An Improved Correlation of Compressibility Factor Prediction of Variable CO₂-Content Condensate Gases

Abstract

The injection of carbon dioxide (${\mathrm{CO}}_2$) into gas reservoirs has become an important way to enhance gas recovery and reduce ${\mathrm{CO}}_2$ emissions. Large discrepancies are observed when predicting natural gas compressibility factors with high ${\mathrm{CO}}_2$ content by several well-known empirical correlations. An explicit correlation is proposed to improve the prediction accuracy in the estimation of compressibility factors on condensate gases with variable ${\mathrm{CO}}_2$ contents. The analysis of the results is carried out on the basis of 202 experimental data from 9 various mixtures of natural gases. The results show that relative deviations of compressibility factors predicted by conventional empirical correlations increase with the increase in ${\mathrm{CO}}_2$ mole fraction with an average error of 8%. The average error of the new method is less than 4%. The effect of compressibility factors on the estimations of dynamic reserves is studied and the compressibility factor causes a 3% reduction in dynamic reserves estimation. The proposed correlation has fewer uncertainties and more accurate results than other correlations that involve the iterative process in calculating compressibility factors of natural gases with variable ${\mathrm{CO}}_2$ contents.

1. Introduction

Development of natural gas reservoirs associated with the continuous changes of temperature and pressure. Determination of pressure–volume–temperature (PVT) properties of natural gas is important. Among these thermodynamic properties, the gas compressibility factor (Z-factor) is a basic and crucial parameter. It is of importance in calculating the material balance equation, evaluating gas reserves, simulating gas reservoir development, etc. [1,2]. In addition, the injection of carbon dioxide (${\mathrm{CO}}_2$) into gas reservoirs has become an important approach to improve natural gas recovery and reduce ${\mathrm{CO}}_2$ emissions to alleviate climate change [3]. However, the full miscibility of ${\mathrm{CO}}_2$ and natural gas affects the predictive accuracy of natural gas compressibility factors. Experimental studies show that the compressibility factor has significant differences when the non-hydrocarbon components are in the natural gas system [4,5]. The compressibility factor of ${\mathrm{CO}}_2$ is sensitive to temperature and pressure where low temperature reduces the inflection pressure of compressibility factor curve. Traditional methods are found to have large discrepancies in predicting the gas compressibility factor when the ${\mathrm{CO}}_2$ contents are high in the natural gas system. The motivation of this study is to introduce an explicit correlation to accurately estimate compressibility factors of natural gases with different ${\mathrm{CO}}_2$ contents.

1.1. Overview of the Current Methods

Experimental measurements, equation of state (EOS) method, empirical correlations, and artificial intelligence (AI) related methods are the four common approaches to predict compressibility factor. Experimental measurement is the most accurate and reliable method and the Standing and Katz chart (SKC) is the most commonly used benchmark [6,7]. Secondly, thermodynamic models based on EOS are used to study the phase behaviour of natural gases and predict compressibility factors [8,9]. Several tuning methods are developed to reduce the relative errors observed between simulation and experiments. For example, Shariati et al. [10] and Jaubert et al. [9] reported the research concerning with the adjustment of splitting $C_{7+}$ fractions and correction of binary interaction parameters. Yan et al. [8] presented a correction, originating from the Soave–Redlich–Kwong (SRK) EOS to improve the prediction accuracy of compressibility factors when applied to high-pressure natural gases. The third method of computing compressibility factors is empirical correlations [11,12,13]. There are more than 20 formulae for fast and convenient estimation of compressibility factors, and these formulae are generally applied to mild pressures and temperatures. The fourth method is artificial intelligence (AI)-based methods, such as neural networks and support vector machine methods [14,15], which can be employed to predict the physical properties of hydrocarbons.

Generally speaking, the measured data are customarily used to validate results calculated from empirical correlations because of their reliability and accuracy. However, PVT tests of fluid samples are often unavailable, especially at the early stage of a gas field development. They are also expensive and time-consuming. Thermodynamic models based on EOS are implicit models, while empirical correlations are explicit methods. The latter is relatively simpler and easier in calculating compressibility factors than the former. One reason is that EOS involves numerous calculations and compressibility factors cannot be solved as a root of the EOS. Finally, the AI-based models are accurate and fast in computing compressibility factors, however, the performance of these methods requires a further exploration. As show in

Table 1. The characteristics of current research in the literature of computing compressibility factors.

