An Experimental and Numerical Simulation Study on a Three-Hydraulic-Cylinder Synchronous Steering Offset Actuator Driven by a Drilling Fluid Rotary Valve Distributor

Featured Application

This work presents a novel three hydraulic cylinder synchronous steering offset actuator incorporating a rotary valve distribution system, which is designed for rotary steerable systems (RSSs). It is applicable to directional drilling in deep and ultra-deep oil and gas wells to improve the accuracy of wellbore trajectory control, steering stability, and tool reliability, particularly under harsh downhole conditions. The proposed design and parameter optimization methods also offer a practical reference for the development and engineering application of high-performance RSS drilling tools.

Abstract

The rotary steerable system (RSS) is the core equipment for precise wellbore trajectory control in deep oil and gas drilling, and its performance is directly determined by the coordination and adaptability of the tool’s offset actuator and control platform. To overcome the limitations of complex control architectures and low positioning accuracy of conventional offset actuators for rotary steering drilling tools, a novel three hydraulic cylinder synchronous steering offset actuator driven by a drilling fluid rotary valve distributor, along with its dedicated control strategy, is proposed. Laboratory experiments and numerical simulations are performed to analyze the piston displacement characteristics of the three hydraulic cylinder under different drilling fluid flow rates and rotary valve rotational speeds. The results demonstrate that the proposed actuator exhibits controllable piston displacement behavior. The simulated and experimental data show consistent variation tendencies with a relative error of less than 8%, thus validating the reliability of the proposed numerical model. Increasing the flow rate from 1 to 1.5 L/s increases the cycle-averaged peak-to-peak piston displacement by 14.5 mm, while raising the rotational speed from 60 rpm to 120 rpm reduces it by 25.3 mm, corresponding to a dogleg severity variation of approximately 1.9–3.1°/30 m. Piston displacement deviations are mainly attributed to valve port machining tolerance, drilling fluid compressibility, pipeline pressure loss, and internal leakage, and these discrepancies are exacerbated as the rotary valve speed or flow rate increases. Finally, optimization strategies for improving synchronization performance are proposed, thereby providing theoretical and technical support for the engineering implementation and parameter optimization of the proposed actuator.

1. Introduction

The rotary steerable system (RSS) plays a crucial role in directional drilling technology and acts as a key technological enabler for the efficient development of deep and complex oil and gas resources [1,2,3,4]. As the exploration of unconventional oil and gas resources moves toward deep and ultra-deep formations with complex geological conditions, drilling environments are increasingly characterized by high temperatures, high pressures, intense vibration, and severe impact loads, imposing more stringent demands on steering accuracy, structural reliability, and control stability [5,6,7,8,9]. As a key component of the rotary steerable tool for bit attitude adjustment, the steering offset actuator [10,11] directly determines the wellbore trajectory control accuracy and service life of the tool.

However, existing offset actuators generally suffer from complex configurations, lengthy control chains, limited synchronization performance among multiple actuation units, and significant coupling between weight on bit (WOB) and torque transmission. Under complex downhole conditions, these issues tend to induce response lag and error amplification, thereby restricting further performance improvement of RSS. At present, several international oilfield service companies and research institutions, including Schlumberger, Baker Hughes, and Halliburton, have successively developed various types of RSSs [12,13,14,15,16]. Schlumberger’s Power Drive system [5,17] achieves bit offset by alternately connecting the hydraulic holes of the upper and lower disk valves and using the high-pressure drilling fluid inside the drill string to drive the supporting pad against the wellbore wall, generating a lateral force on the bit. Although its build-up rate remains below 8°/30 m, the system offers superior wellbore quality. Baker Hughes’s Auto Trak system [6] regulates hydraulic pressure applied to three supporting pads to generate a stable offset force, enabling steering with build-up rates reaching 15–18°/30 m. Halliburton’s Geo-Pilot system [7] employs inner and outer eccentric rings to drive the mandrel, achieving bit direction for steering control. The EZ-Pilot [18], another Halliburton design, uses an electrically actuated eccentric cam sleeve that rotates to a predetermined position and locks, including mandrel offset. The Korea Advanced Institute of Science and Technology (KAIST) developed a rotary steerable system prototype that demonstrated a maximum build-up rate of approximately 32.0°/100 ft in cement block drilling measures [8]. The China National Offshore Oil Corporation (CNOOC) controllable eccentric steering drilling tool system achieves steering by independently controlling the extension and retraction of three pads to form an eccentric displacement vector [19]; Sinopec’s modulated rotary steerable drilling system (MRSS) [20] utilizes drilling fluid to regulate the hydraulic disk valve distribution system, which cyclically actuates pads against the borehole wall to generate offset force. Tianjin University developed a system based on a few-tooth-difference eccentric planetary gear train, driven by a motor, to bend and deform the mandrel and achieve bit offset steering [21]. Although existing RSSs have established diverse technical approaches and have been applied in conventional drilling scenarios, push-the-bit systems remain more prevalent than point-the-bit systems and exhibit notable limitations. Among these, mandrel push-the-bit systems impose high demands on structure and material properties due to frequent pushing actions, which tend to reduce overall tool service life. In wall push-the-bit systems, while some configurations incorporate independent hydraulic systems, structural complexity increases, and the actuation process may compromise wellbore quality.

Gonghui Liu et al. from China University of Petroleum proposed a new type of internal push-pointing rotary steerable drilling tool [22,23,24], which takes an innovative swashplate control mechanism as its core. This mechanism achieves dynamic pointing of the bit by precisely controlling the coordinated motion positions of the three hydraulic cylinder. However, the tool involves a complex structure with numerous integrated components, which complicates the structural layout. To address this issue, this paper proposes a rotary valve distribution system that drives three wedge blocks in combined motion to achieve dynamic bit offset. The steering actuation structure and control method differ significantly from those of existing rotary steerable tools.

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In principle, the rotary valve achieves periodic alternating flow distribution, which drives the three hydraulic cylinder to perform sinusoidal motion with equal phase difference. This forms a dual-parameter cooperative control mechanism, wherein the piston displacement amplitude is regulated by flow rate and the motion frequency is regulated by rotation speed. The steering offset actuator characteristics are readily controllable, meeting the demand for precise steering under complex downhole conditions.

(2)

Structurally, an integrated configuration is adopted, comprising rotary valve fluid distribution and coordinated motion of three wedge blocks with a drum key. This configuration overcomes the structural limitations of conventional push-the-bit and point-the-bit steering systems, achieves decoupling of weight on bit and torque transmission, simplifies the hydraulic control circuit, and enhances tool reliability under high-temperature, high-pressure, and strong-vibrational conditions.

To verify the feasibility of the proposed actuator, this paper conducts experiments and simulation studies on the response characteristics of the rotary valve distribution around its core system, focusing on the influence of flow rate and distribution speed on the synchronous movement accuracy of the three cylinders. At the theoretical level, it breaks through the structural limitations of traditional push-the-bit steering, enriches the decoupling design theory for force and energy transmission in rotary steerable tools, and provides theoretical support for the research and development of new steering execution mechanisms. At the engineering level, the combined structure of rotary valve distribution and wedge blocks, together with the drum-shaped spline structure, simplifies the steering system layout, which can significantly improve the tool service life and wellbore quality, reduce the occurrence of drilling failures in complex formations, and provide a novel technical solution for efficient drilling in complex blocks, such as deep and ultra-deep formations.

2. Materials and Methods

2.1. Control Mechanism of Three Hydraulic Cylinder Synchronous Steering

Based on the above analysis, the proposed three hydraulic cylinder synchronous steering offset actuator, driven by the drilling fluid rotary valve distribution system, is illustrated in Figure 1. The tool primarily consists of a bit offset system and a rotary valve distribution system. The bit offset system enables arbitrary orientation of the drilling plane via sinusoidal motions with a constant phase difference, in coordination with a drum-type key support mechanism. The resulting dynamic trajectory aligns with that of the adjustment rod circle in the figure. The rotary valve distribution system comprises a valve core, a valve sleeve, and a valve body, achieving synchronized position control of the hydraulic cylinders through coordinated flow distribution among the three hydraulic cylinder.

