There is a great deal of low-dip bedding in the shale formation of Jiaoshiba, and the strength of the shale formation is significantly reduced by the weak bedding surface, which shows obvious strength anisotropy. Meanwhile, Longmaxi shale is easy to hydrate [19], and oil-based drilling fluid is typically used for drilling in the field. However, long-term immersion of drilling fluid tends to reduce the shale strength. In order to better understand the influence of oil-base drilling fluid on the strength of shale with different drilling angles, the underground core of the Longmaxi Formation was used to carry out a triaxial mechanics experiment. Furthermore, the strength of shale at different angles, before and after drilling fluid immersion, were tested in order to provide parameters for the study of borehole wall stability in shale formation.

In order to study the rock mechanics parameters of bedding shale formation, only three bedding angles were taken, due to the difficulty in obtaining underground core. According to GB 23561.1-2009 “Determination of Physical Properties of Coal and Rock”, the rock samples were made into specimens with a length and diameter of 50 mm × 25 mm where the included angle between the coring direction and the normal direction of the bedding surface of (a), was 90°, the included angle of (b) was 60°, and the included angle of (c) is 0° (as shown in Figure 1). All samples were dried in an oven at 60°C for 24 h and then set aside.

After drilling the core, the end face should be ground with a grinding machine, and the parallelism error of the two end faces should not exceed 0.5 mm. After the sample preparation, the triaxial mechanics experiment was carried out using a GCTS RTR-1000 (high temperature and high pressure) triaxial mechanics test machine. During the test, when the deformation rate signal feedback by the rock specimen was inconsistent with the predetermined signal, the servo controller would generate a corresponding comparison signal. This signal pushed the servo valve to move and increase or decrease the oil supply quantity of the loading device. Furthermore, the deformation rate of rock specimens was always controlled within an appropriate range. At the moment the rock specimen ruptured, its bearing capacity decreased, and its deformation rate increased. Then, the servo controller would actively close the servo valve, reduce the supply of oil source, and play the role of “letting pressure”, so that the rock specimen could overcome the “burst” phenomenon. Next, the deformation data, after the peak value and before the peak value, can be obtained. Finally, the stress and strain data can be obtained in the process of rock deformation and failure under the formation conditions. In this experiment, each sampling angle was required to carry out experiments under two different confining pressures. The experimental loading confining pressures were 0 and 25 Mpa. The core was taken from the JY1HF well with vertical depth of 2450 m, so the confining pressure was set as 25MPa, and the confining pressure of 0 MPa was selected as the comparison. All cores were divided into two groups, A and B. Group A were dry cores, and Group B were immersed in an on-site oil-base drilling fluid for 30 days to simulate the downhole environment. It was found that in Group A, the compressive strength of the samples is the lowest when the included angle is 60°, and the strength of the samples in the vertical stratification direction is the highest (as shown in Table 1). The rock samples are shown in Figure 2. In addition, the damaged samples are shown in Figure 3. It was found that shear failure occurs at an angle of 0°, shear slip failure along the bedding plane occurs at an angle of 60°, and longitudinal cleavage failure occurs at an angle of 90°. Therefore, the shear slip failure along the weak bedding plane is the key failure mechanism that leads to the low strength of the bedding shale.

In addition, the cores of Group B were soaked with on-site, oil-base drilling fluid for 30 days, and subjected to triaxial mechanical experiments. The experimental results are shown in Table 2.

The results of three-axis experiments of shale, before and after soaking, are compared (as shown in Figure 4 and Figure 5). It is found that the cohesion and internal friction angle of the shale matrix and bedding decreased when soaking time was increased. Figure 4 shows that after drilling fluid immersion, the compressive strength of shale with different sampling angles decreases at 0 and 25 MPa confining pressure. Among them, at the angles of 0° and 90°, the compressive strength decreases greatly, while at the angle of 60°, the compressive strength only slightly decreases. Because the strength of the rock sample is very low at an angle of 60°, the space for decline is very limited. Figure 5 shows that the shale cohesion and internal friction angle of different sampling angles decrease after immersion in drilling fluid, and cohesion decreases the most when the angle is 90°. Moreover, the compressive strength of some rock samples increases after immersion, which may be due to heterogeneity. The influence of oil-based drilling fluid on shale strength may be one of the key factors of wellbore instability. Therefore, the data fitting of shale strength before and after soaking was carried out (as shown in Figure 6). The fitting correlation coefficient is close to 1, and the fitting regression effect is good. The compressive strength of shale in Jiaoshiba area from different angles can be predicted according to the fitting formula.

It is necessary to establish a wellbore stability analysis model for the anisotropic formation of shale after triaxial mechanics experiments, with different included angles, are performed. Then, combined with the rock failure criterion, the influence of drilling fluid weakening on the stability of shale formation wellbore can be estimated.

