(SDM) for Tunnels Passing Through Large Karst Caves

applied sciences

Article Establishment and Application of the Spatial Decomposition Method (SDM) for Tunnels Passing Through Large Karst Caves

Zhenyue Shi 1 , Qingbiao Wang 1,2,3,4,*, Chuming Pang 5,*, Yueming Yuan 6, Fuqiang Wang 6, Hongxu Song 5, Jichang Liu 7, Zijie Zhang 7, Rongbo Sun 8 and Yan Liu 9

1 College of Safety and Environmental Engineering (College of Safety and Emergency Management), Shandong University of Science and Technology, Qingdao 266590, China; [email protected] 2 State Key Laboratory of Mining Disaster Prevention and Control Co-Founded by Shandong Province and the Ministry of Science and Technology, Shandong University of Science and Technology, Qingdao 266590, China 3 National Engineering Laboratory for Coalmine Backﬁlling Mining, Shandong University of Science and Technology, Tai’an 271019, China 4 College of Resources, Shandong University of Science and Technology, Tai’an 271019, China 5 College of Energy and Mining Engineering, Shandong University of Science and Technology, Qingdao 266590, China; [email protected] 6 College of Civil Engineering and Architecture, Shandong University of Science and Technology, Qingdao 266590, China; [email protected] (Y.Y.); [email protected] (F.W.) 7 No.1 Engineering Development CO., Ltd. of China Railway 14th Bureau Group., Rizhao 276826, China; [email protected] (J.L.); [email protected] (Z.Z.) 8 Jinan Rail Transit Group CO., Ltd., Jinan 250101, China; [email protected] 9 Shandong Zhengyuan Construction Engineering Co., Ltd., Jinan 250101, China; [email protected] * Correspondence: [email protected] (Q.W.); [email protected] (C.P.)

Received: 19 July 2020; Accepted: 13 October 2020; Published: 15 October 2020

Abstract: Karst tunnels commonly pass through large karst caves during their construction and operation. Although treatment technologies are relatively mature, a systematic treatment method to guide the selection of treatment technologies is lacking. To solve this problem, a spatial decomposition method (SDM) of large karst caves is proposed that is based on analyzing the spatial relationship between tunnels and karst caves and summarizing the relevant treatment techniques. In this method, the space between the tunnel and the cavern is divided into eight parts using a space dividing line (SDL), which makes the spatial position relationship between the tunnel and cavern more intuitive. A geometric model of the SDM is established, and the numerical values of each geometric parameter are determined by field surveys and drawings. Constructing a three-dimensional spatial diagram by applying relevant parameters to the geometric model provides a reference for selecting a treatment technology. The SDM of the arch top, arch bottom, and two wings matching the treatment technology is proposed. Seven principles of technical selection—namely, safety, convenience, scientificity, sustainability, economy, feasibility, and openness—are mentioned in order to overcome the difficulty of technology selection due to such factors as technological diversity, materials, equipment, and environment. Finally, the SDM is used to solve the problem of the Shangyuan tunnel passing through a large karst cave. The implementation of the SDM in tunnel construction would represent a significant breakthrough and has important engineering value in solving the problem of tunnel passes through large karst caves.

Keywords: spatial decomposition method; spatial decomposition line; huge karst cave; geometric model; treatment technology

Appl. Sci. 2020, 10, 7204; doi:10.3390/app10207204 www.mdpi.com/journal/applsci Appl. Sci. 2020, 10, 7204 2 of 20

1. Introduction Karst is widely distributed around the world, and it causes great inconvenience to the achievement of engineering construction projects. Regions in the southwest of China, such as Yunnan and Guizhou, as well as other regions of China, are typified by steep hills and plenty of Karst terrain [1–4]. With the continuous improvement of China’s “western development” strategy, the prevalence of karst tunnel construction is increasing rapidly, and various technical problems of karst tunnels are constantly appearing [5,6]. Passing through a large karst cave can be considered as one of the main difficulties in tunnel construction that will impose higher theoretical and technological requirements on karst cave treatment. The study aims to identify and analyze a powerful new method for the implementation of karst tunnels. In this regard, there are many relevant studies and engineering applications carried out by scholars that deal with large karst caves revealed during tunnel construction, the main aspects of which are exemplified as follows: Li et al. [7] treated a karst cave in a tunnel floor using the beam and plate method, and also used backfilling to treat the revealed karst cave. Cui et al. [8] introduced the requirements of karst tunnel treatment, grouting materials, construction procedures, and effect detection. Shi et al. [9] and Wang et al. [10] studied the application of alkali-resistant fiberglass grouting material in karst cave treatment. Fan et al. [11] found that the methods of pile foundation treatment at the bottom of the tunnel, strengthening the surrounding rock of the tunnel by full-section grouting, and supporting the top of the karst tunnel with an anchor net and shotcrete are effective. Chen et al. [12] focused on the Naqiu tunnel hall karst cave project, and used the pile-bearing platform-retaining wall method to control the karst cave. Garasic et al. [13] designed a 58 m bridge through a large karst cave. Yang et al. [14] proposed a new controlled grouting method and grouting material for the pre-reinforcement of the underwater karst area of a shield tunnel. Li et al. [15] showed that the treatment technology of an ultralarge karst cave in the Yawan tunnel, which is treated at the bottom of the tunnel by a continuous composite beam bridge, and the wall of the karst cave is treated by the combined support of an anchor net and spray. Zhang et al. [16] reported the treatment of large karst caves encountered during the construction of the Limushan tunnel by backfilling. Yuan et al. [17] treated a large karst cave in the Shangjiawan tunnel with plugging and crossing methods. Tan [18] analyzed the advantages and disadvantages of longitudinal retaining walls, pile foundation-bearing platforms, and backfilling in tunneling through large karst caves from the perspectives of cost, safety, convenience, and time limitations. He [19] used a roundabout method to turn a revealed large cave into a side hidden cave. Tian [20] and Wang [21] analyzed the safety problems of limestone during mining. Based on the results of the above scholars, in view of the treatment technology of large karst caves revealed by tunnels, researchers have combined the practical projects and summarized the rich experience in treatment technology. In general, various technologies can be adopted to treat large karst caves. However, these existing technologies are lacking in commonality because they are based on specific engineering characteristics. There is a lack of a systematic method to guide the selection of one or more treatment technologies. To guarantee that the tunnels through large karst caves can achieve the purposes of convenience, high efficiency, safety, and systematicity, it is necessary to construct a complete set of technologies in order to achieve the comprehensive treatment of tunnels through large karst caves. To overcome the abovementioned challenges, this paper describes a guiding method for tunnels through large karst caves and a complete set of technical measures. In order to achieve this goal, the following steps will be taken: (1) propose a spatial decomposition method (SDM); (2) establish a geometric model of SDM; (3) determine the numerical value of each parameter of the geometric model through field survey (surveying, drilling, mapping); (4) analyze the complete set of treatment technologies; (5) combined with the practical engineering of karst cave in the Shangyuan tunnel, SDM and its treatment technology are applied systematically. The research design of this paper is illustrated in Figure1. Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 20

Figure 1. Research design.

Appl.2. Establishment Sci. 2020, 10, 7204 of SDM for Large Karst Caves 3 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 20 2.1. General Situation When large karst caves are found in tunnel construction, the spatial location relationship between the tunnels and karst caves should be determined, and then a specific construction scheme must be formulated. The spatial relationship cannot provide detailed evidence of the preparation of construction technology, because we also need to satisfy certain necessary conditions such as karst cave filling, water content, and surrounding rock stability. Then, the SDM is proposed by taking into account the treatment process and technology of the bottom, top, and two sides of the cavern through the tunnel. The SDM divides the cave space into different spatial parts when a tunnel passes through a large karst cave, which provides technical solutions for subsequent cavern management.

