Quantitative Inversion of Water-Inrush Incidents in Mountain Tunnel beneath a Karst Pit

Quantitative inversion of accidents is an important work of finding the cause of accidents and avoiding their recurrence. However, quantitative inversion of accidents is difficult due to the lack and limitation of accidents monitoring information. Focusing on water-inrush incidents of Jiguan Mountain tunnel, this paper proposes a set of workflows to find out the missing conditions and quantitative inversion of accidents by flow analysis and structural safety analysis on the basis of investigating the rain capacity and water outflow in water-inrush incidents. First, hydraulic boundary in water-inrush incidents is acquired by analyzing the relationship of catchment, infiltration, and accumulation of rainwater in karst pit using the flooding algorithm of ArcGIS and the topographic mapping of UAV photogrammetry. Second, the permeability coefficients of karst infiltration zone and tunnel surrounding rock are acquired by two-step decoupling and inverse analyzing the water inflow, flow rate, and interval time between rainfall and water inrush. Third, tunnel accidents of the overload of tunnel lining induced by the catchment and infiltration of karst pit under extreme rainfall conditions are numerically simulated by using FLAC. The results indicate that quantitative inversion of water-inrush incidents reveals the process and cause of accidents and provides the safety index of tunnel structure. Not only is the water-inrush incidents of karst tunnel controlled by hydrogeology conditions, but also the rainfall recharge should not be ignored.


Introduction
Karst pit is a typical karst geomorphic unit, the hydrogeological conditions of which are significantly controlled by the recharge from the rainwater catchment of karst pit. In a tunnel pass through beneath a karst pit, when instantaneous rainstorms occur, rainfall-runoff is collected by karst pit and quickly drained through a doline, local groundwater skyrockets in a short time and breaks through the discharge capacity of conventional waterproof and drainage designs. It gives rise to the lining to bear short-term ultrahigh external water pressure in rainy season. In the construction period of Jiguan Mountain tunnel, multiple accidents of water-inrush incidents related to rainfall occurred to cause geological hazards and damage to the newly built tunnel lining.
Geological disasters, such as water inrush, mud inrush, and collapse, are often encountered in the construction process of various underground projects [1][2][3][4][5]. Leaking water can damage tunnel installations and form icicles that may cause danger to traffic or other problems for the tunnels activities [6,7]. To divert leaking water in tunnels without lining, conventional drainage is today installed [8]. Conventional drainage consists of foamed polyethylene (PE) mats which are suspended on rock mounted threaded rods.
e PE mats are assembled with different steel fittings and placed approximately 5-30 cm from the rock wall. e entire drainage structure is then covered with conventional reinforced shotcrete for mechanical protection and reduces the polyethylene mats' fire risk. Waterproofing and drainage system plays an important role in the control of external water pressure on the lining. e waterproof lining can increase the external water pressure [9], but a drainage system can decrease it [8]. Many researchers investigated how the external water pressure affected the lining's mechanical characteristics by analytical analysis, numerical simulations, and physical tests [10][11][12]. In the engineering practice, some strategies, including grouting and pin-hole drain method, have been proposed based on these research studies [13][14][15][16]. Many studies focus on the waterproofing and drainage system for tunnels, which significantly affects linings' external water pressure. Yuan et al. [17] systematically summarized the waterproofing requirements and measures in different special tunnels in China. Jang et al. [18] measured various types of geocomposites in a laboratory to evaluate their filtration and discharge capacity characteristics for their potential application in tunnels. Yoo [19] presented an investigation into the effect of the decrease in drainage capacity by the hydraulic deterioration of tunnel geosynthetic drainage systems on tunnel linings' structural performance. Other researchers explored the mechanism of seepage-induced water inrush and the mitigation of water inrush [20][21][22][23][24][25][26]. All the above studies indicate that the precondition of safety evaluating the drainage system design is an accurate estimation of the external water pressure on lining induced by groundwater.
Quantitative inversion of accidents is an important work of finding the cause of accidents and avoiding their recurrence. However, quantitative inversion of accidents is difficult due to the lack and limitation of accidents monitoring information. Focusing on water-inrush incidents of Jiguan Mountain tunnel, although the rain capacity and water outflow in water-inrush incidents are monitored, the main causes of damaging tunnel structure lining, such as the pressure on tunnel structure, the permeability of surrounding rock, and the hydraulic boundary of rainfall, cannot be directly known. is paper proposes a set of stepby-step workflows to obtain the main causes and quantitative inversion of accidents by flow analysis and fluidstructure interaction analysis on the basis of investigating the rain capacity and water outflow in water-inrush incidents.

