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This paper aims to estimate the stability of the water-resistant strata between the tunnel and the small-medium-sized concealed cavity filled with high-pressurized water or other fillings at optional position around tunnel through solving the double-hole problem. The analytical method to identify the critical water-resistant thickness is proposed based on the Schwarz alternating method and Griffith strength criterion, and the program to calculate the critical thickness was prepared according to this method using mathematical software. Parametric study of the critical thickness indicates that the critical water-resistant thickness will increase with the buried depth of the tunnel and cavity size; the lateral pressure coefficient has more complicated influence on the critical thickness, which is affected by cavity position; when the cavity is filled with sand or gravel, the critical water-resistant thickness will decrease with the increase of the filling pressure; and when the cavity is filled with the high-pressurized water, the critical thickness will decrease as the water pressure initially and increase afterwards. The analytical result of the critical thickness is consistent with that obtained by numerical simulation using the user-defined program based on FLAC^{3D}, demonstrating the rationality and feasibility of the proposed method in this study.

Construction activities of a tunnel in the karst area require engineering measures to prevent water inrush due to exposed cavities with rich water and high pressure. Karst cavities that are not exposed directly in the tunnel excavation, but that are nonetheless impacted by tunnel construction, are referred to as concealed cavities [^{5} m^{3} and that of mud of about 7.0 × 10^{4} m^{3}. The water inrush flooded the 3152 m parallel heading and 2508 m main tunnel in the vicinity of exit and caused the damages of a large number of equipment and machinery with economic losses over RMB 10 million [

The study on the stability of water-resistant strata and its thickness is of great significance for preventing water inrush in the karst tunnel. Researchers have done many useful works about this serious problem on karst tunnel construction. Zhao Mingjie et al. studied the effects of karst cavities above the tunnel on the displacement and stress of surrounding rock using numerical simulation [

Although plenty of research results have been achieved on stability of the water-resistant strata and its thickness in the past years, many problems still exist to accurately identify the water-resistant thickness. Further research must be completed to address issues related to (1) the reasonableness of water-resistant strata between the tunnel and small-medium-sized concealed cavity being simplified into the elastic beam and plate model; (2) the influence of fillings in the karst cavity on the water-resistant strata stability; and (3) examining the stability of strata between the concealed cavity at irregular positions (except directly above, positively lateral to, and directly below the tunnel) and the tunnel. In response to the above issues, the analytical method for analyzing stability of the water-resistant strata in this study was proposed according to the Schwarz alternating method and Griffith strength criteria. Then, the parametric study of the critical water-resistant thickness is conducted based on this method. Finally, considering the Dazhiping tunnel on Yichang-Wanzhou Railway in China, the reliability and rationality of this method is verified through comparison of the critical thickness calculated analytically, the practical thickness, and that obtained by numerical simulation using FLAC^{3D}.

The karst cavity is divided into the “open-field” mechanical model, which ignores the corrosion mechanical effect, and the mechanical model of the tunnel, which takes into account the corrosion effect. Song Zhanping studied the stress distribution rule in surrounding rock mass in two models and observed that the differences of the two models are slighter and may be ignored in engineering application when the thickness of the water-resistant strata is more than 2 m [

In this study, the concealed cavity around the tunnel is analyzed via the mechanical model of the tunnel. This model consists of the following properties: the cavity existed before tunnel excavation, the corrosion effect on the formation of the cavity is similar to tunnel excavation, and the stress concentration in the surrounding rock mass around the cavity is obvious. Therefore, the tunnel and the small-medium-sized concealed cavity (smaller than 15 m or two times of tunnel span) around it can be simplified as a double-hole problem in the plane strain state. The instability mechanism of the water-resistant strata is then studied by solving the double-hole problem, and then the critical water-resistant thickness between the tunnel and the concealed cavity is identified under the condition that prereinforcement measures are not adopted.

The Schwarz alternating method simplifies the multiconnected regions into a series of simply connected regions when the problem of multiconnected regions is solved [_{1} is the coordinate in the _{1}_{1}_{1} coordinate system, and it is also the coordinate in the global coordinate system. As shown in Figure _{2} is the coordinate in the _{2}_{2}_{2} coordinate system;

Mechanical model of double holes at optional position used in the Schwarz alternating method.

The basic procedure to solve the double-hole problem by the Schwarz alternating method is as follows. (1) The stress solution for an infinite region after no. 1 hole excavation under the initial stress condition can be obtained by the Cauchy Integral method. No. 1 hole excavation causes redundant surface traction on the boundary of no. 2 hole, which is not excavated at this time. The redundant surface traction can be obtained by the above stress solution. To balance the redundant surface traction, the reverse surface traction is applied at the edge of no. 2 hole. (2) Another single-hole problem is solved loaded by the reverse surface traction, when no. 2 hole is excavated in the infinite region. This stress solution also creates a nonzero surface traction on the boundary of no. 1 hole. The surface traction is also redundant and should be obtained again through a new stress solution. (3) The third single-hole problem that should be solved applies corresponding reverse surface traction at the edge of no. 1 hole. The resulting redundant surface traction on the boundary of no. 2 hole can then be calculated. The iterative process is continued until the redundant surface traction on the two holes’ boundaries equals zero. The final stress solution is the linear superposition of the stresses for all the single-hole problems during the iterative calculation.

