A Computing Method for Sand Inrush Quantity through a Borehole in Longde Coal Mine

A quicksand disaster through a borehole occurred in Longde coal mine. A lot of aeolian sand, the volume of which is between 310,000m and 380,000m, has submerged into the underground space in about 70.5 h. *e volume flux of quicksand cannot be calculated accurately by the empirical method. Based on the method of fluid mechanics, an all-purpose computing method for quicksand disaster through a borehole was proposed.*e result shows that the inrush volume of sand into underground space was between 310,000m and 350,000m, which was consistent with the actual result. To apply and popularize this method, the impact laws of water yield properties of an aquifer on the volume flux were discussed. *e all-purpose computing method can be suitably used for the volume flux calculation of quicksand disaster through the borehole.

ere are vast deserts [5] in the ground and abundant coal resources underground [6] in Shanxi, Shaanxi, Inner Mongolia, Sinkiang, and Qinghai provinces in China.e hidden dangers of quicksand exist extensively in Western China during mining [7].Quicksand disaster happened occasionally with the water inrush, which have caused serious economic losses [8][9][10][11].
ere are two types of quicksand disaster.Firstly, the quicksand disaster caused by the aeolian sand submerging into underground space through caved zone as shown in Figure 1(a).When the mining depth is shallow, the mininginduced fractures may go through the rock layers to the sand bed, or even to the ground surface.In this situation, the water and sand can move to the goaf through the fractures and granular rock mass, which results in the quicksand disaster [14].Secondly, the quicksand disaster caused by a borehole from ground surface to underground.For example, a disaster of water inrush and quicksand caused by a borehole occurred in Longde coal mine.
e borehole directly transfixed the underground space under the condition of no casing pipe, which resulted in the disaster of water inrush and quicksand, as shown in Figure 1(b).A lot of aeolian sand submerged into the underground space.e sand cleaning work lasted for almost one year, which caused serious economic losses and mining difficulties.e quicksand disasters caused by the borehole have been reported for many times in papers [15,16], but this phenomenon has not been explained reasonably until now.
Flow in the borehole (i.e., in the circular pipe) has been researched by researchers, and a lot of empirical formulas have been put forward.
Manning improved the Manning coefficient based on Chery's research [17].e Chery formula is where v is the velocity, R is the hydraulic radius, J is the hydraulic slope, and n is the Manning coefficient.Hazen and Williams [18] proposed the Hazen-Williams formula for transition region: where d is the diameter of circular pipe and C h is the empirical coefficient of roughness height.
Based on the research of wood-stave pipe, Scobey [19] put forward the Scobey formula: where h f is the frictional head loss, L is the length of circular pipe, Q is the volume ux, and K s is the empirical coe cient.
In addition, there are many other empirical formulas such as Blasius formula [20] and Churchill formula [21].Blasius formula is suitable for the Reynolds number below 10 5 based on a number of experiment results.Churchill formula can be used in both laminar ow and turbulence ow. e volume ux of Longde quicksand disaster was calculated by using the empirical formula with the common value of empirical parameters, and the result is shown as Table 1.e actual volume ux of quicksand is 4,400-5,400 m 3 /h (Section 2.3).As shown in Table 1, the calculation results of Chezy formula and Scobey formula are far from the actual volume ux, and that of the Hazen-William formula contains the actual ux.e simple form of these empirical formulas contributes to the convenient usage in the engineering.However, there are two shortages: on the one hand, the empirical formulas are hard to select under di erent conditions of disasters.e Hazen-William formula can be used to calculate the volume ux in Longde quicksand disaster.However, if the geological conditions of pressure, water head, or diameter of borehole di er from that of Longde disaster, the calculation results of the Hazen-William formula can also appear with errors.On the other hand, each formula has an empirical parameter which is hard to determine in the calculation process.
erefore, an all-purpose computing method was proposed for quicksand through the borehole in this paper.e Bernoulli formula [22], Darcy-Weisbach formula [17], semiempirical Nikuradse formula [23,24], and Colebrook-White formula [25,26] were combined to calculate the volume ux of quicksand.Besides, the volume concentration, density, and viscosity of sand-water mixture were tested for the method used in similar engineering.e all-purpose computing method for the calculation of volume ux can be used in all quicksand disasters though the borehole, which has a certain guiding signi cance in engineering.

