# Productivity equation of low-permeability condensate gas well considering the influence of multiple factors

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## Abstract

Establishment of productivity equation is an important premise for rational and efficient development of low-permeability condensate gas reservoir and accurate analysis of production performance. Based on two-phase seepage mechanism of gas and condensate oil in formation, the productivity equation is established considering threshold pressure, stress sensitivity, slippage effect, retrograde condensate effect and high-velocity non-Darcy effect. The integrals of pseudo-pressure, turbulence factor and pseudo-threshold pressure are calculated with numerical method to resolve the productivity equation. Example calculation results indicate that with the decrease in bottom hole pressure, the increase in condensate gas well production rate is near linear first and then the increase is getting slower and slower. Slippage effect increases apparent permeability and condensate gas well productivity. Retrograde condensate effect, stress sensitivity and threshold pressure decrease apparent permeability and condensate gas well productivity. Based on orthogonal experiment design, the sensitive parameters influencing the productivity of low-permeability condensate gas well are analyzed. Various factors have different influence degrees on gas well productivity in different ranges of bottom hole pressure. Research results lay a theoretical foundation for taking measures to improve gas well productivity in the gas field.

## Keywords

Low permeability Condensate gas reservoir Threshold pressure Stress sensitivity Productivity equation## Introduction

Most of the condensate gas reservoirs are discovered in deep and tight formations with low permeability (Mohammadi et al. 2013). Complex physical and chemical phenomena in low-permeability condensate gas reservoir result in the change in gas well productivity (Zhang et al. 2014). When reservoir pressure is lower than dew point pressure, generation of condensate oil will decrease the gas relative permeability and well productivity (Li and Firoozabadi 2000; Rahimzadeh et al. 2016). In low-permeability reservoir, the absolute permeability is often sensitive to the stress and pore pressure (Sun et al. 2009; Shar et al. 2017). As effective stress increases, the absolute permeability will obviously decrease (Moghadam and Chalaturnyk 2016). Because of the influence of slippage effect, the apparent permeability of low-permeability reservoir will increase (Li et al. 2014; Wang et al. 2018). Furthermore, many experiments found that threshold pressure gradient existed in low-permeability reservoir (Guo and Wu 2007; Gao et al. 2008; Dou et al. 2014). Condensate blockage and relative permeability are research focuses in condensate gas reservoir (Farhoodi et al. 2019; Gholampour and Mahdiyar 2019). Slippage effect, stress sensitivity and threshold pressure gradient are usually considered in low-permeability and tight gas reservoir. However, all these phenomena including retrograde condensation, slippage effect, stress sensitivity and threshold pressure gradient which can influence gas well productivity are rarely considered simultaneously in low-permeability condensate gas reservoir.

Productivity analysis of condensate gas well is mainly based on the method of conventional dry gas well, including binomial and exponential productivity equation as well as empirical modified productivity equation based on them (Yan et al. 2006, 2007; Yuan et al. 2009; Xue et al. 2014). In recent years, productivity equation considering multiphase seepage has been paid more and more attention by scholars (Mokhtari et al. 2013; Huang et al. 2018; Li et al. 2018). With the further study of seepage mechanism of condensate gas reservoir, productivity analysis of condensate gas wells began to take into account the effects of many factors, such as phase change based on experiment and flash calculation, and adsorption effect of porous media (Shi et al. 2006, 2015; Qi et al. 2011; Lu et al. 2014; Jia et al. 2017). Seepage mechanism of low-permeability condensate gas reservoir is complicated because of its characteristics of low porosity, low permeability and phase change. Productivity analysis of low-permeability condensate gas well is not only different from conventional dry gas reservoirs but also different from conventional condensate gas reservoirs.

