#### Abstract

As developments in natural gas pipelines increasingly incorporate higher grades of steel, larger diameters, and higher pressures, the consequences of an accident caused by leakage, explosion, or ignition become progressively more severe. Currently, major technical obstacles include the quantification of the impact of an explosion shock wave of a high-strength, large-diameter natural gas pipeline, and the selection of a reasonable shock wave overpressure model appropriate to the operating conditions. In this paper, six models of shock wave overpressure theories, namely, the TNT equivalent method, the TNO method, the multienergy method, the British Gas method, the Shell method, and the Lee formula, were compared and analyzed to determine their applicability. A shock wave model adapted to the characteristics of a full-scale test was proposed, and the model verification of a full-scale blasting test was conducted on pipelines with diameters of 1422 mm and 1219 mm, respectively. Subsequent results indicated that the modifications to the TNT equivalent and the test parameters correlated with changes in the suitability of the model. Henrych’s formula calculation model of the British Gas method was found to correspond strongly with the measured value, in which the absolute value of the relative error was less than 30% and the absolute error within the range of 78 m to 800 m was no more than 0.05 MPa. Thus, the Henrych formula was adopted as the primary model formula for the shock wave overpressure calculations in this study. To further correct the error of the model, the trend between the curve obtained by the Henrych formula and the fitting curve of the measured value was compared and analyzed. The positive and negative compensations of the shaded area before and after the intersection point were carried out, and the new error correction overpressure model formula was obtained by fitting, with the error controlled within 15%.

#### 1. Introduction

The shock wave hazard caused by a natural gas pipeline explosion is assessed through the present calculation and evaluation model primarily utilizing the explosive test data [1]. Different criteria should be taken into account when evaluating the damaging effect of an explosive shock wave, and there are six commonly used models for assessing the hazardous impacts of a vapor cloud explosion: the TNT equivalent method, the TNO method, the multi-energy method, the British Gas method, the Shell method, and the Lee formula [2]. These are briefly described as follows.

The general principle of the TNT equivalent method is the conversion of the mass of the vapor cloud into its equivalent in TNT. In general, however, since only a small portion of the heat from fuel burning is expressed in the form of shock waves, this method assumes that only a certain proportion (1% to 10%) of the fuel contributes to the formation of overpressure [3]. The TNO method signifies the explosion of hemispherical gas clouds on the ground, calculating their combustion energy to obtain the explosive overpressure at a certain distance by the typical explosion length [4]. The third, multienergy method emphasizes the effect of obstacles and boundary constraints on the consequences of the explosion. This method divides the vapor cloud into several volumes according to the degree of gas limitation, with each volume corresponding to an explosion strength index (1to10) [5]. An explosion strength index of 1 indicates an unconfined or unobstructed vapor cloud explosion, while 10 correspond to vapor cloud detonation. The British Gas method improves on the TNT equivalent method [6] by calculating the explosion consequences of nondetonation natural gas clouds (methane gas with a content of ethane below 5%). This method accepts that an overpressure of 400 kPa will be generated in the identified explosion area, the efficiency coefficient of TNT equivalent method will be increased to 0.2, and the coefficient of the energy ratio will be 10. The fifth, Shell method suggests that the previously mentioned methods underestimate the effect of an explosion in the far field. The general principle of the Shell method is the same as that of the multienergy method, although it estimates the explosion overpressure using an inverse proportional attenuation law [7]. Lee investigated the mechanism involved in flame acceleration and subsequently proposed a formula for calculating the peak value of overpressure when a gas cloud detonates on the ground [8].

