A Deviation Correction Technique Based on Particle Filtering Combined with a Dung Beetle Optimizer with the Improved Model Predictive Control for Vertical Drilling

Abstract

The following study will look at the issue of the dealignment of the trajectory when drilling vertically (a fact), where measurement and process errors are still the primary source of error that can easily lead to the inclination angle having overshot the desired bounds. The current methods, such as the Extended Kalman Filters (EKFs), can incorrectly estimate non-Gaussian noises, unlike the classical particle filters (PFs), which are unable to handle significant measurement errors appropriately. We will solve these problems by creating a new deviation correction mechanism using a dung beetle optimizer particle filter (DBOPF) with a superior Model Predictive Controller (MPC). The DBOPF makes use of the prior knowledge and optimization process to enhance the precision of state estimation and is superior in noise reduction to traditional filters. The improved MPC introduces flexible constraints and weight adjustments in the form of a sigmoid function that enables solutions when the inclination angle exceeds the threshold, and priorities are given to control objectives dynamically. The simulation outcomes indicate that the approach is more effective in the correction of the trajectory and control of inclination angle than the conventional MPC and other optimization-based filters, such as the PSO and SSA, in the presence of the noisy drilling environment.

1. Introduction

1.1. General Background

Drilling in a vertical direction is not a recent practice and began with the discovery of the first oil well in 1859, when Edwin L. Drake drilled a well in Titusville, Pennsylvania. This became the classical approach during the earlier years of the production of oil and gas. Although the technique of vertical drilling remained rather primitive in the 19th century, in the past few years, much has been done, with the 20th-century drilling methods being substituted by the new ones [1,2]. It remains a conventional technique for deep drilling in geological research and resource exploration [3]. However, it established the foundation for the sophisticated techniques employed today. Among others, technology has enhanced drilling rigs, mud systems, and drill bits, enabling one to drill deeper wells where security is assured [4]. Vertical drilling is carried out in a straight line through a healthy head along a plumb drain to reach the desired formation and enhance the quality of the drilling process. The deviation correction is a crucial indicator for drilling performance. Complicated geological formations and man combine to produce skyrocketing changes in their variability, particularly in geological drilling. The condition produces a high demand for deviation correction in x-rayed drilling in order to ensure accuracy and performance in downward drilling [5,6].

1.2. Literature Review

Vertical drilling operations face a significant obstacle from measurement noise, which requires reliable filtering solutions to address. The Extended Kalman Filter (EKF) remains a widely used solution because it provides effective noise management for Gaussian distributions in non-linear systems [7]. The EKF shows reduced performance when operating with non-Gaussian noise, which commonly exists in advanced drilling systems. The Maximum Corr entropy Kalman Filter (MCKF) is a new approach that applies information-theoretic learning to manage non-Gaussian noise without requiring any information about noise statistics [8,9]. The process of deviation correction in geological drilling uses two main approaches, which include passive and active methods. The implementation of passive anti-deviation techniques becomes challenging when dealing with unstable rock formations and steeply inclined strata. The real-time adjustment of drilling parameters depends on active methods that include Proportional-Integral-Derivative (PID) and fuzzy logic controllers [10,11]. The methods require operator experience to function, but they become less efficient when operating in difficult situations. The development of kinematic models for directional drilling has led to improved control precision through path adjustments made by model reorientation, which enhances accuracy in unpredictable drilling conditions.

In the filtering techniques, the particle filter (PF) has gained prominence for its ability to handle non-linear and non-Gaussian systems. The PF employs Sequential Monte Carlo methods to approximate the posterior distribution of system states [12]. The Particle Flow Gaussian Sum Particle Filter (PFGSPF) represents a contemporary algorithm that implements invertible particle flow to prevent weight degeneracy and enhance state estimation for intricate systems [13]. The industrial control method, the Model Predictive Control (MPC), serves as a vital deviation correction technique because it handles both control and state constraints during real-time operations [14]. Research studies show that MPC technology proves successful in minimizing wellbore spiraling issues caused by feedback delays, according to Liu, who created an MPC controller with delay compensation [15]. Zhang, in his work, used the Model Predictive Control (MPC) to enhance the three-dimensional trajectory tracking through the adjustment of steering instructions in rotary steerable systems (RSS) and mud motors [16]. In a more recent workup, some studies have demonstrated that it was possible to utilize what he refers to as a Gaussian fitting with the aid of hybrid bat algorithms to design intelligent compensation tools to regulate the process of the uncertainty in geological structures. Current MPC techniques do have a drawback, though, when the inclination angle is too large, typically due to measurement errors. The machine learning techniques have found increasing applications in other engineering fields to facilitate real-time optimization of the vertical drilling processes. Reinforcement Learning software (python) and Deep learning systems have been researched as a solution to real-time control problems, primarily the problems of trajectory correction and noise cancellation. The application of such techniques has proven to have great potential in improving the automated drilling practices and accuracy, even in complex and inherently noisy environments [17,18].

In this study, the main gaps in the literature that still need answers regarding redressing the deviations when undergoing vertical drilling shall be addressed, with a particular focus on the mistakes associated with measurements and process noise. The process of vertical drilling serves essential functions in geological exploration, yet achieving straight drilling becomes difficult because of complicated subsurface geology and multiple outside interference factors. The current methods for deviation control through passive anti-deviation technologies depend on manual experience and PID controllers and fuzzy control systems, which prove inadequate for unstable formations and high-dip angle strata correction. The current MPC deviation correction methods fail to address infeasible solutions when measurement errors cause the inclination angle to surpass its desired threshold. The current research lacks sufficient studies about developing filtering techniques for vertical drilling operations, which must handle non-Gaussian processes and measurement noises. The research establishes new filtering and control methods to improve vertical drilling precision and reliability when operating in complex environments.

1.3. The Hypothesis of This Study

Being the thesis of the study, DBOPF and a more precise MPC model will assist in enhancing the correction of path deviation of a vertical drilling operation, despite the availability of noisy measurements or nocturnal measurements. The logic behind this assumption is that the availability of the reduction of measurement noise capacity and accuracy enhancement of the state estimates made by DBOPF proposes that the MPC can better regulate the inclination angle.

2. Assumptions and Limitations in Research

2.1. Assumptions

This study assumes that measurement noise is Gaussian, a common assumption for thermal noise in electronic devices. This assumption is made on the characteristics of the clinometer used to measure inclination and azimuth angles. The inclination angle is assumed to be small $\left(0^{\circ }\le \alpha \le 5^{\circ }\right)$ to validate linear approximations in Equation (9).

The rate of process noise is modelled as a Gamma process, because this is observed in real data of build-up rate (r) at actual drilling sites. This choice can also be justified by a field observation that accumulation roles are given by a Gamma distribution.

The experiment follows that the control actions issued will have to follow drilling a distance that is usually relative to a single drill pipe. This imitates the process of drilling, where measurements and corrections are made at set intervals.

The improved MPC builds on the assumption that constraint flexibility can be incorporated into the decision-making process such that constraints can be made active or inactive online to guarantee feasibility when the inclination angle becomes too large. It is assumed that the system is able to operate dynamically and adapt to variations in the operating conditions.

2.2. Limitations

In this study, process noise is modelled as a Gamma distribution; however, actual drilling conditions can be more complex, and the noise characteristics modelled may not necessarily represent this to the fullest extent. This may limit the accuracy of the filter in certain circumstances.

In the linearization of the trajectory extension model, small inclination angles are assumed. Where the inclination angle is large, the linear approximation may be in error, reducing the accuracy of the model.

The flexible constraints in MPC are designed to handle cases where the inclination angle reaches the limit. However, this approach may not consider all possible deviations, especially in highly unpredictable drilling environments.