Illustration of Water Inrush and
Quicksand in Longde Coal Mine 1. e Process of the Disaster.Longde coal mine is located in southwest of Yulin in Shaanxi province in China, and the shape of this eld is similar to a pistol, as shown in Figure 2. is area is vulnerable to long-term e ects of polar continent.As a result, the continental climate is remarkable, and the ground is dominated by desert and hills in this area.
A water inrush and quicksand disaster through a borehole happened in Longde coal mine at 14:30, September 17, 2012.A geological team undertook the construction of cable borehole for the central water pump room.
e wrong statistics of drilling depth and the cable borehole directly getting into the underground space without the protection measures such as steel casing and seal ring caused immediate disappearance of the drill tower in sand seam and submergence of the water pump room by sand.en, the central substation could not deliver electricity normally.As a result, the whole coal mine was submerged by sand.
From September 17 to September 20, although emergency measures had been taken by Longde coal mine, a lot of sand owed into the railway, even main shaft, and auxiliary shaft, as shown in Figure 3.
e sandline elevation of 2 Advances in Civil Engineering quicksand position, main shaft, and auxiliary shaft was 1,227.1 m, 1,081 m, and 1,100 m, respectively.However, the waterline was only 1,029 m less than the sandline elevation of main and auxiliary shaft.It was mainly because that the aeolian sand has become a uid with the lubrication action of water.

Sand Volume Flux Analysis.
e total volume of the underground space was 488,000 m 3 in Longde coal mine.
e statistical data of sand cleaning presented that the actual volume of sand was 380,000 m 3 .e 77.9% volume of the underground space was submerged by sand.After the disaster, in order to protect the infrastructure such as the air  Advances in Civil Engineering shaft square, protective measures were taken; for instance, the gravity dam was set up with stones and concrete.And based on geological conditions, the sand subsidence area did not present a cone type but an irregular cone type (prismoid), as shown in Figure 4. e in ection points of sand subsidence area were monitored after the disaster happened 70.5 h. Figure 4 is the aerial view of the aeolian sand subsidence area.e in ection points were recorded by Xi'an 80 coordinate [27].For the sake of illustration, the origin point was transformed to C1 point, as shown in Figure 4.

Disaster Duration.
e coordinate of 7 in ection points is shown in Figure 4.It is assumed that two adjacent points are in a straight line, and the area of aerial view is calculated by the following equation: where A is the area of aerial view, x is the longitude (east) coordinate, and y is the latitude (north) coordinate.e calculation result shows that the area of aerial view is 22,019.75m 2 until 70.5 h after the disaster, but it only increases 40 m 2 from 70.5 h to 238.5 h.e area percent change only increases 0.18%, which is almost negligible.erefore, it can be con rmed that the disaster mainly happens within 70.5 h.

e Volume of Sand Submerged in Underground Space.
It is necessary to calculate the volume of sand subsidence area for the obtainment of quicksand volume.Consequently, the irregular cone type or prismoid type can be assumed for presenting the sand subsidence area based on the method in [15].Considering that the friction coe cient of the sand is constant, the friction coe cient is tested, as shown in Figure 5. e friction force is generated by the gravity and contact force among particles.e equation of the friction coe cient is k tan θ, where θ can be obtained as shown in Figure 5, and the average value of friction coe cient k is 0.617.
To calculate the volume of sand subsidence area, the pro le plane equation of each plane is established rstly.e uniform form of plane equation is as follows [28]: where a, b, c, and e are calculating parameters of plane equation.e plane equation parameters of each pro le plane are given in Table 2. e volume is calculated by the method [29].e total volume of sand subsidence area is 310,000 m 3 , which is less than the statistical data of 380,000 m 3 from the sand cleaning.It can be explained that the sand volume varies easily under di erent con ning pressure.e conning pressure is nearly zero when the cleaning sand work starts.But the con ning pressure of the ground sand increases with depth.e compaction part of sand causes the volume statistical data of subsidence area bigger than the sand cleaning.In conclusion, the aeolian sand is submerged into the underground space of Longde coal mine in 70.5 h.
e volume of sand is between 310,000 m 3 and 380,000 m 3 , and the volume ux is between 4,400 m 3 /h and 5,400 m 3 /h.

