A Novel Mathematical Model Considering Real Gas PVT Behavior to Estimate Inflow Performance Relationship of Gas Well Production

Abstract

Inflow performance relationship (IPR) is one of the most important methods for the analysis of the dynamic characteristics of gas reservoir production. The objective of this study was to develop a model to improve the accuracy of the IPR for evaluating and predicting the production of gas reservoirs. In this paper, a novel mathematical model, taking into account the real gas PVT behavior, is developed to accurately estimate the inflow performance relationship. By introducing a pseudo-pressure function and a real gas properties database, this model eliminates the error caused by the linearization method and improves the calculation accuracy. The results show that more than 90% of the energy in the flow field is consumed by inertial forces, which leads to significant high-velocity non-Darcy effects in the gas reservoir. The reservoir permeability, original reservoir pressure, stress sensitivity coefficient, and skin factor have a great impact on the inflow performance relationship of gas reservoir production. This model predicts gas IPR curves with excellent accuracy and high efficiency. The high-precision gas well inflow performance relationship lays a solid foundation for dynamic production analysis, rational proration, and intelligent development of the gas field.

1. Introduction

Gas well productivity plays a critical role in gas reservoir engineering. The inflow performance relationship (IPR) of a well is the relationship between its production rate and the flowing bottom hole pressure. The evaluation and prediction of the inflow performance of gas wells are vital steps in developing new gas fields, optimizing the production performance of existing wells, and designing new wells.

Based on previous studies on oil flow in porous media, numerous gas flow models were developed from those of oil flow [1,2,3,4,5,6,7,8]. For oil wells, the fluid flow rate is usually assumed to be proportional to the difference between reservoir pressure and well bottom hole pressure [9], and this assumption leads to a linear relationship for the steady-state flow of incompressible single-phase fluid. Many scholars have studied many factors to improve the IPR accuracy of oil wells. Dietz [10] proposed a variety of shape factors to describe the asymmetrical positioning of a well. Economides [11] presented two correlations for the skin effect in steady- and pseudo-state in a circular drainage area. However, there are notable differences between gas flow and liquid flow in terms of physical characteristics. For example, gas density varies with pressure, while crude oil density is usually treated as a constant. These significant differences have caused some simplifications for oil flow to be ineffective in gas flow.

To accurately characterize the influence of real gas properties on gas flow, many scholars have proposed their linearization methods as follows: (1) Pressure square method [11,12,13,14]—for pressures less than 2000 psi, the product of the compressibility factor Z and viscosity μ is a constant. The real gas pseudo-pressure m(p) shows a linear relationship with the square of pressure. (2) Pressure approximation method [15,16,17,18,19]—when the bottom hole flowing pressure is higher than 3000 psi, the product of the compressibility factor Z and viscosity μ is roughly proportional to the pressure. The real gas pseudo-pressure m(p) shows a linear relationship with pressure. (3) Pseudo-pressure method [20,21,22,23]—this method can be applied in a wide range of pressures without any error. Al-Hussainy et al. [24] defined the pseudo-pressure to derive a quadratic gas deliverability equation for the boundary-dominated flow period. Raghavan et al. [25] defined the pseudo-pressure to obtain solutions to nonlinear flow equations in which reservoir and fluid properties are pressure dependent. Samaniego et al. [26] studied the application of the m(p) method to drawdown, buildup, injection, and falloff testing. Samaniego et al. [27] presented a performance-prediction procedure based on the drainage radius concept and a material-balance equation.

The pressure square method and the pressure approximation method can only be applied under their constraints. According to the limited gas well test data, the gas flow equation is linearized to obtain the analytical gas well IPR whose coefficients are strongly dependent on pressure. It is unreasonable to use the absolute open flow rate extrapolated by IPR to guide the gas reservoir production. The linearization method reduces the degree of nonlinearity of the gas flow equation, which is convenient for solving nonlinear equations analytically. However, the linearization method is an approximate method, which will produce inevitable errors in the process.