2.2. Establishment of SDM Figure 1. Research design. Owing to the diversity and complexity of cave development, the edges of caves are irregular. 2. Establishment of SDM for Large Karst Caves 2.We Establishment assume an irregular of SDM karst for Largecave asKarst shown Caves in Figure 2, and take as the analysis object one of the 2.1.typical General sections, Situation labeled typical section I, as shown in Figure 3. 2.1. GeneralAccording Situation to Figures 2 and 3, the surrounding wall of the karst cave space is the cave outline. In When large karst caves are found in tunnel construction, the spatial location relationship between fact, Whenit is a three-dimensionallarge karst caves (3D)are closedfound surfacein tunnel of thecons cavetruction, contour. the Here,spatial typical location section relationship I (Figure the tunnels and karst caves should be determined, and then a specific construction scheme must between2) is selected, the tunnels such that and the karst cave caves contour should line be is adete closedrmined, curve, and as then shown a specific in the middleconstruction line in scheme Figure be formulated. The spatial relationship cannot provide detailed evidence of the preparation of must3. Owing be formulated. to the erosion The spatialof the surroundingrelationship cannotrocks inprovide the process detailed of evidencekarst development, of the preparation a certain of construction technology, because we also need to satisfy certain necessary conditions such as karst constructionthickness of antechnology, unstable structurebecause we layer also could need be to formed satisfy certainin the surrounding necessary conditions rock of the such karst as cave.karst cave filling, water content, and surrounding rock stability. Then, the SDM is proposed by taking into caveThe tunnel filling, outline water content, (line g) andis the surrounding shape of the rock outermost stability. protective Then, the layer SDM in is theproposed design. by At taking the top into of account the treatment process and technology of the bottom, top, and two sides of the cavern through accountthe tunnel the profile, treatment we drawprocess an and artificial technology tangent of ca thelled bottom, the tunnel top, andtop linetwo (linesides c), of whilethe cavern at the through bottom the tunnel. The SDM divides the cave space into different spatial parts when a tunnel passes through a theof the tunnel. tunnel The contour SDM divides we draw the a cave tangent space called into dithfferente tunnel spatial bottom parts line when (line a d). tunnel There passes is typically through a large karst cave, which provides technical solutions for subsequent cavern management. acertain large karstthickness cave, of which fillings provides in karst technical caves, which solutions may befor gravel, subsequent clay, cavernsand, etc., management. and the treatment of 2.2.fillings Establishment is also an important of SDM part of karst treatment; therefore, the filling line (line e) should be drawn. 2.2.Water Establishment is the most ofcommon SDM filling material in karst and has a significant impact on tunnel construction, whichOwing is represented to the diversity by the and water complexity level (line of cavef). Lines development, a–g are collectively the edges ofreferred caves areto as irregular. spatial Owing to the diversity and complexity of cave development, the edges of caves are irregular. Wedecomposition assume an irregularlines (SDLs). karst SDLs cave are as continuous shown in Figure surfac2es, andrather take than as closed the analysis curves object or line one segments. of the We assume an irregular karst cave as shown in Figure 2, and take as the analysis object one of the typicalFor example, sections, the labeled water level typical (line section f) represents I, as shown the water in Figure surface3. in the project. typical sections, labeled typical section I, as shown in Figure 3. According to Figures 2 and 3, the surrounding wall of the karst cave space is the cave outline. In fact, it is a three-dimensional (3D) closed surface of the cave contour. Here, typical section I (Figure 2) is selected, such that the cave contour line is a closed curve, as shown in the middle line in Figure 3. Owing to the erosion of the surrounding rocks in the process of karst development, a certain thickness of an unstable structure layer could be formed in the surrounding rock of the karst cave. The tunnel outline (line g) is the shape of the outermost protective layer in the design. At the top of the tunnel profile, we draw an artificial tangent called the tunnel top line (line c), while at the bottom of the tunnel contour we draw a tangent called the tunnel bottom line (line d). There is typically a certain thickness of fillings in karst caves, which may be gravel, clay, sand, etc., and the treatment of Appl.fillings Sci. is2020 alsoFigure, 10 ,an x 2.FOR important Schematic PEER REVIEW partdiagram of karst of the treatment; spatial relation relationship therefore,ship between the filling the karstline (line cave ande) should tunnel. be drawn.4 of 20 Water is the most common filling material in karst and has a significant impact on tunnel construction, which is represented by the water level (line f). Lines a–g are collectively referred to as spatial decomposition lines (SDLs). SDLs are continuous surfaces rather than closed curves or line segments. For example, the water level (line f) represents the water surface in the project.

Figure 3. SchematicSchematic diagram diagram of of the the SDM (spatial decomposition method).

UsingFigure SDLs, 2. theSchematic cave space diagram can of be the divided spatial relation into eightship betweenspace parts: the karst the caveupper and space tunnel. of the cave (①), the tunnel cavity space (②), the karst cave space on the two wings of the tunnel (③), the arch bottom cave space (④), the water space (⑤), the filling space (⑥), and the unstable surrounding rock space (⑦). The karst cave space on the two wings of the tunnel can be divided into the left and right space parts, respectively, ③-L and ③-R. The specific space numbers are shown in Figure 3 ①–⑦.

2.3. Geometric Model of SDM To facilitate the application of SDM, the geometric model of SDM should be established. Because the section of the tunnel and the karst cave along the direction of tunnel excavation are in a closed ring, a cylindrical coordinate system is chosen for the mathematical model, as shown in Figures 2 and 4. The central axis of the tunnel is taken as the Z-axis, and the intersection of the tunnel entrance and the karst cave is set as the origin Z = 0. Additionally, a cartesian coordinate system is set up. The reason for setting up two coordinate systems is that lines a, b, and g in Figure 3 are circular closed curves, which are more conveniently represented in a cylindrical coordinate system, whereas lines c and d and other straight lines are more conveniently represented in a cartesian coordinate system. The two established coordinate systems are shown in Figure 5.

Figure 4. Schematic diagram of the cylindrical coordinate system.

Let M = (x, y, zm) be any point in the surrounding rock of the karst cave, and convert it into a cylindrical coordinate system, which can be expressed as follows [22]: x = ρ cosθ  y = ρsinθ , (1)  = z zm where the geometric meaning of ρ is polar diameter, which is the length of OM’, but, in practice, the cave is not a regular cylinder, such that the value of ρ changes. The geometric meaning of θ is the Appl. Sci. 2020, 10, 7204 4 of 20

Appl. AccordingSci. 2020, 10, xto FOR Figures PEER REVIEW2 and3 , the surrounding wall of the karst cave space is the cave outline.4 of 20 In fact, it is a three-dimensional (3D) closed surface of the cave contour. Here, typical section I (Figure2) is selected, such that the cave contour line is a closed curve, as shown in the middle line in Figure3. Owing to the erosion of the surrounding rocks in the process of karst development, a certain thickness of an unstable structure layer could be formed in the surrounding rock of the karst cave. The tunnel outline (line g) is the shape of the outermost protective layer in the design. At the top of the tunnel profile, we draw an artificial tangent called the tunnel top line (line c), while at the bottom of the tunnel contour we draw a tangent called the tunnel bottom line (line d). There is typically a certain thickness of fillings in karst caves, which may be gravel, clay, sand, etc., and the treatment of fillings is also an important part of karst treatment; therefore, the filling line (line e) should be drawn. Water is the most common filling material in karst and has a significant impact on tunnel construction, which is represented by the water level (line f). Lines a–g are collectively referred to as spatial decomposition lines (SDLs). SDLsFigure are continuous 3. Schematic surfaces diagram ratherof the SDM than (spatial closed decomposition curves or line method). segments. For example, the water level (line f) represents the water surface in the project. UsingUsing SDLs,SDLs, the the cave cave space space can can be be divided divided into into eight eight space space parts: parts: the the upper upper space space of the of cavethe cave (O1 ), the(①), tunnel the tunnel cavity cavity space space (O2 ), the (② karst), the cave karst space cave onspace the on two the wings two ofwings the tunnel of the (tunnelO3 ), the ( arch③), the bottom arch cavebottom space cave (O4 space), the water(④), the space water (O5 ),space the filling (⑤), the space filling (O6 ), space and the (⑥ unstable), and the surrounding unstable surrounding rock space rock (O7 ). Thespace karst (⑦). cave The space karst oncave the space two wingson the of two the wings tunnel of can the be tunnel divided can into be divided the left andinto rightthe left space and parts, right respectively,space parts, respectively,O3 -L and O3 -R. ③ The-L and specific ③-R. space The numbersspecific space are shown numbers in Figure are shown3 O1 – Oin7 . Figure 3 ①–⑦.

2.3.2.3. GeometricGeometric ModelModel ofof SDMSDM ToTo facilitatefacilitate thethe applicationapplication ofof SDM,SDM, thethe geometricgeometric modelmodel ofof SDMSDM shouldshould bebe established.established. BecauseBecause thethe sectionsection of of the the tunnel tunnel and and the the karst karst cave cave along along the th directione direction of tunnel of tunnel excavation excavation are in are a closedin a closed ring, aring, cylindrical a cylindrical coordinate coordinate system system is chosen is chosen for the for mathematicalthe mathematical model, model, as shownas shown in Figuresin Figures2 and 2 and4. The4. The central central axis axis of theof the tunnel tunnel is taken is taken as theas theZ-axis, Z-axis, and and the the intersection intersection of the of tunnelthe tunnel entrance entrance and and the karstthe karst cave cave is set is as set the as origin the origin Z = 0. Z Additionally, = 0. Addition aally, cartesian a cartesian coordinate coordinate system system is set up. is set The up. reason The forreason setting for upsetting two up coordinate two coordinate systems systems is that linesis that a, b,lines and a, gb, in and Figure g in3 Figureare circular 3 are closedcircular curves, closed whichcurves, are which more are conveniently more conveniently represented represented in a cylindrical in a cylindrical coordinate coordinate system, system, whereas whereas lines c andlines d c andand otherd and straight other straight lines are lines more are conveniently more convenient representedly represented in a cartesian in a cartesian coordinate coordinate system. The system. two establishedThe two established coordinate coordinate systems are systems shown are in shown Figure 5in. Figure 5.

FigureFigure 4.4. SchematicSchematic diagramdiagram ofof thethe cylindricalcylindrical coordinatecoordinate system.system.

where the geometric meaning of ρ is polar diameter, which is the length of OM’, but, in practice, the cave is not a regular cylinder, such that the value of ρ changes. The geometric meaning of θ is the Appl. Sci. 2020, 10, x FOR PEER REVIEW 5 of 20 polarAppl. Sci. angle,2020, which10, 7204 is the angle between OM’ and the positive X -axis. The geometric meaning of5 Z of is 20 the plane parallel to plane XOY.

FigureFigure 5. 5. MathematicalMathematical model model of of the the SDM. SDM.

