To obtain the wellbore stress, it is necessary to establish the conversion relationship between the geodetic coordinates and the stress coordinates, the wellbore orthogonal polar coordinates, and the layer attitude coordinates. The relationships between the coordinates are illustrated in Figures 1, 2. In these figures, αs represents the angle between the horizontal maximum principal stress and the north direction, while βs represents the angle between the vertical stress and the direction of the axis Z_e. The angle between the wellbore axis and the axis Z_e direction is the wellbore inclination angle β_b. The angle between the projection of the wellbore lowest point on the horizontal plane and the north direction is the wellbore azimuth angle α_b. Additionally, α_bp+π/2 represents the strike of the bedding plane, and β_bp is the angle between the normal of the bedding plane and the plumb direction.

Based on the in situ stress coordinates, the distribution of in situ stress around the well in the Cartesian coordinate system can be obtained through a series of coordinate transformations, which can be expressed as follows (Huang and Zhang, 1998; Tang, 2011; Li et al., 2012; Wen, 2015; Ma, 2015; Huang et al., 2023; Qiu et al., 2023a), σ x b σ y b σ z b τ x z b τ y z b τ z x b = cos 2 ⁡ α b ⁡ cos 2 ⁡ β b sin 2 ⁡ α b ⁡ cos 2 ⁡ β b sin 2 ⁡ β b sin 2 ⁡ α b cos 2 ⁡ α b 0 cos 2 ⁡ α b ⁡ sin 2 ⁡ β b sin 2 ⁡ α b ⁡ sin 2 ⁡ β b cos 2 ⁡ β b − 0.5 ⁡ sin 2 α b cos ⁡ β b 0.5 ⁡ sin 2 α b cos ⁡ β b 0 − 0.5 ⁡ sin 2 α b sin ⁡ β b 0.5 ⁡ sin 2 α b sin ⁡ β b 0 0.5 ⁡ cos 2 ⁡ α b ⁡ sin 2 β b 0.5 ⁡ sin 2 ⁡ α b ⁡ sin 2 β b − 0.5 ⁡ sin 2 β b σ H σ h σ v (1)

After obtaining the distribution tensor of the in situ stress in the wellbore orthogonal coordinate system, the wellbore stress components caused by the pressure of the drilling fluid column inside the well and the stress components caused by the hydration between the drilling fluid and the formation are superimposed. Considering the Biot’s effective stress theory, the equation for the wellbore stress in the well column coordinate system is derived as Eq. 2 (Ma et al., 2016; Liu et al., 2020; Aslannezhad et al., 2020; Qiu et al., 2023; Yang et al., 2023a; Ma et al., 2024a), σ r = σ x b + σ y b 2 1 − r w 2 r 2 + σ x b − σ y b 2 1 − 4 r w 2 r 2 + 3 r w 4 r 4 cos ⁡ 2 θ + τ x y b 1 − 4 r w 2 r 2 + 3 r w 4 r 4 sin ⁡ 2 θ − I m R T V ln α w m α w s h + P w r w 2 r 2 − α P p σ θ = σ x b + σ y b 2 1 + r w 2 r 2 − σ x b − σ y b 2 1 + 3 r w 4 r 4 cos ⁡ 2 θ + τ x y b 1 + 3 r w 4 r 4 sin ⁡ 2 θ − I m R T V ln α w m α w s h − P w r w 2 r 2 − α P p σ z = σ z b − 2 ν σ x b − σ y b r w 2 r 2 cos ⁡ 2 θ − 4 ν τ x y b r w 2 r 2 sin ⁡ 2 θ − α P p τ r θ = σ x b + σ y b 2 sin ⁡ 2 θ + τ x y b ⁡ cos ⁡ 2 θ 1 + 2 r w 2 r 2 − 3 r w 4 r 4 τ r z = τ y z b ⁡ sin ⁡ θ + τ x z b ⁡ cos ⁡ θ 1 − r w 2 r 2 τ θ z = − τ x z b ⁡ sin ⁡ θ + τ y z b ⁡ cos ⁡ θ 1 + r w 2 r 2 (2)

In which, σ_θ, σ_r, σ_z, τ_rθ, τ_rz, and τ_θz represent the components of the wellbore stress in the well column coordinate system, MPa. r_w is the radius of the wellbore, r is the distance from any point around the well to the wellbore axis, m. P_p is the formation pressure, MPa. Α is the Biot effective stress coefficient, dimensionless. θ is the angle around the wellbore, measured counterclockwise from the axis X_b, °; I _m is the membrane efficiency, T is the temperature of the shale formation, V is the partial molar volume of water, R is the gas constant, α_wm is the activity of drilling mud, and α_wsh is the activity of shale.