Combining the phase change characteristics of condensate gas reservoirs with the results of productivity studies of conventional dry gas reservoirs (Wu et al. 2008; Liao et al. 2012), according to steady seepage theory of condensate gas reservoir, the productivity equation of gas well in low-permeability condensate gas reservoir is established, which considers threshold pressure, stress sensitivity, slippage effect, high-speed non-Darcy effect and retrograde condensate effect. Because of nonlinear characteristics of the productivity equation, pseudo-pressure *m*(*p*), turbulence coefficient *B* and pseudo-threshold pressure coefficient *C* are calculated with numerical method to resolve the productivity equation. There are many factors influencing productivity of gas well in low-permeability condensate gas reservoir. Various factors have different influence degrees on gas well productivity. Orthogonal experiment design method is used to analyze the sensitivity, and the order of productivity influencing factors in low-permeability condensate gas well under different bottom hole pressures is known; thus, the corresponding measures can be taken to improve the productivity of gas well.

## Establishment of productivity equation of low-permeability condensate gas well considering the influence of multiple factors

In the process of depletion development of condensate gas reservoir, once formation pressure is lower than dew point pressure of condensate gas, the change in phase behavior will happen and condensate oil will appear, which will result in the change in seepage rule in formation. Based on steady seepage theory of condensate gas, the following assumptions are made. ① Seepage of gas and condensate oil in formation abides by non-Darcy rule. For the gas, high-velocity turbulence effect near the well, threshold pressure gradient effect, stress sensitivity and slippage effect are considered. For the condensate oil, threshold pressure gradient and stress sensitivity are considered. ② The stress-sensitive relationship between reservoir permeability and effective stress is exponential. ③ The threshold pressure gradient of gas phase and condensate oil phase is the same. ④ The influence of capillary pressure is ignored. ⑤ Formation temperature is constant in the process of seepage. According to these assumptions, productivity equation of low-permeability condensate gas well considering the influence of multiple factors is established.

*p*

_{g}is gas pressure, MPa;

*r*is formation radius, m;

*λ*is threshold pressure gradient, MPa/m;

*k*

_{g}is gas permeability at pressure

*p*

_{g}, μm

^{2};

*μ*

_{g}is gas viscosity, mpa∙s;

*v*

_{g}is gas velocity, m/s;

*ρ*

_{g}is gas density, kg/m

^{3};

*β*

_{g}is gas inertial resistance coefficient describing the effect of turbulence in porous media, 1/m.

*p*

_{o}is oil pressure, MPa;

*k*

_{o}is oil permeability at pressure

*p*

_{o}, μm

^{2};

*μ*

_{o}is oil viscosity, mpa s;

*v*

_{o}is oil velocity, m/s.

*k*

_{∞}is equivalent liquid permeability or absolute permeability at pressure

*p*, μm

^{2};

*k*

_{i}is permeability at pressure

*p*

_{i}, μm

^{2};

*α*is stress sensitivity coefficient, 1/MPa.

*k*and absolute permeability

*k*

_{∞}is as follows

*b*is slippage factor, MPa; \(\bar{p}\) is average pressure

*b*and permeability is

*k*

_{rg}is gas relative permeability, fraction, dimensionless.

*k*

_{ro}is oil relative permeability, fraction, dimensionless.

*Q*

_{g}and oil phase

*Q*

_{o}into Eqs. (7) and (8), the following equation is obtained

*β*

_{g}can be calculated as follows (Shi et al. 2006):

*Q*

_{o},

*Q*

_{g}are mass flow rate of oil and gas, kg/s;

*h*is reservoir thickness, m.

*p*, in order to resolve the equation, the function

*f*(

*p*) is introduced, whose expressions can be written as

*ρ*

_{sc}is density of condensate gas under standard conditions, kg/m

^{3};

*Q*

_{sc}is total production rate of condensate gas well under standard conditions, m

^{3}/day.

## Solution of productivity equation of low-permeability condensate gas well considering the influence of multiple factors

To calculate gas well productivity by Eq. (19), the key is to solve three integral terms including pseudo-pressure *m*(*p*), turbulence coefficient *B* and pseudo-threshold pressure coefficient *C*.

### Solution of pseudo-pressure *m*(*p*)

### Solution of coefficients *B* and *C* in the productivity equation

*B*and

*C*in the productivity equation are integrals of radius, and the variation of function value with radius in integral term is caused by the change in formation pressure. Therefore, the exact relation between pressure

*p*and radius

*r*must be obtained first, and then, the integral can be calculated. According to steady seepage theory, the relationship between reservoir pressure and radius under different bottom hole pressures can be obtained.

