Measurement Method of Whole Mechanical Automatic Vertical Drilling Tools

Abstract

The inaccurate positioning of a plat valve is a crucial factor affecting the efficiency and safety of drilling operations. Because of the complicated structure of vertical drilling, it is difficult to obtain the actual position of a plat valve during the working process. This work effectively addresses the challenge of accurately measuring the deflection angle of a plat valve in mechanical automatic vertical drilling. Initially, a comprehensive mathematical model was proposed for determining the angle range of the plat valve, which was founded on the thrust generated by vertical drilling. Furthermore, this work proposed a precise measurement methodology for acquiring the crucial parameters of the MVDT. The actual deviation angle range for the N-MVDT and O-MVDT vertical drilling was calculated using data obtained from measurements of the tools. At the same time, the influence of TID on the performance of the MVDT was determined. Then, a simulation was carried out to compare the calculation results. From this, it was determined that the deflection angle range of the O-MVDT at a higher rotational speed was [86°, 58°], and the deflection angle range of the N-MVDT was [−14°, 14°]. By comparing the two methods, the error in the angle range was derived. The error of the N-MVDT was less than 16.67%, and the error of O-MVDT was less than 9.43%, proving that the measurement method and the mathematical model can accurately estimate the deflection angle range of a plat valve for mechanical automatic vertical drilling. The experimental and mathematical model results consistently showed that the deflection angle $\phi$ of the N-MVDT was significantly smaller than that of the O-MVDT, so the TID significantly improved the stability of the MVDT.

1. Introduction

In recent years, the demand for scientific research on drilling machinery has been on a steady rise because of the increase in drilling hole depth and the corresponding demand for supporting technologies such as ultra-high-speed drilling. Up until now, several studies have been conducted on the challenge of well trajectory control, primarily focusing on the optimization of tool design and the strategic utilization of those tools [1,2,3,4,5]. The mechanical automatic vertical drilling tool (MVDT) is widely used in drilling operations. Since the introduction of automated drilling, scholars have analyzed [6,7,8,9,10] the motion pattern and the rectification principle of the tool, as well as the variation law of the thrust force and its influencing factors [11,12,13,14]. Numerous prior investigations have demonstrated that there is friction in the internal mechanical mechanism of the drilling tool, resulting in an angular deviation in the resting position of the eccentric block subsequent to deviation. This angular deviation, denoted as the critical angle $\phi$ [15], significantly impacts the vertical drill’s rectifying efficacy.

The MVDT structure is shown in Figure 1.

Previous studies showed that friction is mainly caused by bearings and disc valves, and it was pointed out that the friction at the upper disc valve is the main factor affecting the accuracy of the eccentric system [16]. Because of the structure of drilling tools, $\phi$ cannot be eliminated, but it can be reduced. Based on the above problems, many scholars have studied the optimization of the plate valve based on the relationship between the deflection angle $\phi$ and the friction force of the plate valve. Baolin Liu [17] developed a circular plate valve to replace traditional plate valves, which reduced the pressure of drilling fluid on the plate valve working surface, effectively reducing the friction resistance moment between plate valves and improving the sensitivity of drilling tool correction. Yuanbiao Hu [18] designed an optimized design scheme for a small-diameter plate valve with a guide groove. The new plate valve had a guide groove corresponding to the piston chamber flow channel of the push block, which significantly reduced the friction resistance moment between the plate valves and improved the correction performance of the tools. For a further reduction in the deflection angle, a new type of mechanical automatic vertical drilling tool (N-MVDT) was designed by a team at the China University of Geosciences [19], which differed from the traditional mechanical automatic vertical drilling tool (O-MVDT). The difference between the two tools is shown in Figure 2a. Although the N-MVDT has a similar working principle to the O-MVDT [20], it is equipped with a torque-increasing device (TID) internally, which can be used in a closed-loop manner to more effectively reduce and improve the deviation control accuracy of drilling tools.

To further understand the performance of the vertical drill, later generations of researchers verified the degree of reduction in the plate value of the above three design patents in engineering practice. Lin Chai et al. [16] and Rulin Bai [21] studied the factors influencing the friction torque of the plate valve through simulation and theoretical derivation, thus establishing the MVDT trajectory model. Lixin Li [22] studied the relationship between the deflection angle $\phi$ and the friction force of a plate valve by combining numerical simulations and experiments. Long Zhang [23] analyzed the force of a palm with the shape of the thrust force test experiment and verified the influence of the angle of the plate valve on the thrust force. Chaoqun Ma et al. [24] optimized the structure of an adjusting device, which exhibited superior performance through a combination of simulation and empirical methods. However, owing to limitations in the current measurement techniques, a quantitative analysis of the precise optimization level within the angular range of this novel forcing-strengthening stable platform remains elusive.

In the existing body of research, numerous studies are subject to the issue of inadequate experimental verification. Regarding the measurement method of the internal deflection angle of the vertical drill casing, for the present experiment, a simplified device is used instead of the whole machine measurement. And the measurement technique ignores the influence of other parts of the whole machine on the experimental results. In past optimization designs based on drag reduction in the upper and lower plate valves, researchers found that because of the complex internal structure of the vertical drill, it was difficult to measure the deflection angle of the plate valve. They also found that it was difficult to determine the actual deflection angle before and after optimization, which hindered the progress of scientific research. Therefore, it is necessary to study its corresponding performance testing methods. Based on the above problems, many scholars have conducted relevant research on the methods used to test the plate valve of the MVDT.

Some test methods have been developed for mechanical drilling. Yaopeng Zhang [25] developed a method that measured internal pressure, push force, discharge, and response speed, as well as other parameters. The test system was suitable for evaluating the output parameters of drilling. It could not test the specific working state of the internal disc valve under the working state of drilling. Jin Wang et al. [26] proposed a theoretical model for measuring friction and designed a measuring device to study the friction characteristics of the internal mechanism of a vertical drill. In their study, the authors used an experimental bench to test the relationship between the deflection angle of the vertical drill and the length of the weighted block. However, to facilitate angle measurement, the authors removed the outer barrel of the vertical drill during the experiment. A stable platform and an inclination meter were used to verify the theoretical derivation.

The aforementioned method for acquiring parameters can solely assess the effectiveness of drilling; it fails to establish a correlation between the output performance of drilling and the operational state of the internal plate valve. Furthermore, since the sleeve exerts a certain influence on the deflection and potentially even the retraction performance of the plate valve, there persists a degree of inaccuracy when comparing this method to measurements taken by a vertical drilling machine. From a different perspective, the accurate measurement of underground drilling performance is crucial, yet its field implementation can be exceedingly costly.

Summarizing the status of the above research, the current research on this problem lacks a measurement scheme for the deflection angle of the footwall valve in the actual working state of the vertical drilling machine. At the same time, according to engineering experience, the internal flow path will also show different switching states for different deflection ranges of biased blocks. At present, there is no systematic analysis of this problem.

Therefore, this paper presents a method to measure the angle of the control performance of the internal stable platform by measuring the thrust force generated by the drilling tool in experiments. This method avoids the difficulty of internal measurement of the entire machine. The relationship between the push force and the internal angle is established, and the specific range of the internal angle in the working process is obtained. In a future study of vertical drill structure optimization, the a comparison of the method before and after optimization will be provided. At the same time, the deflection angle can also be used as the basis for designing the opening angle of vertical drilling upper arc holes. This work first introduces the working principle of drilling tools and analyzes the relationship between stable platforms and push-back force. By establishing a mathematical model, the range interval expression of the deflection angle under different connection states is obtained, which is universal for various MVDTs. To obtain the specific range of vertical drilling deviation angles for the O-MVDT and the N-MVDT considered in this article, experiments are conducted to determine the mathematical model with the specific number of times the two vertical drilling angles were connected in each cycle. Finally, the accuracy of the model is verified through simulation.

