Size Effect of the Length-to-Diameter Ratio of Pipelines on the Methane/Air Explosion Pressure

Abstract

To overcome the shortcomings inherent in hydraulic fracturing technology, including water wasting and contamination, in situ methane explosive fracturing technology has been put forward. Yet the realization of this technology strongly depends on the creation of high explosion pressure. In this work, the relationship between the length-to-diameter (L/D) ratio of pipelines and the maximum explosion pressure of methane/air has been numerically explored and experimentally verified. The results show that there exists an optimal L/D ratio that can significantly increase the explosion pressure of methane/air by about 50%. For instance, under atmospheric (reference) pressure, the explosion pressure has been enhanced from 0.787 to 1.193 MPa by varying the L/D ratio from 10 to 20. Additionally, both the optimal L/D ratio and the maximum explosion pressure are dependent upon the initial pressure and basically follow a logarithmic and linear function of initial pressure by proposing two fitted equations. After the initial pressure has increased to 10.1 MPa, an optimal L/D ratio of 120 can cause an explosion pressure of up to 124.287 MPa. The methods of optimizing the pipeline L/D ratio applicable in the in situ methane/air explosive fracturing are thus proposed, which can provide theoretical insight for fracturing shales.

1. Introduction

Shale gas is an unconventional natural gas that is stored in shale-dominated rock formations. Compared with the conventional natural gas, the development of shale gas has the advantage of a long mining life and production cycle. − Currently, shale gas is mainly exploited by hydraulic fracturing technology, which utilizes high-pressure water streams doped with chemicals, thus resulting in large amounts of waste and pollution of water resources. − Several fracturing technologies have been developed, and each has its own pros and cons. A newly emerging in situ methane explosive fracturing technology, which relies on the instantaneous high pressure generated by the explosion of in situ methane in the fracture, is expected to attack this problem. However, the challenge with this technology lies in understanding the dominant influencing factors of enhancing the explosion pressure of the methane. Compared with other waterless fracturing technologies, in situ methane explosive fracturing also has a lot of advantages, similar to liquid nitrogen fracturing, including environmental friendliness, independence of water resources, high efficiency, etc. Moreover, liquid nitrogen fracturing suffers from disadvantages such as the high cost of storage and transport of liquid nitrogen similar to hydraulic fracturing technology.

The explosion pressure of methane/air can be critically dependent upon the additive contents. However, most of the previous works have paid attention to the inhibition or reduction of the explosion pressure of methane/air by adding inert substances since methane is readily susceptible to leakage and explosion during transport. For example, Chen has reduced the explosion pressure of methane by 28.01% using 7 wt % N₂ and 13 wt % CO₂. Pei has found that N₂ or CO₂ and ultrafine water mist can decrease the explosion pressure of methane by 69.8 and 79.4%, respectively. A new multicomponent powder formulation has been proposed and the amount of multicomponent powder for complete inhibition has dropped by 41.67% by combining it with 8 wt % N₂. These works have effectively reduced the methane/air explosion pressure by adding various types of inert substances, ,− thereby reducing the risk of methane leakage or explosion. However, in striking contrast with the inhibition, the motivation of the in situ methane explosive fracturing technology lies in creating sufficient explosion pressure to fracture the shale, and consequently new approaches are required to enhance the methane/air explosion pressure.

Nowadays, several effective approaches have been discovered to enhance the explosion pressure including varying the initial pressure and temperature, increasing the oxygen content, and adding combustion improvers. For example, increasing the initial pressure can significantly increase the explosion overpressure of methane, while decreasing the initial temperature also can slightly increase the pressure. , An extreme high pressure initial conditions of 373 K and 15 MPa can lead to a methane explosion overpressure as high as 100 MPa at the optimal methane concentration. Moreover, the purity of oxygen can also influence the methane explosion pressure, which can exceed 24 MPa in pure oxygen (3.4 times higher than the pressure in air). − Furthermore, energetic materials have been employed to enhance the explosion pressure. Li has substantially increased the methane explosion pressure to 30.64 MPa by adding the combustion improver Al and potassium permanganate at ambient temperature and pressure. On the whole, all of these methods can effectively increase the methane explosion pressure, , thus facilitating the fracturing effect of in situ methane explosion on shale.

