Research on Optimization of Intermittent Production Process System

Abstract

As gas wells enter later production stages, the formation pressure decreases and liquid accumulates at the bottom of the gas well. The formation pressure is insufficient to lift the accumulated liquid from the bottom of the well to the surface. At this time, a large number of gas wells need to undergo intermittent production to maintain their production capacity. This article focuses on the four stages of intermittent production in gas wells, considering the changes in slip gas holdup, pressure, and gas–liquid flow in and out of tubing and casing, and establishes a transient mathematical model for intermittent production in gas wells in stages. By using the dynamic tracking technology of moving liquid slugs to divide the wellbore grid and solve it in stages, the optimal shut-in time for intermittent production of gas wells was obtained. The transient mathematical model developed for intermittent gas well production achieved a high historical fit accuracy of over 90%. This indicates that the simulation results are in line with the actual situation of gas well intermittent production and can effectively guide intermittent production. The optimized intermittent production system of gas wells has a higher cumulative gas production compared to the original system, achieving the optimization of intermittent production system. This method is beneficial for guiding efficient production of gas wells in low-pressure formations.

1. Introduction

A large number of gas wells have been exploited for a long time, and the formation pressure is insufficient to lift the bottom hole fluid to the surface, seriously affecting production. At this time, intermittent production is needed to maintain normal gas well production [1]. The gas well enters the intermittent production stage. During the intermittent production process of gas wells, the changes in the dynamic liquid level of the tubing and the liquid level of the casing annulus are both transient processes, and the gas–liquid velocity and pressure vary greatly over time. The commonly used steady-state models are no longer suitable for describing this process. At present, the optimization of the intermittent production system for gas wells is mostly explored in practice, on-site statistics, and empirical judgment, and there are still significant shortcomings. The theory of intermittent production for gas wells has not yet been formed, and there is a great deal of blindness in the exploration of the system [2,3]. Therefore, establishing a geological wellbore coupled unsteady mathematical model that reflects the characteristics of intermittent production in gas wells is of great academic significance and engineering practical value for guiding the optimization of intermittent production systems in gas wells.

Many domestic scholars have optimized the intermittent production system of gas wells. Xiang et al. [4] established a single well numerical model based on the statistical changes in formation pressure and permeability of intermittent and continuous production wells. This model can achieve optimal selection of different intermittent production systems, but cannot determine the optimal intermittent production system by calculating the precise well opening and closing time. Liu et al. [5] analyzed the impact of different switch production systems on gas production in intermittent gas wells, explored the production laws of exhaust wells, and continuously optimized the production system of intermittent wells. Du et al. [6] proposed a method to improve the intermittent production measures of gas wells by evaluating and classifying the adaptability of intermittent wells. This method judges the adaptability of intermittent measures based on on-site experience, but lacks theoretical support in the optimization process of intermittent systems. Li et al. [7] and Zhai et al. [8] used real-time production data and other information from intermittent production wells as the basis for intermittent production system management. Based on the critical flow rate, the pressure range of the switch well is set, which can achieve multiple switch well modes such as remote constant pressure, timing, and constant flow. This method cannot adjust the switch well time in real time and cannot effectively guide the formulation of fine management system for intermittent mining. Based on the current research status [9,10,11], most intermittent production systems are developed through analysis of historical production systems of gas wells, and a very small number are established through mathematical model analysis based on theoretical laws such as pressure and water production. However, this numerical simulation is only based on production dynamics and lacks scientific and effective theoretical support. Due to the lack of a reasonable and scientific mathematical model as a basis, the application effect on site is unsatisfactory, and the scope of use of the intermittent production system developed is limited and lacks certain practicality. Therefore, it is urgent to establish a transient mathematical model that fully considers the inflow and outflow characteristics of gas and water during the intermittent production process of gas wells, optimize the intermittent production system of gas wells, and achieve efficient production of intermittent gas wells.

This article divides the intermittent production period of gas wells into four flow stages: liquid accumulation formation, liquid accumulation outflow, gas–liquid eruption, and liquid reflux. Sensitivity parameters such as slip gas holding rate, pressure, and gas–liquid inflow and outflow status of tubing and casing are considered for each of the four flow stages [12,13]. A transient mathematical model for intermittent production of gas wells is established to calculate the optimal shut-in time and optimize the intermittent production system with the highest cumulative gas production.

2. Mathematical Model

2.1. Model Establishment

The transient mathematical model of intermittent production in gas wells established in this article mainly includes the variation laws of slip gas holding rate, pressure, and gas–liquid inflow and outflow of tubing and casing during the production process of gas wells.

In the process of establishing this model, it is necessary to consider the fluid flow in the wellbore as one-dimensional flow and make some assumptions about gas–liquid flow:

• The gas–liquid flow velocity and pressure distribution in the wellbore are solved using a transient model;
• The steady-state two-phase flow friction and holdup calculation are also applicable to transient conditions.

Among various transient flow simulation methods, three-dimensional flow can consider all transient characteristics and internal flow distribution inside the complete wellbore. However, three-dimensional flow requires a large number of computational grid nodes to accurately solve the three-dimensional flow distribution. Especially for complex wellbore systems several kilometers deep, this method requires too many computational grid nodes, and the existing computer performance is almost unbearable. The one-dimensional transient flow simulation method can study the flow situation in the wellbore through momentum and mass equations [14]. This method can simplify the gas–liquid flow in the wellbore, although it may lose some accuracy, it greatly improves the calculation speed.

The widely used steady-state multiphase flow models currently include the Beggs and Brill model and the Petalas and Aziz mechanism model [15]. These steady-state models are mainly one-dimensional homogeneous flow models, which assume uniform distribution between gas–liquid phases, the same two-phase velocity, and no slip. This is different from the true flow state of gas–liquid two-phase in the wellbore, resulting in a significant gap between the use of steady-state models for calculating the pressure drop of multiphase flow in the wellbore and the actual situation. We used the liquid holdup calculation models proposed by Duns et al. [15], Beggs et al. [16], Govier et al. [17], Mukherjee et al. [18] and Orkiszewski et al. [19] to calculate pressure drop, and compared them with the measured wellbore pressure distribution on site. We found that the Mukherjee liquid holdup calculation model considering slippage phenomenon had the best performance in calculating wellbore pressure distribution, as shown in Figure 1. The transient model used in this article fully considers the relative motion between gas and water phases, and introduces the Mukherjee holdup calculation model to describe the uneven distribution of gas and water phases.

According to the physical phenomena in the intermittent production process of gas wells, the intermittent production period is divided into four flow stages: liquid plug formation stage, liquid plug outflow stage, liquid–gas eruption stage, and liquid reflux stage. The corresponding periods are T₁, T₂, T₃, T₄, and the entire period is T = T₁ + T₂ + T₃ + T₄. The opening time corresponding to each intermittent production period is t_on = T₂ + T₃, and the closing time t_off = T₁ + T₄. When conducting intermittent production, the flow pattern in the tubing exhibits periodic changes and the gas–liquid flow patterns are not consistent in each flow stage. Therefore, the transient mathematical model of intermittent production is different in all four flow stages. The following are discussed separately (Figure 2):

2.1.1. Initial Condition

In the process of intermittent production of gas well fluid accumulation, the fluid accumulation in the tubing mixes with the gas to form a gas–liquid column, and the liquid holding capacity of the column decreases with depth. Near the wellhead, it is a pure gas column; the upper end of the annular space is a stationary pure gas column, and the lower end is a stationary pure liquid column, with a clear gas–liquid interface.

