A deviation correction strategy based on particle filtering and improved model predictive control for vertical drilling

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Research article

A deviation correction strategy based on particle filtering and improved model predictive control for vertical drilling ∗

Dian Zhang, Min Wu , Chengda Lu, Luefeng Chen, Weihua Cao School of Automation, China University of Geosciences, Wuhan 430074, China Hubei Key Laboratory of Advanced Control and Intelligent Automation for Complex Systems, Wuhan 430074, China Engineering Research Center of Intelligent Technology for Geo-Exploration, Ministry of Education, Wuhan 430074, China

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Article history: Received 23 June 2020 Received in revised form 24 November 2020 Accepted 25 November 2020 Available online xxxx Keywords: Geological exploration Vertical drilling Deviation correction Particle filter Improved model predictive control

a b s t r a c t This paper is concerned with the correction of trajectory deviation in vertical drilling. Note that the accuracy of correction control will be reduced significantly by measurement and process noises, which finally leads to that the inclination angle exceeds beyond a tolerable limit. To deal with such noises and take into account practical constraints, a deviation correction strategy is developed for vertical drilling based on particle filtering and improved model predictive control in this paper. Firstly, the distributions and characters of the measurement and process noises in vertical drilling process are analyzed, and their approximate prior probability distributions are obtained. Based on the analysis, the structure of the deviation correction strategy is provided, including a particle filter and an improved model predictive controller which introduces a flexible constraint and an adjustable weight. The particle filter is effective to reject the measurement noises, and the improved model predictive controller plays an important role in achieving a small inclination of the drilling trajectory. Finally, two groups of simulations are carried out to illustrate the effectiveness of the proposed correction strategy. © 2020 ISA. Published by Elsevier Ltd. All rights reserved.

1. Introduction Deep drilling is of great significance in geological exploration and oil/resource prospect [1,2], and vertical drilling process is one of the most common process adopted in deep drilling. In order to reach the target formation and increase the quality of drilling trajectory, the purpose of vertical drilling is to maintain a straight drilling trajectory along with the wellhead’s plumb line. The position deviation and inclination angle are the main indicators for measuring verticality and quality of drilling trajectory. However, due to complex stratum property and human factors, the position deviation and inclination angle of drilling trajectory are easily increasing in practice, especially in geological drilling [3,4]. Hence, deviation correction strategy is the key point in vertical drilling [5]. For conventional strategies of geological drilling, there are mainly two ways to deal with the deviation. One way is by using passive anti-deviation technologies. But passive anti-deviation technologies are difficult to work well in unstable formations or stratums with high dip angle structures, as these formations ∗ Corresponding author at: School of Automation, China University of Geosciences, Wuhan 430074, China. E-mail addresses: [email protected] (D. Zhang), [email protected] (M. Wu), [email protected] (C. Lu), [email protected] (L. Chen), [email protected] (W. Cao).

are often met in geological drilling [6,7]. Some engineers applied directional drilling control technologies and methods to vertical drilling for deviation correction, these technologies are known as the ‘‘directional deviation correction’’. However the ‘‘directional deviation correction’’ heavily relies on manual experience and compensation methods in practice, such as PID and fuzzy control method based on the deviation vector theory [8], resulting that the performance of correction is limited. So deviation correction strategies with advanced control methods are urgently needed for vertical drilling in geological exploration. There are many constructive results for deviation correction control strategies in academic research. Panchal et al. and Kremers et al. established the kinematics model and kinetics model of bottom hole assembly (BHA) for directional drilling respectively and corrected trajectory deviation by controlling the attitude of BHA [9,10]. For improving the accuracy of control under uncertainties, Bayliss et al. discussed the mixed uncertainty stability of drilling, and designed an attitude-hold controllers [11]. Furthermore, Cai et al. designed a compensation control strategy to deal with the problem that the real parameters of drilling trajectory are unmeasurable [12]. However, to the best of our knowledge, the effect of the measurement noise on deviation correction is ignored in these literatures, which actually could greatly reduce the control accuracy in vertical drilling. Besides, as the inclination angle could far exceed the angle limit during the deviation correction due to the measurement noise, it is

https://doi.org/10.1016/j.isatra.2020.11.023 0019-0578/© 2020 ISA. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: D. Zhang, M. Wu, C. Lu et al., A deviation correction strategy based on particle filtering and improved model predictive control for vertical drilling. ISA Transactions (2020), https://doi.org/10.1016/j.isatra.2020.11.023.