In practice, the reaction vessels or pipelines of the in situ methane explosion are of different sizes, and some efforts have been conducted to explore the size or shape effect of the pipelines, but contradictory results have been obtained. For instance, Phylaktou has discovered that when the L/D ratio of a pipeline is reduced from 21.6 to 18.4, the explosion overpressure will increase from about 4.9 to 7.5 bar. In another work, it has been observed that the explosion index of hydrocarbon fuel–air mixtures is related to the pipeline L/D ratio and the explosion pressure increases with decreasing L/D ratio. , Di Benedetto has further concluded that the effect of the pipeline L/D ratio on the methane explosion pressure varies at different oxygen contents. These works indicate that the pipeline L/D ratio is negatively correlated with the explosion pressure of combustible gases within a certain range, and the role of the pipeline L/D ratio could be coupled with other influencing factors. However, some works reveal that the L/D ratio of pipelines is positively correlated to the explosion pressure of combustible gases. For instance, Cui has studied hydrogen-enriched methane explosions in two pipelines with L/D ratios of 75 and 120 and found that when the initial pressure is 0.156 MPa, the oxygen-rich methane will explode in the pipeline of 120 L/D ratio with a peak pressure of 3.536 MPa whereas in the pipeline of 75 L/D ratio with a peak pressure of only 0.181 MPa. And a work of propane explosions in pipelines with L/D ratios of 3, 5, and 7 has found that high-frequency overpressure oscillation occurs under an L/D ratio of 7, which leads to a pressure surge of 900 kPa.

Overall, the above studies emphasize that the explosion vessel indeed influences the explosion characteristics of combustible gases − and give a hint that there exists an optimized L/D ratio to maximize the explosion pressure. However, the influence of the pipeline L/D ratio on the explosion pressure of combustible gases solely or jointly depends on the type of combustible gas, the L/D ratio range, and other factors. The size effect of the reaction pipelines on the methane explosion pressure has not been fully understood. Therefore, a set of pipelines with a wide-span range of L/D ratios will be employed to fully understand the role of the size effect. In the studies mentioned above, the maximum aspect ratio is 120, so this work will simulate the methane explosion with L/D ratios ranging from 1 to 140 coupled with different pressures and temperatures to systematically explore the influence of the L/D ratio on the methane/air explosion pressure. It will be experimentally expensive to fabricate so many pipelines by changing the L/D ratio. So the numerical simulation of methane explosion in multisize pipelines has been carried out using the FLACS hydrodynamic simulation software. Multisize pipeline models are first established, and a variety of initial pressures and temperatures are set to investigate the influence of the pipeline L/D ratio on the methane explosion pressure. Then, the simulation data are analyzed to find the pipeline L/D ratio that can maximize the methane explosion pressure. Finally, the method of selecting an optimal pipeline L/D ratio under different initial pressures is proposed.

2. Models and Methods

2.1. Geometric Models

FLACS is a computational fluid dynamics (CFD) simulation software widely used in the field of natural gas explosions. It uses the finite volume method to solve the conservation equations of mass, momentum, enthalpy, turbulent kinetic energy, turbulent kinetic energy dissipation rate, fuel mass fraction, and mixture fraction on a 3D structured Cartesian grid. On this basis, the standard k-epsilon turbulence model is further used to simulate the turbulent flow, and the diffusive and convective fluxes are solved by a second-order scheme and a second-order kappa scheme, respectively.

Based on the mathematical model, geometric models were established. As shown in Figure , the models were simple pipeline structures, closed at both ends, with a wall thickness of one tenth of the diameter, and the ignition location was in the center of the end of the pipeline. The boundaries were all set to Eulerian boundaries to match the explosion simulation in a closed environment. The time steps of the simulation were set to default values (CFLC = 5 and CFLV = 0.5), which could ensure the stability of the explosion. And at each time step, the FLACS software outputted the maximum values of explosion pressure and flame velocity over the entire computational domain. It was able to observe the pressure at a specific location by setting a limited number of monitors. The total simulation time was contingent on the time when the explosion pressure does not change significantly. In addition, the geometric models could be divided into two categories according to the simulation variables. The first type of models had a fixed L/D ratio of 10, numbered sample 1–5, whereas the second type of models maintained the diameter of the pipeline as 10 cm, numbered sample 2, 6–17. Detailed model parameters are presented in Table .

1.

Geometric models.

1. Conditions of Fuels and Geometric Models.

sample	L (cm)	D (cm)	L/D	initial pressure (MPa)	initial temperature (K)
1	20	2	10	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
2	100	10	10	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
3	200	20	10	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
4	400	40	10	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
5	800	80	10	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
6	10	10	1	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
7	50	10	5	0.1, 0.6, 1.1, 2.1, 4.1, 8.1	293, 313, 333, 353, 373
8	150	10	15	0.1	293, 313, 333, 353, 373
9	200	10	20	0.1, 0.6, 1.1, 2.1, 4.1, 8.1, 10.1	293, 313, 333, 353, 373
10	250	10	25	0.1	293, 313, 333, 353, 373
11	300	10	30	0.1, 0.6	293, 313, 333, 353, 373
12	400	10	40	0.1, 0.6, 1.1, 2.1, 4.1, 8.1, 10.1	293, 313, 333, 353, 373
13	450	10	45	0.6	293, 313, 333, 353, 373
14	500	10	50	0.6	293, 313, 333, 353, 373
15	550	10	55	0.6	293, 313, 333, 353, 373
16	600	10	60	0.6, 1.1, 2.1	293, 313, 333, 353, 373
17	800	10	80	0.6, 1.1, 2.1, 4.1, 8.1, 10.1	293, 313, 333, 353, 373
18	1000	10	100	1.1, 2.1, 4.1, 10.1	293, 313, 333, 353, 373
19	1200	10	120	2.1, 4.1, 8.1, 10.1	293, 313, 333, 353, 373
20	1400	10	140	4.1, 8.1, 10.1	293, 313, 333, 353, 373
21	1600	10	160	10.1	293, 313, 333, 353, 373