When the gas well begins to accumulate liquid, the height of the casing liquid column is close to 0. Assuming that the height of the casing accumulation is 0 and there is a pure gas column in the casing [20], according to the static pressure calculation formula, the bottom pressure of the annular air column is calculated based on the casing pressure, which is the bottom hole flowing pressure. In the following text, the pressure at the connection between the tubing and casing is considered as the bottom hole flowing pressure.

$$p_{wf}=p_{cg}=p_c{e}^s=p_c{e}^{0.03415{\gamma }_{g}H/\left(T_c{Z}_c\right)}$$

(1)

According to the principle of U-shaped tubing [21], the pressure at the connection point between the tubing and the annulus is equal. Calculate the bottom hole flowing pressure when the casing begins to accumulate fluid and the height of the liquid column in the tubing.

$$p_{wf}=p_t+\Delta p_{u1}+\Delta p_{b1}=p_c+\Delta p_{u2}+\Delta p_{b2}$$

(2)

Simultaneous Equations (1) and (2), the height of the accumulated fluid in the tubing during the initial stage of fluid accumulation can be obtained.

$$p_c{e}^{0.03415{\gamma }_{g}H/\left(\overline{T{Z}_c}\right)}=p_t{e}^{0.03415{\gamma }_g\left(H-h_t\right)/\left(\overline{T{Z}_t}\right)}+{\rho }_l\mathrm{g}{h}_t$$

(3)

2.1.2. Formation Inflow

According to the theory of material balance, in order to obtain dynamic control reserves of a single well, it is necessary to first calculate the formation pressure at different times. According to the two-phase gas well productivity model, the formation pressure can be expressed as:

$$\Psi \left(p_r\right)-\Psi \left(p_{wf}\right)=a{q}_{\mathrm{sc}}+b{q}_{\mathrm{sc}}^2$$

(4)

Using the known wellhead pressure, the bottomhole flow pressure P_wf of the productivity model is calculated by Equation (1). The definition of quasi-pressure is:

$$\Psi (p)={\displaystyle {\int }_{p_{wf}}^{p_r}\left(\frac{K_{\mathrm{rg}}{\rho }_{\mathrm{g}}}{{\mu }_{\mathrm{g}}}+\frac{K_{\mathrm{rw}}{\rho }_{\mathrm{w}}}{{\mu }_{\mathrm{w}}}\right)}\mathrm{d}p$$

(5)

Without considering the influence of non-Darcy seepage, Jokhio et al. [22] proposed a method for solving the relative permeability ratios of each item, from which the representation of K_rg/K_rw under the condition of two-phase seepage can be obtained.

$${\displaystyle \frac{K_{\mathrm{rg}}}{K_{\mathrm{rw}}}}={\displaystyle \frac{B_{\mathrm{g}}{\mu }_{\mathrm{g}}}{B_{\mathrm{w}}{\mu }_{\mathrm{w}}}}\left[{\displaystyle \frac{R_{\mathrm{sgw}}-R_{\mathrm{pgw}}}{R_{\mathrm{pgw}}{R}_{\mathrm{swg}}-1}}\right]$$

(6)

Using the empirical formula of the relative permeability curve, the relationship between K_rg/K_rw and saturation S_g and S_w can be obtained [23].

$${\displaystyle \frac{K_{\mathrm{rg}}}{K_{\mathrm{rw}}}}={\displaystyle \frac{S_{\mathrm{g}}^{m_{\mathrm{g}}}}{{\left(S_{\mathrm{w}}-S_{\mathrm{wi}}\right)}^{m_{\mathrm{w}}}}}{S}_{\mathrm{w}}+S_{\mathrm{g}}=1$$

(7)

In Equations (6) and (7), R_pgw is a known quantity, and μ_w, μ_g, B_g, B_w, R_swg, and R_sgw can be directly obtained by the pressure p using the phase theory. On the other hand, K_rg/K_rw is a function of saturation S_g and S_w, so K_rg/K_rw can be obtained from the pressure p, and then the values of S_g and S_w can be obtained according to the phase permeability curve, and finally K_rg and K_rw can be obtained. Finally, both K_rg and K_rw in the two-phase seepage can be expressed as a function of the pressure p, and the expression of the quasi-pressure can be obtained through the definite integral, so as to solve the gas–water two-phase seepage productivity equation.

The dynamic reserves can be calculated by the mass balance equation of the water intrusion gas reservoir.

$${\displaystyle \frac{p}{Z}}[1-{\displaystyle \frac{\left(C_{\mathrm{w}}{S}_{\mathrm{wi}}+C_{\mathrm{f}}\right)\left(p_{\mathrm{i}}-p\right)}{1-S_{\mathrm{wi}}}}-{\displaystyle \frac{W_{\mathrm{e}}-W_{\mathrm{p}}{B}_{\mathrm{w}}}{G{B}_{\mathrm{gi}}}}]={\displaystyle \frac{p_{\mathrm{i}}}{Z_{\mathrm{i}}}}\left(1-{\displaystyle \frac{G_{\mathrm{p}}}{G}}\right)$$

(8)

At this point, the following model equations can be obtained:

$$\left\{\begin{array}{l}{\displaystyle \frac{p}{Z}}\left[1-{\displaystyle \frac{\left(C_{\mathrm{w}}{S}_{\mathrm{wi}}+C_{\mathrm{f}}\right)\left(p_{\mathrm{i}}-p\right)}{1-S_{\mathrm{wi}}}}-\right.\hfill \\ \left.{\displaystyle \frac{W_{\mathrm{e}}-W_{\mathrm{p}}{B}_{\mathrm{w}}}{G{B}_{\mathrm{gi}}}}\right]={\displaystyle \frac{p_{\mathrm{i}}}{Z_{\mathrm{i}}}}\left(1-{\displaystyle \frac{G_{\mathrm{p}}}{G}}\right)\hfill \\ \Psi (p)=a{q}_{\mathrm{gsc}}+b{q}_{\mathrm{gsc}}^2\hfill \\ \Psi (p)={\displaystyle {\int }_{p_{\mathrm{wf}}}^{p_{\mathrm{e}}}\left(\frac{K_{\mathrm{rg}}{\rho }_{\mathrm{g}}}{{\mu }_{\mathrm{g}}}+\frac{K_{\mathrm{rw}}{\rho }_{\mathrm{w}}}{{\mu }_{\mathrm{w}}}\right)}\mathrm{d}p\hfill \end{array}\right.$$

(9)

Thus, the dynamic transformation of the control reserves G and the formation inflow coefficients a and b of a single well is determined.