These six empirical models differ significantly in the near field but tend to be consistent in the far field. Consequently, specific test conditions were established for a preliminary screening of these models of this study: the gas cloud produced in this explosion test will be approximately spherical and relatively high, and the sensors erected will also be relatively high from the ground and, therefore, the ground reflection pressure is not a consideration. Meanwhile, as the TNO model describes the explosion of a hemispherical gas cloud, in which the ground reflection pressure needs to be considered, it is regarded as unsuitable for the objective of this test. The explosion created in this test will be a gas cloud explosion in an open space, with no obstacles or local constraints set in the test field. The multienergy and Shell models both emphasize the effects of obstacles and boundary constraints on the consequences of the explosion and are consequently both considered unsuitable for the theoretical analysis of this test data [9]. This screening, based on the specific test conditions, initially identified three theoretical models for the calculation of the detonation overpressure: the TNT equivalent model, the British Gas model based on the TNT equivalent method, and the Lee model [10]. The results of the full-scale test will, therefore, be verified and the data regression analysis conducted according to the algorithms of these three models.

#### 2. Full-Scale Experimental Design

The site of the test explosion is located in Hami, Northwest China. The experiment was conducted in cooperation with the PetroChina West Pipeline Company. This company is responsible for the development of the natural gas cutting and ignition linkage device, and the CHDL-I digital electronic detonator initiation system employed in performing the blasting test. For the test, FPG, and FPT type piezoelectric pressure sensors were selected (manufacturer: Wuxi Yutian Technology Co., Ltd., origin: Wuxi, Jiangsu), and STYV-2 low-noise cable (manufacturer: Beijing Kunxing Shengda Electronic Technology Co., Ltd., origin: Beijing, similarly hereinafter) was used as the pressure test signal line. An SYV-5 coaxial cable was used for the trigger signal line and a KWR cable for the power supply. Damage to the cables in the harsh environment was prevented by burying them as protection against elements like gravel, heat radiation, and mechanical vehicles. The entire data acquisition system includes a sensor, signal lines, a signal conditioner, a data collector, instrument communication lines, and other ancillary equipment, as shown in Figure 1.

A three-dimensional space field test scheme was adopted for this test. The explosion center of the pipeline served as a circular point from which the sensor mounting rod array was arranged at intervals of 90° in four directions. The sensors were installed in each direction at respective distances of 50 m, 100 m, 150 m, 200 m, 250 m, 300 m, and 400 m, from the explosion center, as shown in Figure 2. Sensor mounting rods were installed at the height of 30 m and higher in order to eliminate ground reflection and to test the shockwave overpressure produced by the deflagration of the gas cloud at a high altitude. The arrangement of the sensors and mounting rods is illustrated by Figure 2.

In summary, Figures 3, 4, and 5 represent the overall fitting curves of the OD1422 X80 12 MPa pipeline, the OD1422 X80 13.3MPa pipeline, and the OD1219 X90 12MPa pipeline, according to the data collected from the four directions, namely, southeast, northeast, southwest, and northwest, respectively.

#### 3. Parameter Determination of the Typical Explosion Overpressure Model

##### 3.1. TNT Equivalent Method

The TNT equivalent method is typically used in simulations of a vapor cloud explosion [11–13]. The principle of this model is to promote the equilibrium between the explosion shock wave energy produced by the unconfined gas cloud explosion and the TNT of equivalent energy. This method was used to verify the test data initially. The calculation formula is as follows:

where represents the detonation heat of TNT in MJ/kg, generally valued at 4.2 MJ/kg; is the combustion heat of methane in MJ/kg, the verified value of which is 55 MJ/kg; is the total leakage of fuel in kg; is the energy release rate, valued between 0.02% to 14.9%; and A is the ground explosion reflection coefficient, which is usually 1.8. However, since the position of the test point was relatively high off the ground, the reflection effect could be ignored, subsequently valuing the reflection coefficient at 1.

The incident shock wave overpressure caused by an explosion can be expressed by scaled distance as

where represents the distance between the test point and the explosion center in meter.

