There are three kinds of commonly used vertical drilling BHAs, namely pendulum BHA, bent-housing motor BHA and vertical drilling system (VDS). In a pendulum BHA, a stabilizer is placed 15–22 m above the bit, such as Φ215.9 mm bit + Φ158.8 mm drill collar*2 + Φ215 mm stabilizer + Φ158.8 mm drill collar*19. Bent-housing motor BHA used with two stabilizers can also reduce well deviation, such as: Φ215.9 mm bit + Φ172 mm bent-housing motor (0.75° bent angle) + Φ158.8 mm short drill collar*1 + Φ214 mm stabilizer + Φ158.75 mm drill collar*17. VDS is a special version of rotary drilling system (RSS), and is often considered as the most powerful vertical drilling tool, but its cost is far greater than that of pendulum and bent-housing motor BHA.

The mechanical model of pendulum BHA can be found in many textbooks, VDS is assumed to be always successful in correcting deviation, and thus only the mechanical model of the bent-housing motor is derived in this section. The beam-column theory is used to establish the mechanical analysis model of the bent-housing motor BHA. Figure 1 is a simplified mechanical model of the bent-housing BHA, which consists of a lower stabilizer, a bent-housing motor, a short collar and an upper stabilizer.

In the mechanical model, the bent-housing motor is cut off at the bend, and there are therefore five beam sections with linear buoyant weight q ₁∼q ₅, length L ₁∼L ₅ and moment of area I ₁∼I ₅. In Figure 1, there are five bending moments M ₁, M ₂, M ₃, M ₄ and M _T. The equilibrium equations can be obtained by setting two rotation angles equal to each other at the joint.

The three-moment equation between the first and second beam can be written as: 2 M 1 [ Y ( u 1 ) + I 1 L 2 L 1 I 2 Y ( u 1 ) ] + L 2 I 1 L 1 I 2 Z ( u 2 ) M 2 − 6 E I 1 L 1 L 2 Y 2 = − 1 4 q 1 L 1 2 X ( u 1 ) − q 2 I 1 L 2 3 4 I 2 L 1 X ( u 2 ) − 6 E I 1 Y 1 L 1 ( 1 L 1 + 1 L 2 ) , (1) Where Y ₁ and Y ₂ are the deflections at the lower stabilizer and bend point, respectively; E is the elasticity modulus; X(u), Y(u) and Z(u) are three rotation amplification factors.

The rotation amplification factors can be calculated by the following equations, u = L 2 P E I . (2) X ( u ) = 3 u 3 ( tan ⁡ u − u ) . (3) Y ( u ) = 3 2 u ( 1 2 u − 1 tan ( 2 u ) ) . (4) Z ( u ) = 3 u ( 1 sin ( 2 u ) − 1 2 u ) . (5)

The bent-housing motor is cut off at the bend point, which results in two new unknown variables, namely the internal bending moment M ₂ and the deflection Y ₂. According to the continuity of rotation angle and shear force at the bend, two three-moment equations can be obtained. M 1 Z ( u 2 ) + 2 M 2 [ Y ( u 2 ) + I 2 L 3 I 3 L 2 Y ( u 3 ) ] + 6 E I 2 Y 2 L 2 ( 1 L 2 + 1 L 3 ) + L 3 I 2 L 2 I 3 Z ( u 3 ) M 3 − 6 E I 2 L 2 L 3 Y 3 = − 1 4 q 2 L 2 2 X ( u 2 ) − q 3 I 2 L 3 3 4 I 3 L 2 X ( u 3 ) + 6 E I 2 L 2 ( Y 1 L 2 − γ ) . (6) M 1 L 2 − ( 1 L 3 + 1 L 2 ) M 2 + ( P 3 L 3 + P 2 L 2 ) Y 2 + M 3 L 3 − P 3 Y 3 L 3 = − q 3 L 3 + q 2 L 2 2 + P 2 Y 1 L 3 , (7) Where P ₂ and P ₃ are the axial forces in the second and third beams, respectively; γ is the bend angle; Y ₃ is the deflection at the upper end of the motor.