3. Process Analysis with Problem Description

This study illustrates the vertical drilling system in Figure 1 by applying the deviation correction method in the geological drilling practice area. The system consists of the house of the driller, an industrial computer (IPC) located within the drilling room, a table, a clinometer, drilling pipes, a drilling tool for guidance, and a drilling bit. The procedure for rectifying directional deviation can be articulated as follows. The trajectory variables will be provided to the industrial computer in the driller’s room. Thereafter, the industrial computer produces command directives via the control algorithm. The downhole motor or rotary table is rotated as directed until a predetermined distance is drilled [19].

By changing the system into the trajectory mode, the vertical rotary cycle of drilling control is accomplished. In this phase, only the steering motor is operating, though the table is stationary. In vertical rotary drilling, the process does its job when it has the joint presence of the downhole and rotating table. The accumulation rate is an at-will rate and is also changed by the successive individual periods in which the system runs in each of these modes. The mud pressure powers the downhole motor and must remain constant during rotary drilling; it is utilized to modify the process’s operational mode by regulating the rotary table’s rotation state. The method clarification indicates that regulating deviation correction reduces positional tilt and deviation angle. The system inputs consist of the reference positions of the drilling trajectory defined by the specified plan [20]. The key parameters that may be adjusted or dislodged are the driving ratio and the type of angle formed by the tool face. In geological drilling, measuring criteria and limits stipulate that path parameters are determined exclusively upon the termination of drilling after a designated distance, usually corresponding to the length of the drilling pipe [21,22].

Moreover, only the borehole depth, inclined angle, and azimuth angle are evaluated due to the limited range of measurement instruments. Moreover, the inclination angle varies during positioning adjustments, and a sharper inclination enables rapid correction. The angle inclination must not go beyond the maximum limit ($\alpha$ max) that the drilling plan outlines in order to ensure that the drilling path is as high as possible in terms of quality. Once the culmination point of this maximal angle is achieved, the primary goal behind deviation correction is to reduce the inclination to a low value. However, the rate of Bottom Hole Assembly (BHA) accretion (r) is minimal, and in vertical drilling, these shorter inclination angles cause a highly increased magnification of the measurement errors. Consequently, it is imperative to implement a filter. To precisely determine the inclined angle and azimuth, it is crucial to examine the properties of both the evaluation and processing noise in the vertical boring operation [23].

Point 1.

Well depth and inclination measurements face errors during the measurement process. As the depth of the well increases and drilling conditions become more severe, the measurement errors associated with the inclination angle and azimuth also escalate. Under the mentioned conditions, the error measure of the inclination angle value is about 1.5°, and conversely, the error measure of the azimuth angle value is about 4°. The Clinometer system, with the application of accelerometers and magnetometers, is subject to interference that introduces short-run bursts of random noise. This sound will have minimal bearing on the geological formation that is the subject of the drill, but again, this noisiness is primarily a result of variation in temperature of the parts that have been involved in the electronics, and generally speaking, the sound has an assumed and widely assumed Gaussian movement. Figure 2a employs a scatter diagram to indicate the rate of accumulation of wells in the so-called training drilling site in central China, which is composed of 18 well sections within the training site. Such build-up rates are themselves variable, not because there are stratigraphic uncertainties, which contribute more strongly to controlling their behavior. These have been documented to have a low of 1.4°/30 m and a high of 9.2°/30 m, averaging 5.1°/30 m. The build-up rate values are grouped into nine, which is illustrated in Figure 2b, to compare the statistical values with a more precise depiction of their statistical properties. The distribution obtained is akin to Gamma, a distribution which is G(3, 2).

Point 2.

Process disturbance and process noise in vertical drilling. The complex nature of vertical drilling operations makes process disturbances an unavoidable occurrence. These disturbances are often modelled as process noise, which typically follows a non-Gaussian distribution. As demonstrated in the earlier analysis, the build-up rate of the healthy trajectory fluctuates between 1.4° and 9.2° per 30 m, with the trend generally adhering to a Gamma distribution Γ(3, 2). When considering the build-up rate as a fixed value, the process disturbance can be assumed to follow this Gamma distribution. The highest observed process error is approximately 4° per 30 m. Therefore, to effectively correct deviations in vertical drilling, it is essential to minimize both the inclination angle and positional deviation.

3.1. Problem Description

Figure 3 is the result of the analysis of the deviation correction. This has demonstrated how the Bottom Hole Assembly (BHA) functions and also depicted the 3-D course that the drill follows. In this display, the Z-axis runs downwards, the X-axis runs towards the east, and the Y-axis spreads out towards the north. To stretch the trajectory from which Equation (1) originates, and when modelling the drilling path, a minimum-curvature method is taken to represent the path. The approach will also assist in enhancing accuracy in the drilling process since it will possess a clear corporate picture of the BHA performance and the complete drilling path.

$$\left\{\begin{array}{l}\tan {\alpha }_x=\tan \alpha \sin \beta \\ \tan {\alpha }_y=\tan \alpha \cos \beta \\ {\dot{S}}_z=\dot{S}\cos \alpha \\ {\dot{S}}_x=\dot{S}\tan {\alpha }_x\\ {\dot{S}}_y=\dot{S}\tan {\alpha }_y\\ {\dot{\alpha }}_x={\omega }_x+{\mu }_x=r{\omega }_{SR}\sin {\tilde{\theta }}_{tf}+{\mu }_x\\ {\dot{\alpha }}_y={\omega }_y+{\mu }_y=r{\omega }_{SR}\cos {\tilde{\theta }}_{tf}+{\mu }_y\end{array}\right.$$

(1)

The model includes three variables, where $\alpha$ represents the angle of inclination, $\beta$ represents the azimuth angle, and $\dot{S}$ represents the rate of penetration. The components ${\dot{S}}_x,{\dot{S}}_y$, and ${\dot{S}}_z$ correspond to the x-, y-, and z-components of the rate of penetration, respectively. The components S are used to define ${\alpha }_x$ and ${\alpha }_y$, which represent the angular components of the inclination angle $\alpha$. These components$,{\dot{S}}_x$ and ${\dot{S}}_y$, are employed to determine the magnitude of positional deviation, while ${\alpha }_x$ and ${\alpha }_y$ are used to assess the angular deviation. To improve the model, virtual control parameters ${\omega }_x=r{\omega }_{SR}\mathrm{s}\mathrm{i}\mathrm{n}\left({\theta }_{tf}\right)$ and ${\omega }_y=r{\omega }_{SR}\mathrm{c}\mathrm{o}\mathrm{s}\left({\theta }_{tf}\right)$ are implemented. Additionally, ${\mu }_x$ and ${\mu }_y$ represent process noises, which are modeled as following a Gamma distribution, as described in Equation (2), with parameters $a_{\mathrm{pex}},b_{\mathrm{pex}},a_{\mathrm{pey}}$, and $b_{\mathrm{pey}}$.

$$\left\{\begin{array}{l}{a}_{\mathrm{pex}}{\mu }_x+b_{\mathrm{pex}}\sim \mathsf{\Gamma }(3,2)\\ a_{\mathrm{pey}}{\mu }_y+b_{\mathrm{pey}}\sim \mathsf{\Gamma }(3,2)\end{array}\right.$$

(2)

