Research on the Multi-Layer Optimal Injection Model of CO₂-Containing Natural Gas with Minimum Wellhead Gas Injection Pressure and Layered Gas Distribution Volume Requirements as Optimization Goals

Abstract

The separate-layer gas injection technology is a key means to improve the effect of refined gas injection development. Currently, the measurement and adjustment of separate injection wells primarily rely on manual experience and automatic measurement via instrument traversal, resulting in a long duration, low efficiency, and low qualification rate for injection allocation across multi-layer intervals. Given the different CO₂-containing natural gas injection rates across different intervals, this paper establishes a coupled flow model of a separate-layer gas injection wellbore–gas distributor–formation based on the energy and mass conservation equations for wellbore pipe flow, and develops a solution method for determining gas nozzle sizes across multi-layer intervals. Based on the maximum allowable gas nozzle size, an optimization method for multi-layer collaborative allocation of separate injection wells is established, with minimum wellhead injection pressure and layered injection allocation as the optimization objectives, and the opening of gas distributors for each layer as the optimization variable. Taking Well XXX as an example, the optimization process of allocation schemes under different gas allocation requirements is simulated. The research shows that the model and method proposed in this paper have high calculation accuracy, and the formulated allocation schemes have strong adaptability and minor injection allocation errors, providing a scientific decision-making method for formulating refined allocation schemes for separate-layer gas injection wells, with significant theoretical and practical value for promoting the refined development of oilfields.

1. Introduction

During the critical phase of China’s “14th Five-Year Plan” to “15th Five-Year Plan” to increase oil and gas reserves and production, the development of low-grade reserves, stable production from mature gas fields, and offshore ultra-shallow gas reservoirs is all constrained by technological bottlenecks. Intelligent separate injection technology, by precisely regulating inter-layer gas injection rates, has become a core means to enhance oil recovery. For instance, Liaohe Oilfield plans to add 304 new intelligent separate injection wells and link injection and production in 70 well groups by 2025. Jilin Oilfield, through the optimization of separate injection processes, has achieved a 10% increase in the qualification rate of effective water injection and an 8.7% reduction in electricity costs. However, the complexity of natural gas separate injection is far greater than that of water injection. The compressibility of gas, changes in multiphase flow patterns, and high-temperature formation responses urgently require the establishment of more sophisticated coupled simulation and dynamic allocation models.

Against the backdrop of the expanding development of low- and ultra-low-permeability reservoirs in China, these reservoirs are characterized by large, significant permeability differences, high initial pressure, and substantial vertical heterogeneity. These characteristics have gradually reduced the adaptability of traditional commingled gas injection development models, making it challenging to achieve efficient reservoir drainage. Technologies such as supercritical CO₂ flooding and separate-layer gas injection have emerged as essential pathways to improve oil recovery in low-permeability reservoirs by enhancing crude oil properties, optimizing mobility ratios, and supplementing formation energy, with positive progress in practical applications [1]. However, commingled gas injection tends to cause the injected gas to enter high-permeability layers, leading to gas channeling preferentially. Consequently, medium- and low-permeability intervals struggle to effectively absorb gas, resulting in significant disparities in the degree of reservoir drainage [2]. At the same time, insufficient corrosion protection of the injection string and limited precision in zonal control further limit improvements in development effectiveness. Therefore, the development of an efficient and reliable separate-layer gas injection technology is of great significance for achieving refined reservoir development.

To address the issues mentioned above, scholars have conducted research in various aspects surrounding separate-layer injection technology. The core of separate-layer injection lies in the coupled flow process of wellbore pipe flow, gas distributor throttling, and formation seepage. Most traditional models focus on a single link and fail to form a full-system collaborative calculation framework. For example, some scholars applied nodal analysis to optimize the operating parameters of separate injection wells, establishing mathematical models for vertical wellbore flow and formation seepage, which enable the prediction of a reasonable injection rate for a specific layer and provide a theoretical basis for parameter design. However, this method does not fully consider the dynamic coupling among wellbore flow, gas distributor throttling, and formation seepage, nor does it involve a collaborative optimization mechanism across multiple intervals, resulting in limitations in handling multi-layer separate injection [3].

