#### Abstract

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.

#### 1. Introduction

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 [1], 61% of natural gas reserves which were discovered worldwide in the past decade are distributed in deep zones and are characterized by strong ground stress, high heterogeneity, temperature, and pressure. High pressure gas reservoirs have downward curving behavior. Precise estimation of the reserves of such gas reservoirs is crucial for reservoir engineers. However, reserve estimation’s basic method is the material balance method [2]. The formation compressibility is one of the key parameters in the material balance equation of a high pressure gas reservoir but challenging to determine; the error in estimates could even exceed 100% [3]. For many years, the petroleum industry has relied on Hall’s correlation [4] for estimating pore volume compressibility. This correlation was developed from measurements on seven consolidated limestone and five consolidated sandstone samples. Compressibilities, however, are highly affected by reservoir type and overburden conditions.

According to the classical theory, the material balance curve of a high pressure gas reservoir has a two-segment characteristic [5], with the second straight line segment occurring late (apparent formation pressure drops by 6%-38%). However, for high pressure and ultrahigh pressure gas reservoirs at the early stage of development, even if the production test time is up to 1 year and the drawdown scope reaches 3%-38% of the initial formation pressure or even higher, the starting point deviating from the early straight line segment would still not occur, and thus, the starting condition of using the material balance method to calculate reserves cannot be met. If conditions are not fully met, arbitrary applications of this method will result in serious deviation in the calculation [6]. In recent years, tremendous efforts have been spent in designing methods independent of formation compressibility, including the type curve match analysis and nonlinear regression methods. Table 1 indicates the scope of various types of information on this subject, listing the relevant papers that have been published in the petroleum literature.

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.

#### 2. Gas Production of Cumulative Unit Pressure Drop

##### 2.1. Isothermal Compressibility of Gas

In general, the isothermal coefficient of compressibility of a material is defined as
where denotes isothermal compressibility of gas in 1/MPa, with denoting pressure in MPa and denoting volume in 10^{8} m^{3}.

##### 2.2. Gas Production of Cumulative Pressure Drop

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 (1) is applied to a gas reservoir,

When separation variables are integrated,
where is the gas production of cumulative unit pressure drop, 10^{8} m^{3}/MPa. When the formation pressure drops to 0.101325 MPa, there is
i.e., when the formation pressure drops to normal atmospheric pressure, the dimensionless gas production of the cumulative unit pressure drop is the reciprocal of the initial formation pressure.

##### 2.3. Variation Law of Gas Production of Cumulative Unit Pressure Drop

The material balance equation of a volumetric gas reservoir can be expressed as
where is the pressure, MPa; is the cumulative gas production, 10^{8} m^{3}; is the reserves, 10^{8} m^{3}; and the subscript i represents the initial state. Equation (5) is deformed, and the dimensionless cumulative gas production is

When the pressure drops to, the dimensionless cumulative gas production is

When the pressure drops to, the dimensionless gas production is

The dimensionless gas production per unit pressure drop (assuming ) is

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 1.

Figure 2 shows the variation of gas production of cumulative unit pressure drop with the drop of formation pressure at different temperatures, when the formation pressure is 40 MPa and the gravity of gas is 0.6. The relation curve of gas production of cumulative unit pressure drop and cumulative pressure drop is linear at early and middle stages; when the linear relation is prolonged to intersect with the vertical axis, the ratio *a* of gas production of cumulative unit pressure drop to intersection value is approximately equal to the minimum deviation factor. In Figure 2(a), the temperature is 333.15 K, *a* =0.025/0.0284 = 0.88 ≈ 0.85, and the relative error is 3.6%; in Figure 2(b), the temperature is 373.15 K, *a* =0.025/0.0264 = 0.95 ≈ 0.94, and the relative error is 1.1%; in Figure 2(c), the temperature is 413.15 K, *a* =0.025/0.0267 = 0.94 ≈ 0.96, and the relative error is 2.1%.

**(a) 333.15 K**

**(b) 373.15 K**

**(c) 413.15 K**

##### 2.4. Minimum Deviation Factor of Natural Gas

The minimum deviation factor of natural gas depends on the pseudocritical property [22], as shown in Figure 3. When the pseudoreduced temperature is more than 1.9, the minimum deviation factor of natural gas is above 0.90.