In terms of measurement noise, the predicted values ${\hat{S}}_x,{\hat{S}}_y,{\hat{\alpha }}_x$, and ${\hat{\alpha }}_y$ can be derived from the measurements $\bar{\alpha }=\alpha +v_{\alpha }$ and $\bar{\beta }=\beta +v_{\beta }$, where $v_{\alpha }$ and $v_{\beta }$ represent the measurement noise associated with the inclination and azimuth angles, respectively. It is observed that ${\alpha }_x$ and ${\alpha }_y$ can be derived from $\alpha$ and $\beta$, as shown in Equation (1). When both $\alpha$ and $v_{\beta }$ are small, the terms $\alpha \mathrm{s}\mathrm{i}\mathrm{n}\beta$ and $\alpha \mathrm{c}\mathrm{o}\mathrm{s}\beta$ can be approximated as following a Gaussian distribution. Therefore, ${\alpha }_x$ and ${\alpha }_y$ are assumed to also follow a Gaussian distribution $N\left(0,{\sigma }_v\right)$. The maximum errors in ${\alpha }_x$ and ${\alpha }_y$, denoted as $v_{\alpha ,x}$ and $v_{\alpha ,y}$, respectively, are smaller than the maximum measurement error $v_{\alpha }$, as indicated in Equation (1). Regarding operational constraints, control instructions are typically generated only after drilling a specified distance. The state constraint is defined as $\hat{\alpha }\le {\alpha }_{\max }$, while the input limitation is $r{\omega }_{SR}\le r$. Effective constraint management is essential for optimizing the performance of vertical drilling systems. In comparison to other control systems, Model Predictive Control (MPC) excels in handling constraints and mitigating noise, making it a preferable choice.

This research aims to address this issue by reducing the deviations of ${\hat{S}}_x,{\hat{S}}_y,{\hat{\alpha }}_x$, and ${\hat{\alpha }}_y$ to zero. This is achieved by adjusting the magnetic tool face angle ${\omega }_{SR}$ and the steering ratio ${\omega }_{SR}$ while adhering to the constraints on both system states and inputs. The inherent trajectory parameters are established using the measurements $\bar{\alpha }$ and $\beta$, with the objective of improving the precision of deviation correction in the drilling process [24].

The present study is distinct from previous studies in that a new approach, dung beetle optimizer particle filter (DBOPF), is combined with an improved Model Predictive Control (MPC) with flexible constraints and weight adjustment for the specific application in vertical drilling with large measurement errors, significant noise and strict limitations on the inclination angle, to achieve better accuracy and robustness in deviation correction.

Model Robustness to Small-Angle Approximation

This model relied on minimal angles of inclination to utilize the linear approximation used in Equation (1). This is, however, no longer true at angles steeper than this small-angle limit. A sensitivity analysis must be carried out to determine the influences of the increased angles on the accuracy, and make changes so as to bring it to a linear form in the worst-case scenario.

4. Deviation Correcting Technique

Here, the design of a type of control system capable of being employed in the deviation correction, which is to be employed during the remaining vertical drilling efforts, is provided. It contains the specifics of the particle filter and the establishment of how it is used in this application of drilling. The implementation of the system is carried out through the CMA Model Predictive Controller (MPC), which allows the soft, though constrained management, as well as adjusting the factors of weighting. The above features ease the operation of the system, especially when the system is in the vicinity of the periphery of the inclination angle.

4.1. Design of Control System

A control scheme is developed in an attempt to fix the deviation in the presence of the measurement noise. The new setup is an integration of two control loops (analogous to Figure 4). Here, $r_{in}$, of the desired drilling path and angles are used in reference to and are ${\hat{S}}_x,{\hat{S}}_y,{\hat{S}}_z$ of the approximate coordinates of the funnel path drilled. These approximated values of inclinations and azimuth angles are characterized by the shortened term, $\hat{\alpha },\hat{\beta }$. The depth (measured) is denoted by the letter S, and the noise factors of the process are referred to as ${\mu }_x,{\mu }_y$. Noise $v_{\alpha ,x},v_{\alpha ,y}$ is the movement noise.

The system of control functions in two loops, with the inner loop and the outer one. The inside loop is used to run the rotary table with respect to using the real-time instructions, e.g., tool face angle, steering ratio. The outer loop handles the deviation correction and includes a number of blocks: a model of the trajectory extension, a filter, the MPC, and a model for calculating trajectories. The flow of vertical drilling is modeled in terms of a trajectory extension model as illustrated in Figure 1. By then, the filter can only reduce noise in measurements that are hypothesized to follow a Gaussian distribution due to the process of the vertical drilling measurements, which is not necessarily purely Gaussian. The particle filter is utilized to address the filtering issue, thereby enhancing the accuracy of measurement and control. In contrast to existing MPC approaches, the proposed method focuses on angle constraints, incorporating a modular restriction and a variable weight to adjust control priorities. These additions make the MPC more realistic, enabling it to keep the inclination angle within the requested limits, even in cases where the inclination angle would tend to exceed the limit. Since the only specified inputs are the depth of the well, inclination, and azimuth angle, the coordinates of the drilling path are computed with the minimum curvature method, which is the agreed industry standard in reporting of the drilling trajectory design, as it was adopted by the American Petroleum Institute in 1985. The MPC then implements these parameters of the previous trajectories to correct deviation, accompanied by the predicted values from the particle filter. It also makes decisions based on this information to determine the necessary control input of the magic tool face angle and steering ratio to enable it to enter the efficient and appropriate drilling activity [25].

The inclination and azimuth clinometer measurements normal error under this study is approximately 0.5 deg and 4 deg, respectively, which are caused by the Gaussian noise on inclination and azimuth measurements. Meanwhile, a Gamma distribution records the process noise, which is very similar to the variability given to the fluctuations in the build-up rates. To solve these gray areas, the dung beetle optimizer particle filter is deployed (DBOPF). This is extremely helpful to reduce the effect of measurement and process noise and to obtain increased precision in state estimation, enabling trajectories to be corrected online.

4.2. Design of Dung Beetle Optimizer Particle Filter

The dung beetle optimizer is a bio-inspired optimization algorithm, which is based on the foraging and rolling behavior of the dung beetles. With these strategies, which are natural, the DBO will be effectively searching the search space to identify the best solutions [26]. The algorithm can be described mathematically as follows:

$$\left(t+1\right)=X\left(t\right)+\sigma \cdot \left(X_{\mathrm{best}}-X\left(t\right)\right)+\Psi \cdot \left(X_{\mathrm{rand}}-X\left(t\right)\right)$$

(3)

where $X\left(t\right)$ is the current location of the dung beetle at iteration $t$, $X_{\mathrm{best}}$ is the best solution, $X_{\mathrm{rand}}$ refers to a position that is randomly chosen from the population, and $\sigma$ and $\Psi$ are coefficients refer to how the movement behaves.

Particle filters are attaining unending popularity because they are capable of calculating the time dependence of the condition of a system in nonlinear and non-Gaussian systems. The idea here is to develop a set of particles (samples), and each of the particles represents one of the ways the system can be, at a given time. Mathematically, the mathematical description of the particle filter is as follows [27]:

4.2.1. Prediction Step

$$x_t^{\left(i\right)}=f\left(x_{t-1}^{\left(i\right)}\right)+w_t^{\left(i\right)}$$

(4)

where $x_t^{\left(i\right)}$ refers to the state of the $i$-th particle at time $t$, $f\left(x_{t-1}^{\left(i\right)}\right)$ refers to the state transition model, and $w_t^{\left(i\right)}$ represents the process noise.

4.2.2. Update Step

$$w_t^{\left(i\right)}={\displaystyle \frac{p\left(y_t\mid x_t^{\left(i\right)}\right)}{\sum _{j=1}^N p\left(y_t\mid x_t^{\left(j\right)}\right)}}$$

(5)

This notation, $x_t^{\left(i\right)}$, means the state of particle number i at time $t$, and $p\left(y_t\mid x_t^{\left(i\right)}\right)$ means the model of the state transition $y_t$ given $x_t^{\left(i\right)}$, $w_t^{\left(i\right)}$ means the noise of the process. The process is effective when it comes to the characterization of complex systems, thereby enhancing the accuracy of the state estimates.