Ramey [4] developed a model for the variation in wellbore temperature with depth and time using approximate solutions for steady-state wellbore heat transfer and unsteady-state radial heat conduction. By incorporating the overall heat transfer coefficient and formation thermal properties, this model enables the quantitative estimation of wellbore heat loss and the prediction of temperature distribution, providing a fundamental theoretical framework for predicting the variation in fluid temperature with depth and time during separate-layer gas injection.

Yasunami et al. [5] considered natural convection in the annulus, heat conduction through insulated tubing, and unsteady-state temperature changes in the formation to establish a prediction model for wellbore temperature and phase behavior during CO₂ injection. They also proposed operational strategies to maintain supercritical conditions by adjusting injection rates or adding electrical heating, providing important references for phase control under high-temperature, high-pressure wellbore conditions.

Wu Xiaodong et al. [6] proposed a coupled temperature–pressure calculation model for CO₂ injection wells that accounts for CO₂ phase changes. They used a combined PR-EXP equation of state to determine the phase state of CO₂ in the wellbore accurately and established a coupled wellbore temperature–pressure model based on the conservation of energy, momentum, and mass equations.

Dou Liangbin et al. [7] used the Span–Wagner equation of state, based on Helmholtz free energy, to couple and iteratively calculate wellbore heat transfer, pressure, and CO₂ physical properties, thereby establishing a mathematical prediction model for the temperature and pressure distributions in CO₂ injection wellbores.

Zhang Yonggang et al. [8] applied heat transfer theory to establish a coupled model for the temperature and pressure distribution in CO₂ injection wellbores by analyzing the wellbore heat transfer process. They then studied the temperature and pressure distribution in CO₂ injection wellbores in the Honghe Oilfield by integrating actual injection parameters. Also, they explained the gas channeling problems observed in field tests.

Liu Zili et al. [9] combined Ramey’s wellbore temperature calculation method with the vertical pipe flow energy equation to establish a temperature–pressure model bounded by the bottom-hole miscible pressure and developed application software. They analyzed the effects of injection rate, temperature, and pressure on the bottom-hole temperature and pressure.

Paterson et al. [10] developed a numerical model for wellbore pressure and temperature that incorporates phase behavior for both CO₂ injection and production wells. This model, based on high-precision equations of state such as the Span–Wagner model, analyzes the phase distribution and thermodynamic response of CO₂ across static, production, injection, and adiabatic conditions.

Yang et al. [11] coupled one-dimensional axial non-isothermal flow inside the pipe with one-dimensional radial unsteady heat conduction between the wellbore and the formation, establishing a two-dimensional unsteady-state model for simulating transient flow and heat transfer behavior in CO₂ fracturing wellbores. This model successfully predicts the thermal-hydraulic parameters of the wellbore and the regularity of CO₂ phase evolution.

Research has also been conducted in the field of multi-component gas injection. While some studies have constructed coupled models of the wellbore temperature field, pressure field, and fluid properties, emphasizing the importance of multi-physics field collaborative calculations for improving prediction accuracy [12], they do not address issues related to gas distributor flow and coordinated separate injection.

Lu et al. [13] fully coupled the thermodynamic isothermal flash process with transient flow equations, establishing a transient thermal wellbore flow model for multi-component CO₂ mixtures. This model enables the simulation of transient flow in wellbores for these mixtures, accounting for phase changes.

Chen Huan et al. [14] combined visual experiments with the PR equation of state to establish a regression model for the physical parameters of high-CO₂-content multi-component gases. This model can accurately predict the variation in key physical parameters, such as density and viscosity, of the mixed gas in a two-phase state over a wide range of temperatures and pressures.

Yang Wenjun et al. [15] described wellbore pipe flow using the Bernoulli equation. They calculated the dynamic feedback between orifice throttling (solved using the Navier–Stokes equations) and formation seepage, thereby preliminarily achieving optimal analysis for multi-layer water injection.

Liu Yonghui et al. [16] established a coupled wellbore temperature–pressure model for separate-layer CO₂ injection wells. By integrating orifice throttling and formation dynamics, they achieved coupling among the wellbore, orifice, and formation, providing a preliminary description of the matching relationship between orifice size and gas injection rate. However, this model is developed only for pure CO₂ gas and cannot meet the calculation requirements for the actual field injection of natural gas with varying CO₂ content.

