To determine the stress distribution characteristics around the wellbore, it is first necessary to transform the in situ stresses into the global coordinate system. As illustrated in Figure 1, X_G-Y_G-Z_G represents the global coordinate system, where X_G points north, Y_G points east, and Z_G is vertical. X_S-Y_S-Z_S represents the in situ stress coordinate system, corresponding to the maximum horizontal stress, minimum horizontal stress, and vertical stress, respectively. To transform the in situ stresses into the global coordinate system, three coordinate transformations are required. First Step, rotate the global coordinate system around the Z_G axis by an angle α₁, resulting in the X′-Y′-Z′ coordinate system. Second Step, rotate the X′-Y′-Z′ coordinate system around the Y′ axis by an angle β₁, yielding the X″-Y″-Z″ coordinate system. Third Step, rotate the X″-Y″-Z″ coordinate system around the X″ axis by an angle β₁. This series of transformations converts the X_G-Y_G-Z_G coordinate system into the X_s-Y_s-Z_s coordinate system. This transformation process enables the accurate determination of the stress distribution around the wellbore in relation to the global geographic orientation.

Based on geometric knowledge, a transformation matrix R ₁ is established, as shown in Equation 1, R 1 = cos ⁡ α 1 ⁡ cos ⁡ β 1 sin ⁡ α 1 ⁡ cos ⁡ β 1 − sin ⁡ β 1 cos ⁡ α 1 ⁡ sin ⁡ β 1 ⁡ sin ⁡ γ 1 − sin ⁡ α 1 ⁡ cos ⁡ γ 1 sin ⁡ α 1 ⁡ sin ⁡ β 1 ⁡ sin ⁡ γ 1 + cos ⁡ α 1 ⁡ cos ⁡ γ 1 cos ⁡ β 1 ⁡ sin ⁡ γ 1 cos ⁡ α 1 ⁡ sin ⁡ β 1 ⁡ cos ⁡ γ 1 − sin ⁡ α 1 ⁡ sin ⁡ γ 1 sin ⁡ α 1 ⁡ sin ⁡ β 1 ⁡ cos ⁡ γ 1 + cos ⁡ α 1 ⁡ sin ⁡ γ 1 cos ⁡ β 1 ⁡ cos ⁡ γ 1 (1)

Using the stress tensor σ _G to represent the components of stress in the global coordinate system and σ _S to represent the components of stress in the stress coordinate system, the transformation relationship between them can be expressed as shown in Equation 2, σ G = R 1 T σ S R 1 (2) Where σ S = σ H , 0 , 0 ; 0 , σ h , 0 ; 0 , 0 , σ v , σ _H represents the maximum horizontal stress, measured in MPa; σ _h represents the minimum horizontal stress, measured in MPa; and σ _v represents the vertical stress, measured in MPa.

After transforming the stress into the global coordinate system, it should be further transformed into the wellbore coordinate system for analysis. The transformation relationship between the global coordinate system (GCS) and the wellbore coordinate system (BCS) is shown in Figure 2, where X_B-Y_B-Z_B represents the wellbore coordinate system, α₂ represents the angle between the wellbore inclination and the true north direction, usually referred to as the wellbore azimuth angle; β₂ represents the angle between the wellbore axis and the vertical direction, typically referred to as the well deviation angle. Using σ_B to represent the stress tensor in the wellbore coordinate system, according to the study by Lee et al. (2012), the transformation relationship between σ_B and σ _G is expressed as shown in Equation 3, σ B = R 2 σ G R 2 T (3)

The transformation matrix R ₂ is represented as shown in Equation 4. R 2 = cos ⁡ α 2 ⁡ cos ⁡ β 2 sin ⁡ α 2 ⁡ cos ⁡ β 2 sin ⁡ β 2 sin ⁡ α 2 cos ⁡ α 2 0 cos ⁡ α 2 ⁡ sin ⁡ β 2 sin ⁡ α 2 ⁡ sin ⁡ β 2 cos ⁡ β 2 (4)

