A Dual-Layer Cooperative Feedback Control Method with Improved DBO-PID for Managed Pressure Drilling: Two-Phase Flow Experimental Verification

Abstract

During managed pressure drilling (MPD), gas influx intensifies the nonlinear relationship between choke valve opening degree and wellhead back pressure, causing conventional PID controllers to suffer from prolonged settling time and excessive overshoot. This paper proposes an automatic wellhead back pressure control method based on pressure–opening degree dual-layer cooperative feedback. The outer layer rapidly positions the choke valve near the target opening degree through a pressure drop–opening degree mapping model. The inner layer employs a PID controller tuned by an improved Dung Beetle Optimizer (DBO) for fine pressure regulation. The improved DBO introduces Logistic chaotic map initialization and an adaptive inertia weight to enhance global search capability, and adopts a comprehensive fitness function integrating the ITAE criterion with engineering safety constraints. Simulation results show that, compared with the Ziegler–Nichols (Z-N) method, the improved DBO-tuned PID reduces overshoot by 83.9% and settling time by 78.0%. Gas–liquid two-phase flow laboratory experiments were conducted with gas void fractions of 0–46.6%. Using manual control (average settling time of 50 s) as the benchmark, the dual-layer system equipped with the improved DBO-PID reduces settling time to 25 s (a 50% reduction), maximum overshoot absolute error to 0.009 MPa (a 74% reduction compared with Z-N-tuned PID), and achieves a mean absolute error of 0.004 MPa during continuous pressure tracking with zero overshoot. Both simulation and experimental results confirm that the synergy between the dual-layer control architecture and the improved DBO-PID enables rapid regulation and stable tracking of wellhead back pressure under gas–liquid two-phase flow conditions.

1. Introduction

With the continued advancement of deep-earth and deep-sea oil and gas resource exploration and development, the formation conditions encountered in drilling operations have become increasingly complex. The safe density window between formation pore pressure and lost circulation pressure has narrowed considerably, making it difficult for conventional drilling methods to maintain bottomhole pressure within the safe range at all times, which readily triggers well kick or lost circulation incidents [1,2,3]. Managed pressure drilling (MPD) technology applies an adjustable back pressure at the wellhead to regulate the bottomhole equivalent circulating density in real time, maintaining it precisely within the formation pressure window, and has become a key technical approach for addressing the challenges of narrow density window drilling [4,5].

In MPD systems, the choke valve is the core actuator for implementing wellhead back pressure regulation [6]. Under constant bottomhole pressure (CBHP) conditions, the system adjusts the choke valve opening degree based on the target pressure magnitude, changing the annular flow cross-sectional area at the wellhead to control wellhead back pressure [4]. Currently, in field operations, choke valve opening degree is primarily adjusted manually by engineers based on the target pressure and real-time back pressure feedback [6,7]. However, during MPD operations, formation gas influx transforms the wellbore fluid from single-phase liquid flow to gas–liquid two-phase flow, and the compressibility of gas causes severe wellhead back pressure fluctuations. Meanwhile, under gas–liquid two-phase flow conditions, the relationship between choke valve opening degree and wellhead back pressure exhibits significant nonlinearity—in the small opening degree range, wellhead back pressure is highly sensitive to opening degree changes, and minor opening adjustments can cause abrupt pressure changes [8,9]. Under the combined influence of gas–liquid two-phase flow pressure fluctuations and choke valve nonlinear characteristics, wellhead back pressure during manual control exhibits large fluctuations and prolonged settling time, making it difficult for control accuracy and response speed to meet the safe drilling requirements under narrow density window conditions [10]. Therefore, developing automatic choke valve control methods to replace manual operation has become an urgent need for MPD technology development [11,12].

Regarding the automatic regulation of MPD wellhead back pressure, researchers have conducted studies from various perspectives. PID controllers, owing to their simple structure and low dependence on controlled object models, are currently the most widely applied method in MPD back pressure control [4,5,6,7,8,9,10,11,12,13]; however, conventional fixed-parameter PID struggles to adapt to the strong nonlinear characteristics of choke valves and is prone to overshoot and response lag when operating conditions change [14,15]. To overcome the limitations of PID, Gorjizadeh et al. [16] designed an MPD choke valve fuzzy state feedback controller based on T-S fuzzy modeling, which outperforms conventional PID under actuator constraints, but the fuzzy rule design relies on experience. Sule et al. [17] and Park et al. [18] applied nonlinear model predictive control (NMPC) to MPD gas kick scenarios respectively, achieving high pressure tracking accuracy, but the dependence on real-time flow models constrains the convenience of engineering implementation. Liang et al. [19] applied an improved genetic algorithm to optimize MPD back pressure fuzzy PID parameters. Long et al. [20] introduced the standard DBO into MPD back pressure PID parameter tuning and verified its feasibility through single-phase flow experiments. Regarding control architecture, Wang [21] proposed a three-level feedback regulation method that improves regulation efficiency through phased coordination of rapid opening degree positioning and PID fine-tuning, but the PID parameters still rely on manual tuning and lack verification under gas–liquid two-phase flow conditions. Beyond these approaches, adaptive control methods have also been explored for MPD pressure regulation. Carlsen et al. [22] demonstrated automated well control through a combined PI-IMC-MPC strategy, and Siahaan et al. [23] proposed an adaptive PID switching controller capable of automatic parameter adjustment across operating conditions, though both involve complex control structures that complicate engineering deployment. Ribeiro et al. [24] and Sheikhi et al. [25] introduced neural network-based and nonlinear predictive controllers respectively, demonstrating adaptive capability in pressure regulation but relying on accurate process models and validated primarily under single-phase flow conditions. In general, existing MPD pressure control methods either depend on accurate physical models, involve complex control architectures, or lack experimental validation under gas–liquid two-phase flow conditions.

The above review suggests that architecture-oriented studies tend to rely on manual PID tuning, and improving parameter tuning quality represents a direct path to better control performance. Although the classical Ziegler–Nichols (Z-N) tuning method is operationally simple, its reliance on linear model assumptions leads to large overshoot and pronounced oscillation in nonlinear systems [26,27,28]. With the development of metaheuristic optimization algorithms, intelligent algorithms such as genetic algorithm (GA), particle swarm optimization (PSO), and grey wolf optimizer (GWO) have been widely used for PID parameter tuning due to their gradient-free nature and applicability to nonlinear multimodal optimization problems [29,30]. The Dung Beetle Optimizer (DBO), proposed in 2023 as a novel metaheuristic algorithm, has demonstrated convergence speed and solution accuracy superior to PSO, GWO, and other algorithms in multiple benchmark tests [31]. However, the standard DBO suffers from uneven initial population distribution and insufficient balance between global search and local exploitation [32]. Researchers have improved DBO by introducing Lévy flight, opposition-based learning, dynamic boundary contraction, and other techniques, and have verified the effectiveness in PID parameter tuning in other industrial domains [33,34,35]. Nevertheless, the aforementioned studies on improved DBO for PID tuning are all oriented toward single-phase, deterministic industrial systems and have not yet been applied to the complex scenario of MPD with strong nonlinear characteristics of gas–liquid two-phase flow, nor have they been experimentally validated. More recently, Cai et al. [36] applied an adaptive DBO to optimize fuzzy PID parameters for nutrient solution temperature control in hydroponic systems, further confirming DBO’s applicability to nonlinear closed-loop control problems in process engineering. Similarly, Yan et al. [37] applied a variable universe fuzzy PID strategy to regulate hydraulic system pressure in anchor drilling machines under nonlinear load disturbances, demonstrating that adaptive adjustment of the fuzzy universe can substantially reduce pressure tracking error compared to conventional PID control. These recent advances in intelligent optimization and adaptive strategies across various engineering domains provide strong motivation for developing a tailored, metaheuristic-optimized control architecture for the complex MPD pressure regulation process.