Over time, countless scholars have outlined numerous prediction formulas pertaining to the relationship between the peak overpressure and the scaled distance of a shockwave [14, 15]. For this study, two sets of prediction formulas were selected after screening [16], namely, that of Henrych and Mills. Henrych obtained the following peak overpressure formula for a test shock wave:

By combining the similarity theory and numerical simulation, Mills obtained this expression of the overpressure of a TNT explosion shock wave:

where is the scaled distance. Certain essential differences should be noted between a gas cloud explosion and a TNT explosion. First, the volume of the explosive source can be ignored when TNT explodes, while the volume of a gas cloud is too large to be ignored, and the volume of the explosion source increases as the explosion proceeds. Second, the energy released by TNT is instantaneous, while the energy release rate is limited during the process of a gas cloud explosion. Third, the shock wave strength of a TNT explosion process is large and the attenuation fast. Conversely, a gas cloud explosion is mostly a deflagration process, which demonstrates a decreased positive pressure action time as opposed to an extended negative pressure time [17]. Therefore, the TNT equivalent method can only be applied to particularly strong gas cloud explosions. The deviation is small for describing the far field, while it is large for describing the near field.

##### 3.2. British Gas Model Based on TNT Equivalent Model

The British Gas method is used to calculate the explosion consequences of non-detonation natural gas clouds (methane gas with a content of ethane below 5%), mainly for natural gas media. It is considered an improvement on the TNT equivalent method. The method considers that an overpressure of 400 kPa will be generated in the identified explosion area, the efficiency coefficient of the TNT equivalent method will be increased to 0.2, and the coefficient of the energy ratio will be 10. The Henrych shockwave peak overpressure test formula and the Mills method, which combines the similarity theory and numerical simulation, can be effectively utilized for shockwave prediction.

##### 3.3. Lee Calculation Model

Lee’s experimental study on the mechanism of flame acceleration led to a formula for calculating the peak value of overpressure when a gas cloud detonates on the ground [18–20]. The Lee formula can be described as follows.

If ,

Alternatively, if ,

where is the peak overpressure value in bar;

where is the contrast distance and* R*_{S} is the distance to the explosion center in m; is the total energy released in J, and is the local atmospheric pressure in bar.

#### 4. The Comparison and Analysis of the OD1422mm Pipeline Blasting Test Results and Subsequent Theoretical Calculations

##### 4.1. OD1422mm X80 12MPa Pipeline Blasting Test

Assuming that all natural gas in the pipeline, calculated to be 80 thousand cubic meters, is leaked into the atmosphere, the total amount of fuel would be approximately 57,143 kg. This energy release rate is taken as 3%, which is based on data from a large number of gas cloud explosion accidents that reported the energy release rate to be mostly between 3% and 4%. The TNT equivalent calculation was employed in combination with the Henrych and Mills overpressure calculation formulae. This process subsequently allowed for the overpressure values to be obtained under different TNT equivalents at varying positions in relation to the explosion center.

The overall pressure values were compared with the calculation results of the TNT equivalent method, as shown in Figure 6.

This comparison shows the measured value to be higher than the calculated value, indicating that the TNT equivalent method is not applicable. In contrast, the results obtained by using the British Gas method based on the TNT equivalent model were found to be relatively ideal to calculate the explosion consequences of natural gas cloud (methane gas with a content of ethane below 5%). The explosion of gas clouds is a non-ideal explosion source. As a result, no fixed relationship is evident between the total energy of the explosion source and the explosion wave effect of the open space gas cloud explosion. Thus, only a portion of the energy is applied to produce the explosion effect. In general, the energy contribution rate of a gas cloud explosion is between 0.1% and 20%, while the British Gas method considers that the efficiency coefficient in the TNT equivalent method increases to 0.2, and the coefficient of energy ratio become 10 if the overpressure of 400 kPa is produced in the identified explosion area.

In this study, the TNT equivalent of the first OD1422 pipeline blasting test was calculated to be approximately 151,020 kg. Since the test point at 50 m was just within the range of the fireball, it was considered to be inside the identified explosion area, and the overpressure value of the test point was more than 400 kPa, which complies with the calculation requirements of prescribed by the British Gas method. A comparison with the measured pressure was conducted a second time, as shown in Figure 7.