Similarly, two equations can be obtained at the joint of motor and short collar: M 2 Z ( u 3 ) − 6 E I 3 Y 2 L 3 2 + 2 M 3 [ Y ( u 3 ) + I 3 L 4 I 4 L 3 Y ( u 4 ) ] + 6 E I 3 Y 3 L 3 ( 1 L 3 + 1 L 4 ) + L 4 I 3 L 3 I 4 Z ( u 4 ) M 4 = − 1 4 q 3 L 3 2 X ( u 3 ) − q 4 I 3 L 4 3 4 I 4 L 3 X ( u 4 ) + 6 E I 3 Y 4 L 3 L 4 . (8) M 2 L 3 − P 3 Y 2 L 2 − ( 1 L 3 + 1 L 4 ) M 3 + ( P 3 L 3 + P 4 L 4 ) Y 3 + M 4 L 4 = − q 3 L 3 + q 4 L 4 2 + P 4 Y 4 L 4 , (9) Where P ₄ is the axial force at the short collar; Y ₄ is the deflection at the upper stabilizer.

A three-moment equation can be obtained at the upper stabilizer, M 3 Z ( u 4 ) − 6 E I 4 Y 3 L 4 2 + 2 M 4 [ Y ( u 4 ) + I 4 L 5 I 5 L 4 Y ( u 5 ) ] = − 1 4 q 4 L 4 2 X ( u 4 ) − q 5 I 4 L 5 3 4 I 5 L 4 X ( u 5 ) − L 5 I 4 L 4 I 5 Z ( u 5 ) M T + 6 E I 4 L 4 ( Y 5 − Y 4 L 5 − Y 4 L 4 ) , (10) Where Y ₅ is the deflection at the upper tangent point, and can be calculated according to collar diameter and wellbore diameter; M _T is the moment at the upper tangent point, and can be calculated according to known wellbore curvature.

In addition, at the upper tangent point, the three bending moment equation can be obtained: 1 4 q 5 L 5 4 X ( u 5 ) + L 5 2 [ 2 M 5 Y ( u 5 ) + M 4 Z ( u 5 ) ] = 6 E I 5 ( ∑ i = 1 5 L i K L 5 − Y 5 + Y 4 ) . (11)

After solving Eqs 1–11, getting M ₁ and Y ₁, the bit side force N _b and bit tilt angle A _α can be calculated: N b = − ( P b Y 1 L 1 + q 1 L 1 2 + M 1 L 1 ) . (12) A α = q 1 L 1 3 24 E I 1 X u ( 1 ) + M 1 L 1 6 E I 1 Z u ( 1 ) − Y 1 L 1 . (13)

It should be pointed out that the new model does not consider drill string buckling, because there are two stabilizers and the drill string between them is too short to buckle.

Bit side force or bit tilt angle alone calculated by the BHA mechanics model cannot determine the drilling direction of the bit, and formation properties must be also taken into account. In this section, the drilling trend angle method (Shi et al., 2017) is adopted to predict build rate, but some equations in Shi’s paper are erroneous, and the correct solution of drilling trend angle is presented.

Figure 2A shows the drilling trend of bit in formation. A bottomhole coordinate system O₁-X₁Y₁Z₁ is established at the bit, The actual drilling direction of the bit is along the X₁ axis, the direction of the wellbore high side is the direction of the Z₁ axis, and the direction of the Y₁ axis is determined by the right-hand rule. The vector e _r represents the drilling trend direction of the bit, and the angle between the vector e _r and the X₁ axis is defined as the drilling trend angle A _r.

Assuming a wellbore curvature K ₀, and calculating A _r, if A _r comes to zero, the drilling trend is just balanced, and K ₀ can be equivalent to BHA build rate. Otherwise, the drilling trend is not balanced, and the hole curvature must be re-assumed and the drilling trend angle A _r must be calculated until it comes to zero. To sum up, the basic idea of drilling trend method predicting the BHA build rate is that the borehole curvature when the drilling trend angle is close to zero is equivalent to BHA build rate.