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imperative to increase the abilities of controllers in dealing with constraints. Filtering is the primary method to deal with measurement noise [13–16]. For the situation that the distributions of noises are unknown and their boundary can be acquired, H∞ filter is the common way to be used. Besides, Cheng et al. constructed a novel partially mode-dependent filter to deal with the filtering problem where the signal transmission of a filter model occurred randomly [17,18]. And as for the situation that the distributions of noises are known, extended Kalman filter (EKF) is one of the most common filters used in industry for its high performance of the filtering in non-linear systems subjected to Gaussian measurement noise [19,20]. However EKF has its limit in systems subjected to nonGaussian noise. Particle filter (PF) utilizes sequential Monte Carlo theory and has a wider range of applications than EKF [21,22]. Basic particle filter is easily realized and has higher speed, it relies on historical data and prior probability distribution, not measurements, to estimate the real states. As different applicative background, there are many variants of the particle filter, including particle filter with optimization algorithms or other filters. The typical filters are particle swarm optimization particle filter (PSOPF), extended Kalman particle filter (EKPF) and their variants [23,24]. The extended Kalman particle filter uses extended Kalman filter to acquire the one step prediction value of each particle for particle filter, by this way, the particle can adjust the distribution of particles based on the measurements before particle filtering. The extended Kalman particle filter will have higher filtering accuracy when the noise obeys a Gaussian distribution. Different from the extended Kalman particle filter, the optimization particle filter utilizes optimization algorithm to adjust the distribution of particles based on the measurements after particle filtering, and it does not required the noise obeyed the Gaussian distribution, but it will have higher filtering accuracy when the measurement noise is smaller. But for deviation control in vertical drilling, there is still no research on filter design. As model predictive control (MPC) has the strong ability to cope with constraints on controls and states in an explicit and optimal way, it is widely used to solve practical problems in industry [25–29]. For vertical drilling, dealing with constraints is the foundation for actual application, so MPC provides a good way for the deviation correction problem. In recent years, MPC is applied to deviation correction process gradually and has obtained good results in practice. Martin et al. provided a MPC scheme for drilling to deal with system delay [30]. Demirer et al. considered the long-range behavior of BHA and built the MPC tracking controller considering curvature limit and the smoothing constraint for directional drilling, and had applied successfully in field tests [31]. Zhang et al. established a model predictive controller based on a trajectory extension model considering angle limit and build up rate constraints in vertical drilling [32]. However, there is no feasible solution to the optimization problem of MPC if inclination angle far exceeds the limit due to the measurement noise, and these existing MPC methods are not applicable to vertical drilling subject to this situation. This paper builds on the authors’ earlier work on deviation correction control for vertical drilling [32]. Bearing the above discussions in mind, although there are many researching results in many other areas, few works pay special attention on noise analysis and filter design for vertical drilling process, which motivates us to fill this gap. Meanwhile there is no feasible solution to the optimization problem of MPC if inclination angle far exceeds the limit due to the measurement noise, and the existing MPC methods are not applicable to vertical drilling subject to this situation, to provide an effective MPC for vertical drilling by addressing such issue is the second motivation of

this paper. Therefore, the objective of this work is to establish a filter resisting the measurement noise and find a way to enhance the deviation correction performance when the inclination angle exceeds the limit. The contribution of this study is highlighted as follows: (a) A particle filter is established to acquire quality estimated parameters of the drilling trajectory for model predictive controller and increase system’s ability of resisting measurement noise. (b) The flexible constraint is used to change the constraint equations of MPC to ensure there is feasible solution to the optimization problem if inclination angle far exceeds the limit due to the measurement noise. (c) The adjustable weight used sigmoid function is introduced for model predictive controller to change the priority of control objective according to the inclination angle. It is worth mentioning that, as the existence of the measurement noise and the small inclination angle in vertical drilling process, the historical data and prior information are more reliable than measurements, which makes the particle filter to be a good choice for our study. And by adjusting the flexible constraint and the weight of optimization problem in MPC online, not only feasible solution can be ensured for the optimization problem of MPC, but also the inclination angle could return within the normal range of value quickly once the inclination exceeds the limit. The remainder of this paper is organized as follows. In Section 2, the demands and constraints of geological drilling process are analyzed, and the problems of the deviation correction strategy are described based on the process. In Section 3, the structure of deviation correction strategy is provided based on the process. At the same time, the particle filter for vertical drilling and an improved model predictive controller which uses a flexible constraint and an adjustable weight are designed. In Section 4, simulations are carried out to validate the validity of the particle filter and the strategy. Some conclusions are presented at the end of this paper. 2. Process analysis and problem description This section describes the deviation correction process of geological vertical drilling, and the objective and characteristics of the process are discussed. Then, control problems are discussed based on the trajectory extension model given by [32]. 2.1. Process analysis According to the directional deviation correction process of the practice geological drilling field, the vertical drilling system considered in this paper is shown in Fig. 1. The drilling system mainly consists of a driller house, an industrial personal computer (IPC) in the drilling house, a clinometer, a rotary table, drill pipes, a screw drilling tool as a steering tool and a drill bit. The directional deviation correction process can be described as follows. The inclination angle and azimuth are measured by the clinometer, and the well depth is measured on the ground. These trajectory parameters will be sent to the IPC in driller house. Then, the IPC generates control instructions by the control algorithm. Finally the rotation of the downhole motor or the rotary table is adjusted based on the instructions until drilling for a certain distance. At this time, one control cycle of vertical drilling is completed. It makes the system work in the direction mode by stopping the rotary table and rotating the downhole motor of steering tool alone, and rotating the downhole motor and rotary table together 2

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Fig. 2. Build up rate r of a practice drilling field.

may be around 1.5◦ and azimuth may be around 4◦ [34]. The clinometer is equipped with accelerometers and magnetometers, and the interference on its measurement curves is the random noise with short cycle, the noise is generally independent of the formation [35]. So like other measurement systems, the measurement noise mainly comes from the thermal noise of electronic devices which usually obeys a Gaussian distribution. Fig. 1. Vertical drilling system.

Fig. 2 shows the build up rate r acquired from the data of a practice drilling field in central China. As (a) shows, the values of 18 well sections are given. As the influence of stratigraphic uncertainty, the build up rate r is not a constant. The maximum value of these data is 9.2◦ /30 m, and the minimum value is 1.4◦ /30 m, the average value of them is 5.1◦ /30 m. In order to investigate the distribution characteristics, the data of build up rate r is divided into 9 groups, as shown in (b). It can be seen that the distribution of the r is similar to Gamma distribution Γ (3, 2).

makes the system work in the vertical drilling mode. By adjusting the proportion of time that system works in these two modes, the system provides various of build up rates. In practice, as the downhole motor is driven by the pressure of mud and it cannot be stopped during drilling, so it is used to switch the working mode of the system by adjusting the rotation state of the rotary table. According to the process description, the control objective of the deviation correction is to reduce both the position deviation and the inclination angle. System’s inputs are the reference coordinates of drilling trajectory which are set according to the well plan. And adjustment parameters are the magnetic tool face angle and the steering ratio. The magnetic tool face angle is the direction of deflection, and it refers to the angle between the tool face of the steering tool and the north. The steering ratio denotes the scaled magnitude of steering force applied in deflecting direction, and it is always adjusted by changing the rotating time of the rotary table in one control cycle. For measurement demands and constraints of geological drilling, trajectory parameters are measured only when the drilling stops after drilling for a certain distance, generally equals to the length of a drill pipe. Meanwhile, due to the limited categories of measurement tools, only the well depth, the inclination angle and the azimuth angle are measurable. In addition, the inclination angle is not zero during the position deviation correcting, and higher inclination facilitates rapid correction. However, in order to increase the quality of drilling trajectory, it is generally necessary to keep the inclination angle less than αmax , as the amplitude of αmax is depending on demands of drilling plan. When the inclination angle exceeds the angle limit αmax , the priority of the deviation correcting is to reduce the inclination angle. At the same time, the build up rate r of BHA is limit [33]. As the inclination angle is small during vertical drilling, the influence of the measurement noise cannot be ignored, so establishing a filter is important. And for the filter, in order to estimate the inclination angle and azimuth accurately, it is necessary to analyze characters of both the measurement and the process noise in vertical drilling process.