In this paper, one cycle of iteration is defined as the process of solving every single-hole problem in succession. The loading condition of the reverse surface traction induced by the former single hole problem is applied along the hole boundary, except for the first stress solution after no. 1 hole excavation under the initial stress condition.

The tunnel and small-medium-sized concealed cavity around it can be simplified as an infinite elastic plane containing two holes under the initial uniform in situ stress of

Position relationship of tunnel and concealed karst cavity and their simplified mechanical model.

Assume _{1}(_{1}) and _{1}(_{1}) are the complex stress functions in _{1}_{1}_{1} (Figure _{2}_{2}_{2} coordinate system are expressed as _{2}(_{2})and _{2}(_{2}). Before and after the coordinate translation, the conversion relationship between the complex stress functions is described as follows [

To distinguish the complex stress functions computed in two iterations, the complex functions computed in two iterations in the _{1}_{1}_{1} coordinate system are denoted by _{11}(_{1}), _{11}(_{1}), _{12}(_{1}), _{12}(_{1}), _{13}(_{1}), and _{13}(_{1}). Also, _{21}(_{2}), _{21}(_{2}), _{22}(_{2}), _{22}(_{2}), _{23}(_{2}), and _{23}(_{2}) are the complex stress functions in the _{2}_{2}_{2} coordinate system computed in two iterations. When computing the complex stress functions for hole 1 under the initial stress or the boundary surface force, the second subscript of the complex stress functions is an odd number, but it is an even number for hole 2.

The Cauchy Integral method is used to solve the mechanical problems of the simply connected region because this paper uses the equivalent method for the cross section of the concealed cavity and tunnel in the Schwarz alternating method. Based on the conversion equations (_{1}, _{2}, and _{13}(_{1}) and _{13}(_{1}), the process to deduce _{13}(_{1}) and _{13}(_{1}) ignores the items concerned with water pressure or filling pressure _{13}(_{1}), refer to the appendix.

After two iterations are completed, the complex stress function of surrounding rock mass around the tunnel is [

Therefore, the stress of surrounding rock mass can be obtained via the following equations:_{r}, _{θ}, and _{rθ} represent radial stress, circular stress, and shear stress in surrounding rock mass, respectively.

The relationships of each stress component between the rectangular and polar coordinate system are described as follows [

Through (_{1}, _{3}, and _{t} represent the major principal stress, the minor principal stress, and tensile strength, respectively. The stability criterion for the water-resistant strata based on (

Using the above equations, we can calculate the safe factor

It is complicated to manually identify the critical water-resistant thickness using the above deductive process of solving the stress state in the water-resistant strata. However the complex stress function is explicit after a certain number of iterations, and it is easy to automatically implement the solving process via programming on mathematical software. The specific flow chart to identify the critical water-resistant thickness with mathematical software is shown in Figure

Flow chart to identify critical water-resistant thickness.

From (

The Schwarz alternating method does not consider the influence of gravity in computing the stress state of the water-resistant strata. The mechanical model is under a two-dimensional stress state, and the equivalent method is used to process the cavity and tunnel section. Therefore, the model is under the left and right, up and down symmetric condition. This paper discusses only the case in which the cavity is located in the upper-right side of the tunnel. In the computation, we assume that the vertical in situ stress ^{3} [_{2} of tunnel cross section remains unchanged. Based on the typical cross-section size of single-track tunnels on the Yu-huai railway and Yi-Wan railway which were constructed in karstic terrain, _{2} = 4.25 m. When the influence of cavity size is not discussed, _{1} is 3 m; the tensile strength of water-resistant strata is 1 MPa [

Relationship between the critical water-resistant thickness and buried depth of tunnel.

Relationship between the critical water-resistant thickness and lateral pressure coefficient.

Relationship between the critical water-resistant thickness and size of the karst cavity.

Figures _{1}.

The lateral pressure coefficient has complex influence on the critical water-resistant thickness. When

The concealed cavity around the tunnel is often filled with the high-pressurized water or sand and gravel. Therefore, it is necessary to analyze the influence of the water pressure or filling pressure in the karst cavity on the stability of the water-resistant strata. The effect of water pressure or filling pressure is illustrated by the example of the concealed cavity directly above the tunnel in this paper. The calculated parameters are the same as them in Section

Relationship between the critical water-resistant thickness and

As seen in Figure

Influence of

Xi has analyzed the influence of water pressure on the critical water-resistant thickness by UDEC and found that the relationship between the critical water-resistant thickness and water pressure is the same as that in Figure

Schematic relationship between the critical thickness and water pressure or filling pressure in the cavity.

DK137+540–DK137++800 of the Dazhiping tunnel is situated in the west wing of the Yangchang River anticline. The solution and gash breccia, limestone, and dolomite strata of Jialingjiang formation in the lower Triassic are uncovered. The rock masses are broken, and the grade of rock masses is III based on

Cavity size and position relationship of tunnel and cavity.