As summarized in

Table 1. Qualitative comparison of representative MPD pressure control approaches across key engineering performance dimensions.

Method	Requires Accurate Physical Model	Computational Cost	Two-Phase Flow Exp. Validation	Engineering Implementability	Large-Deviation Rapid Response
NMPC	High	High	No	Difficult	Yes
Fuzzy PID	Partial	Low	No	Moderate	Partial
PSO/GA PID	No	Medium	No	Moderate	No
Adaptive Control	Partial	Medium	No	Difficult	Partial
This Work	No	Low	Yes	Easy	Yes

, existing MPD pressure control methods each present notable limitations in at least one of the following aspects: dependence on accurate physical models, high computational cost, lack of two-phase flow experimental validation, or limited large-deviation handling capability. The proposed method addresses these gaps simultaneously.

The main contributions include: (1) establishing a pressure–opening degree dual-layer cooperative feedback control method, in which two closed-loop feedback layers operate in a switching manner to achieve the synergy between rapid opening degree coarse adjustment and fine pressure regulation, without requiring simultaneous operation of both loops as in conventional cascade control. (2) improving the standard DBO algorithm by incorporating chaotic initialization and adaptive weighting strategies, and designing a comprehensive fitness function to enhance PID parameter tuning effectiveness for the MPD pressure control objective. (3) providing systematic simulation and gas–liquid two-phase flow laboratory experimental validation of the proposed method, demonstrating its effectiveness across a gas void fraction range of 0–46.6%.

2. Methods

2.1. Pressure–Opening Degree Dual-Layer Cooperative Feedback Control Method

2.1.1. Control Logic

Conventional PID controllers rely solely on pressure error to compute the control output for adjusting the choke valve opening degree. However, under gas–liquid two-phase flow conditions, the strong nonlinearity between opening degree and wellhead back pressure in the small opening degree range necessitates conservative gain settings to avoid abrupt pressure fluctuations. Consequently, when the target pressure undergoes a large change and the corresponding opening degree adjustment stroke is substantial, the limited per-cycle adjustment magnitude constrained by the conservative gain settings causes the controller to require multiple control cycles to gradually approach the target opening degree [4]. During this adjustment process, the integral term continuously accumulates error in the large deviation interval, readily causing overshoot and oscillation. Under gas–liquid two-phase flow conditions, the compressibility of gas further exacerbates the nonlinearity of the pressure–opening degree response, making the above problems more pronounced.

Previous studies have demonstrated that incorporating choke valve opening degree information into the feedback loop, through phased coordination of rapid opening degree pre-positioning and fine pressure regulation, can shorten the settling time in the large deviation interval [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38]. Building upon this foundation, this paper proposes a pressure–opening degree dual-layer cooperative feedback control method, with the control logic illustrated in Figure 1. In the diagram, the upper layer is the opening degree feedback loop, which achieves rapid coarse adjustment of the choke valve opening degree; the lower layer is the pressure feedback loop, where the PID controller performs fine regulation of wellhead back pressure. The system automatically switches between the two loops based on whether the opening degree deviation exceeds a preset threshold.

The execution process of this method is as follows: First, based on the choke valve pressure drop–opening degree characteristic relationship, the target pressure is mapped to the target choke valve opening degree and compared with the actual opening degree measured by the displacement sensor to obtain the opening degree deviation by Equation (1):

$$\Delta \theta ={\theta }_{\mathrm{t}}-{\theta }_{\mathrm{a}}$$

(1)

where Δθ is the opening degree deviation, mm; θ_t is the target opening degree, mm; θ_a is the actual opening degree, mm.

When the absolute opening degree deviation |Δθ| exceeds the preset threshold, the system enters the rapid opening degree adjustment phase, driving the choke valve toward the target opening degree. When |Δθ| decreases to within the threshold range, the system switches to the PID fine regulation phase, using the deviation between target pressure and actual wellhead back pressure as the control input by Equation (2):

$$e_{\mathrm{P}}=P_{\mathrm{t}}-P_{\mathrm{a}}$$

(2)

where e_P is the pressure deviation, MPa; P_t is the target pressure (i.e., target wellhead back pressure), MPa; P_a is the actual wellhead back pressure, MPa.

The PID controller continuously fine-tunes the choke valve opening degree based on the pressure deviation to achieve stable tracking of wellhead back pressure to the target pressure. The above control process executes cyclically at a set period, with choke valve opening degree and wellhead back pressure signals fed back to the respective comparison nodes, forming a pressure–opening degree dual-layer closed-loop control structure. The target opening degree calculation method and the opening degree deviation switching threshold selection are described in Section 2.1.2 and Section 2.1.3, respectively.

2.1.2. Target Opening Degree Calculation Model

During the rapid opening degree adjustment phase, the target choke valve opening degree needs to be determined based on the target pressure. For any type of choke valve, the relationship between choke valve pressure drop and flow rate and opening degree can be expressed as Equations (3) and (4) [8,9]:

$$P_c={\left({\displaystyle \frac{Q}{C_{\mathrm{V}}\theta }}\right)}^2{\rho }_m$$

(3)

$${\rho }_{\mathrm{m}}=\left(1-\alpha \right){\rho }_{\mathrm{l}}+\alpha {\rho }_{\mathrm{g}}$$

(4)

where P_c is the pressure drop across the choke valve, MPa; Q is the flow rate, m³/h; C_V is the flow coefficient, dimensionless; θ is the choke valve opening degree, mm; ρ_m is the fluid density, kg/m³; α is the gas void fractions, dimensionless; ρ_l is the liuquid density, kg/m³; ρ_g is the gas density, kg/m³.

Given the low operating pressure range of this study (gauge pressure 0–0.6 MPa), the pressure-dependent variation in air density is small relative to the liquid density of water, and the ideal gas approximation is sufficient to characterize air properties across the experimental conditions.

From Equation (3), a functional relationship exists between choke valve opening degree, flow rate, and pressure drop. However, the flow coefficient C_V is difficult to determine accurately due to its dependence on choke valve geometry and flow conditions; therefore, it is necessary to establish a target opening degree calculation model through characteristic experiments.

Taking the needle-type choke valve used in the laboratory experiments of this study as an example, with both clean water and gas–liquid two-phase flow at varying gas void fractions as the experimental fluids, the corresponding relationship between choke valve pressure drop and opening degree was measured under different flow rate conditions. Based on the experimental data, a quadratic polynomial regression using the least squares method was performed [38] to establish an explicit relationship between the target choke valve opening degree and pressure drop and flow rate given by Equation (5):

$${\theta }_{\mathrm{t}}{=b}_0+b_1{P}_c+b_{2}Q+b_3{P}_c^2+b_4{Q}^2+b_5{P}_{c}Q$$

(5)

where the regression coefficients are: b₀ = 7.87; b₁ = −25.72; b₂ = 0.4; b₃ = 37.59; b₄ = 0.03; b₅ = 1.11.