Assuming the formation is completely heterogeneous, and on the basis of only considering the elastic deformation of the material line, according to the generalized Hooke’s law the stiffness matrix contains 21 elastic constants. Because the Longmaxi Formation belongs to bedding strata, the bedding strata can be equated with transverse isotropic materials. At this point, the flexibility matrix can be simplified, including five independent flexibility coefficients, and the constitutive equation can be expressed as Eq. 1: [ ε x ε y ε z γ y z γ z x γ x y ] = [ S 11 S 12 S 13 S 21 S 22 S 23 S 31 S 32 S 33 S 44 S 55 S 66 ] [ σ x σ y σ z τ y z τ z x τ x y ] (1) Among them, the parameter is specific, as expressed in Eq. 2: { S 11 = S 22 = 1 / E h S 12 = S 21 = − ν h / E h S 13 = S 31 = S 23 = S 32 = − ν ν / E ν S 33 = 1 / E ν S 44 = S 55 = 1 / E h + 1 / E ν + 2 ν ν / E h S 66 = 2 ( 1 + ν h ) / E h (2) Under the bedding plane coordinate system, the rock equilibrium equation and the strain coordination equation are represented in Eq. 3 and Eq. 4, respectively, according to the assumed conditions, when the rock around the well meets the plane strain assumption. { ∂ σ x ∂ x + ∂ τ y x ∂ y = 0 ∂ σ y x ∂ x + ∂ σ y ∂ y = 0 ∂ τ z x ∂ x + ∂ τ z y ∂ y = 0 (3) { ∂ 2 ε x ∂ y 2 + ∂ 2 ε y ∂ x 2 = ∂ 2 γ x y ∂ x ∂ y ∂ γ z x ∂ y − ∂ γ y z ∂ x = 0 ∂ 2 ε z ∂ y 2 = ∂ 2 ε z ∂ x 2 = ∂ 2 ε z ∂ x ∂ y (4) In order to solve the problems listed, the stress function F ( x , y ) and Ψ ( x , y ) is introduced, so that each stress in the balance equation can be expressed as Eq. 5: σ x = ∂ 2 F ∂ y 2 , σ y = ∂ 2 F ∂ x 2 , τ x y = − ∂ 2 F ∂ x ∂ y , τ x z = − ∂ Ψ ∂ y , τ y z = − ∂ Ψ ∂ x (5) Since it is a plane strain problem, it can be obtained from the physical equation. The axial stress σ z can be solved by Eq. 6, as follows: σ z = − 1 S 33 ( S 31 σ x + S 32 σ y + S 34 τ y z + S 35 τ z x + S 36 τ x y ) (6) For the exact solution of stress concentration in anisotropic rock, two stress functions about stress components are defined, and the general expressions of stress function are given, as shown in Eq. 7: { F = 2 Re [ F 1 ( z 1 ) + F 2 ( z 2 ) + F 3 ( z 3 ) ] Ψ = 2 Re [ λ 1 F 1 ′ ( z 1 ) + λ 2 F 2 ′ ( z 2 ) + F 3 ′ ( z 3 ) λ 3 ] (7) Setiawan [23] introduced a new complex variable function on complex variables, which was expressed as Eq. 8: ϕ 1 ( z 1 ) = F 1 ′ ( z 1 ) , ϕ 2 ( z 2 ) = F 2 ′ ( z 2 ) , ϕ 3 ( z 3 ) = 1 λ 3 F 3 ′ ( z 3 ) (8) By using the above formula, Eq. 9 of the stress component in bedding wells is given: { σ x , h = 2 Re [ μ 1 2 ϕ 1 ′ ( z 1 ) + μ 2 2 ϕ 2 ′ ( z 2 ) + μ 3 2 λ 3 ϕ 3 ′ ( z 3 ) ] σ y , h = 2 Re [ ϕ 1 ′ ( z 1 ) + ϕ 2 ′ ( z 2 ) + λ 3 ϕ 3 ′ ( z 3 ) ] τ x y , h = − 2 Re [ μ 1 ϕ 1 ′ ( z 1 ) + μ 2 2 ϕ 2 ′ ( z 2 ) + μ 3 2 λ 3 ϕ 3 ′ ( z 3 ) ] τ x z , h = 2 Re [ λ 1 μ 1 ϕ 1 ′ ( z 1 ) + λ 2 μ 2 ϕ 2 ′ ( z 2 ) + μ 3 λ 3 ϕ 3 ′ ( z 3 ) ] τ y z , h = − 2 Re [ λ 1 ϕ 1 ′ ( z 1 ) + λ 2 ϕ 2 ′ ( z 2 ) + ϕ 3 ′ ( z 3 ) ] σ z , h = − 1 S 33 ( S 31 σ x + S 32 σ y + S 34 τ y z + S 35 τ z x + S 36 τ x y ) (9) Finally, the formula of stress components around borehole is expressed in Eq. 10. { σ x ′ = σ x x + 2 Re [ μ 1 2 ϕ 1 ′ ( z 1 ) + μ 2 2 ϕ 2 ′ ( z 2 ) + μ 3 2 λ 3 ϕ 3 ′ ( z 3 ) ] σ y ′ = σ y y + 2 Re [ ϕ 1 ′ ( z 1 ) + ϕ 2 ′ ( z 2 ) + λ 3 ϕ 3 ′ ( z 3 ) ] τ x y ′ = τ x y − 2 Re [ μ 1 ϕ 1 ′ ( z 1 ) + μ 2 2 ϕ 2 ′ ( z 2 ) + μ 3 2 λ 3 ϕ 3 ′ ( z 3 ) ] τ x z ′ = τ x z + 2 Re [ λ 1 μ 1 ϕ 1 ′ ( z 1 ) + λ 2 μ 2 ϕ 2 ′ ( z 2 ) + μ 3 λ 3 ϕ 3 ′ ( z 3 ) ] τ y z ′ = τ y z − 2 Re [ λ 1 ϕ 1 ′ ( z 1 ) + λ 2 ϕ 2 ′ ( z 2 ) + ϕ 3 ′ ( z 3 ) ] σ z ′ = σ x x − 1 S 33 ( S 31 σ x + S 32 σ y + S 34 τ y z + S 35 τ z x + S 36 τ x y ) (10) The borehole stability model, Eq. 11, under polar coordinates, can be obtained by plugging into the borehole polar coordinate formula: { σ r = σ x ′ + σ y ′ 2 ( 1 − r i 2 r 2 ) + σ x ′ − σ y ′ 2 ( 1 − 4 r i 2 r 2 + 3 r i 4 r 4 ) cos ⁡ 2 ⁡ θ + τ x y ′ ( 1 + 3 r i 4 r 4 − 4 r i 2 r 2 ) sin ⁡ 2 ⁡ θ + r i 2 r 2 p w − α p p σ θ = σ x ′ + σ y ′ 2 ( 1 + r i 2 r 2 ) − σ x ′ − σ y ′ 2 ( 1 + 3 r i 4 r 4 ) cos ⁡ 2 ⁡ θ − τ x y ′ ( 1 + 3 r i 4 r 4 − 4 r i 2 r 2 ) sin ⁡ 2 ⁡ θ − r i 2 r 2 p w − α p p σ z = σ z ′ + μ [ σ x ′ + σ y ′ − 2 ( σ x ′ − σ y ′ ) r i 2 r 2 cos ⁡ 2 ⁡ θ + 4 τ x y ′ ⁡ sin ⁡ 2 ⁡ θ ] − α p p τ r θ = τ x y ′ ( 1 + 3 r i 4 r 4 − 4 r i 2 r 2 ) cos ⁡ 2 ⁡ θ τ θ z = τ y z ′ ( 1 + r i 2 r 2 ) cos ⁡ θ − τ x z ′ ( 1 + r i 2 r 2 ) sin ⁡ θ τ r z = τ x z ′ ( 1 − r i 2 r 2 ) cos ⁡ θ + τ y z ′ ( 1 − r i 2 r 2 ) sin ⁡ θ (11) where, σ r , σ θ , σ z , τ r θ , τ θ z and τ r z is the stress component under the borehole column coordinates; r i is the borehole radius; r is the radial distance between surrounding rocks and borehole axis; μ is the Poisson’s ratio of rock; and θ is the well circumference angle. According to the formula listed above, the stress distribution of the surrounding rock, under the coordinates of any inclined shaft, can be obtained when the original ground stress and the liquid column pressure in the wellbore are acted upon.

