# Analysis and optimization of control algorithms for RSSTSP for horizontal well drilling

• Li Gang Zhang
• G. R. Liu
• Wei Li
• Shi Bin Li
Open Access
Original Paper - Exploration Engineering

## Abstract

Steering control algorithm plays an important role in a rotary steerable system for horizontal well drilling, including the determination of the well trajectory, vibrations, stability, durability among other variables. This work develops a control algorithm for three static push-the-bit rotary steerable systems (RSSTSP) (TSP is the abbreviation of “three static push-the-bit”). Based on the structure, mechanism, and working process of the RSSTSP, mechanical and mathematical models are proposed to determine the required steering force (amplitude and direction) to move the drill bit from a point to another. Additional equations are constructed to overcome the non-uniqueness and then compute the optimal forces for the three pads to achieve the required steering force. Moreover, a new control algorithm of RSSTSP is developed, considering the steerability, stability, durability, favorable area, unfavorable area, maximum usable magnitude of steering force. The proposed control algorithm is also applied to a new RSSTSP and tested on a GU-693-P102 well for validation. It is found that each pad force changes smoothly, the drilling tool is stable, and the well trajectory is consistent with the design, demonstrating that our proposed control algorithm is robust and effective for RSSTSP for horizontal well drilling.

## Keywords

Horizontal well drilling Rotary steerable system (RSSTSPControl algorithm Steering force Pad force Dogleg

## Structure and work process of RSSTSP

RSSTSP has three support pads: pad 1, 2, and 3, which are spaced 120° apart from each other, as shown in Fig. 1a, with coordinated movements for rotating spindle, drill string, and bit rotating. The pads are relative static to the rotating outside sleeve. The pads are driven by drilling fluid and stretched out to press against the well bore, thereby causing the bit to press on the opposite side causing a direction change. The support force for each pad is expressed as F1, F2, F3, respectively, as shown in Fig. 1b. The resultant force F of three pad forces forms the total steering force, leading to desired changes in inclination and azimuth, as shown in Fig. 1c.

The RSSTSP works as follows: The value and direction of steering force are first determined based on the current actual point and the expected point on the ground. The steering force is then transmitted downhole. Based on steering force, each pad force is next determined in accordance with predetermined control algorithms downhole. Lastly, the pads are pushed out by applying hydraulic pressure, and the expected steering force and well trajectory are realized. In the whole process, a robust algorithm for steering force and pad force is the key factor for achieving the desired control effects.

## Maximum usable steering forces (Amax)

For a given hydraulic pressure applied, the static bias pad has a maximum or minimum support force (Fmax or Fmin). Because the three pads are 120° apart, the range of the steering force is in a hexagon, as shown in Fig. 2a, b. The hexagon shall have an inner circle and an outer circle, as shown in Fig. 2c.
For a given position of the three pads, if the maximum usable magnitude of steering force (Amax) is within the inner circle and outer circle touching the hexagon, it can only be realized at a point on the hexagon, as shown in Fig. 3. If the steering force is within the inner circle, it can be achieved everywhere. In the actual operation, in order to achieve a steering force in all directions at any time, the steering force must be adjustable in 360°. Therefore, the steering force should be limited in the range within the inner circle. Thus, the maximum usable magnitude of steering force is not Fmax, but Amax shown in Fig. 3.
The maximum usable magnitude of steering force can be expressed as follows:
$$A_{\hbox{max} } = \frac{\sqrt 3 }{2} \times (F_{\hbox{max} } - F_{\hbox{min} } ),$$
(1)
where Fmax is the maximum support force of the pad, Fmin is the minimum support force of the pad, and Amax is the maximum usable magnitude of steering force. It is necessary to consider the constraint of Amax in the control algorithm, so as to achieve a desired dogleg, build rate, or walk rate in the actual operations.

## Amplitude and direction of the required steering force

Consider a RSSTSP used in a drilling operation. The amplitude and its direction of the required steering force should be determined based on the current point and orientation of the drill bit and the target point and orientation one would want the drill bit to be.