2. Methods

2.1. Measuring Object

As a result of optimization and improvement, there are many kinds of vertical drills. To verify the test method accurately, this work used a comparative analysis of multiple tools and selected two common vertical drills of typical VDS tools as test objects. The O-MVDT and the N-MVDT are two vertical drills widely used in the drilling industry. Before testing, the performance of the drills was analyzed according to their working principles. In this section, the working principle of the MVDT before and after optimization is introduced, and the direct relationship between the angle of the heavy block and the thrust in the stable platform is analyzed. A schematic diagram of the MVDT stable platform is shown in Figure 2b, and the connection states are shown in Figure 2c.

The structure of the MVDT is shown in Figure 2b. The partial block drives the upper disk valve to rotate synchronously. The upper plate valve is designed to have an arc hole (upper arc hole) opposite the offset block. When the offset block stops at the bottom side of the well, the upper arc hole stops at the theoretical hole height position. The lower plate valve rotates synchronously with the vertical drill housing and is connected to the actuator flow path and the upper arc hole through three lower arc holes. The actuator does not rotate with the drilling tool; it is mainly composed of several pushing fins and corresponding flow channels. When the flow path is open, high-pressure drilling fluid is pushed out through the flow path against the wing rib.

When the drilling tool tilts, the heavy block in the stable platform deflects under the action of gravity and finally stops at the lower side of the theoretical hole, and the upper arc hole stops at the higher side of the theoretical hole. With the rotation of the foot disk valve, the lower arc hole passes through the upper arc hole successively, and the drilling fluid flows into the channel through the upper and lower arc holes. The wing rib (LSR) near the low side is not pushed out, but the wing rib (USR) near the high side is pushed out to the wall. Then, the drilling tool is subjected to the wall reaction force, and the drilling tool attitude is corrected.

With the turn-on and turn-off caused by the rotation of the lower disc valve, the vertical drill presents two different opening states. In Figure 2c, the position relationship of the MVDT structure is represented by the projection of views B-C and B-D. When the upper arc hole is connected to the lower arc hole, this state can be recorded as the L state. After the upper arc hole is parked on the high side of the hole, the passage near the high side of the hole is also connected to the lower arc hole by the rotation of the lower arc hole. The drilling fluid enters the rotating lower arc hole through the upper arc hole and then enters the channel and pushes out against the wing rib. The state in which the lower arc hole, the upper arc hole, and the channel are connected is called state H. States H and L both occur several times during a cycle (L) of drill pipe rotation. When the channel is in the L state, part F of the corresponding fan blade of the channel presents a low thrust with multiple periodic changes, which is denoted as $F_L$ (because of the gap between the footer valve and the channel, a small amount of liquid still enters the channel under the L state, and the discharge of the actuator is delayed, so the thrust at this time is not 0). When the channel is in the H state, the F part of the sector corresponding to the channel has a higher thrust ($F_H$) and has multiple periodic changes.

Figure 3A-A presents a detailed profile of the relative position of the upper orifice, encompassing both the pink external cylinder valve and the blue internal cylinder valve. Figure 3B-B, on the other hand, provides a detailed depiction of the relative position of the lower orifice. The angular placement of the limiting slot is illustrated in Figure 3C-C. Lastly, Figure 3D-D displays the position diagram of both the upper and lower plate valves.

The bias blocks can be classified according to different motion states. When the biased block is completely stable at a certain position, the state is recorded as State 1. When the biased block is not completely stable but always moves within a certain range and does not affect the normal operation of the drill, the state is recorded as State 2. When the partial block is completely unstable, or cannot be stabilized within a certain range, it is denoted as State 3. In State 3, the drill tool cannot be calibrated normally and $\phi$ continues to change irregularly. The angle of the final stable position of the eccentric block can be expressed as ${\phi }_s$, which is called the critical deflection angle. Because of the structure of the drill, ${\phi }_s$ cannot be eliminated, it can only be reduced.

When the heavy block appears in State 1 or State 2, the LSR hardly pushes out, their $F_i$ values appear periodically, and each $F_L$ is very similar. The difference is that in State 1, the USR is pushed out almost continuously and the F of the USR appears periodically similar to $F_H$.

The TID consists of an eccentric block and an adjusting device, resulting in the N-MVDT having two eccentric blocks—the upper and lower eccentric blocks (the lower eccentric block is equivalent to the single eccentric block in the O-MVDT). For simplicity, we refer to the (lower) eccentric block as EB 1 and the upper eccentric block as EB 2. Moreover, the adjusting device mainly includes an internal cylinder valve, an external cylinder valve, and a turbine.

EB 1 is fixedly connected to the upper plate valve; they continuously rotate when the wellbore is tilted and finally stop at a certain position. EB 2 exists in the low-resistance oil chamber, so its instantaneous position is very close to the theoretical lower side of the wellbore. When EB 1 and EB 2 rotate simultaneously, the angle between them is $\phi$. When they stop simultaneously, the angle between them is ${\phi }_s$. The TID can generate an additional torque to EB 1 in real time and in a closed loop by monitoring the change in $\phi$. This additional torque balances the resistance from the upper plate valve and ultimately reduces ${\phi }_s$. Afterward, EB 1 outputs more accurate inclination information to improve the inclination accuracy of the N-MVDT.

Furthermore, EB 2 is attached to the internal cylinder valve, whereas EB 1 is securely fastened to the external cylinder valve. Consequently, $\phi$ represents the instantaneous angular deviation between these two valves. Both the internal and external cylinder valves incorporate numerous upper and lower orifices, designed strategically. As $\phi$ undergoes variation, it triggers an inverse adjustment in the flow areas of the upper and lower orifices. The upper orifices’ flow area expands while the lower orifices’ flow area contracts, and the opposite holds true. The turbine, firmly affixed to the external cylinder valve, harnesses the torque it generates and transfers it to EB 1, thereby balancing the frictional torque effectively.

When $\phi$ increases clockwise, the flux area of the upper orifices becomes larger and the flux area of the lower orifices becomes smaller. As a result, the drilling fluid flows more easily from the upper orifices, which then impacts the turbine to generate the adjusting torque [27].

After the adjusting torque is generated, it balances the friction torque on EB 1, causing $\phi$ to decrease. Once the rotating direction of the shell and the turbine blades has been determined [28], the directions of the adjusting torque and the friction torque are also determined, with the adjusting torque opposing the friction torque. As $\phi$ decreases, the flux area of the upper orifices becomes smaller and the flux area of the lower orifices becomes larger. The drilling fluid flows more easily from the lower orifices and the adjusting torque decreases. The above process is a continuous cycle. The adjusting device provides closed-loop adjusting torque to balance the friction torque on EB 1, enabling the adjustment of $\phi$ and reduction in ${\phi }_s$.

Furthermore, there is a limiting slot on the internal cylinder valve that restricts the maximum range of $\phi$. If the combined moment of the EB 1 cannot reduce $\phi$ even when it reaches the limit, EB 2 will rotate simultaneously with EB 1, and its gravity will be used to adjust $\phi$. The regulation mechanism diagram is shown in Figure 4.

The position of the steering rib is determined by EB 1, and its parked position directly affects the control accuracy of the MVDT. Therefore, the magnitude of ${\phi }_s$ can be used to evaluate the effectiveness of the MVDT. Large values of ${\phi }_s$ indicate poor effectiveness, including reduced stability and deviation control accuracy. (The position of the plate valve is determined by EB 1, and its stopping position directly affects the control accuracy of the MVDT. Therefore, the magnitude of ${\phi }_s$ can be used to evaluate the performance of the MVDT. A large value ${\phi }_s$ indicates poor performance, including reduced stability and deviation in control accuracy.)