In order to consider the effects of bit anisotropy, formation anisotropy, formation dip and other factors on drilling trend angle, the rock-bit interaction model by Ho (1987) is adopted. This model assumes that the formation is isotropic in transverse direction, and the drilling direction can be expressed as: r N e r = I b I r e f + I r ( 1 − I b ) cos ⁡ A a f e a + ( 1 − I r ) r N ⁡ cos ⁡ A r d e d , (14) Where I _b and I _r are the bit and formation anisotropy factors; A _af is the angle between the resultant bit force direction and the axial direction of the bit; A _rd is the angle between the drilling direction and the normal direction of the formation bed; r _N is the drilling efficiency under general condition; e _r, e _f , e _a and e _d are the unit vectors of drilling trend direction, bit force direction, bit axis direction, and formation normal direction, respectively.

To solve A _r, it is necessary to establish the transformation relationship between bit parameters and formation parameters. As shown in Figure 2B , a surface coordinate system O-XYZ is established, in which the X-axis points to the north direction, the Y-axis points to the east direction, the Z-axis points to the gravity direction, and a formation coordinate system O₂-X₂Y₂Z₂ is also established, in which the X₂ axis points to the downward dip direction of the formation, the downward normal direction of the formation is the direction of the Z₂ axis, and the Y₂ axis is determined by the right-handed rule.

The transformation matrix from surface coordinate system to bottomhole coordinate system is as follows: [ X 1 Y 1 Z 1 ] = [ sin ⁡ α ⁡ cos ⁡ φ sin ⁡ α ⁡ sin ⁡ φ cos ⁡ α sin ⁡ φ − cos ⁡ φ 0 cos ⁡ α ⁡ cos ⁡ φ cos ⁡ α ⁡ sin ⁡ φ − sin ⁡ α ] [ X Y Z ] = [ K 1 ] [ X Y Z ] , (15) Where, α and φ represent inclination and azimuth at the bit, respectively.

The transformation matrix from surface coordinate system to formation coordinate system is as follows: [ X 2 Y 2 Z 2 ] = [ sin ⁡ β ⁡ cos ⁡ θ sin ⁡ β ⁡ sin ⁡ θ cos ⁡ β − sin ⁡ θ cos ⁡ θ 0 − cos ⁡ β ⁡ cos ⁡ θ − cos ⁡ β ⁡ sin ⁡ θ sin ⁡ β ] [ X Y Z ] = [ K 2 ] [ X Y Z ] , (16) Where β and θ represent dip angle and dip orientation, respectively (dip orientation is defined as the azimuth of downdip direction of formation bed).Setting, [ K ] = [ K 1 ]   * [ K 2 ] − 1 , (17) getting, ( X 1   Y 1   Z 1 ) T = [ K ] ( X 2   Y 2   Z 2 ) T . (18)

Defining ( e 1 ′ ,   e 2 ′ ,   e 3 ′ ) as the basis vectors of the bottomhole coordinate system, α _f, β _f and γ _f are the angles between the bit resultant force vector e _f and the coordinate axes X₁, Y₁, Z₁, respectively. α _a , β _a and γ _a are the angles between the bit axial direction vector e _a and the coordinate axes X₁, Y₁, Z₁, respectively.