### Amplitude of the required steering force (A k )

Assume that the current drill bit is at point A (XA, YA, ZA), and the actual hole inclination angle and azimuth angle are αA and βA. The targeted position is at point C (XC, YC, Z C ), and the expected inclination angle and azimuth angle are αC and βC. The tangent lines at points A and C intersect at point D, and the normal lines of points A and C intersect at point O, as shown in Fig. 4.
Let the changes in inclination angle and azimuth angle be ΔαAC and ΔβAC, and the average inclination angle of points A and C be α0. The γ of points A and C can be calculated as follows:
$$\gamma = \frac{{30 \times 360\left| {\sin \frac{{\sqrt {\Delta \alpha_{\text{AC}}^{2} + \Delta \beta_{\text{AC}}^{2} \cdot \sin \alpha_{0}^{2} } }}{2}} \right|^{{}} }}{{\pi \sqrt {(X_{\text{C}} - X_{\text{A}} )^{2} + (Y_{\text{c}} - Y_{\text{A}} )^{2} + (Z_{\text{C}} - Z_{\text{A}} )^{2} } }},$$
(2)
where $$\Delta \alpha_{\text{AC}} = \alpha_{\text{C}} - \alpha_{\text{A}}$$, $$\Delta \beta_{\text{AC}} = \beta_{\text{C}} - \beta_{\text{A}}$$, $$\alpha_{0} = \frac{{\alpha_{\text{A}} + \alpha_{\text{C}} }}{2}_{{}}$$, γ is the dogleg, αA is the inclination angle of point “A,” αc is the inclination angle of point “C,” and αo is the average inclination angle.
The γ max of RSSTSP is usually known and supplied by manufactures under Amax. The work efficiency of RSSTSP is defined as follows:
$$A_{k} = \frac{\gamma }{{\gamma_{\hbox{max} } }} \times 100\% ,$$
(3)
where γmax is the dogleg when working under Amax.

If the value of A k is greater than 100%, it would not be drilled to the target point. In this situation, it is necessary to redesign well trajectory and redetermine target point until the value of A k is less than 100%.

It can also be expressed using the amplitudes of the steering forces in relation to its maximum usable magnitude, as follows:
$$A_{k} = \frac{F}{{A_{\hbox{max} } }} \times 100\% ,$$
(4)
where F is the steering force.

Therefore, once A k is obtained, the amplitude of the steering force F can be calculated using Eq. (4).

### Direction of steering force (α k )

The direction angle of steering force (α k ) is defined as clockwise rotation angle from high side to the direction of steering force in the bottomhole plane. The steering force could be broken up into build force and walk force, as shown Fig. 5.

The “+” indicates increase in “build force” or “walk force,” and the “−” indicates decrease in “build force” or “walk force.” Build rate and walk rate are also determined by steering force and the direction angle of steering force α k .

With the current point (A) and expected point (C) shown in Fig. 4, the expected build rate (∆α) and the walk rate (∆β) can be obtained using
$$\Delta \alpha = 30 \times \frac{{\alpha_{\text{c}} - \alpha_{\text{A}} }}{{\Delta D_{m} }},$$
(5)
$$\Delta \beta = 30 \times \frac{{\beta_{c} - \beta_{A} }}{{\Delta D_{m} }},$$
(6)
where Δα is the expected build rate, βA is the azimuthal angle of point “A,” βc is the azimuthal angle of point “A,” βo is the average azimuthal angle, and Δβ is the expected walk rate.
The relations between dogleg γ and α K , Δα and Δβ (Lapeyrouse et al. 2002) are shown in Fig. 6.
Figure 6 shows graphic relationship between α K Δβ, α0, and Δα. According to the changes in the inclination and azimuth, α K can be divided into following nine cases:
$$_{\text{k}} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {0,\left( {\Delta \alpha > 0\text{,}\Delta \beta = 0} \right), \left( 1 \right)} \\ {\begin{array}{*{20}c} {\arctan \left( {\frac{\Delta \beta \times \sin \alpha 0}{\Delta \alpha }} \right),\left( {\Delta \alpha > 0\text{,}\Delta \beta > 0} \right), \left( 2 \right)} \\ {90, \left( {\Delta \alpha = 0\text{,}\Delta \beta > 0} \right), \left( 3 \right)} \\ {\arctan \left( {\frac{\Delta \beta \times \sin \alpha 0}{\Delta \alpha }} \right) + 180,\left( {\Delta \alpha < 0,\Delta \beta > 0} \right),\left( 4 \right)} \\ \end{array} } \\ {180,\left( {\Delta \alpha < 0\text{,}\Delta \beta = 0} \right),\left( 5 \right)} \\ {\arctan \left( {\frac{\Delta \beta \times \sin \alpha 0}{\Delta \alpha }} \right) + 180,\left( {\Delta \alpha < 0\text{,}\Delta \beta < 0} \right),\left( 6 \right)} \\ \end{array} } \\ {270,\left( {\Delta \alpha = 0\text{,}\Delta \beta < 0} \right), \left( 7 \right)} \\ {\arctan \left( {\frac{\Delta \beta \times \sin \alpha 0}{\Delta \alpha }} \right) + 360,\left( {\Delta \alpha < 0\text{,}\Delta \beta < 0} \right), \left( 8 \right)} \\ {None,\left( 9 \right)} \\ \end{array} } \right)$$
(7)
1. 1.