Remark 2. For process noise, as the vertical drilling is a complex process, the process noise of drilling always obeys a non-Gaussian distribution. As the analysis above, the build up rate r changes between 1.4 ∼ 9.2◦ /30 m, and the change trend of the r mainly follows the Gamma distribution Γ (3, 2). So if r is assumed to be a fixed value, the process noise follows a Gamma distribution. The maximum value of the process error is around 4◦ /30 m. As a conclusion, the deviation correction of vertical drilling is to reduce both the position deviation and the inclination angle by adjusting the rotating state of the rotary table. For practical application, the constraints of the inclination angle and the build up rate should be taken into account, and the measurement noise should be dealt with. 2.2. Problem description According to the process analysis, the schematic of the deviation correction process can be described as Fig. 3. The schematic shows the movement of the BHA and the formation of drilling trajectory in perspective under ground, the Z -axis is pointing in the direction of the gravity, the X -axis is due the East, and the Y -axis is due the North. The trajectory extension model is given as (1) [32]:

⎧ tan αx = tan α sin β ⎪ ⎪ ⎪ ⎪ tan αy = tan α cos β ⎪ ⎪ ⎪ ⎪ ⎪ S˙z = S˙ cos α ⎪ ⎨ S˙x = S˙ tan αx ⎪ ⎪ ⎪ ⎪ S˙y = S˙ tan αy ⎪ ⎪ ⎪ ⎪ α˙ x = ωx + µx = r ωSR sin θ˜tf + µx ⎪ ⎪ ⎩ α˙ y = ωy + µy = r ωSR cos θ˜tf + µy

Remark 1. With the increase of the well depth and harsh conditions, the maximum measurement error of inclination angle 3

(1)

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Fig. 4. Configuration of deviation correction control system.

Fig. 3. Schematic of deviation correction process.

subject to the process and measurement noises µx , µy , υα,x and υα,y with distribution mentioned above. It is also important to deal with the problem that there is no feasible solution for the optimization problem of MPC when αˆ > αmax + ωα max .

where α is inclination, β is azimuth angle, S˙ is the rate of penetration (ROP). S˙x , S˙y , and S˙z are x-component, y-component and z-component of ROP S˙ respectively, αx and αy are components of α , as Sx and Sy are used to calculate the scale of position deviation, αx and αy are used to calculate the scale of angular deviation. Magnetic tool face angle θ˜tf and steering ratio ωSR are controllable variables. Build up rate r is the maximum deflection capability of BHA, and r ωSR ∈ [0, r ] denotes the real deflection capability that BHA provided. The virtual controls ωx = r ωSR sinθ˜tf , ωy = r ωSR cosθ˜tf are introduced to simplify the model. µx and µy are process noises. As the process analysis, the process noise µx or µy follows the Gamma distribution written as (2), where apex , bpex , apey , bpey are parameters for adjusting the scale and expectation.

{

apex µx + bpex ∼ Γ (3, 2) apey µy + bpey ∼ Γ (3, 2)

3. Deviation correction strategy In this section, a control system configuration of the deviation correction strategy is given. Then the principle and steps of particle filter in vertical drilling process are discussed. Finally the model predictive controller is provided, the flexible constraint and the adjustable weight are introduced to enhance the control feasibility when the inclination far exceeds the angle limit. 3.1. Control system configuration A control configuration of the deviation correction control strategy is given to solve the deviation correction problem in the presence of the measurement noise, two control loops are included, as shown in Fig. 4. rin indicates the reference of coordinates and angles of the drilling trajectory, Sˆx , Sˆy , Sˆz are estimated coordinates of the drilling trajectory acquired from the particle filter; αˆ , βˆ are estimated inclination and azimuth angle respectively; S is the well depth; µx , µy are process noises; υα,x , υα,y are measurement noises. The inner loop is mainly used to control the rotation of the rotary table under control instructions named tool face angle and steering ratio in real time. The out loop is mainly responsible for deviation correction control, and it is mainly composed of the trajectory extension model, the filter, the model predictive controller and the trajectory calculation model. The trajectory extension model is proposed for describing the vertical drilling process as (1) shows. And the filter is utilized to deal with the measurement noise. As analysis above, the measurement noise always obeys the Gaussian distribution, although the vertical drilling is always a non-Gaussian process. In order to solve this filtering problem, the particle filter is adopted in this strategy to improve the measurements quality and finally leads to higher control accuracy. For higher robust of control and ability of dealing with constraints, a model predictive controller is adopted. Different from other MPC methods, our method mainly considers the problem of angle limit, and introduces a flexible constraint and an adjustable weight. The advantages of the improvements are that the flexible constraint enhances the control feasibility of MPC during vertical drilling, and the adjustable weight changes the priority of control objective to make the inclination angle can return within the normal range of value quickly once the inclination exceeds the limit. In addition, as only the well depth, the inclination angle and the azimuth angle are measurable, the coordinates of drilling

(2)