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

##### 2.5. Calculation Method of Reserves

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 2), and the product of the intersection ordinate value and the minimum deviation factor gives the reserve.

#### 3. Case Analysis

##### 3.1. Small Ultrahigh Pressure Gas Reservoir: Anderson “L” Gas Reservoir

The basic parameters of the American Anderson “L” gas reservoir [23] are as follows: the buried depth is 3404.5 m, the pressure coefficient is 1.907 MPa/100 m, the initial pressure is 65.55 MPa, the temperature is 403 K, the gravity of natural gas is 0.656, and the volumetric reserves are , with the producing history listed as in Table 2.

The relation curve of the cumulative pressure drop and the gas production of the cumulative unit pressure drop is given in Figure 4; linear regression is carried out for the data points, the straight line is prolonged to intersect with the vertical axis, and the intersection ordinate value is , which is multiplied by the initial formation pressure 65.548 MPa, and the reserve is obtained as . The minimum deviation factor is 0.935 based on Figure 3. Therefore, the reserves of the gas reservoir is estimated as , i.e., . The reserves calculated by this method are 3.5% smaller than that by the volumetric method. The curve method, which gave an estimation of , is shown in Figure 5.

##### 3.2. Medium Ultrahigh Pressure Gas Reservoir: Louisiana Offshore Gas Reservoir

The basic parameters of the American Louisiana offshore ultrahigh pressure gas reservoir [23] are as follows: the buried depth is 4055 m, the pressure coefficient is 1.95 MPa/100 m, the initial pressure is 78.903 MPa, the temperature is 401.55 K, the gravity of the natural gas is 0.60, the pseudoreduced temperature is 1.96, the initial water saturation is 0.22, and the reserves are , with the producing history listed as in Table 3.

The relation curve of the cumulative pressure drop and gas production of the cumulative unit pressure drop is given in Figure 6. The intersection ordinate value is , which is multiplied by the initial formation pressure 78.903 MPa leading to the reserves of . The pseudoreduced temperature of the gas reservoir is 1.96, and the minimum deviation factor is 0.93. Therefore, the reserve of the gas reservoir is , i.e., . The reserves calculated by this method are highly consistent with that calculated by the dynamic method in the references [9]. If the reserves are calculated directly by the curve method, the calculated results are , as shown in Figure 7, which are much larger than the values calculated by the new method.

##### 3.3. Large High Pressure Gas Reservoir: X Gas Reservoir

The basic parameters of the X gas reservoir [16] are as follows: the middle buried depth is 3750 m, the pressure coefficient is 2.0 MPa/100 m, the initial formation pressure is 74.48 MPa, the minimum deviation factor is 0.93, and the reserves calculated by the volumetric method are , with the producing history listed as in Table 4.

The relation curve of the cumulative pressure drop and gas production of the cumulative unit pressure drop is given in Figure 8. The intersection ordinate value is , which is multiplied by the initial formation pressure 74.48 MPa leading to the reserves of . The minimum deviation factor is 0.93. Therefore, the reserve of the gas reservoir is , i.e., , which is 2.8% lower than the reserves calculated by the volumetric method. If the reserves are calculated directly by the method, the calculated result is , as shown in Figure 9, which is 17.6% higher than the reserves calculated by the static method.

##### 3.4. Comparison with Calculated Results of Other Methods without Considering Formation Compressibility

As seen in Table 5, the calculated results of the method introduced in the paper are close to those of the nonlinear regression method and the semilogarithmic type curve match method, and the three cases further verify the correctness of the method. When the gas reservoir is in the early stage of development and there is no obvious inflection point in the curve, this method can be used to approximately estimate the reserves of the gas reservoir.

#### 4. Conclusions and Suggestions

(1)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(2)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(3)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(4)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

#### Data Availability

Previously reported data were used to support this study and are available at [doi:10.2118/10125-MS; doi:10.1016/j.ijhydene.2017.04.190; doi:10.2118/2938-PA]. These prior studies (and datasets) are cited at relevant places within the text as references [9, 16, 23].

#### Conflicts of Interest

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

#### Acknowledgments

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).