Combining DBO and PF allows the utilization of DBO’s optimization capabilities to improve the state estimation process of the particle filter as follows:

Employ the dung beetle optimizer to ascertain an estimate for the particles’ circumstances. This technique aids in identifying the initial particle configuration, allowing the filter to commence with a well-informed or nearly optimal assumption.

$$X_{\mathrm{initial}}=\mathrm{D}\mathrm{B}\mathrm{O}\left(f\right(x),p(y\mid x\left)\right)$$

(6)

where $X_{\mathrm{initial}}$ represents the optimized particles obtained from the DBO.

A particle filter within the scope of a moving average framework, Prediction Step, prediction (meaning model). Particles are initially propagated via the system dynamics, updating their weight in the update step, based on the observed measurements.

$$x_t^{\left(i\right)}=f\left(x_{t-1}^{\left(i\right)}\right)+w_t^{\left(i\right)}$$

(7)

Utilize the optimization (DBO) to fine-tune the hyperparameters of the particle filter (PF), including adjusting the process noise and the number of particles used. The DBO algorithm is capable of modifying the PF parameters to enhance the accuracy of observations or reduce errors in estimating the state during the process.

$$w_t^{\left(i\right)}={\displaystyle \frac{p\left(y_t\mid x_t^{\left(i\right)}\right)}{\sum _{j=1}^N p\left(y_t\mid x_t^{\left(j\right)}\right)}}$$

(8)

$${\theta }_{\mathrm{optimal}}=\mathrm{D}\mathrm{B}\mathrm{O}\left(f\left(x_t\right),p\left(y_t\mid x_t\right),\mathrm{Particle}\mathrm{Filter}\mathrm{Parameters}\right)$$

where ${\theta }_{\mathrm{optimal}}$ is the optimized PF variables (number of particles, process noise).

The noise of measurement generally follows the Gaussian distribution. The clinometer’s measurement noise is primarily Gaussian due to thermal fluctuations in its electronic components. As observed in industries, sensor errors [controlled environment] are usually distributed normally. One example of inferential techniques is the dung beetle optimization particle filter (DBOPF), a sequential Monte Carlo inferential algorithm that estimates the posterior probability distribution by sampling and weighing samples, and is relatively easy to apply and takes less time. The error in the DBOPF estimation is minimal in the situation in which the two distributions are close. This is typically practiced in the DBOPF to tune and maximize the coverage of a set of its particles [28] in fine detail using available information to correct the process.

4.2.3. Comparison with Bayesian Filtering Methods

DBOPF differs from conventional Bayesian filters because it uses prior knowledge from neighboring wells to initialize particle distributions, which decreases dependence on real-time measurement noise. The Monte Carlo method in the DBOPF allows it to handle non-Gaussian noise without the linearization errors that occur in Gaussian Bayesian methods like Kalman filters. The DBOPF roulette-wheel resampling method operates at $O\left(G\right)$. complexity, which is more efficient than Bayesian sampling methods like Markov chain Monte Carlo (MCMC), which operate at $O\left(G^2\right)$ complexity when used for real-time drilling control [29]. The capability of the DBOPF is high regarding control of non-Gaussian noise when prior data of the adjacent wells is used; however, this depends on the quality of the information and its reliability. Poor quality of the former data causes the filter to be less effective, and in the more complicated geological formations, which do not follow the assumptions of the distributions, this is particularly true. The accuracy of the initial particle distribution can be increased with the assistance of the added contribution of adjacent wells and sensors, which leads to an improved inference drawn. Also, formulating the adaptive filtering methods by using either prior information or the reliance on measurements available could, possibly, enhance the usefulness of DBOPF performance in extreme situations.

Point 3.

Despite all the distinct possibilities offered by different types of particle filters, the DBOPF proved to be highly helpful in terms of vertical drilling. The largest discrepancies in measurement are usually associated with the actual value of the glory angle; the difference in the packet assembly based on a measurement movement does not necessarily indicate that the household persistently puts up against the posterior probability vase in any genuine probability gadget. Increased mistakes will occur during EKF and EKPF filtering due to their dependence on measurements. DBOPF predominantly depends on prior knowledge; utilizing such knowledge, including logging configurations, is feasible to enhance estimation accuracy. This work presents the pseudo-code for the dung beetle optimizer, combined with particle filtering, applied to vertical drilling A particle filter is implemented where the input of the measured values ​​are the slope and azimuth, and the output is the estimated values. Before a process of filtering is undertaken, prior probability distributions of the process and the measurements must be defined. Such distributions can be on BHA parameters, well data near violations, and logging settings. Data are transformed to ${\bar{\alpha }}_x,{\bar{\alpha }}_y$ by Equation (2) in order to utilize it in a calculation. Resampling can be simplified through the sampling importance resampling (SIR) algorithm, which assumes that importance density is equal to the probability distribution after conditioning on the actual states at each step k, which makes the algorithm preferable to the actual states; there, SIR enters into a particle filter.

The resampling techniques are known to determine the computation time of particle filtering. The Algorithm 1 involves particle reselection by the roulette wheel method, which has to run in its worst-case in O(G) time. Overall, in the G particle, the filter can be said to be of $O\left(G^2\right)$ average time complexity. Since the size of the arrays on which the filter is being applied is at most G, the size of the bits it is consuming is also O(G).

Algorithm 1 DBOPF for vertical drilling
1:	Require: $\bar{\alpha },\bar{\beta }$ values of measurement
2:	Ensure: $\hat{\alpha },\hat{\beta }$ estimated variables
3:	(a) Initializing:
4:	Transform angles from measurements by (2)
5:	$\left({\bar{\alpha }}_0,{\bar{\beta }}_0\right)\to \left({\bar{\alpha }}_{x,0},{\bar{\alpha }}_{y,0}\right)$
6:	Establish prior probability distributions for process and measurement noise.
7:	$a_{\mathrm{pex}}{\mu }_x+b_{\mathrm{pex}}\sim \mathsf{\Gamma }(3,2),a_{\mathrm{pey}}{\mu }_y+b_{\mathrm{pey}}\sim \mathsf{\Gamma }(3,2)$
8:	${\alpha }_x\sim \mathrm{G}\left(0,{\sigma }_{vx}\right),{\alpha }_y\sim \mathrm{G}\left(0,{\sigma }_{vy}\right)$ (10)
9:	Initialize a dung beetle set from the prior probability distribution of process
10:	$\left({\hat{\alpha }}_{x,0}^i,{\displaystyle \frac{1}{G}}\right),i=1,2,\cdots G\sim p\left({\hat{\alpha }}_{x,0}\right)$ (6)
11:	$\left({\hat{\alpha }}_{y,0}^i,{\displaystyle \frac{1}{G}}\right),i=1,2,\cdots G\sim p\left({\hat{\alpha }}_{y,0}\right)$ (7)
12:	for $l=1\to l_{\mathrm{m}\mathrm{a}\mathrm{x}}$ do
13:	(b) Importance sampling
14:	One step prediction from prediction function (1)
15:	$\left\{\left({\tilde{\alpha }}_{x,l}^i,{\displaystyle \frac{1}{G}}\right),i=1,2,\cdots G\right\}\sim q\left({\tilde{\alpha }}_{x,l}^i{\hat{\alpha }}_{x,l-1}^i\right)$
16:	$\left\{\left({\tilde{\alpha }}_{y,l}^i,{\displaystyle \frac{1}{G}}\right),i=1,2,\cdots G\right\}\sim q\left({\tilde{\alpha }}_{y,l}^i\mid {\hat{\alpha }}_{y,l-1}^i\right)$
17:	(c) Updating the weights
18:	convert the angles from measurement by (2)
19:	$\left({\bar{\alpha }}_l,{\bar{\beta }}_l\right)\to \left({\bar{\alpha }}_{x,l},{\bar{\alpha }}_{y,l}\right)$
20:	$w_{x,l}^i=w_{x,l-1}^{i}p\left({\bar{\alpha }}_{x,l}\mid {\tilde{\alpha }}_{x,l}^i\right)$
21:	$w_{y,l}^{i,}=w_{y,l-1}^{i}p\left({\bar{\alpha }}_{y,l}\mid {\tilde{\alpha }}_{y,l}^i\right)$
22:	${\tilde{j}}_{x,l}^i={\displaystyle \frac{j_{\chi ,l}^i}{\sum j_{x,l}^i}}$
23:	${\tilde{j}}_{y,l}^i={\displaystyle \frac{j_{y,l}^i}{\sum j_{y,l}^i}}$
24:	(d) Resampling and estimating
25:	${\hat{\alpha }}_{x,l}=\sum {\tilde{\alpha }}_{x,l}^i{\tilde{j}}_{x,l}^i$
26:	${\hat{\alpha }}_{y,l}=\sum {\tilde{\alpha }}_{y,l}^i{\tilde{j}}_{y,l}^i$
27:	convert the angles to inclination and azimuth by (2)
28:	$\left({\hat{\alpha }}_{x,l},{\hat{\alpha }}_{y,l}\right)\to \left({\hat{\alpha }}_l,{\hat{\beta }}_l\right)$(11)
29:	end for