As the existence of the measurement noise, only estimated values Sˆx , Sˆy , αˆ x and αˆ y can be acquired from measurements α¯ = α + υα and β¯ = β + υβ by the filter, where υα and υβ are measurement noises. It notes that αx and αy can be acquired from α and β as (1) shows. When α and υβ are small, α sin β or α cos β approximately obeys the Gaussian distribution, so it is accounted as that αx or αy also obeys the Gaussian distribution N(0, σv ). The maximum error υα,x or υα,y of αx or αy is less than maximum error υα of α as (1) shows. For constraints, the control instructions should only be generated after drilling for a certain distance, generally equals to the length of a drill pipe. The state constraint is αˆ ≤ αmax , and the input constraint is r ωSR ≤ r. It is worth mentioned that, dealing with constraints is the foundation for practical application in vertical drilling. Comparing with other control methods, MPC has strong ability of dealing with constraints and resisting noises, so it is a good choice for deviation correction. However, the inclination angle α could not meet angle constraint αˆ ≤ αmax in the presence of the measurement noise during deviation correcting. When αˆ > αmax + ωα max , there is no feasible solution for the optimization problem of MPC, resulting that MPC cannot generate right control instructions. ωα max is defined as the maximum change range of angle which is provided by system in one control circle. To end this section, the problem to be addressed in this paper is described as: It is necessary to reduce Sx , Sy , αx and αy to be zeros by adjusting the steering ratio ωSR and the magnetic tool face angle θ˜tf under the state and input constraints. In order to increase the accuracy of deviation correction, the real states of trajectory parameters are estimated from measurements α¯ and β¯ 4

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trajectory are obtained by the minimum curvature method which is defined as the industry standard for trajectory calculation by the American Petroleum Society in 1985 [36]. Designed trajectory parameters and the estimated values from the particle filter are selected as inputs of the model predictive controller for the deviation correction, and the magnetic tool face angle and the steering ratio are obtained to be control instructions.

Algorithm 1 Particle filter for vertical drilling Require: α, ¯ β¯ measurement values Ensure: α, ˆ βˆ estimated values 1: (a) Initialization: 2: Transform angles from measurements by (2) 3: 4:

3.2. Design of particle filter 5:

As the analysis above, the measurement noise always obeys the Gaussian distribution, although the drilling process is always the non-Gaussian process. EKF has its limit in this situation. Particle filter utilizes sequential Monte Carlo to approximate posterior distribution by using a set of weighted samples. Particle filter is easily realized and has higher speed, and it is widely used in industry. It is known that the accuracy of the particle filter depends on the distribution of the particle set. When the distribution of the particle set is similar to the posteriori probability distribution, the estimation error of the particle filter is small. The particle filter mainly relies on the prior knowledge to adjust the distribution of its particle set, other improved algorithms of particle filter mainly rely on measurements, such as the PSOPF and the EKPF.

6: 7:

8: 9: 10: 11: 12: 13: 14: 15: 16: 17: 18:

Remark 3. Different particle filters have their own advantages, but the particle filter has its superiority in vertical drilling. Firstly, the maximum measurement error nearly equals to the real inclination angle, that means adjusting the distribution of the particle set based on measurements will not make the distribution approach to the posteriori probability distribution. More errors will be introduced during filtering in PSOPF and EKPF as they mainly rely on measurements. Secondly, as particle filter mainly relies on prior knowledge, it is possible to utilize prior knowledge, such as logging parameters, to improve the estimation accuracy.

19:

Set prior probability distributions of process and measurement noise apex µx + bpex ∼ Γ (3, 2), apey µy + bpey ∼ Γ (3, 2) αx ∼ N(0, σvx ), αy ∼ N(0, σvy ) Initialize a particle set from the prior probability distribution of process ( i 1) (αˆ xi ,0 , N1 ) , i = 1, 2, · · · N ∼ p(αˆ x,0 ) αˆ y,0 , N , i = 1, 2, · · · N ∼ p(αˆ y,0 ) for k = 1 → kmax do (b) Importance sampling One step prediction from prediction function (1) {( i 1 ) } ( ) α ˜ , , i = 1, 2, · · · N ∼ q α˜ xi ,k |αˆ xi ,k−1 x , k N {( i 1 ) } ( ) α˜ y,k , N , i = 1, 2, · · · N ∼ q α˜ yi ,k |αˆ yi ,k−1 (c) Update the weights Transform the angles from measurement by (2)

( ) ( ) α¯ k , β¯ k → α¯ x,(k , α¯ y,k ) wxi ,k = wxi ,k−1 p (α¯ x,k |α˜ xi ,k ) wyi ,k = wyi ,k−1 p α¯ y,k |α˜ yi ,k wxi ,k ∑ i wx,k

20:

w ˜ xi ,k =

21:

w ˜ yi ,k =

22:

(d) Resampling and estimating

23: 24: 25:

The pseudo code of particle filtering for vertical drilling is given in algorithm 1. The inputs of the particle filter are the measurement values of the inclination angle and azimuth, as the outputs are the estimated values of these two angles. It is worth mentioning that the prior probability distributions of process and measurement should be set before filtering, these prior probability distributions can be acquired from parameters of BHA, data of adjacent well and logging parameters. In order to simplify the calculation, it is necessary to transform the measurements to α¯ x , α¯ y using (2). Meanwhile, as it assumes that the importance density is the same as the posteriori probability distribution and applies resampling at each instant k, the sampling importance resampling (SIR) can simplify the calculation of the particle’s weights and make the filter be closer to real states, so the SIR is applied to the particle filter here [37–39]. In addition, the time complexity of the proposed particle filtering is mainly defined by the complexity of resampling. In our algorithm, the roulette wheel method is selected to reselect one particle, whose complexity is O(N), considering the worst situation. As there are N particles in the filter, the time complexity of the proposed filter is O(N 2 ). Besides, as the maximum length of the array used in the filter is N, the space complexity is O(N).