TheLet MSDL= (line(x, y, a–g)zm) bein Figure any point 3 is inconverted the surrounding into a polar rock coordinate of the karst system, cave, as and shown convert in Figure it into 5. a Thecylindrical following coordinate formulae system, are obtained: which can be expressed as follows [22]: ρ = θ  k K( ) , (2)  x = ρ cos θ   ρy ==ρ sinθ θ , (1)  r R( ) , (3)  z = zm ρ = θ t T ( ) . (4) where the geometric meaning of ρ is polar diameter, which is the length of OM’, but, in practice, the caveρk is isthe not geometric a regular expression cylinder, such of the that cave the valuecontour of (lineρ changes. a), ρr is The the geometric geometric meaning expression of θ ofis the the surroundingpolar angle, whichrock stability is the angle edge between line (line OM’ b), ρ andt is the the geometric positive X expression -axis. The geometricof the tunnel meaning contour of lineZ is (linethe plane g), and parallel θ is the to angle plane betweenXOY. the counterclockwise direction of the polar coordinate axis ρ and the XThe-axis. SDL (line a–g) in Figure3 is converted into a polar coordinate system, as shown in Figure5. The followingAccording formulae to the cartesian are obtained: coordinate system: ρ ==K(θ), (2) kY y1 , (5)

ρr ==R(θ), (3) Y y2 , (6) ρt ==T(θ). (4) Y y3 , (7) ρk is the geometric expression of the cave contour= (line a), ρr is the geometric expression of the Y y4 , (8) surrounding rock stability edge line (line b), ρt is the geometric expression of the tunnel contour line where(line g), y1 andis theθ isgeometric the angle expression between the of counterclockwisethe tunnel top line direction (line c), ofy2 theis the polar geometric coordinate expression axis ρ and of tunnelthe X-axis. bottom line (line d), y3 is the geometric expression of the water level line (line f), and y4 is the geometricAccording expression to the of cartesian the filling coordinate line (line system: e).

3. Determination of the Geometric Model ParametersY = y1, of SDM (5) After the geometric model of SDM is established, many parameters are involved in the model, Y = y2, (6) such as the cave contour (ρk), the surrounding rock stability edge line (ρr), the tunnel contour line (ρt), = and the tunnel top line (y1). These parametersY ycan3, be obtained by field investigations and (7) measurements, and then input to the geometric model to draw a 3D spatial diagram of the karst cave Y = y4, (8) and tunnel. The figure can also describe the comprehensive development of the surrounding rock crushing,where y1 iswater-bearing, the geometric and expression filling space of the of tunnelthe tunnel, top linethereby (line providing c), y2 is the a geometricspatial reference expression for the of selectiontunnel bottom of subsequent line (line treatment d), y3 is the methods. geometric expression of the water level line (line f), and y4 is the geometric(1) Determination expression of of the the ﬁlling tunnel line central (line axis, e). contour line, tunnel top, and bottom line. The tunnel central axis is the reference line of the geometric model for the SDM. The central axis and the outline of the tunnel can be determined by the method of backward intersection. Starting Appl. Sci. 2020, 10, 7204 6 of 20

3. Determination of the Geometric Model Parameters of SDM After the geometric model of SDM is established, many parameters are involved in the model, such as the cave contour (ρk), the surrounding rock stability edge line (ρr), the tunnel contour line (ρt), and the tunnel top line (y1). These parameters can be obtained by field investigations and measurements, and then input to the geometric model to draw a 3D spatial diagram of the karst cave and tunnel. The figure can also describe the comprehensive development of the surrounding rock crushing, water-bearing, and filling space of the tunnel, thereby providing a spatial reference for the selection of subsequent treatment methods. (1) Determination of the tunnel central axis, contour line, tunnel top, and bottom line. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 20 The tunnel central axis is the reference line of the geometric model for the SDM. The central axis and the outline of the tunnel can be determined by the method of backward intersection. Starting from the from the tunnel entrance, measured sections are acquired at intervals of D1, as shown in Figure 6, tunnel entrance, measured sections are acquired at intervals of D , as shown in Figure6, with Z Zi, with Z1 − Zi, where D1 ≤ 5 m and the total of the cave is not less1 than five measured sections.1 −The where D 5 m and the total of the cave is not less than five measured sections. The tunnel top tunnel top1 ≤line (line c) and the tunnel bottom line (line d) were determined by the engineering survey line (line c) and the tunnel bottom line (line d) were determined by the engineering survey method. method. Finally, each line (line c) could be connected to the top surface and each line (line d) to the Finally, each line (line c) could be connected to the top surface and each line (line d) to the bottom surface. bottom surface.

FigureFigure 6. LocationLocation measurement measurement of of ea eachch line line of of the the tunnel tunnel body.

(2) Determination of the cave contour line (line a). A cavity cavity formed formed in in a arock rock mass mass is also is also called called a cave. a cave. There There are many are many ways waysto measure to measure the space the ofspace the ofcave the development, cave development, such as such 3D laser as 3D scanning laser scanning technology technology [23–26] [and23–26 laser] and ranging. laser ranging. In this project,In this project, an angle-laser an angle-laser rangefinder rangefinder was used was which used can which display can both display the angle both and the distance. angle and As distance. shown inAs Figure shown 7, in when Figure laser7, when ranging laser is ranging performed is performed on each monitoring on each monitoring point of section point of Z sectioni, the laserZi, theranging laser equipmentranging equipment cannot be cannot conveniently be conveniently installed installed at the corresponding at the corresponding O point O pointof the of central the central axis axisof the of tunnel,the tunnel, and and thus thus a tripod a tripod is used is used to toset set the the laser laser equipment equipment at at the the bottom bottom of of the the cave, cave, directly directly below below point O. O. By By adjusting adjusting the the laser laser measuring measuring angle, angle, the the circular circular distance distance measuring measuring point point density density can can be adjusted,be adjusted, and and the the value value of ofthe the angle angle can can be be adjusted adjusted ba basedsed on on the the size size of of the the cave cave to to ensure ensure that that each section has at least 12 points. Finally, thethe measurement measurement points points of of the theZ –ZZ1 section–Zi section are plotted are plotted in a cylindrical in a cylindrical coordinate coordinate system, 1 i system,and the pointsand the of points the same of the section same are section joined are to form joined rings. to form The measurementrings. The measurement points (i-1)–( pointsi-n) of ( sectioni-1)–(i- n)Zi ofare section shown Z ini are Figure shown7. The in measurementFigure 7. The pointsmeasurement of di fferent points sections of different are connected sections by are linear connected segments, by linearas shown segments, in Figure as7 shown. The measurement in Figure 7. The points measurement (1–2)–( i-2), points where (1–2)–( “i” is thei-2), section where number,“i” is the form section the number,3D structure form of the the 3D karst structure cave. of the karst cave. When laser ranging is carried out in section Zi, the parameters θ and ρk of the geometric model cannot be directly measured unless a geometric calculation is employed. Therefore, the geometric relationship is established, as shown in Figure8. Laser ranging equipment can read the distance ( d2) between the laser (point E) and the proposed measurement point (point M) directly. The measurement angle can be read (α). d3 is the height of the laser equipment tripod, which can be directly measured in the field.

Figure 7. Schematic diagram depicting the method for determining the cave contour. Appl. Sci. 2020, 10, x FOR PEER REVIEW 6 of 20 from the tunnel entrance, measured sections are acquired at intervals of D1, as shown in Figure 6, with Z1 − Zi, where D1 ≤ 5 m and the total of the cave is not less than five measured sections. The tunnel top line (line c) and the tunnel bottom line (line d) were determined by the engineering survey method. Finally, each line (line c) could be connected to the top surface and each line (line d) to the bottom surface.

Figure 6. Location measurement of each line of the tunnel body.

(2) Determination of the cave contour line (line a). A cavity formed in a rock mass is also called a cave. There are many ways to measure the space of the cave development, such as 3D laser scanning technology [23–26] and laser ranging. In this project, an angle-laser rangefinder was used which can display both the angle and distance. As shown in Figure 7, when laser ranging is performed on each monitoring point of section Zi, the laser ranging equipment cannot be conveniently installed at the corresponding O point of the central axis of the tunnel, and thus a tripod is used to set the laser equipment at the bottom of the cave, directly below point O. By adjusting the laser measuring angle, the circular distance measuring point density can be adjusted, and the value of the angle can be adjusted based on the size of the cave to ensure that each section has at least 12 points. Finally, the measurement points of the Z1 –Zi section are plotted in a cylindrical coordinate system, and the points of the same section are joined to form rings. The measurement points (i-1)–(i- n) of section Zi are shown in Figure 7. The measurement points of different sections are connected by Appl.linear Sci. segments,2020, 10, 7204 as shown in Figure 7. The measurement points (1–2)–(i-2), where “i” is the section7 of 20 number, form the 3D structure of the karst cave.

Appl. Sci. 2020, 10, x FOR PEER REVIEW 7 of 20

When laser ranging is carried out in section Zi, the parameters θ and ρk of the geometric model cannot be directly measured unless a geometric calculation is employed. Therefore, the geometric relationship is established, as shown in Figure 8. Laser ranging equipment can read the distance (d2) between the laser (point E) and the proposed measurement point (point M) directly. The measurement angle can be read (α). d3 is the height of the laser equipment tripod, which can be directly measured in the field. d1 is the vertical distance from the laser equipment to the central axis. The process for determining the value of d1 consists of three processes. First, the total station is used to set a point O’ on the wall of the karst cave with an equal elevation to that of the central axis. Second, the distance between points E and O’ and the measuring angle η are measured using the laser equipment. Third, trigonometricFigureFigure formulae 7.7. Schematic are used diagram to obtain depicting the value the method of d1. forfor determiningdetermining thethe cavecave contour.contour.