The accuracy of the real gas PVT behavior is critical to all analytical and numerical gas flow models. Some empirical formulas, algorithms, and software are introduced to describe the real-gas PVT behavior accurately. Dranchuk et al. [28] used the BWR equation of state to fit the compressibility factor of the natural gas. Lee et al. [29] used eight samples to fit the viscosity of natural gas at high pressure. However, these empirical formulas fitted from limited test data cannot cover all situations, and the errors in the fitting process cannot be eliminated. The model presented in this paper uses the Reference Fluid Thermodynamic and Transport Properties (REFPROP) database as real gas properties data. The REFPROP database program is widely used in the refrigeration industry for the calculation of refrigerant properties. REFPROP is an international authoritative working substance physical property calculation software developed by the National Institute of Standards and Technology (NIST) (Version 9.0, Boulder, CO, USA) [30]. Judging from the previous studies, REFPROP is used by many research projects as a physical data source or as a reference data source for the accuracy of calculation results [31].

Furthermore, several specific factors different from those in oil flow should be considered for gas flow in porous media. The high-velocity non-Darcy effect characterized by Forchheimer’s equation has a great influence on the gas flow. In 1901, Forchheimer [32] found that Darcy’s law was inadequate to describe high-velocity gas flow in porous media and added an additional pressure drop predicted by Darcy’s law in order to account for the discrepancy. Aronofsky and Jenkins [12] developed the solution for a vertical well at steady state from the solution of the differential equation for gas flow through porous media by using the non-Darcy factor. Smith [33] and Swift and Kiel [34] indicated that a non-Darcy flow of gases leads to an additional pressure drop near the wellbore, which can be treated as a rate-dependent skin factor. Jones [13] proposed a correlation that includes the Forchheimer factor β. Odeh [14] derived the pseudo-steady-state flow equation with a noncircular drainage area. Economides [11] suggested a steady-state flow equation for horizontal wells. However, the Forchheimer factor β and turbulence coefficient D calculated at the well bottom condition are used as constants to describe the degree of the non-Darcy effect. The influence of the non-Darcy effect on the entire flow field is not considered. Moreover, the impact of stress sensitivity on the inflow performance should be regarded as in the fractured gas reservoir. Yang et al. [35], Li et al. [36], and Zhang et al. [37] took the features of high-speed non-Darcy flow into consideration but neglected the characteristics of media formation. Clarkson et al. [38] investigated the impact of both stress-dependent matrix permeability and fracture-conductivity changes on rate-transient signatures and derived reservoir and hydraulic-fracture properties.

In this paper, taking into account the real gas PVT behavior, a novel high-velocity non-Darcy flow model is proposed by introducing a pseudo-pressure function with high nonlinearity to accurately estimate the inflow performance relationship of gas reservoir production. Then, the REFPROP database is introduced as a physical property data source to determine the real gas PVT behavior that can adapt to a wide range of reservoir conditions, which greatly improves the accuracy of the physical property parameters of the model. On this basis, the proposed model is verified by field data, and compared with the traditional model using linear approximate solution. Finally, the effects of reservoir permeability, original reservoir pressure, initial gas saturation, stress sensitivity coefficient, and skin factor on the inflow performance relationship of gas reservoir production are analyzed by this model. This research highlights the enormous potential of replacing traditional linear models and predicting the inflow performance relationship with higher accuracy in the field.

2. Model Description

A gas well is the basic unit of gas reservoir development, and accurate evaluation of gas well productivity is the key to efficient and reasonable development of gas reservoirs. The conceptual illustration of a vertical gas-producing well in a carbonate reservoir is shown in Figure 1.

2.1. General Assumptions

In order to study the gas well inflow performance relationship, a schematic diagram of gas reservoir model is shown in Figure 2, and some basic assumptions are made as follows:

• The reservoir is disk shaped, with its center at the wellbore, and it is a homogeneous and isotropic formation with uniform thickness. The upper and lower boundaries of the reservoir are closed, and the supply boundary pressure is constant at P_e.
• The formation has the same permeability and porosity. Both properties are stress-sensitive.
• The only flowing phase is gas phase with unsteady state.
• The well completion method is open hole completion.
• Gravitational forces are negligible.

