Numerical simulation on the leakage and diffusion of the natural gas of the underground pipeline in the soil

For the transportation pipeline with the diameter 1.219 m and the internal pressure 3-12 MPa, the leakage and diffusion of the natural gas of the pipeline in the soil was studied by numerical simulation. First, the finite model containing both pipeline and soil was established, and the porous media was used to simulate the real soil environment. And, the leakage amount of the natural gas was calculated at the cases with different porosities, the pressures inside the pipeline and the diameters of leakage port. Based on the classical theoretical leakage model of small hole in the air and simulation results analysis, the formula of the leakage amount was modified by the soil coefficient α to be suitable for soil environment. Then, the variation trends of the diffusion concentration of methane along different directions in the soil were analyzed by simulation, the influences of internal pressure of pipeline, the diameter of leakage and the porosity of soil were also discussed. Besides, the relationship of the safety distance with time was obtained.


Introduction
During the transportation process of natural gas, the leakage and diffusion of natural gas are the major accident related to personnel safety, especially for the situation of high-pressure and large-diameter pipelines. Hence, the regularity of the leakage and diffusion of natural gas have been attracted the high attention of scholars.
Assuming that the flow process was isentropic adiabatic and reversible, Ahn et al. [1] established the leakage models of hole and pipe, which distinguished the differences between the critical and noncritical flow inside the pipe and at the hole, and proposed the calculate formulas of the leakage amount. Wang et al. [2] divided the accident of the transportation pipeline into three stages, i.e., leakage, diffuse, combustion/explosion, and calculated the leakage amounts of small hole, large hole and pipe by the assumption of the isentropic adiabatic flow. It was found that when the ratio d/D between with the diameters of leakage hole and pipe was smaller than 0.2, the prediction accuracy of small hole model was well; for d/D>0.8, the prediction of the pipe model was also in good agreement with the experiment data, and the prediction accuracy of the larger hole model was well when d/D is only between 0.2 and 0.8. Feng et al. [3] studied the effect of the pressure stability on the leakage of natural gas, and it was found that when leakage hole was enough small, the leakage did less influence on the parameters inside the pipe, so the leakage process was considered as stable. Zhang et al. [4] studied the diffusion behaviors of refined oil in soil, and the effects of different soil properties were taken in account. Li et al. [5] researched the relationship between the axial velocity and the jet distance where the leakage occurred at the aerial transportation pipeline, the effects of gravity and wind speed were also discussed.
Many great progresses have been made in the field of the leakage at aerial pipeline and the following diffusion of natural gas, but there are few reports about the leakage and diffusion of the natural gas of the buried pipelines in the soil, especially those pipelines with high internal pressure and large diameter. In this paper, the regularity of leakage and diffusion of the natural gas of the buried pipeline with high pressure and large diameter was studied using the numerical simulation method, whose results are great significance for prevention and control of leakage accident.

The theoretical model of the leakage of small hole
When the transportation pipelines work under the normal atmospheric condition, there is classical theoretical model for the leakage amount of small hole. Figure.1 shows the leakage of the transportation pipelines, the point 1 is the upstream of the pipeline, the point 2 is the leakage port, and the point 3 is on the central axis of the pipeline corresponding to the leakage port. The leakage amount of small hole is below: where, A is the area of leakage port, M is Molar mass, k is adiabatic index, n is the compact factor of natural gas [6], and C0 is the modified coefficient. Generally, C0 can be taken as 1.0, 0.95 and 0.90 for the different shape of the leakage port, i.e., circular, triangle, rectangle respectively. By differentiating Eq. (1), the pressure P2c at critical flow state is: Then, the leakage volume of this port is expressed as:  Figure.2 shows the geometry model containing the transportation pipeline and the soil, whose length and wide are both 8 m, and the pipeline with diameter D = 1.2 m is 1.5 m below the surface of the soil. The normal of the leakage port is along the Y-axis positive direction, and the location of this port is at the center of the pipeline. Duo to the pressure gradient, the diffusion behaviors of gas mainly process along the Y-axis positive direction, and the diffusion along the negative direction is slow and little at initial short time. Besides, the leakage port is away 5 m from the positive boundary (y=5 m), which is enough long to avoid the influence of the boundary. In order to balance the calculation accuracy and computing cost, the maximum size of element is 256 mm, as shown in Figure.3. The maximum element size of the port is 1/3 of its diameter, and the height of the boundary layer at the leakage port is growth at a ratio of 1.1 to ensure the accuracy of the simulation calculation.