2.1.3. Liquid Plug Formation Stage

When the liquid level in the space below the tubing shoe reaches the position of the tubing shoe, the tubing space and the casing annulus are separated by the liquid level. Most of the gas and most of the liquid enter the tubing space, and the gas carries a small amount of liquid and is discharged through the tubing. The retained liquid raises the gas–liquid interface of the tubing, increases the bottom hole flowing pressure, and forces the accumulated liquid to move towards the annulus, thereby raising the gas–liquid interface of the annulus. At the same time, a small amount of gas enters the annulus and is stored in the upper space of the annulus, as shown in Figure 2a. At this point, the space below the tubing shoe is a gas–liquid column, and the tubing space is divided into two sections. The upper section is a low-speed flow with liquid mist, which can be regarded as a pure gas column, and the lower section is a gas–liquid column; the casing annulus is divided into two sections, the upper section is a static gas column, and the lower section is a static liquid column.

The gas flow rate and liquid flow rate into the wellbore are determined according to the bottom inflow Equation (9).

According to the conservation of matter, the amount of gas–liquid flowing out of the formation during the shut-in stage is equal to the amount of gas–liquid stored in the wellbore. The amount of material flowing into the tubing and casing gas in the formation is divided by area ratio:

$$q_{sc}=q_{tg}+q_{cg}={\displaystyle \frac{A_t}{A_t+A_c}}{q}_{sc}+{\displaystyle \frac{A_c}{A_t+A_c}}{q}_{sc}$$

(10)

According to the gas state equation, the gas flow velocity at any flow state (P, T) can be expressed as:

$$v_{cg}={\displaystyle \frac{q_{cg}{B}_g}{A_c}}=\left({\displaystyle \frac{q_{cg}}{86400}}\right)\left({\displaystyle \frac{T}{293}}\right)\left({\displaystyle \frac{0.101}{p_{cavg}}}\right)\left({\displaystyle \frac{4Z_c}{\pi \left(D^2-d^2\right)}}\right)$$

(11)

$$v_{tg}={\displaystyle \frac{q_{tg}{B}_g}{A_t}}=\left({\displaystyle \frac{q_{tg}}{86400}}\right)\left({\displaystyle \frac{T}{293}}\right)\left({\displaystyle \frac{0.101}{p_{tavg}}}\right)\left({\displaystyle \frac{4Z_t}{\pi d^2}}\right)$$

(12)

Therefore, the flow rate of fluid flowing into the tubing and annulus of the formation is as follows:

$$q_w=A_t{\displaystyle \frac{d{h}_t}{dt}}+A_c{\displaystyle \frac{d{h}_c}{dt}}=A_t{v}_{tw}+A_c{v}_{cw}$$

(13)

For casing, the gas state equation can be obtained:

$$p_{cavg}{V}_{cg}=n_c{Z}_{c}R{T}_c, p_{cg}=2p_{cavg}-p_c$$

(14)

During the process of gas compression in space, the volume does not undergo significant expansion, and the temperature change amplitude is relatively small, which can be regarded as a constant [24]. Therefore, taking the derivative of Equation (14) over time yields:

$$V_{cg}{\displaystyle \frac{d{p}_{cavg}}{dt}}+p_{cavg}{\displaystyle \frac{d{V}_{cg}}{dt}}={\displaystyle \frac{Z_{c}RT}{M}}{\displaystyle \frac{d{m}_g}{dt}}={\displaystyle \frac{Z_{c}RT{\rho }_g{A}_c}{M}}{v}_{cg}$$

(15)

Due to the continuous inflow of liquid phase fluid, the height of liquid accumulation in the casing will increase, and the volume change rate of the compressed space of the casing gas can be expressed by the following equation:

$${\displaystyle \frac{d{V}_{cg}}{dt}}=-A_c{\displaystyle \frac{d{h}_c}{dt}}=-A_c{v}_{cw}$$

(16)

Therefore, the average gas pressure change rate of the casing can be expressed as follows:

$${\displaystyle \frac{d{p}_{cavg}}{dt}}={\displaystyle \frac{\frac{Z_{c}RT}{M_c}\frac{d{m}_g}{dt}-p_{cavg}{A}_c\frac{d{h}_c}{dt}}{\left(H-h_c\right)A_c}}={\displaystyle \frac{\frac{Z_{c}RT{\rho }_g}{M}{v}_{cg}-p_{cavg}{v}_{cw}}{\left(H-h_c\right)}}$$

(17)

The gas–liquid flow process inside the casing is similar during the stages of liquid plug formation, liquid plug outflow, liquid–gas eruption, and liquid reflux. The average gas pressure change rate of the casing can be represented by Equations (11) and (17). The gas–liquid flow process inside the tubing is relatively complex at each stage, so the following mainly introduces the control equations of gas–liquid flow inside the tubing.

Similarly, the average gas pressure change rate of the tubing can be expressed as follows:

$${\displaystyle \frac{d{p}_{tavg}}{dt}}={\displaystyle \frac{\frac{Z_{t}RT}{M}\frac{d{m}_g}{dt}-p_{tavg}{A}_t\frac{d{h}_t}{dt}}{\left(H-h_t\right)A_t}}={\displaystyle \frac{\frac{Z_{t}RT{\rho }_g}{M}{v}_{tg}-p_{tavg}{v}_{tw}}{\left(H-h_t\right)}}$$

(18)

Meanwhile, according to the principle of U-tube, the pressure of gas compression space in the casing can be represented by the pressure of the tubing:

$$p_{cg}+{\rho }_{l}g{h}_c=p_{tg}+{\rho }_{l}g{h}_t$$

(19)

The amount of fluid flowing into the formation is distributed according to the principle of equal pressure at the tubing and casing connection point. By combining Equations (6), (9), (10) and (13)–(15), the control equation for the formation stage of fluid accumulation can be obtained:

$${\displaystyle \frac{2\left(p_{cavg}+\frac{d{p}_{cavg}}{dt}\right)}{1+\frac{1}{e^{sc}}}}+{\rho }_{l}g\left(h_c+{\displaystyle \frac{d{h}_c}{dt}}\right)={\displaystyle \frac{2\left(p_{tavg}+\frac{d{p}_{tavg}}{dt}\right)}{1+\frac{1}{e^{st}}}}+{\rho }_{l}g\left(h_t+{\displaystyle \frac{d{h}_t}{dt}}\right)$$

(20)

Among them: $sc={\displaystyle \frac{0.03415{\gamma }_g\left(H-\left(h_c+\frac{d{h}_c}{dt}\right)\right)}{T{Z}_c}}, st={\displaystyle \frac{0.03415{\gamma }_g\left(H-\left(h_t+\frac{d{h}_t}{dt}\right)\right)}{T{Z}_t}}$ the same below.

The initial condition for the formation of fluid accumulation is $h_c=0$ and $p_{cg}=p_{tg}+{\rho }_{l}g{h}_t=p_{wf}$, at which point the fluid has just reached the bottom inlet of the casing. The termination condition for the liquid plug formation stage is that the bottom hole flowing pressure is equal to the formation pressure, $p_{wf}=p_r$. At this point, the casing fluid column is no longer rising. Velocity of liquid flowing into the formation $v_{cw}=0$, $v_{tw}=0$; gas velocity flowing into the formation $v_{cg}=0$, $v_{tg}=0$.