To ensure the accuracy of the simulation calculations, it is also necessary to set the grid. In gas combustion and explosion simulations, the same cubic mesh must be used in the combustion region without stretching or refining it inside the combustion region, and the mesh can be stretched appropriately outside the combustion region. Considering that the geometric models were closed pipelines and the whole pipelines were combustion regions, the models were meshed with a homogeneous cubic mesh covering the entire pipeline, as shown in Figure . The edge length of the cubic mesh was a constant value of 0.5 cm, and the number of meshes varied with the volume of the models, ranging from 11520 to 1843200. And it was ensured that each model has at least 12 meshes in the smallest direction of the pipeline and at least 20 meshes in any direction inside the gas, even for a slender pipeline with an L/D ratio as high as 160.

2.

FLACS mesh model diagram.

2.2. Fuels and Parameters

A methane–air mixture with an equivalence ratio of 1.0 was chosen as the fuel to create a uniform distribution of the gas cloud throughout the pipeline area. The whole pipeline model has no heat exchange with the outside. The initial pressure and temperature settings for the two types of models were as follows: For the first type of models, six initial pressures of 0.1, 0.6, 1.1, 2.1, 4.1, and 8.1 MPa and five initial temperatures of 293, 313, 333, 353, and 373 K were set. The aim of these settings was to investigate the effect of a constant pipeline L/D ratio and changing diameter on the methane explosion pressure. The second type of model conditions consist of seven initial pressures of 0.1, 0.6, 1.1, 2.1, 4.1, 8.1, and 10.1 MPa and five initial temperatures of 293, 313, 333, 353, and 373 K. The aim of these settings is to investigate the effect of varying the L/D ratio on the methane explosion pressure as comprehensively as possible with a small simulation workload. The detailed conditions are listed in Table .

2.3. Reliability Verification

As the geometric models established in this work included numbers of multiscale pipelines, it was difficult to match them exactly with the models used in previous works. To ensure the reliability of the simulation methodology, methane explosions were simulated within pipelines under L/D ratios of 18.4 and 21.6. The working conditions were consistent with Phylaktou and Andrews. The monitoring points were also set in the center of the end of the pipeline. As shown in Figure , the maximum pressures are 0.752 and 0.538 MPa, corresponding to L/D ratios of 18.4 and 21.6, respectively; the maximum difference from Phylaktou’s results are limited to 0.002 and 0.048 MPa, respectively, with errors of 0.3 and 9.8%, respectively. In general, the deviation of the results is within acceptable limits. As the L/D ratio increased from 18.4 to 21.6, the methane explosion pressure decreased. A similar conclusion to Phylaktou’s work was reached.

3.

Pressure time history of the monitors with an initial pressure under different pipeline ratio of L/D = 18.4 (a), and 21.6 (b).

In order to verify some of the conclusions obtained in this paper, a series of explosion experiments at different initial pressures of 0.11, 0.13, 0.15, 0.17, and 0.19 MPa were also set up. As shown in the Figure a, the experiment mainly consists of a pipeline, igniter, data acquisition system, gas cylinder, vacuum pump, and controller. The length of the pipeline is 4 m, the inner diameter is 168 mm, and the L/D ratio is 23.8. The ignition location is the end of the pipeline, the delay is 500 ms, and the ignition time is 1500 ms. The sensors are set at 1.25, 1.75, 2.25, 2.75, 3.25, and 3.75 m from the end with a range of 0–100 MPa, a response frequency of 100 kHz, and a measurement accuracy of ± 0.25% F.S (Full Scale). The fuel is a methane/air mixture with an equivalent ratio of 1. The sequence control is shown in Figure b.

4.

Schematic diagram (a) and sequence control (b) of a methane pipeline explosion setup with a L/D ratio of 23.8.

3. Results and Discussion

3.1. The Maximum Explosion Pressure under an Optimal L/D Ratio

In situ methane explosion fracturing technology is mainly used to increase the explosion pressure of in situ methane in shale formation by adding combustion improvers to the pipeline and other methods. However, the shape of the pipelines may affect the methane explosion pressure. To guarantee the explosion effect of in situ methane, our work established two types of models to study the effect of pipeline shape on the methane explosion pressure.