Hang Lai et al. [17] employed the Refprop 9.0 software to calculate the physical properties of CO₂ mixtures containing impurities (air, alkanes) and established a temperature–pressure coupled model considering gas composition, including CO₂. This model incorporates axial heat conduction, convective heat transfer, and the Joule–Thomson effect, enabling accurate prediction of the wellbore temperature and pressure profiles during impure CO₂ injection. The model’s prediction accuracy is significantly improved compared to the traditional Ramey model.

Pardeep Kumar et al. [18], targeting the non-ideal fluid characteristics of CO₂, proposed two novel and efficient methods for solving gas–liquid thermodynamic equilibrium within the Homogeneous Equilibrium Model (HEM) framework. These methods substantially reduce the computational cost for simulating transient multiphase flows, such as decompression, in CO₂ pipelines and wellbores, making them suitable for simulating the transportation and injection of CO₂-containing natural gas.

In summary, previous studies have made significant progress in simulating wellbore temperature–pressure fields, CO₂ phase behavior, and single-component fluid injection, laying a crucial foundation for layered gas injection technology. However, the following limitations remain:

• Limited Applicability: Most models are designed for pure CO₂, exhibiting insufficient applicability to the more commonly encountered multi-component natural gas containing CO₂ in actual production. Furthermore, there is a lack of targeted research on the potential two-phase gas–liquid flow within the wellbore and formation layers.
• Insufficient Coupling: Research has predominantly focused on simulating isolated processes (either the wellbore or the formation). Consequently, there is an inadequate investigation into the dynamic coupling mechanisms of the complete “Wellbore–Distributor–Formation” system.
• Lack of Optimization: Traditional methods emphasize post-construction simulation calculations for nozzle size adjustments. There is a scarcity of multi-layer collaborative optimization research aimed at engineering objectives, such as minimizing wellhead pressure and meeting gas distribution requirements.

Therefore, to meet the field requirements for the precise layered injection of CO₂-containing natural gas, it is urgent to establish a model that simultaneously accounts for multi-component fluid properties, full-system coupled flow, and multi-layer collaborative optimization. The innovations of this study are primarily reflected in two aspects:

• Expanded Scope: The simulation object is extended from pure CO₂ to multi-component natural gas with varying CO₂ content, significantly enhancing the model’s field applicability. Furthermore, we overcome the limitations of traditional single-segment simulation by establishing a dynamic coupling flow model for the “Wellbore–Distributor–Formation” system, incorporating calculation models for potential two-phase gas–liquid flow across different segments.
• Optimization Methodology: Building upon this, a multi-layer collaborative gas distribution optimization method is constructed. This method takes the minimum wellhead injection pressure and layer-wise gas distribution requirements as optimization objectives, thereby addressing the shortcomings of traditional research—namely, simple composition, single flow phase, and localized simulation scope.

2. Materials and Methods

Although existing separate-layer gas injection technologies include several new integrated measurement and adjustment techniques, they face problems such as gas injection rate deviation as the number of injection intervals increases, reliance on manual experience for measurement and adjustment, and low efficiency during long-duration multi-layer allocation. Furthermore, the physical models and solution methods for pipe flow in separate injection wells have shortcomings, including the lack of consideration for energy losses from the wellhead to the bottom-hole, the disregard for energy changes as gas flows through the nozzle into the formation, and limitations in accuracy and speed. In view of this, this paper establishes a coupled flow simulation model for the separate-layer gas injection wellbore–nozzle–formation system based on the energy and mass conservation equations of wellbore pipe flow. According to the maximum allowable nozzle size, a solution method for the gas nozzle sizes of multiple injection intervals is constructed based on the principle of minimum wellhead pressure, as illustrated in Figure 1 and Figure 2 below.

The main reasons based on the maximum size limit and the minimum wellhead gas injection pressure target are as follows: (1) to inject gas as accurately as possible, the gas nozzle size should be as small as possible, and the gas injection error is relatively small, so there is a maximum gas nozzle size limit target; and (2) reduce the wellhead gas injection pressure as much as possible, and use ground pipelines with lower pressure conditions under the condition that the injection can be achieved, which can save the investment budget and provide safety for ground gas injection.

2.1. Calculation Method for PVT Properties of CO₂-Containing Natural Gas

2.1.1. Experimental Testing Scope

Laboratory physical simulation experiments were conducted on the experimental equipment shown in Figure 2 below, within the following temperature and pressure ranges:

Temperature: 0–120 °C (25 °C (room temperature), 30 °C, 35 °C, 40 °C, 45 °C, 50 °C, 60 °C, 65 °C, 70 °C, 75 °C, and 80 °C).