The model has a coefficient of determination R² = 0.8196, a root mean square error of 1.51 mm, and a mean absolute error of 1.14 mm. The R² does not reach 0.9, primarily for two reasons. First, although the quadratic polynomial regression form in Equation (5) was derived from the single-phase flow pressure drop–opening degree relationship, the experimental dataset used for regression includes data points obtained under both clean water single-phase flow and gas–liquid two-phase flow conditions with varying gas void fractions. Under two-phase flow conditions, the variations in fluid density, flow pattern, and slip velocity introduce additional scatter that a single quadratic polynomial cannot fully capture. Second, the flow coefficient C_V in Equation (3) is itself a function of flow conditions and valve geometry, yet the regression model implicitly treats it as absorbed into fixed polynomial coefficients, which limits the model’s ability to accommodate the full range of operating conditions. Nevertheless, as shown in Figure 2, the predicted absolute errors of the target opening degree are mostly less than 2 mm, accounting for approximately 5% of the full stroke of the needle-type choke valve (0–44 mm). This level of accuracy is sufficient for the intended purpose: the target opening degree model is not required to meet the final control accuracy; its role is solely to rapidly position the choke valve near the target opening degree during the coarse adjustment phase, with the remaining deviation eliminated by the PID controller during the fine regulation phase.

2.1.3. Opening Degree Deviation Switching Threshold

The setting of the opening degree deviation switching threshold needs to balance two requirements. If the threshold is too large, the choke valve opening degree still has a significant deviation from the target value when the rapid opening degree adjustment phase ends, and the adjustment amount per control cycle during the PID fine regulation phase is limited, requiring more cycles to achieve stable tracking of the target pressure, thereby prolonging the settling time. If the threshold is too small, the system tends to switch frequently between the two regulation phases, affecting control stability.

To ensure that the choke valve opening degree enters the effective regulation range of the PID controller after the rapid opening degree adjustment is completed, and that the PID controller operates within the small deviation interval from the moment of intervention, the opening degree deviation switching threshold was set to 5 mm, which exceeds the predicted absolute errors of the target opening degree model in Section 2.1.2 (mostly less than 2 mm). This threshold is approximately 11% of the full stroke of the needle-type choke valve (0–44 mm), larger than the model prediction error yet smaller than the opening degree adjustment stroke corresponding to typical target pressure changes, balancing coarse adjustment accuracy and switching stability. For other types of choke valves, the target opening degree regression model needs to be re-established based on their characteristic experimental data, and the switching threshold should be adjusted accordingly.

2.2. Transfer Function of the Choke Valve Control Process

This paper employs the improved DBO algorithm for intelligent tuning of PID controller parameters (see Section 2.3). The tuning process requires evaluating the control performance of different parameter combinations based on the transfer function of the controlled object in a simulation environment; therefore, it is necessary to establish the transfer function of the choke valve control process during the PID fine regulation phase.

When the opening degree deviation decreases to within the threshold range, the system switches to the PID fine regulation phase, with its control loop shown in Figure 3. The PID controller calculates the pressure deviation according to Equation (2) and outputs a control signal based on the following control Equation (6):

$$u(t)=K_{\mathrm{p}}\left[e_{\mathrm{p}}(t)+{\displaystyle \frac{1}{T_{\mathrm{i}}}}{\displaystyle {\int }_0^i{e}_{\mathrm{p}}(t)\mathrm{d}t+T_{\mathrm{d}}\frac{d{e}_{\mathrm{p}}\left(t\right)}{\mathrm{d}t}}\right]$$

(6)

where u(t) is the output signal value; t is time, s; K_p is the proportional gain; T_i is the integral time constant; T_d is the derivative time constant.

The control signal is amplified by a signal amplifier and then input to an electro-hydraulic proportional valve, which adjusts the hydraulic oil flow rate according to the input signal, driving the hydraulic cylinder actuator to move the choke valve spool, thereby achieving continuous adjustment of the choke valve opening degree and consequently causing changes in wellhead back pressure [39]. A pressure sensor collects wellhead back pressure in real time and feeds it back to the PID controller, forming a closed-loop regulation process. The above components are connected in series, and their overall transfer characteristic constitutes the controlled object transfer function G required for PID parameter tuning.

According to the control loop shown in Figure 3, the transfer process from the control signal acting on the actuator to the wellhead back pressure change can be divided into four components: the signal amplifier G_a, characterizing the linear amplification of the input signal; the electro-hydraulic proportional valve G_v, converting the electrical signal to hydraulic oil flow rate; the hydraulic cylinder actuator G_h, converting hydraulic drive to spool displacement; and the choke valve opening degree–wellhead back pressure transfer component G_p, describing the pressure response caused by opening degree changes. These components are connected in series given by Equation (7):

$$G\left(s\right)=G_a\left(s\right)\times G_v\left(s\right)\times G_h\left(s\right)\times G_p\left(s\right)$$

(7)

where s denotes the Laplace transform complex variable.

Among these, G_a, G_v, and G_h all belong to the choke valve drive system, whose dynamic characteristics mainly represent signal amplification, inertia, and integral characteristics [39]. The differences among various types of choke valves are mainly reflected in G_p, i.e., the regulation characteristics of opening degree changes on wellhead back pressure. Previous studies have shown that the series transfer process of the MPD choke valve control system can be simplified to a second-order model with pure time delay [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]. While preserving the main dynamic characteristics, the above series components are simplified to an equivalent low-order model by Equation (8) [19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40]:

$$G(s)={\displaystyle \frac{K{\mathrm{e}}^{-\delta \mathrm{s}}}{(T_{1}s+1)\times (T_{2}s+1)}}$$

(8)

where K is the pressure–opening degree gain coefficient, characterizing the degree of influence of choke valve opening degree changes on wellhead back pressure; T₁ and T₂ are the equivalent time constants, describing the inertia characteristics of the actuator and pressure transfer process, s; δ is the equivalent time delay, characterizing the time delay of control signal transmission and wellhead back pressure change, s.

The model structure of Equation (8) is applicable to choke valves of different structural forms and flow characteristics, with differences reflected only in the parameter values, which need to be re-identified according to the specific choke valve type in practical applications. Furthermore, the applicability of this transfer function is ensured by the dual-layer control structure: the rapid opening degree adjustment phase has already adjusted the choke valve opening degree to near the target value, and the PID controller only intervenes for fine regulation after the opening degree deviation enters the threshold, at which point the opening degree adjustment range is limited, and the second-order inertia transfer function with time delay can meet the model accuracy requirements for PID parameter tuning.

2.3. PID Parameter Tuning Method Based on the Improved DBO Algorithm

2.3.1. Standard DBO Algorithm

The Dung Beetle Optimizer (DBO) achieves a dynamic balance between global search and local exploitation by simulating the cooperative behaviors of four types of individuals: ball-rolling dung beetles, brood dung beetles, small dung beetles, and thief dung beetles [31]. In the PID parameter tuning problem, each dung beetle’s position corresponds to a PID parameter vector by Equation (9):

$$X_i=(K_{\mathrm{p}}, T_{\mathrm{i}}, T_{\mathrm{d}})$$

(9)

where X_i represents the position vector of the i-th individual in the parameter space. When the population size is N, the ball-rolling dung beetles, brood dung beetles, small dung beetles, and thief dung beetles account for 30%, 20%, 30%, and 20% of the population, respectively.