Methodologies of Evaluation
e workflow of quantitative inversion is divided into four steps in the study, as illustrated by the flowchart in Figure 1: Step 1: the information of site conditions is collected, including topography, meteorology, and hydrology. e DEM of topography in karst pit is surveyed by UAV photogrammetry, and the geological model of karst pit is acquired by electrical geophysical prospecting.
Step 2: the watershed division of karst pit and the relationship of catchment, infiltration, and accumulation of rainwater in karst pit are acquired by hydrologic analysis tool and flooding algorithm of ArcGIS according to the rain capacity, water inflow, and flow rate in water-inrush incidents during construction.
Step 3: the permeability coefficients of karst infiltration zone and tunnel surrounding rock are acquired by twostep decoupling and inverse analyzing the water inflow, flow rate, and interval time between rainfall and water inrush.
Step 4: on the basis of the relationship of catchment, infiltration, and accumulation of rainwater in Step 2 and the permeability coefficients of karst infiltration zone and tunnel surrounding rock in Step 3, the drainage system of tunnel is evaluated by numerically simulating the overload of tunnel lining induced by the catchment and infiltration of karst pit under extreme rainfall conditions using FLAC. is paper is organized as follows: Section 3 presents site conditions including topography, meteorology, and hydrology; Section 4 presents rainwater catchment and accumulation analysis; Section 5 presents inverse analysis of permeability coefficients; Section 6 presents safety evaluation of drainage system; and Section 7 concludes this study.

Site Conditions
Jiguan Mountain tunnel is located in Lushan Town, Weining County, Liupanshui City, Guizhou Province, China, as shown in Figure 2 and is built for crossing the watershed between Lexi and Qinggou stretched from northeast to southwest, which is an extra-long separated tunnel where the left line is 3772 m long and the maximum burial depth is about 178 m, and the right line is 3800 m long and the maximum burial depth is about 150 m. e tunnel passes through a karst pit in the range between ZK72 + 220 and ZK72 + 340, the roof of which is 79 m below the bottom of the karst pit as shown in Figure 3. e geological features of Jiguan Mountain tunnel belong to the karst tectonic of peak cluster and the uvula landform. A series of karst pit, uvula, and karst funnels are distributed along the boundary between Qixia Formation and Liangshan Formation in pure limestone. A karst pit overlying Jiguan Mountain tunnel has a diameter of 120 m, a depth of 50 m, and the shape of an irregular circle. ere is a doline located at the bottom of karst pit with a depth of more than 15 m by visual inspection. Photos of the karst pit and doline are shown in Figure 4. e setting of Jiguan Mountain tunnel is located in the subtropical monsoon zone with abundant rainfall. According to the meteorological data of Weining County, rainfall is mainly concentrated from May to October, the annual average rainfall is 960.6 mm, and the maximum daily rainfall is 166 mm. In rainy season, sheet flows are gathered along the grooves of slope surface and imported into karst funnels, uvula, and karst pit to recharge groundwater.

Rainwater Catchment and
Accumulation Analysis 4.1. UAV Photogrammetry to DEM. UAV photogrammetry system is used to obtain 3D point cloud data for constructing the DEM data describing the terrain of karst pit. UAV system includes the four-axis aerial UAV of DJI M210 and the camera of DJI ZENMUSE X5S. e surveying area is about 1.577 km 2 , a rectangle with 1200 m from north to south and 1300 m from east to west. e workflow of UAV photogrammetry consists of eight steps: (1) determining the survey area, (2) UAV assembly and inspection, (3) flight route planning, (4) flight parameters setting, (5) flight mode planning, (6) aerial survey flight, (7) grounding marking of control points, and (8) GPS mapping of control points, as shown in Figure 5(a). UAV photogrammetry acquires a total of 338 orthographic images, some of which are shown in Figure 5(b). Pix4Dmapper software is used to solve 3D point cloud data, DOM, and DSM describing the terrain of karst pit from 338 orthographic images and GPS coordinates of controlling points by aerotriangulation, dense point clouds, constructing TIN data, and texture mapping. DOM data have a specific color difference due to weather conditions. 3D point cloud data is imported into ArcGIS and transformed into DEM data with the resolution of 0.5 m using the tool of LAS point Statistics as Raster tool according to the sampling mode of PULSE_COUNT. DOM data, DSM data, point cloud data, and DEM data are shown in Figure 5(c).  Advances in Civil Engineering respectively, as shown in Figure 6(a). e confluence flow is calculated based on the value of the water flowing at each point in the flow direction data, as shown in Figure 6(b).