To facilitate calculations, most strength criteria typically use the form of principal stresses. Therefore, it is necessary to convert the wellbore stress into the form of principal stresses. By substituting the components of wellbore stress, as shown in Eq. 2, into Eq. 3, we can obtain the values of principal stresses around the wellbore, as shown in Eq. 3 (Ding et al., 2022; Ma et al., 2024b), σ 1 , 2 = σ θ + σ z / 2 ± σ θ + σ z 2 + 4 τ θ z 2 / 2 σ 3 = σ r (3)

Based on the experimental research in this paper, it is known that the Jaeger single weak plane criterion, which is expressed as Eq. 4, can better reveal the strength of different shale layers with varying bedding angles (Ma, 2015; Ma and Chen, 2015; Yang et al., 2023b; Yang et al., 2023c; Yu et al., 2022). This study adopts this theory as the criterion for determining the lower limit of the wellbore safety density window. σ 1 − σ 3 = 2 c o + σ 3 ⁡ tan ⁡ φ o 1 + ⁡ tan 2 ⁡ φ o + ⁡ tan ⁡ φ o   β < β 1 o r β > β 2 σ 1 − σ 3 = = 2 c bp + σ 3 ⁡ tan ⁡ φ bp 1 − ⁡ tan ⁡ φ bp ⁡ tan ⁡ β sin ⁡ 2 β   β 1 < β < β 2 (4)

In Eq. 4, c_o and φ_o represent the cohesion and internal friction angle of the shale matrix, measured in MPa and degrees, respectively. c_bp and φ_bp represent the cohesion and internal friction angle of the shale bedding plane, measured in MPa and degrees, respectively. β denotes the angle between the normal of the bedding plane and the applied load, measured in degrees. β₁ and β₂ represent the critical angles at which the rock fails along the bedding plane, measured in degrees.

Shale formations typically exhibit a set of parallel bedding planes, leading to significant anisotropy in their strength. The failure of wellbore confinement pressure will result in shear failure along the matrix and shear sliding failure along the bedding planes. The influence of bedding planes on the wellbore collapse pressure polar plot is depicted in Figure 4. In the figure, the polar plot transitions from blue to purple, indicating a gradual increase in collapse pressure. The circumferential direction represents the wellbore azimuth angle ranging from 0° to 360°, where 0° denotes the direction of minimum horizontal stress and 90° represents the direction of maximum horizontal stress. The radial direction indicates the wellbore inclination angle, where 0° represents vertical wells and 90° represents horizontal wells.

Figure 4A illustrates the polar plot of wellbore collapse pressure without considering bedding planes. The range of collapse pressure varies from 0.85 to 1.1 g/cm³ and symmetrically distributes along the directions of minimum horizontal stress and maximum horizontal stress. Lower collapse pressure is observed when drilling in the direction of minimum horizontal stress, indicating better wellbore stability. Additionally, within the range of 35°–45° wellbore inclination, the collapse pressure reaches its minimum value along the direction of minimum horizontal stress. Taking into account the influence of bedding planes, assuming the development of horizontal bedding in the formation, Figure 4B depicts the polar plot of wellbore collapse pressure. The polar plot still symmetrically distributes along the directions of minimum horizontal stress and maximum horizontal stress, but the collapse pressure significantly increases, ranging from 1 to 2 g/cm³. Wellbores with inclination angles below 30° exhibit the lowest collapse pressure, indicating the optimal drilling trajectory. It is evident that considering the effect of bedding planes leads to significant changes in collapse pressure and the optimal wellbore trajectory. Therefore, the influence of bedding planes should be considered in the stability analysis of shale formations.