(1)

Ball-rolling dung beetles. Ball-rolling dung beetles simulate the behavior of dung beetles pushing food balls in a straight line along the light source direction, performing wide-area search in the PID parameter space. Their position update is given by Equation (10):

$$\left\{\begin{array}{c}{X}_i\left(k+1\right)=X_i\left(k\right)+\alpha \times H\times X_i\left(k-1\right)+b\times \Delta X\\ \Delta X=\left|X_i\left(k\right)-X_{\mathrm{w}}\right|\end{array}\right.$$

(10)

where k denotes the current iteration number; X_i(k) is the position of the i-th dung beetle at the k-th iteration; α is a natural coefficient with values of −1 or 1; H ∈ (0, 0.2] represents the deflection coefficient; b is a constant with b = 0.3; ΔX simulates the change in light intensity; X_w represents the global worst position.

When a dung beetle encounters an obstacle, it adjusts its direction through dancing behavior, which is simulated using the tangent function. The position update is given by Equation (11):

$$X_i\left(k+1\right)=X_i\left(k\right)+\tan \phi \left|X_i\left(k\right)-X_i\left(k-1\right)\right|$$

(11)

where φ is the deflection angle after the dung beetle dances, φ ∈ [0, π]. When φ equals 0, π/2, or π, the second term on the right side of Equation (10) becomes zero, and the dung beetle’s position is not updated.

(2)

Brood dung beetles. The search range of brood dung beetles is centered on the global best position X_b and gradually contracts with iterations. Their position update is given by Equation (12):

$$\left\{\begin{array}{c}{X}_i\left(k+1\right)=X_{\mathrm{b}}+B_1\times \left(X_i\left(k\right)-L\right)+B_2\times \left(X_i\left(k\right)-U\right)\\ L=\max \left(X_{\mathrm{b}}-\mathrm{R}\left(U_{\mathrm{b}}-L_{\mathrm{b}}\right), L_{\mathrm{b}}\right)\\ U=\min \left(X_{\mathrm{b}}+\mathrm{R}\left(U_{\mathrm{b}}-L_{\mathrm{b}}\right), U_{\mathrm{b}}\right)\\ R=1-{\displaystyle \frac{k}{N_{max}}}\end{array}\right.$$

(12)

where B₁ and B₂ are uniform random numbers in (0, 1); L and U are the dynamic search boundaries centered on the global best position; L_b and U_b are the lower and upper bounds of the PID parameter search range, respectively; R is the contraction factor, which decreases linearly with iterations.

(3)

Small dung beetles. Small dung beetles perform fine search within the foraging area. Their position update is given by Equation (13):

$$X_i\left(k+1\right)=X_i\left(k\right)+C_1\left[X_i\left(k\right)-L\right]+C_2\left[X_i\left(k\right)-U\right]$$

(13)

where C₁ is a random number following the standard normal distribution; C₂ is a uniform random vector in (0, 1). Unlike brood dung beetles that use the global best position X_b as the basis, small dung beetles search based on their own current position X_i(k), performing fine foraging near the current position through normally distributed random perturbations.

(4)

Thief dung beetles. Thief dung beetles perform perturbation search around the position of a randomly selected individual in the population to increase solution diversity. Their position update is given by Equation (14):

$$X_i\left(k+1\right)=X_{\mathrm{j}}+S\times g\left(\left|X_i\left(k\right)-X^{*}\right|+\left|X_i\left(k\right)-X_{\mathrm{b}}\right|\right)$$

(14)

where X_j is the position of a randomly selected individual in the population; S is a constant with S = 0.5; g represents a random vector following the normal distribution; X^* is the current global best position.

When X^* = X_b, the two absolute value terms |Xi(k) − X^*| and |Xi(k) − X_b| in Equation (14) become identical, and their sum reduces to 2*|Xi(k) − X_b|. In this case, the perturbation step size of the thief dung beetle is entirely determined by twice the distance between the current position and the global best position, effectively doubling the weight of the attraction toward the global optimum in the perturbation update.

The four types of dung beetles continuously update their positions through the above cooperative search mechanism, ultimately outputting the globally optimal PID parameter combination.

2.3.2. Two Improvements

The standard DBO algorithm has two shortcomings when applied to choke valve PID parameter tuning in managed pressure drilling [19,32]: first, the random initialization method tends to cause uneven population distribution in the three-dimensional PID parameter space, potentially missing potential optimal regions; second, the fixed search step size strategy lacks adaptive adjustment capability, making it difficult to balance global search and local exploitation. To address these shortcomings, the following two improvements were implemented.

Improvement 1: Population initialization based on Logistic chaotic map.

To improve the distribution uniformity of the initial population in the PID parameter space, the Logistic chaotic map is adopted to replace random initialization for generating the initial population by Equation (15) [41]:

$$z_{k+1}=\mu \times z_k\times (1-z_k)$$

(15)

where z_k is the chaotic variable at the k-th iteration, z_k ∈ (0, 1); μ is the chaotic sequence control parameter.

The Logistic map is in a fully chaotic state at μ = 4, and the generated sequence possesses ergodicity and aperiodicity. After mapping the chaotic sequence to the PID parameter search range, the initial population individuals can cover the entire parameter space more uniformly, reducing the risk of missing potential optimal regions.

Improvement 2: Adaptive inertia weight factor. To achieve a dynamic balance between global exploration and local fine exploitation during the search process, an adaptive inertia weight factor that decreases linearly with the iteration number is introduced by Equation (16):

$$\omega ={\omega }_{\max }-{\displaystyle \frac{{\omega }_{\max }-{\omega }_{\min }}{N_{\max }}}\times k$$

(16)

where ω is the inertia weight; ω_max is the maximum inertia weight; ω_min is the minimum inertia weight.

The adaptive inertia weight factor ω is applied to the position update processes of ball-rolling dung beetles, small dung beetles, and thief dung beetles, while the position update mechanism of brood dung beetles remains unchanged. The improved position updates are given by Equations (17)–(19).

For the ball-rolling dung beetle, the ω term acts on the direction increment term, and the updated position is as follows:

$$\left\{\begin{array}{c}{X}_i\left(k+1\right)=X_i\left(k\right)+\omega \times \left(\alpha \times H\times X_i\left(k-1\right)+b\times \Delta X\right)\\ \Delta X=\left|X_i\left(k\right)-X_{\mathrm{w}}\right|\end{array}\right.$$

(17)

Compared with the Formula (10) of the standard DBO, ω simultaneously scales the inertia term and the light intensity change term, acting on the entire search step length. In the early stages of iteration, ω is larger, and the contribution of the previous position X_i(k−1) to the new position is greater, resulting in a larger search step length, which is conducive to conducting large-scale global exploration in the PID parameter space; as the iteration progresses, ω gradually decreases, the search step length shrinks, and the algorithm enters the stage of fine local optimization.

For the small dung beetle, ω scales the normally distributed perturbation term, and the updated position becomes:

$$X_i\left(k+1\right)=X_i\left(k\right)+\omega \times C_1\times \left[X_i\left(k\right)-L\right]+C_2\times \left[X_i\left(k\right)-U\right]$$

(18)

Compared with the standard DBO Equation (13), for the small dung beetle, ω causes the normal random search range of the small dung beetle to further shrink in the later stages of the iteration, which is conducive to conducting fine local optimization near the current position.

