^{1}

^{2}

^{1}

^{3}

^{3}

^{3}

^{2}

^{1}

^{2}

^{3}

There is high uncertainty in reserve estimation during the early development of deep ultrahigh pressure gas reservoirs, largely because it remains challenging in accurately determining the formation compressibility. To overcome this, starting from the definition of compressibility, a novel gas production of cumulative unit pressure drop analysis method was established, of which the effectiveness was proven by applications in calculating the reserves of three gas reservoirs. It has been found that, in the limiting case, i.e., when the formation pressure dropped to the normal atmospheric pressure, the dimensionless gas production of the cumulative unit pressure drop was the reciprocal of the initial formation pressure. Besides, the relationship curve of the dimensionless gas production of the cumulative unit pressure drop and pressure drop was a straight line in the medium term, extending the straight line and intersecting the vertical line passing through the original formation pressure point, and the reserves can be determined according to the intersection point and the initial formation pressure. However, due to the influence of natural gas properties, the value needs further correction, and the correction coefficient depends on the pseudocritical temperature of natural gas. Specifically, when the pseudocritical temperature is given, the correction coefficient would be close to the minimum value of the natural gas deviation factor. When the pseudocritical temperature is more than 1.9 and less than 3.0, the minimum deviation factor would be between 0.90 and 1.0, and the higher the pseudocritical temperature, the closer the ratio is to 1.0.

The proportion of natural gas in primary energy consumption has been continuously increasing since the beginning of the 21^{st} century, demonstrating its importance as a global strategic resource and a livelihood material. According to existing reports [

According to the classical theory, the material balance

Published studies of reserve estimate problem of high pressure gas reservoir.

No. | Ref. | Year | Author | Solution method | Need pore volume compressibility | Advantages and disadvantages | ||
---|---|---|---|---|---|---|---|---|

1 | [ | 1971 | Hammerlindl | Polyline analysis method | Hammerlindl method | Average formation compressibility | ✓ | |

2 | Material balance | ✓ | ||||||

3 | [ | 1983 | Chen | Chen Yuanqian method | ✓ | Volumetric gas reservoirs, pressure coefficient at the transition point is 1.2~1.3 | ||

4 | [ | 2001 | Gan | Gan-Blasingame | Х | No pore volume compressibility is required, and it can be calculated whether the curve inflection point appears or not | ||

5 | [ | 1981 | Ramagost | Linear regression method | Ramagost-Farshad | ✓ | Volumetric gas reservoirs | |

6 | [ | 1981 | Roach | Roach | Х | Pore volume compressibility can be calculated, but it is sensitive to the original pressure data | ||

7 | [ | 1994 | Poston | Improved Roach method | Х | It is suitable for water drive gas reservoirs and can calculate the reserves before water invasion, the size of water invasion, and the effective compressibility coefficient | ||

8 | [ | 1993 | Becerra-Arteaga | Becerra-Arteaga | Х | Need gradient | ||

9 | [ | 1963 | Havlena | Havlena-Odeh | Х | Suitable for water intrusion gas reservoirs, sensitive to original pressure and early data; if it is a closed gas reservoir, the rock compressibility coefficient can be calculated | ||

10 | [ | 1993 | Yuanqian | Nonlinear regression method | Binary regression | Х | Knowing the variation law of natural gas deviation coefficient; this method can calculate dynamic reserves and compressibility | |

11 | [ | 2008 | Gonzalez | Nonlinear regression method | Quadratic production model | Х | Dimensionless linear coefficient is less than 0.4 | |

12 | [ | 2011 | Qin | Trinomial approximation | Х | The shape of the curve is not easy to control and may cause large errors | ||

13 | [ | 2017 | Jiao | Limit form | Х | Volumetric gas reservoirs | ||

14 | [ | 2019 | Sun | Power function form | Х | Volumetric gas reservoirs | ||

15 | [ | 1991 | Ambastha | Type curve matching | Ambastha method | Х | Volumetric gas reservoirs, original pressure, and temperature are known in type curve matching, strong multisolution | |

16 | [ | 1998 | Fetkovich | Fetkovich method | Х | Considering the influence of nonreservoir dissolved gas, inversely calculate the compressibility and the size of the water body | ||

17 | [ | 2008 | Gonzalez | Gonzalez method | Quadratic production model | Х | Mainly used to verify the results of other methods | |

18 | [ | 2020 | Sun | Single log match analysis | Х | Volumetric gas reservoirs | ||

19 | [ | 2001 | Marhaendrajana | Multiwell modern production decline analysis method | ✓ | Without considering the impact of water invasion, only reserves of connected well groups can be calculated | ||

20 | [ | 1998 | Walsh | Trial analysis method | ✓ | It can be used to calculate the volume of water bodies and calculate the water-soluble gas reserves |

This has provided new solutions to the reserve estimate problem of a high pressure gas reservoir when the starting condition is met in the middle and late stages of development. However, a simple but practical method is still missing for the reserve estimate in the early development stage.