During drilling, the tool rotates integrally under the drive of the upper drill string, while the rotary valve rotates in the opposite direction relative to the drill string driven by the control motor. When the flow rate to the rotary valve is kept constant, three uniformly distributed built-in wedges, driven by the piston rod, can undergo axial linear motion relative to the outer shell; supported by the drum key, the wedges act on the front-end adjustment mechanism by pushing the adjustment rod, causing the rod to achieve an omnidirectional small-angle deflection around its rotation center, thereby generating a steering adjustment. By controlling the movement law of the wedges, the direction of the adjustment mechanism center relative to the tool axis can be accurately adjusted; adjusting the inlet flow rate of the rotary valve can adjust the radial offset distance between the drilling direction line and the drill string center line, thus achieving directional steering.

The tool’s steering is controlled by the rotary valve distribution system, and its driving force is provided by a double-acting hydraulic cylinder based on the pressure difference of drilling fluid; the valve sleeve is fixedly connected to the hydraulic cylinder sleeve and rotates synchronously with the tool shell and drill string; the valve core is driven by the motor of the upper control system, and as shown in Figure 2, its working process can be divided into the following three stages:

• High-Pressure Distribution Stage: High-pressure drilling fluid enters the valve core from its left side and synchronously delivers fluid to the same side chamber (piston chamber or rod chamber) of the three double-acting hydraulic cylinders through the high-pressure side flow channels a1, b1, and c1, driving the pistons to displace synchronously in accordance with the preset motion law. At the same time, the low-pressure drilling fluid in the opposite side chambers of the hydraulic cylinders is discharged from the low-pressure side valve ports through the flow channels a2, b2, and c2, forming a complete drilling fluid circulation path.
• Valve Core–Valve Body Distribution Structure Action Stage: The valve core is structurally designed to include two distribution valve ports and four fluid inlet ports along the radial direction. Among them, the two distribution valve ports do not intersect each other and are symmetrically arranged at 180° along the axial direction; the four fluid inlet ports are uniformly arranged along the radial direction, with a 90° angle included between adjacent ports. The valve body is correspondingly provided with 6 drilling fluid channels along the radial direction, which are, respectively, connected one-to-one with the rod chambers and piston chambers of the three double-acting hydraulic cylinders, realizing the double-chamber control of the three hydraulic cylinders.
• Periodic Circulation, Distribution, and Steering Realization Stage: When the rotary valve is driven by the control motor to rotate, high-pressure drilling fluid enters the valve core through the fluid inlet ports and is simultaneously delivered to the valve body flow channels a1, b1, and c1 through the high-pressure side distribution valve port. At the same time, the other distribution valve port is connected to the low-pressure flow channel, enabling the valve body flow channels a2, b2, and c2 to be connected to the low-pressure return channel. With the continuous rotation of the rotary valve, the high- and low-pressure distribution channels switch alternately periodically, realizing the periodic synchronous driving of the three hydraulic cylinder. The hydraulic cylinder pistons drive the three wedge blocks to perform periodic movement with equal phase differences, thereby realizing bit attitude adjustment and steering offset. By adjusting the angular position of the high-pressure port of the rotary valve, the precise control of the tool face angle can be further achieved.

2.2. Test Design and Equipment

2.2.1. Test System Design and Function

In this test system, the inlet flow rate of the rotary valve and its rotational speed are selected as the controllable input variables, which are precisely regulated using a variable frequency pump control system and a servo motor, respectively. The piston displacements of the three hydraulic cylinder serve as the primary output response variables, while auxiliary parameters, including flow rate, pressure, torque, and rotational speed, are synchronously acquired to characterize the actuation performance and identify potential error sources. The control unit, actuation unit, and data acquisition unit are integrated into a unified platform, enabling real-time parameter adjustment and multi-channel synchronous data acquisition. The overall system configuration and signal transmission pathway are illustrated in Figure 3.

All systems achieve collaborative operation through mechanical connections and signal interaction, thereby ensuring the controllability of the test process and the accuracy of data acquisition. The specific functions of each system are as follows:

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Variable Frequency Pump Control System

Composed of a variable frequency motor, a hydraulic pump, a variable frequency controller, and related pipelines, the system is illustrated in Figure 3 (hydraulic supply system). The system controls the rotational speed of the hydraulic pump by adjusting the output frequency of the frequency converter, thereby changing the fluid supply flow rate and system pressure and simulating the precise regulation of the inlet flow rate of high-pressure fluid. This module is mainly used to simulate the impact of drilling fluid displacement changes on system dynamic response and control accuracy, and it is a key component for achieving controllable input.

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Rotary Valve System

As the core component for realizing periodic flow distribution control, the system is mainly composed of a rotary valve body, a valve core assembly, a housing, a sealing unit, and a drive coupling, as illustrated in Figure 3 (rotary valve flow distribution system). The system drives the rotary valve to rotate via a servo motor, altering the connection relationship between the high- and low-pressure channels, and it realizes periodic high–low pressure switching and flow distribution. By adjusting the rotary valve speed, different distribution frequencies and working conditions downhole can be simulated, thereby studying the influence of flow distribution speed on the piston displacement output characteristics of the three hydraulic cylinder.

(3)

Three Hydraulic Cylinder Actuation System

The three hydraulic cylinder actuation system comprises three groups of hydraulic cylinders, piston assemblies, and connecting rod structures, and it is the main output execution unit of the system. Under the action of the rotary valve distribution, the hydraulic cylinders perform alternating fluid inlet and return movements, converting the hydraulic energy of drilling fluid into piston displacement. Piston movement data is collected in real time using displacement sensors, which act as the key indicators for evaluating the system’s control accuracy and synchronization performance.

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Measurement and Data Acquisition System

The measurement and control system consists of two parts: a data acquisition subsystem and a control subsystem. As illustrated in Figure 3, the data acquisition system includes displacement/pressure sensors and real-time data acquisition devices, while the control system comprises servo drivers, servo motors, and control interface modules. A unified platform is adopted to achieve real-time parameter monitoring, multi-variable synchronous acquisition, and data storage, as well as precise control of the flow distribution frequency, enabling coordinated regulation with the variable frequency pump-controlled system. The system can replicate various typical operating conditions to investigate the stability and dynamic characteristics of the system under different input conditions, providing data support for subsequent error analysis and control algorithm optimization.

2.2.2. Test Equipment and Platform Construction

According to the test system block diagram, the corresponding components were selected and configured, including a rotary valve drive motor, controller, data acquisition instrument, etc. Detailed specifications of the main components in the system are listed in

Table 1. Parameters of the main components of the test system.

Component Name	Parameter Value
Rotary Valve Drive Motor	Speed: 400 rpm
	Power: 0.3 kW
	Rated Torque: 7.3 N·m
Controller	Processor: ARM Cortex A-72
	Input Power: 24 VDC@ 1.25 A
	Local (I/O): 6 digital inputs, 2 digital outputs
Data Acquisition Instrument	16-bit 8-channel synchronous acquisition
	Input Voltage: 0~5 V
	Operating Temperature: 0~40 °C
	External Power Supply: +24 V
Pressure Sensor	Range: 0–7 MPa
	Output Voltage: 0–5 V
Displacement Sensor	Input: 24 V
	Output: 0–5 V
	Linear Error: <±0.25%
	Operating Temperature: −20–+85 °C
	Range: 100 mm
Three-Phase AC Asynchronous Motor	Speed: 1460 rpm
	Power: 15 kw
Frequency Converter	Input Voltage: 380 V
	Frequency: 50 Hz
	Rated Input Capacity: 2.4 kVA
	Input: 0–10 V voltage/0–20 mA current
Pump	Speed: 1450 rpm
	Flow Rate: 150 L/min
	Pressure: 100 bar
Hydraulic Cylinder	Diameter: Φ25 mm
	Piston Rod Diameter: Φ15 mm
	Maximum Piston Displacement: 100 mm

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The displacement sensor used in the test system has a manufacturer-specified accuracy of ±0.1% of full scale. For a typical displacement measurement of 14.5 mm, the corresponding maximum possible error is ±0.0145 mm. This error is negligible compared to the observed displacement changes reported in the Results Section. Flow rate and rotational speed measurements are within manufacturer-specified errors, with accuracies of ±0.5% of reading and ±5 rpm, respectively. The overall measurement uncertainty is considered acceptable for the purpose of this study, as the observed trends significantly exceed the sensor accuracy limits.