Pressure: 0–30 MPa (2 MPa, 4 MPa, 6 MPa, 8 MPa, 10 MPa, 15 MPa, 20 MPa, 25 MPa, and 30 MPa).

Physical property parameters were obtained for five formulated natural gas mixtures with varying CO₂ contents (see

Table 1. Composition of CO₂-containing mixed gases.

Component/Mole Fraction	C1	C2	C3	iC4	nC4	iC5	nC5	C6+	N2	CO2
1	52.1	10.27	8.19	2.81	3.82	1.79	1.45	1.45	8.09	10.03
2	40.52	7.99	6.37	2.18	2.97	1.39	1.13	1.13	6.29	30.03
3	28.94	5.71	4.55	1.56	2.12	0.99	0.81	0.81	4.49	50.02
4	17.36	3.42	2.73	0.93	1.27	0.59	0.48	0.48	2.69	70.05
5	5.78	1.14	0.91	0.31	0.42	0.19	0.16	0.16	0.89	90.04

below).

2.1.2. Experimental Equipment

The PVT experimental equipment is shown in Figure 3.

Table 2. Main equipment parameters.

No.	Device Name	Device Model	Main Technical Indicators
1	High-temperature and high-pressure sampler	PY-2 (Hai’an County, Jiangsu Province, China)	Effective volume: 1000 mL Maximum pressure resistance: 100 MPa Maximum temperature resistance: 200 °C Temperature control accuracy: 0.1 °C Stirring speed: 10 rpm/min Stirring angle: 180°
2	High-pressure displacement pump	BY100-II (Hai’an County, Jiangsu Province, China)	Effective volume of single pump body: 500 mL Maximum pump pressure: 100 MPa Pressure control accuracy: 0.1% Flow range: 0.001–30 mL/min Applicable temperature: normal temperature Mode: constant voltage, constant current, and quantitative
3	Kinematic viscosity tester	LY-ND-01 (Hai’an County, Jiangsu Province, China)	Working temperature: normal temperature ~150 °C Maximum pressure resistance: 69 MPa
4	Gas meter	QL-I (Hai’an County, Jiangsu Province, China)	Effective volume: 1000 + 1000 mL Volume accuracy: 1 mL
5	Electronic balance	BSA423 (Beijing, China)	Max = 420 g d = 0.001 g
6	Gas chromatograph	Agilent 6890 (Beijing, China)	Measurement components: CO2, N2, O2, C1–C8
7	Gas chromatograph	Agilent 7890 (Beijing, China)	Measurement components: C1-C50
8	Viscometer	LQ-III (Hai’an County, Jiangsu Province, China)	Test pressure: 0.1–70 MPa Test temperature: room temperature −200 °C Test angle: 23°, 40°, 70° Test time: 0–9999 s

presents the main equipment parameters. Figure 4 shows the other important experimental equipment.

2.1.3. Experimental Process

The experiment adopts the “constant temperature and pressure reduction” mode: the experimental temperature is fixed, the pressure is slowly reduced from 30 MPa, the gas-phase components at each pressure point are observed, and the dew point pressure (bubble point pressure) is recorded.

(1)

Assemble a high-temperature and high-pressure reactor and evacuate the reactor to prevent air from causing gas impurity.

(2)

Introduce the gas from the gas tank into the high-temperature and high-pressure reaction kettle.

(3)

Place the high-pressure piston pump into the high and low temperature alternating experimental chamber and adjust the temperature to room temperature.

(4)

Keep the temperature constant, slowly reduce the pressure, and observe the phase change until the first droplet or bubble appears. Record the pressure at this time as the dew point pressure (bubble point pressure) at the temperature.

(5)

Increase the temperature and repeat steps 4 to 5 to finally obtain the dew point pressure (bubble point pressure) at different temperature points.

2.1.4. Phase Diagram Drawing and Calculation Formula Fitting

Phase diagrams were plotted based on the experimental data, as shown in Figure 5 below.

According to the experimental data, the calculation methods for different physical property parameters are fitted and plotted. The fitting was performed in MATLAB R2021a using various formulas, and the formula with the best fit was selected as the final formula. The following formulas for physical property parameters related to pressure and temperature were obtained. For example, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 show natural gas containing 90% CO₂.