In the bottomhole coordinate system, e _f and e _a calculated according to the bit side force and bit tilt angle can be expressed as: e f = cos ⁡ α f e 1 ′ + cos ⁡ β f e 2 ′ + cos ⁡ γ f e 3 ′ = P b P b 2 + N b α 2 + N b φ 2 e 1 ′ − N b φ P b 2 + N b α 2 + N b φ 2 e 2 ′ + N b α P b 2 + N b α 2 + N b φ 2 e 3 ′ . (19) e a = cos ⁡ α a e 1 ′ + cos ⁡ β a e 2 ′ + cos ⁡ γ a e 3 ′ = cos ⁡ A α ⁡ cos ⁡ A φ e 1 ′ + sin ⁡ A φ e 2 ′ + sin ⁡ A α ⁡ cos ⁡ A φ e 3 ′ (20)

Defining ( e 1 ″ ,   e 2 ″ ,   e 3 ″ ) as the basis vector of the formation coordinate system,α _r , β _r and γ _r represent the angle between the drilling direction vector e _r and the coordinate axes X₂, Y₂ and Z₂, respectively, which are the parameters needed to be solved. e _r can be expressed by the basis of the formation coordinate system as follows. e r = [ cos ⁡ α r , cos ⁡ β r , cos ⁡ γ r ] [ e 1 ″ e 2 ″ e 3 ″ ] = cos ⁡ α r e 1 ″ + cos ⁡ β e 2 ″ + cos ⁡ γ r e 3 ″ . (21)

The unit vector e _d of the formation normal direction can be expressed as: e d = e 3 ″ = [ K ] [ 0 0 1 ] = [ K 13 , K 23 , K 33 ] [ e 1 ′ e 2 ′ e 3 ′ ] . (22)

The unit vector systems in Eqs 19–22 were converted to the bottomhole coordinate system and substituted into rock-bit interaction model. The following results could be obtained: K 11 ⁡ cos ⁡ α r + K 12 ⁡ cos ⁡ β r + I r K 13 ⁡ cos ⁡ γ r = S 1 r N . (23) K 21 ⁡ cos ⁡ α r + K 22 ⁡ cos ⁡ β r + I r K 23 ⁡ cos ⁡ γ r = S 2 r N . (24) K 31 ⁡ cos ⁡ α r + K 32 ⁡ cos ⁡ β r + I r K 33 ⁡ cos ⁡ γ r = S 3 r N , (25) Where S ₁ = I _b I _r cosα _f +I _r (1–I _b ) cosA _af cosα _a ; S₂ = I _b I _r cosβ _f + I _r (1–I _b ) cosA _af cosβ _a ; S ₃ = I _b I _r cosγ _f + I _r (1–I _b) cosA _af cosγ _a .

The drilling trend angle A _r can be obtained by substituting the bit side force and bit tilt angle into Eqs 23–25, and converting the calculated results to the bottomhole coordinate system, which can be expressed as follows: A r = arccos ( [ [ K 1 ] * [ K 2 ] − 1 ] − 1 I b I r ⁡ cos α f + I r ( 1 − I b ) cos ⁡ A a f ⁡ cos α a r N ) . (26)

The factors that affect the anti-deviation ability of BHA can be grouped into two types, namely formation factors and adjustable factors. The formation factors can hardly be changed, including formation dip, formation anisotropy and hole enlargement rate. Adjustable factors include WOB and BHA structure parameters, which can be controlled. The optimal design of BHA is the process of changing adjustable factors to satisfy the requirement of formation factors and to drill a vertical well.

The optimal design of BHA includes two interactive processes, that is, the selection of BHA type and the design of specific structural parameters. On the one hand, optimized BHA should have good anti-deviation ability, and on the other hand the selected BHA should keep drilling cost as low as possible. Therefore, this paper applies a drilling cost prediction method to calculate the expected cost of different types of vertical drilling BHA in a bit life cycle, and selects the type and structure of the BHA based on the principle of minimizing the expected cost.

The cost of a vertical drilling BHA over the bit life can be expressed by the following formula (Si et al., 2009), c = [ P j ( c m 1 + c m 2 ) + ( 1 − P j ) c m 3 ] t , (27) Where c _m1 is the daily drilling cost, include labor cost, depreciation cost, management cost, material consumption cost and maintenance cost; t is the drilling time; c _m2 is the cost of the BHA; c _m3 is the extra cost of drilling at a small WOB if the designed vertical drilling BHA fail; P _j is the probability of effective anti-deviation of the BHA.