( ) ( ) α¯ 0 , β¯ 0 → α¯ x,0 , α¯ y,0

26: 27:

wyi ,k ∑ i wy,k

∑ i i αˆ x,k = α˜ w ˜ ∑ xi ,k xi ,k αˆ y,k = α˜ y,k w ˜ y,k Transform the (angles )to inclination and azimuth by (2) ) (

αˆ x,k , αˆ y,k → αˆ k , βˆ k

end for

the estimated values Sˆx , Sˆy , αˆ x and αˆ y acquired from the particle filter are selected to be the feedback signals of model predictive controller. Linearizing and dispersing the trajectory extension model is the first step of building the predictive equation of MPC. For linearizing, as the inclination angle is small during vertical drilling, tan αˆ x ≈ αˆ x and tan αˆ y ≈ αˆ y , the linear trajectory extension model is given as (3):

⎡ ⎤ ˙ ⎡ Sˆx 0 ⎢˙ ⎥ ⎢αˆ x ⎥ ⎢0 ⎢˙ ⎥ = ⎣ 0 ⎣ Sˆy ⎦ 0 ˙αˆ y

S˙ 0 0 0

0 0 0 0

⎤⎡ ⎤

0 0 Sˆx 0⎥ ⎢ α ˆ x⎥ ⎢1 ⎢ ⎥ +⎣ ⎦ 0 S˙ ⎣ Sˆy ⎦ 0 0 αˆ y

⎡

0 [ ] 0⎥ ωx 0⎦ ωy 1

⎤

(3)

For dispersing, the sampling period is set to be T . In order to denote the deviation between the real trajectory and the well plan, it is assumed that there is another BHA drilling along the reference trajectory, which is vertical in vertical drilling process, as Fig. 3 shows. Each point of the real trajectory has the same true vertical depth (TVD) h with its corresponding reference. As the reference kinematics model is the same as (3), the error system in discrete time relative to reference trajectory can be written as (4):

3.3. Design of improved model predictive controller The objective of the model predictive controller is to reduce Sx , Sy , αx and αy to be zeros by adjusting the steering ratio ωSR and the tool face angle θ˜tf . In order to improve the control accuracy, 5

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⎡

⎤

Sˆex (k + 1) 1 ⎢αˆ ex (k + 1)⎥ ⎢0 ⎢ ⎥=⎣ ⎣ Sˆey (k + 1) ⎦ 0 0 αˆ ey (k + 1)

⎡

˙ ST 1 0 0

0 0 1 0 ⎡ 0 ⎢T +⎣ 0 0

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⎤⎡

⎤

0 Sˆex (k) ⎥ 0 ⎥⎢ α ⎢ ˆ ex (k)⎥ ˙ ⎦ ⎣ Sˆey (k) ⎦ ST 1 αˆ ey (k) ⎤ 0 [ ] 0⎥ ωex (k) 0 ⎦ ωey (k) T

(4)

˙ is equal to the length of a drill pipe Lpipe , it means that where ST the distance between k and k + 1 equals to the length of a drill pipe. Sˆex (k) = Sˆx (k) − Srx (k) and Sˆey (k) = Sˆy (k) − Sry (k), they are estimated deviations of BHA on x-axis and y-axis respectively relative to the reference trajectory at k. aˆ ex (k) = aˆ x (k) − arx (k) and aˆ ey (k) = aˆ y (k) − ary (k) are projections of αˆ on XOZ plane and YOZ plane relative to the reference trajectory at k. The control instructions are ωex (k) = ωx (k) − ωrx (k) and ωey (k) = ωy (k) − ωry (k). As for vertical drilling, Srx (k), Sry (k), arx (k), ary (k) and ωrx (k), ωry (k) are zeros. Using the above model (4), and p is predictive horizon, c is control horizon, the prediction equation of model predictive controller can be established as follow: Yˆ (k) = Ξk xˆ (k|k) + Θk W (k)

Fig. 5. Weight of MPC Q.

introduced to solve this problem.

⎧    ⎪ αˆ (k + n) ≤ αmax (if αˆ (k) ≤ αmax ) ⎨     αˆ (k + n) ≤ αˆ (k) (if αˆ (k) > αmax ) ⎪ ⎩ n = 1, . . . , p As α is always small, αˆ can also equal to

αˆ x2 + αˆ y2 . Trans-

ferring into incremental form gives the deviation correction optimization problem:

(5)

(

)

min J Yˆ (k), W (k) = Yˆ (k)T Q Yˆ (k) + W (k)T RW (k)

where W (k) is a incremental control signal relative to the reference steering ratio, Yˆ (k) is a incremental state variable relative to parameters of the designed trajectory, and the parameter matrixes can be expressed as:

⎧ 2 2 ⎪ ≤ α2 ⎪ ⎪(αrx (m) + αˆ ex (m)) + (αry (m) + αˆ ey (m))   max ⎪ ⎪ ⎪ (if αˆ (k) ≤ αmax ) ⎪ ⎪ ⎪ ⎨ 2 (αrx (m) + αˆ ex (m)) + (αry (m) + αˆ ey (m))2 ≤ (αˆ (k))2 s.t.   ⎪ (if αˆ (k) > αmax ) ⎪ ⎪ ⎪ ⎪ ⎪(ωrx (m) + ωex (m))2 + (ωry (m) + ωey (m))2 ≤ r 2 ⎪ ⎪ ⎪ ⎩ m = k + 1, . . . , k + p

⎤ Sˆex (k + 1|k) ⎡ ω (k|k) ⎤ ⎢aˆ ex (k + 1|k)⎥ ex ⎢ ⎥ ⎢Sˆey (k + 1|k)⎥ ω ey (k|k) ⎥ ⎢ ⎢ ⎥ ⎢ωex (k + 1|k)⎥ ⎢aˆ ey (k + 1|k)⎥ ⎢ ⎥ ⎥ ⎥ ˆY (k) = ⎢ ... ⎢ ⎥ W (k) = ⎢ ⎢ωey (k + 1|k)⎥ ⎢ˆ ⎥ ⎢ ⎥ ... ⎢ Sex (k + p|k) ⎥ ⎣ ⎦ ⎢aˆ (k + p|k)⎥ ω (k + c | k) ex ⎢ ex ⎥ ⎣ˆ ⎦ ωey (k + c |k) Sey (k + p|k) aˆ ey (k + p|k) [ ]T xˆ (k|k) = Sˆex (k|k) aˆ ex (k|k) Sˆey (k|k) aˆ ey (k|k) ⎡