Based on the working principle of the vertical drill described above, since many torques caused by friction affect the movement of EB 1, it is difficult to analyze the relationship between the state of the deflection angle $\phi$ and the push force F by existing methods. Because of the short time of the new vertical study, there is no theoretical model based on the above difficulties for the plate valve and flow channel, so we need to analyze the real-time state of the plate valve in the working process.

2.2. Analysis of the Operating Status of the Plate Valve

Based on the structure of the MVDT, a two-dimensional projection of the upper and lower arc holes and channel can be constructed on a plane (the N-MVDT is the same as the O-MVDT), as illustrated in Figure 5. The center of the shell serves as the origin, with each channel evenly distributed around it. The upper and lower arc holes are also rotated around the origin. An angular coordinate system is established using a straight line passing through the origin and parallel to the direction of the upper side of the wellbore as the axis. The theoretical upper side of the well bore is set at 0° (or 360°). To analyze the effect of the TID on the stability improvement in the N-MVDT quantitively, a mathematical model is established for the relationship between the state of the pushing by force F and $\phi$.

In the model, $\mathsf{\omega }$ is the angular velocity of the shell rotation, assuming it is constant. Then, the rotation period $T$ of the lower plate valve can be expressed as a function of $\omega$ and $n$:

$$T={\displaystyle \frac{2\pi }{\omega }}={\displaystyle \frac{1}{n}}$$

(1)

For channel $i$ (equivalent to steering rib $i$), assuming the position of channel $\mathrm{i}$ remains unchanged, we can define the angle between channel $\mathrm{i}$ and the theoretical upper side of the wellbore (i.e., 0°) as ${\theta }_i \left(i=1,2,3,4,5\right)$. The angle between the center line of lower arc hole $j$ and 0° can be denoted as ${\gamma }_j \left(j=1,2,3\right).$ The flow passage opening of the lower arc hole $j$ is symmetric along its centerline, the maximum opening angle can be denoted as $\beta$, and the flow passage range of the lower arc hole $j$ is $\left[{\gamma }_j-\beta /2,{\gamma }_j+\beta /2\right]$. The position of channel 1 ($i$ = 1) and lower arc hole 1 ($j$ = 1) are marked in Figure 5.

Additionally, since the center line of the upper arc hole is at $\mathsf{\phi }$ from 0° and the opening of the upper arc hole is symmetrically distributed along its centerline, the maximum opening angle of the upper arc hole can be defined as $\alpha$. Therefore, the flow range of the upper arc hole can be expressed as $\left[\phi -\alpha /2,\phi +\alpha /2\right]$.

Next, we denote time. Assuming that the connection between lower arc hole 1 and channel 1 was just established at $t$ = 0, then we have ${\theta }_1={\gamma }_1+\beta /2$, as shown in view A in Figure 5. Considering the periodicity in the rotation of each structure, let $k \left(k\in N\right)$ be the periodic coefficient. Therefore, the variation in ${\gamma }_j$ concerning $t$ can be obtained as shown in Figure 6. Their relationship can be represented by:

$${\gamma }_j\left(t\right)={\theta }_1-{\displaystyle \frac{\beta }{2}}+\omega t-{\displaystyle \frac{2\pi }{3}}\left(j-1\right)-2k\pi$$

(2)

When the lower arc hole $j$ is connected to channel $\mathrm{i}$, we have:

$${\gamma }_j\left(t\right)-{\displaystyle \frac{\beta }{2}}<{\theta }_i<{\gamma }_j\left(t\right)+{\displaystyle \frac{\beta }{2}}$$

(3)

where $t\left(i,j\right)$ resents the time when channel $i$ and lower arc hole $j$ are in status L, which is calculated using inequality (3). The connection condition at status L is illustrated in Figure 7.

$$t\left(i,j\right)\in \left[{\displaystyle \frac{2\pi }{5\omega }}\left(i-1\right)+{\displaystyle \frac{2\pi }{3\omega }}\left(j-1\right)+kT,{\displaystyle \frac{2\pi }{5\omega }}\left(i-1\right)+{\displaystyle \frac{\beta }{\omega }}+{\displaystyle \frac{2\pi }{3\omega }}\left(j-1\right)+kT\right]$$

(4)

$${\gamma }_j\left[t\left(i,j\right)\right]={\theta }_1-{\displaystyle \frac{\beta }{2}}+\omega t\left(i,j\right)-{\displaystyle \frac{2\pi }{3}}\left(j-1\right)-2k\pi$$

(5)

Moreover, for status H of channel $i$ and lower arc hole $j$, we have:

$$\phi \left[t\left(i,j\right)\right]-{\displaystyle \frac{\alpha }{2}}<{\gamma }_j\left[t\left(i,j\right)\right]<\phi \left[t\left(i,j\right)\right]+{\displaystyle \frac{\alpha }{2}}$$

(6)

The solution of (6) is:

$$\mathsf{\phi }\left[t\left(i,j\right)\right]\in \left[{\theta }_1-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{\alpha }{2}}+\omega t\left(i,j\right)-{\displaystyle \frac{2\pi }{3}}\left(j-1\right)-2k\pi ,\hspace{1em}\hspace{1em}{\theta }_1-{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{\alpha }{2}}+\omega t\left(i,j\right)-{\displaystyle \frac{2\pi }{3}}\left(j-1\right)-2k\pi \right]$$

(7)

To simplify the expression, we replace the term $\phi \left[t\left(i,j\right)\right]$ in Equation (7) with ${\lambda }_{\left(i,j,k\right)}$, which yields:

$$\phi \left[t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)}$$

(8)

For steering rib $i$ to have only one $F_H$ in a period, the following condition must be satisfied:

$$\phi \left[t\left(i,j\right)\right]=\left\{\begin{array}{c}\exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=1\\ \forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=2\\ \forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=3\end{array}\right.$$

(9)

If $F_i$ has only one $F_H$ in a period, the range of $\phi \left[t\left(i,j\right)\right]$ can be obtained by solving (9).

Conversely, if there are two $F_H$ in a period of $F_i$, the range can be obtained by solving:

$$\phi \left[t\left(i,j\right)\right]=\left\{\begin{array}{c}\exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=1\\ \exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=2\\ \forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=3\end{array}\right.$$

(10)

If there are three $F_H$ in one period of $F_i$, then it satisfies:

$$\phi \left[t\left(i,j\right)\right]=\left\{\begin{array}{c}\exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=1\\ \exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=2\\ \exists t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\in {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=3\end{array}\right.$$

(11)

If there is no $F_H$ in one period of $F_i$, then it satisfies:

$$\phi \left[t\left(i,j\right)\right]=\left\{\begin{array}{c}\forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=1\\ \forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=2\\ \forall t\left(i,j\right),\hspace{1em}\hspace{1em}\phi \left[ t\left(i,j\right)\right]\notin {\lambda }_{\left(i,j,k\right)},\hspace{1em}\hspace{1em}j=3\end{array}\right.$$

(12)

This model shows the relationship between the MVDT’s vertical drilling push performance and the deflection angle of the internal plate valve, which is also applicable to general MVDS structures. To simplify this study, the model ignores the influence of the flow path on the fluid and the delay of the push-hand motion and assumes a constant speed of the foot wall valve. These factors are also major sources of model error. However, since the solution of the model is not an exact value but rather a range of $\phi$, this error is acceptable.