Rainwater Catchment
e catchment threshold is determined to be 100. After trial and error, the confluence river networks are calculated, as shown in Figure 6(c). e watershed segmentation is Advances in Civil Engineering divided using the hydrological tools of ArcGIS, as shown in Figure 6(d). It is divided into 12 watersheds' segmentation (5 of which are inflow areas and 7 of which are outflow areas). e largest watershed area is 0.1919 km 2 , and the smallest one is 0.0307 km 2 . e watershed attributes are shown in Table 1.

Rainwater Accumulation Analysis by ArcGIS.
Water-inrush incidents in tunneling construction are closely related to intensive rainfall. When the discharge flow rate of doline at the bottom of karst pit is less than rainfall intensity, it begins rainwater accumulation at the bottom of karst pit. Rainwater accumulation of a karst pit is determined by the runoff of catchment and the discharge flow of doline at the bottom of karst pit. One part of the discharge flow pours into the drainage system of tunnel during tunnel operation or causes the water inrush during tunneling construction, and the other part of the discharge flow goes into the groundwater system. Due to the fact that the part of the discharge flow going into the groundwater system is inestimable, it can be supposed to be a specific water-inrush ratio in the tunnel. So the discharged flow QI can be expressed as where η is the coefficient of flowing into the groundwater system and Q g is the yield of water inrush. Given Jiguan Mountain tunnel beneath a karst pit, η is regarded as the minimum value under extreme rainfall conditions. In other projects, the value range of η is 0.2∼0.5 according to the relative position between tunnel and karst pit.

Advances in Civil Engineering
Rainwater accumulation of karst pit can be roughly obtained by the rainfall capacity minus the discharge flow, as shown in the following equation: where Q a is the amount of rainwater accumulation, w is the rainfall intensity, t is the duration of rainfall, S is the area of catchment, and Q I is the rate of discharge flow. e karst pit overlying Jiguan Mountain tunnel is the No. 7 watershed catchment in Table 2, with a total area of 108492 m 2 . According to the measured water inflow in three water-inrush incidents, the rainwater accumulation of each water-inrush incident is calculated using equation (1) as shown in Table 3. e height of rainwater accumulation at karst pit is calculated by the flooding algorithm of ArcGIS. e iterative calculation process is shown in Figure 7 and Table 2. e initial height of rainwater accumulation is set to 2110 m due to the fact that the lowest altitude of the karst pit is 2108 m. e amplitude of water head is set to 1 m in the first iteration and 0.1 m in the second iteration. e results of the iterative calculation on the flooding algorithm are shown in Table 4. Under the rainfall condition of 166 mm, it iterates the highly accurate solutions of the rainwater accumulation in other inner basins by formula (3). e calculation results are shown in Table 5.
where Q g is the daily water inflow; α is the rainfall infiltration coefficient, α � 0.7 (annual) or α � 0.5 (daily); W is the rainfall capacity; and S is the catchment area (km 2 ).

Rainwater Accumulation on Estimation
Method. e geometric shape of karst pit can be generalized into two geometric models as cone and truncated cone, as shown in Figure 8. e cone or truncated cone's height is based on the difference between the lowest elevation on the erosion peak and the elevation on the bottom around a karst pit. e top diameter of cone or truncated cone is based on the approximate diameter of a karst pit at the lowest elevation of erosion peak. e bottom diameter of truncated cone is based on the approximate diameter of karst pit at its bottom. e above dimensions can be easily obtained from the corresponding topographic map of karst pit.
According to the geometric shape of karst pit, the two types of geometric model as cone and truncated cone are discussed, respectively.

Cone Shape.
Based on the cone-shaped volume calculation formula, the water depth h can be expressed as where Q is the actual amount of rainwater accumulation (m 3 ); θ is the slope of karst pit, tan θ � H/R, and H is the model height of karst pit with cone shape (m), which is the radius on the top of karst pit with cone shape.