It is evident in Figure 7 that the calculated results obtained by the British Gas method are similar to the measured values, but the results obtained using the Mills formula revealed a closer relationship to the measured values. The connection between the calculated values and the measured values was further analyzed by fitting the curves of the measured values, the values calculated by Henrych formula, and the values calculated by Mills formula and is presented in Figure 8.

It is clear that the curve fitted using the Mills formula is more consistent with the variation trend of the measured value. The variation trend of the curve tends to be more consistent over a longer distance. The fitted curve obtained through the Henrych formula intersects with the curve fitted by the measured value. The prediction of the Mills formula curve is substantial, whereas the Henrych formula curve is small in the near field. Based on the measured values of the test, the errors in both the Mills formula and the Henrych formula were analyzed, as presented in Figure 9.

**(a) Errors of Mills formula and Henrych formula of the OD1422mm X80 12MPa pipeline**

**(b) Errors of Mills formula of the OD1422mm X80 12MPa pipeline**

**(c) Errors of Henrych formula of the OD1422mm X80 12MPa pipeline**

This analysis highlighted the errors in the calculation results obtained by applying the two formulas as being relatively large at both ends, with a maximum relative error as high as 35%. The outcome is attributable to the fact that the measured pressure value at the far end was relatively small, and the relative error could not directly reflect the rising trend between the fitted curve of the calculated value and the fitted curve of the measured value. However, the results of the first blasting test revealed that the prediction results calculated using the British Gas model based on the TNT equivalent model were more consistent with the measured results. Furthermore, the calculation model based on the Lee formula offered a certain amount of applicability. Therefore, the two models were still applied for comparative analysis and further correction in the second and third tests.

##### 4.2. OD1422 X80 13.3MPa Pipeline Blasting Test

The TNT equivalent in the second OD1422 pipeline blasting test was approximately 166,079 kg. As the test point at 50m was just within the range of the fireball, it was considered to be inside the identified explosion area. The overpressure value of the test point was 611 kPa, larger than 400 kPa, thus complying with the calculation requirements of the British Gas method. Backstepping was conducted for the TNT equivalent to eliminate interference from inaccurately measured values inside the fireball. The process was managed according to the value obtained at the far field measurement point, based on the principle that the peak value of the shock wave overpressure is the same at an equidistant point. All values measured at a 250 m horizontal distance from the explosion center were selected for calculation, and an average overpressure of 0.0338 MPa was obtained. The TNT equivalent obtained by backstepping was 142,000 kg, according to the calculation formula. When the actual distance between the test point and the fireball center was taken into account, the height of the fireball center was 90 m, the height of the test point was 30 m, and the actual distance was 257 m. The recalculated TNT equivalent was 153,500 kg, and the corresponding efficiency coefficient was 0.185, which is very close to 0.2. Therefore, the selection of the efficiency coefficient was considered reasonable for this method and corresponded with the maximum safety range principle. A comparison with the measured pressure was conducted, presented in Figure 10.

This comparison indicates that the results obtained by the British Gas method correspond with the measured values. To further analyze the degree of similarity between the calculated values and the measured values, the curves of the measured values, the values calculated by Henrych formula, and the values calculated by Mills formula were fitted as shown in Figure 11.

The curve fitted according to the Henrych formula for this test is more consistent with the variation trend of the measured values, especially in the near field, and intersects with the curves fitted by the measured values employing the Mills formula. The results obtained utilizing the Mills formula indicated a more substantial trend than those of the Henrych formula, which is smaller than that of the measured value in the near field. There is little difference in the far field and shows a closer resemblance to the measured overpressure value.

##### 4.3. Comparison and Analysis of the Lee Calculation Model and Test Results

To further analyze the law of overpressure vibration in the gas cloud explosion, the theoretical model was tested, and it was ascertained whether the gas cloud has a detonation phenomenon. The explosion overpressure was calculated using the Lee calculation formula when the gas cloud detonated.

