Research on the Optimization of Self-Injection Production Effects in the Middle and Later Stages of Shale Gas Downdip Wells Based on the Depth of Pipe String

Abstract

In the final phases of casing production, shale gas horizontal wells with a downward slope frequently find it difficult to sustain self-flow production. The ideal tubing insertion depth for self-flow production in gas wells has not been thoroughly studied, even though the timely adoption of tubing production can successfully prolong the self-flow production period. Using a fully dynamic multiphase flow simulation program, the ideal tubing depth for gas well self-flow production was ascertained. A wellbore structural model was built using a particular well as an example. By altering the tubing depth, the formation pressure limit values necessary to sustain gas well self-flow production at various tubing depths were simulated. The appropriate tubing depth for gas well self-flow production was examined, along with the well’s cumulative gas output at various tubing depths. Using the example as a case study, it was discovered that the critical formation pressure for gas well self-flowing production dropped to 7.8 MPa when the tubing was lowered to 2600 m. This effectively increased cumulative production by 56.19 × 10⁶ m³ and extended the self-flow production time by roughly 135 days. The study’s findings offer strong evidence in favor of maximizing shale gas wells’ self-flow production performance in later phases of production.

1. Introduction

One of the most important energy sources created in China is shale gas, which has a short, high high-production period, a long, low-pressure and low-production self-injection stable production mining period, and a problem with difficult discharge. Therefore, one of the main challenges in the middle and late stages of shale gas well exploitation is how to further optimize discharge mining process technology so that shale gas wells can achieve the maximum development benefits. Relying on formation pressure, China’s shale gas wells are currently in the middle and late stages of exploitation. Failure occurs to some degree when the energy of the gas well is not enough to generate production.

Scholars have carried out various studies on drainage and extraction effects in shale gas reservoirs. Peng et al. conducted a systematic review of water invasion mechanisms and identification, and the dynamic prediction of gas reservoirs. The technical adaptability and applicable scope of different methods for enhancing oil recovery were summarized and their application effects were evaluated [1]. Among the studies on the actual drainage and extraction effects of gas well production, Bai et al. proposed a new type of foaming agent to increase gas well production by analyzing the process characteristics of low-producing gas wells [2]. Na L et al. proposed a practical application model and strategy for drainage gas extraction technology by analyzing the water production patterns, production capacity details, and diagnostic patterns of wellbore fluid accumulation in shallow gas fields [3]. Zhi Y et al. modeled intelligent diagnosis and optimization of the plunger process to improve the effectiveness of the gas well process by 15% and reduce economic costs by 70% [4]. Wang et al. combined theoretical and indoor tests to develop a fluid-carrying model based on different tubing depths in horizontal wells [5]. Xu et al. developed a shale gas well production prediction model, which was modified by introducing a material balance equation [6]. Leli et al. predicted the dynamic gas production of gas wells by summarizing the characteristics of different production capacity equations and production pressure variations [7]. He et al. conducted a systematic review of Fuling shale gas well drainage technology in response to the current mismatch between the current drainage process and the actual situation of gas field development [8]. Gao et al. optimized the sulfur-resistant velocity column in terms of column dimensions and tubing materials, selected appropriate diameters and tubing materials, and developed a new type of broken disc velocity column plug, thereby achieving stable and continuous liquid carryover in gas wells [9]. Guo et al. used a horizontal well indoor simulation test rig to simulate the flow state and liquid accumulation of gas–liquid two-phase fluids in the wellbore before and after loading the internal vortex tool. They also used numerical simulation methods to study the flow conditions and pressure drop amplitude of gas-liquid two-phase fluids under different gas–liquid flow conditions [10]. Zhang et al. used horizontal well, multiphase flow, experimental apparatus to apply external drainage, vortex tools, and Venturi acceleration vortex tools to horizontal pipes, and analyzed the effects of different gas flow rates and liquid flow rates on the spiral flow length and pressure drop generated by the three tools [11]. Wu Ruidong et al. considered the characteristics of atomization-assisted production and the dynamic equilibrium principle of gas–liquid two-phase flow in the wellbore, established a gas phase liquid-carrying droplet model, and studied the solution methods for the upstream and downstream driving forces of droplet flow [12]. Li Haitao et al. studied and analyzed the liquid and solid carrying problems of vertical wellbores under foam circulation purging conditions, revealing the relevant liquid carrying laws and solid carrying laws [13]. Bin Huang et al. analyzed the distribution characteristics of foam in a wellbore during the foam drainage gas recovery process through experiments and introduced the concepts of foaming efficiency and foam-carrying efficiency to calculate the liquid discharge time and liquid accumulation time of foam drainage gas recovery in the wellbore [14]. Jianda Chen et al. started from the production characteristics of liquid-producing gas wells, analyzed the productivity of gas wells, wellbore pressure distribution, and critical liquid-carrying flow rate, and proposed a design method for pumping depth in the drainage and gas production process of shale gas wells by using the node analysis method [15].