It is important to note that through theoretical analysis, we can only infer Equations (9)–(12). This means that we can currently obtain the relationship between the status and range of $\phi$. To obtain the specific range of the deflection angle of the test object, we also need to obtain the specific occurrences of the two connected states in the working cycle of the two test objects. Therefore, we have to test the corresponding push force $F_H$ and $F_L$ in the two connected states.

2.3. Discussion of the Measurement Principles

An illustrative diagram of the measuring method is shown in Figure 8, which is based on the above requirements for parameter measurement. The measurement method is discussed to determine the following measurement scheme. First, an experimental machine was built to measure the occurrence of F_H and F_L in the same period of two devices as the known amount to solve the mathematical model. Second, multibody dynamics and CFD simulation were used to obtain $\phi$, ${\phi }_s$, and $t$, which were compared with the results obtained by the mathematical model. The accuracy of the measurement method was guaranteed.

2.4. Measurement Experiment

As described in Section 2.2, this work determined the connection status of the TID in actual operation and obtain the data required for a mathematical model. To obtain the specific deflection angle of the vertical drilling plate valve, we also obtained the real situation of the pushing force of two vertical drillings during a working cycle, so the experiment was designed based on vertical drilling with the O-MVDT and the N-MVDT. In this section, we describe the experimental design used to test the performance of the O-MVDT and the N-MVDT.

2.4.1. Experimental Measurement Rig

The main body of the drilling tool and the experimental machine were processed by Shandong Baohang Machinery Co., Ltd., Jinan, China. A schematic diagram of the test system and the physical diagram of the test bench is depicted in Figure 9. The test bench includes an incline adjustment system, mud circulation system, rotary drive system, and a push-force measurement and acquisition system. The inclination adjustment system can simulate the inclination by adjusting the inclination angle of the drill tool through the support platform. The inclination can be measured using an inclination meter. It includes a clinometer to measure and adjust the tilt angle during the test. To simulate the tilt angle of the drill tool when encountering an inclination within the borehole, the mud circulation system simulates the circulation of fluid inside the drill tool through a multistage centrifugal pump, aiming to simulate the down hole mud environment. The rotary drive system realizes the drill tool rotation by installing a motor on the top of the bench and adjusts the drill tool speed by a frequency converter, aiming to simulate the rotating rock breaking condition when the drill tool carries the drill bit. The push force measurement and acquisition system measure the push force of each hand when the drilling tool is tilted through the spoke sensor in contact with the palm, and it collects data through the oscilloscope. The test rig has high practicability and can test both the common vertical drilling experimental prototype with the common stable platform inside and the new vertical drilling experimental prototype with the new forced-stabilized platform inside. It is also suitable for the vertical drilling of other structures.

The O-MVDT and N-MVDT prototypes, which were introduced in Section 2.2, can be fixed on the experimental platform, as shown in Figure 10. The tools were in accordance with Chinese Industrial Standards [26]. The conventional surface accuracy achieved minimum roughness detection values below 6.3 µm. Some contact surfaces with higher accuracy, such as the end face of the plate valve, achieved minimum roughness detection values below 1.6 µm. Coordination tolerance of the key part, according to multiple design patents, such as the gap between the plate valves of 0.2 mm, was guaranteed by the PDC block between the two plate valves. The assembly tolerance of the inner tube and the thread was IT9. The tolerance in the radial and axial direction of the tool was determined by the national standards and met the functional requirements of drilling tools.

To meet the testing requirements, the measurement device performed the following functions:

The thrust measurement and acquisition system has a force sensor with an accuracy of 0.0001 F.S. The parameters of the force sensor are shown in

Table 1. Parameters of the force sensor.

Parameters	Range/Value
Capacity (N)	0–20,000 N
Accuracy (F. S.)	0.0001
Error	±1%
Environment temperature (°C)	−20–60
Service voltage (V)	10

.

The accuracy means that the resolution of the sensor is 0.0001 F.S. It also means the accuracy of the measured push force size.

When the test bed control tool tilts, it causes EB 1 and EB 2 to deflect under the action of gravity. Based on the working principle of the drilling tool described in Section 2, the biased block is connected to the curved hole corresponding to the upper and lower plate valve, and the high side of the hole palm points out. In this experiment, the actuator has 5 slaps, each of which is independently equipped with a sensor to monitor the thrust size. The number of slaps and the corresponding sensor is set as $i$, and the corresponding push force measured by the sensor is $\mathrm{F}$; thus, the push force measured by the sensor is $F_i$ ($i$ = 1, 2, 3, 4, 5). Since the circumferential position of the hand in the well is random, to simplify the experiment, the position of hand No. 1 is fixed, and the drill tool is tilted towards hand No. 1, so the position of hand No. 1 is the high side of the hole. In addition, because hands 1, 2, and 5 are near the high side of the hole, we describe them as “high hands”, while hands 3 and 4 are near the low side of the hole, so we describe them as “low hands”, as shown in Figure 11.

The design parameters of the experimental rig are shown in

Table 2. Design parameters of the experimental rig stand.

Parameters	Range/Value
Drilling inclination angle (°)	0–15
Pump pressure (MPa)	0–1.5
Motor speed (rpm)	0–300
Thrust force (N)	0–20,000

.

2.4.2. Experimental Method and Process

In the experimental measurement in this paper, the inclination angle pump pressures and rotational speeds affect ${\mathrm{F}}_{\mathrm{i}}$ and ${\phi }_s$. The experimental parameters were selected as the well inclination angle $\left(\delta \right)$, pump pressure $\left(P\right)$, and rotational speed of the shell $\left(\mathrm{n}\right)$, which were used to determine the most favorable parameters for the experiment and subsequent analysis. The influence of the three variables on the drilling performance of automatic vertical drilling were also determined. The effectiveness of the O-MVDT and the N-MVDT were evaluated using ${\mathrm{F}}_{\mathrm{i}}$. The parameter values are shown in

Table 3. Experimental operating parameters of the two prototypes.

Parameters	Value	Total Data Group	Evaluation Indicator
$\delta$ (°)	0; 1; 2; 3; 5	2 × 5 × 3 × 2 =60	$F_i$ (N)
$P$(Mpa)	0.5; 1; 1.5
$n$ (rpm)	90; 120

.

Because of the unsatisfactory reduced deviation control effect of the N-MVDT when $\mathsf{\delta }$ is low ($\mathsf{\delta }<1$), the main focus of this paper was on the degree of performance improvement in the MVDT by the TID. The range of delta in this experiment was limited to 1~5°. The weight on the bit was selected as 0.5~1.5 MPa, according to the actual working condition.

Because of the large volume of data, the reliability of the experiment was validated using a subset of the data. This is a commonly used metric in practical applications since it is affected by both the pressure difference between the two sides of the steering ribs ($\Delta {\mathrm{p}}_{\mathrm{i}}$) and the effective contact area of the steering ribs.

$$\overline{F_i}=s\Delta {\overline{p}}_i$$

(13)

2.4.3. Reliability of the Experiment

This experiment was designed to be conducted above ground, following the testing standards and operational procedures outlined [29]. As there was no wellbore or rock present, the vibrations caused by the steering ribs pushing against the wellbore and the cutting of the rock by the drill bit affected the experiment. Additionally, as the sensors and steering ribs were fixed, the influence of the pushing angle of the steering ribs on the pushing force was ignored.

It should be noted that there may be experimental errors due to the absence of rock debris in the circulating fluid medium and the inability of the above-ground experimental environment to simulate the effect of temperature on fluid flow as it exists in the downhole. Moreover, experimental errors can also arise from the rusting of parts that are exposed to the fluid environment.

Since $\overline{F_i}$ varies with time, in order to study the variation in $\overline{F_i}$ more clearly, we defined $\overline{F_i}$ as the average value of $\overline{F_i}$. $\overline{F_i}$ represents the equivalent pushing force that can be generated by each steering rib against the well wall.