For the case of thief dung beetles, ω scales the perturbation step size, and the updated position becomes:

$$X_i\left(k+1\right)=X_{\mathrm{j}}+\omega \times S\times g\left(\left|X_i\left(k\right)-X^{*}\right|+\left|X_i\left(k\right)-X_{\mathrm{b}}\right|\right)$$

(19)

Compared with the Formula (14) of the standard DBO, ω reduces the perturbation amplitude of the thief dung beetle in the later stages of the iteration.

In the early iterations, ω is large, and the search step size of ball-rolling dung beetles is large, facilitating wide-range global exploration in the PID parameter space to rapidly locate parameter regions that meet the wellhead back pressure regulation speed requirements. As iterations progress, ω gradually decreases and the search step size shortens correspondingly, with the algorithm transitioning to the fine local exploitation phase, which facilitates further optimization of PID parameters within candidate optimal regions, reducing wellhead back pressure overshoot and steady-state fluctuations. This adaptive strategy can accommodate the dual requirements of managed pressure drilling for both rapidity and stability of wellhead back pressure regulation.

2.3.3. Fitness Function

The design of the fitness function directly affects the PID parameter tuning results, as different fitness functions will guide the optimization algorithm to search for different optimal solutions in the parameter space. Managed pressure drilling requires wellhead back pressure to converge rapidly to the steady state after a target pressure change, while allowing a certain transient deviation during the initial regulation period. Based on these requirements, this paper selects the integral of time-weighted absolute error (ITAE) as the basic performance index by Equation (20) [29]:

$$J_0={\displaystyle {\int }_0^{T}t\times \left|e_p(t)\right|}dt$$

(20)

where e_p(t) is the pressure deviation at time t; T is the simulation time window, s.

ITAE applies an increasing penalty to residual errors in the later regulation period through the time-weighting factor t, biasing the optimization process toward selecting parameter combinations with fast late-stage convergence and high steady-state accuracy, while exhibiting higher tolerance for transient deviations during the initial regulation period. This characteristic matches the control requirements of managed pressure drilling, which allows moderate initial deviations but demands rapid convergence.

The ITAE index alone only measures the time-accumulated characteristics of pressure deviation without imposing independent constraints on overshoot and settling time. Therefore, a dynamic performance penalty term P₁ and a safety constraint penalty term P₂ are introduced on the basis of ITAE to form a comprehensive fitness function by Equations (21)–(23):

$$f={\displaystyle \frac{1}{J_0+P_1+P_2}}$$

(21)

$$P_1={\lambda }_1\times \max \left(\sigma -{\sigma }_0,0\right)+{\lambda }_2\times T_{\mathrm{ac}}+{\lambda }_3\times T_{\mathrm{r}}$$

(22)

$$P_2={\mu }_1\times \max \left(\sigma -{\sigma }_{\max },0\right)+{\mu }_2\times \max \left(e-e_{\max },0\right)$$

(23)

where, σ represents the percentage overshoot of the unit step response, %; σ₀ is the overshoot penalty threshold, and a penalty is applied when σ > σ₀; T_ac is the settling time, s; T_r is the rise time, s; λ₁, λ₂, and λ₃ are the penalty coefficients for overshoot and rise time respectively; σ_max is the maximum allowable overshoot, %; e is the steady-state error percentage, %; e_max is the maximum allowable steady-state error, %; μ₁ and μ₂ are the safety constraint penalty coefficients.

The assignment of penalty coefficients in Equation (21) reflects the engineering safety priorities of managed pressure drilling. The overshoot penalty coefficient λ₁ is greater than the settling time penalty coefficient λ₂, because the engineering consequences of the two are not equivalent: a prolonged settling time means that wellhead back pressure temporarily deviates from the target value, whereas overshoot means that wellhead back pressure momentarily exceeds the target value, causing bottomhole pressure to deviate from the safe pressure window. The safety constraint penalty coefficients μ₁ and μ₂ are much larger than λ₂, serving as hard constraint boundaries in the optimization space. When the overshoot exceeds σ_max or the steady-state error exceeds e_max, P₂ increases sharply, causing the fitness value of this parameter combination to approach zero, and thus being eliminated by the optimization algorithm. The simulation parameter settings are given in

Table 2. Parameter settings of the improved DBO algorithm.

Parameter	Symbol	Value
Population size	N	50
Maximum iterations	Nmax	200
PID search range	[Kp, Ti, Td]	[0.001, 20]
Simulation time window	T	20 s
Simulation step size	—	0.01 s
Inertia weight range	ωmax/ωmin	0.9/0.1
Logistic control parameter	μ	4
Overshoot penalty threshold	σ0	5%
Overshoot penalty coefficient	λ1	7
Settling time penalty coefficient	λ2	0.6
Rise time penalty coefficient	λ3	0.2
Max allowable overshoot	σmax	20%
Max allowable steady-state error	emax	2%
Safety constraint penalty coefficients	μ1/μ2	200

.

2.3.4. Parameter Tuning Procedure

Based on the improved DBO algorithm and the comprehensive fitness function, the overall procedure for intelligent PID parameter tuning is shown in Figure 4. (a) Initialize algorithm parameters, including population size N and maximum number of iterations N_max; (b) input the choke valve control process transfer function model G(s) as the controlled object; (c) generate the initial population using the Logistic chaotic map; (d) perform closed-loop system step response for each candidate PID parameter combination; (e) calculate the comprehensive fitness value f according to Equation (21); (f) update positions according to the improved DBO algorithm; (g) determine whether the current iteration number k has reached N_max; (h) output the optimal PID parameter combination (K_p^*, T_i^*, T_d^*).

3. Results and Discussion

3.1. Simulation Verification

Based on the choke valve control process transfer function established in Section 2.2, the PID parameter tuning effectiveness of the improved DBO algorithm was verified through unit step response simulation. The transfer function used in the simulation was determined according to the type of needle-type choke valve used in the laboratory experiments, incorporating the transfer processes of the signal amplifier, electro-hydraulic proportional valve, hydraulic cylinder, and choke valve, while considering the actuator response time and pressure transmission delay. The transfer function parameters were identified through the step response characteristics of the needle-type choke valve in preliminary experiments [39]: under constant flow rate conditions, multiple step inputs of different amplitudes were applied to the choke valve opening degree, and the wellhead back pressure time response curves after each step adjustment were recorded. The graphical method was used to extract the static gain K (the ratio of steady-state pressure change to opening degree change), equivalent time constants T₁ and T₂ (fitted based on the rising segment of the response curve), and equivalent time delay δ (the initial response delay) from the response curves. The resulting transfer function is by Equation (24):

$$G(s)={\displaystyle \frac{0.85{\mathrm{e}}^{-{0.3}_s}}{(0.8s+1)(0.2s+1)}}$$

(24)

This transfer function also serves as the controlled object model for the PID controller in the subsequent laboratory experiments. Using the step response performance of the PID parameters tuned by the Z-N trial-and-error method (K_p = 1.8, T_i = 1, T_d = 0.8) as the benchmark, the comparison of PID parameters and control performance indices for the three tuning methods is presented in

Table 3. Comparison of PID parameters and control performance between Z-N tuning and improved DBO tuning.

Controller	Kp	Ti	Td	Overshoot (%)	Settling Time (s)
Z-N PID	1.80	1	0.8	24.8	8.2
Standard DBO-PID	2.03	0.92	0.56	11.5	4.6
Improved DBO-PID	2.36	1.07	0.25	4.0	1.8

, and the comparison of unit step response curves is shown in Figure 5.