Most scholars’ research on shale gas wells has focused on analysis of the advantages and disadvantages of drainage and production processes as well as practical application models. There are relatively few in-depth studies on the optimization of production effects in the middle and later stages of shale gas well production, and even fewer studies on shale gas down-inclined wells. Therefore, in order to analyze the influence of tubing depth on the self-injection production effect in the later stages of shale gas downpour well production, this paper uses an example to study the critical formation pressure of self-injection in gas wells with different tubing depths and analyze the cumulative gas production of gas wells with different tubing depths. This paper discusses the problems encountered in the middle and later stages of production in horizontally inclined shale gas wells, especially the difficulty of self-injection production. By using the full dynamic multiphase flow simulation program, the ideal tubing depth for gas well self-injection production was established and, taking a certain well as an example, a wellbore structure model was established. By changing the depth of the tubing, the limit values of formation pressure required to support the self-injection production of gas wells at different tubing depths were simulated, thereby investigating the appropriate tubing depth for the self-injection production of gas wells and cumulative gas production at different tubing depths.

2. Mathematical Model

2.1. Principles of GLV Model Simulation

The GLV gas lift model in OLGA 2022.1.0 software is used to simulate the flow behavior of gas–liquid–viscous multiphase flows (e.g., containing highly viscous crude oil, non-Newtonian fluids, etc.). The model injects gas from the annulus or casing into the tubing, defines the corresponding inflow–dynamic relationships according to different reservoirs, and calculates the mass flow rate between the reservoir and the wellbore using a binomial capacity equation [16]. At the same time, the model can simulate the production effect of a gas well by adjusting the downstream depth of the tubing at a constant injection pressure (formation pressure). Therefore, a gas lift model is selected in this paper to simulate the production effect of gas wells. The mass conservation equation of gas–liquid two-phase flow is shown in Equation (1), the energy conservation equation of gas–liquid two-phase mixing is shown in Equation (2), and the mass flow rate calculation equation is shown in Equations (3) and (4). The descriptions of each parameter are shown in

Table 1. Meaning of each equation abbreviation.