The representation of stress tensors around the wellbore in polar coordinates is more convenient. Therefore, after transforming the stress from the global coordinate system to the wellbore coordinate system, the stress tensor σ _B, i.e., [σ_x, σ_y, σ_z, τ_xy, τ_xz, τ_yz], needs to be further transformed into polar coordinates. Due to the disturbance of the original stress state by drilling through the formation, the stress redistributes around the wellbore and creates stress concentration effects. According to Biot’s effective stress theory, the equation for the effective stress around the wellbore at any inclination angle is expressed as shown in Equation 5, σ r = σ x + σ y 2 1 − r w 2 r 2 + σ x − σ y 2 1 − 4 r w 2 r 2 + 3 r w 4 r 4 cos ⁡ 2 θ + τ x y 1 − 4 r w 2 r 2 + 3 r w 4 r 4 sin ⁡ 2 θ + P w r w 2 r 2 − α P p σ θ = σ x + σ y 2 1 + r w 2 r 2 − σ x − σ y 2 1 + 3 r w 4 r 4 cos ⁡ 2 θ + τ x y 1 + 3 r w 4 r 4 sin ⁡ 2 θ − P w r w 2 r 2 − α P p σ z = σ z z − 2 ν σ x − σ y r w 2 r 2 cos ⁡ 2 θ − 4 ν τ x y r w 2 r 2 sin ⁡ 2 θ − α P p τ r θ = σ x + σ y 2 sin ⁡ 2 θ + τ x y ⁡ cos ⁡ 2 θ 1 + 2 r w 2 r 2 − 3 r w 4 r 4 τ r z = τ y z ⁡ sin ⁡ θ + τ x z ⁡ cos ⁡ θ 1 − r w 2 r 2 τ θ z = − τ x z ⁡ sin ⁡ θ + τ y z ⁡ cos ⁡ θ 1 + r w 2 r 2 (5)

In the equation, r_w represents the wellbore radius, measured in meters; r represents the distance from any point around the wellbore to the wellbore axis, measured in meters; P_w represents the bottomhole fluid column pressure, measured in MPa; Pp represents the pore pressure of the formation, measured in MPa; θ represents the azimuth angle around the wellbore, i.e., the angle, measured in degrees, from a point around the wellbore rotating clockwise to the X_b coordinate axis; α represents the Biot’s effective stress coefficient, which ranges from 0 to 1 and is dimensionless.

Once the stress distribution around the wellbore is determined, these values must be incorporated into rock strength criteria to evaluate the formation’s stability. Because these criteria are typically expressed in terms of principal stresses, the wellbore stresses need to be transformed accordingly. Equation 6 shows how to convert stresses from polar coordinates into principal stresses, σ i = σ r σ j = σ θ + σ z / 2 + σ θ + σ z 2 + 4 τ θ z 2 / 2 σ k = σ θ + σ z / 2 − σ θ + σ z 2 + 4 τ θ z 2 / 2 (6)

The relative magnitudes of the three principal stresses at any point around the wellbore vary with changes in the bottomhole fluid column pressure. To accurately distinguish between the three principal stresses around the wellbore, substitute the expression σ i , σ j , σ k obtained from Equation 6 into Equation 7, σ 1 = max σ i , σ j , σ k σ 3 = min σ i , σ j , σ k σ 2 = σ i + σ j + σ k − σ 1 − σ 3 (7)

To differentiate between types of tensile failure in the wellbore wall, it is necessary to quantitatively analyze the relationship between the three principal stresses around the wellbore. Additionally, it is important to compare the minimum principal stress around the wellbore with the tensile strength of the rock, as shown in Equation 8, σ 3 ≤ − σ t (8) When σ_r equals σ₃ and Equation 8 holds true, drilling-induced fractures occur around the wellbore. If the hydrostatic pressure exceeds the formation pressure, it may lead to drilling fluid loss. The relationship between the angle of induced fractures and the well axis stress is expressed as shown in Equation 9, tan ⁡ 2 ω = 2 τ θ z σ z − σ θ (9)