As shown in Table 3 and Figure 5, the PID controller tuned by the Z-N method has an overshoot of 24.8% and a settling time of 8.2 s, with the response curve undergoing multiple oscillations near the target value before stabilizing. The PID controller tuned by the standard DBO algorithm has an overshoot of 11.5% and a settling time of 4.6 s, representing reductions of 53.6% and 43.9% compared with the Z-N method, respectively, but the response curve still exhibits noticeable oscillation. The PID controller tuned by the improved DBO algorithm has an overshoot of 4.0% and a settling time of 1.8 s, with the step response curve rising smoothly and converging rapidly to the target value without noticeable oscillation. Compared with the Z-N method, the overshoot is reduced by 83.9% and the settling time is shortened by 78.0%; compared with the standard DBO, the overshoot is further reduced by 65.2% and the settling time is shortened by 60.9%.

Figure 6 presents the fitness convergence curves of the standard DBO and improved DBO over 200 iterations. The improved DBO reaches a fitness value of 0.642 by iteration 10 and converges to its final value of 1.341 by approximately iteration 100, after which no further improvement occurs. In contrast, the standard DBO achieves only 0.380 at iteration 100 and reaching 0.597 at iteration 200—still below the improved DBO’s converged value. The faster convergence and superior final fitness of the improved DBO originate from the synergistic effect of the two improvements and the fitness function.

The above performance improvements originate from the synergistic effect of the two improvements and the fitness function. The Logistic chaotic map initialization improves the coverage uniformity of the initial population in the three-dimensional PID parameter space, reducing the risk of the algorithm falling into local optima. The adaptive inertia weight enables the search step sizes to dynamically adjust with iterations—larger step sizes in the early phase for global exploration and contracted step sizes in the later phase for improved local exploitation accuracy. Furthermore, the overshoot penalty coefficient λ₁ = 7 in the comprehensive fitness function is much larger than the settling time penalty coefficient λ₂ = 0.6, causing the optimization process to prioritize overshoot suppression, which is consistent with the low overshoot of only 4.0% achieved by the improved DBO-PID. Based on the PID parameters obtained from the above simulation tuning, the subsequent laboratory simulation experiments were conducted.

3.2. Two-Phase Flow Experimental Verification

3.2.1. Experimental Apparatus and Evaluation Indices

To verify the control performance of the wellhead back pressure automatic control system, a laboratory simulation experimental apparatus was designed and constructed using the needle-type choke valve commonly used in drilling operations as the actuator, with reference to the field choke valve–wellbore structure, as shown in Figure 7.

The apparatus mainly consists of four components: the control unit (a laptop computer equipped with automatic control software), the execution unit (an automated choke valve control box, hydraulic lines, and a needle-type choke valve), the fluid circulation unit (a simulated wellbore, a plunger pump, an air compressor, and an air storage tank), and the signal acquisition unit (a displacement sensor, pressure sensors, and flow meters). Sensor signals are transmitted to the control unit via optical fiber, which together with the downlink channel for control commands forms a complete control–execution–feedback closed-loop structure. The main technical parameters of the apparatus are as follows: plunger pump liquid flow rate 0–20 m³/h, air compressor gas flow rate 0–20 m³/h, choke valve opening degree 0–44 mm, and wellhead back pressure range 0–0.6 MPa.

Within the technical parameter range of the above apparatus, five groups of two-phase flow experimental schemes were designed, as shown in

Table 4. Two-phase flow laboratory experimental schemes.

Number	Name	Objective
A	Manual control experiment	Manually adjust the choke valve opening degree, record settling time and steady-state pressure fluctuations, serving as the benchmark for automatic control performance evaluation
B	Conventional PID control experiment	Test the control performance of the Z-N-tuned PID controller under different target pressures
C	Automatic control system experiment	Test the control performance of the wellhead back pressure automatic control system and compare with Experiments A and B
D	Disturbance rejection experiment	Abruptly change the gas flow rate to simulate gas kick disturbance, and test the system’s disturbance rejection capability at different back pressure levels
E	Continuous pressure tracking experiment	Import a target pressure–time curve to test the system’s tracking capability for continuously changing target pressure

. Experiments A–C were conducted under constant flow rate conditions with a liquid flow rate of 14 m³/h and a gas flow rate of 6 m³/h (gas void fraction of 30%), using manual control, conventional PID control, and the automatic control system, respectively. Experiment D tested the system’s disturbance rejection capability by abruptly changing the gas flow rate. Experiment E tested the system’s tracking capability for the target pressure under continuously varying gas void fraction conditions. Among these, Experiment B used PID parameters tuned by the Z-N method, while Experiments C, D, and E used PID parameters tuned by the improved DBO algorithm (all shown in Table 3).

To quantitatively evaluate the control performance, the following three evaluation indices were selected: settling time T_ac, defined as the time required for the actual wellhead back pressure to reach and remain within the steady-state allowable range from the moment of target pressure change, s; maximum overshoot absolute error OAE_max, defined as the maximum absolute deviation between actual back pressure and target value before the system enters steady state, MPa; mean absolute error MAE, defined as the mean value of the absolute deviation between actual back pressure and target value after reaching steady state, MPa. The steady-state criterion is that the absolute error remains less than 0.005 MPa for five consecutive sampling instants (sampling period of 2 s).

3.2.2. Two-Phase Flow Constant Target Pressure Control Experiments

According to the schemes of Experiments A, B, and C in Table 4, three groups of experiments were conducted under constant target pressure conditions using manual control, conventional PID control, and the automatic control system, respectively. The comparison of wellhead back pressure response curves is shown in Figure 8, and the summary of key performance indices is presented in

Table 5. Summary of performance indices for three methods under two-phase flow constant target pressure.

Evaluation Index	Manual Control (Experiment A)	Conventional PID (Experiment B)	Automatic Control System (Experiment C)	Experiment C vs. Experiment A	Experiment C vs. Experiment B
Average Tac (s)	50	50.2	25	50% reduction	50% reduction
Maximum OAEmax (MPa)	—	0.035	0.009	—	74% reduction
Steady-state MAE (MPa)	0.01–0.05	<0.008	<0.005	—	40% reduction

.

(1)

Manual control (Experiment A). The average T_ac for manually adjusting the choke valve under two-phase flow conditions was 50 s (Figure 8a). After manually adjusting the wellhead back pressure to the target value and holding the choke valve opening degree constant, the MAE reflects the inherent pressure fluctuation level of the experimental system under that operating condition. The MAE in the low-to-medium pressure range (0.1–0.3 MPa) was 0.01–0.02 MPa; when the target pressure increased to 0.5 MPa, the MAE increased to 0.05 MPa, because this pressure approaches the upper limit of the air storage tank injection pressure, causing the gas–liquid two-phase flow to become unstable.

