the curve (which does not exist when P = 0) and reveals a corresponding relationship with the position of the corresponding crushed rock. The larger the crushed rock content, the more frequent the curve fluctuation, indicating that crushed rock has a pronounced impact on the stress distribution within the fault. Specifically, the magnitude and direction of stress near the crushed rock fluctuate, resulting in opposite patterns of stress variation under P = 0 conditions when compared to the stress change in the absence of fracture in the fault.

Through qualitative understanding and numerical experimental analysis, it is evident that faults significantly disrupt the direction of local stress fields and exhibit consistent changes based on factors such as the angle between the maximum principal stress direction and the fault plane, the modulus ratio of faults to the surrounding rock, and the proportion of principal stress. When utilizing measured point data affected by faults in inversion calculations, it is crucial to consider the disturbance pattern to avoid distorted and non-uniform inversion stress fields. Therefore, understanding the disturbance patterns described in this section can serve as a basis for identifying abnormal measured in situ stress data influenced by faults prior to inversion analysis.

When faults slide under the compression of the stress field, they reach an equilibrium stress state, which represents the initial stress field following the fault disturbance. Since faults primarily impact the local stress field, the stress field with faults can be considered as a local adjustment to the regional stress field after the fault occurrence. The investigation conducted in Section 3 can also be interpreted as examining changes in the local stress field before and after the occurrence of a fault. It is evident that accurate load conditions must be employed to describe the regional stress field before the fault occurrence during the inversion calculation of small fault-inclusive models. The stress field primarily arises from self-weight and tectonic actions. The key to the secondary inversion calculation of small fault-inclusive models lies in extracting appropriate structural load conditions from the results of the primary inversion analysis. Given the complex regional structures, valley topography, and other factors, common submodel methods extract boundary load conditions from a single inversion result and simulate regional tectonic effects using boundary forces. However, this approach can lead to local stress distortion and reduced accuracy. During the process of large-scale model inversion, it is necessary to apply equivalent structural forces to the element nodes to achieve accurate equilibrium. When conducting secondary calculations, applying boundary loads can result in an internal stress field in the small model that does not conform to the stress field law of the larger model area. To address this issue, a secondary inversion analysis method based on an equivalent structural action load is proposed, building upon the large-scale model inversion. This method involves selecting the crucial fault segments within the larger model and creating a refined small model that encompasses these segments. The scope of the small model should consider various requirements, such as the engineering calculation range and the range of fault disturbances. Next, the stress field obtained from the inversion of the large model is interpolated into the small model based on the centroid of each element. In accordance with the method described in Section 3, the equivalent structural load of each node in the small model is calculated. This load can be viewed as a regular force field representing the regional structure before the fault occurrence. Finally, by applying the equivalent structural loads and self-weight loads to the small model, secondary inversion calculations based on elastic-plastic theory can be performed. This approach enables the determination of the initial stress field following the disturbance caused by the fault. By utilizing this method, the internal stress field of the small model is consistent with the stress field law of the larger model area, ensuring more accurate results in the secondary inversion analysis.

4. Engineering Verification

This article validates the stress distribution conclusions presented by selecting a certain tunnel as the engineering support. The tunnel’s entrance and exit mileage are DK1217 + 793 and DK1255 + 758, respectively, with a total length of 37,965 m. The tunnel is buried at a maximum depth of 1687.85 m and traverses thick layers of Quaternary Holocene and Upper Pleistocene alluvial-proluvial cohesive soil and gravel layers at the entrance, exit, and valley. The tunnel stratum consists mainly of Himalayan granite, diorite, and Caledonian granite, with the exit section featuring the gneiss stratum of the Zhenba Formation of the Nianqing Tanggula Group of the Middle Neoproterozoic. The tectonite mainly consists of fault breccia and crushed rock. The tunnel body passes through approximately 33,895 m of granite and diorite, 3300 m of gneiss, and 770 m of the Quaternary stratum. The cross-section is presented in Figure 10.

The tunnel body intersects with four regional faults, namely F4-3, F4-5, F61, and F4-6. Among them, the F4-3 fault strikes nearly northwest and intersects the tunnel at a large angle at DK1230 + 100~DK1230 + 350 section. The width of the fault fracture zone is about 250 m. The hanging wall and footwall strata are Himalayan granite, and the fault zone material is mainly structural cataclasite; The F4-5 fault strikes nearly south-north, passing through the tunnel at a small angle at DK1244 + 750. The hanging wall and footwall strata are Himalayan granite, and the fault zone material is structural cataclasite; The strike of the F6-1 fault is nearly south-north, passing through the tunnel at a small angle at DK1246 + 275~DK1246 + 575. The hanging wall is Himalayan Diorite, and the footwall is Himalayan granite. The fault zone material is also tectonic cataclasite; The strike of the F4-6 fault is nearly northwest. The hanging wall and footwall strata are Caledonian granite, and the fault zone material is structural cataclasite.

Table 3. Main fault structural characteristics in the Sejila Mountain tunnel site area.