As shown in Figure , with initial pressures of 0.1 and 8.1 MPa, with the increase of pipeline diameter from 2 to 80 cm, the maximum explosion pressure did not change significantly (specific dimensions of pipelines are shown in Table , samples 1–5). When the initial pressure was 0.1 MPa, the largest explosion pressure variation ratio was 0.9%; at the initial pressure of 8.1 MPa, the largest explosion pressure variation ratio was 0.06%. In addition, for other initial pressure conditions, the maximum explosion pressure variation was also small. Thus, it could be considered that in the first type of model conditions, only the change of the pipeline diameter on the methane explosion pressure was not significant. It is noted that the lower the initial temperature, the higher the maximum explosion pressure. A similar phenomenon has been observed elsewhere. The temperature increase determines the increase in the laminar burning velocity and the decrease in the maximum explosion pressure. At higher temperatures, the explosion reaction rate is higher, but the total chemical energy and mass of CH₄ in the mixtures will decrease with the initial temperature. The lower fuel gas content could result in a lower explosion pressure. As the temperature rises, the heat release speed and flame speed of gas combustion and explosion increase, so the explosion pressure declines during the combustion process.

5.

Maximum explosion pressure as a function of pipeline diameter under different temperatures and different initial pressures of p ₀ = 0.1 MPa (a), 8.1 MPa (b).

Figure illustrates the pressure time history under pipeline diameters of 2, 40, and 80 cm. The maximum explosion pressures are all around 0.8 MPa, while the explosion time at D = 80 cm is 62.96 times longer than that at D = 2 cm. Their pressure trends remain basically the same, and no pressure surges occur after reaching the maximum value. This might be because, in an adiabatic environment, changing the diameter of the pipeline within a certain range does not change the concentration of combustible gas per unit volume nor does it change the aggregation effect of the shock wave caused by the explosion on the combustible gas. The phenomenon that the pressure remains constant after reaching its peak is inconsistent with the actual physical laws in practical engineering scenarios because of the ideal adiabatic condition. However, our attention has been focused on the peak pressure value only. The duration of the high overpressure cannot last, and this leads to limited direct guiding significance for engineering applications.

6.

Pressure time history with an initial pressure of p ₀ = 0.1 MPa and an initial temperature of T ₀ = 293 K under different pipeline ratios of D = 2 (a), 40 (b), and 80 cm (c).

The change in the L/D ratio can trigger a surge in the methane explosion pressure. As shown in Figure , under an initial pressure of 0.1 MPa and a temperature of 293 K, when the L/D ratio is not greater than 10, the pipeline L/D ratio has no significant effect on the maximum explosion pressure; when the L/D ratio reaches 20, the maximum explosion pressure surges; and when the L/D ratio is 40, the maximum explosion pressure decreases slightly (specific dimensions of pipelines are shown in Table , samples 2, 8, 9, 10, 11, 12). This surge indicates that changing the L/D ratio affects the methane explosion pressure, but further research into the causes of this phenomenon is needed in a bid to boost the explosion pressure in this manner.

7.

Maximum explosion pressure as a function of different L/D ratios under different initial temperatures. The initial pressure is fixed at p ₀ = 0.1 MPa.

For a given volume of fuel and temperature, the flame can contact the side wall of high aspect ratio much earlier than those with a low L/D ratio, thus leading to a higher explosion pressure. However, with a further increase of the L/D ratio, the turbulence intensity of the flame front increases and the flame propagation speed becomes faster, thus resulting in an increase of the explosion pressure. After reaching the optimal L/D ratio, the inner space of the pipeline becomes much narrower, the resistance to flame propagation increases, the flame front is easily cooled by the pipe wall, and the combustion reaction rate decreases. On the other hand, in a confined space, heat and combustion products are difficult to disperse effectively, which may lead to incomplete combustion in some areas. In extreme cases, it could even cause the flame to be extinguished, ultimately weakening the intensity of the gas expansion and reducing the explosion pressure. The optimal L/D ratio is dependent upon the temperature and increases with increasing the temperature. As the temperature rises, the heat release speed and flame speed of gas combustion and explosion increase, so the explosion pressure declines during the combustion process and it takes a longer time and a larger L/D ratio to reach the maximum explosion pressure.