The established test system for the three hydraulic cylinder synchronous control based on rotary valve flow distribution is shown in Figure 4. The test bench is arranged at the core of the test system, where the rotary valve synchronous steering offset mechanism is fixed by brackets. The rotating shaft at one end of the system is connected to the rotary valve control system, and its fluid port is connected to the input of the pump-controlled system. The other end is connected to sensors, with the fluid port connected to the fluid circuit. Based on the test block diagram, the whole test procedure is divided into four steps, as detailed below.

• Before the test starts, the measurement and control system is calibrated for all sensors and initializes the target input parameters, including the variable frequency pump frequency (corresponding to the supply flow rate) and the servo motor speed (corresponding to the rotary valve distribution frequency). Then, the variable frequency pump control system is started, and the hydraulic pump delivers high-pressure drilling fluid at the set flow rate under the control of the frequency converter, which is delivered to the fluid inlet chamber of the rotary valve via the high-pressure pipeline.
• As the servo motor drives the valve core to rotate, the relative position between the valve core and the valve sleeve changes periodically, sequentially connecting the high-pressure channel to the corresponding hydraulic ports of the three hydraulic cylinder, thereby realizing alternating high–low pressure switching and sequential distribution. The pressurized drilling fluid enters the corresponding chamber of the hydraulic cylinder, pushing the piston to generate axial displacement; at the same time, the other hydraulic cylinders are in either fluid-return or pressure-relief mode. With the continuous rotation of the valve core, the three hydraulic cylinder performs periodic alternating actions according to the set phase relationship, thus generating synchronous or quasi-synchronous displacement output. During this process, the displacement sensor records the piston displacement change in the hydraulic cylinder in real time, and the data is centrally collected and synchronously stored by the measurement and control system, which is used to analyze the rotary valve flow distribution performance, hydraulic cylinder dynamic response characteristics, and system dynamic stability.
• When the frequency of the variable frequency pump is changed, the system input flow rate changes, and the response amplitude of the hydraulic cylinder is adjusted accordingly (the rotary valve speed is independently controlled); when the speed of the servo motor is changed, the distribution cycle changes, thereby affecting the actuation frequency and phase matching relationship of the three hydraulic cylinder. By testing different input combination working conditions, the synchronous control performance of the rotary valve distribution system under different flow-rate coupling conditions can be comprehensively evaluated.
• After the test, the pump frequency and motor speed are gradually reduced to achieve stable pressure relief, and the drilling fluid is returned to the water tank through the hydraulic circuit to complete fluid recycling and conclude the entire test procedure.

3. Results

Based on the previously established rotary valve distribution synchronous control test system, the periodic variation characteristics, phase consistency, and amplitude stability of the displacement curves of the three hydraulic cylinder were tested under the condition of fixed inlet flow rate of the rotary valve, and test data on the influence of rotational speed on the actuation frequency of the three hydraulic cylinder and the steering response characteristics were obtained. The motor speed was stabilized at 90 rpm through PID control [25] to keep the rotary valve distribution period constant. Only the inlet flow rate of the rotary valve was changed; the amplitude variation, phase consistency, and stability characteristics of the displacement curves of the three hydraulic cylinder under different displacement conditions were compared; and the influence of supply flow rate on piston displacement, as well as the improvement in steering performance, was analyzed. The test results are as follows.

3.1. Displacement Response Under Constant Rotational Speed and Fluid Supply Rate

In this group of tests, the frequency converter was set to 20 Hz, corresponding to a pump flow rate of approximately 1 L/s, thereby maintaining a constant supply flow rate into the rotary valve. Only the rotary valve speed was varied to analyze the influence of speed variation on the movement characteristics of the three hydraulic cylinder and the steering response characteristics.

Figure 5 illustrates the time-domain piston displacement responses of the three hydraulic cylinder at a rotary valve speed of 60 rpm and a flow rate of 1 L/s. To quantitatively characterize the synchronization performance of the three hydraulic cylinder actuation system, three critical indices were computed: phase lag (in degrees), amplitude variation, and cycle-to-cycle RMS deviation. The pairwise comparative results are summarized in

Table 2. Pairwise comparison results.

Hydraulic Cylinder Pair	Max Phase Lag	Cylinder-to-Cylinder Amplitude Variation	Cycle-to-Cycle RMS Deviation
No. 2 vs. No. 1	2.9°	4.5%	6.2%
No. 3 vs. No. 1	3.2°	4.8%	6.5%
No. 3 vs. No. 2	3.1°	4.2%	5.8%

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As shown in Table 2, the maximum phase lag between any two cylinders is less than 3.2°, signifying minimal temporal discrepancy. The peak inter-cylinder amplitude variation is maintained below 4.8%, verifying consistent stroke amplitudes among the three hydraulic cylinder. Additionally, the maximum cycle-to-cycle RMS deviation is measured at 6.5%, which validates exceptional motion repeatability. Overall, the three hydraulic cylinder exhibit superior synchronous behavior under the sequential fluid distribution delivered by the rotary valve assembly.

3.2. Displacement Response Under Increased Rotational Speed and Fluid Supply Rate

Figure 6 depicts the time-domain piston displacement responses of the three hydraulic cylinder at a rotary valve speed of 90 rpm and a flow rate of 1.5 L/s. To quantitatively characterize the synchronization performance of the three hydraulic cylinder actuation system, three critical indices were computed: phase lag (in degrees), amplitude variation, and cycle-to-cycle RMS deviation. The pairwise comparative results are summarized in

Table 3. Pairwise comparison results.

Hydraulic Cylinder Pair	Max Phase Lag	Cylinder-to-Cylinder Amplitude Variation	Cycle-to-Cycle RMS Deviation
No. 2 vs. No. 1	2.8°	4.3%	6.0%
No. 3 vs. No. 1	3.0°	4.6%	6.3%
No. 3 vs. No. 2	2.9°	4.0%	5.6%

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As revealed in Table 3, the maximum phase lag between any two cylinders is less than 3.0°, signifying minimal temporal discrepancy. The peak inter-cylinder amplitude variation is maintained below 4.6%, verifying consistent stroke amplitudes among the three hydraulic cylinder. Additionally, the maximum cycle-to-cycle RMS deviation is measured at 6.3%, which validates exceptional motion repeatability. Overall, the three hydraulic cylinder exhibits superior synchronous behavior under the sequential fluid distribution delivered by the rotary valve assembly.

It can be seen from the figure that the displacement curves of the three hydraulic cylinder maintain stable periodic changes, and their actuation frequency is consistent with that of the 90 rpm working condition with no observable frequency drift, indicating a high control accuracy of the rotary valve speed under PID control; compared with the low displacement working condition, the peak piston displacement of the three hydraulic cylinder is significantly improved, and the overall amplitude of the displacement curves is increased, indicating that the increase in supply flow rate directly improves the effective piston displacement within a single cycle. A stable phase difference is still maintained among the three hydraulic cylinder, indicating that the system’s synchronization performance remains stable with the increase in flow rate.

Further comparison reveals:

• At constant rotational speed, the periods of the displacement curves of the three hydraulic cylinder are essentially consistent, indicating that the actuation frequency is determined by the rotary valve speed and is independent of the supply flow rate.
• The displacement amplitude increases with the increase in supply flow rate and the peak-to-peak displacement difference among the three hydraulic cylinder increases, indicating a positive correlation between the tool offset and the supply flow rate.
• As the supply flow rate increases, the slight displacement discrepancy among the three hydraulic cylinder is amplified, indicating that the flow distribution discrepancy becomes more significant at higher flow rates. This may be attributed to the valve port throttling characteristics, pressure loss in the flow channels, and increased internal leakage.