2.1.4. Liquid Plug Outflow Stage

As production progresses, the accumulation of liquid in the tubing causes the gas–liquid interface inside the tubing to continuously rise, and the bottom hole flowing pressure continues to increase, forcing the accumulated fluid to continue moving towards the casing. The gas in the casing is also constantly accumulating. When enough gas compression energy is gathered in the casing, the tubing is started for production. The compressed gas in the tubing is produced, and the gas production of the tubing is greatly increased. The compressed gas in the casing expands, and the flow changes in the casing, and the extruded liquid flows out of the casing and into the tubing, further raising the gas–liquid interface in the tubing; the tubing space is divided into two sections, the upper section is a low-speed liquid mist flow, the lower section is a gas-containing liquid column, and the tubing annulus is divided into two sections, the upper section is a static gas column and the lower section is a static liquid column, as shown in Figure 2b.

The pressure in the tubing is calculated by the flowing pressure formula:

$$p_{tg}=\sqrt{p_t^2{e}^{2st}+1.32\times {10}^{-18}f{\left(q_{tsc}{T}_t{Z}_t\right)}^2\left(e^{2st}-1\right)/d^5}$$

(21)

At this time, the pressure behind the wellhead throttle valve is constant, $p_2= \mathrm{Const}$. According to the mouth flow formula [25], the gas outflow and the gas outflow speed are obtained.

$$\begin{array}{l}{q}_{tsc}={\displaystyle \frac{0.408p_t{d}^2}{\sqrt{{\gamma }_{g}TZ}}}\sqrt{\left({\displaystyle \frac{k}{k-1}}\right)\left(R^{{\displaystyle \frac{2}{k}}}-R^{{\displaystyle \frac{k+1}{k}}}\right)}\\ R=\left\{\begin{array}{l}{\displaystyle \frac{p_2}{p_t}},{\displaystyle \frac{p_2}{p_t}}\le {\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}\hfill \\ {\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}},{\displaystyle \frac{p_2}{p_t}}>{\left({\displaystyle \frac{2}{k+1}}\right)}^{{\displaystyle \frac{k}{k-1}}}\hfill \end{array}\right.\end{array}$$

(22)

$$v_{tg}={\displaystyle \frac{q_{tg}{B}_g}{A_t}}=\left({\displaystyle \frac{q_{tg}}{86400}}\right)\left({\displaystyle \frac{T}{293}}\right)\left({\displaystyle \frac{0.101}{p_{tavg}}}\right)\left({\displaystyle \frac{4Z_t}{\pi d^2}}\right)$$

(23)

$$v_{tgout}={\displaystyle \frac{q_{tsc}{B}_g}{A_t}}=\left({\displaystyle \frac{q_{tsc}}{86400}}\right)\left({\displaystyle \frac{T}{293}}\right)\left({\displaystyle \frac{0.101}{p_{tavg}}}\right)\left({\displaystyle \frac{4Z_t}{\pi d^2}}\right)$$

(24)

Substituting Equations (22)–(24) into the gas state equation, the average gas pressure change rate of the tubing can be expressed as follows:

$${\displaystyle \frac{d{p}_{tavg}}{dt}}={\displaystyle \frac{\frac{Z_{t}RT}{M}\frac{d{m}_g}{dt}-p_{tavg}{A}_t\frac{d{h}_t}{dt}}{\left(H-h_t\right)A_t}}={\displaystyle \frac{\frac{Z_{t}RT{\rho }_g}{M}\left(v_{tg}-v_{tgout}\right)-p_{tavg}{v}_{tw}}{\left(H-h_t\right)}}$$

(25)

According to the equality in the pressure formula at the connection between the tubing and the casing, the control equations of the effusion stage can be obtained by simultaneous (10), (13), (14), (17), (21), (22), and (25):

$$\begin{array}{l}{\displaystyle \frac{2\left(p_{cavg}+\frac{d{p}_{cavg}}{dt}\right)}{1+\frac{1}{e^{sc}}}}+{\rho }_{l}g\left(h_c+{\displaystyle \frac{d{h}_c}{dt}}\right)={\rho }_{l}g\left(h_t+{\displaystyle \frac{d{h}_t}{dt}}\right)\\ +\sqrt{{\left({\displaystyle \frac{2\left(p_{tavg}+\frac{d{p}_{tavg}}{dt}\right)}{1+e^{st}}}\right)}^2+1.32\times {10}^{-18}f{\left(q_{tsc}{T}_t{Z}_t\right)}^2\left(e^{2st}-1\right)/d^5}\end{array}$$

(26)

The initial condition of liquid plug outflow stage is the termination condition of liquid plug formation stage. At this moment, $v_{cw}=0$, $p_2= \mathrm{Const}$. The termination condition of the liquid plug outflow stage is that the liquid plunger in the casing is pushed to the connection position at the bottom of the tubing and casing by gas. At this time, there is a pure gas column in the casing, $h_c=0$.

2.1.5. Liquid–Gas Eruption Stage

When the air-liquid interface of the casing descends to the tubing shoe, the tubing is reconnected with the casing, and the gas stored in the casing enters the tubing, so that the gas and liquid of tubing are ejected. The gas accumulated in the casing enters the tubing, increasing the gas production of the tubing. The gas flow velocity is greater than the critical liquid carrying velocity of the tubing, and the accumulation of liquid at the bottom of the well is identified. The bottom hole flowing pressure decreases, and the production increases. In a short period of time, the gas well production is high, and the casing pressure drops to the lowest level. At this point, the casing annular space is a static gas column, and the space below the tubing shoe is divided into two sections. The upper section is a ring mist flow or gas–liquid column, and the lower section is a gas–liquid column, as shown in Figure 2c.

The casing contains a pure gas column, and the pressure at the bottom of the gas column is the bottomhole flowing pressure:

$$P_{cg}=P_{wf}$$

(27)

In the gas–liquid ejection stage, there are two processes: liquid plug ejection and segmental plug ejection [26]. During the process of liquid plug eruption, the long liquid plug inside the tubing flows out of the tubing under the push of gas. When the liquid section moves to the wellhead position, the wellhead begins to produce liquid, and the outflow of the liquid plug can be represented by the flow theory of the gas nozzle [27].

$$q_{lo}=1.106d^2{C}_d\sqrt{{\displaystyle \frac{p_t-p_2}{\left(1-{\beta }^4\right){\rho }_l}}}$$

(28)

For speed, engineering focuses more on the outflow velocity of the liquid plug, $v_{twout}={\displaystyle \frac{q_{lo}}{A_t}}$, and the average velocity at the outlet of the tubing [13]. Considering the incompressibility of liquids [28], $v_{tout}=v_{twout}$. During the process of slug ejection, there is no longer a long liquid plug in the tubing, and the average velocity inside the tubing is expressed as $v_{tout}=v_m=v_{tg}+v_{tw}$.