Truncated Cone Shape.
e relationship between the karst pit with truncated cone shape and the rainwater accumulation volume is that where r is the radius on the bottom of karst pit with the truncated cone shape (m); tan θ � H/(R − r); and R is the radius on the top of karst pit with truncated cone shape. Formula (5) can be organized to get a cubic equation of one variable as h; formula (6) can use Sheng-jin formula to solve a unary cubic equation, which can obtain the depth of rainwater accumulation.
e estimation method can be verified based on the water-inrush incidents. e geometric dimension of karst pit model overlaying tunnel is shown in Table 6. ese geometric dimensions are put into the formulas for calculating the height of rainwater accumulation, and the results are compared with the accurate solution based on flooding algorithm of ArcGIS. e calculation results are shown in Table 7. e results show that the truncated cone's estimated results are 0.2 m, 0.0 m, and 0.2 m away from the ArcGIS iterative inversion results. When the ground in the area of karst pit overlaying tunnel is relatively flat, the prediction result of the truncated generalization model is close to the exact solution based on flooding algorithm of ArcGIS.
Under the rainfall condition of 166 mm, the height of rainwater accumulation is estimated and compared by three methods as shown in Tables 8 and 9. When the bottom of karst pit is steep, the topography of No. 6 and No. 8 basins is similar to the cone shape. When the bottom of karst pit is relatively flat, the topography of the No. 7, No. 9, and No. 10 basins is more similar to the truncated cone shape. It is evident that the generalized conical model results are more accurate when the bottom of karst pit is steep; when the bottom of karst pit is relatively flat, the height of rainwater accumulation of the truncated conical generalized model is closer to that predicted by flooding algorithm of ArcGIS.

Electrical Geophysical
Prospecting. e high-density electrical geophysical prospecting is used to detect the distribution of karst carve and karst infiltration zone in the tunnel surrounding rock, which adopted a four-pole electrode arrangement for 2-dimensional measurement. ree measuring lines with a length of 1.2 km were arranged for crossing the karst pit and following the tunnel path. e three measuring lines have 120 electrodes with 360 V voltage and spacing of 5 meters and their probing depth is about 70 meters. e results of probing and interpreting are shown in Figure 9, which indicates that there are two large faults with SE22°inclination angle of 52°a nd SE23°inclination angle of 59°which are simplified to construct a numerical geological model as shown in Figure 10.

Two-step Decoupling.
e controlling factors of rainwater infiltration mainly include the permeability coefficients of karst infiltration zone and tunnel surrounding rock, except the boundary of rainwater accumulation in Section 4. e rainwater infiltration path and time between rainfall time and starting time of water inrush are mainly controlled by the permeability coefficient of karst infiltration zone. e rainwater infiltration amount and rate are mainly controlled by the combined effect of the permeability coefficients of karst infiltration zone and tunnel surrounding rock. erefore, the ideas of two-step decoupling can be carried out as follows: (1) the interval time between the occurrence of rainfall and the occurrence of starting water inrush is used as a judgment scale to initially determine the permeability coefficient of karst infiltration zone; (2) the flow rate of water inrush inside the tunnel is used as a judgment scale to determine the relative value of the permeability coefficients of karst infiltration zone and tunnel surrounding rock. Decoupling indexes of three water-inrush incidents are listed in Table 10.

Inversion Analysis.
In inversion analysis, the difference in the flow rate of water inrush in the left line of tunnel between numerical analysis and water-inrush incidents is constructed as the objective function, and the step-by-step scanning method is used to iteratively find     Advances in Civil Engineering 9 where N is the total number of samples; x t is the infiltration time of numerical analysis in the No. t water-inrush incident; X t is the starting time of water inrush in the No. t actual water-inrush incident; y t is the flow rate of water inrush in numerical analysis in the No. t water-inrush incident; and Y t is the flow rate per unit length of water inrush in the No. t actual water-inrush incident. Unsteady flow analysis on FLAC3D is carried out to numerically simulate the process that rainwater infiltrates into the tunnel through a doline at the bottom of karst pit as to cause the water inrush of tunnel in three water-inrush incidents.

Karst Infiltration Zone.
Taking the No. 1 water-inrush incident as an example, the baseline value of the permeability coefficient in karst infiltration zone is 1 × 10 −3 cm/s, and the step size of searching is 1 × 10 −4 cm/s. e relationship between k 1 and f(k 1 ) and the curve of water pressure at the monitoring point of tunnel are shown in Figure 11(a). e permeability coefficient in karst infiltration zone can be finally determined to be 1.3 × 10 −3 cm/s. In the No. 2 and No. 3 water-inrush incidents, the relationship between k 1 and f(k 1 ) and the curves of water pressure at the monitoring point of tunnel are shown in Figures 11(b) and 11(c). e permeability coefficients in karst infiltration zone are 1.4 × 10 −3 cm/s and 4.5 × 10 −3 cm/s. Hence, the permeability coefficient of karst infiltration zone is set as 2.4 × 10 −3 cm/s, which is the average permeability coefficient of karst infiltration zone in three water-inrush incidents as shown in Table 11.