The Lee model was used to calculate the comparison between the values. The distribution law of the overpressure scatter of the measured values is shown in Figure 12. It is evident that the calculated value is very close to the measured value at 100 m, and the calculated value is evidently higher and differs significantly from the measured value after a distance of 100 m.

The fitted curves of the calculated values and test values were further compared, as presented in Figure 13.

From the pressure curve of the calculated values and the measured values shown in Figure 13, it is clear that the values calculated using the Lee formula in the far field are closer to those measured by the Mills formula. However, the values measured by the Lee formula in the near field are notably different from the measured values and it is apparent that this formula is based on the occurrence of gas detonation. This test presented no detonation, but rather a Chapman Jouguet (C-J) deflagration. Therefore, the curve fitted by the measured value may be considered most reasonable.

##### 4.4. Error Analysis and the Comparison Results of the Theoretical Models

Further analysis attempted to promote a better understanding of the above results and the applicability of the model. An error analysis of the values calculated by the Mills formula, those calculated by the Henrych formula in the British Gas, and the Lee formula calculation model-based calculated values was conducted based on test measurement values. This process is represented by Figures 14(a), 14(b), 14(c), and 14(d).

**(a) Errors of Mills formula, Lee formula and Henrych formula of the OD1422 X80 13.3MPa pipeline**

**(b) Errors of Mills formula of the OD1422 X80 13.3MPa pipeline**

**(c) Errors of Lee formula of the OD1422 X80 13.3MPa pipeline**

**(d) Errors of Henrych formula of the OD1422 X80 13.3MPa pipeline**

To maintain consistency with the distance from the test point to the explosion center in the 1219mm pipeline blasting, the error analysis was started at 78m. From the error analysis, it is evident that the error for results calculated by Mills formula in the British Gas method is relatively large in the near field. Therefore, it is indicated that the TNT equivalent method overestimated the gas explosion in the near field. However, the error decreases rapidly as the distance increases. The error acquired through the Henrych formula in the British Gas method is less than 30% in both the near and far fields in this blasting test. Nevertheless, the prediction value in the near field is lower than the measured results, which is inappropriate for hazard range assessment and definition. The relative error measured by the Lee formula is smaller in the near field, increasing as the distance increases, which is the opposite of the variation trend of error calculated by the Mills formula and is too large in the far field. The comprehensive analysis conclusively shows that the results of this test are in proper compliance with the requirements of the Henrych formula [21].

#### 5. Comparison and Analysis of OD1219 Pipeline Blasting Test Results and Theoretical Calculation

##### 5.1. Comparison and Analysis of the British Gas Model and Test Results

The British Gas method based on the TNT equivalent model was used to calculate the explosion consequences of a natural gas cloud (methane gas with an ethane content below 5%). When the same method was applied to verify the coefficient of efficiency and the coefficient of energy in OD1422, they were determined to remain consistent at 0.2 and 10, respectively. Testing revealed the TNT equivalent to be approximately 87,127 kg.

The calculated value by the British Gas model is compared with the measured pressure, and the scatter distribution is shown in Figure 15.

The calculated results obtained using the British Gas method corresponds with the measured values, as evident in Figure 15. Further investigation was conducted into the degree of similarity between the calculated values and the measured values. The curves of the measured values, the values calculated by the Henrych formula, and the values calculated by the Mills formula were all fitted and are shown in Figure 16.

A pronounced difference is evident in the variation trend between the curves fitted by the two formulas and those of the measured values, both in the far and near fields. The predicted values by the Henrych formula are lower in the near field than the measured values, while the predicted values by the Mills formula are higher in the near field than the measured values. Their predicted values in the far field are both higher than the measured values, although the differences between them are not very distinct. The measured values were closest matched by the Henrych formula values.