It should be noted that, due to the limitations of laboratory test conditions and equipment performance, it is difficult to accurately reproduce some key operating parameters on the existing test platform. For example, it is difficult to systematically anaylze the transient pressure fluctuations under the coupling condition of high rotary valve speed and high flow rate, the influence of minor geometric errors of the valve port on the flow distribution characteristics, the dynamic impacts of internal leakage on the synchronization performance of the three hydraulic cylinder, and the response under extreme operating conditions with different combinations of structural parameters through multi-parameter coupling using a single test method. In addition, the test rig has certain limitations in terms of pressure level, continuous operation time, and measurement resolution, which limit the in-depth characterization of the high-frequency dynamic characteristics and the sources of small displacement deviations.

Therefore, it is necessary to establish a corresponding numerical model based on the experimental results and perform supplementary investigations on the rotary valve flow distribution process and the motion characteristics of the three hydraulic cylinder via parametric modeling and multi-case simulation analyses. On the one hand, it can expand the high-pressure, high-speed, and multi-parameter coupling operating conditions that are difficult to achieve in experiments; on the other hand, it enables the sensitivity analysis of key structural and physical parameters, provides theoretical support for the optimization of the system’s synchronization control accuracy and the design of its structural parameters, and lays a foundation for the comparison and verification of subsequent test results.

3.3. Numerical Simulation

3.3.1. Numerical Model

To systematically investigate the factors influencing the piston displacement of the three hydraulic cylinder through numerical simulation, based on the aforementioned tests, an integrated simulation model of the offset steering system for the rotary steerable tool was established on the AMESim simulation platform [26] to further analyze the control characteristics and key influencing factors of the offset steering mechanism. The overall physical modeling architecture of the simulation model is shown in Figure 7. The system comprises four core subsystems: a rotary valve model, a rotary valve control model, a pump-controlled displacement model, and a hydraulic cylinder and load model. The functions of each module are described as follows:

• The rotary valve model simulates the hydraulic connection relationships and flow distribution characteristics of its internal flow distribution structure, and it is a key component for achieving pressure switching and flow regulation among multiple chambers. This model describes the on/off state between the fluid inlet port, fluid return port, and each working port via an equivalent valve port structure, and it dynamically adjusts the effective flow area in real time according to the rotational angle of the valve core to achieve flow distribution among the different working chambers. Its core function is to realize the distribution and switching of hydraulic energy among the actuation chambers, which is the basis for establishing system pressure and regulating flow.
• The rotary valve control model simulates the control logic and actuation process for driving the rotary valve. Based on the input control signal (e.g., an electrical signal or a target angle command), this module outputs a driving signal through computation by the control algorithm to drive the rotary valve to reach the target position. The model is used to reflect the rotary valve’s response and control accuracy during actual control, which exerts a significant influence on the dynamic performance of the system.
• The pump-controlled displacement model describes the flow regulation mechanism of a variable displacement pump and its coupling with system pressure. This module adjusts the pump’s output flow rate in real time in response to the control signal, thereby controlling the system flow and pressure to achieve an on-demand fluid supply. The dynamic characteristics of flow rate regulation are incorporated into the model, enabling the simulation results to reflect the energy-saving performance and pressure response of the pump-controlled system.
• The hydraulic cylinder and load model is used to simulate the kinematic and dynamic characteristics of the actuation unit. The hydraulic cylinder model generates a driving force according to the pressure difference between the two chambers and calculates the piston motion state, taking into account the piston area, frictional resistance, and internal leakage; the load model describes the mass, damping, and stiffness characteristics of the external mechanical load. The coupling of these two models accurately reflects the changes in the actuation unit’s displacement, speed, and output force under different working conditions, providing an important basis for evaluating the system’s control accuracy and dynamic response performance.

The above four subsystems are coupled logically according to the power transmission path and signal control flow, and mechanical coupling relationships in actual operation, ensuring consistency between the simulation model and the physical test system. Among these, the rotary valve distribution model and the pump-controlled flow rate model are core modules for simulation analysis. Their modeling processes are carefully designed, incorporating the actual structural parameters and working principles of the prototype. The specific construction methods of their physical models will be elaborated below.

By clarifying the function of each module and the parameter matching characteristics, this simulation model is designed to meet the actual engineering requirements. Via simulation, the pressure response characteristics under different combinations of rotary valve speed and pump flow rate are analyzed to verify the rationality of the system’s flow distribution and pressure matching relationship. After unified calibration and parameter adjustment of each module, a numerical simulation analysis covering the entire working cycle can be conducted, providing a theoretical basis and data support for the physical prototype testing, system structural improvement, and control strategy optimization, thereby reducing test costs and improving the reliability of engineering applications.

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Simulation Model of the Flow Distribution System

Since no dedicated rotary valve model is available in the AMESim [27] component library, a functionally equivalent spool valve module is adopted as a simulation substitute based on the actual working principle and circumferential flow distribution characteristics of the rotary valve [28,29]. As shown in Figure 8, it is the equivalent modeling structure of a single group of flow distribution valves. During the modeling process, a single flow distribution unit is constructed by combining four custom valve port modules, which simulate the on/off switching logic relationships of the fluid inlet (port P), two working fluid ports (port a1, port b1), and the fluid return port (port T), respectively. The periodic angular displacement signal generated by motor rotation is converted into the reciprocating linear displacement input of the valve core, and through the coordinated adjustment of the valve core displacement control variable and the motor speed, the three sets of spool valves act in sequence following the 120° phase difference rule, thus realizing the functional equivalent simulation of the circumferential flow distribution process of the rotary valve.

To accurately replicate the flow distribution law of the rotary valve, the correlation data between valve core displacement and the effective flow area of the valve port is incorporated into the model, and the processes of valve port opening/closing and throttling area variation are described using a data-driven approach, thus achieving a refined characterization of the flow distribution characteristics. In the equivalent model, the P ports at both ends are fluid supply ports, and the entire flow distribution process follows the logic of alternating high-and low- pressure switching: when the high-pressure flow distribution valve port (a1) is connected to the fluid supply, the low-pressure flow distribution valve port (b1) is closed, and the fluid return path is opened at the same time; when the low-pressure flow distribution valve port (b1) is connected to the fluid supply, the high-pressure flow distribution valve port (a1) is synchronously closed and connected to the fluid return path. The above high- and low-pressure switching process constitutes a complete flow distribution cycle that corresponds exactly to one rotation of the motor, and its operating frequency is accurately controlled by the motor speed.

One reciprocating linear displacement cycle of the spool valve corresponds to driving the hydraulic cylinder piston to complete one reciprocating motion. Therefore, the motor speed directly determines the spool valve switching frequency, thereby determining the frequency and dynamic response characteristics of the hydraulic cylinder movement. Using this equivalent modeling method, the dynamic coupling between the rotary valve flow distribution process and the motion of the three hydraulic cylinder is simulated.

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Simulation Model of the Pump-Controlled System

As the core control unit of the motor drive system, a simulation model of the frequency converter is established based on the voltage frequency (V/F) control principle, and an electromagnetic mechanical coupling model is established by combining the motor electromagnetic torque equation and the mechanical motion equation. During the modeling process, a linear mapping relationship between the 0–10 V standard analog input and the output frequency of 0~50 Hz is first established to convert the control voltage signal into a target frequency command; then, the functional relationship between the power supply frequency and the motor speed is established according to the asynchronous speed formula of the asynchronous motor: n ≈ 60f/p. Meanwhile, the electromagnetic torque slip characteristic equation and the mechanical dynamics equation (including moment of inertia, damping, and load torque) are incorporated to form a closed-loop coupling model. The main challenges in this modeling process lie in the frequency variation process, which is accompanied by the synchronous adjustment of the voltage amplitude. It is necessary to ensure that the magnetic flux remains approximately constant to avoid model distortion and that there is dynamic coupling between the motor speed and the main pump load torque; abrupt load changes will lead to motor speed fluctuations. Therefore, the model must reasonably introduce moment of inertia and damping parameters to ensure simulation stability and engineering consistency.