2.5. Numerical Simulation

2.5.1. Numerical Simulation of Torque

Because of the difficulty in measuring the specific value of the disc valve deflection angle in the experiment, simulation auxiliary experiments were conducted in this section to jointly verify the accuracy of the mathematical model. Firstly, the fluid working condition was simulated by the CFD method, and $\phi$ was taken as the independent variable to obtain the curve of M changing with $\phi$, which was briefly denoted as $\mathrm{M}\left(\phi \right)$. Then, the working process of the regulating device under the action of fluid was studied. For the double-biased block in the N-MVDT, its motion was simulated by multibody dynamic simulation. It was input as an influencing factor to determine the influence of adding $M$ on $\phi$, and then the ranges of two kinds of vertical drilling deflection angles $\mathsf{\phi }$ were obtained.

2.5.2. Control Equation

A continuous equation was applied in this work based on the finite volume method. The inner flow field of the tool, which is three-dimensional incompressible flow, was described by the Navier–Stokes equation. The turbulent model adopted the k-ε turbulence model, which modifies turbulent viscosity considering the average flow of rotation and rotation flow and can better handle the high strain rate and degree of streamline curvature flow.

In the standard k-ε turbulence model, k and ε are two basic unknowns, and the corresponding transport equation is

$${\displaystyle \frac{\partial \rho }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_i}{{\delta }_k}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+G_k+G_b-\rho \epsilon -Y_m+S_k$$

(14)

$${\displaystyle \frac{\partial \rho }{\partial t}}\left(\rho k\right)+{\displaystyle \frac{\partial }{\partial x_i}}\left(\rho k{u}_i\right)={\displaystyle \frac{\partial }{\partial x_j}}\left[\left(\mu +{\displaystyle \frac{{\mu }_i}{{\sigma }_{\epsilon }}}\right){\displaystyle \frac{\partial k}{\partial x_j}}\right]+C_{1\epsilon }{\displaystyle \frac{\epsilon }{k}}\left(G_k+G_{3\epsilon }{G}_b\right)-C_{2\epsilon }\rho {\displaystyle \frac{{\epsilon }^2}{k}}+S_{\epsilon }$$

(15)

where $G_k$ is the generation term of turbulent kinetic energy k caused by the average velocity gradient, is the generation term of turbulent kinetic energy k caused by buoyancy, is the contribution of pulsation expansion in compressible turbulent pulsation, $C_{1\mathsf{\epsilon }}$, $C_{2\mathsf{\epsilon }},$ and $C_{1\mathsf{\epsilon }}$ are empirical constants, ${\sigma }_k$ and ${\sigma }_{\epsilon }$ are the Prandtl numbers corresponding to turbulent kinetic energy k and dissipation rate ε, respectively, and $S_k$ and $S_{\epsilon }$ are user-defined source terms.

2.5.3. Calculation Model

To investigate the impact of the newly developed enhanced stability platform, it was imperative to initially simulate the regulating device using computational fluid dynamics (CFD). The three-dimensional model of the regulating device was established, which mainly included the inner cylinder valve, the outer cylinder valve, and the turbine. Then, the fluid domain model of the 3-D model of the regulating device was inversely calculated, and this model was used as the simulation object of the CFD simulation. The process is shown in Figure 12.

2.5.4. Boundary Condition Settings

In the computing module, following grid division, the boundary conditions for the fluid domain of the regulator were established. These conditions and parameters were selected based on similar research studies. Specifically, the acceleration of gravity was set to 9.8 m/s², and the realizable turbulence model in k-ε was selected. To simplify this study and enable a comparison with the experiment, pure water (water liquid) was selected as the fluid medium. Since the pump displacement in geological drilling is generally small, the inlet was set as the flow inlet, and the flow value in the initial simulation was 8 kg/s. The dynamic mesh was set to smoothing, the diffusion parameter was set to 1.5, and the mesh cell height was set to 0.002. The torque exerted on the turbine wall was monitored when $\mathsf{\phi }$ changed from the maximum value to the minimum value. The other major boundary conditions are shown in

Table 4. CFD simulation of the main boundary conditions.

Boundary Conditions	Value
The scope of φ (°)	[−15, 15]
Flow inlet (kg/s)	8
Pressure outlet (MPa)	0
Fluid density (kg/m3)	998.2
Hydrodynamic viscosity (N·s/m2)	0.013

. After the settings were complete, all zones were initialized (all zones) and a transient simulation was performed in a 0.01 s time step for a total of 40 steps, as shown in Table 3.

2.5.5. Grid Independence Test

The fluid domain model of the regulator was imported into ANSYS fluent 18.0, and the mesh division operation was carried out to obtain the finite element model of the fluid domain of the regulator, as shown in Figure 13. The size of the global mesh was selected as 1mm. Considering that the turbine wall involves a curved surface, the mesh near it had higher quality requirements, so the mesh near the turbine wall was added with a curved surface function to generate a dense mesh. The total number of statistical grids was 4, 221, 547. Since the grid density has a great influence on the accuracy of numerical simulation, the grid independence of the calculation area was analyzed in this work and the torque at 1 wall at the deflection angle $\phi =-3°$ was used as the index for the evaluation of grid independence, as shown in Figure 14. When the grid was increased to 4.2 million, the torque fluctuated steadily with the increase in the number of grids, and the change range was less than 1%, which met the requirement of grid independence.

2.5.6. Multibody Dynamic Simulation Model of the Partial Block

For a stable platform, $\phi$ changes instantaneously with time under the action of multiple moments in EB 1, and EB 1 would stabilize at ${\phi }_{\mathrm{s}}$ after the stable time $t_p$. These three parameters ($t_p$, ${\mathsf{\phi }}_{\mathrm{s}},$ and the range of $\phi$) were obtained by multibody dynamics simulation [30] and were used to evaluate the results of the simulation. The stability of EB 1 was better when $t_p$ was shorter and the range of $\phi$ was smaller. The stability was also reflected when ${\phi }_s$ was closer to zero.

As there was only one eccentric block in the O-MVDT and two eccentric blocks in the N-MVDT, the single eccentric block 3-D model multibody dynamics simulation was simplified as the SEBD simulation, and the double eccentric block 3-D model multibody dynamics simulation was denoted as the DEBD simulation for brevity. The simulation parameters are shown in

Table 5. Relevant simulation parameters.

Parameter	Value	Parameter	Value
Adjusting torque (N)	$M_I$	Damping factor	500
Inclination angles (°)	5	Penetration depth (mm)	$1.0\times {10}^{-9}$
Angular speed of lower plate valve (rad/s)	720	Coefficient of static friction	0.05
Thrust of upper plate valve (N)	5200	Coefficient of kinetic friction	0.03
Rigidity	$1.0\times {10}^5$	Static translational speed (m/s)	$1.0\times {10}^{-6}$
Eccentric block radius (mm)	34	Frictional translational speed (m/s)	$5.0\times {10}^{-6}$
Eccentric block length (mm)	950	Material of plate valve	45 steel
Material of infilling of eccentric block	Plastics	Material of outermost shell of eccentric block	40 CrMo

.

3. Results and Discussion

3.1. Factors Influencing the Performance of Thrust

In this section, the effects of $\delta$, $n$, and $P$ on push force are analyzed based on the experimental results. The parameters that make the push force output more stable are selected and used to verify the accuracy of the measurement method in this paper.