2.2. Mathematical Model of Radial Flow in Gas Reservoir

To study the non-Darcy flow effect, the equation proposed by Forchheimer [32] is used in this model. The flow law equation in SI units is given as follows:

$$-\frac{\partial P}{\partial r}=\frac{{\mu }_g}{K}{v}_g+\beta {\rho }_g{v}_g^2$$

(1)

where ρ is the gas density, μ is the gas viscosity, K is the permeability, and v_g is the gas velocity. At high flow rate, the gas flow is dominated by the inertial forces, and the magnitude of the viscous force is constant. In the laminar flow situation, there is no inertial force item in Equation (1).

Equation (1) can be rewritten as:

$$-\frac{\partial P}{\partial r}=a{v}_g+b{v}_g^2$$

(2)

where

$$a=\frac{{\mu }_g}{K}$$

(3)

$$b=\beta {\rho }_g$$

(4)

The parameter β is given by three relations, in field units (K in mD, β in 1/ft):

• Geertsma relation [39]:

  $$\beta =\frac{48511.34}{{\left(K{K}_{rp}\right)}^{0.5}{\left(\varphi S_p\right)}^{5.5}}$$

  (5)
• Frederick and Graves’ first relation [40]:

  $$\beta =\frac{2.11\times {10}^{10}}{{\left(K{K}_{rp}\right)}^{1.55}{\left(\varphi S_p\right)}^{1.0}}$$

  (6)
• Frederick and Graves’ second relation [40]:

  $$\beta =\frac{7.89\times {10}^{10}}{{\left(K{K}_{rp}\right)}^{1.6}{\left(\varphi S_p\right)}^{0.404}}$$

  (7)

The above three relations can be applicable to the single gas phase system and Frederick and Graves’ first relation is used to describe the degree of non-Darcy effect in this paper. In order to characterize gas flow accurately, the stress sensitivity is also taken into account in the model. The stress sensitivity of the permeability and porosity can be expressed as:

$$K=K_0{e}^{-\alpha \left(P_{ref}-P\right)}$$

(8)

$$\varphi ={\varphi }_a·\left[1+C_p\left(P_{ref}-P\right)\right]$$

(9)

Substituting Equations (7), (8) and (9) into Equation (1), the flow law equation in field units can be rewritten as:

$$\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}\frac{\partial P}{\partial r}=11.5741\frac{{\rho }_g\left|v_g\right|}{K_0}+1.3396\times {10}^{-13}\frac{\beta {\rho }_g^2{\left|v_g\right|}^2}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}$$

(10)

The gas velocity can be expressed as:

$$v_g=\frac{Q_g}{2\pi rh}$$

(11)

Based on the material balance, the mass flow rate M_r towards the well is given by:

$$M_r={\rho }_g{Q}_g={\rho }_{gsc}{Q}_{gsc}$$

(12)

Substituting Equations (11) and (12) into Equation (10), Equation (10) in field units can be rewritten as:

$$\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha (P_{ref}-P)}\frac{\partial P}{\partial r}=1.8421\frac{{\rho }_{gsc}}{K_{0}rh}{Q}_{gsc}+1.1133\times {10}^{-14}\frac{\beta {\rho }_{gsc}^2}{{\mu }_g{r}^2{h}^2}{e}^{-\alpha (P_{ref}-P)}{Q}_{gsc}^2$$

(13)

According to the national standard SY/T 5440-2009 in China, the variation of a steady-state gas well bottom hole pressure should not exceed 0.5% of the current steady flow pressure in the next 8 h. The pressure of a steady-state well is only related to the spatial location, and Equation (13) can be expressed as:

$$\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}\frac{dP}{dr}=1.8421\frac{{\rho }_{gsc}}{K_{0}rh}{Q}_{gsc}+1.1133\times {10}^{-14}\frac{\beta {\rho }_{gsc}^2}{{\mu }_g{r}^2{h}^2}{e}^{-\alpha \left(P_{ref}-P\right)}{Q}_{gsc}^2$$

(14)