After establishing that bedding planes have a significant impact on wellbore collapse pressure in shale formations, this study investigated the influence of bedding plane attitudes on wellbore collapse pressure. When the bedding plane strikes at 0°, the polar plots of wellbore collapse pressure under different bedding plane dip angles are shown in Figure 5. For a bedding plane dip angle of 0°, the range of wellbore collapse pressure under any wellbore trajectory varies from 1 to 2 g/cm³. When the bedding plane dip angle is 30°, the range of wellbore collapse pressure varies from 1 to 2.2 g/cm³. With a bedding plane dip angle of 40°, the range of wellbore collapse pressure varies from 0.8 to 2.4 g/cm³. At a bedding plane dip angle of 90°, the range of wellbore collapse pressure varies from 1 to 2.2 g/cm³. Comparison reveals that when the bedding plane dip angle is 30°, the distribution range and upper limit of wellbore collapse pressure are the largest, indicating that wellbore is more prone to shear sliding failure along the bedding planes at this inclination. Furthermore, as the bedding plane dip angle changes, there are significant variations in the distribution range of optimal wellbore trajectories. When the bedding plane dip angle is 0°, wellbore collapse pressure is lower for wellbore inclination angles below 30°. Conversely, when the bedding plane dip angle is 90°, high-angle wellbores drilled along the direction of minimum horizontal stress exhibit lower wellbore collapse pressure. This indicates that in the design of wellbore trajectories in shale formations, the attitude of bedding planes should be determined. Accordingly, wellbore trajectories should be adjusted based on the orientation of bedding planes to enhance wellbore stability.

When the bedding plane dip angle is fixed at 30°, the polar plots of wellbore collapse pressure under different bedding plane strikes (ranging from 0° to 360°) are shown in Figure 6. Comparison reveals that regardless of the bedding plane strike, the distribution range of wellbore collapse pressure remains unchanged at 1–2.2 g/cm³. When the bedding plane strike is at 0°, wellbores with azimuth angles between 130° and 230° and inclination angles between 0° and 40° exhibit lower wellbore collapse pressure. Similarly, when the bedding plane strike is at 60°, wellbores with azimuth angles between 190° and 290° and inclination angles between 0° and 40° also experience lower wellbore collapse pressure. This suggests that with each 60° increase in bedding plane strike, the distribution area of optimal wellbore trajectories rotates clockwise by 60°. Therefore, while changes in bedding plane strike do not affect the numerical value of collapse pressure, they do influence the distribution area of optimal wellbore trajectories.

One of the main characteristics of shale formations is the delayed instability of the wellbore, which is a phenomenon known as time-dependent instability. Accurately understanding the quantitative relationship between the wellbore’s safe drilling fluid density window and the drilling time in shale formations can assist in the development of wellbore stability techniques and efficient drilling construction measures. This study investigates the influence of drilling time on the polar plot of wellbore collapse pressure in shale formations.

Without considering the influence of bedding planes, the polar plots of wellbore collapse pressure at the initial stage of formation drilling, 2 days after formation drilling, 4 days after formation drilling, and 6 days after formation drilling are shown in Figure 7. At the initial stage of formation drilling, the maximum collapse pressure for any trajectory wellbore is 1.1 g/cm³, while the minimum is 0.85 g/cm³. After 2 days of formation drilling, the distribution range of collapse pressure for any trajectory wellbore ranges from 1.1 to 1.45 g/cm³. After 4 days of formation drilling, the distribution range of collapse pressure expands to 1.2–1.5 g/cm³. Subsequently, after 6 days of formation drilling, the distribution range of collapse pressure further increases to 1.2–1.55 g/cm³. It is evident that with increasing formation drilling time, the weakening of strength due to shale hydration leads to a gradual increase in collapse pressure.

Comparatively, it can be observed that the distribution of wellbore collapse pressure polar plots remains unchanged for different drilling times, symmetrically distributed along the directions of maximum and minimum horizontal stresses. Only the numerical values of collapse pressure change. This indicates that shale hydration does not affect the distribution of optimal wellbore trajectories in the block. However, with prolonged hydration time, the lower limit of drilling fluid density required to maintain wellbore stability gradually increases.

Assuming horizontal bedding in the formation and considering the influence of bedding planes, the variation of wellbore collapse pressure polar plots with formation drilling time in shale formations is illustrated in Figure 8. Analysis reveals that when considering the influence of bedding planes, the change in collapse pressure for low-angle wellbores is relatively small, while for high-angle wellbores, the collapse pressure significantly increases. This indicates that high-angle wellbores experience a notable increase in collapse pressure around the wellbore due to shear sliding failure along the bedding planes. Furthermore, with the extension of formation drilling time, the collapse pressure for low-angle wellbores also significantly increases. This suggests that after shale hydration, the strength of the bedding planes further decreases, making the surrounding rock more susceptible to shear sliding failure along the bedding planes.