In which, ω represents the angle between the drilling-induced fractures and the wellbore axis, measured in degrees, as shown in Figure 3. This angle represents the orientation between the maximum principal stress around the wellbore and the wellbore axis. By incorporating in situ stress, wellbore trajectory, and fluid column pressure, the principal stresses at different points around the wellbore can be determined. This enables the identification of tensile failure types and the calculation of the angle at each point relative to the wellbore axis, ultimately predicting the morphology of drilling-induced fractures under these conditions.

It is worth noting that this study assumes the formation rock to be a linear elastic medium and does not consider the effects of temperature, fluid interactions, chemical influences, or formation exposure time on the stress distribution around the wellbore. Additionally, given the characteristics of rock being strong in compression but weak in tension, as well as the natural development of fractures, it is generally assumed that cracks will form when the rock experiences tensile stress.

Based on the MATLAB 2022a platform, a wellbore-induced fracture prediction model was developed and implemented. To verify the accuracy of the program, the data from Table 1 was used as input for the model. The program was then employed to calculate the circumferential, radial, and vertical stresses around a vertical wellbore. The results were compared with those obtained using Equation 5 to ensure the correctness of the developed program.

The circumferential, radial, and vertical stresses on the wellbore wall of a vertical well, as calculated using the program developed in this study, are shown in Figure 4. Similarly, for a vertical well in the direction of the minimum horizontal in situ stress, the calculation process and results for wellbore circumferential stress, based on Equation 5, are presented in Equation 10, σ r = σ H + σ h 2 × 0 + σ H − σ h 2 × 0 × cos ⁡ 2 θ + 0 × 0 × sin ⁡ 2 θ + P w × 1 − P p = 5 MPa σ θ = σ H + σ h 2 × 2 − σ H − σ h 2 × 4 ⁡ cos ⁡ 2 θ + 0 × 4 ⁡ sin ⁡ 2 θ − P w × 1 − P p = 35 MPa σ z = σ v − 2 ν σ H − σ h × 1 × cos ⁡ 2 θ − 4 ν × 0 × 1 × sin ⁡ 2 θ − α P p = 25 MPa (10)

Comparing the results obtained from Equation 5 with those shown in Figure 4 reveals a perfect match, confirming the accuracy of the developed model and calculation program, including the coordinate transformation equations and principal stress calculations.

This study investigates the locations and morphologies of drilling-induced fractures in a normal fault environment. Assuming a fluid column pressure of 50 MPa in the wellbore, the remaining parameters from Table 1 are used to analyze the effects of borehole trajectory and pressure on the characteristics of drilling-induced fractures. Utilizing the drilling-induced fracture prediction model established in this study and inputting the parameters from Table 1, the variation trends of the principal stresses on the wellbore wall and the angles between induced fractures and the wellbore axis along the circumferential angle of the wellbore are depicted in Figure 5–20. The dual vertical axes in the diagrams respectively quantify two critical parameters: the left axis measures principal stress magnitudes, while the right axis (purple crosses) tracks angular deviation between drilling-induced fractures and the wellbore axis. Rock mechanics analysis reveals these fractures initiate when tangential stress at the borehole wall exceeds the tensile strength threshold. Specifically, fracture propagation commences as the minimum principal stress transitions from compressive to tensile domain, a fundamental principle derived from tensile failure criteria in wellbore stability studies.

From Figure 5a, it can be observed that the first and second principal stresses around the wellbore do not vary with the circumferential angle. The third principal stress reaches its maximum value in the direction of minimum horizontal in situ and its minimum value in the direction of maximum horizontal in situ. The angle ω between induced fractures and the wellbore axis remains 0 throughout. This indicates that a vertical fracture is generated in the direction of maximum horizontal in situ, as depicted in Figure 5b.