To solve Equation (14), separation of variables method is used to process the gradient term. Integrating Equation (14) over the radius of the well control zone yields Equation (15):

$${\displaystyle {\int }_{P_{wf}}^{P_e}\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}dP}={\displaystyle {\int }_{r_w}^{r_e}\left(1.8421\frac{{\rho }_{gsc}}{K_{0}rh}{Q}_{gsc}+1.1133\times {10}^{-14}\frac{\beta {\rho }_{gsc}^2}{{\mu }_g{r}^2{h}^2}{e}^{-\alpha \left(P_{ref}-P\right)}{Q}_{gsc}^2\right)dr}$$

(15)

The simplified mathematical model is expressed as follows:

$$\begin{array}{ll}{\displaystyle {\int }_{P_{wf}}^{P_e}\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}dP}& =1.8421\frac{{\rho }_{gsc}}{K_{0}h}{Q}_{gsc}\ln \left(\frac{r_e}{r_w}\right)+\\ & 1.1133\times {10}^{-14}\frac{{\rho }_{gsc}^2}{h^2}{Q}_{gsc}^2{\displaystyle {\int }_{r_w}^{r_e}\left(\frac{\beta }{{\mu }_g{r}^2}{e}^{-\alpha \left(P_{ref}-P\right)}\right)dr}\end{array}$$

(16)

If the skin effect is considered, Equation (16) can be changed as follows:

$$\begin{array}{ll}{\displaystyle {\int }_{P_{wf}}^{P_e}\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}dP}& =1.8421\frac{{\rho }_{gsc}}{K_{0}h}{\ln }^{\left(\frac{r_e}{r_{wr}}\right)}{Q}_{gsc}+\\ & 1.1133\times {10}^{-14}\frac{{\rho }_{gsc}^2}{h^2}{Q}_{gsc}^2{\displaystyle {\int }_{r_{wr}}^{r_e}\left(\frac{\beta }{{\mu }_g{r}^2}{e}^{-\alpha \left(P_{ref}-P\right)}\right)dr}\end{array}$$

(17)

where $r_{wr}=r_w{e}^{-S}$, and S is the skin factor.

Defining the function of gas-phase pseudo-pressure:

$$\phi \left(P\right)={\displaystyle {\int }_{P_m}^P\frac{{\rho }_g}{{\mu }_g}{e}^{-\alpha \left(P_{ref}-P\right)}dP}$$

(18)

The simplified mathematical model is expressed as follows:

$$\Delta \phi =A{Q}_{gsc}+B\left(P\right)Q_{gsc}^2$$

(19)

where:

$$\{\begin{array}{l}\Delta \phi =\phi \left(P_e\right)-\phi \left(P_{wf}\right)\\ A=1.8421\frac{{\rho }_{gsc}}{K_{0}h}\ln \left(\frac{r_e}{r_w}\right)\\ B\left(P\right)=1.1133\times {10}^{-14}\frac{{\rho }_{gsc}^2}{h^2}{\displaystyle {\int }_{r_w}^{r_e}\left(\frac{\beta }{{\mu }_g{r}^2}{e}^{-\alpha \left(P_{ref}-P\right)}\right)dr}\end{array}$$

(20)

Once the flow parameters are obtained, the surface gas flow rate, Q_gsc, at any other flow pressure is obtained from Equation (19):

$$Q_{gsc}=\frac{-A+\sqrt{A^2+4B·\Delta \phi }}{2B\left(P\right)}$$

(21)

Thus, the mathematical model to estimate the inflow performance relationship is established.

3. Solving Method of the Mathematical Model

Coefficient B in Equation (19) is variable with pressure, so it is necessary to describe the pressure profile in the entire gas reservoir. In order to get the distribution of pressure along the radial direction, the radial length is logarithmically divided into N segments, as shown in Figure 3.