Figure shows the pressure time history of p ₀ = 0.1 MPa and T ₀ = 293 K under different L/D ratios derived from the simulation. It is shown that when the L/D ratio is 10, the explosion pressure surge does not occur and the smallest pressure value is reached, and the flame speed to reach the peak time becomes shorter; when the L/D ratio increases to 20, the pressure surge and the pressure value reach the largest, the pressure and the flame speed to reach the peak happen at the same time; when the L/D ratio reaches 40, the pressure surge phenomenon becomes weaker, and the pressure to reach the peak of the time required turns longer. The possible reason for the surge phenomenon is that the ignition of the gas mixture may produce a flame front and shock wave, thus driving the fuel to the rear end of the pipeline and resulting in an increase in the concentration of methane and consequently the explosion pressure.

8.

Pressure time history with an initial pressure of p ₀ = 0.1 MPa and an initial temperature of T ₀ = 293 K under different pipeline ratios of L/D = 10 (a), 20 (b), and 40 (c).

The underlying mechanism of the surge phenomenon can be better analyzed through the pressure and combustion product cloud maps, as shown in Figures and , respectively. Methane explosions in high L/D pipelines can be divided into four stages. In the first stage at 0–24 ms (Figures a–c and a–c), the fuel was ignited, and the flame covered 3/5 of the pipeline. However, the pressure in the pipeline did not change obviously, suggesting that the shock wave drove most of the unburnt fuel to the end of the pipeline and created more pressure there. In the second stage at 28–41 ms (Figures d–h and d–h), the flames continued toward the end of the pipeline, and the shock wave was reflected in the pipeline, causing pressure oscillations and movement of the high-pressure zones. And at 41 ms (Figures h and h), a high-pressure zone of nearly 0.7 MPa was formed at the front end of the pipeline, which would drive the gas at the front toward the end and increase the pressure there. In the third stage at 42 ms (Figures i and i), the return of gas at the front created a high-pressure zone at the end, while the remaining fuel was ignited at a high pressure, creating a pressure surge of up to 1.4 MPa. In the last stage at 44 ms (Figures j and j), all of the fuel had been burned and the resulting shock wave pushed the gas to move toward the front, also forming a high pressure of about 1.4 MPa. However, if the L/D ratio of the pipelines is too small, the shock wave will not have a significant effect on gas aggregation, which will not meet the conditions of the pressure surge. And if the L/D ratio of the pipelines is too large, the pressure wave will oscillate over a longer period, resulting in multiple high-pressure zones and the dispersion of pressure, which will allow aggregated fuel to explode at a lower pressure and will not meet the conditions of the pressure surge, either. It can be expected that there is a specific L/D ratio in a certain range that allows the fuel to collect in the rear part of the pipeline and explode at a high pressure, thus creating the highest overpressure, i.e., the optimal L/D ratio for methane explosions in pipelines.

9.

Pressure cloud map with p ₀ = 0.1 MPa and T ₀ = 293 K under a pipeline ratio of L/D = 20 at different times of (a) t = 0, (b) 16, (c) 24, (d) 28, (e) 33, (f) 35, (g) 39, (h) 41, (i) 42, and (j) 44 ms.

10.

Combustion product cloud map with p ₀ = 0.1 MPa and T ₀ = 293 K under a pipeline ratio of L/D = 20 at different times of (a) t = 0, (b) 16, (c) 24, (d) 28, (e) 33, (f) 35, (g) 39, (h) 41, (i) 42, and (j) 44 ms.

To further analyze the optimal L/D ratio and explosion pressure, the L/D ratios of the pipeline corresponding to the maximum explosion pressures were defined as the optimal L/D ratio (denoted as (L/D)_opt). As presented in Figure , the (L/D)_opt ratio was 20 for different initial temperatures of 293, 313, and 333 K at an initial pressure of 0.1 MPa. And the pressure rise value of the maximum explosion pressure at the (L/D)_opt ratio with respect to that at L/D ratios where no explosion pressure surge occurs is defined as RP. To quantitatively characterize the rate of pressure rise, an indicator of RP has been defined by eq

$$\mathit{RP}=\frac{p_{\mathrm{opt}}-p_i}{p_i}\times 100\%$$ 1

where p _opt and p _i are the maximum explosion pressures at the (L/D)_opt ratio and the L/D ratio where no explosion pressure surge occurs, respectively.

3.2. Effect of Initial Pressures and Temperatures on (L/D)_opt Ratio

Since pipelines at different depths in the shale fracturing environment are exposed to different initial pressures, the effect of the L/D ratio on the explosion pressure at different initial pressures must be further investigated. As shown in Figure , a methane explosion pressure surge occurs at all initial pressures, while the (L/D)_opt ratio of the explosion varies with the initial pressure but is almost independent of the temperature.

11.

Maximum explosion pressure as a function of L/D ratio under different initial temperatures at initial pressures of p ₀ = 0.6 (a), 1.1 (b), 2.1 (c), 4.1 (d), 8.1 (e), and 10.1 MPa (f).