Figure 8. GeometricGeometric relationship of th thee laser ranging parameters.

d1 is the vertical distance from the laser equipment to the central axis. The process for determining According to the cosine formula, the geometric expression of the cave contour (line a) ρk is as follows:the value of d1 consists of three processes. First, the total station is used to set a point O’ on the wall of the karst cave with an equal elevation to that of the central axis. Second, the distance between points E and ρ = 2 + 2 − β O’ and the measuring angle η are measuredk d using1 d 2 the2 laserd1d 2 equipment.cos . Third, trigonometric formulae(9) are used to obtain the value of d1. As can be seen from the sine formula: According to the cosine formula, the geometric expression of the cave contour (line a) ρk is as follows: d sin β γq= 1 arcsin2 2 , (10) ρk = d1 + d2 2ρd1d2 cos β. (9) − k As canπ be seen from the sine formula: where β =−α ; formula (10) can be further derived as follows: d1 sin β 2 γ = arcsin , (10) ρk ρ = 2 + 2 − α k d1 d 2 2d1d2sin . (11) where β = π α; Formula (10) can be further derived as follows: 2 − Similarly, formula (11) can be further derived as follows: q 2 2 ρk = d1 + d2 d cos2d1dα2 sin α. (11) γ = arcsin −1 ρ . (12) k

Substituting θα=−γ yields the following: d cosα θα=−arcsin 1 ρ . (13) k ρ = θ Therefore, formula (2) k K( ) can be derived as follows: Appl. Sci. 2020, 10, 7204 8 of 20

Similarly, Formula (11) can be further derived as follows:

d cos α γ = arcsin 1 . (12) ρk

Substituting θ = α γ yields the following: − d cos α θ = α arcsin 1 . (13) Appl. Sci. 2020, 10, x FOR PEER REVIEW − ρk 8 of 20

Therefore, Formula (2) ρk = K(θ) can be derived as follows: d cosα ρ =− αα1 ! =+−q 22 k Kddddarcsind cos α 12 2 12 sin . (14) = 1 ρ = 2 + 2 ρk Kα arcsin k d1 d2 2d1d2 sin α. (14) − ρk −

Finally, the measurement points of section Zi are plotted in a 3D diagram to form the cave Finally, the measurement points of section Zi are plotted in a 3D diagram to form the cave contour contour (line a). (line a). (3) Determination of the surrounding rock stability edge line (line b). (3) Determination of the surrounding rock stability edge line (line b). Section Zi in Figure 7 is selected to display the surrounding rock stability edge line (line b), as Section Zi in Figure7 is selected to display the surrounding rock stability edge line (line b), shown in Figure 9. Line b is located inside the rock mass and is not visible; thus, it cannot be measured as shown in Figure9. Line b is located inside the rock mass and is not visible; thus, it cannot be directly. In order to accurately estimate the thickness of the unstable surrounding rock, a coring measured directly. In order to accurately estimate the thickness of the unstable surrounding rock, survey method is adopted [27–31]. The same section is selected for the coring survey and laser a coring survey method is adopted [27–31]. The same section is selected for the coring survey and ranging, but it is more difficult and time-consuming to acquire high-density coring survey points laser ranging, but it is more diﬃcult and time-consuming to acquire high-density coring survey points compared to performing laser ranging, and the detection accuracy is limited. Therefore, the core compared to performing laser ranging, and the detection accuracy is limited. Therefore, the core points points can be adjusted according to the size of the cave, with no fewer than six core points for each can be adjusted according to the size of the cave, with no fewer than six core points for each section. section. Each core point should be drilled into the medium-weathered rock mass to a depth of no less Each core point should be drilled into the medium-weathered rock mass to a depth of no less than than 2 m [32]; that is, dp ≥ 2 m. du is the coring depth of the unstable surrounding rock. The du depths 2 m [32]; that is, dp 2 m. du is the coring depth of the unstable surrounding rock. The du depths of of each core point are≥ connected to draw a closed stable line b, as shown in Figure 9. Then, line b can each core point are connected to draw a closed stable line b, as shown in Figure9. Then, line b can be be expressed as: expressed as: ρ ==d ρrr duu.. (15) (15)

Figure 9. Determination of the surrounding rock stability edge line (line b).

(4) Determination ofof thethe fillingfilling lineline (line(line e).e). The fillingfilling line (line e) of sediments such as gravel,gravel, sand, and silt in the cave was measured by the engineeringengineering measurementmeasurement method, method, which which is tois measureto measure the distancethe distance between between the filling the surfacefilling andsurface the tunneland the center tunnel point center O with point a total O with station. a total This station. distance This was distance set to y4 ,was and set the to value y4, and of y 4thein Formulavalue of y (9)4 in was formula determined. 9 was determined. Because df may Because be below df may or abovebe below the pointor above O, d thef satisfies point O, Formula df satisfies (16). Lineformula e adopts 16. Line the cartesiane adopts coordinatethe cartesian system coordinate with the system central with axis the of the central tunnel ax asis theof the origin. tunnel It should as the origin. It should be noted that, when determining the coring survey of line b, the thickness df of the filling material can be determined simultaneously, as shown in Figure 10. Then, line e can be expressed as follows: π y = d − K(− ) . (16) 4 f 2 (5) Determination of the water level (line f). The water level (line f) of the cave is spatiotemporally dependent, such that the water level should be a change interval. The variation range of the water level is obtained by observing the historical trace of the water level on the karst cave wall. The value of y3 is the maximum over the years, and the water depth is set as dw, as shown in Figure 10. Again, because dw may below or above point O, dw satisfies formula 18. Then, the water level line can be expressed as follows: Appl. Sci. 2020, 10, 7204 9 of 20

beAppl. noted Sci. 2020 that,, 10 when, x FORdetermining PEER REVIEW the coring survey of line b, the thickness df of the ﬁlling material9 of can 20 be determined simultaneously, as shown in Figure 10. Then, line e can be expressed as follows: ππ y4 = dd f −KK((− ).) . (17) (16) 3 w − − 22

Appl. Sci. 2020, 10, x FOR PEER REVIEW 9 of 20

π FigureFigure 10.10. Determination of y the= ﬁllingfillingd − K lineline(− (line(line) . e)e) andand waterwater levellevel (line(line f).f). (17) 3 w 2 (5)(6) DeterminationGeometric drawing of the of water SDM. level (line f). TheAfter water the geometric level (line parameters f) of the cave of is the spatiotemporally SDL are determined, dependent, the spatial such that and the structural water level diagram should of bethe a karst change cave interval. and tunnel The variation can be drawn, range ofas theillustrated water level in Figure is obtained 11. From by observing this structural the historical diagram, trace the ofspatial the water position level relation on the karst between cave wall.the cave The and value the of ytunnel3 is the is maximum immediately over clear, the years, thus andrealizing the water the depthrealistic is settransformation as dw, as shown of the in SDM. Figure 10. Again, because dw may below or above point O, dw satisﬁes Formula (18). Then, the water level line can be expressed as follows: π y3 = dw K( ). (17) − − 2 Figure 10. Determination of the filling line (line e) and water level (line f). (6) Geometric drawing of SDM. After the(6) geometric Geometric drawing parameters of SDM. of the SDL are determined, the spatial and structural diagram of the karst caveAfter and the geometric tunnel canparameters be drawn, of the asSDL illustrated are determined, in Figure the spatial 11 and. From structural this structuraldiagram of diagram, the spatialthe position karst cave relation and tunnel between can be drawn, the cave as illustrated and the in tunnelFigure 11. is From immediately this structural clear, diagram, thus the realizing the spatial position relation between the cave and the tunnel is immediately clear, thus realizing the realistic transformationrealistic transformation of the ofSDM. the SDM.

Figure 11. Geometry of the SDM.

4. Complete Set Technology of SDM

4.1. General Situation After the establishment of the SDM, it is necessary to take corresponding technical measures to

deal with the difficult problem of tunneling through a large karst cave. According to the SDM, the FigureFigure 11. 11.Geometry Geometry of ofthe the SDM. SDM. tunnel and the karst cave are divided into different spatial parts, and corresponding measures are 4.taken Complete to treat4. Complete Set the Technologydifferent Set Technology karst of spaces. SDM of SDM In order to ensure safety, the SDM is managed in the sequence of the cave spaces at the top, bottom, and two wings of the tunnel. 4.1. General Situation 4.1. General Situation 4.2. TreatmentAfter Technology the establishment of Karst of Cave the SDM, Space it atis necessarythe Top ofto Tunneltake corresponding technical measures to Afterdeal the with establishment the difficult problem of the of SDM, tunneling it is thro necessaryugh a large to karst take cave. corresponding According to the technical SDM, the measures to dealIn withordertunnel the to and ensure di thefficult karst the problem cave safety are dividedof of the tunneling upper into different spac throughe spatial duringa parts, large the and tunnel karst corresponding cave.constr Accordinguction, measures the are toupper the SDM,space theof the tunnel cavetaken and should to the treat karstbe the treated different cave arefirst. karst divided spaces. In into order di tofferent ensure spatial safety, the parts, SDM andis managed corresponding in the sequence measures are of the cave spaces at the top, bottom, and two wings of the tunnel. takenThe to treat cavern the dispacefferent at karstthe top spaces. of the In tunnel order to contains ensure safety,the upper the SDM space is managedof the cave in the(①) sequence and the ofunstable the cave surrounding4.2. spaces Treatment at Technology the rock top, space bottom, of Karst (② Cave), and as Space shown two at wingsthe in Top the of of Tunnel Figure the tunnel. 3. According to the height of the upper space of the caveIn order and to theensure stability the safety of of the the brokenupper spac surroundinge during the tunnel rock constr massuction, of the the unstable upper space surrounding rock space,of the methods cave should such be treatedas supporting first. column reinforcement, increasing parapet reinforcement, and anchor netThe shotcrete cavern space support at the topare of generally the tunnel adopted. contains the upper space of the cave (①) and the unstable surrounding rock space (②), as shown in the Figure 3. According to the height of the upper space of the cave and the stability of the broken surrounding rock mass of the unstable surrounding rock space, methods such as supporting column reinforcement, increasing parapet reinforcement, and anchor net shotcrete support are generally adopted. Appl. Sci. 2020, 10, 7204 10 of 20