Fault	Position	Occurrence (°)		Width	Quality
		Inclined	Inclination η
F4-3	DK1230 + 085~DK1230 + 355	30	295	295	Inverse faults
F4-5	DK1244 + 695~DK1244 + 795	270	61	61	Inverse faults
F61	DK1246 + 275~DK1246 + 575	270	235	235	Inverse faults
F4-6	DK1254 + 615~DK1254 + 715	45	99	99	Inverse faults

summarizes the distinctive structural characteristics of these faults based on geological exploration. Furthermore, to determine the initial crustal stress distribution in the tunnel area, hydraulic fracturing was employed to measure the maximum horizontal principal stress value and direction of deep boreholes, as presented in

Table 4. Statistics table of deep hole crustal stress measurement data of Sejla Mountain tunnel.

Drill Hole Number	Depth (MPa)	SH (MPa)	Maximum Horizontal Principal Stress Direction
SJLSZ-1	136	18.17~19.46	N 78° E
SJLSZ-2	288.4	10.49~19.68	N 54° E
A5KSZ-21	712.45	6.89~28.23	N 67°~83° E
SJLSZ-3	928	6.99~35.03	N 50° E
SJLSZ-5	630.3	6.81~30.10	N 64°~82° E
SPSZ-2	764.45	25.57~30.09	N 69°~81° E

.

Borehole A5KSZ-21 is located within fault F4-3, borehole SJLSZ-5 is situated in the strata adjacent to fault F61, and borehole SPSZ-2 is located inside fault F4-6, as shown in Table 4. Due to the presence of crushed rock in the fault fracture zone, the measured stress direction exhibits some variation. A plan view in Figure 11 illustrates the location of each borehole and its corresponding measured maximum horizontal principal stress direction.

Based on Table 4 and Figure 11, it is evident that the tunnel site area is subject to tectonic stress, with the main direction of maximum horizontal principal stress being N 50° E. Upon encountering fault F4-3, where the angle between the σ₁ direction and the vertical fault strike direction is α < 45°, the principal stress inside the fault deviates towards the direction parallel to the fault, which is consistent with the principal stress direction N 67°–83° E measured at point A5KSZ-21. When encountering faults F61 and F4-5, as the fault trend is north-south and the angle between the σ₁ direction and the vertical fault direction is α < 45°, the main stress of the strata near the fault will slightly deflect towards the direction perpendicular to the fault direction. This finding is consistent with the main stress direction N 64°–82° E measured at SJLSZ-5 point. When encountering fault F4-6, similar to fault F4-3, the main stress inside the fault deflects in the direction parallel to the fault, which is consistent with the main stress direction N 69°–81° E measured by SPSZ-2. In conclusion, the findings of this paper regarding the disturbance law of faults on the crustal stress distribution are in agreement with the actual measurements of the project, indicating a high degree of reliability.

5. Conclusions

The article focuses on analyzing the impact of fault structure and the degree of fault fragmentation on the distribution of ground stress through numerical simulations. The results of the study reveal the following findings:

(1)

Presence of Fault Fracture Zone: When a geological area is influenced by tectonic stress and contains a fault fracture zone, the magnitude and direction of the main stress are disturbed by the fault. The degree of disturbance depends on factors such as the differences in deformation characteristics between the fault and ordinary strata, the degree of fault fragmentation, and the angle between the fault strike and the main stress direction.

(2)

Deformation Characteristics and Fault Fragmentation: The differences in deformation characteristics between faults and ordinary strata primarily affect the degree of stress change. Larger differences result in greater degrees of change. The degree of fault fragmentation affects the distribution of crustal stress within the fault, causing fluctuations from the original state. The angle between the fault strike and the main stress direction directly influences the direction and magnitude of the deviation of the main stress.

(3)

Angle α Less Than 45°: When the angle α between the maximum horizontal principal stress direction and the vertical fault direction is less than 45°, the internal σ₁ and σ₃ within the fault increase, while the surrounding rock area near the fault slightly decreases. Moreover, the direction of σ₁ inside the fault deflects parallel to the fault, while σ₁ near the outside of the fault experiences slight deflection in the direction of the vertical fault. When α = 0°, the direction of principal stress remains unchanged.

(4)

Angle α Greater Than 45°: When the angle α between the maximum horizontal principal stress direction and the vertical fault direction exceeds 45°, the σ₁ and σ₃ inside the fault decrease, whereas σ₁ and σ₃ in the surrounding rock area near the fault increase. Additionally, the σ₁ direction inside the fault deflects perpendicular to the fault trend, while the σ₁ deflection direction near the outside of the fault is the opposite. When α = 90°, the direction of principal stress remains unchanged.

(5)

Project Considerations: The variation of ground stress caused by faults should be considered in engineering projects. In high-stress hard rock tunnels, stress concentration at fault edges may increase the likelihood of rock bursts. Therefore, measures should be taken to strengthen the affected areas and minimize the risk to ensure the safety of the construction process.