4.3. Design of Model Predictive Controller

4.3.1. The Objective of the Model Predictive Controller

Measures of the control objective include $S_x$, $S_y,{\alpha }_x,$ and ${\alpha }_y$ have to be equal to zero broadened through changing steering ratio ${\omega }_{SR}$and angularity of the tool-face ${\tilde{\theta }}_{tf}$. The particle filter is used as a performance improvement technique of performing control with estimation of ${\hat{S}}_x,{\hat{S}}_y,{\hat{\alpha }}_x,$ and ${\hat{\alpha }}_y$ Fed back into the Model Predictive Controller. The development of the prediction equation that is involved in Model Predictive Control (MPC) begins with the process of linearization and discretization of the model of the trajectory extension. In making linearization, where the inclined angle is restrained, following vertical drilling, and tan ${\hat{\alpha }}_x\approx {\hat{\alpha }}_x$ and tan ${\hat{\alpha }}_y\approx {\hat{\alpha }}_y$ depict an extension model of the trajectory based on a linearized Schubert model in Figure 5.

$$\left[\begin{array}{c}{\dot{\hat{S}}}_x\\ {\dot{\hat{\alpha }}}_x\\ {\dot{\hat{S}}}_y\\ {\dot{\hat{\alpha }}}_y\end{array}\right]=\left[\begin{array}{llll}0& \dot{S}& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& \dot{S}\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{\hat{S}}_x\\ {\hat{\alpha }}_x\\ {\hat{S}}_y\\ {\hat{\alpha }}_y\end{array}\right]+\left[\begin{array}{ll}0& 0\\ 1& 0\\ 0& 0\\ 0& 1\end{array}\right]\left[\begin{array}{l}{\omega }_x\\ {\omega }_y\end{array}\right]$$

(9)

The sampling period is designated as T for dispersion. To illustrate the discrepancy between the actual trajectory and the intended design, it is recommended that an additional BHA be drilled vertically along the reference trajectory, as shown in Figure 3. Every point in the actual trajectory possesses a corresponding true vertical depth (TVD) h together with its relevant reference. The error system in discrete time concerning the reference trajectory can be articulated as (4), given that the reference kinematics model corresponds to Figure 3.

$$\begin{array}{c}\left[\begin{array}{c}{\hat{S}}_{ex}(l+1)\\ {\hat{\alpha }}_{ex}(l+1)\\ {\hat{S}}_{ey}(l+1)\\ {\hat{\alpha }}_{ey}(l+1)\end{array}\right]=\left[\begin{array}{cccc}1& \dot{S}T& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& \dot{S}T\\ 0& 0& 0& 1\end{array}\right]\left[\begin{array}{c}{\hat{S}}_{ex}\left(l\right)\\ {\hat{\alpha }}_{ex}\left(l\right)\\ {\hat{S}}_{ey}\left(l\right)\\ {\hat{\alpha }}_{ey}\left(l\right)\end{array}\right]\\ +\left[\begin{array}{ll}0& 0\\ T& 0\\ 0& 0\\ 0& T\end{array}\right]\left[\begin{array}{c}{\omega }_{ex}\left(l\right)\\ {\omega }_{ey}\left(l\right)\end{array}\right]\end{array}$$

(10)

$\dot{S}T$ illustrates that a drill pipe is a length equal to the difference between the values of $L_{\mathrm{pipe}}$ signifies that the interval between $l$ and $l+1$ corresponds to the length of a drill pipe ${\hat{S}}_{ex}\left(l\right)={\hat{S}}_x\left(l\right)-S_{rx}\left(l\right)$ and ${\hat{S}}_{ey}\left(l\right)={\hat{S}}_y\left(l\right)-S_{ry}\left(l\right)$; these denote the estimated deviations of the BHA along the x-axis and y-axis, respectively, with relation to the reference trajectory $l.{\hat{a}}_{ex}\left(l\right)={\hat{a}}_x\left(l\right)-a_{rx}\left(l\right)$ and ${\hat{a}}_{ey}\left(l\right)={\hat{a}}_y\left(l\right)-a_{ry}\left(l\right)$are the projections of $\hat{\alpha }$ onto the XOZ and YOZ planes, respectively, with relation to the reference trajectory at $l$. The control inputs are defined as ${\omega }_{ex}\left(l\right)={\omega }_x\left(l\right)-{\omega }_{rx}\left(l\right)$ and ${\omega }_{ey}\left(l\right)={\omega }_y\left(l\right)-{\omega }_{ry}\left(l\right)$. In the context of vertical drilling, $S_{rx}\left(l\right),S_{ry}\left(l\right),a_{rx}\left(l\right),a_{ry}\left(l\right)$and ${\omega }_{rx}\left(l\right),{\omega }_{ry}\left(l\right)$are all equal to zero. Utilizing the previously mentioned model (4), whereby p denotes the predictive horizon and c signifies the control horizon, the equation for predicting the Model Predictive Controller can be formulated as follows: ($\hat{Y}\left(l\right)={\mathsf{\Xi }}_k\hat{x}(l\mid l)+{\mathsf{\Theta }}_{k}J\left(l\right)$).

where $J\left(l\right)$ denotes the overall control signal about the reference steering ratio, $\hat{Y}\left(l\right)$ signifies an incremental state variable about the elements of the proposed trajectory, and the parameter matrices can be expressed as follows:

$$\begin{array}{rr}& \hat{Y}(l)=\left[\begin{array}{c}{\hat{S}}_{ex}(l+1\mid l)\\ {\hat{a}}_{ex}(k+1\mid k)\\ {\hat{S}}_{ey}(l+1\mid l)\\ {\hat{a}}_{ey}(l+1\mid l)\\ \dots \\ {\hat{S}}_{ex}(k+p\mid k)\\ {\hat{a}}_{ex}(l+p\mid l)\\ {\hat{S}}_{ey}(l+p\mid l)\\ {\hat{a}}_{ey}(l+p\mid l)\end{array}\right]W(l)=\left[\begin{array}{c}{\omega }_{ex}(k\mid l)\\ {\omega }_{ey}(l\mid l)\\ {\omega }_{ex}(l+1\mid l)\\ {\omega }_{ey}(l+1\mid l)\\ \dots \\ {\omega }_{ex}(k+c\mid k)\\ {\omega }_{ey}(k+c\mid k)\end{array}\right]\\ & \hat{x}(l\mid l)={\left[\begin{array}{llll}{\hat{S}}_{ex}(l\mid l)& {\hat{a}}_{ex}(l\mid l)& {\hat{S}}_{ey}(l\mid l)& {\hat{a}}_{ey}(l\mid l)\end{array}\right]}^{\mathrm{T}}\end{array}$$