4.2. Treatment Technology of Karst Cave Space at the Top of Tunnel In order to ensure the safety of the upper space during the tunnel construction, the upper space of the cave should be treated ﬁrst. The cavern space at the top of the tunnel contains the upper space of the cave (O1 ) and the unstable surrounding rock space (O2 ), as shown in the Figure3. According to the height of the upper space of the cave and the stability of the broken surrounding rock mass of the unstable surrounding rock space, Appl. Sci. 2020, 10, x FOR PEER REVIEW 10 of 20 methods such as supporting column reinforcement, increasing parapet reinforcement, and anchor net shotcrete support(1) When are generally the surrounding adopted. rock stability of the karst cave is poor, and the spatial height of the (1) Whenupper the space surrounding of the cave is large: rock stability of the karst cave is poor, and the spatial height of the Generally, the unstable rock space has weak stability at the top of the tunnel, and rubble may upper space ofAppl. the Sci. 2020 cave, 10, isx FOR large: PEER REVIEW 10 of 20 fall or even collapse during the construction and operation of the tunnel. Moreover, the damage to Generally, the unstable rock space has weak stability at the top of the tunnel, and rubble may the tunnel(1) Whenbody increases the surrounding in accordance rock stab withility the of theincr karsteased cave height is poor, in the and upper the spatialspace of height the cave, of the with fall or evenhigher collapseupper gravitational space during of the cavepotential the is construction large: energy and greater and operation converted kinetic of the energy. tunnel. Therefore, Moreover, the unstable the damage to the tunnelsurrounding bodyGenerally, increases rock theshould in unstable accordance be cleaned, rock space after with has which weak the the stability increased rock surface at the height shouldtop of the be in tunnel,supported the upper and withrubble space an may anchor of the cave, with highernet gravitationalfall to eliminateor even collapse the potential danger during posed energythe construction by the and unstable greater and operationsurrounding converted of the rock kinetictunnel. space. Moreover,energy. the Therefore, damage to the unstable theSecond, tunnel bodythe supporting increases in column accordance reinforcement with the incr metheasedod height is adopted; in the upper the bottom space of of the the cave, supporting with surroundingcolumn rockhigher goes should gravitational deep be into cleaned, potential the complete energy after surrounding and which greater the rock,converted rock the surface supporting kinetic shouldenergy. column Therefore, be top supported is installedthe unstable withwith a an anchor net to eliminateprefabricatedsurrounding the danger “U”-type rock should posed concrete be bycleaned, theslab, after unstable and which the “U”-type the surrounding rock surfacegroove should is rock filled be space. supportedwith low-grade with an concrete anchor to Second,stopnet the the to supporting topeliminate of the thekarst danger column cave posedand reinforcement prevent by the unstable the roof surroundingof method the karst is caverock adopted; space.from sagging, the bottom as shown of in theFigure supporting 12. Second, the supporting column reinforcement method is adopted; the bottom of the supporting column goescolumn deepinto goes deep the completeinto the complete surrounding surrounding rock, rock, the the supporting supporting column column top is installed top is with installed a with a prefabricatedprefabricated “U”-type concrete“U”-type concrete slab, and slab, the and “U”-typethe “U”-type groove groove is filled ﬁlled with with low-grade low-grade concrete concrete to to stop the top of thestop karst the cavetop of the and karst prevent cave and the prevent roof the of roof the of karst the karst cave cave from from sagging, sagging, as shown as shown in Figure in Figure 12. 12.

Figure 12. Schematic diagram of the supporting column treatment technology.

(2) When the surrounding rock stability of the karst cave is good , and the spatial height of the Figure 12. Schematic diagram of the supporting column treatment technology. upperFigure space of 12. theSchematic cave is large: diagram of the supporting column treatment technology. When the surrounding rock of the karst cave is relatively stable, the surrounding rock cannot be (2) When the surrounding rock stability of the karst cave is good, and the spatial height of the treated. Instead, the anchor net shotcrete support is directly adopted to reinforce the surrounding (2) Whenupper the space surrounding of the cave is rock large: stability of the karst cave is good, and the spatial height of the rock at the top directly, as shown in Figure 13. For example, reference [15] a takes a similar approach. upper space of theWhen cave the issurrounding large: rock of the karst cave is relatively stable, the surrounding rock cannot be treated.(3) When Instead, the distancethe anchor between net shotcrete the tunnel support vault is anddirectly the caveadopted wall to above reinforce is short: the surrounding When therockIn surrounding atthis the condition, top directly, rockthe as height shown of the of in theFigure karst parapet 13. cave For is example, isgenerally relatively re ferenceincreased stable, [15] to a takesmake the a surroundingthesimilar parapet approach. directly rock cannot be treated.contact Instead, (3)with When the or anchor eventhe distance go netdeep between shotcrete into the the complete tunnel support va ultrock and is at directly the the cave top wall adoptedof the above cave is to short:wall, reinforce and lower-grade the surrounding rock at theconcrete top directly,In is this poured condition, as showninto thethe upperheight in Figure ofspace the 13parapetabove. For the is example, generallystructural increased referenceprotection, to make as [15 shown] the a takesparapet in Figure a directly similar 14. For approach. example,contact referencewith or even [12] gotakes deep a similarinto the approach. complete rock at the top of the cave wall, and lower-grade concrete is poured into the upper space above the structural protection, as shown in Figure 14. For example, reference [12] takes a similar approach.

Figure 13. Schematic diagram of the treatment technology of anchor net spraying. Figure 13. Schematic diagram of the treatment technology of anchor net spraying. Figure 13. Schematic diagram of the treatment technology of anchor net spraying. Appl. Sci. 2020, 10, 7204 11 of 20

(3) When the distance between the tunnel vault and the cave wall above is short: In this condition, the height of the parapet is generally increased to make the parapet directly Appl. Sci. 2020, 10, x FOR PEER REVIEW 11 of 20 contact with or even go deep into the complete rock at the top of the cave wall, and lower-grade concrete is poured into the upper space above the structural protection, as shown in Figure 14. For example, reference [Appl.12] Sci. takes 2020, a10 similar, x FOR PEER approach. REVIEW 11 of 20

FigureFigure 14. FigureSchematicSchematic 14. Schematic diagram diagram diagram of of treatment treatment of treatment technique technique technique for for in in increasingcreasingcreasing the heightthe height of the protectiveof the protective wall. wall.

4.3. Treatment Treatment4.3. TreatmentTeachnology Teachnology ofof thethe KarstofKarst the Karst CaveCave Cave atat theatthe the BottomBottom Bottom of of the thethe Tunnel TunnelTunnel According to the SDM, there are four spatial parts in the karst cave space at the bottom of the Accordingtunnel: to the the arch SDM, bottom there cave spaceare four(④), thespatial water parts space (in⑤ ), the the karstfilling spacecave (⑥space ), and at the the unstable bottom of the tunnel: the thesurrounding arch arch bottom bottom rock cave cavespace space space(⑦). First, ( (④O4 the),), the the water water water in the spacespace water space( (⑤O5 ),), (the ⑤) shouldfilling filling be space space removed, ( (⑥O6 ),the and backfill the unstable surroundingsuch rock rock as silt space and gravel ( ⑦O7 ). inFirst, First, the backfill the the water water space inshould in the the bewater water removed, space space and ( (⑤ Othen5 ) should the following be removed, construction the backfill backfill such as silt judgment and gravel should in be the made. backfill backfill space should be removed, and then the following construction (1) When the height of the karst cave at the bottom of the tunnel is large: judgmentjudgment should When be the made. cavern space at the bottom of the tunnel is relatively high and the cave is water-rich, (1) When generally, the height a beam of or the arch karstis adopted cave to at cross the the bottom cave sp oface, the which tunnel can also is large:protect the karst water, Whenas the shown cavern in Figure space 15. atFor at the example, bottom references of of the the [13,15] tunnel tunnel take is is arelatively relativelysimilar approach. high high and and the the cave cave is is water-rich, water-rich, generally, aabeam beam oror archarch isis adoptedadopted toto crosscross thethe cavecave space,space, which can also protect the the karst karst water, water, as shown in Figure 1515.. For example, references [13,15] [13,15] take a similar approach.

Figure 15. Schematic diagram of the treatment technique of span mode (vertical profile).

(2) When the height of the karst cave at the bottom of the tunnel is small: When the height of the karst cave space at the bottom of the tunnel is small, grouting reinforcement and foundation replacement can be adopted, as shown in Figure 16. It is important to note that the height of the protective wall should be greater than that of the maximum water level, and the excess height should be no less than 2 m to prevent karst water from infiltrating above the Figuretunnel.Figure For15. example,SchematicSchematic reference diagram diagram [7] oftakes of the the a treatment treatmentsimilar approach. tec techniquehnique of span mode (vertical profile). proﬁle).