Substituting Equations (21)–(25) into the gas state equation, the average gas pressure change rate of the tubing can be expressed as follows:

$${\displaystyle \frac{d{p}_{tavg}}{dt}}={\displaystyle \frac{\frac{Z_{t}RT}{M}\frac{d{m}_g}{dt}-p_{tavg}{A}_t\frac{d{h}_t}{dt}}{\left(H-h_t\right)A_t}}={\displaystyle \frac{\frac{Z_{t}RT{\rho }_g}{M}\left(v_{tg}-v_{tgout}\right)-p_{tavg}\left(v_{tw}-v_{twout}\right)}{\left(H-h_t\right)}}$$

(29)

The pressure inside the tubing is calculated using the flow pressure Formula (21), and the acceleration of the gas–liquid column is calculated using Newton’s second law:

$$\begin{array}{r}{p}_{cg}{A}_t-p_{tg}{A}_t-{\rho }_{m}g{A}_t{h}_{tg}-\rho g{A}_t{h}_t-f_m{\displaystyle \frac{{\rho }_m{v}_m^2}{2d}}{A}_t{h}_t\\ =\left({\rho }_{m}g{A}_t{h}_{tg}+\rho g{A}_t{h}_t\right)a\end{array}$$

(30)

During the gas–liquid injection stage, flow parameters such as velocity, flow rate, pressure, and gas content are not uniformly distributed and constantly change inside the pipeline [29,30]. Therefore, it is necessary to study the changes in flow pattern characteristic parameters inside the tubing from a macroscopic perspective, and use the Mukherjee et al. [31] correlation to determine the flow pattern and calculate the slip holdup rate:

$$H_l=\exp \left[\left(c_1+c_2\sin \beta +c_3{\sin }^2\beta +c_4{N}_1^2\right){\displaystyle \frac{N_{vg}^{c_5}}{N_{vl}^{c_6}}}\right]$$

(31)

The gas phase drift velocity of bubbly flow is represented by Harmathy et al. [32] equation for the ascending velocity of small bubbles in static liquid:

$$v_0=1.53{\left(g\sigma \left({\rho }_l-{\rho }_g\right)/{\rho }_l^2\right)}^{0.25}\sqrt{\sin \theta }$$

(32)

The gas phase drift velocity of slug flow is represented by Bendisken et al. [33] equation for the rising velocity of Taylor bubbles in static fluid:

$$v_0=\left(0.35\sin \theta +0.54\cos \theta \right)\sqrt{gd\left({\rho }_l-{\rho }_g\right)/{\rho }_l}$$

(33)

The rising speed of the first gas bomb in the tubing determines the duration of the liquid plug eruption process, which only depends on the outflow velocity of the liquid plug in front of it. This value can be obtained from the Nicklin formula [34]:

$${\displaystyle \frac{d{h}_{tg}}{dt}}=v_{tbg}=C_0{v}_m+v_0$$

(34)

In the formula, $C_0$ is the flow coefficient. When the flow is bubbly flow, $C_0=1.2$, $v_0=1.53{\left(g\sigma \left({\rho }_l-{\rho }_g\right)/{\rho }_l^2\right)}^{0.25}\sqrt{\sin \theta }$; When the flow is a slug flow, $C_0=1.2$, $v_0=\left(0.35\sin \theta +0.54\cos \theta \right)\sqrt{gd\left({\rho }_l-{\rho }_g\right)/{\rho }_l}$; When the flow is an agitated flow, $C_0=1$, $v_0=0.28\sqrt{gd\left({\rho }_l-{\rho }_g\right)/{\rho }_l}$; When the flow becomes annular, gas bubbles no longer exist [35], and $v_{tbg}$ is the apparent velocity of the gas, $v_{tbg}=v_{tg}$.

According to the fact that the pressure at the connection between the tubing and the casing is always equal, combining Equations (10), (13), (14), (17), (21)–(31), and (34) can obtain the control equation for the stage of liquid accumulation outflow.

$$\begin{array}{c}{\displaystyle \frac{2\left(p_{cavg}+\frac{d{p}_{cavg}}{dt}\right)}{1+\frac{1}{e^{sc}}}}=a\left({\rho }_{l}g\left(h_t+{\displaystyle \frac{d{h}_t}{dt}}+{\displaystyle \frac{d{h}_{tg}}{dt}}\right)+{\rho }_{m}g\left(h_{tg}+{\displaystyle \frac{d{h}_{tg}}{dt}}\right)\right)\\ +{\rho }_{l}g\left(h_t+{\displaystyle \frac{d{h}_t}{dt}}+{\displaystyle \frac{d{h}_{tg}}{dt}}\right)+\left({\rho }_{m}g+f_m{\displaystyle \frac{{\rho }_m{v}_m^2}{2d}}\right)\left(h_{tg}+{\displaystyle \frac{d{h}_{tg}}{dt}}\right)\\ +\sqrt{{\left({\displaystyle \frac{2\left(p_{tavg}+\frac{d{p}_{tavg}}{dt}\right)}{1+\frac{1}{e^{st}}}}\right)}^2+1.32\times {10}^{-18}f{\left(q_{tsc}{T}_t{Z}_t\right)}^2\left(e^{2st}-1\right)/d^5}\end{array}$$

(35)

Wherein, ${\rho }_m={\rho }_g\left(1-H_l\right)+{\rho }_l{H}_l$.

The initial condition for the gas–liquid eruption stage is the termination condition for the liquid accumulation outflow stage. The height of the casing liquid column decreases to the connection between the tubing and the casing, $h_c=0$. The termination condition of gas–liquid eruption stage is that when the pressure inside the casing decreases to a point where it is not enough to push the liquid in the tubing to continue flowing out, that is, when the gas flow velocity in the tubing is less than the critical liquid carrying velocity, the droplets in the tubing begin to fall, $v_{tg}=v_{cr}$.

2.1.6. Liquid Reflux Stage

The depletion of gas energy in the casing annular space leads to a rapid decrease in the amount of gas ejected, causing the liquid in the tubing that has not yet been discharged to slide down to the bottom of the well and form liquid accumulation again. At this time, there is a low-speed flow with liquid mist inside the wellbore tubing, and the casing annular space is a static gas column. The space below the tubing shoe is divided into two sections, with the upper section being a low-speed flow with liquid mist and the lower section being a gas–liquid column, as shown in Figure 2d.