At initial pressures of 0.6 and 8.1 MPa, the (L/D)_opt ratios are almost the same at all temperatures, whereas at initial pressures of 1.1, 2.1, 4.1, and 10.1 MPa, the maximum explosion pressures experience several maxima concurrent with one local minimum after increasing sharply with increasing the ratio. However, the ratios corresponding to either the maximum or minimum always decrease with increasing the temperature. Generally speaking, the optimized (L/D)_opt ratios increase with increasing the initial pressure from around 50 at 0.6 MPa, 60 at 1.1 MPa, 100 at 2.1 MPa to 120 at 8.1 and 10.1 MPa. But there exists one exception at 4.1 MPa as shown in Figure d, where the (L/D)_opt ratio at p ₀ = 4.1 MPa varies with changing the temperature. This fluctuating change might be caused by the limited simulated working conditions.

The explosion pressure increases upon increasing the initial pressure, while the increase in temperature reduces its value. It can be understood that in a definite amount of gas mixtures, at elevated pressures, gas molecules collide more frequently under high pressure, and the molecular distance is compressed. Therefore, heat transfer between molecules and combustion reactions is more likely to occur, while heat loss can be reduced, all of which promote the gas explosion. When using a cylindrical vessel to carry out an end-wall ignition explosion experiment, the unburned CH₄–air mixtures cannot be entirely consumed by the spreading spherical flame when it approaches the vessel’s inner wall. The explosion pressure obtained rises nearly linearly with the initial pressure. It is also noticeable that the relative explosion pressure generated at low temperature is higher than that at high temperature under similar conditions.

It has been discovered that the pressure rise rate is seemingly restricted by two factors based on the study of Gieras et al. The acceleration of the explosion rate is due to the rise of the CH₄–air mixtures temperature in the vessel and the deceleration of the rate generated by a reduction of the CH₄ mass during the explosion. Likewise, the pressure rise can also be influenced by the explosion rate. High temperature and high pressure can accelerate the explosion rate, whereas the high temperature can reduce the amount of CH₄, thus decelerating the explosion rate. Therefore, Figure d shows anomalous fluctuations with temperature, suggesting that temperature has a small-scale nonlinear coupling effect with pressure.

Considering that the initial pressure at different positions of the same pipeline in the shale fracturing environment may also vary with its depth. The initial pressure is within a certain range, and the (L/D)_opt ratio cannot be determined. To solve this problem, the RP of methane explosion in pipelines with different L/D ratios at each initial pressure needs to be explored. As observed from Figure , for a given initial pressure and given temperature, the maximum pressure initially does not change, followed by a gradual increase to a peak value and then a sudden drop. What’s more, for a given initial pressure, the L/D ratios corresponding to the maximum explosion pressure at different temperatures almost are independent of temperature but contingent upon the initial pressure only. Therefore, the RP at different initial pressures and L/D ratios can be obtained as per eq and listed in Table . For example, for a given initial pressure of 0.6 MPa and temperature of 293 K, the explosion pressures when L/D ratios fall within 0–40 are almost constant, around 4.75 MPa, but the maximum explosion pressure attains 7.17 MPa, and the corresponding RP is calculated to be around 50.93%. Correspondingly, the RP values at different initial pressures and L/D ratios are summarized in Table . And the data with RP greater than 40% are marked in red and bold to represent a significant effect of the methane explosion pressure increase. As shown in Table , the L/D ratio of 80 pipelines at the initial pressure of 1.1, 2.1, and 4.1 MPa could make the methane explosion pressure increase by more than 46.95%; the L/D ratio of 120 pipelines at the initial pressure of 2.1, 4.1, 8.1, and 10.1 MPa could make the methane explosion pressure increase by more than 47.15%. Consequently, in the initial high-pressure range of 1.1 ∼ 4.1 and 2.1 ∼ 10.1 MPa, respectively, the choice of L/D ratio of 80 and 120 pipelines is expected to increase the explosion pressure of methane by more than 40%.

2. RP under Different L/D Ratios and Initial Pressures at 293 K.

p 0 (MPa)	0.1	0.6	1.1	2.1	4.1	8.1	10.1
L/D
20	51.56%	0	0	0	0	0	0
40	16.70%	27.83%	2.10%	0	0	0	0
50		50.93%					
60		13.11%	62.32%	22.53%	0	0	0
80			48.10%	51.82%	46.95%	20.05%	0.00
100			60.39%	62.33%	34.69%		20.18%
120			30.66%	47.91%	53.67%	47.15%	50.47%
140					34.12%	38.40%	44.55%
160							35.46%

In addition, the (L/D)_opt ratio of the methane explosion increased significantly with increasing initial pressure. As shown in Figure , the (L/D)_opt ratios at a temperature of 293 K are 20, 50, 60, 100, 120, 120, and 120 when the initial pressure is 0.1, 0.6, 1.1, 2.1, 4.1, 8.1, and 10.1 MPa, respectively. Because increasing the initial pressure will increase the initial energy of the shock wave, it may still be able to provide enough energy to cause a surge in the explosion pressure in the pipeline with a larger L/D ratio.