(2) When the height of the karst cave at the bottom of the tunnel is small: When thethe height height of theof the karst karst cave spacecave atspace the bottomat the ofbottom the tunnel of the is small, tunnel grouting is small, reinforcement grouting reinforcementand foundation and replacement foundation can replacement be adopted, can as be shown adopted, in Figure as shown 16. Itin is Figure important 16. It tois noteimportant that the to noteheight that of the protectiveheight of the wall protective should be wall greater should than be that greater of the than maximum that of the water maximum level, and water the excesslevel, andheight the should excess be height no less should than 2 be m tono prevent less than karst 2 m water to prevent from inﬁltrating karst water above from the infiltrating tunnel. For above example, the tunnel.reference For [7 example,] takes a similar reference approach. [7] takes a similar approach. Appl. Sci. 2020, 10, 7204 12 of 20 Appl. Sci. 2020,, 10,, xx FORFOR PEERPEER REVIEWREVIEW 12 of 20

Figure 16. FoundationFoundation replacement and gr groutingouting treatment technology.

4.4.4.4. Treatment TechnologyTechnology ofof KarstKarst CaveCave SpaceSpace on on Two Two Wings Wings of of Tunnel Tunnel AccordingAccording to thethe SDM,SDM, thethe twotwo wingswings of thethe tunneltunnel include three spaces and two space parts: thethe karstkarstkarst cavecavecave space spacespace on onon two twotwo wings wingswings of ofof tunnel tunneltunnel (O 3((③) and)) andand the thethe unstable unstableunstable surrounding surroundingsurrounding rock rockrock space spacespace (O7 ). ((⑦ For).). ForFor the treatmentthethe treatmenttreatment of the ofof thethe two twotwo wings wingswings of theofof thethe tunnel, tunnel,tunnel, the thethe distance distancedistance between betweenbetween the thethe tunnel tunneltunnel and andand the thethe wall wallwall of ofof the thethe cavecavecave shouldshould bebe considered,considered, andand thethe supportingsupporting methodmethod shouldshould bebe adoptedadopted accordingly.accordingly. (1)(1) WhenWhen oneone sidesideside ofofof thethethe tunneltunnetunnell isisis closecloseclose tototo thethethe cavecavecave wall:wall:wall: IfIf thethe distancedistancedistance betweenbetween oneone side side of of thethe tunneltunnel and and the the wall wall of of the the karst karst cave cave is is relatively relatively short, short, whichwhich isis generallygenerally lessless thanthan thethe maximummaximum designdesign thicknessthicknessickness ofof thethe protectiveprotective wall,wall, thethe partialpartial anchoranchor shouldshould bebe appliedapplied toto strengthenstrengthen thethe wallwall ofof thethe karstkarst cave,cave, andand thenthen the side should be backfilled backﬁlled with slurryslurry slatesslates oror pumpedpumped concrete,concrete, asas shownshown inin FigureFigure 1717.. When the two wings of thethe cavecave spacespace areare lessless thanthan thethe thicknessthickness ofofof thethe parapet,parapet, slurryslurry slatesslatesslates oror concreteconcrete backﬁllbackfill cancan bebe appliedappliedapplied tototo thethethe twotwotwo wings.wings. ForFor example,example, referencereference [[18]18] takestakestakes aaa similarsimilarsimilar approach.approach.approach.

Figure 17. Treatment technologytechnology ofof thethe karstkarst cavecave spacespace in a short distance.distance.

(2)(2) WhenWhen thethe distancedistancedistance betweenbetweenbetween thethethe twotwotwo wingswingswings ofofof thethethe tunneltunneltunnel andandand thethethe wallwallwall ofofof thethethe cavecavecave isisis large: large:large: WhenWhen thethe spacespace ofof the two wings of the cave is is large,large, itit generallygenerally exceedsexceeds thethe maximummaximum thickness thickness ofof the parapet. parapet. It It is is unnecessary unnecessary to to consider consider the the support support of the of the unstable unstable surrounding surrounding rock rock of the of two the twowings, wings, and the and protective the protective wall is wall used is to used protect to protect the two the sides two directly, sides directly, asas shownshown as showninin FigureFigure in 18.18. Figure ItIt mustmust 18. Itbe must emphasized be emphasized that if the that surrounding if the surrounding rock of rock the ofkarst the cave karst wall cave is wall relatively is relatively poor, poor, the thickness the thickness and andheight height of the of theprotective protective wall wall should should be beappropriately appropriately increased increased to to improveimprove improve thethe the protectionprotection protection ofof thethe tunneltunnel body.body. Appl. Sci. 2020, 10, 7204 13 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 13 of 20

Figure 18. Treatment techniquetechnique whenwhen thethe spacespace ofof thethe two-wingtwo-wing karstkarst cavecave isis large.large.

TheThe above treatment treatment techniques techniques have have solved solved the theproblem problem of karst of karstspaces spaces in the inupper, the upper,lower, lower,and two and wings two of wings the tunnel. of the tunnel.However, However, in practical in practicalapplications, applications, the above thetechnologies above technologies should be shouldanalyzed, be analyzed,improved, improved, supplemented, supplemented, and updated and updated according according to the specific to the specific situation situation to make to makethem themmore moreconsistent consistent with with the field the field practice. practice. At the At the same same time, time, the the complete complete set set technologies technologies for for the karstkarst cavecave atat thethe top,top, bottom,bottom, andand twotwo wingswings of the tunneltunnel are not independent from each other, butbut ratherrather areare interrelatedinterrelated systems,systems, and the relations among thethe three should be considered comprehensively. InIn orderorder toto achieveachieve thethe purposespurposes of safety, scientificscientific soundness,soundness, systematicity, andand economyeconomy of SDM, itsits applicationapplication shouldshould followfollow thethe followingfollowing principles:principles: (1)(1) High safety. The safety principle is the most basicbasic principleprinciple ofof construction;construction; thethe activitiesactivities cancan proceed smoothly and orderly if safety andand securitysecurity areare ensured.ensured. (2)(2) Good flexibility.flexibility. AlthoughAlthough there there are are fixed fixed construction construction sets sets and and methods methods for the for treatment the treatment of karst, of karst,specific specific treatment treatment measures measures should should be formulated be formulated according according to the torelationship the relationship between between the theactual actual karst karst state andstate the and rock the mass’s rock physical,mass’s ph mechanical,ysical, mechanical, geotechnical, geotechnical, engineering engineering geological, geological,hydrogeological hydrogeological characteristics, characteristics, among others. among others. (3)(3) Rationality and scientificity.scientificity. InIn short, the treatmenttreatment ofof karst should not only comply with various laws andand regulationsregulations but but also also consider consider the the feasibility feasibility of treatmentof treatment methods methods and and whether whether they they can canachieve achieve practical practical effects. effects. (4)(4) Sustainability. The treatment of karst should notnot only be aimedaimed toto achieveachieve successfulsuccessful treatmenttreatment but also take into accountaccount thethe smoothsmooth conversionconversion ofof constructionconstruction techniquestechniques andand proceduresprocedures before and after the treatment, the continuity of the operation before and after the treatment, and the long-term safety and reliability of the tunnel operation. (5)(5) Economy. Under Under the the premise premise of of ensuring ensuring construc constructiontion safety safety and and project project quality quality as aswell well as sustainability,as sustainability, every every effort effort should should be bemade made to tosave save costs costs and and reduce reduce input. input. (6) Realization of technology. A perfect construction plan may be designed, but it could be restricted (6) Realization of technology. A perfect construction plan may be designed, but it could be restricted by by subjective and objective conditions such as advanced equipment, materials, and construction subjective and objective conditions such as advanced equipment, materials, and construction conditions, conditions, such that the perfect plan cannot be implemented. Therefore, the formulation of the such that the perfect plan cannot be implemented. Therefore, the formulation of the scheme scheme should also consider its feasibility. should also consider its feasibility.

5.5. Engineering Application

5.1.5.1. General Situation WhenWhen thethe construction construction of theof the Shangyuan Shangyuan tunnel tunn in theel in Guizhou the Guizhou section ofsection the Shanghai–Kunming of the Shanghai– high-speedKunming high-speed railway reached railwayD1K760 reached+ 965D1K760, a large + 965 ﬁlling-type, a large filling-type karst cave waskarst discovered. cave was discovered. The ﬁlling materialThe filling was material soft and was plastic-like soft and clay,plastic-like and there clay, was and a smallthere amountwas a small of water amount in the of cave.water Through in the cave. the analysisThrough of the the analysis geological of the prediction geological results prediction in advance, results no otherin advance, karst caves no other developed karst caves around developed the cave. Figurearound 19 the shows cave. the Figure initial 19 state shows of the the karst initial cave state discovery. of the karst This cave project discover adoptedy. This the SDMproject to treatadopted the karstthe SDM cave. to The treat main the karst application cave. The process main of applicat the SDMion is process shown of in the Figure SDM 20 is. shown in Figure 20. Appl. Sci. 2020, 10, 7204 14 of 20 Appl. Sci. 2020,, 10,, xx FORFOR PEERPEER REVIEWREVIEW 14 of 20

FigureFigure 19.19. Development diagram of thethe karstkarst cave.cave.

FigureFigure 20.20. Application process of the spatialspatial decompositiondecomposition method.method.