*B*and

*C*. Because the influences of pseudo-threshold pressure and turbulence term on fluid seepage are small, coefficients

*B*and

*C*can also be solved by using the approximate method of definite integral, which are shown as follows

*Q*

_{sc}is calculated to be

## Example calculation

### Analysis of single factor impact

^{3}. The data are from flash experiment test result of condensate gas system samples in formation. It is seen that the contents of intermediate and heavy hydrocarbon are low, which is a condensate gas system with low medium content of condensate oil and high gas–oil ratio. Thickness of gas pay

*h*is 36.20 m, drainage radius

*r*

_{e}is 500 m, temperature of gas reservoir

*T*

_{i}is 165.3 °C, original formation pressure

*p*

_{i}is 38.64 MPa, dew point pressure

*p*

_{d}is 35.92 MPa, absolute permeability

*k*

_{i}is 1.23 × 10

^{−3}μm

^{2}, porosity

*ϕ*is 0.09, stress sensitivity coefficient

*α*is 0.059/MPa, irreducible water saturation

*S*

_{wc}is 0.302, and threshold pressure gradient

*λ*is 0.004 MPa/m. The density and viscosity of gas and liquid phase of the condensate gas system under different pressures in Table 2 are from phase equilibrium calculation based on fitting of PVT experiments including flash separation experiment, equal composition expansion experiment and constant volume depletion experiment through PVTi software of Eclipse.

Tested components and compositions of condensate gas

Component | Molar composition (%) | Component | Molar composition (%) |
---|---|---|---|

CO | 5.734 | NC | 0.506 |

N | 0.354 | C | 0.804 |

C | 69.943 | C | 0.839 |

C | 12.119 | C | 0.926 |

C | 4.178 | C | 0.534 |

IC | 0.758 | C | 0.307 |

NC | 1.163 | C | 1.042 |

IC | 0.792 | ∑ | 100 |

Statistical table of gas–liquid parameters under different pressures of fluid in a condensate gas reservoir

Pressure (MPa | Condensate oil viscosity (mPa s) | Gas viscosity (mPa s) | Condensate oil density (kg/m | Gas density (kg/m | Condensate oil saturation (%) |
---|---|---|---|---|---|

0.11 | 0.594 | 0.015 | 925.1 | 0.7 | 4.80 |

1.8 | 0.216 | 0.015 | 729.9 | 12.6 | 5.27 |

3.8 | 0.196 | 0.015 | 696.9 | 27.0 | 5.62 |

5.8 | 0.192 | 0.016 | 681.5 | 42.1 | 5.90 |

7.8 | 0.190 | 0.016 | 671.7 | 58.0 | 6.07 |

9.8 | 0.190 | 0.017 | 665.7 | 74.5 | 6.19 |

11.8 | 0.193 | 0.018 | 663.0 | 91.3 | 6.07 |

13.8 | 0.198 | 0.019 | 663.2 | 108.3 | 5.86 |

15.8 | 0.204 | 0.020 | 665.1 | 125.0 | 5.56 |

17.8 | 0.210 | 0.021 | 667.3 | 141.3 | 5.23 |

19.8 | 0.215 | 0.022 | 668.9 | 156.9 | 4.84 |

21.8 | 0.219 | 0.024 | 669.1 | 171.8 | 4.39 |

23.8 | 0.221 | 0.026 | 667.8 | 186.2 | 3.88 |

25.8 | 0.220 | 0.027 | 664.9 | 200.0 | 3.22 |

27.8 | 0.219 | 0.029 | 660.7 | 213.4 | 2.49 |

29.8 | 0.215 | 0.031 | 655.3 | 226.5 | 1.50 |

31.8 | 0.211 | 0.033 | 648.9 | 239.4 | 0.76 |

33.8 | 0.206 | 0.036 | 641.5 | 252.3 | 0.39 |

35.9 | – | 0.038 | – | 265.1 | 0 |

38.6 | – | 0.039 | – | 273.2 | 0 |

Comparison table of productivity calculation results of a condensate gas well considering different factors