It is necessary to select appropriate strength criterion for accurately evaluating the stratigraphic stability. According to the test results, when the included angle of the rock sample is 0°, the rock is considered a single shear failure. Rock failure shear force on the shear surface must overcome rock cohesion and friction, as shown in Eq. 12: τ = c + σ ⁡ tan φ (12) M-C criterion can be expressed as Eq. 13 in the form of principal stress, as shown in Figure 7. σ 1 = σ 3 ⁡ cot 2 ( 45 0 − φ 2 ) + 2 c ⁡ cot ( 45 0 − φ 2 ) (13)

In this model, the X axis is consistent with the north geographical pole, and the Y axis is aligned with the east geographical pole. Weak plane occurrence is defined by inclination direction (α) and inclination angle (β). In order for the weak plane to be studied in the wellbore stability analysis model, the inclination angle is assumed to be 30° and 60°, and the study is carried out under different inclinations. The experimental samples, in this paper, are the underground core of Longmaxi Formation, with a sampling depth of 2,450 m. According to the field hydraulic fracturing and logging data, the geomechanics parameters of the target interval were obtained (as shown in Table 3). Then, according to laboratory rock mechanics experiments, rock mechanics parameters, before and after drilling fluid immersion, are obtained (as shown in Table 4). All these parameters are input parameters for the lower critical mud weight analysis of the Longmaxi shale formation in Fuling.