12.

Maximum explosion pressure as a function of the L/D ratio pipeline under different initial pressures at an initial temperature of T ₀ = 293 K.

3.3. Selection of (L/D)_opt Ratio in an Explosive Fracturing Environment

By qualitatively analyzing the effects of initial pressure and temperature on the (L/D)_opt ratio, it was concluded that an increase in initial pressure would increase the (L/D)_opt ratio, while a change in temperature could cause a slight change in the (L/D)_opt ratio. To further predict the (L/D)_opt ratio in an explosive fracturing environment, the effect of initial pressure on the (L/D)_opt ratio and RP at 293 K was quantitatively analyzed. As shown in Figure , the (L/D)_opt ratio increases as the initial pressure increases, but the rate of increase slows and shows a logarithmic function trend. In addition, the rate of pressure increase varies at different initial pressures, but the difference is not large, with a maximum value of 62.32% and an average value of 54.06%. This also further indicates that by selecting the (L/D)_opt ratio of the pipeline, the maximum explosion pressure of methane could steadily increase by about 50%.

13.

(L/D)_opt ratio and RP as a function of initial pressures.

The rate of pressure rise can be fitted to an exponential function of the L/D ratio whereas the adiabatic explosion pressure is generally a linear function of the initial pressure. In addition, the intercepts of the fitting functions of the (L/D)_opt ratio and p _opt are all 0. Therefore, the (L/D)_opt ratio and the p _opt can be predicted by logarithmic and linear functions, respectively. The fitted correlation coefficients are 0.986 and 0.999, as shown in eqs and and Figure . Applying these two equations, it is now possible to predict the (L/D)_opt ratio and the maximum explosion pressure from the initial pressure, which may effectively enhance the explosion pressure of in situ methane.

$${(L/D)}_{\mathrm{opt}}=120\left[1-0.95\exp (-0.08\frac{P_o}{P_{\mathrm{ref}}})\right]$$ 2

$$p_{\mathrm{opt}}=12.247\times p_0$$ 3

where p ₀ is the initial pressure in MPa and p _ref is the reference pressure, p _ref = 0.1 MPa.

14.

(L/D)_opt ratio as a function of normalized pressures of P ₀/P _ref (a) and P _opt as a function of P ₀ (b).

In order to verify the equations obtained in this work, a series of explosion experiments with an L/D ratio of 23.8 were also set up. Through eq , it can be calculated that the initial pressure corresponding to the (L/D)_opt ratio of 23.8 is 0.157 MPa. As shown in Figure , the experiments resulted in a maximum p _max of 0.727 MPa and a maximum p _max/p ₀ of 4.279 at a pressure of 0.17 MPa. This indicates that at 0.17 MPa, the pipeline with an L/D ratio of 23.8 has a significant aggregation effect on methane and produces a strong high-pressure zone in the second half of the pipeline. And 0.17 MPa is the initial pressure corresponding to the pipeline of the (L/D)_opt ratio of 23.8 obtained by the experiment with only 8.3% difference from the predicted value. If the initial pressure is increased from 0.13 to 0.17 MPa, then the p _max/p ₀ will increase by a factor of 67.4% from 2.556 to 4.279. This shows that in the experiment the increase of explosion pressure by selecting the (L/D)_opt ratio could be also more than 50%.

15.

p _max and p _max/p ₀ under different initial pressures obtained from the experiment (a) and the pressure history under 0.17 MPa (b).

To further verify the equations, methane explosion at 0.17 MPa was simulated, too. The maximum pressure value obtained by the corresponding monitor point in the simulation is 1.056 MPa (Figure a), while the value obtained by the experiment is 0.727 MPa. The reason for difference may be that the monitor point cannot be set on the boundary in the simulation, but inside the pipeline. The maximum pressure of the whole computational domain of simulation is 1.853 MPa (Figure b). Through eq , it can be calculated that the maximum pressure under the (L/D)_opt ratio of 23.8 is 1.923 MPa with a 3.6% ratio difference from 1.853 MPa. However, this still cannot fully verify the accuracy of the equation of maximum explosion pressure in this work, because the sensor cannot be placed in the whole pipeline in the experiment, and the monitor points cannot be set at the grid boundary in the simulation. To sum up, the prediction formula for the (L/D)_opt ratio in this work is accurate at the initial pressure of 0.11–0.19 MPa. The percentage difference of the estimated initial pressure given the (L/D)_opt ratio between the prediction value of 0.157 MPa and the experimental value of 0.17 MPa is 8.3%. However, the prediction formula of the maximum explosion pressure is difficult to prove completely due to the limitations of sensors and observation points.