Abbreviation	Abbreviation Meaning	Unit
GLV	Gas lift valve	\
${\rho }_g$	Gas phase density	kg/m3
${\alpha }_{\mathrm{g}}$	The volume fraction of the gas	\
${\upsilon }_g$	Gas phase velocity	m/s
$M_g$	The mass flow rate of the gas phase	kg/s
${\rho }_l$	Liquid phase density	kg/m3
${\alpha }_l$	Liquid volume fraction	\
${\upsilon }_l$	Liquid phase velocity	m/s
$M_l$	The mass flow rate of the liquid phase	kg/s
$A_t$	The cross-sectional area of the passage	m2
$M_i$	The relative molecular mass of component i	\
$N_i$	The mass transfer rate of component i, i = 1, 2 … n	kg/(m3·s)
$m_g$	Gas phase mass	kg
$E_{\mathrm{g}}$	Gas phase internal energy	J
${\upsilon }_g$	Gas phase velocity	m/s
$H_g$	Enthalpy of the gas phase	J/kg
${\mathrm{m}}_l$	Liquid phase mass	kg/m3
$E_l$	Liquid phase internal energy	J
${\upsilon }_l$	Liquid phase velocity	m/s
$H_l$	Enthalpy of the liquid phase	J/kg
$m_D$	The quality of the particle phase	kg
$E_D$	The internal energy of the particle phase	J
${\upsilon }_D$	The velocity of the particle phase	m/s
$H_D$	The enthalpy of the granular phase	J/kg
$\mathrm{h}$	height	m
$g$	Gravitational acceleration	m/s2
${\mathrm{H}}_{\mathrm{s}}$	The enthalpy value that the source possesses	J/kg
$U$	The heat of the pipe wall	J
$W_{TOT}$	The total mass flow rate	\
$W_G$	Gas mass flux	kg/(m2s)
${\rho }_{mix}$	Average bulk density	kg/m3
${\rho }_G$	Gas density	kg/m3
$W_P$	The mass flow rate of each phase flow	\
${\chi }_P$	The mass fraction corresponding to the phase flow	\

.

$$({\rho }_l{\alpha }_l{\upsilon }_l{A}_t{\displaystyle \frac{x_i{M}_i}{M_l}}+{\rho }_g{\alpha }_g{\upsilon }_g{A}_t{\displaystyle \frac{y_i{M}_i}{M_g}})+N_i{M}_i{A}_{t}dz={\displaystyle \frac{\partial }{\partial t}}({\rho }_l{\alpha }_l{A}_t{\displaystyle \frac{x_i{M}_i}{M_l}}+{\rho }_g{\alpha }_g{A}_t{\displaystyle \frac{y_i{M}_i}{M_g}})$$

(1)

$$\begin{array}{l}{\displaystyle \frac{\partial }{\partial t}}[m_g(E_g+{\displaystyle \frac{1}{2}}{v}_g^2+gh)+m_l(E_l+{\displaystyle \frac{1}{2}}{v}_l^2+gh)+m_D(E_D+{\displaystyle \frac{1}{2}}{v}_D^2+gh)]\\ =-{\displaystyle \frac{\partial }{\partial z}}[m_g{v}_g(H_g+{\displaystyle \frac{1}{2}}{v}_g^2+gh)+m_l{v}_l(H_l+{\displaystyle \frac{1}{2}}{v}_l^2+gh)+m_D{v}_D(H_D+{\displaystyle \frac{1}{2}}{v}_D^2+gh)]+H_s+U\end{array}$$

(2)

The total mass flow rate is expressed as

$$W_{TOT}=W_G\sqrt{{\displaystyle \frac{{\rho }_{mix}}{{\rho }_G}}}$$

(3)

The mass flow rate of each phase flow (oil, gas, water, etc.) is expressed as

$$W_P={\chi }_P\cdot W_{TOT}$$

(4)

2.2. Gas Well Production Equation

The binomial productivity equation calculates the relationship between pressure and production derived from the seepage theory of gas reservoirs. After calculating the production capacity index coefficient through Equations (6) and (7), OLGA calculates the relationship between flow rate and pressure drop based on the input production capacity index coefficient and the seepage parameters set by the software. Its expression is as follows and the descriptions of each parameter are shown in

Table 2. Meaning of each equation abbreviation.

Abbreviation	Abbreviation Meaning	Unit
$B$	Laminar coefficient	\
$C$	Turbulence coefficient	\
$p_r$	Stratigraphic pressure	MPa
$p_{wf}$	Bottoming-out pressure	MPa
$q_g$	Stabilized production from gas wells	104 m3/d
$T$	Reservoir temperature	°
$z$	Gas compression factors for reservoir conditions	\
$r_e$	Drive radius	m
$r_w$	Wellbore radius	m
$s$	Mechanical skin factor	\
$D$	Non-Darcy flow coefficient	\
$k$	Effective penetration rate	mD
$h$	Effective thickness	m