To bridge the gap, starting with the definition of formation compressibility and the material balance equation of volumetric gas reservoir, we established an analysis method for the reserve estimate based on gas production of cumulative unit pressure drop and evaluated its performance in three gas reservoirs.

In general, the isothermal coefficient of compressibility of a material is defined as
^{8} m^{3}.

Gas production per unit pressure drop is defined as cumulative gas production per unit drop of average formation pressure. Gas production of cumulative unit pressure drop is defined as the ratio of cumulative gas production to cumulative drawdown of formation pressure from the initial formation pressure condition to the current formation pressure condition.

When equation (

When separation variables are integrated,
^{8} m^{3}/MPa. When the formation pressure drops to 0.101325 MPa, there is

The material balance equation of a volumetric gas reservoir can be expressed as
^{8} m^{3}; ^{8} m^{3}; and the subscript i represents the initial state. Equation (

When the pressure drops to

When the pressure drops to

The dimensionless gas production per unit pressure drop (assuming

The dimensionless gas production of cumulative unit pressure drop is

The theoretical calculation results show that with the depletion of formation pressure, affected by the deviation factor, the gas production per unit pressure drop first increases and then decreases, while the dimensionless gas production of cumulative unit pressure drop increases gradually. When the formation pressure drops to atmospheric pressure, its value is the reciprocal of the initial formation pressure, as shown in Figure

Gas production per unit pressure drop and dimensionless gas production of cumulative unit pressure drop (formation pressure is 40 MPa, formation temperature is 373.15 K, and gravity of gas is 0.6).

Figure

Effect of temperature on gas production of cumulative pressure drop.

333.15 K

373.15 K

413.15 K

The minimum deviation factor of natural gas depends on the pseudocritical property [

Minimum deviation factor at different pseudoreduced pressures.

When the pseudoreduced temperature is unknown, it can be calculated based on the gravity of natural gas, as given by

To sum up, the Cartesian plot of gas production of the cumulative unit pressure drop and the cumulative pressure drop is drawn, the linear regression is carried out, the straight line is prolonged to intersect with the vertical axis (as shown in Figure

The basic parameters of the American Anderson “L” gas reservoir [

Producing history of Anderson “L” gas reservoir (Duggan, 1972).

MPa | 10^{8} m^{3} | MPa | MPa | 10^{8} m^{3}/MPa |

65.548 | 0.000 | 45.52 | 0 | 0.0000 |

64.066 | 0.118 | 45.18 | 1.482 | 0.0796 |

61.846 | 0.492 | 44.59 | 3.702 | 0.1329 |

59.260 | 0.966 | 44.09 | 6.288 | 0.1536 |

57.447 | 1.276 | 43.65 | 8.101 | 0.1575 |

55.220 | 1.647 | 43.07 | 10.328 | 0.1595 |

52.421 | 2.257 | 42.31 | 13.127 | 0.1719 |

51.062 | 2.620 | 41.92 | 14.486 | 0.1809 |

48.277 | 3.146 | 41.05 | 17.271 | 0.1822 |

46.340 | 3.519 | 40.40 | 19.208 | 0.1832 |

45.057 | 3.827 | 39.98 | 20.491 | 0.1868 |

39.741 | 5.163 | 37.92 | 25.807 | 0.2001 |

32.860 | 6.836 | 33.63 | 32.688 | 0.2091 |

29.613 | 8.389 | 31.91 | 35.935 | 0.2334 |

25.855 | 9.689 | 29.02 | 39.693 | 0.2441 |

22.387 | 10.931 | 26.21 | 43.161 | 0.2533 |

The relation curve of the cumulative pressure drop and the gas production of the cumulative unit pressure drop is given in Figure

Reserves of Anderson “L” gas reservoir calculated by the method of gas production of cumulative unit pressure drop.

Reserves of Anderson “L” gas reservoir calculated by the

The basic parameters of the American Louisiana offshore ultrahigh pressure gas reservoir [

Producing history of Louisiana offshore gas reservoir (Ramagost, 1981).

