As shown in Figure 1, the actual wellpath of the horizontal section drilled by a slide steering drilling system is always in a wavy or tortuous shape (Samuel et al., 2005; Noshi and Schubert, 2019). Because the inclination and azimuth commonly fluctuate slowly under compound drilling, the slide drilling must be alternated to decrease (or increase) the inclination and azimuth and control the wellpath within the target area once the drill bit is near the boundary lines of the horizontal target area. For a long horizontal section, two drilling modes may alternate multiple times. Then, the actual horizontal section has more control cycles. In one wavy cycle or control cycle, two drilling modes alternate once.

Suppose that the compound drilling tends to increase inclination and azimuth simultaneously, and then take the rectangular horizontal target area and the first wellpath control cycle as an example. The target half-width is set as a, and the target half-height is b; for the compound drilling interval, the starting point is No. 0 point, the lowest point on the vertical projection plot (the far left point on the horizontal plot) is No. 1 point, and the endpoint is No. 2 point; and for the slide drilling interval, the starting point is No. 2 point, the highest point (the far right point on the horizontal plot) is No. 3 point, and the last point is No. 4 point. To reduce the alternate times of slide and compound drilling, let the inclinations of the highest and lowest points on the vertical projection plot be equal to the target inclination and the azimuths of the far left and right points on the horizontal plot be equal to the target azimuth. All node parameters of the actual horizontal section are presupposed and shown in Figure 1. For the compound drilling interval, the key parameters include the compound drilling length Δ L r ( Δ L r = Δ L r 1 + Δ L r 2 ), the inclination change rate K α , and azimuth change rate K ϕ ; for the slide drilling interval, the key parameters include the slide drilling length Δ L s ( Δ L s = Δ L s 1 + Δ L s 2 ) and build-up rate K s ; and for these two intervals, the key parameters include the allowable inclination bias Δ α T and azimuth bias Δ ϕ T .

According to the wellpath calculation methods (Guan et al., 2021), the actual shapes of compound and slide drilling intervals are closest to the cylindrical spiral line and the inclined arc line, respectively. The cylindrical spiral method (curvature radius method) and inclined arc method (minimum curvature method) are applied alternatively to calculate all wellpath parameters of the actual horizontal section. According to the presupposed node parameters in Figure 1, the first wellpath control cycle from No. 0 point to No. 4 point can be divided into four intervals, and then all vertical depth increments Δ D i (i = 1–4), N-coordinate increments Δ N i (i = 1–4), E-coordinate increments Δ E i (i = 1–4), and dog-leg angles γ i can be calculated using Eqs 1–4, respectively: Δ D i = 1 K α sin ⁡ α i − sin ⁡ α i − 1   i = 1,2 , 1 K s tan γ i 2 cos ⁡ α i − 1 + cos ⁡ α i   i = 3,4 , (1) Δ N i = 1 K ϕ ⋅ Δ α T cos ⁡ α i − 1 − cos ⁡ α i sin ⁡ ϕ i − sin ⁡ ϕ i − 1   i = 1,2 , 1 K s tan γ i 2 sin ⁡ α i − 1 ⁡ cos ⁡ ϕ i − 1 + sin ⁡ α i ⁡ cos ⁡ ϕ i   i = 3,4 , (2) Δ E i = 1 K ϕ ⋅ Δ α T cos ⁡ α i − 1 − cos ⁡ α i cos ⁡ ϕ i − cos ⁡ ϕ i − 1   i = 1,2 , 1 K s tan γ i 2 sin ⁡ α i − 1 ⁡ sin ⁡ ϕ i + sin ⁡ α i ⁡ sin ⁡ ϕ i   i = 3,4 , (3) cos ⁡ γ i = cos ⁡ α i − 1 ⁡ cos ⁡ α i + sin ⁡ α i − 1 ⁡ sin ⁡ α i ⁡ cos ϕ i − ϕ i − 1   i = 1 ∼ 4 . (4)

As shown in Figure 1, while drilling the horizontal section, the actual horizontal wellpath must be controlled within the target area, and then the actual wellpath of the horizontal section must meet the inequality constraint as follows: Δ D 2 + Δ D 3 sin ⁡ α T ≤ 2 b , Δ N 2 + Δ N 3 sin ⁡ ϕ T + Δ E 2 + Δ E 3 cos ⁡ ϕ T ≤ 2 a , (5.a) or Δ D 1 + Δ D 4 sin ⁡ α T ≤ 2 b , Δ N 1 + Δ N 4 sin ⁡ ϕ T + Δ E 1 + Δ E 4 cos ⁡ ϕ T ≤ 2 a . (5.b)

If the designed target inclination α T ≠ 90 ° and/or target azimuth ϕ T ≠ 90 ° , the next derivation process will be more complex. However, you can imagine that the horizontal section can be rotated in the horizontal direction on the vertical projection plot and in the east direction on the horizontal plot. After the rotation transformation, the shape of the actual horizontal section has not changed. Next, the node parameters in Figure 1 are substituted into Eqs 1–4 and then into Eq. 6. Finally, Eq. 6 can be deduced as follows: 1 − cos ⁡ Δ α T K α + tan γ 2 sin ⁡ Δ α T K s ≤ 2 b , sin ⁡ Δ α T 1 − cos ⁡ Δ ϕ T K ϕ ⋅ Δ α T + tan γ 2 cos ⁡ Δ α T ⁡ sin ⁡ Δ ϕ T K s ≤ 2 a , cos ⁡ γ = cos ⁡ Δ α T ⁡ cos ⁡ Δ ϕ T . (6)