Suppose the discrete reference block in space is the ith block, and the discrete reference point in time is the (n + 1)th time step. For the ith block, the equation is shown as follows:

$$A_2{P}_{i-1}^{n+1}-\left(A_1+A_2+B_1\right)P_i^{n+1}+A_1{P}_{i+1}^{n+1}=-B_1{P}_i^n$$

(22)

where

$$\{\begin{array}{l}{A}_1=\frac{r_{i+\frac{1}{2}}{\left(\rho \delta \frac{K}{\mu }\right)}_{i+\frac{1}{2}}^{n+1}}{r_i·\Delta r_{i+\frac{1}{2}}·\left(r_{i+\frac{1}{2}}-r_{i-\frac{1}{2}}\right)}\\ A_2=\frac{r_{i-\frac{1}{2}}{\left(\rho \delta \frac{K}{\mu }\right)}_{i-\frac{1}{2}}^{n+1}}{r_i·\Delta r_{i-\frac{1}{2}}·\left(r_{i+\frac{1}{2}}-r_{i-\frac{1}{2}}\right)}\\ B_1=\frac{{\left(\frac{\partial \rho }{\partial P}\varphi +{\varphi }_0{C}_p\rho \right)}^n}{t^{n+1}-t^n}\end{array}$$

(23)

As the physical quantity $\left(\rho \delta \frac{K}{\mu }\right)$ in Equation (23) is a function of space and time, the upwind scheme is used to assign values to physical quantities on the grid points. The upwind scheme is shown as follows:

$${\left(\rho \delta \frac{K}{\mu }\right)}_{i+\frac{1}{2}}^{n+1}=\{\begin{array}{l}{\left(\rho \delta \frac{K}{\mu }\right)}_i^n \mathrm{If} \mathrm{the} \mathrm{fluid} \mathrm{goes} \mathrm{from} i \mathrm{to} i+1\\ {\left(\rho \delta \frac{K}{\mu }\right)}_{i+1}^n \mathrm{If} \mathrm{the} \mathrm{fluid} \mathrm{goes} \mathrm{from} i+1 \mathrm{to} i \end{array}$$

(24)

After the upwind scheme processing, Equation (23) can be rewritten as:

$$\{\begin{array}{l}{A}_1=\frac{r_{i+\frac{1}{2}}{\left(\rho \delta \frac{K}{\mu }\right)}_{i+1}^n}{r_i·\Delta r_{i+\frac{1}{2}}·\left(r_{i+\frac{1}{2}}-r_{i-\frac{1}{2}}\right)}\\ A_2=\frac{r_{i-\frac{1}{2}}{\left(\rho \delta \frac{K}{\mu }\right)}_i^n}{r_i·\Delta r_{i-\frac{1}{2}}·\left(r_{i+\frac{1}{2}}-r_{i-\frac{1}{2}}\right)}\\ B_1=\frac{{\left(\frac{\partial \rho }{\partial P}\varphi +{\varphi }_0{C}_p\rho \right)}^n}{t^{n+1}-t^n}\end{array}$$

(25)

Then, the pressure distribution can be calculated from the well bottom to the reservoir boundary by using Gaussian elimination with scaled partial pivoting. The detailed solution process of the model is shown as a flowchart in Figure 4.

4. Results and Discussion

4.1. Model Validation

To validate this new model, the calculation results of this model are compared with simulated results, as shown in Figure 5. The key parameters obtained from the geological model for model validation are listed in

Table 1. The key parameters obtained from the geological model for model validation.

Parameters	Value	Unit
Reservoir thickness	45	m
Porosity	0.07	-
Permeability	1	mD
Original reservoir pressure	58	MPa
Reservoir temperature	116	°C
Initial gas saturation	0.4	-
Drainage radius	500	m
Wellbore radius	0.0762	m

.

Figure 5 shows that the simulated results and the calculation results of this model have a very good match. Therefore, this model can provide a reasonable prediction of gas production rate in the reservoir.

Then, a comparative study of the non-linear theoretical model results and the linearization method results was conducted at three different pressure conditions (Figure 6). Figure 6a shows that the results of pressure square method at low pressure have a good match with the predicted results of this model. This linearization method is acceptable for use in the prediction of inflow performance. Figure 6b shows that the deviation between the non-linear theoretical model and the pressure approximation method became greater within the pressure of 25–30 MPa. When the test pressure was high, the error of 22.58% indicates that the linearization method is not reliable for dealing with nonlinear equations, as shown in Figure 6c.