Therefore, a control structure is established based on the idea of vector minor closed-loop control, as shown in Figure 9. A speed command is set at the input end, and a frequency control signal is generated after proportional–integral (PI) regulation; the frequency module outputs the corresponding alternating current (AC) frequency value, and the voltage amplitude is synchronously generated via the V/F proportional function. The motor module calculates the instantaneous rotational speed based on the output frequency and load torque. Then, a closed-loop adjustment is formed via speed feedback to achieve dynamic speed stabilization control. The motor output shaft is rigidly coupled with the main pump’s moment of inertia via a coupling model, so that the motor speed is directly mapped to the main pump speed, thereby completing the simulation of stepless speed regulation for the main pump. By setting different input voltage signals (0–10 V), the main pump speed can be continuously adjusted in the range of 0–1450 r/min, and load disturbances can be superimposed to analyze the system’s dynamic response characteristics under acceleration, deceleration, and impact load conditions.

3.3.2. Control Equations

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Control Equation of the Flow Distribution Valve

To ensure that the valve core opening shape meets the requirements for continuously variable dynamic output characteristics, this study optimizes the valve port shape parameters by combining the pressure–flow rate relationship with the feasibility of mechanical processing [30]. The valve port shape of the flow distribution valve core must ensure that the effective flow area $A_i\left(i=1,2,3\right)$ of the three hydraulic cylinder varies periodically, and its variation law must ensure that the piston displacement of the cylinder conforms to the simple harmonic motion law to ensure the stability and accuracy of the coordinated pushing of the wedge blocks. The geometric relationship of the circumferential rotation fit between the valve core and the valve sleeve is mapped to a plane, as shown in Figure 10. The flow channel port of the valve sleeve is designed as a rectangular opening with a fixed width, and the valve core opening adopts a boundary function $f\left(x\right)$ envelope structure, which effectively adapts to the dynamic flow distribution requirements.

$$A_i={\int }_x^{x+b}f(x)dx$$

(1)

$${\displaystyle \frac{d{A}_i}{dt}}={\displaystyle \frac{d{A}_i}{dx}}Rw=c_1\mathrm{s}\mathrm{i}\mathrm{n}(wt+c_2)$$

(2)

Based on the above valve port design and the working principle of rotary valve flow distribution, the positioning and sealing structures are designed to form an integrated valve assembly. The effective flow area corresponding to the valve core displacement is mapped to the geometric opening stroke, which is then converted into the area variation of the rotary valve port using a conversion relationship [31]. The opening area is shown in Figure 11.

(2)

Control Equation of Variable Frequency Pump Control

To achieve accurate regulation of the rotary valve flow distribution, it is necessary to establish governing equations of the hydraulic system. The lumped-parameter method, implemented via the AMESim platform, is employed to solve these equations. The system hydraulics are governed by two fundamental conservation laws: the continuity equation (mass conservation) and the momentum equation (momentum conservation).

• Continuity Equation (Mass Conservation)

For each control volume, the net flow rate equals the rate of volume change plus the rate of pressure-induced compression due to fluid compressibility. The continuity equation in lumped-parameter form is given by:

$$\sum Q_{in}-\sum Q_{out}-\sum Q_{leak}={\displaystyle \frac{dy}{dt}}+{\displaystyle \frac{V}{{\beta }_e}}{\displaystyle \frac{dp}{dt}}$$

(3)

The effective bulk modulus accounts for both fluid compressibility and the compliance of the hydraulic lines:

$${\displaystyle \frac{1}{{\beta }_e}}={\displaystyle \frac{1}{{\beta }_f}}+{\displaystyle \frac{V_{line}}{V}}·{\displaystyle \frac{1}{E_{line}}}$$

(4)

• Momentum Equation (Momentum Conservation)

For flow through hydraulic elements (e.g., valves, orifices, and pipes), the pressure drop balances the inertial, viscous, and orifice effects. The momentum equation in lumped-parameter form is expressed as:

$$∆p={\displaystyle \frac{\rho }{2C_d^2{A}^2}}{Q}^2+R*Q+L{\displaystyle \frac{dQ}{dt}}$$

(5)

The relationship between the control voltage input to the frequency converter, the motor input voltage, and the flow rate [32,33,34] allows the quantitative expression of flow rate control and can be derived as follows:

$$Q=f(u_c,U_1)$$

(6)

$$U_1=K_u·K_f·u_c=22u_c$$

(7)

Therefore, the relationship between these two voltages ($U_1,u_c$) can be expressed by a proportional link. Based on the motor working principle, the electromagnetic torque of the three-phase AC asynchronous motor is given by [35]:

$$T_e={\displaystyle \frac{3n_p{U}_1^2{R}_r^{\prime }/s}{{\omega }_1[{\left(R_s+\frac{R_r^{\prime }}{s}\right)}^2+{\omega }_1^2{(L_{ls}+L_{lr}^{\prime })}^2]}}$$

(8)

When the slip ratio $s$ is small, the formula can be simplified as:

$$T_e={\displaystyle \frac{3n_p{U}_1^{2}s}{{\omega }_1{R}_r^{\prime }}}={\displaystyle \frac{3n_p{U}_1^{2}s}{2\pi f_1{R}_r^{\prime }}}$$

(9)

Without considering low-voltage compensation, substituting the motor slip ratio into the above formula yields:

$$T_e={\displaystyle \frac{3n_p}{2\pi R_r^{\prime }}}{K}_f{U}_1-{\displaystyle \frac{n_p^2}{40\pi R_r^{\prime }}}{K}_f^2{n}_p=K_1{U}_1-K_2{n}_p$$

(10)

The torque balance equation of the motor system is:

$${\displaystyle \frac{2\pi }{60}}{J}_T{\displaystyle \frac{dn}{dt}}=T_e-T_{LT}-{\displaystyle \frac{2\pi }{60}}{B}_{T}n$$

(11)

After rearrangement, the equation is obtained as:

$${\displaystyle \frac{dn}{dt}}=(T_e-T_{LT}-T_{df}-K_{4}n)·K_5$$

(12)

Calculation formulas for the axial piston pump [36,37]:

$$q={\displaystyle \frac{V_g·n·v}{1000}}$$

(13)

$$T_{LT}={\displaystyle \frac{D_{p}P}{{\eta }_{pm}}}=K_{3}P$$

(14)

3.3.3. Parameter Setting

To systematically analyze the dynamic characteristics and steering performance of the three hydraulic cylinder synchronous steering control system under different operating parameters, the boundary conditions were established based on the overall physical model developed in the previous section. The inlet flow rate is set to 1.0, 1.25, and 1.5 L/s, corresponding to the pump flow rate, while the return pipeline pressure is maintained at a constant 0.1 MPa (atmospheric reference pressure). The flow passage area of the rotary valve is modeled using a data-driven function derived from the geometric opening curve, as shown in Figure 11, with the discharge coefficient defined for the orifice. Simulations are performed using the AMESim standard solver (ODT) with the following settings: a variable-step ordinary differential equation solver employing the DASSL algorithm, a simulation duration of 12 s, and a data sampling interval of 0.001 s. The key assumptions of the numerical model are as follows.

Given the predominantly turbulent flow within the valve orifice, the drilling fluid is treated as a Newtonian fluid with a constant viscosity. Temperature variations are neglected, and both the bulk modulus and viscosity are assumed to be constant. Distributed parameter effects in the pipelines are approximated using lumped resistance, inertia, and capacitance elements. The fluid pressure throughout the simulation domain remains above the vapor pressure at all times. The valve core and hydraulic cylinders are regarded as rigid bodies with no elastic deformation, with only viscous friction considered for the cylinders and Coulomb friction neglected. All model parameters are derived from the actual experimental prototype, as shown in

Table 4. Values of the coefficients of the system equation.