Water Volume Flux Analysis.
e water level of auxiliary shaft was monitored after the disaster happened 12 h.4 Advances in Civil Engineering the total volume ux was 3.014 times of auxiliary shaft.After the disaster happened 12 h, the total volume ux was 37.05 m 3 /h, while it was only 1.51 m 3 /h after the disaster happened 30 h. ere would also be a small amount of water ow near to 1.51 m 3 /h in the mine without disaster.erefore, the water volume ux variation caused by the disaster could be negligible.
In conclusion, the volume ux of sand was between 4,400 and 5,400 m 3 /h, while water ow was only 37.05 m 3 /h.e volume ux of sand was much bigger than that of water.Consequently, it can be con rmed that the disaster of water inrush and quicksand is nearly the ow of solid particle in Longde coal mine.Jaeger et al. [30] and MiDi [31] have also presented that tiny solid particle can show the characteristics of uid.

Semiempirical Method for Quicksand in
Longde Coal Mine e photograph of the borehole and chamber after cleaning is shown in Figure 7.It can be clearly seen that the borehole was not damaged in the whole process of quicksand disaster, and the drill pipe remained in the borehole all the time.e pro le of the quicksand process is shown in Figure 8: (1) ere is aeolian sand layer with a thickness of 30 m in the ground.e lithology is silty ne sand with a distinct thickness variation.e sand seam constitutes a uni ed aquifer together with the Sara Wusu group aquifer below.Before nding the methods used for quicksand in Longde coal mine, the uid condition of sand-water mixture ow has to be judged rstly, as di erent formulas need to be applied for the calculation of the sand-water mixture ow in di erent uid conditions.For instance, the ow is easy to obtain with an accurate analytic solution in laminar ow, while the turbulence ow is complicated, which has to be calculated by semiempirical method.e uid condition can be judged by Reynolds number: where Re is the Reynolds number, ρ is the density, v is the velocity, d is the diameter, equivalent diameter d 4R (in this paper), and μ is the dynamic viscosity.
In general, Re of laminar ow is below 2,300 and Re of turbulence ow is above 2,300.Nikuradse [23] divided the turbulence ow into e ectively smooth ow, transition region, and rough ow.
When the ow is in e ectively smooth ow, the range of Re is 4000 < Re < 26.98 where k s is the roughness height.When the ow is in the transition region, the range of Re is When the ow is in rough ow, the range of Re is e velocity of sand-water mixture ow is calculated by viscous Bernoulli formula [22].e equation is where h w is the head loss, z is the third Cartesian coordinate of cross-sectional area, as shown in Figure 9, p is the relative pressure (that the absolute pressure subtracts the atmosphere pressure) of cross-sectional area, and α 1 and α 2 are correction factors, which are related to the velocity of crosssectional area.e 1-1 area and 2-2 area (shown as Figure 9) are the same, so α 1 is approximately equivalent to α 2 .
From the continuity equation of uid mechanics [22], it can be known that there is a relationship between the volume ux Q of 1-1 and 2-2 cross-sectional area as follows: For the borehole drilled with steel casing all the while, the sectional area S of borehole remains unchanged, and v 1 is same to v 2 .e sand-water mixture ow contacts with air in 2-2 area.e relative pressure of 2-2 area is near to zero.e 2-2 area is taken as the base level, and the third Cartesian coordinate z 2 is zero.Equation ( 10) is simpli ed to e head loss h w is constituted by frictional head loss h f and local head loss h j .From Figure 9, it can be seen that the borehole is a long straight pipe with a constant sectional area.ere are only low local head loss h j in both inlet and outlet.Due to the value of h j is much less than h f , h j is negligible.So, h w is approximately equal to h f .e h f can be expressed as [22] where l is the distance between 1-1 area and 2-2 area and λ is the ow resistance ratio and it can be expressed with different equations in di erent ow conditions.In laminar ow, the equation is [33] In e ectively smooth ow, the equation is [24] 1 In the transition region, the equation is [25]

Advances in Civil Engineering
In rough ow, the equation is From Equations ( 15)-( 17), it can be known that λ is only related to Re in e ectively smooth ow.λ is only related to k s in rough ow.λ is related to both λ and k s in the transition region.
Put Equation ( 13) into Equation (12), and the velocity v can be expressed as follows: In conclusion, the computational model diagram is drawn as Figure 10.e velocity will be calculated under di erent ow conditions after all ow parameters have been determined.en, Re will be obtained by the calculating result of velocity.Finally, the ow condition will be veri ed with Re one by one until the right velocity is worked out.