(11)

${\hat{S}}_{ex}\left(l+1∣l\right)$ indicates the planned amount at the time $(l+1)$ derived from the estimated value at $l$- time, equivalent to the coefficient matrices ${\mathsf{\Xi }}_l$ and ${\mathsf{\Theta }}_l$ that can be obtained from Model Predictive Control (MPC) theory. They apply to the variables $p$ and $c$. To achieve optimal control performance, $p$ and $c$ should be configured within the range of $2\sim 5$ due to measurement noise. According to the estimation equation, the MPC objective function can be formulated to reduce the gap between the actual and intended trajectories while maximizing the control signal, as follows:

$$\begin{array}{l}\mathrm{m}\mathrm{i}\mathrm{n}J\left(\hat{Y}\right(l),J(l\left)\right)=\hat{Y}\left(l{)}^{\mathrm{T}}Q\hat{Y}\right(l)+J(l{)}^{\mathrm{T}}RW\left(l\right)\\ \mathrm{s}.\mathrm{t}.\left\{\begin{array}{c}\Vert \hat{\alpha }(l+n)\Vert \le {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\\ ‖r{\omega }_{SR}(l+n)‖\le r\\ n=1,\dots ,p\end{array}\right.\end{array}$$

(12)

During vertical drilling, the inclination angle can easily be reached due to measurement noise; a fixed constraint indicates no feasible method to resolve the MPC optimization problem. To address this difficulty, the following limitation is employed:

$$\left\{\begin{array}{l}\Vert \hat{\alpha }(k+n)\Vert \le {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\left(\mathrm{if}\Vert \hat{\alpha }\left(l\right)\Vert \le {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\\ \Vert \hat{\alpha }(k+n)\Vert \le \hat{\alpha }\left(l\right)\left(\mathrm{if}\Vert \hat{\alpha }\left(l\right)\Vert >{\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}\right)\\ n=1,\dots ,p\end{array}\right.$$

(13)

$\alpha$ remains small, $\hat{\alpha }$about $\sqrt{{\hat{\alpha }}_x^2+{\hat{\alpha }}_y^2}$. Changing into an incremental form provides the deviation correction optimization problem:

$$\begin{array}{rr}& \mathrm{m}\mathrm{i}\mathrm{n}P\left(\hat{Y}\right(l),J(k\left)\right)=\hat{Y}\left(l{)}^{\mathrm{T}}Q\hat{Y}\right(l)+J(k{)}^{\mathrm{T}}RJ\left(l\right)\end{array}$$

(14)

In view of the vertical drilling requirements, the MPC can either decrease the inclination angle when the angle $\alpha$ reaches to the maximum angle ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ or it can attempt to minimize the directional deviation when the angle is below $\alpha$ is less than ${\alpha }_{\max }.$In this study, the weight vector $Q=\left[q_{S,x}{q}_{\alpha ,x}{q}_{S,y}{q}_{\alpha ,y}\right]$ is used, and the priority of the MPC is rearranged mainly through the adjustment of $q_{\alpha ,x}$ and $q_{\alpha ,y}$ coefficients depending on the angle of inclination at the measurement point. As a measure of the connection between weights $q_{\alpha ,x}$ and $q_{\alpha ,y}$. This is because it should be used for three reasons.

1. Firstly, it has been performed with the sigmoid, which has been useful in showing inclination angle changes at a given scope.

2. Second, its parameters are simple and can be commonly understood and applied in practice; therefore, their application is common.

3. And finally, the rate of change is controllable via the sigmoid and, consequently, the weights of $q_{\alpha ,x}$ and $q_{\alpha ,y}$are dynamically managed on the basis of the inferences of the inclination angle.

The sigmoid function is written as follows:

$$f\left(x\right)={\displaystyle \frac{b_Q}{1+e^{-a_Q\left(\hat{\alpha }-c_Q\right)}}}+d_Q$$

(15)

4.3.2. Incorporating Real-World Errors

Measurement Tool Alignment Errors

The effect of misalignment is to introduce bias in inclination and azimuth measurements

$$\bar{\alpha }=\alpha +v_{\alpha }+b_{\alpha },\bar{\beta }=\beta +v_{\beta }+b_{\beta }$$

(16)

where $b_{\alpha }$ and $b_{\beta }$ are alignment biases (e.g., constant or depth-dependent).

Local Gravity Variations

The gravity anomalies affect accelerometer readings, leading to inclination errors.

$${\alpha }_{\mathrm{measured}}=\alpha +\Delta g\cdot \mathrm{s}\mathrm{i}\mathrm{n}\left(\theta \right),\theta =\mathrm{angle}\mathrm{between}\mathrm{tool}\mathrm{and}\mathrm{gravity}\mathrm{vector}.$$

where $\Delta g$ is the local gravity deviation.

Magnetic Field Bias

Magnetic interference distorts azimuth measurements.

$${\beta }_{\mathrm{measured}}=\beta +\Delta B\cdot \mathrm{c}\mathrm{o}\mathrm{s}\left(\varphi \right),\varphi =\mathrm{angle}\mathrm{between}\mathrm{tool}\mathrm{and}\mathrm{magnetic}\mathrm{field}.$$

where $\Delta B$ is the magnetic bias (e.g., from nearby casing or formations).

4.3.3. The Modified MPC

State-space update: extend the state vector to include bias terms:

$$\mathbf{x}={\left[\begin{array}{llllll}{S}_x& {\alpha }_x& S_y& {\alpha }_y& b_{\alpha }& b_{\beta }\end{array}\right]}^T$$

Measurement model: update (1) to include biases:

$$\left[\begin{array}{l}{\bar{\alpha }}_x\\ {\bar{\alpha }}_y\end{array}\right]=\left[\begin{array}{l}\mathrm{t}\mathrm{a}\mathrm{n}\left(\alpha +b_{\alpha }\right)\mathrm{s}\mathrm{i}\mathrm{n}\left(\beta +b_{\beta }\right)\\ \mathrm{t}\mathrm{a}\mathrm{n}\left(\alpha +b_{\alpha }\right)\mathrm{c}\mathrm{o}\mathrm{s}\left(\beta +b_{\beta }\right)\end{array}\right]+\mathbf{v}$$

(17)

Process noise: augment ${\mu }_x,{\mu }_y$ to include bias dynamics:

$${\dot{b}}_{\alpha }=w_{\alpha },{\dot{b}}_{\beta }=w_{\beta }$$

where $w_{\alpha }\mathrm{and}{w}_{\beta }$ are small Gaussian noises.

Implementation steps:

Filter Enhancement

Use DBOPF to estimate biases $b_{\alpha },b_{\beta }$ alongside states.

Modify Algorithm 1 to include bias terms in particle initialization and updates.

MPC Constraints

Add bounds for estimated biases to the constraint set in (12):

$$\left|b_{\alpha }\right|\le b_{\alpha ,\mathrm{m}\mathrm{a}\mathrm{x}},\left|b_{\beta }\right|\le b_{\beta ,\mathrm{m}\mathrm{a}\mathrm{x}}$$

(18)

Sigmoid Weight Adjustment:

Extend (15) to penalize bias-induced deviations when $\alpha$ nears ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$:

$$q_{\alpha ,x}=f\left(\alpha \right)+\lambda \left|b_{\alpha }\right|,q_{\alpha ,y}=f\left(\alpha \right)+\lambda \left|b_{\beta }\right|$$

(19)

where $\lambda$ scales bias impact.