4.2. Effect of Reservoir Permeability

The effect of permeability on gas production was analyzed, as shown in Figure 7. Figure 7a shows the inflow performance relationship curves at different reservoir permeabilities. It can be seen that the higher reservoir permeability greatly increased the absolute open flow rate. With the increase of gas reservoir permeability, the slope of the curves gradually decreases, indicating that the unit production pressure difference corresponds to a higher productivity. In ultra-low-permeability reservoirs (K < 0.1 mD), a high production pressure difference will not help the production. Further analysis of the bottom hole flow characteristics was conducted. In Figure 7b, the straight pink line represents the relationship between well bottom hole pressure and flow rate in the Darcy flow condition and the black curve indicates that flow has non-Darcy characteristics. The gas flow is mainly controlled by viscous force (v term) and inertial force (v² term). Figure 7b shows that due to the increased permeability, the proportion of flow controlled by viscous force continues to decrease, from 9.34% to 2.96%. At the same time, inertial force is always in the dominant position, controlling 90% of the flow. The high flow rate enhances the influence of inertial resistance. For the high-velocity flow, the v² quadratic term tends to the limit, so a third term of the velocity form is necessary to describe the high-velocity non-Darcy effect.

4.3. Effect of Original Reservoir Pressure

Since the gas isothermal coefficient of compressibility is greater than the rock compressibility, the original reservoir pressure determines the elastic energy of the gas reservoir. The effect of original reservoir pressure on the gas inflow performance relationship was analyzed, and the results are presented in Figure 8.

Figure 8a shows that for the same bottom hole pressure, wells drilled in reservoirs with the higher original reservoir pressure have the higher production rate. From Equation (1), one would expect that a high original reservoir pressure is equivalent to a high gas pseudo-pressure. For the same flow resistance, this means high productivity.

Figure 8b shows that a high original reservoir pressure increases the gas production rate, and enhances the flow controlled by viscous force and inertial force to different degrees. With the increase of the original reservoir pressure, the pressure losses caused by both viscous force and inertial force increase at the well bottom. The portion of the viscous flow influenced by the high production rate decreases from 16.33% to 5.01%. Because of the high velocity at the well bottom, the inertial force is still dominant.

4.4. Effect of Initial Gas Saturation

In the whole seepage process, the porosity directly determines the reserves. The effect of initial gas saturation on the gas inflow performance relationship was analyzed, and the results are presented in Figure 9. From Equation (7), the porosity has a slight influence on the β coefficient and the inertial force, and will eventually have an impact on the absolute open flow rate (Figure 9a). Figure 9b shows that with the increase of pressure loss caused by the viscous force, the pressure loss caused by inertial force decreases from 94.82% to 94.26%. The high-velocity non-Darcy effect cannot be ignored. The inertial force is dominant.

4.5. Effect of Stress Sensitivity Coefficient

The reservoir permeability is the key parameter affecting the inflow performance relationship, as shown in Figure 10. Figure 10a shows that a higher stress sensitivity coefficient results in a lower absolute open flow rate and a faster decreasing trend of production rate. As the stress sensitivity coefficient increases, the stress sensitivity of the reservoir increases. This shows that under the same bottom hole pressure condition, the permeability of the near-wellbore will drop sharply. In Figure 10b, with the increase of stress sensitivity coefficient, gas velocity decreases, and pressure losses caused by viscous force and inertial force increase. The rate of seepage resistance increased by the drop in permeability is much greater than the rate of pressure gradient change. More than 94.82% of the pressure gradient is consumed by inertial force, and the inertial force is still dominant.

4.6. Effect of Skin Factor

For the wells with imperfect hydrodynamics, the radius of an imperfect well can be replaced with that of a hypothetical perfect well. The effect of skin factor on the gas inflow performance relationship was analyzed, and the results are presented in Figure 11. Figure 11a shows that a lower skin factor results in a higher production rate. Even though a field has a high permeability, the productivity of a well with a high skin factor cannot be increased without taking measures. In Figure 11b, with the increase of skin factor, absolute open flow rate decreases, and gas velocity and pressure losses caused by viscous force and inertial force increase. The converted radius of the imperfect well has a great influence on the gas flow velocity. Therefore, the minimum value of flow velocity at the bottom hole corresponds to the maximum value of the absolute flow rate. When the skin coefficient is equal to 3, the gas velocity is the greatest, and the proportion of pressure caused by inertial force can reach 99.07%. In the case of a skin coefficient equal to −3, the gas velocity is the lowest, and the proportion of pressure caused by inertial force can reach 70.64%. The inertial force is still dominant.