Methods	Brief Introduction	Advantages	Limitations	Research Gaps
Experiments	Carried out in PVT apparatus	Accurate; Reliable; Benchmarks	Expensive; Time consuming	High-temperature and high-pressure conditions are limited [16]; Unable to measure all compositions.
Equation of state methods	Using EOS to study the volumetric and phase behaviours of reservoir fluids	Theoretical calculation; Accurate; Phase equilibrium calculation	Implicit; Numerous calculations; complicated	Adjustment or regression of EOS parameters, splitting and characterization of ${\mathrm{C}}_{7+}$ fraction [10]; Correction of binary interaction parameters [9]
Empirical correlations	Defined as a function of reduced pressure and temperature	Efficient; Easy; Explicit; Mostly used methods	Limited to certain conditions; Depends on the range of data the correlation originating from.	High-temperature and high-pressure conditions [8]; Non-hydrocarbon components (${\mathrm{CO}}_2$, ${\mathrm{H}}_2\mathrm{S}$, ${\mathrm{N}}_2$) [17]; Tuning methods to reduce relative errors
AI-based methods	Model is developed based on machine learning neural network techniques	New approach; Fast; Accurate	Slow convergence; Over-fitting; Poor generalization performance	The amount of source data is limited; Different types of algorithms and methods needed to be combined [14,18]

, comprehensive comparisions are provided to better elaborate on the characteristics of available methods in the literature from both aspects of advantages and limitations.

1.2. Statement of the Problem

In this paper, the empirical correlations are adopted to calculate compressibility factors of natural gases in a condensate gas reservoir with variable ${\mathrm{CO}}_2$ contents. Because empirical correlations are the most efficient and simple methods for industrial applications. However, large discrepancies are found by using these empirical correlations [2,11,12]. These methods perform well in traditional natural gases, whereas the high inclusion of ${\mathrm{CO}}_2$ greatly reduces the prediction accuracy. Therefore, a new correlation is developed in this paper. Similar to the classical correlations, the application of the new one is limited to a certain range of pressures and temperatures. However, the new method has some novelties. First, the proposed correlation can deal with natural gas samples that contain variable ${\mathrm{CO}}_2$ contents. Second, it is applied to condensate gas samples that have relatively large amounts of components than dry and wet gases.

The structure of this paper is outlined below. Section 2 is the methodology that introduces the general procedures to compute compressibility factors and three basic empirical correlations in the literature. The development of the new correlation is also introduced in this section. Section 3 shows the results and discussions, which firstly compares the compressibility factors computed from these correlations in a realistic natural gas reservoir. Then, six natural gas mixtures taken from the literature are utilized to validate the accuracy and quantify the uncertainties of the proposed correlation. In addition, the effect of compressibility factors on the estimation of dynamic reserves is studied. Finally, conclusions are drawn in Section 4.

2. Methodology

This section introduces the natural gas samples used in the laboratory and the general procedures to compute compressibility factors. Three empirical correlations from the literature and the correlation developed in this paper are demonstrated.

2.1. Data Introduction

The natural gas samples used in this paper were taken from a gas field in the south of China. All samples were used without further purification. The JEFRI-PVT apparatus made by the Schlumberger company (Canada) was used to conduct the PVT experiments. Constant composition expansion experiments were performed to determine the gas volume and calculate compressibility factors.

The compressibility factor z is defined from the EOS:

$$Z=\frac{PV}{nRT}$$

(1)

where P and T are the measured pressure and temperature in the PVT cell; V is the gas volume; n is the moles of gas; and R is the universal gas constant. Compressibility factor is a measure of the amount of gas that deviates from its perfect behaviour and is calculated from experiments using:

$$Z=\frac{P_e{V}_e·T_a}{T_e·P_a{V}_a}$$

(2)

in which the subscripts e and a denote the experiment and ambient conditions. The ambient pressure used here is 0.101325 MPa and the temperature is 20 °C.

Three condensate gas samples with different ${\mathrm{CO}}_2$ contents were utilized and studied.

Table A1. Compositions of three condensate gas mixtures from three different wells in the H2 field, where the unit is the mole fraction (percentage).