##### 5.2. Comparison and Analysis of the Lee Calculation Model and Test Results

The theoretical model was tested to verify whether a detonation phenomenon existed within the gas cloud, to allow for further examination regarding the law of overpressure vibration in the gas cloud explosion. The explosion overpressure generated from the gas cloud detonation was calculated by the Lee calculation model [22, 23]. A comparison concerning the distribution law of overpressure of measured values is presented in Figure 17.

Furthermore, the fitting curves of the calculated values and test values were compared, as shown in Figure 18.

The pressure curves of the calculated values and the measured values, as shown in Figure 18 indicate that the Lee formula values in the near field are closer to the measured values than the Mills formula values. In the far field the Lee formula values are clearly more extensive, and the Mills formula values are closer to the measured value. Moreover, these results confirm that instead of detonation the test displayed the C-J deflagration [24, 25].

##### 5.3. Error Analysis and Comparison Results of the Theoretical Model

To better analyze the above fitting results and the applicability of the model, an error analysis was performed on the values calculated by the Mills formula, the Henrych formula, and those calculated by the Lee formula. The error analysis was based on the test measurement values, as shown in Figure 19.

**(a) Errors of Mills formula, Lee formula, and Henrych formula of OD1219 pipeline**

**(b) Errors of Mills formula of OD1219 pipeline**

**(c) Errors of Lee formula of OD1219 pipeline**

**(d) Errors of Henrych formula of OD1219 pipeline**

From the error analysis, it can be seen that the results calculated by the Mills formula and the Henrych formula, based on the British Gas method, correlate with the measured overpressure values. However, the average relative error between this test data and the results calculated by the Henrych formula is smaller, with a maximum relative error no larger than 30%, while the Mills formula error is more significant in both the near and far fields. As with the comparison results of the OD1422 mm blasting test, the prediction values in the near field are lower than the measured values, making this result unfavorable to the damage consequences assessment because it does not conform to the maximum safety range principle, despite the Henrych formula being coincidentally better. An error of nearly 30% in the far end can be considered relatively large. This result can be attributed to the fact that the measured pressure values in the far end are reasonably small, and the relative error cannot directly reflect the approaching trend between the fitting curves of the calculated values and the measured values. It remains necessary to further improve the calculation model in order to reduce the overall relative errors.

Additional reasons for the errors between the measured overpressure values and the theoretical calculation values include the following:

(1) As the measured overpressure is usually obtained by data processing software, such as Origin and Matlab, a certain amount of errors exists in the consequent readings. In particular, the error will be higher when the value of the overpressure is smaller.

(2) Data errors can be caused by a multitude of factors including a complex flow field that can severely impede data collection. Moreover, the explosion range is vast with multipoint phenomena present in the explosion process that exerts influence on the gas cloud. Moreover, since there are also variations in the data obtained from the test, systematic errors are inevitable in the data fitting.

(3) The TNT equivalent method is an empirical formula based on the explosion of the condensed phase explosive, and the British Gas method is based on the TNT equivalent empirical formula, in which the explosion of the condensed phase explosive can be approximated to a point explosion, while the explosion of the gas cloud is an explosion in volume. As there are considerable differences in their physical mechanisms and hazard effects, errors are likely to occur, especially in the far field. Therefore, the theoretical calculation methods of overpressure require further data correction [26–28], as will be discussed in the next section.

#### 6. Correction and Determination of Theoretical Calculation Model

##### 6.1. The Selection of a Correction Formula

From the data analyses and theoretical model verifications in the two OD1422 blasting tests and one OD1219 blasting test, it was found that the consistency of the models applied were altered along with changes in the TNT equivalent and the test parameters. In the first OD1422 blasting test, the Mills formula in the British Gas method was found to be more suitable for predicting the measured results of TNT equivalent, although the consistency of the Henrych formula was also relatively sound. In the second OD1422 blasting test and the OD1219 pipe blasting test, the British Gas method combined with the Henrych formula was found to be congruent with the measured values. The absolute value of relative error was not higher than 30%, and the absolute error from 78 m to 800 m was not higher than 0.05 MPa. The only shortcoming was the intersection with the fitting curve of the measured values. However, the Henrych formula in the British Gas method was more stable in the overall analysis of the three tests, making it the most appropriate for correction.