5.2.5.2. Supplementary Geological Survey AccordingAccording toto thethe geometric geometric model model parameter parameter determination determination method method of of the the SDM, SDM, the the central central axis axis of theof the tunnel tunnel is located is located by means by ofmeans the engineering of the engi measurementneering measurement method. The method. longitudinal The developmentlongitudinal lengthdevelopment of the karstlength cave of the is approximatelykarst cave is approxim 51 m alongately the 51 tunnelm along direction, the tunnel and direction, survey sectionsand survey are setsections up every are 7–8set mup along every the 7–8 central m along axis, suchthe thatcentra sevenl axis, survey such sections that seven are constructed—namely, survey sections are Zconstructed1–Z7 in Figure—namely, 21a–g. Z The1-Z7 distancein Figure of 21a–g. the measurement The distance point of the on measurement the wall of the point cave on is the measured wall of bythe acave laser is rangefinder.measured by Aa laser core rangefinder. investigation A is core carried investigation out on the is carried surrounding out on rock the surrounding of the karst cave,rock of as the shown karst cave, in Figure as shown 22a,b. in Figure Section 22a,b.Z4 Sectionis taken Z4 as isis takentaken an example, asas anan example,example, as shown asas shownshown in Figure inin FigureFigure 22c. y = 774 cm and y = 356 cm can be obtained by surveying the filling and karst water in the 22c.1 y−1 = −774 cm and2 y2 =− −356 cm can be obtained by surveying the filling and karst water in the karst karstcave. cave.From Fromthe measurement, the measurement, the distance the distance between between the measured the measured point O point and the O and filling the line filling is 1190 line iscm 1190 (ρk). cm There (ρk). Thereis a small is a small amount amount of blocks of blocks and and silt siltin inthe the cave, cave, and and the the sediment sediment thickness thickness isis d y approximately 30–50 cm ( f ), i.e., 44 = (1220–1240) cm. The maximum water flow is approximately approximately 30–50 cm (dff), i.e., y = −(1220–1240)− cm.. TheThe maximummaximum waterwater flowflow isis approximatelyapproximately 11 1 L/s, the water level is essentially zero, and the highest historical water level is approximately 2.4 m L/s, the water level is essentially zero, and the highest historical water level is approximately 2.4 m (dw)—i.e., y3 = 950 cm. According to the coring survey, the crushing depth of the bottom surrounding (dw)—i.e., y3 = −−950 cm. According to the coring survey, the crushing depth of the bottom surrounding rock is approximately 70 cm (du), and we also obtain ρk = 1260. Here, we determine the spatial rock is approximately 70 cm (du), and we also obtain ρk = 1260. Here, we determine the spatial relationship of section Z4. According to the field survey results of each section, spatial diagrams of relationship of section Z4.. AccordingAccording toto thethe fieldfield surveysurvey resultsresults ofof eacheach section,section, spatialspatial diagramsdiagrams ofof sections Z1–Z7 were drawn, as shown in Figure 21. Then, each section is drawn into a continuous sections Z1-Z7 werewere drawn,drawn, asas shownshown inin FigureFigure 21.21. Then,Then, each section is drawn into a continuous 3D 3D spatial diagram to realize the modeling of geometric parameters and directly reflect the spatial spatial diagram to realize the modeling of geometric parameters and directly reflect the spatial relationship, as shown in Figure 23. relationship, as shown in Figure 23. 5.3. Treatment Measures of Karst Cave According to the treatment technology of SDM discussed in Section4 of this paper, the comprehensive treatment method is formulated based on the site situation of this project:

(1) The arch bottom cave space (O4 ) treatment: manually clean the tunnel bottom gravel, using a 120 cm reinforced concrete pile reinforcement treatment, with pile transverse spacing of 5.0 m, vertical spacing of 8.0 m, pile bottom depth bedrock no less than 3.0 m, pile upper section set to a 1 m thick C35 reinforced concrete cap, and a cap bearing platform for the C30 concrete ﬂoor.

(a) D1K760+961D1K760+961(Z1) (b) D1K760+954(Z2) (c) D1K760+947(Z3) (d) D1K760+940(Z4) Appl. Sci. 2020, 10, x FOR PEER REVIEW 14 of 20

Figure 19. Development diagram of the karst cave.

Figure 20. Application process of the spatial decomposition method.

5.2. Supplementary Geological Survey According to the geometric model parameter determination method of the SDM, the central axis of the tunnel is located by means of the engineering measurement method. The longitudinal development length of the karst cave is approximately 51 m along the tunnel direction, and survey sections are set up every 7–8 m along the central axis, such that seven survey sections are constructed—namely, Z1-Z7 in Figure 21a–g. The distance of the measurement point on the wall of

the cave is measured by a laser rangefinder. A core investigation is carried out on the surrounding rock of the karst cave, as shown in Figure 22a,b. Section Z4 is taken as an example, as shown in Figure 22c. y1 = −774 cm and y2 = −356 cm can be obtained by surveying the filling and karst water in the karst cave. From the measurement, the distance between the measured point O and the filling line is 1190 cm (ρk). There is a small amount of blocks and silt in the cave, and the sediment thickness is

4 approximately 30–50 cm ( df ), i.e., y = −(1220–1240) cm. The maximum water flow is approximately 1 L/s, the water level is essentially zero, and the highest historical water level is approximately 2.4 m (dw)—i.e., y3 = −950 cm. According to the coring survey, the crushing depth of the bottom surrounding Appl. Sci. 2020, 10, 7204 15 of 20 rock is approximately 70 cm (du), and we also obtain ρk = 1260. Here, we determine the spatial relationship of section Z4. According to the field survey results of each section, spatial diagrams of (2)sectionsProtective Z1-Z7 were wall: drawn, C35 concrete as shown is adoptedin Figure for 21 the. Then, protective each section wall. Theis drawn thickness into a of continuous the protective 3D spatialwall diagram is not less to realizethan 2 m the and modeling the slope of rate geometric of the outer parameters wall is 1:0.2. and directly reflect the spatial (3)relationship,The upper as shown space of in the Figure cave 23. (O1 ).

Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 15 of 20

(a) D1K760+961(Z1) (b) D1K760+954(Z2) (c) D1K760+947(Z3) (d) D1K760+940(Z4)

5 6 7 (e) D1K760+933(Z 5) (f) D1K760+926(Z 6) (g) D1K760+919(Z 7) (e) D1K760+933(e) D1K760+933(Z5)(Z ) ((ff)) D1K760+926((Z6)) ( g ) (D1K760+919g) D1K760+919(Z )( Z7) Figure 21. Spatial relationship between the tunnel and karst cave. Figure 2 21.1. SSpatialpatial relationship relationship between the tunnel and karst cave.

(a) Measurement points (b) Wall rock coring (c) D1K760 + 940 section (unit: cm) ((a)) Measurement points ( (b)) Wall rock coring ( (c)) D1K760 + 940 section (unit: cm) Figure 22. Supplementary geological survey and section drawing. Figure 2 22.2. SSupplementaryupplementary geological survey survey and section drawing.

(a) Front 3D view (b) Left 3D view (c) Right 3D view (d) Best Angle of 3D view ((a)) Front 3D view ((b)) Left 3D view ((c)) Right 3D view ((d)) Best Angle of 3D view Figure 23. 3D3D spatial spatial image. image. Figure 23. 3D spatial image. 5.3. Treatment Measures of Karst Cave 5.3. TTreatmentreatment Measures of Karst Cave According to the treatment technology of SDM discussed in Section 4 of this paper, the According to the treatment technology of SDM discussed in Section 4 of this paper, the comprehensive treatment method is formulated based on the site situation of this project: comprehensive treatment method is formulated basedbased on the site situation ofof thisthis project:project: (1) The arch bottom cave space (④) treatment: manually clean the tunnel bottom gravel, using a 120 (1)(1) The arch bottom cave space (④)) treatment:treatment: manuallymanually cleanclean thethe tunneltunnel bottombottom gravel,gravel, usingusing aa 120120 cm reinforced concrete pile reinforcement treatment, with pile transverse spacing of 5.0 m, cm reinforced concrete pile reinforcement treatmenttreatment,, with pile transverse spacing of 5.0 m, vertical spacing of 8.0 m, pile bottom depth bedrock no less than 3.0 m, pile upper section set to vertical spacing of 8.0 m, pile bottom depth bedrock no less than 3.0 m, pile upper section set to a 1 m thick C35 reinforced concrete cap, and a cap bearing platform for the C30 concrete floor. a 1 m thick C35 reinforced concrete cap, and a cap bearing platform for the C30 concrete floor. (2) Protective wall: C35 concrete is adopted for the protective wall. The thickness of the protective (2)(2) Protective wall: C35 concrete is adopted for tthehe protective wall. TThehe thickness of the protective wall is not less than 2 m and the slope rate of the outer wall is 1:0.2. wall is not less than 2 m and the slope rate of the outer wall is 1:0.2. (3) The upper space of the cave (①). (3)(3) The upper space of the cave (①).). The unstable surrounding rock is cleared from the vault and cleaned up, and the anchor net The unstable surrounding rock isis clearedcleared fromfrom thethe vaultvault andand cleanedcleaned up,up, andand thethe anchoranchor netnet shotcrete support is used in the cleaning area. The body of the tunnel is pumped with C30 concrete shotcrete support is used in the cleaning area. TheThe body of the tunnel is pumped with C30 concrete Appl. Sci. 2020, 10, 7204 16 of 20

Appl. Sci. 2020, 10, x FOR PEER REVIEW 16 of 20 Appl. Sci.The 2020 unstable, 10, x FOR surrounding PEER REVIEWrock is cleared from the vault and cleaned up, and the anchor16 of net 20 shotcrete support is used in the cleaning area. The body of the tunnel is pumped with C30 concrete to to a thickness of 1 m as the structural protection layer and sand to a thickness of 2 m as the buffer toa thickness a thickness of 1of m 1 asm theas the structural structural protection protection layer layer and and sand sand to a to thickness a thickness of 2 mof as2 m the as bu theﬀer buffer layer layer above the structural protection layer, as shown in Figure 24. layerabove above the structural the structural protection protection layer, layer, as shown as shown in Figure in Figure 24. 24.

