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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%.

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 [

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 [

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 [

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

A schematic diagram of the data acquisition system.

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

A schematic diagram of the installation positions of the sensors.

In summary, Figures

Overall data fitting pressure curve of the OD1422 X80 12MPa pipeline.

Overall data fitting pressure curve of the OD1422 X80 13.3MPa pipeline.

Overall data fitting pressure curve of the OD1219 X90 12MP pipeline.

The TNT equivalent method is typically used in simulations of a vapor cloud explosion [

where

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

where

Over time, countless scholars have outlined numerous prediction formulas pertaining to the relationship between the peak overpressure and the scaled distance of a shockwave [

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

where

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.

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 [

If

Alternatively, if

where

where _{S} is the distance to the explosion center in m;

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

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

The comparison of measured data and calculation results of the TNT equivalent method.

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

The distribution of values calculated by the TNT equivalent method combined with the British Gas method.

It is evident in Figure

The comparison of fitted curves of measured pressure with fitted curves of values calculated by the Mills formula and the Henrych formulae.

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

The errors of the theoretical calculation values of the OD1422mm pipeline.

Errors of Mills formula and Henrych formula of the OD1422mm X80 12MPa pipeline

Errors of Mills formula of the OD1422mm X80 12MPa pipeline

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.

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

The distribution of values calculated using the TNT equivalent method combined with British Gas method.

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

The comparison of the fitted curves of measured pressure with the fitted curves of values calculated by Mills formula and Henrych formula.

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.

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

A comparison of the calculated value of overpressure using the Lee formula and the measured value.

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

A comparison of the fitted curves of the values calculated using the Lee formula and the Mills formula, as well as the measured values.

From the pressure curve of the calculated values and the measured values shown in Figure

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

The errors of the theoretical calculation valuesof the OD1422 X80 13.3MPa pipeline.

Errors of Mills formula, Lee formula and Henrych formula of the OD1422 X80 13.3MPa pipeline

Errors of Mills formula of the OD1422 X80 13.3MPa pipeline

Errors of Lee formula of the OD1422 X80 13.3MPa pipeline

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 [

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

Distribution of values calculated by the TNT equivalent method combined with the British Gas method.

The calculated results obtained using the British Gas method corresponds with the measured values, as evident in Figure

A comparison of fitted curves of measured pressure with fitted curves of values calculated by the Mills formula and the Henrych formula.

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.

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 [

A comparison between the values of overpressure calculated by the Lee formula and the measured value.

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

A comparison of the fitting curves of the values calculated by the Lee formula, the values calculated by the Mills formula, and those of the measured values.

The pressure curves of the calculated values and the measured values, as shown in Figure

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

Relative errors of the theoretical calculation values of OD1219 pipeline.

Errors of Mills formula, Lee formula, and Henrych formula of OD1219 pipeline

Errors of Mills formula of OD1219 pipeline

Errors of Lee formula of OD1219 pipeline

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 [

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.

The trend comparison between the curves obtained by the Henrych formula and the fitting curve of measured values, as shown in Figure

A comparison of the trend between the fitting curve of the Henrych formula values and the fitting curve of the measured overpressure values.

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

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

Test I:

Test II:

Test III:

The approximate general error correction formula is obtained by merging the formulae (

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

where

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

Model error analysis after correction.

The measured value of Test 1

The measured value of Test 2

The measured value of Test 3

Scatter distribution of the error value of the three tests

As shown in Figure

(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.

No data were used to support this study.

The authors declare that they have no conflicts of interest.

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.