(1)

Gas-phase volume fraction

Fitting formula:

$$V_{gas}=100-100.53\times \left(1∕\left(1+{\left(T∕25.98\right)}^{93.75}\right)\right)\ast \left(1∕{\left(P∕6.03\right)}^{-40.2}\right)$$

(1)

where $P$ is pressure, MPa; T is temperature, °C; and $V_{gas}$ is gas-phase volume fraction, %.

(2)

Liquid-phase volume fraction

Fitting formula:

$$V_{Liquid}=-0.00008+100.53\times \left(1∕\left(1+{\left(T∕25.98\right)}^{93.75}\right)\right)\ast \left(1∕{\left(P∕6.03\right)}^{-40.2}\right)$$

(2)

where $V_{Liquid}$ is liquid-phase volume fraction, %.

(3)

Gas-phase density

Fitting formula:

$${\rho }_{gas}={\displaystyle \frac{0.075-0.0021T-0.015P+0.0002T^2+0.0006TP}{1-0.043T-0.013P+0.0007T^2+0.0009P^2-0.00016TP}}$$

(3)

where ${\rho }_{gas}$ is gas-phase density, kg/m³.

(4)

Liquid-phase density

Fitting formula:

$${\rho }_{Liquid}={\displaystyle \frac{-36.63-0.192T+20.938P-0.345T^2-0.224TP}{1-4.231T+20.937P+0.512T^2-0.298P^2-0.513TP}}$$

(4)

where ${\rho }_{Liquid}$ is liquid-phase density, kg/m³.

(5)

Gas-phase compressibility factor

Fitting formula:

$$Z_{gas}={\displaystyle \frac{1.148+0.002T-0.183P+0.005T^2+0.002TP}{1-0.003T-0.02P+0.0001T^2+0.006P^2+0.0001TP}}$$

(5)

where $Z_{gas}$ is the gas compression factor.

(6)

Gas-phase viscosity

Fitting formula:

$${Vis}_{gas}=0.0038+0.02\times \left(1+erf\left(\left(T-26.44\right)∕\left(0.147\sqrt{2}\right)\right)\right)\ast \left(1+erf\left(P-12.03\right)∕\left(8.42\sqrt{2}\right)\right)$$

(6)

where ${Vis}_{gas}$ is gas viscosity, cP.

2.2. Wellbore Multiphase Flow Pressure and Temperature Calculation Method

The physical property parameters of CO2-containing natural gas are calculated using the above physical property parameter calculation method based on experimental data fitting. The injection process belongs to downward flow, and the Mukherjee–Brill method [19] is adopted. The formula for downward multiphase flow calculation is as follows:

$${\displaystyle \frac{dp}{dz}}={\rho }_{m}g\sin \theta -f_m{\displaystyle \frac{\rho {v_m}^2}{2D}}+\rho v_m{\displaystyle \frac{d{v}_m}{dz}}$$

(7)

where $p$—pressure in the pipe segment, Pa;

$z$—elevation of the pipe segment, m;

${\rho }_m$—density of the gas–liquid mixture in the pipe segment, kg/m³;

$g$—gravitational acceleration;

$\theta$—angle between the pipe segment and the vertical direction, °;

$v_m$—flow velocity of the gas–liquid mixture, m/s;

$f_m$—flow resistance coefficient of the gas–liquid mixture;

D—inner diameter of the pipeline, m.

It should be noted that the mass flow rate from the gas injection wellhead to the first gas injection interval is the total gas injection rate. From the first interval to the second interval, the gas injection rate in the wellbore is the total gas injection rate minus the gas injection rate of the first interval, and so on. In multiphase flow calculation, the physical properties of natural gas with different CO₂ contents are calculated by interpolation using the physical property calculation method in Section 2.1.