Pressure (MPa) | Calculated production rate (10 | ||||
---|---|---|---|---|---|

Phase change | Phase change + slippage | Phase change + stress sensitivity | Phase change + stress sensitivity + slippage | Phase change + stress sensitivity + slippage + threshold pressure | |

0.1 | 23.74 | 24.81 | 20.84 | 21.78 | 19.82 |

3.8 | 23.65 | 24.71 | 20.74 | 21.68 | 19.72 |

5.8 | 23.34 | 24.39 | 20.45 | 21.36 | 19.41 |

7.8 | 22.85 | 23.86 | 19.96 | 20.85 | 18.90 |

9.8 | 22.17 | 23.14 | 19.31 | 20.16 | 18.20 |

11.8 | 21.30 | 22.23 | 18.49 | 19.30 | 17.34 |

13.8 | 20.27 | 21.14 | 17.53 | 18.28 | 16.32 |

15.8 | 19.09 | 19.90 | 16.43 | 17.12 | 15.17 |

17.8 | 17.76 | 18.51 | 15.21 | 15.85 | 13.89 |

19.8 | 16.32 | 16.99 | 13.90 | 14.47 | 12.52 |

21.8 | 14.76 | 15.36 | 12.51 | 13.02 | 11.06 |

23.8 | 13.13 | 13.66 | 11.07 | 11.51 | 9.55 |

25.8 | 11.43 | 11.88 | 9.58 | 9.96 | 8.00 |

27.8 | 9.65 | 10.03 | 8.05 | 8.36 | 6.40 |

29.8 | 7.82 | 8.12 | 6.48 | 6.73 | 4.77 |

31.8 | 5.92 | 6.14 | 4.88 | 5.06 | 3.11 |

33.8 | 3.96 | 4.11 | 3.25 | 3.37 | 1.41 |

35.8 | 1.98 | 2.06 | 1.62 | 1.68 | 0.00 |

38.64 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

### Multifactor sensitivity analysis based on orthogonal design

_{9}(3

^{4}). Based on the productivity equation of low-permeability condensate gas well under the influence of multiple factors, the productivity of each case is calculated. The results are shown in Tables 5 and 6.

Level table of influencing factors for productivity of a low-permeability condensate gas well

Level | Factor 1 | Factor 2 | Factor 3 | Factor 4 |
---|---|---|---|---|

Threshold pressure gradient | Stress sensitivity coefficient | Slippage factor | Condensate oil saturation | |

1 | 0 | 0 | 0 | 0 |

2 | 0.002 | 0.03 | 0.3 | 3 |

3 | 0.004 | 0.06 | 0.6 | 6 |

Orthogonal experiment result table L_{9}(3^{4}) with 4 factors influencing productivity of a low-permeability condensate gas well (*P*_{wf} = 12 MPa)

Number | Factor 1 | Factor 2 | Factor 3 | Factor 4 | Production rate (10 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 23.15 |

2 | 1 | 2 | 2 | 2 | 21.5125 |

3 | 1 | 3 | 3 | 3 | 19.875 |

4 | 2 | 1 | 2 | 3 | 20.775 |

5 | 2 | 2 | 3 | 1 | 22.625 |

6 | 2 | 3 | 1 | 2 | 18.1975 |

7 | 3 | 1 | 3 | 2 | 21.8875 |

8 | 3 | 2 | 1 | 3 | 17.46 |

9 | 3 | 3 | 2 | 1 | 19.31 |

k1 | 21.5125 | 21.9375 | 19.6025 | 21.695 | |

k2 | 20.5325 | 20.5325 | 20.5325 | 20.5325 | |

k3 | 19.5525 | 19.1275 | 21.4625 | 19.37 | |

R | 1.96 | 2.81 | 1.86 | 2.325 |

Orthogonal experiment result table L_{9}(3^{4}) with 4 factors influencing productivity of a low-permeability condensate gas well (*P*_{wf} = 20 MPa)

Number | Factor 1 | Factor 2 | Factor 3 | Factor 4 | Production rate (10 |
---|---|---|---|---|---|