Component	Sample 1	Sample 2	Sample 3
${\mathrm{CO}}_2$	6.165	10.184	14.893
N2	0.269	0.485	1.034
C1	67.499	70.704	63.68
C2	13.945	11.461	9.899
C3	6.896	4.126	5.855
iC4	1.423	0.855	1.511
nC4	1.858	0.963	1.814
iC5	0.773	0.409	0.675
nC5	0.449	0.255	0.393
C6	0.563	0.374	0.225
C7	0.134	0.146	0.017
C8	0.026	0.038	0.004
Total	100	100	100
Reservoir pressure (MPa)	45.34	46.31	26.57
Reservoir temperature (°C)	150.54	150.23	116.28
$C_{7+}$ relative density	0.82	0.82	0.77
Gas oil ratio	1552	3887	1685

shows the mole fraction of compositions of gas mixtures. Each sample is split into three subsamples to be analysed at different temperatures. Therefore, nine experiments are conducted, and a total of 134 data points are measured. The measured compressibility factors of the three samples are reported in

Table A2. Measured compressibility factors of sample 1 in the H2 field.

P (MPa)	Z (T = 150 °C)	P (MPa)	Z (T = 115 °C)	P (MPa)	Z (T = 80 °C)
33.3	0.9999	32.7	0.8437	31.5	0.8117
35	1.027	34	0.8631	33	0.8388
36	1.0436	35	0.8791	34	0.8567
37	1.0587	36	0.8951	35	0.8739
38	1.0741	37	0.9098	36	0.8914
39	1.09	38	0.9255	37	0.9095
40	1.1054	39	0.9411	38	0.9276
41	1.1214	40	0.9569	39	0.9453
42	1.1371	41	0.9729	40	0.9631
43	1.1532	42	0.9882	41	0.9813
44	1.169	43	1.0039	42	0.9985
45.34	1.1909	44	1.0206	43	1.0164
-	-	45.34	1.042	44	1.0342
-	-	-	-	45.34	1.0579

,

Table A3. Measured compressibility factors of sample 2 in the H2 field.

P (MPa)	Z (T = 150 °C)	P (MPa)	Z (T = 107 °C)	P (MPa)	Z (T = 75 °C)
33.45	1.0604	35.76	1.061	36.47	1.0405
34	1.0692	36	1.0579	37	1.0477
35	1.0804	37	1.0717	38	1.0646
36	1.0921	38	1.086	39	1.0809
37	1.1047	39	1.0998	40	1.0977
38	1.117	40	1.1146	41	1.1163
39	1.128	41	1.1308	42	1.1325
40	1.1409	42	1.1469	43	1.1494
41	1.1533	43	1.1601	44	1.1674
42	1.1665	44	1.1762	45	1.1851
43	1.1807	45	1.1917	46.31	1.2092
44	1.1925	46.31	1.212	-	-
45	1.2057	-	-	-	-
46.31	1.2244	-	-	-	-

and

Table A4. Measured compressibility factors of sample 3 in the H2 field.

P (MPa)	Z (T = 116 °C)	P (MPa)	Z (T = 91 °C)	P (MPa)	Z (T = 66 °C)
9	0.8476	8	0.8168	27	0.8316
10	0.8365	9	0.8001	8	0.7539
11	0.8272	10	0.787	9	0.74
12	0.8196	11	0.7766	10	0.7281
13	0.813	12	0.7681	11	0.717
14	0.8081	13	0.7607	12	0.7076
15	0.8053	14	0.7556	13	0.7012
16	0.8025	15	0.7524	14	0.6968
17	0.8014	16	0.7501	15	0.6937
18	0.802	17	0.75	16	0.693
19	0.8035	18	0.7507	17	0.6942
20	0.8067	19	0.754	18	0.6975
21	0.8105	20	0.7583	19	0.703
22	0.8158	21	0.7641	20	0.7102
23	0.823	22	0.7723	21	0.7186
24	0.8302	23	0.7814	22	0.7303
25	0.8392	24	0.7917	23	0.742
26.576	0.8541	25	0.804	24	0.7558
-	-	26.576	0.8243	25	0.7725
-	-	-	-	26.2	0.7925
-	-	-	-	26.576	0.8006

in Appendix A.

2.2. Empirical Methods Introduction

This section elaborates on the general procedures of compressibility factor calculation by using empirical correlations. Four empirical correlations are introduced.

2.2.1. General Procedures to Compute Z

The procedures to calculate the compressibility factor using empirical correlations are similar. The input parameters include gas compositions, pressure, and temperature.