According to the “Ramey model” [4] and the “Hasan–Kabir model”, the injection wellbore temperature model is

$${\displaystyle \frac{\mathrm{d}{T}_{\mathrm{f}}}{\mathrm{d}z}}=J_t{\displaystyle \frac{\mathrm{d}p}{{\mathrm{d}}_z}}+{\displaystyle \frac{T_{\mathrm{e}\mathrm{i}}-T_{\mathrm{f}}}{M}}+{\displaystyle \frac{g\sin \theta }{C_{\mathrm{p}\mathrm{m}}}}-{\displaystyle \frac{{\nu }_{\mathrm{m}}}{C_{\mathrm{p}\mathrm{m}}}}{\displaystyle \frac{\mathrm{d}{\nu }_{\mathrm{m}}}{\mathrm{d}z}}$$

(8)

$${\displaystyle \frac{\mathrm{d}{T}_{\mathrm{f}}}{\mathrm{d}z}}={\displaystyle \frac{T_{\mathrm{e}\mathrm{i}}-T_{\mathrm{f}}}{M}}+{\displaystyle \frac{g\sin \theta }{C_{\mathrm{p}\mathrm{m}}}}+\left(J_t+{\displaystyle \frac{{\nu }_{\mathrm{m}}{\nu }_{\mathrm{g}\mathrm{g}}}{p{C}_{\mathrm{p}\mathrm{m}}}}\right){\displaystyle \frac{\mathrm{d}p}{dz}}$$

(9)

$$M={\displaystyle \frac{C_{\mathrm{p}\mathrm{m}}G}{2A}}\left({\displaystyle \frac{k_{\mathrm{e}}+T_D{r}_{\mathrm{t}\mathrm{o}}{U}_{\mathrm{t}\mathrm{o}}}{r_{\mathrm{t}\mathrm{o}}{U}_{\mathrm{t}\mathrm{o}}{k}_{\mathrm{e}}}}\right)$$

(10)

where $T_{\mathrm{f}}$ is the fluid temperature, K;

$T_{\mathrm{e}\mathrm{i}}$ is the formation temperature, K;

$J_t$ is the Joule–Thomson coefficient of the mixed fluid;

$C_{\mathrm{p}\mathrm{m}}$ is the constant-pressure specific heat capacity of the mixed fluid, J/(kg K);

$k_{\mathrm{e}}$ is the formation thermal conductivity, W/(m K);

$T_D$ is the transient heat transfer function;

$U_{\mathrm{t}\mathrm{o}}$ is the total wellbore heat transfer coefficient, W/(m² K);

$r_{\mathrm{t}\mathrm{o}}$ is the outer radius of the tubing, m.

2.3. Three-Phase (Oil–Gas–Water) Throttle Calculation Model for Gas Distribution Nozzles

The gas distribution nozzle adopts a three-phase (oil–gas–water) throttle calculation model; see the method in the literature [20]. According to the energy conservation equation:

$$144p_1{\nu }_1+{\displaystyle \frac{V_1^2}{2g_c}}+C_{\nu }(T_1-T_2)=144p_2{\nu }_2+{\displaystyle \frac{V_2^2}{2g_c}}$$

(11)

where $p_1$ is the pressure before throttling, MPa; $v_1$ is the specific volume before throttling, m³/kg; $V_1$ is the flow velocity before throttling, m/s; $g_c$ is the gravitational acceleration constant, m/s²; $C_v$ is the constant-volume specific heat capacity, kJ/(kg·°C); $T_1$ is the temperature before throttling, °C; $T_2$ is the temperature after throttling, °C; $p_2$ is the pressure after throttling, MPa; $v_2$ is the specific volume after throttling, m³/kg; and $V_2$ is the flow velocity after throttling, m/s.

Using the above formula, the relationship between the pressure ratio before and after throttling and the flow rate is further obtained, i.e., the throttle calculation model.

2.4. Formation Injection Capacity Prediction Method

The concepts of gas injection capacity and gas production capacity are similar. The formation injection capacity is determined using the gas injection index method, which can be estimated by fitting the actual injection data or provided directly.

$$Q_g={\displaystyle \frac{774.6Kh(P_{wf}^2-P_r^2)}{T{\mu }_{g}Z\mathrm{l}\mathrm{n}\left(\frac{r_e}{r_w}\right)}}$$

(12)

in the formula,

$Q_g$ is the daily gas injection rate of the gas well, ${\mathrm{m}}^3/\mathrm{d}$;

$P_{wf}$ is the bottom-hole flowing pressure, MPa;

$P_r$ is the average formation pressure, MPa;

$K$ is the effective permeability of formation, $\mathrm{mD}$;

$h$ is the effective gas layer thickness, m;

$T$ is the gas layer temperature, $K$;

${\mu }_g$ is the gas viscosity under average pressure, $\mathrm{mPa}\cdot \mathrm{s}$;

$Z$ is the gas deviation factor (compressibility factor) under average pressure;

$r_e$ is the supply radius, $\mathrm{m}$;

$r_w$ is the wellbore radius, $\mathrm{m}$.