Quantities	Value	Quantities	Value
$K_u$	$5 \mathrm{H}\mathrm{z}/\mathrm{V}$	$v$	0.93
$K_f$	$4.4 \mathrm{V}/\mathrm{H}\mathrm{z}$	$D_p$	$16.4643\times {10}^{-6} {\mathrm{m}}^3/\mathrm{r}\mathrm{a}\mathrm{d}$
$n_p$	2	$n_{pm}$	0.93
$B_T$	$0.01 \mathrm{N}·\mathrm{m}·\mathrm{s}/\mathrm{r}\mathrm{a}\mathrm{d}$	$\rho$	$1000 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$
$K_4$	0.001	$\mu$	$0.27 \mathrm{N}·\mathrm{s}/{\mathrm{m}}^2(40 °C)$
$K_5$	0.001	${\beta }_e$	$1.15\times {10}^9 {\mathrm{P}}_{\mathrm{a}}$
$V_g$	$111.23 {\mathrm{c}\mathrm{m}}^3$	$C_d$	0.62

. This framework enables the investigation of the effects of two key input parameters: rotary valve rotational speed and flow rate. Before solving the model, the coefficient values of the model equations are parameterized, as shown in Table 4.

(1)

Simulation of Three Hydraulic Cylinder Synchronous Characteristics Under Benchmark Working Conditions

A fully symmetric reference working condition model for the three hydraulic cylinder is established under ideal consistent conditions. During the simulation, the structural, load, and control parameters of the three hydraulic cylinder are kept consistent to eliminate the influence of structural differences on the motion results, thus focusing on the system’s flow distribution mechanism and inherent power transmission characteristics. The system boundary conditions are uniformly set as follows: flow rate 1 L/s, rotary valve speed 90 rpm, rotary valve leakage gap 0.3 mm, and hydraulic cylinder piston diameter 38 mm. All the above parameters are derived from the structural design values and experimental calibration data of the actual test prototype and represent typical field operating conditions.

(2)

Influence of Rotary Valve Speed on Steering Performance Under Fixed Displacement Condition

In the simulation, a constant flow input boundary condition is applied, and the flow rate is fixed at 1 L/s to isolate the effects of flow-related variables on the system response, thus allowing the analysis to focus on the rotary valve speed factor. The rotary valve speeds are set to 60 rpm, 90 rpm, and 120 rpm, respectively, covering low-frequency, medium-frequency, and high-frequency flow distribution intervals, which are consistent with the actual adjustability in field applications; the structural parameters of the three hydraulic cylinder are kept consistent, and displacement deviation working conditions are established by assigning different load values to simulate the uneven formation resistance during drilling. For each operating condition, displacement data from the steady-state operation stage are extracted, and the influences of start-up and transition processes are neglected. The displacement distribution characteristics and steering adjustment amplitudes of the three hydraulic cylinder under different speed conditions are compared, thus analyzing the influence law of flow distribution frequency variation on steering sensitivity and system stability.

(3)

Influence of Flow Rate on Steering Performance Under Fixed Speed Conditions

In the simulation, the flow distribution frequency is kept constant by fixing the rotary valve speed at 90 rpm, and a single-factor comparison study is conducted by varying the pump flow rate. The pump flow rates are set to 1 L/s, 1.25 L/s, and 1.5 L/s, respectively, with all other structural and control parameters kept unchanged to isolate the interference from the flow distribution frequency, ensuring the system response is solely affected by the flow rate variation. The above flow rate range covers the rated adjustment range of the main pump and typical field operating conditions for different drilling pressure and rotational speed requirements. Displacement data of the three hydraulic cylinder are extracted from the steady-state stage for each operating condition, and the variation trends of displacement deviation and steering amplitude under different flow rate input conditions are compared to evaluate the influence of flow rate adjustment on steering response capability and synchronization performance.

4. Discussion

4.1. Model Validation and Reliability Analysis

4.1.1. Validation of the Flow Distribution Principle

A simulation is conducted to validate the flow distribution principle at a pump flow rate of 1 L/s with fully consistent structural parameters for the three hydraulic cylinder. Figure 12 shows that the simulated piston rod displacements of the three hydraulic cylinder exhibit stable periodic variation. Each cylinder follows the same displacement variation law, which verifies the correctness of the rotary valve flow distribution logic and the continuity of power transmission within the system.

As observed from the three colored simulated curves in Figure 12, the peak displacement of the three piston rods is essentially consistent. The displacement deviation approaches zero in the steady state, and the curve cycle remains stable; meanwhile, the time intervals between the occurrence of displacement peaks for each cylinder are equal, and the displacement curves exhibit a slight peak clipping characteristic at the peak values.

This characteristic arises mainly from the following mechanisms. First, under the condition of completely consistent structural parameters for the three hydraulic cylinder, each cylinder has the same pressure-bearing area, frictional characteristics, and load mass, with hydraulic driving force uniformly distributed; thus, the motion responses remain synchronous. Second, the effective area of the piston chamber of the double-acting hydraulic cylinder is larger than that of the rod chamber; under the same pressure, the driving force generated during the extension stage is greater than that during the retraction stage, leading to a shorter piston extension time than retraction time and forming an asymmetric cyclic characteristic. Third, when the piston moves to the limit piston displacement position, it is subject to a mechanical limit, which restricts the displacement peak and results in a flat-top characteristic.

In summary, under the condition of fully consistent parameters, the system can achieve stable, periodically uniform synchronous motion, which verifies the correctness of the established simulation model in terms of flow distribution logic and power transmission mechanism and provides a reliable benchmark for subsequent variable influence analysis.

4.1.2. Validation of Consistency Between Simulation and Experiment

Under the operating condition of a pump flow rate of 1 L/s and a rotary valve speed of 60 rpm, distinct displacement responses are induced in the three hydraulic cylinder by applying different push loads to each piston rod, as illustrated in Figure 13a. A comparison between the simulated displacement curves and the test-acquired curves for each hydraulic cylinder is presented in Figure 13b, where the legend “Displacement” represents the measured data and “S_Displacement” denotes the simulated data. The average displacement was derived from five consecutive cycles for each hydraulic cylinder. Cylinder No. 1 exhibits the largest deviation, with a test value of 31.9 mm and a simulated value of 34.5 mm, corresponding to a relative error of 8% and a phase lag of 7.8°. These values fall within the acceptable ranges for amplitude deviation (<10%) and phase offset (within ±5°~±15°), satisfying the controllability requirements for dynamic tool face angle adjustment during asynchronous operation of the three hydraulic cylinder. The overall trends exhibit excellent agreement, confirming that the developed simulation model is highly reliable and well-suited for characterizing the dynamic behavior of the actual system.

As seen from the displacement curves, the displacement cycles of the three piston rods are stable, and the time intervals between the occurrence of displacement peaks are equal, which conforms to the simple harmonic motion law with equal phase differences. However, due to load differences, the motion displacement values of each cylinder differ. Affected by the structural characteristics of the double-acting cylinder, the piston extension displacement during periodic motion is consistently greater than the retraction displacement. When the piston extends to a certain distance, it comes into contact with the hydraulic cylinder sleeve, and the displacement peak is constrained, resulting in a flat-top characteristic. During retraction, the piston does not contact the cylinder sleeve, and the displacement valley exhibits slight fluctuations during dynamic flow redistribution. This characteristic is consistent with the simulation results, further confirming the model’s reliability.

4.1.3. Quantitative Error Evaluation

The root mean square error (RMSE) of the piston displacement for each hydraulic cylinder is selected as the error evaluation index to compare the deviation between simulation and experimental data under various operating conditions. When the pump flow rate is set to 1 L/s and the rotary valve speeds are set to 60 rpm, 90 rpm, and 120 rpm, as shown in Figure 14a, the corresponding RMSE values are 0.267, 1.58, and 1.56, respectively. When the rotary valve speed is set to 90 rpm and the pump flow rates are set to 1 L/s, 1.25 L/s, and 1.5 L/s, as shown in Figure 14b, the corresponding RMSE values are 1.03, 2.07, and 2.20, respectively.

Through a comparison of test and simulation data under two sets of operating conditions, “fixed pump flow rate with variable rotary valve speed” and “fixed rotary valve speed with variable pump flow rate”, the reliability of the AMESim simulation model is fully verified from three aspects: trend consistency, error magnitude, and physical mechanism characterization, demonstrating that the model is free of deviations in its physical mechanism construction.