First, pseudo critical properties are calculated. Several mixing rules can be used, for example, Kay’s mixing rule, the Stewart–Burkhardt–Voo (SBV) mixing rule, the Sutton modification of SBV (SSBV) and the Piper mixing rule. Kay’s mixing rule is the best-known and is used in this study. The pseudo critical properties are calculated as follows:

$$P_{pc}={\displaystyle \sum _{i=1}^{n_c}}{y}_i{P}_{ci}$$

(3)

$$T_{pc}={\displaystyle \sum _{i=1}^{n_c}}{y}_i{T}_{ci}$$

(4)

where $y_i$ is the mole fraction of component i in the gas mixture.

Secondly, the Wichert and Aziz [19] correlation is applied to correct the pseudo critical properties of natural gases to sour gas components (${\mathrm{CO}}_2$ and ${\mathrm{H}}_2\mathrm{S}$). The Wichert and Aziz correlation takes the following form:

$$ϵ=\frac{5}{9}[120(A^{0.9}-A^{1.6})+15(B^{0.5}-B^4)]$$

(5)

$$T_{pc}^{\prime }=T_{pc}-ϵ$$

(6)

$$P_{pc}^{\prime }=\frac{P_{pc}{T}_{pc}^{\prime }}{T_{pc}+B(1-B)ϵ}$$

(7)

where A and B are mole fractions of ${\mathrm{H}}_2\mathrm{S}$ and ${\mathrm{CO}}_2$.

Thirdly, the reduced pressure and temperature are calculated:

$$P_r=\frac{P}{P_{pc}^{\prime }}$$

(8)

$$T_r=\frac{T}{T_{pc}^{\prime }}$$

(9)

Then compressibility factors can be calculated using empirical correlations.

2.2.2. The Empirical Correlations

There are more than twenty correlations available to compute compressibility factors from fitting the Standing–Katz chart. In this research, three well-known empirical correlations are adopted to calculate gas compressibility factors, the Dranchuk and Abu-Kassem method (DAK) [12], the Hall and Yarborough method (HY) [11], and the Granmer method [20]. Details of these methods are listed below. The implementations of these methods includes an iterative method and a direct calculation. The Newton–Raphson iterative method is used to calculate the result [20].

The Dranchuk and Abou-Kassem [12] correlation is a widely used empirical method to calculate the gas compressibility factor:

$$\begin{array}{cc}\hfill Z=& 1+(a_1+\frac{a_2}{T_r}+\frac{a_3}{T_r^3}+\frac{a_4}{T_r^4}+\frac{a_5}{T_r^5}){\rho }_r+(a_6+\frac{a_7}{T_r}+\frac{a_8}{T_r^2}){\rho }_r^2-a_9(\frac{a_7}{T_r}+\frac{a_8}{T_r^2}){\rho }_r^5\hfill \\ \hfill & +a_{10}(1+a_{11}{\rho }_r^2)(\frac{{\rho }_r^2}{T_r^3})e^{(-a_{11}{\rho }_r^2)};\hfill \end{array}$$

(10)

$${\rho }_r=0.27\frac{P_r}{Z{T}_r}$$

(11)

where $a_1=0.3265, a_2=-1.07, a_3=-0.5339, a_4=0.01569, a_5=-0.05165, a_6=0.5475,$$a_7=-0.7361, a_8=0.1844, a_9=0.1056, a_{10}=0.6134, a_{11}=0.721, {\rho }_r$ is the reduced density of gas.

Hall and Yarborough [11] presented an EOS to represent the Standing–Katz chart:

$$Z=-\frac{a}{y}$$

(12)

where y can be found from the following equation using the Newton–Raphson iterative method:

$$a+\frac{y+y^2+y^3+y^4}{{(1-y)}^3}-b{y}^2+c{y}^d=0$$

(13)

Coefficients a, b, c, and d are expressed as follows:

$$a=-0.06125\frac{P_r}{T_r}{e}^{(-1.2{(1-\frac{1}{T_r})}^2)};$$

(14)

$$b=\frac{14.76}{T_r}-\frac{9.76}{T_r^2}+\frac{4.58}{T_r^3}$$

(15)

$$c=\frac{90.7}{T_r}-\frac{242.2}{T_r^2}+\frac{42.4}{T_r^3}$$

(16)

$$d=2.18+\frac{2.82}{T_r}$$

(17)