When the gas well begins to accumulate fluid, the lowest flow velocity in the wellbore is called the critical velocity of liquid carrying in the gas well, which follows Li Min’s ellipsoidal model [36].

$$v_{cr}=2.5{\left[\sigma \left({\rho }_l-{\rho }_g\right)/{\rho }_g^2\right]}^{0.25}$$

(36)

The falling speed of a droplet is affected by the drag force of the gas on the droplet and the settling gravity of the droplet itself, and is calculated by the following equation [37].

$$m_l{\displaystyle \frac{d{v}_{fd}}{dt}}={\displaystyle \frac{\pi }{6}}{d}_l^3\left({\rho }_l-{\rho }_g\right)g-{\displaystyle \frac{\pi }{4}}{d}_l^3{C}_l{\displaystyle \frac{v_{tg}^2}{2}}{\rho }_g$$

(37)

The volume of the falling liquid is given by the following equation:

$$V_f={\displaystyle {\int }_0^H\left(1-H_l\right)}{A}_{t}dh$$

(38)

The pressure caused by the liquid falling back from the bottom of the tubing is as follows:

$$p_f=\rho g{\displaystyle \frac{V_f}{A_t}}$$

(39)

The time of the liquid reflux stage is very short [38]. If the inflow of the formation during this stage is ignored, the control equation of the liquid reflux stage can be greatly simplified. The characteristic parameters of the liquid reflux stage can be directly calculated from Formulas (36)–(39). The height of liquid accumulation that forms a liquid plug when the liquid falls back, $h_0={\displaystyle \frac{V_f}{A_t}}$, The liquid reflux time is $T_4$. To make the calculation results more accurate, the control equation for the liquid reflux stage in this article considers two stages of gas–liquid flow processes: formation inflow and liquid reflux [39]. Substituting Equation (38) into the gas state equation, the average gas pressure change rate of the tubing can be expressed as follows:

$${\displaystyle \frac{d{p}_{tavg}}{dt}}={\displaystyle \frac{\frac{Z_{t}RT}{M}\frac{d{m}_g}{dt}-p_{tavg}{A}_t\frac{d{h}_t}{dt}}{\left(H-h_t-\frac{V_f}{A_t}\right)A_t}}={\displaystyle \frac{\frac{Z_{t}RT{\rho }_g}{M}{v}_{tg}-p_{tavg}{v}_{tw}}{\left(H-h_t-\frac{V_f}{A_t}\right)}}$$

(40)

According to the pressure equation at the connection point between the tubing and the casing, the control equation for the liquid reflux stage can be obtained by combining Equations (6), (9), (10), (13), (15) and (38)–(40):

$${\displaystyle \frac{2\left(p_{cavg}+\frac{d{p}_{cavg}}{dt}\right)}{1+\frac{1}{e^{sc}}}}+{\rho }_{l}g\left(h_c+{\displaystyle \frac{d{h}_c}{dt}}\right)={\displaystyle \frac{2\left(p_{tavg}+\frac{d{p}_{tavg}}{dt}\right)}{1+\frac{1}{e^{st}}}}+{\rho }_{l}g\left(h_t+{\displaystyle \frac{V_f}{A_t}}+{\displaystyle \frac{d{h}_t}{dt}}\right)$$

(41)

The initial conditions for the liquid reflux stage are the termination conditions for the gas–liquid ejection stage. When the gas flow velocity in the tubing is less than the critical liquid carrying velocity, the droplets in the tubing begin to fall, $v_{tg}=v_{cr}$. The termination condition for the liquid reflux stage is that the liquid droplets in the tubing fall back to the bottom of the tubing.

2.2. Model Solution

2.2.1. Grid Division Method

Perform one-dimensional grid partitioning on intermittent production gas wells, as shown in Figure 3a. Due to the fixed position of the tubing shoes and the fact that the tubing shoe position is a critical calculation location, the grid can be divided into n parts from the bottom of the well to the tubing shoes, m parts from the tubing shoes to the wellhead, and a total of n + 1 grids from bottom to top. In severe slug flow simulations similar to intermittent production, the liquid segment interface is usually used as a fixed interface for grid division. As the liquid section moves, the mesh in the wellbore will continuously deform, reducing the quality of the mesh, so it takes some time to redraw the grid. In order to reduce computational complexity, the initial grid is fixed unchanged, which may result in the slug interface not being exactly at the grid boundary [40,41,42]. This requires resegmentation of the grid, as shown in Figure 3b. This segmented interface tracking method only adds a few more computational grids to the model, without the need to constantly redraw the grids during the calculation process.

The position of the upper and lower interfaces of the initial plug is known, and the equation for the position of the plug interface is as follows:

$$\left\{\begin{array}{l}{X_{\mathrm{down}}}^{\mathrm{new}}={X_{\mathrm{down}}}^{\mathrm{old}}+v_{\mathrm{down}}\Delta t\\ {X_{\mathrm{up}}}^{\mathrm{new}}={X_{\mathrm{up}}}^{\mathrm{old}}+v_{\mathrm{up}}\Delta t\end{array}\right.$$

(42)

2.2.2. Boundary Conditions

(1) Boundary conditions during the closing stage

The pressure at the bottom of the well boundary is equal to the bottomhole flowing pressure, and the change in bottomhole flowing pressure needs to be calculated by coupling the inflow of the formation. The gas–liquid flow velocity at the bottom of the well is also calculated based on the inflow of the formation.

Bottom boundary of tubing:

$$\left\{\begin{array}{l}{p}_{\mathrm{t}}\left(0,t\right)=p_{wf}\\ v_l\left(0,t\right)=f\left(p_{wf},p_r\right)\\ v_g\left(0,t\right)=g\left(p_{wf},p_r\right)\end{array}\right.$$

(43)

Top boundary of tubing:

$$\left\{\begin{array}{l}{v}_l\left(n_{\mathrm{t}},t\right)=0\\ v_g\left(n_{\mathrm{t}},t\right)=0\end{array}\right.$$

(44)

Bottom boundary of casing:

$$\left\{\begin{array}{l}{p}_{\mathrm{t}}\left(0,t\right)=p_{wf}\\ v_l\left(0,t\right)=f\left(p_{wf},p_r\right)\\ v_g\left(0,t\right)=g\left(p_{wf},p_r\right)\end{array}\right.$$

(45)

Top boundary of casing:

$$\left\{\begin{array}{l}{v}_l\left(n_{\mathrm{c}},t\right)=0\\ v_g\left(n_{\mathrm{c}},t\right)=0\end{array}\right.$$

(46)

(2) Boundary conditions during the opening stage

The pressure at the bottom of the well boundary is equal to the bottomhole flowing pressure, and the change in bottomhole flowing pressure needs to be coupled with the formation inflow calculation. The bottomhole temperature is known, and the gas–liquid flow velocity at the bottomhole position is also calculated based on the formation inflow.

Bottom boundary of tubing:

$$\left\{\begin{array}{l}{p}_{\mathrm{t}}\left(0,t\right)=p_{wf}\\ v_l\left(0,t\right)=f\left(p_{wf},p_r\right)\\ v_{\mathrm{g}}\left(0,t\right)=g\left(p_{wf},p_r\right)\end{array}\right.$$

(47)

Top boundary of tubing:

$$\left\{\begin{array}{l}{p}_2=Const\\ v_{\mathrm{L}}\left(n_{\mathrm{t}},t\right)=f^{\prime }\left(p_{\mathrm{t}},p_2\right)\\ v_{\mathrm{g}}\left(n_{\mathrm{t}},t\right)=g^{\prime }\left(p_{\mathrm{t}},p_2\right)\end{array}\right.$$

(48)

Bottom boundary of casing:

$$\left\{\begin{array}{l}{p}_{\mathrm{t}}\left(0,t\right)=p_{wf}\\ v_l\left(0,t\right)=f\left(p_{wf},p_r\right)\\ v_{\mathrm{g}}\left(0,t\right)=g\left(p_{wf},p_r\right)\end{array}\right.$$

(49)

Top boundary of casing:

$$\left\{\begin{array}{l}{v}_l\left(n_{\mathrm{c}},t\right)=0\\ v_{\mathrm{g}}\left(n_{\mathrm{c}},t\right)=0\end{array}\right.$$

(50)

2.2.3. Solution Steps

The steps for solving the transient mathematical model of intermittent production are as follows:

Step 1: Initial value calculation. Calculate the wellbore pressure distribution based on the wellhead tubing pressure and casing pressure at the beginning of fluid accumulation. At this time, the casing contains a pure gas column, $h_c=0$. Calculate the initial liquid accumulation height of the tubing, $h_t$, at the time of starting the liquid accumulation.