Tunnel Surrounding Rock.
e baseline value of the permeability coefficient of tunnel surrounding rock is 1.3 × 10 −4 cm/s, and the step size of searching is 0.2 × 10 −4 cm/s. e relationship between k 2 and f(k 2 ) and the curve of peak flow rate of water inrush per length at the monitoring point of tunnel are shown in Figure 12. Hence, the permeability coefficient of tunnel surrounding rock is set as 3.2 × 10 −4 cm/s, which is the average permeability coefficient of tunnel surrounding rock in three water-inrush incidents as shown in Table 11.

External Water Pressure on Lining Structure.
Unsteady flow analysis on FLAC3D is carried out to numerically simulate the process that rainwater infiltrates into the tunnel through a doline at the bottom of karst pit as to cause the external water pressure on lining structures. According to the maximum rainfall intensity of 234 mm in the No. 3 waterinrush incident, the maximum water head of rainwater accumulation in karst pit is 7.3 meters using flooding algorithm of ArcGIS. e drainage system with a central trench was used at Jiguan Mountain tunnel, the water discharge capacity of which is 1.21 m 3 /h per unit length of tunnel that is indicated as the flow boundary of 0.6 m 3 /h at both sides' foot of lining wall in numerical simulation. e permeability coefficients of karst infiltration zone and tunnel surrounding rock are, respectively, set as 2.4 × 10 −3 cm/s and 3.2 × 10 −4 cm/s according to inversion analysis in Section 5. All the above conditions are put into the boundary of numerical simulation. Water pressure at the monitoring point of tunnel lining (left line and right line) is shown in Figures 13(a) and 13(b) which indicates that water pressure achieves the maximum when the curve is smoothly close to the limit along with the infiltration time. e right arch of the left-line tunnel reaches a maximum of about 12 m of water head, and the left arch of the right-line reaches a     Type  Cone  Truncated  cone  Cone  Truncated  cone  Cone  Truncated  cone  Cone  Truncated  cone  Cone  Truncated  cone  Top radius       Actual peak flow rate of water-inrush per length 2.

Safety Evaluation of Lining Structure.
On the basis of acquiring the external water pressure on lining structure, fluid-structure interaction of FLAC is used to numerically simulate the security index of tunnel lining under the external water pressure. In the range between ZK72 + 220 and ZK72 + 340, classification of surrounding rock passing through a karst pit is V grade. e mechanics parameters of surrounding rock and lining structure are shown in Table 12. Safety evaluation of lining structure mainly relies on the safety factor of the lining's key parts to judge whether the lining reaches the limited water head load it can bear. e safety factors of lining structure are shown in Table 13.
When the maximum water head on lining structure is 12 m, the safety factor of right spandrel and right arch waist in the left line of the tunnel is less than 1.0, which indicates that the lining has been damaged.

Conclusions
Based on investigating the rain capacity and water outflow in water-inrush incidents, a synthesized approach of quantitative inversion is proposed to evaluate the safety of tunnel structure on the basis of finding out the key conditions of water-inrush incidents, such as the pressure on tunnel structure, the permeability of surrounding rock, and the hydraulic boundary of rainfall. e pressure on tunnel structure is 12 m water head, the permeability coefficient of tunnel surrounding rock is set as 3.2 × 10 − 4 cm/s, the permeability coefficient of karst infiltration zone is set as 2.4 × 10 − 3 cm/s, and the hydraulic boundary of rainfall is 4.6 m water head. e safety factor of right spandrel and right arch waist in the left line of the tunnel is less than 1.0, where the corresponding position is also the failure location of tunnel structure. Quantitative inversion of accidents is an important work of finding the cause of accidents and avoiding their recurrence. is paper proposes a set of step-by-step workflows to obtain the main causes and quantitative inversion of accidents by flow analysis and fluid-structure interaction    16 Advances in Civil Engineering analysis on the basis of investigating the rain capacity and water outflow in water-inrush incidents. e quantitative inversion method presented in this paper can be effectively applied in a similar karst area.

Data Availability
e data used to support the findings of this study are available from the corresponding author upon request.