e Calculation of Computing Method.
e calculating parameters are shown in Table 4. e density and viscosity of aeolian sand are obtained by experiments.
e obtaining way of the roughness height in Table 4 is as follows: researchers have studied on di erent roughness heights of di erent rocks, as shown in Table 5. e roughness heights of all rocks are between 0.9 and 16.2 mm. e rock in Longde coal mine is mainly claystone and sandstone.e heights of them are within the range from 1.0 to 5.3 mm without quicksand.But the changing of roughness height is di cult to obtain in the process of quicksand.It can be known that the volume ux of quicksand is about 4,400 to 5,400 m 3 /h, and the velocity is about 15 to 19 m/s.e roughness height of borehole will decrease gradually with the high-velocity friction.e phenomenon is just like the grinding wheel sanding samples, and the common speed of grinding wheel is 35 m/s (diameter is 200 mm and rpm is 3,150 r/min) [38].
erefore, the phenomenon in quicksand disaster can be described as follows: the velocity is slow and the roughness height is high at the beginning.en, the roughness height decreases gradually under the continuous impact of aeolian sand while the velocity increases with the decrease of roughness height.
e interaction between velocity and roughness height repeats until they reach a stable value  Advances in Civil Engineering nally.As the velocity is near to the speed of the grinding wheel, the value of roughness height is near to the minimum roughness height of technical standard, as shown in Table 6.In this way, the value of 0.1 mm varies to 0.4 mm, which is the minimum roughness height value taken as the calculation basis of the semiempirical formula in this paper.
e obtaining of the water pressure is as follows: there are 4 hydrological boreholes near by the position of quicksand disaster.
e water level elevation is recorded aperiodically, as shown in Figure 11.From the gure, it can be known that the water level elevation decreases in the underground engineering but has not decreased violently with the quicksand disaster.e decreasing laws of aquifer are similar before and after the quicksand disaster.erefore, the average value of the observation data of 4 hydrological boreholes on October 29 is selected as the water level elevation value of the computing method.And the water level elevation of aquifer h 1 is 1,198.20 m, and the elevation of aquifer roof h 2 is 1,179.15m. e water pressure can be obtained according to the Pascal law: From Equation ( 19), the water pressure of aquifer roof p 1 is 0.187 MPa.
Taking z 1 equal to 157 m, p 1 equal to 0.187 MPa, ρ s equal to 1,652.89kg/m 3 in Equation ( 12), it can be obtained that h w and h f are equal to 168.54 m and the equivalent diameter d 4R is equal to 0.25 m.Taking the above parameters into Equation (18), e velocity is calculated by the simultaneous Equations ( 6), (14), and (20), Equations ( 6), (15), and (20), Equations ( 6), (16), and (20), and Equations ( 17) and ( 20) when the sandwater mixture ow is, respectively, in laminar ow, e ectively smooth ow, transition region, and rough ow. e calculation results are shown in Table 7, from which it can be known that the sand-water mixture ow belongs to transition region, and the velocity and volume are 15.32 to 17.78 m/s and 310,000 to 350,000 m 3 , respectively.e volume result is contained in the actual result 310,000 to 380,000 m 3 .

Discussion
As the coal roof aquifer belongs to low water abundance aquifer in Longde coal eld, the sand-water mixture ow can be seen as the ow of solid particle of aeolian sand.However, there may be high or higher water abundance aquifers (Table 3) even in the desert area.At this point, the volume concentration of sand C in the sand-water mixture will decrease, and the density ρ and viscosity μ in Equation (3) will vary with the change of volume concentration of sand.In order to apply the computing method in this paper expediently, the density ρ and viscosity μ of sand-water mixture with di erent volume concentration of sand are tested.