(6)

Besides, according to demands of the vertical drilling, MPC should prioritize the reduction of the inclination angle when α exceeds αmax , or prioritize the reduction of position deviation when α is less than αmax . It assumes that Q = [qS ,x qα,x qS ,y qα,y ] in this paper, so it change the priority of MPC mainly by adjusting qα,x and qα,y according to the inclination angle at measurement point. The sigmoid function (7) is selected to established the relationship between the weights qα,x , qα,y and inclination angle α . There are three advantages to use the sigmoid function here: Firstly, the sigmoid function can easily contain the change range of inclination angle; Secondly, the parameters of sigmoid function have good interpretability, and are easily implement in practice; Finally, it has adjustable change rate to increase or decrease the weights qα,x and qα,y , that means it can deal with the inclination angle limit more flexible. The sigmoid function is written as follow:

Sˆex (k + 1|k) is the estimated value at (k + 1)-time predicted from the estimated value at k-time, as same as the others. Ξk and Θk are the coefficient matrixes which can be calculated based on MPC theory. They are related to the parameters p and c. For better control performance, p and c should be set within 2∼5 because of the measurement noise. Based on the prediction equation, the objective function of MPC can be established to minimize the error between the actual and the designed trajectory, as well as the increment of control signal. That is:

f (x) =

(

√

)

min J Yˆ (k), W (k) = Yˆ (k)T Q Yˆ (k) + W (k)T RW (k)

bQ ˆ cQ ) 1 + e−aQ (α−

+ dQ

(7)

Fig. 5 shows the means of parameters of the sigmoid function. bQ is the value of qα,x , qα,y when inclination α ≪ αmax , and bQ +dQ is the value of qα,x , qα,y when inclination α ≫ αmax . cQ is the turning point, and it should be around αmax . aQ defines the change rate of the weights qα,x and qα,y when inclination α is around αmax , it is set according to the drilling demands.

⎧    ⎪ ⎨ αˆ (k + n) ≤ αmax s.t. ∥r ωSR (k + n)∥ ≤ r ⎪ ⎩ n = 1, . . . , p During vertical drilling, the inclination angle α is easy to exceed αmax because of the measurement noise, fixed constraint may cause that there is no feasible solution for the optimization problem of MPC. A flexible constraint shown as follow is

Remark 4. Different from other model predictive controllers, a flexible constraint and an adjustable weight are proposed for MPC to enhance the control feasibility when the inclination far exceeds 6

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the angle limit. A flexible constraint is used to ensure there is always feasible solution for the optimization problem of MPC, and the sigmoid function (7) is adopted to set weight of Q to make the controller concentrate on different goals before and after the inclination angle exceeds the constraint. For the complexity analysis, the particle filter and the model predictive controller are independent of each other, which means that the complexity of MPC will not be increased when the number of the particle changes. For the improvements of the proposed controller, the flexible constraint and the sigmoid function are simple numerical computation methods, so the complexity of the proposed MPC will not be increased comparing with the normal MPC algorithm by these improvements. Normally, the whole running time of one control cycle is always under 0.25 s. As outputs of the controller are increments relative to reference control instructions, actual control instructions equal to the sum of reference control instructions and outputs. At the same time, according to the model predictive control law, the first sequence of values ωex (k) and ωey (k) of the optimized calculation sequence W (k) should be taken as the actual control increments. So the actual control instructions can be obtained as:

⎧ ⎪ ⎪ ⎪ ⎨

√

Fig. 6. Filtering results.

(ωrx + ωex )2 + (ωry + ωey )2

ωSR = r ) ( ⎪ ω + ω ⎪ rx ex ⎪ ⎩ θ˜tf = arctan ωry + ωey

Fig. 7. Filtering results with different process and measurement errors.

Finally, as for the stability, let J ∗ (Yˆ (k), W (k)) be the optimal value of J(Yˆ (k), W (k)) as evaluated at time k. Clearly J ∗ (Yˆ (k), W (k)) ≥ 0 only if Yˆ (k) and W (k) are zeros, or the x(k|k) is zero. So the J ∗ (Yˆ (k), W (k)) can be chosen as a Lyapunov function for the control system. When the disturbances µx or µy are set to be zeros, the estimated error of the particle filter will be zeros as its property. It means that when the disturbances is inexistence, Yˆ (k) = Y (k), and the Lyapunov function can be transformed as J ∗ (Y (k), W (k)). Then the control system can be transformed as a simple MPC control system in [32]. So the stability of proposed system can be ensured when MPC control system in [32] is stable, as the stability condition can be acquired from many literatures [40–42].

Table 1 Errors of filtering. Algorithm

PF PSOPF EKF EKPF

αx

αy

MAE

RMSE

MAE

RMSE

0.27373 0.34833 0.74752 0.32082

0.34534 0.42811 0.91005 0.39788

0.28816 0.35194 1.0081 0.35448

0.36065 0.43301 1.2032 0.43345

Table 2 MAE of filtering with υα,x ∼ N(0, 0.49).

4. Simulation and result analysis In this section, simulations are conducted to validate the validity of the particle filter and the deviation correction strategy respectively. The involved parameters are collected from a practical drilling process and existing literature, some parameters are slightly modified to facilitate the simulations.