Full working to advance the inclination, while the azimuth is not changed.

2. 2.

Both the inclination angle and azimuth angle are all advanced.

3. 3.

Full working to increase azimuth, while the inclination is not changed.

4. 4.

The inclination angle decreased, while the azimuth angle increased.

5. 5.

Full working to decrease inclination angle, while the azimuth angle is not changed.

6. 6.

The inclination angle and the azimuth angle all decreased.

7. 7.

Full working to decrease azimuth, while the inclination is not changed.

8. 8.

The inclination angle and the azimuth angle all decreased.

9. 9.

No working. The inclination and the azimuth were controlled by BHA.

## A new algorithm for pad forces (F1, F2, F3)

When the amplitude A k and the direction α k of the required steering force F are determined, a control algorithm is then needed to adjust each pad force to achieve the required steering force. The control variables are these three pad forces. These force vectors constitute a planar concurrent force system in the bottomhole plane, as shown in Fig. 7.
Figure 7 shows two programs which adjust three pad forces (F1, F2, F3) or (F1′, F2′, F3′) to achieve the same required steering force F. F12 is the resultant force of F1 and F2. F12′ is the resultant force of F1′ and F2′. The relation between F and F1, F2, F3 can be expressed as follows:
\left\{ \begin{aligned} &F\cos \alpha_{k} = F_{1} \cos \alpha_{1} + F_{2} \cos (\alpha_{1} + 240) + F_{3} \cos (\alpha_{1} + 120) \hfill \\ &F\sin \alpha_{k} = F_{1} \sin \alpha_{1} + F_{2} \sin (\alpha_{1} + 240) + F_{3} \sin (\alpha_{1} + 120) \hfill \\ \end{aligned} \right.
(8)

The angle (α1) between pad 1 and high side is measured by the RSSTSP system. Once F and α k are determined, there are three unknown parameters F1, F2, and F3. However, there are only two equations in Eq. (8). The solutions are thus not unique. To determine F1, F2, and F3, an additional equation must be established.

When a RSSTSP starts to work, in order to prevent pad damages, forces exerted on each pad should be limited. Initially, the three pads should start simultaneously with the same forces. The initial force can be set as,
$$F_{\text{ini}} = \frac{{F_{\hbox{max} } + F_{\hbox{min} } }}{2},$$
(9)
where Fini is the pad force at the initial working time.
When the RSSTSP starts to steer the drill bit to a target point, we require a steering force F and direction α K . One has to decide on whether each of the pad forces is in a “favorable” or “unfavorable” area. Such a decision is made by assessing the direction of each of the pad forces F1, F2, and F3, in relation to the required steering force. If the angle between a pad force and the required steering force is within (− 30°, 30°), the pad force plays a positive role in achieving the required steering force, as shown in the hexagon in Fig. 8. In such cases, the pad force falls in the “favorable” area, and the force of the pad should be:
$$F_{\text{f}} = \frac{{F_{\hbox{max} } + F_{\hbox{min} } }}{2} + \frac{{F_{\hbox{max} } - F_{\hbox{min} } }}{2}A_{K} ,$$
(10a)
where Ff is the pad force on the favorable area and the second term on the right-hand side of Eq. (10a) is the augment of the pad force. On the other hand, if the angle between the pad force and the opposite direction of steering force is within (− 30°, 30°), the pad force plays a negative role in achieving the required steering force. In such cases, the pad force falls in the unfavorable area. The pad force takes big value in favorable area and takes small value in the “unfavorable” area, and the force of the pad should be:
$$F_{\text{uf}} = \frac{{F_{\hbox{max} } + F_{\hbox{min} } }}{2} - \frac{{F_{\hbox{max} } - F_{\hbox{min} } }}{2}A_{K} ,$$
(10b)
where the second term on the right-hand side of Eq. (10b) is the reduction of the pad force. The Fuf is the pad force on the unfavorable area. The bottomhole plane can now be divided into six areas, as shown in Fig. 8. At any point in time, the steering force (determined in “Maximum usable steering forces (Amax)” section) should fall in one of these six areas. The favorable or unfavorable area for one of the pad forces can then be easily determined.