Based on the above analysis, the consistency of the Henrych formula is considered more stable than the other models, with a crossing point depending on the relative parameters. Therefore, the correction equation is calculated to conduct positive and negative compensations for parameters in the Henrych formula, both in near and far field. This allows for more consistent results and an error reduction to less than 15% in the middle of the measured values curve and the calculated values curve in the three tests.

##### 6.2. Correction Method and Results

The trend comparison between the curves obtained by the Henrych formula and the fitting curve of measured values, as shown in Figure 20, suggests that positive and negative compensations need to be conducted for the shaded area before and after the intersection points of both.

The analysis suggests that the variation of relative error with distance can be applied as a parameter for the correction of the Henrych formula. These specific correction steps are as follows: analysis of the relative errors of each test result; selection of error data fitting methods and analysis of fitting results; analysis of the results corrected by each fitting formula and fine adjustments to the error fitting formula; nondimensionalization of the distance of each error fitting formula, considering the difference of gas parameters; merging of the three error fitting formulas; re-correction of the calculation results of the Henrych formula of each fitting by the merged error fitting formula; error analysis after correction; and obtaining the corrected Henrych formula.

Analysis of the error data indicates that conducting quadratic polynomials for the error values can achieve better results. The error correction formula can be obtained by adding 1 to the error fitting results. By comparing the measured results, the error correction formulae after the fitting by fine adjustment are as follows:

Test I:

Test II:

Test III:

where is the horizontal distance to the explosion center and is the relative error between the measured values and the calculated values at different distances of the test.

To obtain a unified error correction formula that could be applied to all three tests, the above three formulae need to be combined and fitted. However, as the total amount of gas released from the three tests varies, the TNT equivalent is also different. Therefore, the above formulae need to be combined, in which case can be converted to scaled distance , where . The following error correction formulae are obtained:

Test I:

Test II:

Test III:

The approximate general error correction formula is obtained by merging the formulae (10), (11) and (12) as follows:

The error correction formula is introduced into the third item in the Henrych formula to obtain the following model calculation formula:

where and .

##### 6.3. Error Analysis after Correction

The calculated value of the model was compared to the measured value to verify the accuracy of the corrected model. The relative error obtained by the comparison of the three tests is shown in Figure 21.

**(a) The measured value of Test 1**

**(b) The measured value of Test 2**

**(c) The measured value of Test 3**

**(d) Scatter distribution of the error value of the three tests**

As shown in Figure 21, the absolute values of relative errors in the first test are the largest, at approximately 16%, and the absolute value of relative errors in the second and third test are both less than 15%. The correction formula of this model can be considered to have met the expected requirements.

#### 7. Conclusion

(1) The various degrees of applicability of the six traditional models of shock wave overpressure theory were analyzed. A shock wave model method was adapted to suit the characteristics of the full-scale test and presented in line with the properties of natural gas blasting. The TNT equivalent model, the British Gas model, based on the TNT equivalent method, and the Lee model for detonation overpressure were determined to conform to the test situation.

(2) Based on the analysis of the full-scale test data, the Henrych formula was corrected and a new formula pertaining to the calculating model of overpressure theory was obtained.

(3) In an investigation into the correction model of the Henrych formula, it was verified that the absolute value of the relative error of the first test was the largest, at approximately 16%, while the absolute values of the relative errors of each position in the second and third tests were less than 15%. These results confirm that this model correction formula meets the requirements for industrial use and can, therefore, provide value to the field.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study is financially supported by “National Key R&D Program of China (2017YFC0805800)” and National Natural Science Foundation of China (51874322). The help from PetroChina West Pipeline Company by Xihua Min and Jian Liu in the implementation of the experiment is appreciated.