Variance of µx

PF

PSOPF

α¯ x

0.6 ∗ (Γx 0.4 ∗ (Γx 0.2 ∗ (Γx 0.1 ∗ (Γx

0.33933 0.27008 0.21539 0.13465

0.40775 0.34462 0.25969 0.17657

0.55987 0.5771 0.52803 0.55583

− 6)/6 − 6)/6 − 6)/6 − 6)/6

of measurements is about 0.54 and its RMSE is about 0.69. So it can be easily seen that errors of PF, PSOPF and EKPF are smaller than measurements in vertical drilling process. EKF is liable to get divergence because of the linearization errors and the process error with Gamma distribution. Comparing results of PF with PSOPF and EKPF, PSOPF or EKPF loses the prior information at importance sampling stage and adjusts distribution of the particle set based on measurements. As the analysis above, PSOPF and EKPF do not make the particle set approach to their true location because of the measurement noise, so errors of them are larger than PF in vertical drilling process. In order to validate the superiority of particle filter in vertical drilling process further, tests are conduct with different process and measurement errors, comparing with PSOPF, the results are shown as Fig. 7 and Tables 2, 3. As seen in the left figure of Fig. 7 and Table 2, the MAEs of α¯ x are around 0.55, and the performances of filters become better with reducing of the process error. The amplitude of process error is depend on the prior knowledge, such as logging parameters.

4.1. Filtering performance analysis We use the trajectory extension model (1) to verify the effectiveness of the particle filter. Assume that the measurement noise obeys the Gaussian distribution υα,xk , υα,yk ∼ N(0, 0.49), and the process noise obeys the Gamma distribution µx,k = 0.6 ∗ (Γx − 6)/6, µy,k = 0.6 ∗ (Γy − 6)/6, where Γx ∼ Γ (3, 2) and Γy ∼ Γ (3, 2). It means that the maximum measurement error is around 1.5◦ , and the maximum process error is around 4◦ /30 m. Fig. 6 shows the results of several filters in one test, and Table 1 shows filtering results of 100 Monte Carlo tests. According to filtering results, the MAE of PF is about 0.28, and the RMSE is about 0.35. The filtering performance of PSOPF and EKPF is similar, the MAEs of them are about 0.35 and the RMSEs are about 4.3. EKF has the worst filtering performance as its MAE is about 0.9 and the RMSE is about 1.1. Need to add that, the MAE 7

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Table 3 MAE of filtering with µx = 0.6 ∗ (Γx − 6)/6. Variance of υα,x

PF

PSOPF

α¯ x

N(0, 0.49) N(0, 0.16) N(0, 0.04) N(0, 0.01)

0.34632 0.23015 0.13424 0.076094

0.45044 0.23829 0.13759 0.074415

0.55812 0.28727 0.15782 0.08103

Table 4 Simulation parameters. Parameter

Description

S˙ T r

30 m/h 0.3 h 6◦ /30 m 3◦ 5 [50000,50000] 1000 [0.1, −5(α− ˆ 3) + 5,0.1,

αmax

p, c R Q

1+e

1000 ˆ 3) 1+e−5(α−

+ 5]

So the accuracy of particle filter can be further increased by acquiring more prior knowledge during drilling. The right figure in Fig. 7 shows the result with different measurement errors. With the decreasing of measurement error, the accuracy of PSOPF is quickly improved, and it exceeds the accuracy of particle filter when variance of υα,x is 0.1, although the measurement error is also very small. Once again, it confirms that the accuracy of PSOPF depends on the measurement error. As the measurement error is really large in vertical drilling process, the particle filter is more applicable. In conclusion, particle filter has lower estimation error and is more applicable to vertical drilling. As a conclusion, the particle filter has lower estimation error than measurements and other improved particle filters, the filtering error is around 0.28 when the maximum measurement error is around 1.5◦ and the maximum process error is around 4◦ /30 m. As particle filter mainly relies on the prior knowledge, it is possible to utilize logging parameters et al. to improve the estimation accuracy.

Fig. 8. Deviation correction performance comparison. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

4.2. Dynamic response of correction strategy Simulations are carried out to test the deviation correction capacity for the case of vertical drilling under the measurement noise. The parameters of simulation are selected shown in Table 4: According to data from an actual drilling site, the horizontal deviation between actual trajectory and the reference is 8.82 m in XOZ plane at 600 m measured depth (MD), meanwhile the horizontal deviation is 1.51 m in the YOZ plane [32]. In order to show the performance of improved model predictive controller, the inclination angle is set to be 5.83◦ , the azimuth angle is 56.6◦ . According to analysis of Section 2, it is assumed that the measurement noise obeys the Gaussian distribution vk ∼ N(0, 0.49), which means the maximum measurement error of simulation is around 1.4◦ . At the same time, the process noise is assumed to obey the Gamma distribution (12 ∗ µk + 6) ∼ Γ (3, 2), which means that the maximum process error of simulation is around 3.4◦ /30 m. In order to validate the validity of our strategy, simulations of basic MPC [32] and MPC with particle filter are conducted. Fig. 8 shows results of the position and angle deviation correction. The blue line shows the reference positions and inclination angle of the vertical trajectory along with the measurement depth. The purple line, the green line and the red line show the actual states α, Sx , Sy controlled by MPC in [32], MPC with only PF, and strategy proposed in this paper respectively, as well as the Fig. 9, which

Fig. 9. Tool face angle and steering ratio. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

shows the corresponding tool face angle and the steering rate of the simulations. For the control accuracy, as seen in Fig. 8, deviations of the inclination angle and the position have been corrected to some extent by all these three strategies. The inclination angle is going 8

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feasible solution for the optimization problem of MPC when the inclination far exceeds the angle limit, and the adjustable weight changes the priority of control objective and makes the proposed strategy tend to reduce the inclination to be lower than the angle limit, which makes the quality of drilling trajectory more smooth. Future works are trying to consider the parameter uncertainty of the trajectory extension model and developing a corresponding control strategy to further increase the robust of the control system.