Figure 24. Treatment of the karst in section Z4. Figure 24. Treatment of the karst in section Z4. Figure 24. Treatment of the karst in section Z4. 5.4. Effect Analysis by Monitoring Measurement 5.4. Effect Effect Analysis by Monitoring Measurement To evaluate the treatment effects, the tunnel stability is an important factor, which includes To evaluate thethe treatmenttreatment eeffects,ffects, the the tunnel tunnel stability stability isis anan importantimportant factor,factor, whichwhich includes arch settlement, clearance convergence, floor vertical displacement, surrounding rock pressure, arch settlement, clearance convergence, floor floor ve verticalrtical displacement, surrounding surrounding rock rock pressure, pressure, initial support structure stress, and secondary lining stress. Because the tunnel body is not in contact initial support structure stress, and secondary lining stress. Because Because the tunnel body is not in contact with the surrounding rock, its stability is mainly reflected in the settlement of the tunnel body caused with the surrounding rock, its stability is mainly reflectedreflected in the settlement of the tunnel body caused by pile settlement. Therefore, only settlement monitoring is needed for the tunnel. Three monitoring by pile settlement. Therefore, Therefore, only settlement moni monitoringtoring is needed for the tunnel. Three monitoring points (MS-1 to MS-3) were arranged at D1K760+955, D1K760+940, and D1K760+925, as shown in points (MS-1 to MS-3)MS-3) werewere arrangedarranged atatD1K760 D1K760+955+955,, D1K760D1K760+940+940,, and D1K760D1K760+925+925,, as as shown in Figure 25. Figure 2525..

Figure 25. Arrangement of the subsidence monitoringmonitoring points in the tunnel. Figure 25. Arrangement of the subsidence monitoring points in the tunnel. The monitoring data are shown in Figure 2626.. The settlement of the vault is accelerated in three The monitoring data are shown in Figure 26. The settlement of the vault is accelerated in three stages and finallyfinally reaches aa stablestable settlementsettlement state.state. According to the change rule of the accumulatedaccumulated stages and finally reaches a stable settlement state. According to the change rule of the accumulated settlement value of the vault, the settlement trend can be divided into stages I–IV. Stage I is the settlement value of the vault, the settlement trend can be divided into stages I–IV. Stage I is the settlement inin the later stage of lining. In this stage, stage, monitoring monitoring activities are initiated after completing settlement in the later stage of lining. In this stage, monitoring activities are initiated after completing the lininglining constructionconstruction (after (after the the stripping). stripping). The The collected collected data data do do not not include include the the settlement settlement during during the the lining construction (after the stripping). The collected data do not include the settlement during liningthe lining construction construction and and before before the stripping. the stripping. Stage Stage II is theII is settlement the settlement of the of structural the structural protection protection layer, the lining construction and before the stripping. Stage II is the settlement of the structural protection whichlayer, which is due is to due the increasedto the increased load caused load caused by the by concrete the concre structurete structure protection protection layer onlayer the on main the layer, which is due to the increased load caused by the concrete structure protection layer on the bodymain ofbody the tunnel,of the resultingtunnel, resulting in the acceleration in the acceleration of the settlement. of the settlement. The third stage The isthird the constructionstage is the main body of the tunnel, resulting in the acceleration of the settlement. The third stage is the settlementconstruction of settlement the buffer layer. of the In buffer this stage, layer. the In settlementthis stage, accelerationthe settlement changes acceleration owing tochanges the increased owing construction settlement of the buffer layer. In this stage, the settlement acceleration changes owing terminationto the increased caused termination by filling caused the upper by filling part ofthe the up protectionper part of layerthe protection of the concrete layer of structure the concrete with to the increased termination caused by filling the upper part of the protection layer of the concrete coarsestructure sand. with Stage coarse IV is sand. a stable Stage stage. IV Afteris a thestable completion stage. After of all the processes, completion the cumulative of all processes, settlement the structure with coarse sand. Stage IV is a stable stage. After the completion of all processes, the valuecumulative remains settlement within a value stable rema range.ins within a stable range. cumulative settlement value remains within a stable range. Appl. Sci. 2020, 10, 7204 17 of 20 Appl. Sci. 2020, 10, x FOR PEER REVIEW 17 of 20

Figure 26. Monitoring of the tunnel vault settlement.

6. Discussion Although importantimportant discoveries discoveries have have been been revealed revealed by these by studies,these therestudies, are certainthere are limitations. certain Thelimitations. results The show results that theshow SDM that is the mainly SDM usedis mainly for the used tunnel for the crossing tunnel orcrossing partial or crossing partial crossing of large karstof large caves, karst without caves, considering without specialconsidering water environmentalspecial water conditionsenvironmental suchas conditions underground such rivers as andunderground high-pressure rivers dynamic and high-pressure water. When suchdynamic working water. conditions When are such encountered, working the conditions SDM may are not beencountered, applicable the or maySDM need may tonot be be improved applicable for or specific may need working to be conditions.improved for At specific the same working time, karstconditions. development At the same is complex time, karst and variable,development and theis complex treatment and techniques variable, forand karst the cavestreatment are flexibletechniques and for varied, karst ascaves well are as flexible the supporting and varied, techniques as well as for the the supporting SDM. As atechniques result, builders for the should SDM. followAs a result, the principles builders should of safety, follow science, the flexibility,principles andof safety, sustainability science, flexibility, during practical and sustainability applications. Furthermore,during practical the applications. treatment technology Furthermore, lacks technicalthe treatment economic technology consideration lacks technical based on economic the SDM. In general, the technical economy should be compared with other technical schemes, and the safer consideration based on the SDM. In general, the technical economy should be compared with scheme with the lower cost should be implemented in the project. other technical schemes, and the safer scheme with the lower cost should be implemented in the 7.project. Conclusions

7. ConclusionsIn this study, the application and geometric model of the SDM are proposed, regarding the construction and operation of tunnels passing through large karst caves. The geometric parameter determinationIn this study, method the andapplication supporting and treatmentgeometrictechnology model of the would SDM provide are proposed, assurance regarding to select the karstconstruction tunnel implementation.and operation of The tunnels following passing conclusions through large are drawn karst fromcaves. the The results geometric of this parameter work: determination method and supporting treatment technology would provide assurance to select the (1)karst Thetunnel SDM implementation. is proposed. The The SDL following is used toconclusi divideons the are space drawn of the from large the karst results cave of into this eight work: parts. Compared with traditional construction methods, this present method provides a means to (1) The SDM is proposed. The SDL is used to divide the space of the large karst cave into eight parts. conduct a spatial analysis which is more straightforward and clear for tunnels through large Compared with traditional construction methods, this present method provides a means to conduct karst caves. a spatial analysis which is more straightforward and clear for tunnels through large karst caves. (2) The geometric model of the SDM is established. Cartesian and cylindrical coordinate systems (2) The geometric model of the SDM is established. Cartesian and cylindrical coordinate systems are used to describe the geometric parameters of the SDL. A method for the site investigation of are used to describe the geometric parameters of the SDL. A method for the site investigation of tunnels and large karst caves is proposed. The geometric parameters of the SDL are determined tunnels and large karst caves is proposed. The geometric parameters of the SDL are determined to realize the spatial geometric description of the SDM. to realize the spatial geometric description of the SDM. (3) According to the SDM, spatial treatment technologies for the top, bottom, and two wings of the (3) According to the SDM, spatial treatment technologies for the top, bottom, and two wings of the tunnel are proposed. It is emphasized that the overall technology application should follow the tunnel are proposed. It is emphasized that the overall technology application should follow the principles of safety, ﬂexibility, science, sustainability, and economy. principles of safety, flexibility, science, sustainability, and economy.

Author Contributions: Conceptualization, Z.S. and Q.W.; methodology, J.L. and Z.Z.; investigation, R.S. and Y.L.; data curation, Y.Y. and F.W. and H.S.; writing—review and editing, C.P.; All authors have read and agreed to the published version of the manuscript. Appl. Sci. 2020, 10, 7204 18 of 20

Funding: This research was funded by the National Natural Science Foundation of China (NSFC) (No. 51778351) and SDUST Research Fund (No. 2018TDJH101). Conﬂicts of Interest: The authors declare no conﬂict of interest.

Nomenclature

ρ polar diameter θ polar angle Z a plane parallel to plane XOY i a section is numbered i D1 spacing of sections ρk geometric expression of the cave contour ρr geometric expression of the surrounding rock stability edge line ρt geometric expression of the tunnel contour line y1 geometric expression of the tunnel top line y2 geometric expression of tunnel bottom line y3 geometric expression of the water level line y4 geometric expression of ﬁlling line d1 vertical distance from the laser equipment to the central axis d2 distance between the laser equipment and the proposed measurement point d3 height of the laser equipment tripod du depth of the unstable surrounding rock dp depth of a borehole into medium-weathered rock mass df thickness of the ﬁlling in the cave dw depth of the water in the cave α angle between d2 and the horizontal direction β angle between d2 and the vertical direction γ angle between d2 and ρk

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