Taking the following pendulum BHA as an example, the BHA structure is: Ф215.9 mm bit + Ф172 mm straight-housing motor + Ф210 mm stabilizer + Ф158.8 mm collar * 24. The pendulum BHA was run in vertical well A at well depth of 3,480 m, and WOB was controlled within 40–60 kN. However, the well inclination starts to increase from 3,660 m, and reaches 4° at a depth of 4,000 m. Seismic data shows the formation dip in the third spud hole is about 5.5°.

In order to analyze the cause of well deviation, rotary drilling build rates of the pendulum BHA under different drilling parameters and formation parameters are calculated. According to the geological design of the well and the drilling engineering data and formation data of offset wells, the bit anisotropy factor I _b is set to 0.2, the formation anisotropy factor I _r is set to 0.98, the formation dip is set to 6°, the initial inclination is set to 3°, and the WOB is set to 50 kN. The WOB, formation dip and formation anisotropy factor are varied to calculate the BHA build rate.

Figure 3 shows that with the increase of WOB, the pendulum BHA build rate increases and keeps positive at all WOB, which means that the pendulum BHA does not have the ability to prevent well deviation. Figure 4 shows that the pendulum BHA build rate linearly increase with the increase of formation dip. If formation anisotropy factor is equal to 1, the formation has no anisotropy. According to this definition and Figure 5, the pendulum BHA build rate almost linearly increases with the increase of formation anisotropy.

Adjusting the pendulum BHA structure and calculating the build rate, the results are shown in Figures 6, 7. From Figure 6, with the increase of the distance from stabilizer to bit, the build rate first decreases and then gradually tends to be stable, and the anti-deviation ability reaches the maximum when the distance is about 19 m. According to Figure 7, with the increase of stabilizer outer diameter, the build rate decreases, but the values are still positive, which means the pendulum BHA still increase inclination.

The change in the build rate after the wellbore expansion is calculated, and the results are shown in Figure 8. The build rate increases rapidly after the diameter expansion occurs, and when the well diameter expansion rate reaches 100%, the build rate increases by five times. According to caliper logging, the diameter expansion rate can reach 160% in some section. Therefore, diameter expansion rate must be considered during the vertical drilling BHA design.

Based on the calculated results, the reason why the pendulum BHA failed is that the distance from the stabilizer to the bit is too short. In summary, the main factors affecting the anti-deviation ability of pendulum BHA are formation dip, formation anisotropy and well diameter expansion. Besides, the distance from stabilizer to bit, stabilizer diameter and WOB also play important roles.

The structure of the bent-housing motor BHA is as follows: Φ215.9 mm bit + Φ172 mm 0.75° bent-housing motor (with stabilizer) + Φ158.8 mm drill collar*1 + Φ214 mm stabilizer + Φ158.8 mm drill collar*17. This bent-housing motor BHA is used in vertical well B, with drilling fluid density 1.3 g/cm³ and WOB 60 kN. The well inclination increases to 2° at 2000 m and reaches 4° at 2,550 m. The average formation dip of this drilling section is about 3.5°.

In the calculations of rotary drilling build rates, according to data offset wells, the bit anisotropy factor I _b equals 0.2, the formation anisotropy factor I _r equals 0.95, the inclination angle of the well is 3°, and WOB is 60 kN.

Figure 9 shows that the build rate of linearly increases with increasing WOB, but the growth rate is very low, and the build rate only changes 0.1°/30 m within 40–100 kN WOB, which indicates that the effect of WOB is not significant for bent-housing motor BHA.

Figure 10 shows that as the formation dip increases the build rate of the bent-housing motor increases and the growth rate also increases, indicating that the formation dip has a greater influence. Besides, it can be seen that this bent-housing motor BHA cannot correct well deviation in the formation with a dip angle of 3.5°.