(2)

Conventional PID control (Experiment B). The average T_ac of the conventional PID controller was 50.2 s (Figure 8b), essentially comparable to the 50 s of manual control, indicating that the conventional PID controller failed to shorten the settling time under two-phase flow conditions. When the target pressure change amplitude reached 0.3 MPa, T_ac increased to a maximum of 92 s, while OAE_max reached 0.035 MPa. As shown by the choke valve opening degree curve in Figure 8b, the choke valve opening degree adjustment stroke corresponding to multiple target pressure changes exceeded 10 mm. The conventional PID controller only makes small incremental adjustments to the opening degree based on pressure deviation, requiring multiple control cycles to approach the target opening degree in the large stroke interval, thereby prolonging the settling time, while the integral term continuously accumulates in the large deviation interval, leading to overshoot. During the steady-state phase, the PID controller suppressed the MAE to below 0.008 MPa through continuous fine adjustment of the choke valve opening degree, which is lower than the inherent fluctuation level when the choke valve opening degree was fixed in Experiment A, indicating that the continuous feedback of the PID controller provided active compensation for the inherent pressure fluctuations of the system.

(3)

Automatic control system (Experiment C). The average T_ac of the automatic control system was 25 s (Figure 8c), representing a 50% reduction compared with both manual control and conventional PID control; OAE_max was 0.009 MPa, a 74% reduction compared with conventional PID; and MAE was less than 0.005 MPa at all target pressures. Taking the initial phase of the experiment as an example, the choke valve was regulated from the fully open state to a target pressure of 0.1 MPa, with an opening degree adjustment stroke of approximately 25 mm and a Tac of 38 s. This initial pressurization event involved a larger valve stroke than any of the subsequent target pressure switching events and is therefore not included in the statistical summary presented in

Table 6. Individual performance indices for each target pressure switching event in Experiment C (automatic control system, gas void fraction 30%).

Pt (MPa)	ΔP (MPa)	Tac (s)	OAEmax (MPa)	MAE (MPa)
0.2	0.1	32	0.009	0.002
0.3	0.1	22	0.002	0.003
0.4	0.1	14	0.003	0.003
0.1	0.3	32	0.005	0.002
0.3	0.2	24	0.001	0.003
mean ± standard deviation	—	25 ± 7.6 s	0.004 ± 0.003 MPa	0.003 ± 0.001 MPa

, which covers only the five inter-setpoint switching events following the initial pressurization. In Experiment B, a smaller stroke required the PID controller to complete the adjustment through multiple cycles of small incremental adjustments, resulting in a settling time actually longer than the large-stroke adjustment time in Experiment C. In contrast, the automatic control system directly positioned the choke valve near the target opening degree through the rapid opening degree adjustment phase, with the PID controller performing fine regulation only within the small deviation interval. Because the rapid opening degree adjustment phase reduced the initial pressure deviation of the PID fine regulation phase, the PID controller operated in the small deviation interval from the moment of intervention, while overshoot was effectively suppressed.

To further address the repeatability of the control performance, Table 6 presents the individual performance indices for each target pressure switching event in Experiment C. The mean settling time across the five switching events is 25 ± 7.6 s, with the variation in T_ac attributable to differences in both the pressure change magnitude (ΔP) and the absolute target pressure level, which together determine the required opening degree adjustment stroke across switching events: the three events with ΔP = 0.1 MPa yield Tac values of 14–32 s across different target pressure levels (0.2, 0.3, and 0.4 MPa), consistent with the expected dependence of settling time on the required valve stroke, while the target pressure of 0.3 MPa appears twice under different initial conditions (ΔP = 0.1 MPa and ΔP = 0.2 MPa) and yields T_ac values of 22 s and 24 s respectively, demonstrating good consistency under comparable conditions. In contrast, the steady-state MAE shows markedly lower variability across all five events (0.003 ± 0.001 MPa), confirming that the fine regulation performance of the PID controller is consistent and independent of the pressure change magnitude. These results collectively confirm that the observed variation in Tac is physically consistent with the changing operating conditions, and that the steady-state regulation accuracy of the control system is repeatable under the tested two-phase flow conditions.

The above experiments verified the control performance of the automatic control system under constant target pressure conditions. However, in actual managed pressure drilling operations, gas influx and other factors introduce external disturbances, and the target pressure typically changes continuously with the drilling process. Therefore, it is necessary to further examine the system’s disturbance rejection performance and tracking capability for continuously changing target pressures.

3.2.3. Two-Phase Flow Gas Flow Rate Disturbance Experiment

To evaluate the disturbance rejection performance of the automatic control system against abrupt gas flow rate changes under constant target pressure conditions, Experiment D was conducted. The liquid flow rate was 15 m³/h, the target pressure was 0.4 MPa, and the gas flow rate was switched among 0, 12, and 18 m³/h to simulate wellhead pressure disturbances caused by gas influx and gas discharge. The experimental results are shown in Figure 9 and

Table 7. Experimental data of Experiment D: gas flow rate disturbance.

Qg (m3/h)	0	12	0	18	0
ΔQg (m3/h)	-	12	−12	18	−18
Tac (s)	-	24	30	42	24
ΔPd (MPa)	-	0.073	0.083	0.103	0.097
MAE (MPa)	-	0.003	0.005	0.003	0.005

.

The automatic control system exhibited no overshoot in all four gas flow rate step changes, with MAE maintained at 0.003–0.005 MPa during all steady-state phases. When the gas flow rate step change amplitude was 12 m³/h, T_ac was 24–30 s, and the maximum pressure deviation ΔP_d was 0.073–0.083 MPa. When the step change amplitude increased to 18 m³/h, T_ac increased to 42 s and ΔP_d increased to 0.103 MPa, indicating that the larger the disturbance amplitude, the longer the recovery time required by the system and the greater the transient pressure deviation. As shown by the choke valve opening degree curve in Figure 9, after each gas flow rate step change, the rapid opening degree adjustment phase first positioned the choke valve near the opening degree matching the new operating condition, followed by the PID fine regulation phase eliminating the remaining pressure deviation. The coordination of the two phases enabled the system to recover from the maximum gas flow rate disturbance (18 m³/h) within 42 s.

3.2.4. Two-Phase Flow Continuous Pressure Tracking Experiment

To evaluate the tracking capability of the automatic control system under conditions of continuously changing target pressure with simultaneous gas flow rate variations, Experiment E was conducted. The liquid flow rate was 16 m³/h, the gas flow rate was adjusted multiple times within the range of 0–14 m³/h (corresponding to gas void fractions of approximately 0–46.6%), and the target pressure varied continuously within the range of 0.1–0.4 MPa, while gas injection flow rate was adjusted simultaneously to simulate pressure disturbances caused by gas void fraction changes resulting from gas influx during managed pressure drilling. The experimental results are shown in Figure 10 and

Table 8. Experimental data of Experiment E: two-phase flow continuous pressure tracking.

Time (s)	Target Pt (MPa)	Gas Flow (m3/h)	MAE (MPa)	Remarks
52–500	0.1	3–6	0.006	Constant target pressure
500–2452	0.1–0.4	3–12	0.003	Gas influx phase
2454–2770	0.4	6–14	0.005	Constant target pressure
2772–4722	0.4–0.1	14–0.6	0.002	Gas discharge phase

.

The automatic control system exhibited no overshoot throughout the entire continuous tracking process, with an average MAE of 0.004 MPa across the four phases. As shown in Table 8, the MAE during the continuously changing target pressure phases (gas influx and gas discharge phases) was 0.003 MPa and 0.002 MPa, respectively, lower than the two constant pressure phases (0.006 MPa and 0.005 MPa). The reason is that during the constant pressure phases, multiple adjustments of gas flow rate constituted repeated disturbances to wellhead back pressure, while the choke valve opening degree was maintained near the fixed target value, and the system relied solely on the PID fine regulation phase to passively respond to disturbances. In contrast, during the continuously changing target pressure phases, the choke valve opening degree continuously followed the target opening degree adjustments, and the frequent intervention of the rapid opening degree adjustment phase enabled the system to respond more promptly to pressure deviations caused by gas flow rate changes. The operating conditions of Experiment E encompassed the dual effects of continuously changing target pressure and gas void fraction fluctuations, representing the conditions closest to actual managed pressure drilling field conditions among the five experiments, yet the MAE remained below 0.006 MPa.