We set the weight of the bit to 1 MPa and the rotational speed $n$= 120 rpm to test the push force output under different inclination angles. Because of the unsatisfactory reduced deviation control effect of the N-MVDT when $\delta$ is low ($\delta <1$), and because the main focus of this paper is on the degree of performance improvement in the MVDT by the TID, the range of $\delta$ in this experiment was limited to 1~5°. In the tool inclination correlation analysis $\overline{F_i}$, the average value of the output pushing force of each hand was used. The experimental result is shown in Figure 15.

According to Figure 15, when $\delta$ is small ($\mathsf{\delta }$ ≤ 1), the difference between $\overline{F_i}$ in the respective prototypes is similar. However, when $\delta$ is larger ($\mathsf{\delta }$ ≥ 3), the $\overline{F_i}$ values of the USR are similar and significantly larger than the $\overline{F_i}$ values of LSR in the respective prototypes. This indicates that the O-MVDT and the N-MVDT effectively reduce the inclination of the wellbore when $\delta$ is large. Therefore, the reliability of the corrective function of the O-MVDT and the N-MVDT was preliminarily confirmed.

Furthermore, for $\delta$ = 2, there is no notable disparity in the $\overline{F_i}$ value of the O-MVDT, whereas the $\overline{F_{\mathrm{i}}}$ value of the LSR in the N-MVDT is marginally lower than the corresponding value of the USR. This suggests that the N-MVDT has a superior capability to reduce the wellbore inclination than the O-MVDT at $\delta$ = 2.

It should be noted that other MVDT structures can still maintain their normal inclination deviation correction function even at very low values of $\delta$. However, for the N-MVDT, the inclination deviation function is gradually lost when $\delta <2$. Since this paper focuses on studying the improvement effect of the TID on the N-MVDT, experimental data with $\delta$ > 2 were selected to compare the inclination deviation effects between the O-MVDT and the N-MVDT in subsequent studies.

Additionally, the $\overline{F_i}$ values observed in the O-MVDT are marginally higher than their counterparts in the N-MVDT for all $\overline{F_i}$ values. This discrepancy could potentially stem from the N-MVDT’s intricate design, which encompasses multiple elements like the adjusting device. These components contribute to a heightened pressure drop in the drilling fluid, as stated in [31], subsequently resulting in a decreased inlet pressure within the N-MVDT’s channel.

The speed was set to 120 rpm and the inclination was 3°. The push force under three different sets of bit weights was obtained through the experiment based on the principles of the executing agency described in Section 2. After increasing the weight on the bit, since the magnitude of $\overline{F_i}$ is dependent on the pressure difference between the inside and outside of the drilling tool [32], a larger $P$ results in a larger $\overline{F_i}$. According to the data in Figure 16, the $\overline{F_i}$ of the N-MVDT is slightly lower than that of the O-MVDT. Since the difference between the high and low edge $\overline{F_i}$ values is more obvious at a higher bit weight, P = 1.5 MPa was selected as the experimental parameter for use in the subsequent experiment.

When comparing the push force alterations at the well’s upper and lower sides, analyzing the variations in the upper side proves to be more straightforward. Figure 17 illustrates the output curve of the tested drill tool’s high-side hand push force at varying speeds. Notably, the O-MVDT’s force sensor, denoted as $F_i$, underwent two and three complete cycles at rotational speeds of 90 and 120, respectively, indicating a direct correlation between the number of cycles and the increase in speed. However, at slower rotational speeds, EB 1 encountered static friction from the upper plate valve, causing it to rotate in unison with the shell, which poses a potential issue. Consequently, to streamline the subsequent analyses, a rotational speed of $n$= 120 was chosen as the preferred option.

The above regularities are consistent with the experimental data, indicating that the fundamental design objectives were achieved for both prototypes and confirming the reliability of the experimental data. To visually compare the stability differences between the O-MVDT and the N-MVDT, the operating conditions of $\delta$ = 3, $n$ = 120, and $P$ = 1.5 were selected for the experimental parameter.

3.2. The Difference in Performance of the O-MVDT and the N-MVDT

According to the experimental parameters selected in Section 3.1, the push-back performance of the two kinds of vertical drills was tested, and the output curves of the force of the two kinds of vertical drill was obtained under the selection of $\delta$ = 3, $n$ = 120, and $P$ = 1.5. Figure 18(a1,a2) show $F_i$ in the O-MVDT and the N-MVDT, respectively. The graphs reveal that the $F_i$ values of the LSR are similar and lower than the ${\mathrm{F}}_{\mathrm{i}}$ values of the USR in both prototypes, and the data fluctuations are regular. This indicates that both prototypes were in a relatively stable condition during the measurement.

It should be noted that when the drilling tools were tilted, the upper arc hole should have remained stable at the upper side of the well bore. While there should be no drilling fluid passing through the channel of the LSR, the $F_i$ values of the LSR in both prototypes were not zero, as depicted in Figure 18(a1,a2). This is due to the small gap between the two plate valves, which allows for the drilling fluid to enter the channel of the LSR [33], leading to non-zero $F_i$ readings.

The three waves of $F_i$ may exhibit different patterns in a single period, as depicted in Figure 18(c1,c2). When a “rising–falling” or “rising–stable–falling” wave is more prominent within a period, it is referred to as a higher pushing force ($F_H$), while considerably smaller waves are referred to as a lower pushing force ($F_L$). The distinction between $F_H$ and $F_L$ is evident in Figure 18(c1,c2). When a channel, an upper arc hole, and one of the lower arc holes are all connected, this connection status is referred to as status H. During status H, $F_H$ starts to rise as the drilling fluid flows into the channel, and it remains high for the duration of this status. $F_H$ falls when the connection is not in status H. $F_L$ occurs during status L when the drilling fluid flows into the channel through the gap between the upper and lower plate valves, even when the upper arc hole is not connected. At this moment, $F_i$ will slightly increase, resulting in the appearance of $F_L$. Moreover, it is expected that the upper plate valve and EB 1 will be stabilized in their final positions. In this scenario, if $F_H$ occurs, there should be three identical ${\mathrm{F}}_{\mathrm{H}}$ in one period. However, in reality, $F_H$ and $F_L$ exist simultaneously in one period, indicating that EB 1 is not completely static.

The analysis above indicates that the function of EB 1 for detecting well inclination remains effective even in a partially static condition. For simplicity, we define State 1 as when EB 1 or upper plate valve is completely static, State 2 as when EB 1 is not completely static but does not affect the inclination correction of the drilling tool, and State 3 as when EB 1 is completely unstable.

In State 1, $F_i$ will display three identical waves during each period. In State 2, the upper plate valve will not be stationary and will oscillate within a small range because of the slight rotation. Consequently, both status L and status H will occur simultaneously during one period for a given channel, resulting in the presence of both $F_H$ and $F_L$.

The above law applies to both the O-MVDT and the N-MVDT, but there are significant differences between the two prototypes in terms of $F_i$. The magnitude of ${\mathrm{F}}_{\mathrm{i}}$ differs between the prototypes, as shown in Figure 18(a1,a2). In the O-MVDT, the sequence of $F_i$ is $F_1\approx F_2>F_5>F_3\approx F_4$, whereas in the N-MVDT, the sequence of $F_i$ is $F_1\approx F_5>F_2>F_3\approx F_4$. This indicates that the position of the upper arc hole is different in the two prototypes because of the influence of the adjusting device. The different positions of the upper arc hole result in different flows of the channels and change the magnitude sequences of $F_i$.

The statistics of $F_i$ are presented in Figure 18(b1,b2), which shows that in the N-MVDT, the average and maximum values of $F_i$ for the USR are significantly larger than those for the LSR. In contrast, the differences among these values are relatively small in the O-MVDT. This suggests that EB 1 is more stable under the action of the adjusting device in the N-MVDT, and the lower arc hole stays longer at the USR compared with the O-MVDT.