Considering the allowed inclination bias Δ α T and azimuth bias Δ ϕ T , which are characterized by the parameters of the compound drilling interval (inclination change rate K α , azimuth change rate K ϕ , and length Δ L r ) and slide drilling interval (build-up rate K s and length Δ L s ), Eq. 6 can be transformed into Eq. 7. 1 − sin K α Δ L r K α + tan K s Δ L s 2 sin K α Δ L r K s ≤ 2 b , sin K α . Δ L r 1 − cos K ϕ Δ L r K ϕ . K α . Δ L r + tan K s Δ L s 2 cos K α Δ L r sin K ϕ Δ L r K s ≤ 2 a , cos K s Δ L s = cos K α Δ L r cos K ϕ Δ L r . (7)

From Eqs 6, 7, it can be observed that

(1) Once the target half-width a and target half-height b are given, the maximum allowed inclination bias Δ α T and azimuth bias Δ ϕ T of the horizontal section, the maximum allowed compound drilling length Δ L r , and slide drilling length Δ L s in one wellpath control cycle depend on the build-up rate K s of slide drilling and the inclination change rate K α and azimuth change rate K ϕ of compound drilling.

In one wellpath control cycle of the horizontal section, the allowed compound drilling length is Δ L r , the slide drilling length is Δ L s , and the compound drilling proportion c r is defined and calculated using Eq. 8. c r = Δ L r Δ L r + Δ L s . (8)

If the build-up rate K s of slide drilling and the inclination change rate K α , and azimuth change rate K ϕ of compound drilling are given, the maximum allowed compound drilling length Δ L r and slide drilling length Δ L s in one wellpath control cycle can be solved using Eq. 7. In fact, according to the last formula in Eq. 7, another relationship among these five characteristic parameters can be derived as follows: K s ⋅ Δ L s > K α ⋅ Δ L r , K s ⋅ Δ L s > K ϕ ⋅ Δ L r . (9)

According to Eq. 9, the upper limit of the compound drilling proportion c r can be estimated as follows: c r = Δ L r Δ L r + Δ L s < min K s K α + K s , K s K ϕ + K s . (10)

Because the inclination change rate K α and azimuth change rate K ϕ of compound drilling may be positive (the inclination and azimuth increase) or negative (the inclination and azimuth decrease), these two values in Eqs 9, 10 must use their absolute values.

According to Eq. 10, it can be observed that the compound drilling proportion of the horizontal section depends on the build-up rate K s of slide drilling and the inclination change rate K α and azimuth change rate K ϕ of compound drilling and is independent of the target parameters and wavy amplitude of the horizontal section. In order to enhance the compound drilling proportion for faster and more cost-effective drilling, the build-up rate K s of slide drilling should be enhanced properly, and then the inclination change rate K α and azimuth change rate K ϕ of compound drilling should be reduced to the greatest extent.

According to the actual horizontal drilling data, the target area is set at 1.0 m (target half-height) × 6.0 m (target half-width), and then the key parameters of slide and compound drilling are set as follows: the build-up rates K s = 6.0°–9.0°/30 m (6.0°/30 m for 1.0° bent PDM, 7.5°/30 m for 1.25° bent PDM, and 9.0°/30 m for 1.5° bent PDM), the inclination change rate K α = 0.1°–1.5°/30 m, and azimuth change rate K ϕ = 0.3°/30 m. The drilling results are estimated using Eqs 7, 8 and shown in Figures 2A, B. In the legend zone, “SD” and “CD” represent “slide drilling” and “compound drilling,” respectively, and the numbers represent the build-up rate (unit: °/30 m) of slide drilling.

If the inclination change rate of compound drilling increases, the compound drilling length Δ L r decreases significantly, but the slide drilling length Δ L s increases slightly, and then the compound drilling proportion decreases obviously. For a given inclination change rate under compound drilling, if the build-up rate of slide drilling increases, the compound drilling length Δ L r increases slightly, but the slide drilling length Δ L s decreases obviously, and then the compound drilling proportion increases obviously. It is proved that increasing the build-up rate K s of slide drilling and reducing the inclination change rate K α and azimuth change rate K ϕ of compound drilling contribute to increasing the compound drilling length and proportion of the horizontal section.

Taking 1.25° bent PDM as an example, the build-up rate of slide drilling is set at 7.5°/30 m, the inclination change rate of compound drilling is set at 0.5°–1.5°/30 m, and the azimuth change rate is set at 0.3°/30 m, and then the drilling results are estimated using Eqs 7, 8 and shown in Figures 2C, D. In the legend zone, “SD” and “CD” represent “slide drilling” and “compound drilling,” respectively, but the numbers represent the inclination change rates (unit: °/30 m) of compound drilling.

If the build-up rate of slide drilling increases, the compound drilling length Δ L r increases slightly, but the slide drilling length Δ L s decreases obviously, and then the compound drilling proportion increases obviously. For a given build-up rate under slide drilling, if the inclination change rate of compound drilling increases, the compound drilling length Δ L r decreases significantly, but the slide drilling length Δ L s increases slightly, and then the compound drilling proportion decreases obviously. It is proved again that increasing the build-up rate K s of slide drilling and reducing the inclination change rate K α and azimuth change rate K ϕ of compound drilling contribute to increasing the compound drilling length and proportion of the horizontal section.