As can be seen in Figure _{2} = 6.23 m based on the equivalent circle method. The average width of cross section of this karst cavity filled with high-pressurized water is 8.45 m and its height is 6.65 m, so _{1} = 3.78 m. Based on the position relationship of the cavity and tunnel, ^{3}, the overburden pressure _{t} = 1.0 MPa, and the average hydraulic pressure

Relationship of the principal stress and the water-resistant thickness at the middle point of water-resistant strata.

From Figure _{1}–_{3}) will decrease, and stability of this point will enhance. This changeable law meets the actual condition, which indicates that the method that uses the Schwarz alternating method and Griffith strength criteria is reasonable and feasible for assessing stability of the water-resistant strata between the small-medium-sized concealed cavity and tunnel.

The distance from the tunnel vault to the cavity bottom is about 3.4 m–4 m, and the average distance is 3.7 m in this project. The safe factor

The critical water-resistant thickness of the engineering example on Yichang-Wanzhou Railway.

Identification of the failure zone of water-resistant strata by fast Lagrange analysis of continua in three dimensions (FLAC^{3D}) was further developed through the user-defined program in the FISH environment according to the experimental results of mechanical properties and the failure mechanism of karst limestone under natural and saturated states [^{3D}. The developed method is used to assess the stability of water-resistant strata in the Dazhiping tunnel as mentioned above, and the results are shown in Figure

Failure zone scope in the water-resistant strata obtained by the user-defined program on FLAC^{3D}. (a) The water-resistant thickness of 3.5 m. (b) The water-resistant thickness of 4.0 m. (c) The water-resistant thickness of 4.5 m. (d) The water-resistant thickness of 5.0 m.

As shown in Figure

The small-medium-sized concealed karst cavity filled with high-pressurized water is usually distributed around the tunnel. Tunnel excavation often creates stress concentration, and crack initiates and propagates at the location of stress concentration in the water-resistant strata between the tunnel and concealed cavity subjected to karst water pressure. Therefore, accurately calculating the stresses at the edge of the tunnel or cavity is important to assess the stability of the water-resistant strata. At present, some numerical and experimental methods can be used to study the stress distribution in the water-resistant strata and then analyze its safety, but they cannot quickly or correctly obtain the stresses and displacements at any point around the tunnel and cavity. More importantly, one can clearly know the stress and stability of the water-resistant strata in theory by means of an analytical method. Likewise, it is worth noting that the analytical results can be used for the validation and reference of numerical procedures and tests [

The Schwarz alternating method, the earliest known domain decomposition method, was introduced in a seminal paper by Hermann Schwarz in 1870 [

The elastic-plastic solutions for stress and displacement around a single circular or cylindrical hole have received a tremendous amount of scientific attention, and comprehensive research achievements are obtained. The majority of these solutions are based on the linear Mohr–Coulomb (M–C) failure criterion or nonlinear Hoek–Brown (H–B) failure criterion for an elastic-brittle-plastic or elastic-perfectly plastic rock mass [

Obert and Duvall described brittle of a material such as cast iron and brittle rocks to end by fracture at or after the yield stress [

The analytical method to identify the critical water-resistant thickness between the tunnel and cavity is established based on the Schwarz alternate method and Griffith strength criterion. Parametric study of the critical water-resistant thickness is conducted. The feasibility of this method is verified by the numerical simulation through engineering application. The following conclusions can be drawn:

For the small-medium-sized concealed cavity at optional position around the tunnel, a method is proposed to estimate stability of the water-resistant strata between the tunnel and karst cavity filled with water or other fillings and identify the water-resistant thickness using the Schwarz alternate method and Griffith strength criterion in this study. The program to calculate the critical water-resistant thickness was prepared according to this method on mathematical software.

The critical water-resistant thickness will increase with the increase of the buried depth of tunnel and cavity size. When the lateral pressure coefficient is 1.2, the critical water-resistant thickness will increase as the inclination of the connection line between the cavity center and tunnel increases. When lateral pressure coefficient is less than 1, the critical water-resistant thickness will reduce with the increase of inclination.

The lateral pressure coefficient has complex influences on the critical water-resistant thickness. When

When the cavity is filled with sand and gravel, its pressure will not be higher than the in situ stress, and the critical water-resistant thickness will decrease with the increase of the filling pressure; when the cavity is filled fully with the high-pressurized water, the critical thickness will decrease as the water pressure initially and increase afterwards in face, and the water pressure to the turning point is the critical pressure for the hydraulic fracturing.

To verify the theoretical method established in this study, the analytical result of the critical water-resistant thickness for the Dazhiping tunnel on Yichang-Wanzhou Railway is compared to that obtained by the numerical simulation. The critical thickness determined by a user-defined program based on FLAC^{3D} is very close to that calculated by the analytical method, therefore demonstrating the rationality and feasibility of the proposed method.

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This work is financially supported by Chinese National Programs for Fundamental Research and Development (Grant no. 2013CB036003), National Natural Science Foundation of China (Grant nos. 51778215, 50174097), and Doctoral Foundation of Henan Polytechnic University (Grant no. B2012-016).