The Granmer method is the third used to compute the gas compressibility factor [20]:

$$Z=1+(0.31506-\frac{1.0467}{T_r}-\frac{0.5783}{T_r^3}){\rho }_r+(0.5353-\frac{0.6123}{T_r}){\rho }_r^2+\frac{0.6815}{T_r^3}{\rho }_r^2$$

(18)

$${\rho }_r=\frac{0.27P_r}{Z{T}_r}$$

(19)

In addition to the above three methods, an empirical correlation is proposed in this paper. Poettman and Carpenter [21] digitalized the Standing–Katz chart by thousands of data points in different ranges of pressure and temperature. That is how the classical correlations were developed. In this study, the experimental data measured from the real condensate gas reservoir are utilized. Compressibility factors were correlated to reduced pressure and temperature by multiple regression analysis. The tuned coefficients are obtained by minimizing the sum of the squares of the residuals of the compressibility factor formula. The more precise the experimental data, the more accurate the tuned coefficients and the fewer deviations from the predicted results.

$$Z=a0+a_1{P}_r+a_2{P}_r^2+\frac{a_3{P}_r^{a_4}}{T_r^{a_5}}+\frac{a_{6}P{r}^{a_7}}{T_r^{a_8}}+\frac{a_9{P}_r^{a_{10}}}{T_r^{a_{11}}}$$

(20)

where $a_0=0.9704, a_1=0.04751, a_2=-0.0008034, a_3=-0.5986, a_4=0.8204,$$a_5=2.8986, a_6=0.1104, a_7=1.8212, a_8=4.2627, a_9=-0.002545, a_{10}=2.8201,$$a_{11}=5.2625$.

3. Results and Discussions

This section firstly compares the compressibility factors calculated by the correlation proposed in this paper and three classical correlations with the experimental data. Secondly, six natural gas mixtures with ${\mathrm{CO}}_2$ contents ranging from 9% to 89% in the literature are utilized to validate the accuracy of the proposed correlation. Thirdly, dynamic reserves are estimated using the material balance equation and the effect of the calculated compressibility factors on the dynamic reserves are analysed.

3.1. Comparisons with the Other Empirical Correlations

The three condensate gas samples are utilized here. Each sample is split into three subsamples that are tested at different temperatures and a total of 134 data points are obtained.

Table 2. The absolute average error of the compressibility factors computed by the four empirical methods compared with measured results.

Correlation	Sample 1	Sample 2	Sample 3	Total Average
DAK	6.02	8.10	9.86	8.00
HY	7.22	6.57	10.85	8.22
Granmer	5.51	10.64	9.34	8.50
This paper	1.96	3.00	8.63	4.53

reports the absolute average errors of the four correlations compared with the experiments. Table A2, Table A3 and Table A4 in Appendix A report the measured data. The absolute average error is calculated from:

$$E_{aa}=|\frac{Z_{calculated}-Z_{measured}}{Z_{measured}}|\times 100\%$$

(21)

It can be seen that the total average errors of the three empirical correlations are larger than 8% and the errors in some samples even exceed 10% (for example, the HY method in sample 3 and the Granmer method in sample 2). However, the average errors computed by the correlation proposed in this paper are as smaller as 1.96% and 3% for sample 1 and 2, respectively. All four correlations have large errors in sample 3 but the error of the method proposed in this paper is the smallest. The total average error of the proposed method is 4.53%, which is less than the other three empirical correlations.

In order to compare the accuracy of the four correlations, the cumulative frequency versus the absolute average errors of the 134 experimental compressibility factor data points is plotted in Figure 1. The correlation can predict 90% of the data points accurately by assuming an acceptable error of 10%, while the other three methods predict fewer than 80% of the data points. The cross plot of the calculated and measured compressibility factors of the correlation proposed in this paper is shown in Figure 2. Compressibility factors predicted by the correlation proposed in this paper have a reasonably good agreement with the experimental results. Therefore, the correlation proposed in this paper is the most accurate of the four empirical correlations.

In addition, Figure 3 shows the absolute average errors of the four empirical correlations versus the ${\mathrm{CO}}_2$ mole fraction. In general, the errors increase with the increase in ${\mathrm{CO}}_2$ content. Large errors are observed when the ${\mathrm{CO}}_2$ mole fraction is larger than 10% for all four correlations. The correlation proposed in this paper has the smallest error when the ${\mathrm{CO}}_2$ mole fraction is smaller than 10%.