Furthermore, the duration and frequency of $F_H$ in the USR of the N-MVDT are longer and higher, respectively, compared with the O-MVDT. Specifically, $F_H$ of $F_1$ in the N-MVDT appears three times in one period and occupies almost the entire period, while in the O-MVDT, there is only one $F_H$ of $F_1$ in one period with much less duration.

As the lower arc hole rotates at a uniform speed and passes through one channel three times in one period, in State 2, the duration of status H is longer than the duration of status L when the upper arc hole stays in the position of this channel for a longer time. As the lower arc hole rotates at a constant speed, it traverses a channel three times within a single cycle. In State 2, the duration of status H surpasses that of status L because of the upper arc hole’s extended presence within the channel’s position. As a result, more $F_H$ appears, and the duration of $F_H$ is longer. By combining the data on all $F_i$, it can be concluded that the upper arc hole stays longer at the USR, and the value of ${\phi }_{\mathrm{s}}$ comes closer to the theoretical lower side in the N-MVDT than in the O-MVDT.

The adjusting device of the TID operates in a fluid environment within the downhole. The adjusting torque, denoted as $\phi$, can be obtained through computational fluid dynamics simulation (CFD simulation).

First, the three-dimensional fluid domain model of the adjusting device and rotation of the fluid domain of the external cylinder valve were derived to represent its rotation under the influence of EB 1. When the value of $\phi$ is different, the fluid domain of the orifices in the external cylinder valve will rotate to different angles, resulting in different fluid flow rates passing through the turbine. The turbine at different values of $\phi$ is obtained through CFD simulation (as shown in Figure 19), which considers the flow area of the upper and lower orifices, as illustrated in Figure 3. As EB 1 deflects in the direction of the balanced friction torque, the positive direction of $\phi$ is defined as such. Equivalently, the positive orientation of $\phi$ is designated as being opposite to the frictional torque experienced by the upper plate valve.

When $\phi$ is in range 2, the flow area of the upper orifice is smaller than that of the lower orifice, but $M_I$ remains positive. Despite its opposite direction to the friction torque, $\phi$ moves away from zero and reduce negatively at this point.

When $\phi$ is in range 3, the smaller flow area of the upper orifice and the negative $M_I$ produce a combined effect that brings $\phi$ closer to zero, despite the direction of $M_I$ being the same as the friction torque.

When $\phi$ is in range 4, the upper orifice is nearly closed, and the lower orifice is almost completely opened. $M_I$ is approximately zero, and $\phi$ approaches zero under the combined effect of the two torques, without any adjusting torque.

Although $M_I$ briefly becomes negative because of reflux, its magnitude is small, and it helps to bring $\mathsf{\phi }$ from negative values closer to zero as the eccentric block swings because of inertia. Furthermore, since the direction of the friction torque is constant, $\phi$ is generally not negative. Therefore, by outputting the adjusting torque, the adjusting device can be closed-loop to bring $\phi$ closer to zero.

For a stable platform, $\mathsf{\phi }$ changes instantaneously with time under the action of multiple moments in EB 1, and EB 1 stabilizes at ${\phi }_s$ after the stable time ($t_p$). These three parameters ($t_p$, ${\phi }_s$, and the range of $\phi$) can be obtained by multibody dynamics simulation [34] and can be used to evaluate the results of the simulation. The stability of EB 1 is better when $t_p$ is shorter and the range of $\mathsf{\phi }$ is smaller. The stability is also reflected when ${\phi }_s$ is closer to zero.

The curve of the fail over time in the DEBD simulation and the SEBD simulation was obtained after inputting the adjusting torque, as illustrated in Figure 20.

The main difference between the DEBD simulation and the SEBD simulation is that $t_p$ is shorter in the DEBD simulation. Additionally, the range of $\phi$ in the DEBD simulation is much smaller, approximately [0°, 11°], compared with the SEBD simulation, which has a range of approximately [0°, 55°]. Finally, ${\phi }_s$ is approximately 5° and 27° in the DEBD simulation and SEBD simulation, respectively.

In summary, increasing the adjusting torque of EB 1 significantly reduces $t_p$, ${\mathsf{\phi }}_s$, and the range of $\phi$. The simulation results demonstrate that the TID can achieve its design purpose, and increasing the TID will improve the stability and reduce the deviation control accuracy of the N-MVDT. The experimental results in this paper confirm that because the pressure loss caused by the regulating device, the pushing force of the improved N-MNDT decreases compared with that of O-MVDT. However, because the time of the development of the TID is short, there are still some drawbacks. Therefore, the current N-MVDT is suitable for drilling conditions with small pushing force and high accuracy requirements. The flow channel structure can be optimized and improved to solve this problem.

3.3. Measurement Accuracy Analysis

According to the results of the experiment, the fluctuations in $F_i$ showed a regular periodic pattern, as illustrated in Figure 18(c1,c2). Each period of $F_i$ contained three waves. This can be attributed to the design of the lower plate valve, which has three arc holes that align with the channel as the valve rotates. This creates three instances of what is referred to as status L within one period of rotation. During status L, $F_i$ starts to rise and remains elevated until the valve moves out of the status L position, at which point $F_i$ begins to fall again, as shown in Figure 21.

Figure 21. Diagrams and reasons for the appearance of $F_H$ and $F_L$. Table 6 shows all the $F_H$ numbers of $F_i$ in one period.

Figure 21. Diagrams and reasons for the appearance of $F_H$ and $F_L$. Table 6 shows all the $F_H$ numbers of $F_i$ in one period.

Table 6. Comparison of $F_H$ numbers in one period of the O-MVDT and the N-MVDT under $\delta =3, n=120, P=1.5$.

$\mathit{i}$	${\mathit{F}}_{\mathit{H}}$ Numbers of the O-MVDT	${\mathit{F}}_{\mathit{H}}$ Numbers of the N-MVDT
1	1	3
2	1	1
3	0	0
4	0	0
5	2	2

Table 6. Comparison of $F_H$ numbers in one period of the O-MVDT and the N-MVDT under $\delta =3, n=120, P=1.5$.

$\mathit{i}$	${\mathit{F}}_{\mathit{H}}$ Numbers of the O-MVDT	${\mathit{F}}_{\mathit{H}}$ Numbers of the N-MVDT
1	1	3
2	1	1
3	0	0
4	0	0
5	2	2

Specifically, to simplify the calculation, if there are three $F_H$ in one period of $F_i$, and each $F_H$ has a longer duration (almost one-third of the period, as shown by $F_1$ in Figure 18(c2)), it can be assumed that the steering rib $\mathrm{i}$ is in the through flow state (or status H) at any time. At this moment, Equation (11) can be simplified as:

$$\mathsf{\phi }\in \left[{\mathsf{\theta }}_i+{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{\alpha }{2}},{ \mathsf{\theta }}_i-{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{\alpha }{2}}\right]\pm 2k\pi$$

(16)

If there is no $F_H$ in one period of $F_i$, the steering rib i cannot flow through via the upper arc hole at any time. In this case, (12) can be simplified as:

$$\mathsf{\phi }\notin \left[{\mathsf{\theta }}_i-{\displaystyle \frac{\beta }{2}}-{\displaystyle \frac{\alpha }{2}},{ \mathsf{\theta }}_i+{\displaystyle \frac{\beta }{2}}+{\displaystyle \frac{\alpha }{2}}\right]\pm 2k\pi$$

(17)

The range of $\mathsf{\phi }$ can be solved according to $F_i$ using (9)–(12), (16) and (17).