Density of Sand-Water Mixture Flow.
e density test results of di erent C are obtained, as shown in Figure 12.It can be seen that the density of sand-water mixture rstly increases and then decreases with the increase of volume concentration of sand.e max value is 2,095.81kg/m 3 when the volume concentration of sand is 0.78.When the volume concentration of sand is greater than 0.78, the density of sand-water mixture decreases gradually, while the volume concentration of sand tends to 1, the density tends to 1,652.89kg/m 3 which is the dry density of aeolian sand.

e Dynamic Viscosity μ of Sand-Water Mixture Flow.
e dynamic viscosity μ of sand-water mixture is tested, as shown in Figure 13.e max viscosity ratio of sand-water mixture is 9.70, and the viscosity ratio of the dry sand is 9.23.
e test results are tted.When S is below 0.49, the tting equation is where μ 0 is the viscosity of water.When S is between 0.49 and 0.78, the tting equation is  Advances in Civil Engineering e aeolian sand is assumed su cient in the ground.e roughness height is 0.4 mm. e values of water head height z, pressure p, and drilling diameter d are evaluated according to the approach in Section 3.
e calculation results of velocity under the condition of di erent volume concentrations of sand are obtained, as shown in Figure 14.It can be known that the velocity decreases with the increase of volume concentration of sand, and the overall trend is close to a straight line.It can be explained that both the density in the denominator of Equation ( 6) and the viscosity in the numerator of Equation ( 6) increase with the increase of volume concentration of sand.But the increased rate of the viscosity is greater than that of the density; thus, both the Re and velocity tend to decrease with the increase of volume concentration of sand.To sum up, with the same roughness height, the higher the water abundance aquifer is, the smaller the volume concentration of sand will be (Table 3).
erefore, the velocity of sand-water mixture will increase with higher water abundance aquifers, and then larger scale disaster will be caused.

Conclusion
An all-purpose computing method was constructed with Bernoulli formula, Darcy-Weisbach formula, semiempirical Nikuradse formula, and Colebrook-White formula to calculate the volume of quicksand.e variation laws of density and dynamic viscosity under di erent water yield properties were analyzed based on the experimental results.e inuence of water yield properties on volume ux was discussed: (1) It can be seen that the total volume of quicksand is between 310,000 m 3 and 380,000 m 3 , and the quicksand disaster duration is about 70.5 h from the statistical results of sand cleaning and the variation laws of sand subsidence area with time.(2) e value of water volume ux is much less than that of sand volume ux. e sand-water mixture can be regarded as dry sand in Longde water inrush and quicksand disaster.(3) e volume of sand between 310,000 m 3 and 350,000 m 3 , i.e., calculated by all-purpose computing method, is contained with the actual result between 310,000 m 3 and 380,000 m 3 .(4) Both the density and dynamic viscosity of sandwater mixture increase with the increase of sand volume concentration.With the fixed value of water head, roughness height, and diameter of borehole, the velocity decreases with the increase of sand volume concentration, which means that the sand volume concentration will be smaller and the velocity will be faster under a higher abundance level aquifer.

Notation
A ird Cartesian coordinate of cross-sectional area L: e length of circular pipe λ : Flow resistance ratio p: Relative pressure.

Figure 4 :
Figure 4: Aerial view in subsidence area of aeolian sand.
(2) e Sara Wusu group aquifer is the main aquifer of the coal roof.e main lithology is based on silty ne sand, medium-coarse sandstone, and loam.e units-in ow is between 0.0441 and 0.0569L•s −1 •m −1 .eaquifer belongs to low water abundance aquifer, as shown in Table3.(3)e borehole diameter is 325 mm in quicksand disaster, including a built-in drill pipe with the diameter of 75 mm.And the borehole and drill pipe have not been damaged in the disaster.erefore, the cross-sectional area S, wetted perimeter χ, and hydraulic radius R are 0.0785 m 2 , 1.256 m, and 0.0625 m, respectively.

Figure 6 :Figure 7 :
Figure 6: Volume ow rate change curve of auxiliary shaft with time in Longde coal mine.

Figure 8 :
Figure 8: Cutaway view of sediment-water ow model.

8
: e area of sand subsidence ρ: Density C: e volume concentration of sand Q: Volume flux d: Diameter of borehole R: Hydraulic radius h w: Head loss Re: Reynolds number h f: Frictional head loss S: Cross-sectional area h j: Local head loss μ: Dynamic viscosity J: Hydraulic slope v : Velocity k: Friction coefficient χ: Wetted perimeter k s: Roughness height z:

Table 2 :
Parameters of plane equation.

Table 6 :
Max roughness of technical standard in part country.