In shale formations, horizontal wells usually pass through bedding strata with different well angles, and bedding strata also have different tendencies, which affects the formation’s lower critical mud weight, and the choice of the optimal drilling direction. In addition, it is found that bedding inclination angles are mostly low-dip angles in field drilling construction. Therefore, the formation dip angle of 30° and 60° were selected for the study. Based on the experimental data, the influence of drilling fluid on wellbore stability of shale formation is studied. The purpose is to get closer to identifying the best working conditions, and also choose the optimal drilling construction scheme.

When the bedding inclination angle is 30°, it is found that the collapse pressure is generally lower along the direction of bedding tendency, or its relative direction (as shown in Figure 9). However, in the direction where the angle with bedding tendency is 90°, the collapse pressure is generally heightened to the maximum value, which is not suitable for a drilling well trajectory direction. Moreover, with the increase of the azimuth of stratigraphic tendency, the lower critical mud weight of bedding tendency decreases gradually, with a decrease of 0.3 g/cm³. In the direction of bedding inclination, the closer the angle of the well inclination is to the bedding inclination, the lower the collapse pressure. The collapse pressure is minimal when the axial direction of borehole is perpendicular to the bedding. Combined with the laboratory triaxial rock mechanics experiment, it is found that when the loading direction is perpendicular to the bedding, the maximum rock mechanical parameters are taken, so the borehole is more stable. By comparing and analyzing the collapse pressure, before and after drilling fluid immersion, the overall lower critical mud weight increased to a certain extent. The minimum collapse pressure rose to 0.2 g/cm³, while the maximum collapse pressure rose from 1.75 g/cm³ to 1.95 g/cm³. However, the choice of borehole for the azimuth angle is greater than it was before immersion. Drilling is not necessary along the bedding tendency, but within a certain angle to the bedding tendency. The optimal drilling azimuth is changed to an angle of about 30° with the bedding tendency.

When the bedding inclination angle is 60°, the change of collapse pressure is shown in Figure 10. It is found that with the increase of the bedding inclination angle, the figure shows that different bedding plane inclination angles have a significant effect on the collapse pressure density of the shaft wall. When the inclination is 0°, the included angle with the azimuth angle is 30°, which is the optimal drilling direction. The overall lower critical mud weight is around 1.45 g/cm³. When the angle of inclination is 30° and the well bevel is below 40°, the collapse pressure is higher than 1.70 g/cm³. When the inclination angle is 60°, the optional orientation of the wellbore trajectory is smaller, and the collapse pressure is lower, but only when the well is drilled with a high inclination angle in the bedding inclination azimuth. Also, the minimum collapse pressure is 1.1 g/cm³. When the inclination angle is 90°, the lower critical mud weight is smaller, in the range of about 30° of bedding tendency, and the collapse pressure range is 1.1–1.4 g/cm³. The collapse pressure, before and after soaking, is compared and analyzed. Similar to the inclination angle with bedding of 30°, the overall rise amplitude of the collapse pressure is 0.2 g/cm³. However, the choice of borehole orientation is larger, and the collapse pressure is higher, but only in a small range. When the trend is 0°, the larger value is 1.9 g/cm³ when the azimuth angle is 60°. When the trend is 30°, the collapse pressure is higher than 1.9 g/cm³, within the azimuth range of 0°–60° and 210°–240°. When the trend is 60°, the value of collapse pressure is higher, in the azimuth range of 90°–180° and 270°–360°. When the trend is 90°, the optimal drilling direction is 60°–120° and 240°–300°, and the collapse pressure is lower than 1.6 g/cm³.

By comparing and analyzing the bedding inclination angles of 30° and 60°, with the increase of the bedding inclination angle, the optimal well inclination angle also increases. When the borehole axis is perpendicular to the bedding surface, the minimum value of lower critical mud weight is always set. In addition, when the bedding inclination angle is 30°, the optional borehole bearing angle is also more concentrated than when the bedding inclination angle is 60°. When the bedding inclination angle is 60°, the collapse pressure changes are more complex, and the bedding occurrence has a significant influence on the selection of the optimal drilling direction. Finally, the oil-base drilling fluid has a significant effect on shale strength, making the overall minimum drilling fluid density increase by about 0.2 g/cm³, and the maximum collapse pressure rise from 1.75 g/cm³ to 1.95 g/cm³. In other words, the mechanism of shale mechanical failure, bedding occurrence and weakening effect of drilling fluid, are all very important in the wellbore stability analysis of shale formation.