In terms of trend consistency, the variation laws of the simulated and measured data are perfectly consistent with no directional deviation. When the pump flow rate is fixed, as the rotary valve speed increases from 60 rpm to 120 rpm, the average displacement amplitude of the three hydraulic cylinder exhibits a decreasing trend in both datasets, with consistent variation directions of the inter-cylinder deviations; when the rotary valve speed is fixed, as the pump flow rate increases from 1 L/s to 1.5 L/s, the average displacement amplitude exhibits an increasing trend in both, with a highly consistent evolution law for inter-cylinder synchronization. In terms of error magnitude, the RMSE of displacement between the measured and simulated results falls within a reasonable range of 0.267–2.20, without significant abnormal deviations. This indicates that the model can accurately capture the inherent correlation among flow distribution frequency, fluid supply time, and displacement amplitude and simultaneously accurately characterize the physical quantitative relationship between fluid supply volume and piston displacement, which further demonstrates that the core logic of the simulation model is consistent with that of the actual physical system.

4.2. Influence Laws of Key Operating Parameters

4.2.1. Effect of Rotary Valve Speed (Fixed Pump Flow Rate)

Under a fixed pump flow rate of 1 L/s, the displacement characteristic simulation of the piston rods of the three hydraulic cylinder is conducted by gradually adjusting the rotary valve speed (60, 90, and 120 rpm) to investigate the regulation mechanism of rotary valve speed on the system’s steering behavior. The simulation results show that as the rotary valve speed increases, the piston displacement curves of the piston rods of the three hydraulic cylinder maintain periodic variation, the piston displacement amplitude exhibits a decreasing trend, and the motion cycle shortens with increasing rotary valve speed, as shown in Figure 15a, which demonstrates the regulating effect of rotary valve speed on the piston motion cycle.

As shown in Figure 15b, at a rotary valve speed of 60 rpm, the steady-state piston displacement amplitude of a single piston rod is 57.6 mm. As the speed increases to 90 rpm and 120 rpm, the displacement amplitude decreases to 42.0 mm and 32.3 mm, respectively, corresponding to a maximum reduction of 25.3 mm. This indicates a clear trend of decreasing displacement amplitude with increasing rotary valve speed. Although the pressure responses of the three hydraulic cylinder exhibit certain non-ideal characteristics, the piston displacement curves generally maintain the characteristics of simple harmonic motion with approximately equal phase difference, with stable cyclic behavior. Furthermore, the cycle-to-cycle peak displacement coefficient of variation (CV) is less than 1% under all tested conditions, as shown in

Table 5. Coefficient of variation (CV) under 1 L/s operating condition.

Rotary Valve Speed	Mean Peak Displacement	Standard Deviation		CV
$60 \mathrm{r}\mathrm{p}\mathrm{m}$	$57.6 \mathrm{m}\mathrm{m}$	$v$	$0.28 \mathrm{m}\mathrm{m}$	0.49%
$90 \mathrm{r}\mathrm{p}\mathrm{m}$	$42.0 \mathrm{m}\mathrm{m}$	$D_p$	$0.28 \mathrm{m}\mathrm{m}$	0.67%
$120 \mathrm{r}\mathrm{p}\mathrm{m}$	$32.3 \mathrm{m}\mathrm{m}$	$n_{pm}$	$0.27 \mathrm{m}\mathrm{m}$	0.84%

, demonstrating excellent cyclic stability of the system.

The underlying mechanism for this behavior can be summarized as follows: as the rotary valve speed increases, the flow distribution cycle shortens, reducing the fluid supply duration per cycle; this, in turn, decreases the driving force received by each cylinder per cycle, and consequently reduces the piston displacement amplitude. Meanwhile, given the consistent structural parameters of the three hydraulic cylinder, the piston responses remain synchronized, thus ensuring the characteristic of equal phase differences. This enables the adjustment of the tool face angle by varying the piston motion cycle.

4.2.2. Effect of Flow Rate (Fixed Rotary Valve Speed)

Under the condition of a fixed rotary valve speed of 90 rpm (a constant flow distribution speed for each hydraulic cylinder), the motion curves of the three piston rods are collected and simulation analysis is carried out by gradually adjusting the flow rate (1, 1.25, and 1.5 L/s) to investigate the influence of pump flow rate on the system’s steering behavior. As shown in Figure 16a, with increasing pump flow rate, the piston displacement amplitude of the three piston rods increases correspondingly. Although the pressure responses of the three hydraulic cylinder exhibit a non-ideal simple harmonic law with equal phase differences, they still satisfy the characteristic conditions for simple harmonic displacement with equal phase differences, with a stable motion cycle; the motion frequency of the three pistons remains consistent, which is manifested as the tool face angle sweeping around the axis at a fixed frequency.

The analysis of the piston displacement of a single piston rod under different flow rates is shown in Figure 16b. At a flow rate of 1 L/s, the steady-state displacement amplitude of the piston rod is approximately 42.0 mm. As the flow rate increases to 1.25 L/s and 1.5 L/s, the displacement amplitude rises to 44.1 mm and 55.2 mm, respectively, indicating a positive correlation between the flow rate and the piston displacement amplitude. An increase in flow rate leads to a greater fluid supply volume per cycle, which prolongs the piston stroke and consequently increases the piston displacement amplitude. These results confirm that precise regulation of the piston displacement—and thereby the tool bending angle—can be achieved by adjusting the inlet flow rate of the rotary valve inflow. As shown in

Table 6. Coefficient of variation (CV) under 90 rpm operating conditions.

Flow Rate	Mean Peak Displacement	Standard Deviation		CV
$1 \mathrm{L}/\mathrm{s}$	$42.0 \mathrm{m}\mathrm{m}$	$v$	$0.28 \mathrm{m}\mathrm{m}$	0.67%
$1.25 \mathrm{L}/\mathrm{s}$	$44.1 \mathrm{m}\mathrm{m}$	$D_p$	$0.28 \mathrm{m}\mathrm{m}$	0.63%
$1.5 \mathrm{L}/\mathrm{s}$	$55.2 \mathrm{m}\mathrm{m}$	$n_{pm}$	$0.27 \mathrm{m}\mathrm{m}$	0.49%

, the cycle-to-cycle coefficient of variation (CV) of peak displacement remains below 1% under all measured conditions, demonstrating excellent cyclic stability of the system.

4.3. Deviation Sources and Synchronization Optimization

A comparison of the results from the two operating conditions indicates that the flow rate has a more pronounced effect on the synchronization of the three piston rods. The inter-cylinder deviation in piston displacement amplitude increases substantially with either higher rotary valve speeds or larger flow rates. Considering the discrepancies observed between the simulation and experimental results, the key factors influencing this deviation sensitivity are identified as follows:

(1)

Manufacturing and processing deviations: The machining tolerance of the rotary valve port and the geometric error of the flow channel lead to inconsistent flow area of each cylinder, resulting in uneven flow distribution, which is the main source of inter-cylinder amplitude difference.

(2)

Non-ideal characteristics of the hydraulic system: Previous studies have demonstrated that transient pressure pulsations play a significant role in the observed non-ideal response, particularly under high-speed and high-flow-rate conditions [38,39]. Meanwhile, fluid compressibility introduces a time delay in pressure buildup within the cylinders. Furthermore, asymmetries in branched pipeline configurations give rise to differential frictional resistances. Nonlinear pressure drops across local flow resistances—including fittings, bends, and abrupt flow-area variations in valve manifolds—also intensify flow maldistribution among the hydraulic cylinder supply lines with an increasing flow rate. Simultaneously, dynamic clearance leakage at the piston–cylinder interface exhibits time-varying nonlinear characteristics under the coupled effects of lateral forces and pressure gradients, further exacerbating the discrepancy between actual and theoretical flow allocation to each cylinder. Collectively, these factors amplify the displacement deviations among the three hydraulic cylinder.

(3)

Simulation simplification errors: The simulation model does not fully reproduce the actual physical deviations, including manufacturing tolerances, the aforementioned nonlinear characteristics of dynamic leakage, the distributed parameter effects of branched pipelines (such as pressure wave propagation and reflection), and the quadratic pressure drops across local resistance elements. Consequently, the simulation error increases slightly under high rotational speed and large piston displacement conditions.

In view of the above deviation sources, a triple optimization scheme of “structural optimization + parameter correction + experimental condition improvement” is proposed to narrow the gap between simulation and experiment and improve the system synchronization performance and model reliability.