to be small after 825 m, means the main deviation has been eliminated. However, the inclination angle controlled by the basic MPC strategy between 690 m and 800 m changes rapidly around the angle limit, and easily exceeds the angle constraint because of the measurement noise. With the particle filter, the inclination angles controlled by the other two strategies are more controllable as the angle is easier within the angle limit. At the same time, as coordinates of trajectory Sx , Sy are calculated from measurements, it makes the deviation between the measurement and the real trajectory become larger after 870 m, which causes decline of the accuracy and quality of the drilling trajectory seriously in basic MPC strategy. Compared, coordinates of trajectory Sx , Sy controlled by the other two strategies are much closer to the reference after 870 m. For the flexible constraint, comparing our strategy with the MPC with only PF, the main different is the ωSR at 600 m and 645 m. At the beginning of correction, the inclination angle is about 5.83◦ , it is larger than αmax + ωα max , as well as the αˆ at 645 m because a large measurement noise is introduced. The MPC with only PF generates the wrong instructions at these two place, and the ωSR exceeds 100%, they are not valid steering rates in practical drilling. Compared, the proposed strategy has deal with this problem by the flexible constraint and generates the right ωSR which is less than 100%. For the adjustable weight, although the position deviation correction performance of MPC with only PF is similar to our strategy, the inclination angles controlled by our strategy are lower than the angle controlled by MPC with only PF between 715 m and 805 m. Therefore, the drilling trajectory controlled by the proposed strategy is smoother between 715 m and 805 m, as Fig. 8 shows. It means that our strategy tends to reduce the inclination to be lower than angle limit αmax , and this will make the quality of drilling trajectory be better than that of MPC with only PF. In conclusion, our strategy can efficiently correct the deviation in vertical drilling process. Comparing with basic MPC, our strategy can acquire more precise trajectory parameters from the particle filter, and significant increase the control accuracy of inclination angle α and positions Sx , Sy . The flexible constraint ensures there is always feasible solution for the optimization problem of MPC when αˆ > αmax + ωα max , and the adjustable weight makes the proposed strategy tend to reduce the inclination to be lower than angle limit αmax , which makes the quality of drilling trajectory more smooth.

Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61733016, the National Key Research and Development Program of China under Grant 2018YFC0603405, the Hubei Provincial Technical Innovation Major Project under Grant 2018AAA035, the Natural Science Foundation of Hubei Province, China, under Grants 2020CFA031, the 111 project under Grant B17040, and the Fundamental Research Funds for the Central Universities under Grant CUGCJ1812. References [1] Mason CF. Regulating offshore oil and gas exploration: Insights from the deepwater horizon experience in the gulf of Mexico. Rev Environ Econ Policy 2019;13(1):149–54. [2] Lu CD, Wu M, Chen X, Cao WH, Gan C, She JH. Torsional vibration control of drill-string systems with time-varying measurement delays. Inform Sci 2018;467:528–48. [3] Wang WC, Zhan H, Li N, Wang CS, Teng XQ, Zhu WP, Di QF. The dynamic deviation control mechanism of the prebent pendulum BHA in air drilling. J Pet Sci Eng 2019;176:521–31. [4] Huang WJ, Gao DL, Liu YH. Buckling analysis of tubular strings with connectors constrained in vertical and inclined wellbores. SPE J 2018;23(2):301–27. [5] Li H, Wu MF, Wang YQ, Yang B. Research on deviation correction and modeling of geological drilling. Appl Mech Mater 2012;256:220–8. [6] Godhavn JM, Pavlov A, Kaasa GO, Rolland NL. Drilling seeking automatic control solutions. IFAC Proc Vol 2011;44(1):10842–50. [7] Lu YD, Zeng LC, Zeng FL, Kai GS. Dynamic simulation and research on hydraulic guide system of automatic vertical drilling tool. Mater Res Innov 2015;18(s6):170–4. [8] Xue QL, Wang RH, Song WQ, Huang LL. Simulation study on fuzzy control of rotary steering drilling trajectory. Res J Appl Sci Eng Technol 2012;4(13):1862–7. [9] Panchal N, Bayliss MT, Whidborne JF. Attitude control system for directional drilling bottom hole assemblies. IET Control Theory Appl 2012;6(7):884–92. [10] Kremers NAH, Detournay EM, Wouw VD. Model-based robust control of directional drilling systems. IEEE Trans Control Syst Technol 2016;24(1):226–39. [11] Bayliss M, Whidborne J. Mixed uncertainty analysis of pole placement and h controllers for directional drilling attitude tracking. J Dyn Syst Meas Control 2015;137(12):1–8. [12] Cai Z, Lai XZ, Wu M, Chen LF, Lu CD. Compensation control for tool attitude in directional drilling systems. In: Proceedings of the 12th Asian Control Conference (ASCC 2019), Kitakyushu, Fukuoka, Japan; 2019, p. 376-380. [13] Lu CD, Zhang XM, Wu M, Han QL, He Y. Energy-to-peak state estimation for static neural networks with interval time-varying delays. IEEE Trans Cybern 2018;48(10):2823–35. [14] Shen MQ, Park JH. H∞ filtering of Markov jump linear systems with general transition probabilities and output quantization. ISA Trans 2016;63(1):204–10. [15] Vafamand N, Arefi MM, Khayatian A. Nonlinear system identification based on takagi-sugeno fuzzy modeling and unscented Kalman filter. ISA Trans 2018;74(1):134–43. [16] Lu CD, Wu M, Chen LF, Cao WH. An event-triggered approach to torsional vibration control of drill-string system using measurement-while-drilling data. Control Eng Pract 2021;106:104668.

5. Conclusion In this paper, a deviation correction strategy based on particle filtering and improved model predictive control has been proposed for vertical drilling to deal with the measurement noise. Two control loops are included, and the out loop is mainly responsible for deviation correction control. The out loop mainly composes of the trajectory extension model, the particle filter, the model predictive controller and the trajectory calculation model. For filter design, it considers the effect of measurement noise with non-Gaussian process, and has established a particle filter to deal with measurement noise in vertical drilling process. For model predictive controller design, a MPC method based on the trajectory extension model has been proposed for deviation correction problem. And the flexible constraint and the adjustable weight have been introduced for model predictive controller to enhance the control feasibility when the inclination far exceeds the angle limit. Simulation results have shown that the control accuracy is improved significantly when the particle filter is introduced. Furthermore, the flexible constraint can ensures there is always 9

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