Maintaining the drilling direction constant, but varying the well deviation angle, the wellbore’s principal stresses, the angle between induced fractures and the wellbore axis, and the induced fracture orientations for well deviation angles of 30°, 60°, and 90° are illustrated in Figures 6–8, respectively. The mechanical response exhibits distinct patterns across different well orientations. At βb = 30° deviation, circumferential stress analysis shows dual stress minima at 90°/270° azimuths. Stress state transition initiates fracture development with maintained near-orthogonal fracture-wellbore alignment (θ ≈ 90°), producing characteristic feather-textured fracture patterns as documented in Figures 6a,b. Notably, increasing deviation to β_b = 60° induces comprehensive tensile transformation (σ_min<0 throughout azimuths), driving θ fluctuations exceeding 60°. This mechanically governed instability generates multi-axial fracture intersections shown in Figures 7a,b, demonstrating progressive complexity with inclination angle. The horizontal well configuration (β_b = 90°) presents polar stress minima at 0°/180° positions. Theta maintains perfect coaxial alignment (θ≡0°) during wellbore rotation, forming vertically oriented fracture planes that bisect the borehole circumference (Figures 8a,b). This orientation-dependent fracture geometry evolution follows fundamental principles of anisotropic stress redistribution in inclined boreholes.

Keeping the wellbore azimuth angle at 30° and varying the well deviation angle to 0°, 30°, 60°, and 90°, the principal stresses around the wellbore, induced fractures, and their angles with the wellbore axis are depicted in Figures 9–12. When the well deviation angle β_b = 0°, the minimum principal stress around the wellbore reaches its minimum values at 60° and 240° around the wellbore. The angle between induced fractures and the wellbore axis remains 0°, resulting in two vertical fractures at positions 60° and 240° around the wellbore. For a well deviation angle of β_b = 30°, the minimum principal stress around the wellbore transitions to tensile stress in the range of 0°–108° and 181°–289° around the wellbore. Mechanical analysis reveals pronounced angular instability in fracture trajectory deviation (Δθ>45°), yielding distinctive hooked fracture geometries. This bifurcation pattern, encompassing both J-type curvature and mechanical fishhook configurations, arises from critical stress anisotropy conditions documented in Figures 10a,b. The observed morphological bifurcation demonstrates fundamental fracture path selection mechanisms governed by near-wellbore stress tensor reconfiguration. With a well deviation angle of β_b = 60°, the minimum principal stress around the wellbore transitions to tensile stress in the ranges of 0°–82°, 163°–262°, and 323°–360° around the wellbore. Similar to the previous case, the angle between induced fractures and the wellbore axis undergoes significant variations, resulting in induced fracture orientations resembling “J-shaped” fractures. The induced fractures exhibit inflection points at 60° and 240°, forming more pronounced “J-shaped” fractures, as depicted in Figures 11a,b. In horizontal well configurations (β_b = 90°), fracture trajectory angularity demonstrates constrained variability (Δθ<10°), manifesting hybrid fracture morphologies that bridge classical feather-textured patterns and mechanical “fishhook” configurations. This transitional morphology emerges from the unique stress coupling effects inherent to horizontal borehole geometries.

Maintaining the wellbore azimuth angle at 60°, the variations in principal stresses around the wellbore and induced fracture orientations with respect to the circumferential angle for well deviation angles of 0°, 30°, 60°, and 90° are illustrated in Figures 13–16. The mechanical response exhibits systematic dependence on wellbore orientation relative to in situ stress fields. In vertical wells (β_b = 0°), coaxial alignment with the principal in situ stress orientation induces fracture propagation vectors parallel to the borehole axis through geomechanical coupling. At intermediate inclinations (β_b = 30°,60°), stress anisotropy intensification during tensile state transition triggers angular instability (Δθ>30°), forming characteristic J-curvature fractures through stress-dependent path selection mechanisms, as quantitatively verified in Figure 10b. Conversely, horizontal wells (β_b = 90°) demonstrate constrained trajectory deviation (Δθ<10°) under compressive-tensile stress tensor reconfiguration. This transitional regime produces hybrid fracture architectures blending feather-textured discontinuities with incipient hook-shaped geometries, reflecting progressive stress decoupling in high-angle completions.