(1)

Structural optimization: Optimize the shape of the rotary valve port, adopt precision honing technology to reduce the roughness of the flow channel and improve the uniformity of flow distribution, reduce the fit gap between the valve core and the valve body, adopt a combined sealing structure to effectively control internal leakage, strictly control the coaxiality of the flow distribution channel and the tolerance of the flow area to reduce the impact of manufacturing deviations.

(2)

Parameter correction: Incorporate the test-measured leakage coefficient and the fluid compressibility coefficient into the simulation model, optimize the model parameter settings, and improve the simulation accuracy under high-speed and large-piston displacement working conditions.

(3)

Improvement of measurement conditions: Optimize the load control strategy, adopt a piston displacement sensor with a higher sampling frequency to reduce experimental measurement errors, and provide more accurate benchmark data for model validation.

4.4. Engineering Implications

The proposed rotary valve-distributed three hydraulic cylinder synchronized steering offset actuator is designed to address extreme downhole conditions, including high temperature, high pressure, strong vibration, and complex drilling fluid rheology. The core performance of the proposed actuator is reflected in the quantitative conversion from the piston axial displacement to the bit radial eccentricity and further to the RSS build dogleg severity (DLS). The piston displacement peak amplitude (the key output of the actuator) is converted into the wedge block radial displacement, corresponding to a wedge block inclination angle $\theta$, with the relationship:

$$\delta =x·\mathrm{t}\mathrm{a}\mathrm{n}\left(\theta \right)$$

(15)

Three wedge blocks uniformly distributed at 120° along the actuator circumference perform sinusoidal radial motions with equal phase difference, and their synthetic motion yields a constant bit radial eccentricity. Based on the classic RSS bottom hole assembly (BHA) engineering design theory, the bit radial eccentricity is translated into the achievable build rate/DLS (°/30 m) using the following relationship:

$$DLS\approx 2\times \mathrm{a}\mathrm{r}\mathrm{c}\mathrm{s}\mathrm{i}\mathrm{n}({\displaystyle \frac{\delta }{L}})\times {\displaystyle \frac{180}{\pi }}\times 30$$

(16)

Using the BHA parameters ($L=2.8 \mathrm{m},\theta =12°$), the experimentally observed piston displacements of 14.5 mm yield a DLS of 1.9°/30 m, while the displacement of 23.5 mm yields a DLS of 3.1°/30 m. Under the radial offset condition, the estimated DLS reaches approximately 10°/30 m, which is consistent with field requirements for medium- to high-build-rate rotary steerable systems [40].

4.5. Downhole Applicability Analysis

Downhole environments are harsh and complex, imposing strict constraints on the performance of the proposed actuator. This subsection elaborates on key downhole factors and their impacts, providing targeted countermeasures to ensure the actuator’s reliability.

4.5.1. Temperature and Pressure Effects

Downhole temperatures in deep/ultra-deep wells typically reach 150–250 °C and pressures of 20–100 MPa [41], which degrade fluid properties and sealing performance. When the temperature rises from 25 °C to 200 °C, high temperature can reduce the viscosity of downhole hydraulic fluids by 50–80%, accelerate fluid oxidation, and increase compressibility, leading to poor lubrication and leakage. High pressure above 60 MPa may cause plastic deformation of conventional sealing materials, resulting in a 30% increase in leakage rate. To address these issues, we can adopt high-temperature/pressure-resistant fluids and seals (e.g., PTFE composite seals with a temperature resistance of up to 260 °C) and optimize the pressure-bearing design of components.

4.5.2. Non-Newtonian Behavior of Downhole Fluids

Drilling muds (typical downhole fluids) exhibit non-Newtonian behavior (i.e., viscosity varies with shear rate), with the yield stress of common water-based drilling muds ranging from 5 to 25 Pa. This leads to uneven flow and a 15–25% increase in pressure loss in narrow flow channels [42]. To address these issues, we can optimize fluid formulations and flow channel design to adapt to viscosity variations, thereby ensuring stable flow distribution for the three hydraulic cylinders.

4.5.3. Vibration and Shock Loads

Axial, radial, and torsional vibrations, as well as shock loads during drilling (vibration frequency: 5–200 Hz, amplitude: 0.5–3 mm; shock intensity up to 2000 g in extreme cases [43]), cause component damage and valve–port misalignment, thereby affecting control accuracy. To address this issue, shock-absorbing structures, precision assembly, and fatigue-resistant materials (e.g., alloy steel with a fatigue limit of 350 MPa) can be adopted to enhance the actuator’s vibration resistance.

4.5.4. Sealing Durability

The long-term sealing reliability of the rotary valve is critical to avoid leakage and tool malfunction. Under continuous downhole operation, cyclic motion, high temperature/pressure, and fluid corrosion can reduce seal life by 40–60%. To address these issues, we can use wear- and corrosion-resistant seal materials (e.g., ceramic-coated seals with a wear rate of 0.02 mm/1000 cycles [44]), optimize the seal design, and add fluid filtration systems.

4.5.5. Summary of Gaps and Recommendations

The above analysis identifies key downhole challenges; however, several research gaps remain: the lack of quantitative sensitivity analysis on the relationships between temperature/pressure and fluid properties and insufficient experimental validation of the proposed mitigation strategies. To address these gaps, future work should quantify the influences of key factors, test the effectiveness of mitigation measures (e.g., verifying seal durability under simulated downhole conditions for 10,000+ cycles), and refine the actuator’s structural design to further improve its downhole applicability. Notably, compared with conventional approaches that rely on multiple solenoid valves or complex hydraulic circuits, the proposed system employs a single rotary valve to synchronously control three hydraulic cylinders, which, by integrating flow distribution functionality into a single moving component, effectively reduces potential failure points and ensures the resulting actuation efficiency meets engineering requirements. Additionally, the velocity displacement coupling regulation law provides a practical basis for downhole parameter matching and tool control strategy design, and overall, the optimized structure of the actuator contributes to extending the service life and improving the steering reliability of rotary steerable tools in deep and ultra-deep well drilling applications.

5. Conclusions

In this study, a novel three hydraulic cylinder synchronous steering offset actuator driven by a drilling fluid-driven rotary valve distribution was proposed for rotary steerable drilling systems. A systematic investigation was carried out using laboratory experiments and AMESim numerical simulations to reveal the piston displacement characteristics, synchronization performance, and key parameter influence laws of the actuator. The main conclusions are drawn as follows:

(1)

Rotational speed of the rotary valve and inflow rate exhibit significant and stable regulatory effects on the piston displacement output of the three hydraulic cylinder. With a fixed flow rate, the piston displacement amplitude decreases as the rotary valve speed increases. With fixed rotary valve speed, the piston displacement amplitude increases with increasing flow rate. Under both working conditions, the three hydraulic cylinder maintains stable motion with equal phase difference and approximate simple harmonic characteristics, which guarantees reliable and precise steering control.

(2)

The established AMESim numerical model shows consistency with experimental data. The variation trends of the simulation and measurement results are highly consistent, and the root mean square error (RMSE) is within 0.267–2.20. This verifies the reliability of the model and provides an effective tool for multi-condition analysis and parameter optimization of the steering actuator.

(3)

The synchronization deviation among the three hydraulic cylinder is mainly caused by valve port machining tolerance, drilling fluid compressibility, pipeline pressure loss, and internal leakage. The deviation tends to increase with higher rotary valve speed or larger flow rate. A comprehensive optimization strategy, including structural optimization, model parameter correction, and measure condition improvement, is proposed to effectively enhance synchronization performance.

(4)

The proposed actuator simplifies the system layout and improves steering stability. This design provides a new technical solution for wellbore trajectory control in deep and ultra-deep directional drilling and offers an important reference for the development and engineering application of rotary steerable drilling tools.

Future research and development of the rotary valve-structured three hydraulic cylinder synchronous steering offset actuator will focus on addressing its limitations to promote practical application, as it has not yet been put into field use. Specific measures include optimizing its structure and materials, designing a modular structure for better scalability, integrating intelligent control, and conducting relevant measures to lay a foundation for its field application.