:

$${p_r}^2-{p_{wf}}^2=B{q}_g+{C{q}_g}^2$$

(5)

and is calculated in Equations (6) and (7):

$$B={\displaystyle \frac{T{\mu }_{g}z}{0.703kh}}[\ln ({\displaystyle \frac{r_e}{r_w}})-0.75+s]$$

(6)

$$C={\displaystyle \frac{T{\mu }_{g}z}{0.703kh}}D$$

(7)

2.3. Material Balance Equation

The material balance equation, also known as the pressure drop method, is a widely used and relatively accurate method of calculating dynamic reserves. Currently, the main types of gas reservoirs where the material balance equation is applied are fixed-volume confined gas reservoirs, water-driven gas reservoirs, condensate gas reservoirs, and anomalously high-pressure gas reservoirs. The descriptions of each parameter are shown in

Table 3. Meaning of each equation abbreviation.

	Abbreviation Meaning	Unit
$G_p$	Cumulative gas production	m3
$G$	Original geological reserve	m3
$B_g$	Volume factor at current formation pressure	MPa
$B_{gi}$	Original volume factor	\
$T_{sc}$	Temperature of the ground under standard Conditions	K
$p$	Current ground pressure, MPa	MPa
$p_{sc}$	Ground standard pressure	MPa
$p_i$	Original ground pressure	MPa
$Z_i$	Gas compression factor at the original formation pressure	MPa

. In this study, the cumulative gas production of the example wells was calculated using the material balance equation for a fixed volume confined gas reservoir [17]:

$$G_p{B}_g=G(B_g-B_{gi})$$

(8)

$$B_g={\displaystyle \frac{p_{sc}ZT}{p{T}_{sc}}}$$

(9)

$$B_{gi}={\displaystyle \frac{p_{sc}{Z}_{i}T}{p_i{T}_{sc}}}$$

(10)

Then:

$${\displaystyle \frac{p}{Z}}={\displaystyle \frac{p_i}{Z_i}}\left(1-{\displaystyle \frac{G_p}{G}}\right)$$

(11)

2.4. Model Verification

We used this model to simulate a real gas well that is currently in production. The simulation results were highly consistent with the real drainage and gas testing data of this well (Figure 1), demonstrating the authenticity and validity of the model.

3. Model Parameter Acquisition

3.1. Well Structure

According to the well trajectory data of the example well, the well has a measured depth of 3500 m and a vertical depth of 2072.99 m, and the gas well trajectory simulation is built in OLGA numerical simulation software (Figure 2), which can be seen as a typical downward-dipping well. According to the simulation of 2-inch tubing, the inner diameter of the tubing is 50.66 mm and the outer diameter is 60.23 mm. The tubing was used in the early stage of production and then it was changed to annular production and the gas production was improved. At present, the annular air production is still used, the gas production is about 25,000 m³/d, and the liquid production is about 2 m³/d or so.

3.2. Production Equation

The drainage gas test data of the well (Figure 3) were processed by the binomial capacity test well data processing method to obtain the capacity equation for the whole section of the well as

$${{\mathrm{p}}_r}^2-{{\mathrm{p}}_{wf}}^2=27.317q_g+3.0725{q_g}^2$$

3.3. P-Z Relationship for Fixed-Volume Gas Reservoirs

According to the gas components (

Table 4. Fluid component.

Hydrogen (mol%)	Helium (mol%)	Nitrogen (mol%)	CO2 (mol%)	Methane (mol%)	Ethane (mol%)	Propane (mol%)
0	0.03	0.28	0.34	98.9	0.35	0.01

), the DAK8 parameter method was used to calculate the gas compression factor under different formation pressures at the formation temperature of 92 °C [18], the compression factor change curve shown in Figure 4 was obtained, and the specific data of the formation pressure and compression factor are shown in

Table 5. Gas compression factors at different formation pressures.