In conclusion, the experiments and analysis of the O-MVDT and N-MVDT prototypes revealed significant changes in the rotating situation of the upper plate valve. The adjusting torque on EB 1 balanced the friction moment and reduced the value of $\mathsf{\phi }$. In the N-MVDT, the range of $\phi$ was smaller and ${\phi }_s$ was closer to the theoretical lower side. Additionally, in State 2, the N-MVDT had more $F_H$ numbers and a longer duration of $F_H$. Despite reducing the magnitude of $F_i$, the addition of the TID improved the stability of the upper plate valve and $F_i$. These findings demonstrated the successful operation of the TID.

The deviation angle range of the O-MVDT and the N-MVDT obtained from mathematical models was used to obtain the mathematical model of the instantaneous deflection angle and analyze the optimization effect of the TID on the N-MVDT’s stability. In this section, we calculated the range of $\mathsf{\phi }$ based on the experimental data.

Because of the large amount of data, only a portion was selected for analysis. The selected data had the same parameters as in Section 3.1 as follows: $\delta$ = 5, $n$ = 120, and $P$ = 1.5. The parameters in the model for this operating condition are shown in

Table 7. Parameters of the instantaneous deflection angle model.

Parameter	Value
$\theta$ (°)	0
$\alpha$ (°)	100
$\beta$ (°)	72
n (r/min)	120
$\omega$ (rad/s)	4π
T (s)	0.5

.

The wave situation of $F_i$ in one period for the O-MVDT and the N-MVDT is shown in Table 6. For $F_1$ of the N-MVDT, we can obtain the range of $\phi$ through (16):

$$\mathsf{\phi }\left(t\right)\in \left[-14,14\right]$$

(18)

For $F_2$ of the N-MVDT, the range of $\mathsf{\phi }$ can be obtained using Equation (17).

$$\begin{gathered} \mathsf{\phi }\left[t\left(2, j\right)\right]=\left\{\begin{array}{c}\exists t\left(2,1\right),\phi \left[t\left(2,1\right)\right]\in \left[-14,158\right]-360k\\ \forall t\left(2,2\right),\phi \left[t\left(2,2\right)\right]\notin \left[-14,158\right]-360k\\ \forall t\left(2,2\right),\phi \left[t\left(2,3\right)\right]\notin \left[-14,158\right]-360k\end{array}\right., t\left(2,j\right) \\ \in \left\{\begin{array}{c}\left[{\displaystyle \frac{1}{10}},{\displaystyle \frac{1}{5}}\right]+{\displaystyle \frac{k}{2}},j=1\\ \left[{\displaystyle \frac{4}{15}},{\displaystyle \frac{11}{30}}\right]+{\displaystyle \frac{k}{2}},j=2\\ \left[{\displaystyle \frac{13}{30}},{\displaystyle \frac{8}{15}}\right]+{\displaystyle \frac{k}{2}},j=3\end{array}\right. \end{gathered}$$

(19)

Through Equation (10), we can calculate the range of $\mathsf{\phi }$ for $F_5$ of the N-MVDT:

$$\mathsf{\phi }\left[t\left(5,j\right)\right]\left\{\begin{array}{c}\exists t\left(5,1\right),\phi \left[t\left(5,1\right)\right]\in \left[-158,14\right]-360k\\ \exists t\left(5,2\right),\phi \left[t\left(5,2\right)\right]\in \left[-158,14\right]-360k\\ \forall t\left(5,2\right),\phi \left[t\left(5,3\right)\right]\notin \left[-158,14\right]-360k\end{array}\right., t\left(5,j\right)\in \left\{\begin{array}{c}\left[{\displaystyle \frac{2}{5}},{\displaystyle \frac{1}{2}}\right]+{\displaystyle \frac{k}{2}},j=1\\ \left[{\displaystyle \frac{17}{30}},{\displaystyle \frac{2}{3}}\right]+{\displaystyle \frac{k}{2}},j=2\\ \left[{\displaystyle \frac{11}{15}},{\displaystyle \frac{5}{6}}\right]+{\displaystyle \frac{k}{2}},j=3\end{array}\right.$$

(20)

The range of $\mathsf{\phi }$ for $F_3$ and $F_4$ of the N-MVDT can be obtained using Equation (17):

$$\mathsf{\phi }\left(t\right)\notin \left[58,230\right]$$

(21)

$$\mathsf{\phi }\left(t\right)\notin \left[130,302\right]$$

(22)

Using (18) to (22), we can calculate the range of $\mathsf{\phi }$ with respect to $\mathrm{t}$ in the N-MVDT. We can apply the same method to obtain the range of $\mathsf{\phi }$ with respect to $\mathsf{\phi }$ for the O-MVDT, as illustrated in Figure 22.

The range of $\phi$ is more clearly defined when the upper plate valve is in State 2. When it is in State 3, although $\phi$ can still cause flow in the corresponding channel, the movement of the upper plate valve is irregular, and the range of $\phi$ is less accurate. However, the upper plate valve will eventually return to State 2 after some time.

During the N-MVDT’s operation, the upper plate valve remains in State 2 for almost the entire time, allowing for a controlled range of $\phi$ within [−14°, 14°]. On the other hand, during the O-MVDT’s operation, the range of $\phi$ is much larger at [−86°, 58°] when the upper plate valve is in State 2, and the duration of State 2 is much shorter compared with the N-MVDT. The TID effectively decreased the range of $\phi$ and increased duration in State 2, indicating that the MVDT’s stability was improved. Moreover, the range of $\mathsf{\phi }$ was more uniformly distributed in the N-MVDT, providing further evidence that the N-MVDT’s stability is superior to the stability of the O-MVDT, which demonstrates the effectiveness of the TID in achieving the desired design objectives.

In this section, the experimental data was fed into the derived relationship, and the specific angle range of phi was obtained. The accuracy of the mathematical model was validated by the comparison of simulation and experimental results.

Additionally,

Table 8. Comparison of the range of $\mathsf{\phi }$ in the mathematical model and simulation.

Prototypes	$\mathbf{Range} \mathbf{of} \mathit{\phi }$ in Simulation	$\mathbf{Range} \mathbf{of} \mathit{\phi }$ in Model	Relative Error in $\mathbf{Positive} \mathit{\phi }$
O-MVDT	[0°, 53°]	[−86°, 58°]	9.43%
N-MVDT	[0°, 12°]	[−14°, 14°]	16.67%

presents a comparison between the range of $\phi$ obtained from the model and simulation. The main source of error comes from the negative range of $\phi$, which is caused by the inertia of EB 1 during the experiment, leading to rotation in the negative direction of $\phi$. However, the error in the positive range of $\phi$ is small, indicating the reliability of the model.

4. Conclusions

In this paper, a comprehensive mathematical model was introduced aimed at deriving the angular range of a plat valve, utilizing the thrust generated during vertical drilling operations. Furthermore, a precise measurement methodology was devised for acquiring the crucial parameters of the MVDT. The actual deviation angle range of N-MVDT and O-MVDT vertical drilling was calculated using data obtained from measurement and input into the model.

The pushing forces of the N-MVDT and the O-MVDT at different angles, weights on the bit, and rotational speed were measured by the measuring device. Based on experience, this paper quantitatively analyzed the influence of the TID on the stability of the N-MVDT. Based on the experimental findings presented herein, it was verified that the thrust force exhibited by the modified N-MNDT underwent a lesser decrease compared with that of the O-MVDT, which was attributable to the pressure loss incurred by the conditioning apparatus. This issue must be addressed in the future through the optimization and enhancement of the flow structure. Moreover, if the N-MVDT is to be implemented in a field production setting, additional research is imperative to further bolster its anti-deviation performance under smaller inclination angles (specifically, less than 0.5°), thereby emerging as a pivotal direction for future investigations.