5. Field Application

The Bereketli-Pirgui gas field is located on the right bank of the Amu Darya River. The gas-bearing area is 21.72 km² and the geological reserves is 201.97 × 10⁸ m³. The reservoir depth is 3127.63 m. The initial reservoir temperature is 112.619 °C, and the pressure is 53.47 MPa. As of August 2020, there were 22 gas wells in the block, and most of the production wells only produce gas. The inflow performance relationship of the gas wells is unclear, and the production system has not been determined. Therefore, it is significant to develop a model to evaluate the IPR with high precision and establish a rational proration plan.

The target layer was layer XVa1, and the average permeability of this layer was 1.03 mD. In March 2019, a well test was carried out on well Pir-21. The Pir-21 well test data are shown in

Table 2. Pir-21 well test data.

Test Date	Test Phase	Test Time (h)	Daily Surface Gas Production Rate (104 m3/d)	Well Bottom Hole Pressure (MPa)
2019.3.13–3.25	Productivity test	8	10.34	52.5
		8	15.75	51.96
		8	20.29	51.48
		8	27.14	50.71
	Well shut-in	30		53.44

. A production pressure difference of less than 3 MPa is suitable for a well test. Firstly, a small production pressure difference ensures that the well bottom hole pressure will become stable quickly. Second, a small production pressure difference guarantees that the reservoir fluid properties that are strongly dependent on pressure are not changed drastically.

In this context, coefficients A and B of Equation (19) can be treated as constant. For linear regression, Equation (19) can be expressed as:

$$\frac{\Delta \phi }{Q_{gsc}}=A+B{Q}_{gsc}$$

(26)

The values of A and B can be obtained by the fitting relationship between Q_gsc and △φ/Q_gsc as shown in Equation (26), and their values are 6.6608 × 10⁻⁵ and 4.4411 × 10⁻¹¹, respectively.

According to Equation (20), the flowability coefficient K₀h can be calculated, and is equal to 165.2030 mD·m. Based on the well logging test, the value of reservoir effective thickness is 68.78 m. Accordingly, the effective horizontal permeability is 2.4019 mD. The results are satisfying and indicate that this model is an effective approach to obtain reservoir parameters in gas reservoirs. Based on the Pir-21 well test data, the model presented in this paper and traditional linearization methods (e.g., pressure square method and pressure approximation method) were used to predict the inflow performance relationship, as shown in Figure 12.

The results of traditional linearization methods greatly underestimate the productivity of the gas wells, with an error up to 35.93%. It is noticeable that the model in this paper can fully consider the impact of the stress sensitivity, the gas high-velocity non-Darcy effect, and the change of gas properties on gas well productivity.

6. Conclusions

In this study, a novel mathematical model considering real gas PVT behavior was developed. The pseudo-pressure function and gas properties database are introduced to improve the accuracy of this model which estimates the inflow performance relationship of gas well production. On this basis, the effects of reservoir permeability, original reservoir pressure, initial gas saturation, stress sensitivity coefficient, and skin factor on the inflow performance relationship of gas reservoir production were analyzed by this model. Based on the results, several conclusions are obtained as follows:

• A novel mathematical model considering real gas PVT behavior is developed to accurately estimate the inflow performance relationship of gas well production.
• For gas flow equations, the errors caused by traditional linearization methods and empirical formulas lead to the underestimation of gas well productivity.
• The results show that more than 90% of the energy in the flow field is consumed by inertial forces, which leads to significant high-velocity non-Darcy effects in the gas reservoir.
• The reservoir permeability, original reservoir pressure, stress sensitivity coefficient, and skin factor have a great impact on the IPR of gas reservoir production.
• This model predicts gas IPR curves with excellent accuracy and high efficiency. The high-precision gas well inflow performance relationship lays a solid foundation for dynamic production analysis, rational proration, and intelligent development of gas fields.