It can be seen that the gas injection index is

$$J_{g\left(P^2\right)}={\displaystyle \frac{Q_g}{P_r^2-P_{wf}^2}}={\displaystyle \frac{774.6Kh}{T{\mu }_{g}Z\ln \left(\frac{r_e}{r_w}\right)}}$$

(13)

where $J_{g\left(P^2\right)}$ is a constant under the given formation and gas well conditions, m³/d/MPa²

Based on the two-phase seepage permeability calculation formula,

$$K_{real}=K\cdot (K_{rg}(S_g)+K_{rl}(S_g))$$

(14)

Therefore, the gas injection index for the actual formation layer is

$$J_{rg}=K_{real}\cdot {\displaystyle \frac{774.6h}{T{\mu }_{g}Z\ln (\frac{r_e}{r_w})}}=(K_{rg}(S_g)+K_{rl}(S_g))\cdot J_{g(p^2)}$$

(15)

3. Calculation Example

3.1. Basic Parameters of Gas Injection Well

The basic parameters of the example well are shown in

Table 3. Base data of the well.

Well ID		Well XXX
Tubing Inner Diameter	mm	50.6
Tubing Depth	m	3500
Casing Inner Diameter	mm	114.3
Reservoir Temperature	°C	100
Surface Temperature	°C	15
Formation Water Specific Gravity	-	1.02
Gas Specific Gravity	-	1.15
CO2 Molar Content	%	50
Well Depth (MVD/TVD)	m	3843/3843
Maximum Allowable Nozzle Size	mm	2.0

. This well injects gas into multiple layers, and the basic parameters and gas distribution of each interval are shown in

Table 4. Parameters and gas distribution of gas injection intervals.

Well ID	Interval No.	Interval Top (m)	Interval Bottom (m)	Effective Thickness (m)	Gas Saturation (%)	Formation Pressure (MPa)	Gas Injection Index (104 m3/d/MPa2)	Interval Gas Allocation (104 m3/d)
Well 1	1	3000	3200	50	90	10	0.05	1.9
	2	3100	3200	50	90	10	0.03	1.5
	3	3300	3400	50	90	10	0.02	1
	4	3400	3500	50	90	10	0.05	1.5

. It is assumed that the reservoir permeability under different gas containment conditions in each layer is as shown in

Table 5. Phase permeability data of different gas saturations in layers.

Gas Saturation (Sg)	Liquid-Phase Permeability (Kl)	Gas-Phase Permeability (Kg)
100	0	5
90	0.1	4.8
80	0.5	3.8
70	1	3
60	1.3	2.7
50	1.6	2.5
40	2	2.3
30	1.8	2
20	2.2	1.8
10	2.5	1.3
0	2.7	0

.

3.2. Optimization Results of Multi-Layer Gas Injection

Based on the multi-layer separate injection collaborative allocation optimization method with the minimum wellhead pressure, under the condition that the maximum allowable gas nozzle size is 2 mm, the layered gas allocation optimization is carried out for this well according to the gas allocation of each interval, yielding the results shown in Figure 12 and

Table 6. Parameters and gas distribution of gas injection intervals and optimization results.

Well ID	Interval No.	Interval Top (m)	Interval Bottom (m)	Effective Thickness (m)	Gas Saturation (%)	Formation Pressure (MPa)	Gas Injection Index (104 m3/d/MPa2)	Interval Gas Allocation (104 m3/d)	Wellhead Pressure (MPa)	Injection Pressure at Interval (MPa)	Gas Nozzle (mm)	Calculate the Gas Injection Volume Under the Condition of the Gas Nozzle (104 m3/d)	Accuracy of Air Distribution Volume Under Nozzle Conditions (%)
Well 1	1	3000	3200	50	90	10	0.05	1.9	9.80	12.28	1.2	1.9	100%
	2	3100	3200	50	90	10	0.03	1.5	9.80	12.33	2.0	1.5	100%
	3	3300	3400	50	90	10	0.02	1	9.80	12.57	1.0	1	100%
	4	3400	3500	50	90	10	0.05	1.5	9.80	12.70	0.9	1.5	100%

by developed through our own intelligent gas injection management software V1.0. It can be observed that the calculated gas nozzle size for each interval exactly meets the injection allocation requirement under the current gas injection pressure.