3.2. Validation Data from the Literature

In this subsection, six different natural gas mixtures from the literature are utilized to validate the accuracy of the proposed correlations [7,22,23,24].

Table A5. Compositions of the six natural gas mixtures from the literature to validate the four empirical correlations, where samples 1–3 are taken from Bian et al. [7] and samples 4–6 are from Rushing et al. [22], Adisoemarta et al. [23], Simon et al. [24], respectively.

Component	Sample 1	Sample 2	Sample 3	Sample 4	Sample 5	Sample 6
${\mathrm{CO}}_2$	0.0984	0.2886	0.5099	0.2	0.75	0.8994
N2	0.0205	0.0124	0.0083	0	0	0.0004
C1	0.8602	0.6773	0.4654	0.768	0.242	0.0944
C2	0.0167	0.0159	0.0116	0.024	0.007	0.0021
C3	0.0031	0.0042	0.0033	0.008	0.001	0.001
iC4	0.0004	0.0006	0.0005	0	0.001	0
nC4	0.0005	0.0007	0.0006	0	0	0.0006
iC5	0.0001	0.0003	0.0002	0	0	0
nC5	–	0.0001	0.0001	0	0	0.0005
C6	0.0001	–	–	0	0	0.0016
Total	1	1	1	1	1	1

summarizes the details of compositions of the six samples where the ${\mathrm{CO}}_2$ contents vary from 9% to 89%. Some statistical parameters shown below are used to quantify the performance of the four empirical correlations.

Average relative error ($E_{ar}$):

$$E_{ar}=\frac{{\displaystyle \sum _{i=1}^{N_d}}(\frac{Z_i^{calculated}-Z_i^{measured}}{Z_i^{measured}})}{N_d}\times 100\%$$

(22)

where $N_d$ is the number of data points and $i=1,2,3,\dots ,N_d$.

Average absolute relative error ($E_{aar}$):

$$E_{aar}=\frac{{\displaystyle \sum _{i=1}^{N_d}}(|\frac{Z_i^{calculated}-Z_i^{measured}}{Z_i^{measured}}|)}{N_d}\times 100\%$$

(23)

Root mean square error ($E_{rms}$):

$$E_{rms}=\sqrt{\frac{{\displaystyle \sum _{i=1}^{N_d}}{E}_i^2}{N_d}}$$

(24)

Coefficient of variation (CV):

$$CV=\frac{\sqrt{\frac{{\displaystyle \sum _{i=1}^{N_d}}{E}_i^2}{N_d-1}}}{E_{aar}}$$

(25)

Figure 4 illustrates the average relative errors versus different ${\mathrm{CO}}_2$ contents.

Table 3. The statistical results of the four methods when predicting compressibility factors of the six natural gas samples with different contents of ${\mathrm{CO}}_2$.

Parameters	This Paper	DAK	HY	Granmer
$E_{ar}$	1.52	−0.23	−5.55	0.51
$E_{aar}$	6.42	2.74	8.55	2.58
$E_{rms}$	8.64	3.52	18.03	3.42
$CV$	1.41	1.92	2.04	2.01

summarizes the uncertainties of the four empirical correlations. All four correlations perform reasonably well for most samples. Among them, the DAK method and Granmer method are the best two methods for most samples where the relative errors do not exceed 5%. The correlation proposed in this paper has fewer errors in samples with large ${\mathrm{CO}}_2$ contents and has the smallest coefficient of variation. The low value of the coefficient of variation indicates that the data have little variability and high stability. Therefore, the method proposed in this paper is the most stable one among the four empirical correlations. On the contrary, the HY method is the worst and has the largest average relative error and root mean square error in all samples.

3.3. The Influence on Dynamic Reserves Estimation

This section shows the influence of compressibility factors on the estimation of dynamic reserves by using a material balance equation in a real gas reservoir. Dynamic reserve is defined as the initial hydrocarbon in place that is determined by dynamic methods, compared to the static reserve determined by the traditional volumetric method [25].

The material balance equation is able to estimate the initial gas in place [26]. For a constant volume gas reservoir without a water influx, the material balance equation is simplified as [27]:

$$\frac{p}{Z}=\frac{p_i}{Z_i}(1-\frac{G_p}{G})$$

(26)

where $G_p$ is the cumulative gas production, G is the dynamic gas reserve and subscript i is the initial reservoir condition.