Step 2: Grid division. Divide the grid of the tubing and casing based on the position of the connection between the tubing and casing.

Step 3: Calculation of tubing and casing pressure. Based on the transient mathematical model of intermittent production, calculate the pressure and velocity distribution at each position of the tubing and casing at time t.

Step 4: Formation inflow coupling. According to the bottom hole flow pressure at time t, calculate the gas flow rate of the formation into the wellbore.

Step 5: Coupling of segment interface position. Calculate the liquid level position at the next time based on the fluid velocity at the plug interface grid at time t.

Step 6: Flow stage judgment. If $h_c>0$, the wellbore continues to accumulate fluid; Otherwise, the wellbore is opened and enters the stage of fluid accumulation and outflow; The time for the formation stage of fluid accumulation is $t=T_1$. If $h_c=0$, the liquid plug outflow stage ends and enters the gas–liquid ejection stage. The duration of the liquid outflow stage is $t=T_2$; If $h_c=0$ and $v_{tg}<v_{cr}$, the well will be shut in and the gas–liquid injection stage will end, entering the liquid reflux stage. The duration of the gas–liquid spraying stage is $t=T_3$; If $v_{tg}<v_{cr}$ and $t=T_4$, then the liquid reflux stage is over. A complete intermittent production period ends and enters the stage of liquid accumulation formation in the next intermittent production period.

Step 7: Update of boundary conditions. Based on the formation related equations, calculate the inflow gas and liquid volume at time t ~ t + 1, and update the formation pressure. Based on the pressure distribution of the tubing and casing at time t, calculate the outflow gas and liquid volume from time t to t + 1, and update the bottomhole flow pressure, wellhead tubing pressure, and casing pressure.

Step 8: Repeat steps 3 to 6 to calculate the opening time, $t_{\mathrm{on}}=T_2+T_3$, and closing time, $t_{\mathrm{off}}=T_1+T_4$, for each intermittent production period, which will guide the optimization of the intermittent production system.

3. Optimization of Intermittent Production System

3.1. Overview of Intermittent Gas Well Production

Based on the established transient mathematical model of intermittent production, the production dynamic characteristics are obtained through simulation to guide the optimization of intermittent production process system. The reservoirs of the East Sichuan Carboniferous gas reservoirs in the Sichuan Basin generally have the characteristics of low matrix porosity, poor permeability, and strong heterogeneity. Some Carboniferous gas reservoirs produce formation water during development and production. A large number of gas wells in this block are entering the mid to late stage of production and require intermittent production. Taking the Menxi 005-H3 well as an example, the well depth is 5085.1 m, the inner diameter of the tubing is 73 mm, and the inner diameter of the casing is 166 mm. The Menxi 005-H3 well was put into operation in April 2012, with an initial gas production of around 20 × 10⁴ m³/d. After self exploitation, the gas production continued to decrease. By 18 April 2018, the average gas production of the well had dropped to 5 × 10⁴ m³/d, with a water production of 0.6 m³/d and a gas water ratio of 8.33 × 10⁴ m³/m³. The tubing pressure was 4.7 MPa, and at this time, the bottomhole flowing pressure was insufficient to maintain the production of the gas well. As a result, the gas well began intermittent production, and the gas production curves are shown in Figure 4 and Figure 5.

3.2. Intermittent Production History Fitting

Based on the established transient mathematical model of intermittent production, production history fitting was carried out under the original intermittent system of Menxi 005-H3 well. At this time, the on–off system was 144 h of well opening and 21 h of well closing. The pressure behind the valve of the well fluctuates between 1.9 and 2.3 MPa. During the opening period, the casing pressure drops by 5.1 MPa and the tubing pressure drops by 5.1 MPa. At the moment of opening, the instantaneous production at the wellhead reaches 4.51 × 10⁴ m³/d, but the decrease is rapid. The average gas production during the opening period is 1.45 × 10⁴ m³/d. The fitting of casing pressure, tubing pressure, and gas production history for gas wells is shown in the Figure 6, Figure 7 and Figure 8.

The evaluation index for the fitting accuracy of intermittent production history in gas wells adopts the average relative error δ: the ratio of the difference between the actual value and the fitted value is calculated using the actual value and the fitted value [43], as shown in Equation (51):

$$\delta ={\displaystyle \frac{1}{M}}{\displaystyle \sum _{m=1}^M\frac{|y_m-{\widehat{y}}_m|}{y_m}}$$

(51)

In the formula: $M$ represents the total number of production dynamic data during a period of intermittent production in the gas well, $y_m$ represents the fitted value of the mth production dynamic data, and ${\widehat{y}}_m$ represents the actual value of the mth production dynamic data. The smaller the average relative error δ, the better the fitting effect of intermittent production history; The larger the average relative error δ, the worse the fitting effect of intermittent production history. The average relative errors of historical data fitting for wellhead casing pressure, wellhead tubing pressure, and gas production of Menxi 005-H3 gas well are δ_Pcg = 2.84%, δ_Ptg = 2.45%, and δ_qsc = 3.69%, respectively. There is a slight deviation between the simulation results and the actual casing pressure, oil pressure, and gas production in the gas field, which is because the one-dimensional flow assumed in this article does not take into account the complex geometric structure of the wellbore, and cannot fully reflect the gas–liquid two-phase mixed flow state in the wellbore, which inevitably leads to deviation. Meanwhile, as the relative error values are all below 5%, it proves that the simulation results of the transient model for intermittent production are highly accurate and can reflect the real situation of intermittent production in gas wells. When potential future work in testing model with more diverse on-site data, if the gas well already has a fixed intermittent production cycle, the model can be directly used for production history fitting. If the gas well does not have a fixed intermittent production cycle, it is necessary to determine the intermittent production cycle of the gas well based on its historical production data, and carry out intermittent production under this cycle to obtain corresponding production data. Then, the applicability of the model to gas wells was verified by fitting the production data of the gas wells.