### A new algorithm for pad forces

We are now ready to develop a new algorithm for computing the pad forces. For convenience, we define a $$\alpha_{K}^{'} = \alpha_{K} - \alpha_{1}$$, which is the relative direction of $$\alpha_{K}$$ with respect to $$\alpha_{1}$$, as shown in Fig. 9.
As a general convention, α1 and α k are all defined as clockwise positive. The relative steering force direction $$\alpha_{K}^{'}$$ is also defined as clockwise positive with respect to pad 1, and it can be expressed in the following two cases.
$$\left\{ {\begin{array}{*{20}l} {\alpha_{K}^{'} = \alpha_{K} - \alpha_{1} ;} \hfill & {\alpha_{\text{K}} \ge \alpha_{1} } \hfill \\ {\alpha_{K}^{'} = \alpha_{K} - \alpha_{1} + 360;} \hfill & {\alpha_{K} < \alpha_{1} } \hfill \\ \end{array} } \right.$$
(11)

According to the relative status of the steering force, one of the pads can be determined in either the favorable and unfavorable areas, following the procedure detailed in “Direction of steering force (α k )” section. Hence, there are six possible situations.

### Situation 1:

The steering force is in area 1, as shown in Fig. 10.
In Fig. 10, F1F, F2F, and F3F are the force components in the directions of F, respectively, for F1, F2, and F3. F1f, F2f, and F3f are the force components in the normal direction of F, respectively, for F1, F2, and F3. At this situation, $$\alpha_{K}^{'} > 330^{^\circ }$$ or $$\alpha_{K}^{'} \le 30^{^\circ }$$, and pad 1 is in a unfavorable area. Then, supplying F1 = Fuf into Eq. (8), we obtain:
$$\left\{ {\begin{array}{*{20}l} {F_{1} \cos (\alpha_{K}^{'} - 180) + F_{2} \cos (\alpha_{K}^{'} - 300) + F_{3} \cos (\alpha_{K}^{'} - 60) = A_{\hbox{max} } A_{K} } \hfill \\ {F_{1} \sin (\alpha_{K}^{'} - 180) + F_{2} \sin (\alpha_{K}^{'} - 300) + F_{3} \sin (\alpha_{K}^{'} - 60) = 0} \hfill \\ {F_{1} = F_{\text{uf}} } \hfill \\ \end{array} } \right.$$
(12)
Equation (12) now has unique solutions, and the result is as follows:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{uf}} } \hfill \\ {F_{2} = F_{\text{uf}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} - 60)A_{K} } \hfill \\ {F_{3} = F_{\text{uf}} + \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} + 60)A_{K} } \hfill \\ \end{array} } \right.$$
(13)

### Situation 2:

The steering force is in area 2, as shown in Fig. 11.
In this situation, pad 3 is in a favorable area, and $$30 < \alpha_{K}^{'} \le 90^{^\circ }$$, which gives an additional equation of F3 = Ff. Therefore, a unique solution can be given as follows:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{f}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} + 60)A_{K} } \hfill \\ {F_{2} = F_{\text{f}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} )A_{K} } \hfill \\ {F_{3} = F_{\text{f}} } \hfill \\ \end{array} } \right.$$
(14)

### Situation 3:

The steering force is in area 3, as shown in Fig. 12.
In this situation, pad 2 is in a unfavorable area, and $$90 < \alpha_{\text{K}} '\le 150^{o}$$, which gives F2 = Fuf. The unique solution becomes:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{uf}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (60 - \alpha_{k}^{'} )A_{K} } \hfill \\ {F_{2} = F_{\text{uf}} } \hfill \\ {F_{3} = F_{\text{uf}} + \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} )A_{K} } \hfill \\ \end{array} } \right.$$
(15)