Pressure (MPa)	Gas Compression Factor	Pressure (MPa)	Gas Compression Factor
50	1.1921	24	0.9488
48	1.1697	22	0.9383
46	1.1477	20	0.9299
44	1.1260	18	0.9241
42	1.1047	16	0.9210
40	1.0839	14	0.9208
38	1.0637	12	0.9236
36	1.0442	10	0.9296
34	1.0254	8	0.9386
32	1.0076	6	0.9504
30	0.9908	4	0.9648
28	0.9753	2	0.9814
26	0.9612	1	0.9905

.

3.4. Analog Programming

From the production data of the gas well, it is known that the original formation pressure of the well is about 20 MPa. In the simulation process, firstly, the tubing was lowered to the inclination point at 2200 m and the formation pressure was set to drop from 20 MPa to simulate the production dynamics of the gas well when the tubing was lowered to the inclination point under different formation pressure conditions. Subsequently, the tubing depth was changed to simulate the production dynamics of gas wells under different tubing depths and formation pressures, and the critical formation pressure that can maintain gas wells under different conditions was obtained. The specific simulation scheme is shown in

Table 6. Simulation scheme.

Tubing Depth (m)	Bottom Hole Flowing Pressure (MPa)	Tubing Depth (m)	Bottom Hole Flowing Pressure (MPa)	Tubing Depth (m)	Bottom Hole Flowing Pressure (MPa)
2200	9	2400	9	2600	9
	8		8		8
	7.8		7.8		7.8
	7.7		7.7		7.7
	7.6		7.6		7.6
	7.4		7.4		7.4
	7		7		7
	6		6		6
2800	9	3000	9	3200	9
	8		8		8
	7.8		7.8		7.8
	7.7		7.7		7.7
	7.6		7.6		7.6
	7.4		7.4		7.4
	7		7		7
	6		6		6

.

4. Results and Discussion

As the formation gas is continuously extracted, the formation pressure gradually decreases. When the formation pressure drops to a certain critical value, the formation pressure will not be able to maintain the gas well self-blowout production if the formation pressure is further reduced, and this paper refers to this critical value as the critical formation pressure.

According to the simulation results (Figure 5), when the formation pressure is 8 MPa, the fluctuation of gas production is more stable, and there is no obvious downward trend; when the formation pressure drops to 7.8 MPa, the gas production shows a slight downward trend in the late simulation, and the amount of liquid production decreases; when the formation pressure drops to 7.7 MPa, the fluctuation of gas production is frequent in the late stage of production and the downward trend is obvious, the liquids start to accumulate at the bottom of the gas well, and eventually the gas well accumulates liquid and stops production (the final gas well accumulates liquid). Eventually, the gas well accumulated liquid and stopped production (Figure 6). Due to the accumulation of liquid in the casing, gas passes through the liquid accumulation section during the production of the gas well, thus causing fluctuations in gas production. Moreover, as the formation pressure decreases, it can be seen from Figure 4 that the trend in gas production decline does not have a clear boundary and it is difficult to determine the critical formation pressure at this time. In order to determine the critical formation pressure, a straight fit is made to the gas production curve, and the decreasing trend is judged based on the slope (Figure 6), thereby determining the reasonable critical formation pressure. Therefore, the critical formation pressure is 8 MPa when the tubing is lowered to the inclination point and according to the gas components (Table 4), analyzing the gas compression factor under different formation pressures, and determining the geological reserve of 8.24 × 10⁷ m³ based on the production data of the gas wells, the cumulative gas production of the wells at this time is calculated to be 55.26 × 10⁶ m³ by using the equation of the balance of substances.