The diameter control of this well is good, but a certain degree of well diameter expansion occurs in some sections, and the well diameter expansion rate reaches 30% at a depth of about 2,150 m. The effect of the well diameter expansion on the build rate is calculated and the results are shown in Figure 11, which shows that when the well diameter expansion reaches 30%, the build rate increases 6 times compared to 5% expansion. Comparing the inclination data with the well diameter data, it can be found that the actual build rate of is larger in the diameter expanded section.

Changing the distance of the upper stabilizer to bit, calculating the build rate, the results are shown in Figure 12. Figure 12 illustrates that as the distance from the bit to the upper stabilizer increases, the anti-deviation capability of the BHA decreases. This bent-housing motor BHA has a long drill collar connected between the motor and the upper stabilizer, which is not conducive to preventing deviation.

In summary, the main factors affecting the anti-deviation ability of the bent-housing motor BHA are formation dip, formation anisotropy, hole expansion and distance of upper stabilizer to bit, while the influence of WOB is relatively small.

The uncertainties of vertical drilling BHA mainly come from the variation of formation properties, which are difficult to obtain before drilling. Therefore, in this paper, an uncertainty function f (I _r , β, γ) is attached to the formation parameters when evaluating the BHA performance. It is assumed that the three key parameters, formation anisotropy factor I _r , formation dip β and well diameter expansion rate γ, take random values from a specified range. The three parameters are limited to a spatial range, and the parameters are orthogonalized within the range. The frequency of effective anti-deviation of the BHA in the orthogonalized space is equal to the probability of effective anti-deviation of the BHA.

A synthetic case study is presented to illustrate the design method of anti-deviation BHA proposed in this paper. For the case well of Ф215.9 mm hole, the range of formation parameters is selected as follows: formation dip 5–15°, formation anisotropy factor 0.95–0.98, and well diameter expansion rate 0–30%.

Based on the results in Section 3.1 and filed experience, the best pendulum BHA candidate is designed as: Ф215.9 mm bit + Ф172 mm straight-housing motor +Φ158.8 mm drill collar + Ф215 mm stabilizer + Ф158.8 mm collar * 20. Recommended WOB is 60 kN. As shown in Figure 13, the effective anti-deviation probability of this pendulum BHA in 704 orthogonal parameters is 33.4%.

Similarly, based on the results in Section 3.2 and field experience, the best candidate for bent-housing motor BHA is designed as: Φ215.9 mm bit+Φ172 mm bent-housing motor (1.25° bend angle, with Φ215 mm stabilizer) + Φ158.8 mm short drill collar (2 m) + Φ213 mm stabilizer + Φ158 mm drill collar*20. As shown in Figure 14, the effective anti-deviation probability is 12.9% in the orthogonal formation parameters of the 704 group.

The average ROP is set as 4.83 m/h for both straight-motor pendulum BHA and bent-housing motor BHA, and the ROP is assumed as 1.5 m/h when a small WOB is applied in case of the failure of vertical drilling BHA. The motor cost is 4,000 RMB per day and the rig daily cost is 60,000 RMB per day. The expected cost of the two sets of BHAs is calculated. Assuming that the VDS has 100% anti-deviation success rate, the expected cost of the VDS is calculated according to the ROP of 5 m/h, daily cost of 45,000 RMB per day and footage cost of 430 RMB per meter. The expected drilling cost of three vertical drilling BHA at different drilling footage are shown in Table 1.

From Table 1, it can be seen that the straight-housing motor pendulum BHA, with low cost and high anti-deviation probability, is the best choice in the range of 900 m footage. The anti-deviation probability of the bent-housing motor BHA is low and the corresponding expected drilling cost is high due to the expected existence of hole diameter expansion. With the increase of drilling footage, VDS has become the best choice when the footage reaches 1,200 m.

If there is no expected well diameter expansion, the probability calculation is redone. The new calculation results show that the success rate of the pendulum BHA and the bent-housing motor BHA are 36.4% and 100%, respectively. Therefore, it is advisable to use the bent-housing motor BHA in the current situation. At the same time, the importance of hole diameter expansion to the performance of vertical drilling BHA is highlighted.