The boundary condition
The transportation pipeline involved is with high pressure, whose maximum is 12 MPa. The boundary condition of the leakage port is set as the inlet of pressure, and the boundary condition of the pipeline is the solid wall, there isn't any exchange between with inside and outside of the pipeline. The all arounds of this model are set as the pressure outlet, and the initial concentration of methane in this region is zero.

The validation of simulation
All relevant parameters were listed in Table. 2, and brought into the previous theoretical model of small hole to attain the leakage amount Q = 0.1594 kg/s. For the validation of the simulation, the medium in the fluid computing region was firstly set as air, to simulate the leakage and diffusion of the gas in air. And, the simulation result of the leakage amount was 0.1752 kg/s, which differed 9.91% with the previous theoretical value. The comparison showed that this numerical simulation had good reliability.  Figure. 4-7 illustrate the curves of the leakage amount with the pressure inside the pipeline and the diameter of the port at different porosity. It can be found that the leakage amount is approximately liner with the pressure, and the relation between the leakage amount and the diameters can be well described by quadratic function. Besides, the above variation rules were similar to those which the theoretical model of small hole implies.

The spatial and temporal distribution of the diffusion concentration
The leakage port in the model is taken as the origin point, the normal of the leakage port is along the positive direction of Y-axis, the vertical upward is the positive direction of Z-axis, and the direction along the pipeline is X-axis. Monitoring points were set with those coordinates as listed in Table. 3, to obtain the variation of the methane concentration with time. Table 3. The coordinates of the monitoring points The concentration nephogram at different moment during 100-5000 s were shown in Figure.8, and the gas diffusion behaviors in the soil were observed. It can be seen that the methane mainly diffuses in an approximately circular shape along the positive direction of the Y-axis in the early stage. When the diffusion distance exceeded the radius of the pipeline, the methane begins to diffuse along the negative direction of the Y-axis. Besides, as the time goes on, the shape of the nephogram isn't symmetrical on the Y-axis, and has a tendency to move upwards, which is the result of the effect of the buoyancy along the positive direction of the Z-axis.   Figure.9 shows the variation of the concentration with time at different monitoring points, it is found that with monitoring points being further away from the leakage port, the slope of the curve is smaller, so that the diffusion velocity is lower. When the time is less than 5000 s, the diffusion velocity along the X-axis is larger than that along the Y-axis. For the monitoring point X-1, the concentration reaches its maximum 88% at 1580 s, and keeps almost constant in the rear. However, for the point Y-1 with the same distance which the point X-1 is away from the leakage port, the concentration reaches its maximum 87% at 2800 s. Besides, for the diffusion along the Z-axis, the concentration of the point Z-1 continues to rise with the time going on, and finally reaches 73.2%, but that of the point Z-2 does less change. The average velocities of the diffusion V1 and the concentration of accumulation V2 of the points X-1 and X-2 were listed in Table. 4.
The variation of the methane concentration with distance at 5000 s is shown in Figure.10, it is found that the symmetry of concentration along that X-axis keeps well, the maximum distance of the diffusion along the positive direction reaches 2.25 m, and the distance along the negative direction is -2.15 m. However, for the methane diffusion along the negative direction of the Y-axis, owing to the separation of the solid wall condition from the computational region, there are no finite element in the region (from -1.219 m to 0 m) which is inside the pipeline, so that no data of the concentration was measured. For the positive direction of Y-axis, the methane can diffuse to 1.66 m. And, the maximum distance of the diffusion along the positive direction of Z-axis can reach 2.01 m, it can be also significantly found that the distribution of concentration along Z axis isn't symmetry, and the concentration along the positive direction is larger than that along the negative direction, which is result of the effect of buoyancy.