Change the position of the tubing depth to simulate the production dynamics of gas wells at different tubing depths and determine the corresponding critical formation pressure at this time. Set the tubing depth at 2400 m and simulate the production trend of gas wells under different formation pressure conditions (Figure 7 and Figure 8). The results are similar to those of the simulation when the tubing is lowered to the inclination point, when the formation pressure is 7.8 MPa, the gas production shows a slight downward trend, and the liquid production decreases. When the formation pressure drops to 7.7 MPa, the gas production fluctuates frequently in the late stage of production and the downward trend is obvious, the liquids begin to pile up in the bottom of the well, and the well eventually stops production due to the accumulation of liquids, so it is judged that the corresponding critical formation pressure at a depth of 2400 m of the tubing is still 8 MPa. Similarly, it can be determined that the critical formation pressure is 7.8 MPa when the tubing depth is 2600 m (Figure 9), and the critical formation pressure is 8 MPa when the tubing depth is 2800 m, 3000 m, and 3200 m (Figure 10, Figure 11 and Figure 12).

Combining the simulation results under different conditions (Figure 13 and Figure 14), it is concluded that when the tubing is lowered to 2600 m, the critical formation pressure at which the gas well can maintain self-injection production is reduced to 7.8 MPa and at this time, the cumulative gas production of the gas well is 56.19 × 10⁶ m³, which is 0.93 × 10⁶ m³ more than that when the tubing is lowered to the inclination creation point. When the tubing depth is 2600 m and the formation pressure drops to the critical spontaneous injection formation pressure, the cumulative gas production is 56.19 × 10⁶ m³, which is approximately 2.1 × 10⁵ m³ higher than that at a tubing depth of 2200 m. The gas production increase is approximately 1.7 × 10⁵ m³ compared to a depth of 2400 m below the oil pipe, about 2 × 10⁵ m³ compared to a depth of 2800 m below the oil pipe, about 2.8 × 10⁵ m³ compared to a depth of 3000 m below the oil pipe, and about 2.4 × 10⁵ m³ compared to a depth of 3200 m below the oil pipe. Due to the small vertical depth difference of this well, the cumulative gas production of the gas well changes little when the tubing is lowered to different depths. According to the calculation of the simulation software, when the depth of the oil pipe reaches 2600 m, the self-injection production time is extended by approximately 135 days.

Most previous studies on shale gas wells have focused on the drainage effect of wellbore fluid accumulation. The optimal tubing depth obtained from the above simulation and the critical formation pressure of the well can effectively help researchers in the real working environment control the operation effect of the gas well, adjust the production process of the gas well in a timely manner, control the timing and position of tubing insertion, and maximize the production effect of the gas well. This is of great significance for the development of real gas wells.

5. Conclusions

(1) Numerical simulation of the production effect of self-injection in the middle and late stages of production of horizontal wells in shale gas reservoirs under different tubing depth conditions was carried out, and the well structure model was established by OLGA software to determine the production capacity equations for this downward-dipping well.

(2) By using a fully dynamic multiphase flow simulation program and taking a certain well as an example, a wellbore structure model was established. By changing the depth of the tubing, the limit values of formation pressure required to support self-injection production of gas wells under different tubing depths were simulated. The suitable tubing depth for self-injection exploitation of gas wells and the cumulative gas production at different tubing depths were studied. The simulation results can provide effective support for the actual production process for different shale gas wells and help on-site workers take effective measures in a timely manner to solve the problem of gas well injection stoppages in the middle and later stages.

(3) The impacts of formation pressure on the gas well’s spontaneous injection production dynamics under different tubing depth conditions were simulated, the critical formation pressure that can maintain the gas well’s spontaneous injection production at different tubing depths was obtained, and it was concluded that when the tubing was lowered to 2200 m and 2400 m, the critical formation pressure of the gas well was 8 MPa; when the tubing was lowered to 2600 m, the critical formation pressure was lowered to 7.8 MPa; the critical formation pressure was around 8 MPa when the tubing was lowered to 2800 m and 3200 m; and the critical formation pressure was about 7.8 MPa when the tubing continued to be lowered to 3000 m and 3200 m.

(4) According to the material balance equation and original geological reserves, the cumulative gas production of the gas well is 56.19 × 10⁶ m³ when the tubing is lowered to 2600 m, which is 0.93 × 10⁶ m³ more than that when the tubing is lowered to the sloping point (2200 m) and can extend the self-injection production time of the gas well by about 135 days.