Further analysis of the variation patterns of wellbore stability with hydration time reveals the changes in collapse pressure for vertical and horizontal wells, as shown in Figures 9, 10. For vertical wells, collapse pressure is unaffected by bedding planes, meaning they do not experience damage caused by shear sliding along bedding planes. With increasing formation drilling time, the collapse pressure of vertical wells initially sharply increases, then stabilizes after approximately 5 days. However, when considering the influence of bedding planes, the collapse pressure along both the direction of maximum and minimum horizontal stresses significantly increases. Additionally, it increases with hydration time. The difference lies in the fact that when considering the influence of bedding planes, the collapse pressure stabilizes after approximately 5 days of formation drilling, whereas without considering bedding planes, the collapse pressure essentially stops significantly increasing after approximately 2 days of formation drilling. This indicates that the weakening effect of hydration on the strength of bedding planes is more pronounced.

Taking Well X on QY1 Platform in the western part of the deep depression in Meijiayuan Sub-depression of Qintong Depression, Subei Basin as an example, this well serves as an appraisal well. Its drilling objective is to evaluate the adaptability of development technologies for the Upper II shale oil reservoir (sublayers 5-8) in the Fu’er Formation within the QY1 area, assess the production capacity and economically recoverable reserves of the 1500 m horizontal section, and obtain various geological parameters as a basis for reserve certification. Based on neighboring well drilling data, regional geological data, and seismic data, the expected strata encountered while drilling this well from top to bottom are the Dongtai Formation of the Quaternary, Yancheng Formation of the Neogene, Sandou Formation, Dainan Formation, Funing Formation (fourth, third, second sections) of the Paleogene.

To prevent collapse of the wellbore due to shale formation hydration, oil-based drilling fluid was used initially. However, during composite drilling to a depth of 1762.8m, a kick occurred, resulting in a flow rate of 2 m³. Throttling and adjusting the drilling fluid’s properties, the flow rate increased to 2.8 m³ at 17:40, prompting the well to be shut in immediately with a shut-in pressure of 7.9 MPa. After shutting in, personnel conducted circulation and prepared kill mud. During the kill mud injection process, a slight loss of circulation was observed after injecting 20 m³ of kill mud, leading to a decrease in flow rate. The choke was immediately lowered, the choke valve fully opened, and the shut-in pressure dropped to 0 MPa. Rig activity resumed, with a hang-off weight of 135t. Attempts to release the stuck drill string were unsuccessful. Subsequent pump circulation resulted in the removal of thick slurry and clear water, totaling 32m3 of contaminated drilling fluid. Analysis indicated water contamination of the oil-based drilling fluid, causing performance degradation, leading to borehole instability, well leakage, and drill string sticking. Consequently, the drill string had to be side-tracked after explosive unblocking, resulting in significant losses.

To prevent formation water contamination of oil-based drilling fluid, a water-based drilling fluid system was utilized for the sidetracked well. The mud weight was tested at 0.78 activity. At a depth of 1762.8m, the wellbore inclination angle was 12.5°, with a maximum angle of 75° relative to the horizontal maximum stress direction. The formation consists of mud shale with horizontal bedding planes. Using the data in Table 1 and models presented in this paper, different wellbore collapse pressures under various drilling cycles and drilling fluid densities were predicted, as shown in Figure 11.

After designing the drilling fluid density at 1.55 g/cm³, drilling proceeded to 2,290.0 m. Due to elevated torque during lifting, a short downstroke was initiated for safety, requiring the entire section to be reamed. Torque and pump pressure remained stable throughout the reaming process, which efficiently removed a significant number of cuttings and small chunks from the wellbore wall. Overall, the drilling progress during lifting and lowering was normal. After drilling for 5 days into the formation, a blockage was encountered at 1770.0m, leading to frequent pump stalling and top drive stalling events (with a top drive torque setting of 18 kN m, while normal reaming torque ranged from 13 to 15 kN m). Thickening agents with sand were added to the well, resulting in the removal of substantial chunks ranging from 1 to 2cm, with some as large as 3–4 cm. After reaming to the bottom, drilling resumed normally after circulation. Based on the research findings in this paper, the drilling fluid density was increased to 1.86 g/cm³, resulting in smooth drilling operations with significantly reduced chunking and normal cuttings return throughout, without any incidents of sticking.