16.

Pressure history under initial pressure of 0.17 MPa from the monitor of simulation (a) and the whole computational domain of simulation (b).

In Liu ’s study of hydraulic fracturing, it was found that shale fracture formation requires at least a pressure of nearly 22 MPa. In situ methane fracturing technology, as an alternative or a supplement to hydraulic fracturing, must also be able to generate pressure above 22 MPa. For nonoptimal L/D ratio pipelines, only 17.01 MPa pressure can be generated under the initial pressure of 2.1 MPa, whereas by selecting the (L/D)_opt ratio pipeline of 100, a 27.635 MPa pressure surge can be generated (Figure c). If the combustion improver is added at the same time when (L/D)_opt is selected, the explosion pressure of in situ methane will be further increased. In summary, the selection of the (L/D)_opt ratio greatly improves the feasibility of in situ methane explosion fracturing technology and reduces the demand for a high-pressure environment. In addition, by predicting the (L/D)_opt of methane explosions, it is possible to avoid the selection of pipelines that produce a pressure surge, thus reducing the hazard of explosions when methane leaks. It is worth mentioning that both temperature and initial pressure play an important role in the fracturing. Engineering parameters such as well depth and formation, temperature can be correlated by empirical formulas. The temperature–pressure gradient is closely related to the geological environment in different regions. For example, to determine the distribution of methane at different depths, the pressure gradient with depth in the Sichuan Basin was 15 MPa/km, and the temperature gradient was 30 K/km. ,

$$T=T_0+\mathit{\Delta T}·d$$ 4

$$P=P_0+\mathit{\Delta P}·d$$ 5

where T is the temperature (K), T ₀ is the surface temperature (K), ΔT is the temperature gradient with depth (K/km), d is the formation depth (km), P is the pressure (MPa), P ₀ is the initial surface pressure (MPa), and ΔP is the pressure gradient with depth (MPa/km).

4. Conclusions

In this work, the surge phenomena of methane/air explosion pressure in pipelines with specific L/D ratios have been investigated by modeling complex multipipelines with L/D ratios ranging from 1 to 160. The influence of the L/D ratio on the methane/air explosion pressure was discussed under different initial pressures and initial temperatures. On this basis, methods for optimizing the L/D ratio of the pipeline to enhance the methane explosion pressure have been obtained. The main conclusions are as follows:

• (1)There exists an optimal L/D ratio to maximize the methane/air explosion pressure. For instance, for given initial temperature of 293 K and initial pressure of 0.1 MPa, the explosion pressure can be increased from 0.787 to 1.193 MPa by about 51.5% using a pipeline with an L/D ratio of 20.
• (2)The optimized (L/D)_opt ratio for methane explosion is contingent upon and proportional to the initial pressure, but the ratio of explosion pressure rise is maintained at about 50%. For a given initial temperature of 293 K, the corresponding (L/D)_opt ratio at different initial pressures of 0.6, 1.1, 2.1, 4.1, 8.1, and 10.1 MPa are 50, 60, 80, 120, 120, and 120, respectively.
• (3)The (L/D)_opt ratio of the pipeline and the maximum explosion pressure p _opt satisfy the logarithmic and linear relationship with the initial pressure (p ₀), respectively. And the proposed formulas are ${(L/D)}_{\mathrm{opt}}=120\left[1-0.95\exp (-0.08\frac{P_o}{P_{\mathrm{ref}}})\right]$ and p _opt = 12.239 p ₀, respectively. The formulas have been preliminarily verified by experiments.
• (4)For a given constant initial pressure, the two fitting equations can be used to select the (L/D)_opt ratio, which is expected to boost the explosion pressure by about 50%. In detail, when the initial pressure falls within the range of 0.6–4.1 and 2.1–10.1 MPa, the corresponding optimal L/D ratios are 80 and 120, respectively, and it is expected that the methane/air explosion pressure can be enhanced by more than 40%.

Methane/air explosion can generate sufficiently high pressure to fracture the shale, but the duration time is millisecond-level. In order to generate permeability enhancement effect, multistage pulse explosion fracturing technology will be employed to create the complex development of fracture network. On the other hand, in situ methane explosive fracturing technology can be utilized along with hydraulic fracturing technology to initiate the shale fracture and promote the development of the network. The assumption of the ideal adiabatic condition can lead to a phenomenon that is inconsistent with and even contrary to a practical scenario, and care should be taken in utilizing the derived formula in guiding application.

⊥.

Department of Engineering Mechanics, School of Civil Engineering, Southeast University, Nanjing 251119, China

W.S.: Data curation, investigation, funding acquisition, supervision, and review and editing. L.L.: Writingoriginal draft. H.W.: Manuscript editing and revision.

The authors declare no competing financial interest.