MPa | 10^{8} m^{3} | MPa | MPa | 10^{8} m^{3}/MPa |

78.903 | 0.000 | 52.743 | 0.000 | 0.0000 |

73.595 | 2.809 | 51.178 | 5.309 | 0.5291 |

69.851 | 8.104 | 50.000 | 9.053 | 0.8952 |

63.797 | 15.178 | 47.968 | 15.106 | 1.0047 |

59.116 | 21.994 | 46.184 | 19.788 | 1.1115 |

54.510 | 28.719 | 44.317 | 24.394 | 1.1773 |

50.883 | 34.082 | 42.687 | 28.020 | 1.2163 |

47.208 | 41.062 | 40.908 | 31.695 | 1.2955 |

44.044 | 45.485 | 39.255 | 34.860 | 1.3048 |

40.176 | 51.633 | 37.062 | 38.728 | 1.3332 |

37.294 | 55.991 | 35.283 | 41.610 | 1.3456 |

34.474 | 61.068 | 33.372 | 44.430 | 1.3745 |

31.026 | 66.754 | 30.872 | 47.877 | 1.3943 |

28.751 | 69.631 | 29.100 | 50.152 | 1.3884 |

The relation curve of the cumulative pressure drop and gas production of the cumulative unit pressure drop is given in Figure

Reserves of Louisiana offshore gas reservoir calculated by the method of gas production of cumulative unit pressure drop.

Reserves of Louisiana offshore gas reservoir calculated by the

The basic parameters of the X gas reservoir [

Producing history of X gas reservoir.

MPa | 10^{8} m^{3} | MPa | MPa | 10^{8} m^{3}/MPa |

74.48 | 0.00 | 53.06 | 0 | 0 |

74.33 | 2.58 | 53.01 | 0.15 | 17.20 |

72.67 | 34.96 | 52.42 | 1.81 | 19.31 |

68.55 | 119.74 | 50.95 | 5.93 | 20.19 |

63.53 | 226.48 | 49.13 | 10.95 | 20.68 |

58.45 | 338.60 | 47.18 | 16.03 | 21.12 |

53.84 | 451.04 | 45.25 | 20.64 | 21.85 |

49.53 | 555.25 | 43.26 | 24.95 | 22.25 |

45.70 | 659.46 | 41.29 | 28.78 | 22.91 |

41.71 | 763.96 | 39.01 | 32.77 | 23.31 |

38.06 | 868.17 | 36.69 | 36.42 | 23.84 |

34.59 | 972.38 | 34.24 | 39.89 | 24.38 |

31.01 | 1076.59 | 31.47 | 43.47 | 24.77 |

27.83 | 1181.08 | 28.79 | 46.65 | 25.32 |

24.73 | 1285.29 | 25.98 | 49.75 | 25.83 |

21.52 | 1389.50 | 22.89 | 52.96 | 26.24 |

18.46 | 1493.16 | 19.78 | 56.02 | 26.65 |

15.55 | 1593.89 | 16.72 | 58.93 | 27.05 |

The relation curve of the cumulative pressure drop and gas production of the cumulative unit pressure drop is given in Figure

Reserves of X gas reservoir calculated by the method of gas production of cumulative unit pressure drop.

Reserves of X gas reservoir calculated by the

As seen in Table

Comparison of reserves calculated by different methods.

Gas reservoir | Volumetric method reserves (10^{8} m^{3}) | Reserves (10^{8}m^{3}) | |||
---|---|---|---|---|---|

Nonlinear regression method | Semilog type curve match method | Method introduced in the paper | |||

Anderson “L” | 19.68 | 26.6 | 19.9 | 20.0 | 19.0 |

Louisiana offshore | 131.6 | 161.2 | 125.3 | 130.0 | 131.2 |

X | 2091.5 | 2460.4 | 2032.0 | 2000.0 | 2032.55 |

Since it is difficult to accurately determine the formation compressibility in the material balance equation of high pressure gas reservoirs, the method without considering the formation compressibility is recommended for reserve estimation

When the formation pressure drops to the normal atmospheric pressure, the dimensionless gas production of the cumulative unit pressure drop is the reciprocal of the initial formation pressure. The relation curve of the dimensionless gas production of the cumulative unit pressure drop and pressure drop is a straight line in the medium term. The reserve correction coefficient is related to temperature

The method of gas production of cumulative unit pressure drop, the semilog curve match analysis technique, and the nonlinear regression method can be used in turn to calculate the reserves based on the length of production time. If the production time is short, the reserves can be roughly estimated by the method of gas production of the cumulative unit pressure drop. Based on the reserve estimate results, combined with material balance equation, the effective compressibility coefficient can be calculated, and then, the water influx can be calculated

For the water drive gas reservoir, the reserves calculated by the method of gas production of the cumulative unit pressure drop are relatively optimistic; in the future, the evaluation method of water drive gas reservoirs should be further studied to reduce the ambiguity of analysis results

Previously reported data were used to support this study and are available at [doi:

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is financially supported by the project of the Major Science and Technology Project of PetroChina Company Limited “Research and Application of Key Technologies for Development of Deep and Ultra-Deep Gas Fields in Kuqa Depression” (No. 2018E-1803).