### Situation 4:

The steering force is in area 4, as shown in Fig. 13.
In this situation, pad 1 is in a favorable area, and $$150 < \alpha_{K}^{'} \le 210^{^\circ }$$, which supplies F1 = Ff. Then, the solution is obtained as follows:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{f}} } \hfill \\ {F_{2} = F_{\text{f}} + \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (60 - \alpha_{k}^{'} )A_{K} } \hfill \\ {F_{3} = F_{\text{f}} + \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha \alpha_{k}^{'} + 60)A_{K} } \hfill \\ \end{array} } \right.$$
(16)

### Situation 5:

The steering force is in area 5, as shown in Fig. 14.
In this situation, pad 3 is in a favorable area, and $$210 < \alpha_{K}^{'} \le 270^{^\circ }$$, which gives an addition equation of F3 = Fuf. Then, the result is found as follows:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{uf}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (60 + \alpha_{k}^{'} )A_{K} } \hfill \\ {F_{2} = F_{\text{uf}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} )A_{K} } \hfill \\ {F_{3} = F_{\text{uf}} } \hfill \\ \end{array} } \right.$$
(17)

### Situation 6:

The steering force is in area 6, as shown in Fig. 15.
In this situation, pad 2 is in a favorable area, and $$270 < \alpha_{\text{K}} '\le 330^{o}$$, which leads to an addition equation of F2 = Ff. Then, the result becomes:
$$\left\{ {\begin{array}{*{20}l} {F_{1} = F_{\text{f}} - \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (60 - \alpha_{k}^{'} )A_{K} } \hfill \\ {F_{2} = F_{\text{f}} } \hfill \\ {F_{3} = F_{\text{f}} + \frac{2\sqrt 3 }{3}A_{\hbox{max} } \sin (\alpha_{k}^{'} )A_{K} } \hfill \\ \end{array} } \right.$$
(18)

Integrating the steer capability, stability, durability, favorable area, unfavorable area, maximum usable magnitude of steering force, a new control algorithm of RSSTSP can easily be written using Eqs. (12)–(18).

## Field tests

The new control algorithm is applied to a RSSTSP and carried out a field test in a GU-693-P102 well. The RSSTSP has Fmax = 20KN, and Fmin = 0.7KN. If α1 = 30, the pad forces (F1, F2, F3) are changing with the α k based on the new control algorithm, and the outcome is plotted in Fig. 16.
It is shown that each pad force changes smoothly and decreases with decreasing A k , which reduces drill bit vibration and improves the stability and durability of the RSSTSP system. The design track and the well trajectory of GU-693-P102 are shown in Fig. 17, where the blue line is the design track and the red line is the well trajectory achieved using the present control algorithm. A is the target spot, and B is the termination spot.

The results demonstrate the well is consistent with design track as designed. The maximum dogleg rate is 5.92°/30 m, and well trajectory is smooth. It validates that our proposed control algorithm is robust and effective for RSSTSP systems.

## Inclusions

1. 1.

According to the current and targeted build and walk rates, this work establishes a method to calculate the work efficiency (A k ) and direction of the required steering force (α K ). The calculated results offer key instructions to transmit from ground to downhole for a RSSTSP drilling process.

2. 2.

Considering maximum usable magnitude of steering force, steerability, stability, durability, favorable area and unfavorable area, a new algorithm for assigning each pad an optimal force is developed to achieve the required steering force to move the drilling bit from point A to point B.

3. 3.

The present new control algorithm is applied to a new RSSTSP, and a field test is carried out in a GU-693-P102 well for validation. It demonstrates that the new control algorithm is robust and effective for RSSTSP systems.

## Notes

### Acknowledgements

The research is mainly supported by Natural Science Foundation of Hei Long Jiang Province (No. QC2017042).

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## Authors and Affiliations

• Li Gang Zhang
• 1
• 3
• G. R. Liu
• 2
• 3
• Wei Li
• 1
• Shi Bin Li
• 1
1. 1.Department of Petroleum EngineeringNortheast Petroleum UniversityDaqingChina
2. 2.School of Mechanical EngineeringHebei University of TechnologyTianjinChina
3. 3.Department of Aerospace Engineering and Engineering MechanicsUniversity of CincinnatiOhioUSA