1 | 1 | 1 | 1 | 1 | 17.30 |

2 | 1 | 2 | 2 | 2 | 15.8693 |

3 | 1 | 3 | 3 | 3 | 14.4386 |

4 | 2 | 1 | 2 | 3 | 15.2136 |

5 | 2 | 2 | 3 | 1 | 16.455 |

6 | 2 | 3 | 1 | 2 | 13.0143 |

7 | 3 | 1 | 3 | 2 | 15.7993 |

8 | 3 | 2 | 1 | 3 | 12.3586 |

9 | 3 | 3 | 2 | 1 | 13.6 |

k1 | 15.8693 | 16.1043 | 14.2243 | 15.785 | |

k2 | 14.8943 | 14.8943 | 14.8943 | 14.8943 | |

k3 | 13.9193 | 13.6843 | 15.5643 | 14.0036 | |

R | 1.95 | 2.42 | 1.34 | 1.7814 |

In Tables 5 and 6, k1, k2, k3 are average values of production rate at the same level for each factor. R is the difference between the maximum and the minimum of k1–k3. The bigger the R is, the more obvious the influence of this factor is. Results indicate that in the range of threshold pressure gradient *λ* = 0–0.004 MPa/m, stress sensitivity coefficient *α* = 0–0.06/MPa, slippage factor *b* = 0–0.6 MPa, condensate oil saturation S_{o} = 0–6%, when bottom hole pressure is 10–17.5 MPa, the order of productivity influencing factors in low-permeability condensate gas wells is stress sensitivity > retrograde condensate phase change > threshold pressure gradient > slippage effect. When bottom hole pressure is 17.6–24.7 MPa, the order of productivity influencing factors in low-permeability condensate gas wells is stress sensitivity > threshold pressure gradient > retrograde condensate phase change > slippage effect. When bottom hole pressure is 24.8–33.8 MPa MPa, the order of productivity influencing factors in low-permeability condensate gas wells is threshold pressure gradient > stress sensitivity > retrograde condensate phase change > slippage effect. Within different ranges of bottom hole pressure, corresponding measures can be taken to improve the productivity of gas well according to the key sensitive factors affecting the productivity of low-permeability condensate gas well.

## Conclusions

- 1.
In low-permeability condensate gas reservoir, because of condensate banking or blockages near the well bore area, two-phase flow of gas and condensate oil arises in the reservoir. The relative permeability and mobility of each fluid are different, and they compete for flow toward the well. Compared to conventional dry gas reservoir, seepage mechanism is more complicated in low-permeability condensate gas reservoir, where two-phase flow with various non-Darcy effects of gas and condensate oil should be considered simultaneously.

- 2.
The binomial productivity equation with the form of pseudo-pressure in low-permeability condensate gas well is established considering the comprehensive effect of multiple factors including threshold pressure gradient, stress sensitivity, slippage effect, retrograde condensate effect, high-speed non-Darcy effect. The integrals of pseudo-pressure, turbulence factor and pseudo-threshold pressure are calculated with numerical method to resolve the productivity equation.

- 3.
The productivity of an example well is calculated by considering different factors. Results indicate that slippage effect increases apparent permeability; thus, the pseudo-pressure and gas well production rate increase. Retrograde condensate effect decreases gas relative permeability and gas well production. The apparent permeability decreases with the increase in effective stress, so pseudo-pressure and gas well production rate decreases. Threshold pressure gradient reduces the production pressure difference; therefore, the productivity of gas well decreases.

- 4.
Based on orthogonal experiment design, the sensitivity factors affecting productivity of low-permeability condensate gas well are analyzed in different ranges of bottom hole pressure. The research lays a theoretical foundation for taking measures to improve gas well productivity in the gas field.

## Notes

### Acknowledgements

The authors wishes to acknowledge the assistance of the National Natural Science Foundation of China (51574052), the Chongqing Basic Science and Advanced Technology Research Project (cstc2016jcyjA0293), the Science and Technology Research Project of Chongqing Municipal Education Committee (KJ1601319), the University Innovation Team Project of Chongqing Municipal (CXTDX201601033) and the Internal research fund of Chongqing University of Science and Technology (ck2017zkyb006).

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