These variables have mean values in the sigmoid functions, which are plotted in Figure 6. In this case, $b_Q$ is the value of $q_{\alpha ,x},q_{\alpha ,y}$ when the inclination is $\alpha \ll {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, and $b_Q+d_Q$ is the magnitude of $q_{\alpha ,x},q_{\alpha ,y}$ when the inclination is $\alpha \gg {\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}.c_Q$ is the turning point, and it must be adjacent to ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}.a_{\mathrm{Q}}$ defines the evolution rate of the weights $q_{\alpha ,x}$ and $q_{\alpha ,y}$. When inclination $\alpha$ is around ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, it is determined according to the drilling requirement.

Point 4.

The study presents a new method of flexible restriction with adjustable weight to improve the control system’s practicality when the inclination angle approaches its boundary. The optimization problem in Model Predictive Control (MPC) receives feasible solutions through relaxed inclination angle constraints, while the sigmoid function adjusts the weighting factor $Q$ based on inclination angle limits. The DBOPF and MPC act independently, i.e., there is no impact of the amount of particles being used in the filter on the MPC. The approximate computational performance of the MPC is preserved because the approximations provided using the flexible constraints and the use of the sigmoid function provide a simple but efficient mechanism to enhance the performance of the controller. The implemented modifications maintain the same computational complexity level as a standard MPC system. The total time needed to run one control cycle remains below 0.25 s. The actual control operations combine the reference control commands with the suggested control instructions because the control outcomes match the reference commands. The initial values ${\omega }_{ex}\left(l\right)$ and ${\omega }_{ey}\left(l\right)$ of the revised calculation sequence $J\left(l\right)$ serve as actual control increments according to Model Predictive Control principles. The final control instructions can be expressed as follows.

$$\left\{\begin{array}{l}{\omega }_{SR}={\displaystyle \frac{\sqrt{{\left({\omega }_{rx}+{\omega }_{ex}\right)}^2+{\left({\omega }_{ry}+{\omega }_{ey}\right)}^2}}{r}}\\ {\tilde{\theta }}_{tf}=\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{t}\mathrm{a}\mathrm{n}\left({\displaystyle \frac{{\omega }_{rx}+{\omega }_{ex}}{{\omega }_{ry}+{\omega }_{ey}}}\right)\end{array}\right.$$

(20)

with respect to stability, let $P^{*}\left(\hat{Y}\left(l\right),J\left(l\right)\right)$ denote the optimal solution of $P\left(\hat{Y}\left(l\right),J\left(l\right)\right)$ at time step k. It can be observed that. $P^{*}\left(\hat{Y}\right(l),J(l\left)\right)\ge 0$ is true if and only if $\hat{Y}\left(l\right)$ and $W\left(l\right)$ are both zero or if $x(l\mid l)$ is equal to zero. Therefore the $P^{*}\left(\hat{Y}\left(l\right),J\left(l\right)\right)$ can be chosen as a Lyapunov function of the control system. When the fluctuations ${\mu }_x$ or ${\mu }_y$are assumed to be zero, the particle filter’s expected error is also zero due to the properties of the filter. This means that in the presence of disturbances, $\hat{Y}\left(l\right)=Y\left(l\right)$, and the Lyapunov function can be written as $P^{*}\left(Y\right(l),W(l\left)\right)$. Therefore, the control system can be simplified as an MPC control system as presented in this study. Therefore, it is possible to state that the stability of the system can be guaranteed if the MPC control system is considered, and this fact can be referred to the existing literature [30].

5. Discussion of Results

This section presents the analysis of simulations conducted to verify and correct deviations. The effectiveness of the required dung parameters for beetles includes particle-derived filters based on real-life drilling operations and recent research, with certain parameters updated to facilitate the simulation process using the parameters in

Table 1. Parameters of simulation.

Parameter	Description
$\dot{S}$	$30\mathrm{m}/\mathrm{h}$
$T$	0.3 h
$r$	$6^{\circ }/30\mathrm{m}$
${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$	$3^{\circ }$
$p,c$	6
$R$	[40,000,40,000]
$Q$	$\left[0.1,{\displaystyle \frac{1000}{1+e^{-5(\dot{\alpha }-3)}}}+6,0.1,{\displaystyle \frac{1000}{\left.1+e^{-5(\dot{\alpha }-3)}+5\right]}}\right.$
Monte Carlo	1000 iteration, the convergence criterion was set to a maximum difference of 0.01
Dung beetle optimizer	100 particles, 500 iterations, coefficients of $\alpha$= 0.5 and b = 0.3, process noise was adjusted to 0.05

.

5.1. Analysis of Filtering

With the help of this, they tested the dung beetle optimizer particle filter performance based on the trajectory extension model. In this analysis, the given measurement noise was assumed to follow a Gaussian $v_{\alpha ,xk},v_{\alpha ,yk}\sim G\left(0,0.48\right)$; the process noise is assumed to be a Gamma distribution. The means were provided as follows: ${\mu }_{x,k}=0.5 *$ $\left({\mathsf{\Gamma }}_x-5\right)/5,{\mu }_{y,k}=0.5*{\displaystyle \frac{\left({\mathsf{\Gamma }}_y-5\right)}{5}}$, where ${\mathsf{\Gamma }}_x\sim \mathsf{\Gamma }(4,3)$ and ${\mathsf{\Gamma }}_y\sim \mathsf{\Gamma }(4,3)$. The error of measurement was about 1.5 deg and the maximum error was also about 4 deg in a distance of 30 m.

Table 2. Errors of filtering.

Filter	${\mathit{\alpha }}_{\mathit{x}}$		${\mathit{\alpha }}_{\mathit{y}}$
	Mean Absolute Error	Root Mean Square Error	Mean Absolute Error	Root Mean Square Error
DBO	0.26353	0.33564	0.27816	0.35095
PSO	0.35853	0.43881	0.36174	0.42351
SSA	0.36541	0.44032	0.35976	0.43674
EKPF	0.45944	0.47738	0.44664	0.48287
EKF	0.73652	0.92605	1.00715	1.2682

is a summary of 150 Monte Carlo simulations.

As illustrated in Figure 7 and Table 2, the dung beetle optimizer particle filter (DBOPF) demonstrates superior performance in vertical drilling applications compared to other filtering techniques. In contrast, the particle swarm optimizer particle filter (PSOPF) and the sparrow search algorithm particle filter (SSAPF) exhibit comparable performance, each with an MAE of 0.36 and an RMSE of 0.43. The extended Kalman particle filter (EKPF) shows diminished accuracy, with an MAE of 0.44 and an RMSE of 0.48, while the Extended Kalman Filter (EKF) is the least effective, yielding an MAE of 0.73 and an RMSE of 0.92. The poor performance of the EKF is primarily attributed to divergence issues arising from linearization errors and inaccuracies in modeling the noise using a Gamma distribution. Moreover, the EKPF, despite incorporating nonlinear updates, tends to disregard historical data at critical sampling moments and fails under noisy measurement conditions.