4. Discussion

This study developed a coupled “wellbore–gas distribution nozzle–formation” flow model that systematically integrates multiple physical processes, including pipe flow, throttling, and seepage, to accurately simulate the dynamic behavior of natural gas with varying CO₂ content during separate-layer injection. The validation results from the example well show that the model can effectively coordinate the gas injection rates of each layer. The optimized gas nozzle sizes match well with the injection pressures, and the injection allocation errors for each layer are within the engineering tolerance range, verifying the model’s reliability and engineering value under complex working conditions.

The optimization results reveal significant heterogeneity in the reservoir’s gas injectivity. For example, Interval 2 requires the largest gas nozzle (2.0 mm), indicating its poor permeability, where a larger flow area is needed to release its gas-absorption potential. In contrast, Interval 4 has the smallest gas nozzle (0.9 mm), reflecting its higher gas injectivity, and requires a stronger throttling to control the injection rate and avoid single-layer breakthrough. Notably, under a unified, optimized minimum wellhead pressure (9.80 MPa), all intervals achieve precise injection allocation, demonstrating the effectiveness of the proposed minimum wellhead pressure optimization principle. This strategy not only reduces the equipment load and energy consumption of the surface gas injection system but also provides key technical support for the efficient and economical operation of separate injection wells.

Compared with existing studies, the innovation of this model lies in two aspects: first, it extends the simulation range from a single CO₂ component to multi-component natural gas with different CO₂ contents, enhancing the model’s applicability; second, it solves the dynamic coupling problem among wellbore flow, gas distribution nozzle throttling, and formation seepage, and establishes a multi-layer collaborative optimization mechanism, making up for the deficiency of “emphasizing simulation while neglecting optimization” in traditional studies. Especially in complex phase environments with high CO₂ content, the model can accurately capture the influence of fluid property changes on system flow, providing a more scientific basis for field allocation decisions.

However, the model still has limitations at this stage. First, its core assumptions rely on static formation property parameters and fail to consider the feedback effect of dynamic reservoir changes during gas injection on flow, which may lead to deviations in long-term predictions. Second, it treats the gas distributor as a component with constant flow characteristics, neglecting performance degradation due to scaling, corrosion, or hydrate formation during long-term field operation. These shortcomings limit the model’s predictive accuracy and practical value throughout its life cycle.

5. Conclusions

(1)

Based on the mass and energy conservation equations of wellbore pipe flow, a coupled “separate layer gas injection wellbore–gas distribution nozzle–formation” flow simulation model was established. This model can calculate the gas temperature distribution through the radial heat transfer equation and modify flow parameters by combining the property mixing rules of multi-component gases, achieving full-system flow simulation of natural gas with different CO₂ contents under multiphase flow conditions and solving the problems in traditional models where different flow segments are decoupled and only a single gas component (CO₂) is considered.

(2)

Taking the maximum allowable gas nozzle size as the boundary condition and the minimum wellhead pressure as the optimization objective, the optimal combination of gas nozzle sizes for each layer’s gas distributor and the wellhead pressure was obtained through iterative solution. This not only avoids the risk of gas nozzle blockage but also reduces gas injection energy consumption, significantly improving the efficiency and injection allocation accuracy of multi-interval allocation.

(3)

Through the example validation of Well XXX, based on the minimum wellhead pressure multi-layer separate injection collaborative allocation optimization method, under the condition of a maximum allowable gas nozzle size of 2 mm, the layered gas allocation optimization was performed for this well according to the gas allocation of each interval. The calculated gas nozzle size for each interval can meet the injection allocation requirement under the current gas injection pressure condition.

(4)

The results of this study provide a solid theoretical foundation and efficient solution for the refined and intelligent allocation of CO₂-containing natural gas separate injection wells, and have far-reaching theoretical and engineering significance for improving the gas injection development effect in complex reservoirs such as low-permeability and offshore ultra-shallow layers.

(5)

It is recommended to record the gas injection volume and update the cumulative injection volume of the layer in real time with the downhole measuring instrument, and then combine it with the cumulative production volume and production rate of the layer generation reasonably to estimate the new formation pressure changes and gas saturation based on the material balance equation, so as better to calculate the injection capacity parameters of the layer.