3.3. Dynamic Analysis of Intermittent Production Process

(1) Closing stage

Based on the transient mathematical model of intermittent production, the variation laws of pressure, liquid level rise velocity, liquid level height, etc. during the shut in process are calculated. From Figure 9, Figure 10 and Figure 11, it can be seen that during the intermittent production shut in stage, the bottomhole pressure, tubing pressure, and casing pressure initially increase rapidly, and the growth velocity gradually slows down with the increase of shut in time. When the shut in time reaches 11 h, the bottomhole pressure, tubing pressure, and casing pressure stop increasing. Similarly, the velocity of fluid accumulation in the tubing gradually decreases with increasing shut in time, and the height of fluid accumulation in the tubing gradually increases with increasing shut in time. When the shut in time reaches 11 h, the velocity of fluid accumulation in the tubing decreases to 0, and the height of fluid accumulation in the tubing stops increasing. At this point, the fluid accumulation in the tubing ends, and the bottomhole pressure just reaches its maximum value. Therefore, the shortest time required for the bottomhole flow pressure to reach its maximum value after well closure is 11 h.

(2) Opening stage

Based on the transient mathematical model of intermittent production, the variation laws of pressure, liquid level rise velocity, liquid level height, etc. during the well opening process are calculated. From Figure 12, Figure 13 and Figure 14, it can be seen that during the intermittent production well opening stage, the tubing pressure and casing pressure initially decrease rapidly, and the rate of decrease gradually slows down with increasing well opening time. When the well opening time reaches 2 h, the tubing pressure and casing pressure stop decreasing. The velocity of liquid level rise in the tubing increases first and then decreases with the increase in well opening time. When the well opening time is between 0 and 0.18 h, due to the pressure in the casing being higher than that in the tubing, the accumulated fluid in the casing gradually flows into the tubing, and the velocity of liquid level rise slowly increases with time. At this point, it is in the stage of accumulated fluid outflow. When the well opening time is between 0.18 and 2 h, there is no accumulation of fluid in the casing, and the liquid in the tubing separates from the bottom of the tubing, forming a liquid plug and rapidly moving upwards. At this point, it is in the stage of gas–liquid eruption. As the well opening time increases, the bottomhole pressure and casing pressure gradually decrease, resulting in a gradual decrease in the lifting force that drives the upward movement of the liquid plug, and the acceleration of the upward movement of the liquid plug gradually decreases, even becoming negative. At this time, the velocity of the liquid plug first increases and then decreases. When the well opening time reaches 2 h, the velocity of fluid accumulation in the tubing decreases to 0, and the height of fluid accumulation in the tubing stops increasing. At this time, the liquid plug in the tubing has just reached the wellhead.

3.4. Optimization Simulation of Intermittent Production System

According to the transient mathematical model of intermittent production, the shortest time required for the bottomhole flowing pressure to reach its maximum value after well closure is 11 h. Therefore, the optimal shut in time for the intermittent production system is 11 h, and the change curves of wellhead tubing pressure, casing pressure, and gas production corresponding to the intermittent production well under the corresponding system are calculated, as shown in Figure 15, Figure 16 and Figure 17. At this time, the corresponding tubing pressure during well opening is 7.57 MPa, and the casing pressure is 7.60 MPa. The instantaneous production at the wellhead reaches 4.49 × 10⁴ m³/d at the moment of well opening, but it decreases rapidly. The average gas production during well opening is 1.51 × 10⁴ m³/d. Compare the cumulative gas production of the original intermittent production system and the optimized system within one month, as shown in Figure 18. It can be seen that the cumulative gas production of the Menxi 005-H3 well within one month under the original intermittent production system was 3.65 × 10⁵ m³, and after the optimization of the intermittent production system, the cumulative gas production within one month was 3.91 × 10⁵ m³, indicating a higher cumulative gas production. This indicates that the intermittent production system optimization model and method established in this article are effective in achieving the highest cumulative gas production of the intermittent production system optimization.

3.5. Integration of Intelligent Control System for Intermittent Production System

In order to guide the actual production on the oilfield site, an intelligent optimization control system for intermittent production was developed based on the transient mathematical model of gas well intermittent production, which specifically includes the storage unit, processing unit, automatic execution device, and time recording device. The storage unit and processing unit are integrated into the existing production monitoring system, and the automatic execution device and time recording device are installed in the oil pipe production control valve, as shown in Figure 19. Due to the complexity of the transient model established in this article, it is necessary for the production monitoring system to have high computing power to achieve real-time monitoring. A Core i9 13,900 or above CPU, GeForce RTX 4080 or above graphics card, 32 GB or higher memory, and reserve more than 10 GB of available hard disk space are needed. To significantly improve computational efficiency, in the actual calculation process, the calculation time interval designed for this model is 10 s, and the time interval for obtaining production dynamic data on the gas field site is 10–30 s, which significantly improves computational efficiency while ensuring accurate results. The specific implementation of the intelligent optimization control system for intermittent production is as follows:

Step 1: The storage unit automatically retrieves and stores the production dynamic data of the intermediate gap gas well in the production monitoring system, including gas production, water production, wellhead oil pressure, and wellhead casing pressure, and transmits the data to the processing unit;

Step 2: The processing unit includes the transient mathematical model for intermittent production of gas wells established in this article. By solving the model, the opening time t_on = T₂ + T₃ and the closing time t_off = T₁ + T₄ for each intermittent production cycle are obtained;

Step 3: The processing unit receives the time recording device to record the well closing time t₁ and the well opening time t₂. If t₁ = t_off, the well closing phase ends and enters the well opening phase, where the processing unit generates a well opening instruction; If t₂ = t_on, the well opening phase ends and enters the well closing phase, where the processing unit generates a well closing instruction.

Step 4: The automatic execution unit receives the well closing instruction from the processing unit, causing the production control valve of the oil pipe to close. At the same time, the time recording device starts automatically recording the well closing time t₁.

Step 5: The automatic execution unit receives the well opening instruction from the processing unit, causing the production control valve of the oil pipe to open. At the same time, the time recording device starts automatically recording the well opening time t₂.

Step 6: Repeat Steps 1 to 5 to achieve real-time control of the production switch system for intermittent gas wells.

4. Conclusions

• The intermittent production process of gas wells is divided into four stages, and a transient mathematical model for intermittent production is established in stages, considering the variation laws of slip gas holding rate, pressure, and gas–liquid inflow and outflow of tubing and casing;
• The use of dynamic tracking technology for the movement of liquid plugs in the wellbore results in fine mesh division of the plug interface position. The mesh boundary is re finely divided to ensure that the liquid plug interface falls exactly at the mesh boundary, greatly reducing computational complexity and improving computational accuracy;
• The transient mathematical model developed for intermittent gas well production achieved a high historical fit accuracy of over 90%. This indicates that the simulation results are in line with the real situation of gas well intermittent production and can effectively guide intermittent production;
• Based on the transient mathematical model of intermittent production, the boundary point between opening and closing the well is determined, and the variation laws of pressure, liquid level rise velocity, liquid level height, etc. during the closing process are obtained. The optimal closing time for intermittent production in Menxi 005-H3 well is determined to be 11 h;
• The cumulative gas production of Well Menxi 005-H3 within one month under the original intermittent production system was 3.65 × 10⁵ m³. After optimizing the intermittent production system, the cumulative gas production within one month was 3.91 × 10⁵ m³, indicating a higher cumulative gas production. This demonstrates the effectiveness of the intermittent production system optimization model and method established in this paper, achieving the optimization of the intermittent production system and guiding efficient production of gas wells in low-pressure formations.