The comparative analysis between the DBOPF and the Bayesian estimation approach, presented in Figure 8 and

Table 3. Mean absolute error with $v_{\alpha ,x}\sim \mathrm{G}(0,0.48).$

$$\mathbf{Contrast}\mathbf{of}{\mathit{\mu }}_{\mathit{x}}$$	DBOPF	Bayesian Method (Kalman Filter)	$${\stackrel{\mathit{‾}}{\mathit{\alpha }}}_{\mathit{x}}$$
$0.5$ * $\left({\mathsf{\Gamma }}_x-5\right)/5$	0.32822	0.38354	0.54774
$0.3$ * $\left({\mathsf{\Gamma }}_x-5\right)/5$	0.26789	0.33776	0.56223
$0.2$ * $\left({\mathsf{\Gamma }}_x-5\right)/5$	0.21055	0.23565	0.53594
$0.1$ * $\left({\mathsf{\Gamma }}_x-5\right)/5$	0.13326	0.15849	0.54381

and

Table 4. Mean absolute error of filtering with ${\mu }_x$ = $0.5$ * $\left({\mathsf{\Gamma }}_x-5\right)/5$.

$\mathbf{Contrast}\mathbf{of}{\mathit{v}}_{\mathit{\alpha },\mathit{x}}$	DBOPF	Bayesian Method (Kalman Filter)	${\stackrel{\mathit{‾}}{\mathit{\alpha }}}_{\mathit{x}}$
$\mathrm{G}(0,0.48)$	0.33657	0.42846	0.54987
$\mathrm{G}(0,0.15)$	0.22094	0.25547	0.29076
$\mathrm{G}(0,0.03)$	0.13006	0.14309	0.16034
$\mathrm{G}(0,0.01)$	0.06983	0.08123	0.09134

, highlights DBOPF’s capacity to incorporate prior knowledge and historical data, thereby reducing estimation errors and minimizing the risk of divergence, particularly under non-Gaussian noise conditions. Experimental findings also reveal that reducing the process noise significantly enhances filter performance, with a maximum inclination angle error ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ MAE of 0.54. Furthermore, decreasing the measurement noise improves the accuracy of DBOPF, as evidenced by a minimal divergence in $v_{\alpha ,x}$ at 0.1. The DBOPF achieves higher particle distribution accuracy than alternative filters, diminishing the dependency on post hoc error correction. Notably, the integration of the dung beetle optimizer within DBOPF facilitates dynamic tuning of hyperparameters—such as process noise and particle count—thereby enhancing overall filtering accuracy. Collectively, these attributes underscore DBOPF’s effectiveness in delivering reliable and precise state estimation under conditions of elevated process and measurement noise, making it a suitable solution for vertical drilling applications.

5.2. Dynamic Response of Correction Technique

The results in

Table 5. Drilling data.

$\mathbf{Depth}/\mathbf{m}$	${\mathit{S}}_{\mathit{x}}/\mathbf{m}$	${\mathit{S}}_{\mathit{y}}/\mathbf{m}$	α/∘	${\mathit{S}}_{\mathit{x}}/\mathbf{m}$	${\mathit{S}}_{\mathit{y}}/\mathbf{m}$	α/∘
600	9.82	1.55	1.6	9.82	1.52	1.5
700	12.61	1.42	1.7	4.39	0.94	3.1
800	16.26	1.15	2.2	0.13	0.03	0.39

represent the experience of drilling site simulation. The displacement of 9.82 m occurs horizontally at a depth of 600 m and with an offset of the coordinate system of 1.55 m. The origin of the system is 5.52 deg callon inclination, 54.6 deg callon azimuth. Measurement noise is defined as $v_k\sim G\left(0,0.48\right)$ and has a maximum error of 1.4 deg. Meanwhile, a Gamma distribution $\left(12*{\mu }_k+5\right)\sim \mathsf{\Gamma }(3,2)$ is a maximum simulation error of 3.2 deg within a range of 30 m. What intrigues in the proposed method in Figure 9 are the interests in the positional and spatial deviation compared to MPC (in green) and MPCDBOPF (in purple). Also, when comparing the given method to rectifying the trajectory, as presented in Figure 5.

The three methods in Figure 8 show partial correction of inclination angle and position errors which result in better control precision. The inclination angle shows its smallest value at 820 m, which indicates that most of the deviation has been eliminated. The basic Model Predictive Control (MPC) method shows periodic movements between 680 and 800 m while its angle measurements exceed the specified limits because of measurement noise. The DBOPF-based method provides better control stability than the other methods because it keeps the inclination angles inside the defined limits. The basic MPC method shows significant divergence of its $S_x,S_y$ trajectory coordinates from the target path starting at 865 m, which results in decreased precision and quality. The two alternative methods show better performance than the others when maintaining their trajectory coordinates near the target path at this distance.

The proposed method and MPC with DBOPF show different steering rate values at 600 m and 635 m. The initial inclination angle reaches 5.52°, which exceeds both ${\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\omega }_{\alpha \mathrm{m}\mathrm{a}\mathrm{x}}$ and $\hat{\alpha }$ at 640 m because of major measurement noise. The steering rate ${\omega }_{SR}$ from MPC with DBOPF becomes infeasible because it exceeds 100% but the proposed method maintains rates under this threshold through its flexible constraint system. The proposed method generates a more stable path with smaller inclinations than MPC with DBOPF between 713 m and 803 m while keeping its deviations under ${\alpha }_{\max }$. The proposed method uses the dung beetle optimizer particle filter (DBOPF) to improve the precision of α and $S_x\mathrm{and}{S}_y$ while providing MPC optimization solutions when $\hat{\alpha }>{\alpha }_{\mathrm{m}\mathrm{a}\mathrm{x}}+{\omega }_{\alpha \mathrm{m}\mathrm{a}\mathrm{x}}$. The adjustable weight parameters enhance the trajectory minimization process, especially when dealing with angle restrictions.

The method shows excellent resistance to non-Gaussian noise, which makes it particularly useful for vertical drilling operations. The method achieves better estimation accuracy through its optimization of particle distribution using the dung beetle optimizer and its integration of historical data and prior information. The method successfully manages noisy measurements in unclean environments through its adaptable filter structure, which includes resampling and dynamic parameter control. The proposed method delivers precise inclination angle control and minimal deviation while maintaining low computational requirements compared to other filters, which results in efficient and effective performance in vertical drilling operations.

6. Conclusions

In this paper, a groundbreaking solution was provided to the redirection of the deviation and implemented as a process: First, the calculation is imposed on the dung beetle, the theory of particle filtering, and a better control algorithm is applied under the framework of the Model Predictive Control (MPC), also incorporating to improve the performance of vertical drilling in presence of noisy measurement. The primary branch of the outer control loop performs significant corrections, and that is the trajectory dynamics, which involves a trajectory extension model, a particle filter, the MPC, and a trajectory calculation module. The DBOPF achieves superior performance through its unique ability to combine prior information with posterior updates, which outperforms Bayesian filters for non-Gaussian noise and PSO/SSA optimizers for computational efficiency $O\left(G^2\right).$The sigmoid-weighted MPC system maintains controlled inclination angles through measurement noise. The DBOPF achieves superior measurement noise reduction performance with an MAE of 0.26 and RMSE of 0.33 compared to traditional PF and EKF methods, which produce higher errors (MAE of 0.35 for PF and 0.73 for EKF). The system requires precise trajectory estimation for vertical drilling operations. The control system receives deviation correction through the implementation of the MPC technique with the trajectory extension model. The MPC system uses adjustable constraints and variable weights to enhance system controllability when the inclination surpasses its designated boundary. The simulation results demonstrate that the dung beetle optimizer-based particle filter enhances control precision. The flexible constraint enables the MPC optimization problem solution when the inclination exceeds its allowed angle range. The adjustable weight system enables users to define their preferred control objectives. The proposed method reduces the probability of exceeding angle restrictions while maintaining a stable drilling path. Future research will concentrate on resolving trajectory extension model parameter uncertainties while developing robust control methods to enhance system control stability.