As shown in Figure 5, pressure drop-down curves ($\frac{p}{Z}$ vs. $G_p$) are plotted where the intersection point of the curve with the X-axis is the estimated dynamic gas reserve. Gas compressibility factors are calculated from the four empirical correlations.

Well production data are taken from the H2 gas field, a small gas condensate reservoir surrounded by sealing faults. The H2 gas field has 10 wells and the initial reservoir pressure is about 30 MPa. Three wells are selected to calculate dynamic reserves and the production history data of well #1 are shown in

Table A6. Production history data of well #1.

Time	Cumulative Production Volume (108 m3)	Pressure (MPa)
2010.01	0.0008	24
2010.06	0.0585	22.6
2011.01	0.1189	19.79
2011.06	0.1470	17.94
2012.01	0.1798	17.2
2012.06	0.1991	16.55
2013.01	0.2239	14.81
2013.06	0.2399	15.23
2014.01	0.2578	15.24
2014.06	0.2693	14.83
2015.01	0.2844	13.78
2015.06	0.2899	12.52
2016.01	0.3020	13.83
2016.06	0.3094	13.79

. Pressure drop-down curves are plotted in Figure 5.

Table 4. Well-controlled dynamic reserves of three wells by using the material balance method where the unit is 10⁸ m³.

Well	This Paper	DAK	HY	Granmer	Deviation (%)
Well #1	0.6840	0.6644	0.6684	0.6610	3.48
Well #2	0.3150	0.3113	0.3143	0.3065	2.76
Well #3	1.5992	1.5550	1.5590	1.5536	2.94

reports the computed dynamic reserves of three wells. The maximum deviations among the four correlations are calculated:

$$D_{max}=\frac{G_{max}-G_{min}}{G_{min}}\times 100\%$$

(27)

where $G_{max}$ and $G_{min}$ are the estimated maximum and minimum dynamic reserves calculated from the four empirical correlations.

Even though the relative deviations of dynamic reserves estimated by the four correlations are small (about 3%), absolute deviations of dynamic reserves are large, for example, well #3 has $4.7\times 6$ m³ reserve deviation. This indicates that the compressibility factor does influence the estimation of dynamic reserves. In addition, Figure 6 shows the effect of different ${\mathrm{CO}}_2$ contents on the dynamic reserves calculation. There is no direct relationship between the ${\mathrm{CO}}_2$ mole fraction and the estimated dynamic reserves.

4. Conclusions

In this study, a simple correlation is developed to increase the accuracy of calculating compressibility factors of natural gases with different ${\mathrm{CO}}_2$ contents. The advantage of the proposed correlation is that it is an explicit method and does not require an iterative process, which is demanded by the other three classical correlations. Compressibility factors calculated from the new correlation are compared with 202 experimental data measured from 9 natural gas mixtures. The proposed correlation has better statistical parameters and fewer uncertainties than the other three empirical correlations. The average error of the new correlation is less than 4%.

It is found that the relative errors increase with the increase in ${\mathrm{CO}}_2$ mole fraction. Compressibility factors affect the estimation of dynamic reserves and a total 3% reduction in reserve is observed, however, there is no direct relationship between the ${\mathrm{CO}}_2$ mole fraction and the dynamic reserves calculation.

The proposed correlation has limitations. The analysis of the results are only carried out based on the 202 experimental data. Even though the proposed correlation has small relative errors in these samples, it is still not sufficient to conclude that the correlation is accurate in all conditions. The proposed method is applicable to natural gases that contain high ${\mathrm{CO}}_2$, but it cannot be applied to high-pressure and high-temperature conditions, where compressibility factors are generally larger than 2.0. It is clear that more accurate experimental data would give rise to more precisely tuned coefficients and prediction accuracy. Nevertheless, even though this work is not sufficient, it provides an alternative correlation in calculating compressibility factors with fewer uncertainties for engineers in the oil and gas industry.

In recent years, the AI-based methods have been popular in the oil and gas industry. They can easily calculate the basic petrophysical properties, for example, compressibility factors, viscosity, density, etc. Therefore, our future research directions include the development of an AI-based model in predicting thermodynamic parameters of natural gases. In addition, it is important to study the relationship between the fluid flow properties (e.g., permeability) and reservoir formation parameters by AI techniques.