The wellbore azimuth angle is 90°, and the well deviation angles are 0°, 30°, 60°, and 90°. The principal stresses around the wellbore and the induced fracture orientations for these angles are depicted in Figures 17–20. In stress-aligned well configurations, the borehole axis maintains strict parallelism with principal in situ stress orientations: vertical stress correspondence at βb = 0° (Figures 17a,b) and maximum horizontal stress alignment at β_b = 90° (Figures 20a,b). This geometric optimization induces dual fracture initiation with perfect coaxial propagation through stress-controlled failure mechanisms. The resultant fracture planes exhibit diametric symmetry with exact 180° phasing, forming textbook hydraulic fracture patterns through geomechanical resonance. When the well deviation angles are 30° (Figures 18a,b) and 60° (Figures 19a,b) , the minimum principal stress around the wellbore reaches its minimum value at the circumferential angles of 0° and 180°. When the minimum principal stress transitions to tensile stress, the angle between induced fractures and the wellbore axis is close to 0° at the 0° (Figures 17a,b) and 180° (Figures 20a,b) positions, exhibiting minimal variation. As the circumferential angle increases, the variation becomes more significant, resulting in induced fracture orientations resembling sine curves, as predicted in Figures 18b, 19b. This makes them easily mistaken for natural fractures.

When analyzing the induced fractures around the wellbore formed by different trajectories, it is evident that if the wellbore axis is parallel to any principal in situ stress direction, such that no shear stress is induced around the wellbore, the angle between the induced fractures and the wellbore axis will remain at 0°. As a result, two symmetrically distributed fractures parallel to the wellbore axis are typically produced. When the variation in the angle between the induced fractures and the wellbore axis is within 10°, feather-like fractures appear. Once this angular variation exceeds 10°, “J-shaped” fractures are formed. Furthermore, when the angle between the induced fractures and the wellbore axis approaches 0° at the minimum principal stress extremum around the wellbore and then changes sharply either increasing or decreasing, the fractures tend to follow an approximately sine-curve pattern. Moreover, the induced fractures usually occur in pairs symmetrically arranged at 180° intervals.

Bottomhole fluid column pressure is one of the most direct, effective, and controllable measures to maintain wellbore stability. It directly determines complex scenarios such as erosion or shear collapse of the wellbore wall and loss of drilling fluid. The authors studied the influence of bottomhole fluid column pressure on the development of drilling-induced fractures. The induced fracture orientations of a wellbore with azimuth angle α_b = 0° and well deviation angle β_b = 30° under different bottomhole pressures are illustrated in Figure 21. Analysis indicates that feather-like fractures are consistently formed under various bottomhole pressures. As the bottomhole fluid column pressure rises, the angle between the drilling-induced fractures and the wellbore axis progressively diminishes, while the length of these induced fractures gradually increases. The induced fractures are symmetrically distributed within the wellbore, with a spacing of 180° between them.

The variations in drilling-induced fracture orientations of a wellbore with azimuth angle α_b = 30° and well deviation angle β_b = 30° under different bottomhole pressures are depicted in Figure 22. Analysis indicates that feather-like fractures are consistently formed under various bottomhole pressures. As the bottomhole fluid column pressure rises, the angle between the drilling-induced fractures and the wellbore axis progressively diminishes, while the length of these induced fractures gradually increases. The induced fractures are symmetrically distributed within the wellbore, with a spacing of 180° between them.