3.3. Discussion

Section 3.1 has analyzed the sources of performance improvement of the improved DBO algorithm. The following discussion focuses on the performance of the dual-layer control architecture in the two-phase flow experiments.

The core of the dual-layer cooperative feedback control architecture lies in the rapid opening degree adjustment phase, which addresses the problems of sluggish regulation and susceptibility to overshoot of conventional PID in the large deviation interval. Experiment B shows that the settling time and overshoot of the conventional PID increase nonlinearly with the increase in target pressure change amplitude. This is because the cycle-by-cycle small incremental adjustment strategy requires multiple control cycles to approach the target opening degree in the large stroke interval, during which the integral term continuously accumulates, thereby triggering overshoot. In Experiment C, the opening degree controller rapidly adjusted the choke valve to near the target opening degree, enabling the PID controller to operate within the small deviation interval from the moment of intervention, with limited integral accumulation, effectively suppressing both settling time and overshoot.

The relationship between disturbance amplitude and recovery time exhibits directional asymmetry. For gas injection disturbances, increasing the step change amplitude from 12 to 18 m³/h (50%) caused T_ac to rise from 24 to 42 s (75%), reflecting the disproportionately larger target opening degree change and the corresponding increase in regression model prediction error under higher gas flow rates. Conversely, for gas discharge disturbances, T_ac decreased from 30 to 24 s under the same amplitude increase, likely because reducing gas content drives the flow toward single-phase liquid behavior with more predictable choke valve characteristics. This directional asymmetry indicates that the accuracy of the target opening degree regression model under varying flow conditions is a key factor governing the system’s disturbance recovery speed.

In Experiment E, the MAE during the continuously changing target pressure phases was lower than that during the constant pressure phases. The reason is that the rapid opening degree adjustment frequently intervened during the continuous change phases, enabling the system to respond more promptly to gas flow rate variations; whereas during the constant pressure phases, the choke valve opening degree was maintained near the fixed target value, and the system relied solely on PID fine regulation to passively compensate for pressure disturbances caused by gas flow rate fluctuations. This result suggests that in actual drilling operations during constant pressure control phases, if frequent gas disturbances are encountered, feedforward compensation based on flow rate signals could be considered to further improve control performance.

The experimental verification in this paper was completed in a laboratory simulation environment, which differs to some extent from actual drilling field conditions. This limitation is mainly reflected in two aspects: first, the wellhead back pressure range in the laboratory experiments (0–0.6 MPa) is lower than the actual field magnitude (typically 1–5 MPa and above). Under high-pressure conditions involving compressible formation gases such as methane, the transfer function parameter (K, T₁, T₂, δ) will require re-identification, although the second-order-with-delay model structure has been shown to remain applicable across varying pressure conditions [15]; second, the PID parameters in this system remained fixed after simulation tuning, and experiments involving dynamic adjustment of PID parameters in response to gradually changing operating conditions have not yet been conducted. In addition, the adaptability of parameters such as the opening degree regression model and switching threshold under complex multiphase fluid conditions remains to be verified.

4. Future Work

Based on the limitations discussed in Section 3.3, future work may proceed in the following directions:

(1)

Conducting field drilling trial verification to evaluate the applicability of the proposed method under actual wellhead back pressure ranges (on the order of 1–5 MPa) and drilling fluid conditions. Under such conditions, the compressibility of formation gases such as methane becomes non-negligible, which may alter the pressure–opening degree gain K and the equivalent time constants of the transfer function. Re-identification of the transfer function parameters under field conditions will be necessary prior to deployment. In addition, systematic testing across a wider range of gas void fractions will help establish the operational boundary of the dual-layer control architecture under field conditions.

(2)

An online parameter adaptive mechanism can be introduced to enable real-time updating of PID parameters in response to changing operating conditions [14], addressing the problem of transfer function parameter drift during prolonged drilling operations. The convergence behavior demonstrated in Figure 6, where the improved DBO reaches a near-optimal solution within approximately 100 iterations, suggests that a reduced-iteration online implementation may be practically feasible, warranting further investigation.

(3)

It is also worth exploring the integration of the dual-layer control architecture with model predictive control (MPC) [17,18], utilizing the predictive capability of MPC to compensate in advance for foreseeable disturbances such as gas influx, and exploring the application potential of hybrid control schemes in complex and variable drilling environments.

5. Conclusions

Gas influx during managed pressure drilling intensifies the nonlinear relationship between choke valve opening degree and wellhead back pressure, causing prolonged settling time and overshoot in conventional PID control. This paper proposed a pressure–opening degree dual-layer cooperative feedback control method, constructed a PID parameter intelligent tuning method based on the improved DBO algorithm, and conducted systematic evaluation through simulation verification and two-phase flow laboratory experiments with gas void fractions of 0–46.6%. The main conclusions are as follows:

(1)

To address the shortcomings of the standard DBO in terms of uneven initial population distribution and fixed search strategies, Logistic chaotic map initialization and adaptive inertia weight are introduced, and a comprehensive fitness function integrating the ITAE criterion, dynamic performance penalty terms, and engineering safety constraint penalty terms is designed, enabling the algorithm to search for parameter combinations in the PID parameter space that balance response speed and overshoot suppression. Simulation results show that, using the Z-N method as the benchmark, the PID controller tuned by the standard DBO reduces overshoot by 53.6% and shortens settling time by 43.9%; the PID controller tuned by the improved DBO reduces overshoot by 83.9% and shortens settling time by 78.0%, demonstrating significantly superior tuning performance over the standard DBO.

(2)

The pressure–opening degree dual-layer cooperative feedback control architecture directly positions the choke valve near the target opening degree through the rapid opening degree adjustment phase, ensuring that the PID fine regulation phase always operates within the small deviation interval. In gas–liquid two-phase flow laboratory experiments with gas void fractions of 0–46.6%, using manual control (average settling time of 50 s) as the benchmark, the Z-N-tuned PID controller fails to reduce settling time and yields an OAE_max of 0.035 MPa; the dual-layer cooperative feedback control system equipped with the improved DBO-PID achieves zero overshoot across three types of operating conditions—constant target pressure, abrupt gas flow rate changes, and continuous pressure tracking—with average settling time reduced to 25 s (a 50% reduction), OAE_max reduced to 0.009 MPa (a 74% reduction), and steady-state MAE less than 0.005 MPa.

The above simulation and experimental results confirm that the improved DBO algorithm outperforms the standard DBO in parameter optimization capability, and the synergy between the dual-layer control architecture and the improved DBO-PID enables rapid regulation and stable tracking of wellhead back pressure under gas–liquid two-phase flow conditions, providing a reference for the design of automatic choke valve control systems in managed pressure drilling. It should be noted that the experiments in this paper were conducted within a pressure range of 0–0.6 MPa, and further verification is needed for the extension of this method to field high-pressure operating conditions.