Multiphysics Simulation of the Catastrophic Process of Water and Mud Inrush in a Karst Tunnel: A Case Study of Tunnel, Western China

Abstract

This paper investigated the mechanism and dynamic process of a significant water and mud inrush disaster that occurred in the Baiyunshan Tunnel, which crosses a karst fault zone. By integrating multi-source data including geological exploration and geophysical surveys, a three-dimensional geological model characterizing the cave–conduit–tunnel system was developed. A numerical approach coupling the Phase-Field and Particle-Tracking Methods was employed, successfully reconstructing the entire disaster process involving the transport of water-air-mud three-phase flow. Simulation results demonstrated that the dynamic viscosity of the mudflow predominantly controls the dynamic characteristics of the particle, such as transport distance and mudflow velocity. Parameter sensitivity analysis revealed quantitative relationships between key mudflow parameters (transport distance, velocity, and drag force) and the Reynolds number, identifying an exponential decay of drag force with increasing Reynolds number in high-viscosity mudflows. This study establishes a comprehensive methodology from geological identification to numerical simulation, providing a theoretical basis and technical support for precise risk assessment and the design of preventive measures for tunnel water and mud inrush disasters.

1. Introduction

Water and mud inrush is a typical hydrodynamic disaster faced by tunnels when crossing the intersection area of karst and fault fracture zones, which involves a complex geological–mechanical–seepage coupling process. The west section of the Baiyunshan Tunnel on the Yu-Xiang Highway in the area of Chongqing, a city in western China, encountered a sudden water and mud inrush accident at a section during excavation. This resulted in the inflow of up to approximately 150,000 m³ of water and mud into the construction tunnel within an hour and a half. Subsequently, intermittent water inrushes, occasionally accompanied by mud and sand, occurred at the section. The average water inflow was around 20,000 m³, with occasional instances lasting up to three months. This is a typical disaster-causing feature of a fault and karst underground river system, which has had a severe impact on both the construction and subsequent maintenance of the project.

In the past, risk assessments for water and mud inrush disasters were predominantly conducted through evaluations of hazard susceptibility or system vulnerability. Methods such as the Analytic Hierarchy Process (AHP) [1] and the “Three Maps-Two Predictions” method [2], commonly used in coal mine water inrush assessments, typically provided possible causes and consequences in qualitative or semi-quantitative forms. In recent years, significant progress has been made in the research on the mechanism of sudden water and mud inrush in tunnels, especially in the high-distribution mountainous areas of southwest China [3,4,5,6]. Notable advancements have been achieved in field investigations of geological conditions, scale model experiments, and numerical simulations of the mechanisms involved [7,8,9,10]. At some research institutions in China, the development of large-scale experimental platforms has been underway to simulate true 3D loading systems, enabling model tests for complex stress fields and engineering environments [11,12,13]. In terms of risk assessment, Rajat M. Gangrade et al. [14] proposed a probabilistic evaluation method based on geostatistical modeling. They applied the Pluri-Gaussian Simulation (PGS) technique to analyze borehole data from a karst geological tunnel project in Malaysia, quantifying the uncertainty in the occurrence probability, number, frequency, and spatial distribution of karst cavities of various sizes. However, due to limitations in the scale of experimental systems, challenges in replicating complex geological environments, the development of analogous materials, as well as difficulties in field data collection and high safety risks, research breakthroughs are still facing significant challenges. A more accurate assessment, however, requires the establishment of an intuitive and quantitatively explicit evaluation model.

Based on high-precision exploration data, a 3D geological model can be constructed, which characterizes the spatial distribution of fault-karst conduit-groundwater systems, enabling an in-depth understanding of the mechanism of water and mud inrush disasters and improving disaster prevention and control levels [15], whereas various geophysical methods, which rely on the differences in medium properties as the basis for signal acquisition, are significantly constrained in their application range due to the influence of transmitted power. Commonly used methods in engineering, such as tunnel seismic detection, transient electromagnetic (TEM) methods, and ground-penetrating radar (GPR), have limitations in terms of shallow detection depth and are suitable for interpreting geological structures at a local scale, typically used for advanced detection [16,17]. However, for large-scale geological structures involving the entire engineering scale, further exploration is still needed.

Tunnel water inrush and mud inrush disasters are typical chain disaster processes, where the evolution involves the dynamic coupling of multiple factors, including geology, hydrology, and mechanics. Overall, the process can be divided into four stages: disaster background (upstream), triggering mechanism (midstream), disaster evolution (downstream), and disaster consequences (final stage). The chain process of the disaster and the analysis of its key stages are shown in Figure 1. China has witnessed a high output of research on water and mud inrush in tunnels, largely driven by the large-scale construction of high-speed railways and expressways over the past decade. Extensive engineering practice has significantly advanced the understanding of inrush mechanisms, improved prediction accuracy, and enhanced prevention technologies.

While not explicitly framed within sustainable development theory, this research closely aligns with its multidimensional objectives. Environmentally, it clarifies the mechanisms of water and mud inrush and their interaction with groundwater, providing a scientific basis for protecting fragile karst hydrogeological settings during construction and minimizing ecosystem disturbance. Economically, the developed disaster simulation and prediction framework enhances engineering risk prevention, helping to avert substantial financial losses, resource waste, and project delays, thereby improving the life-cycle sustainability of major infrastructure. Socially, it prioritizes the safety of personnel during construction and operation—a people-centered approach—while boosting infrastructure reliability and resilience to support regional connectivity and community well-being.

Methodologically, the integrated use of multi-source data fusion, 3D geological modeling, and multiphysics-coupled numerical simulation constitutes a systematic, preventive analytical framework. This technical approach not only advances the precision and scientific rigor of disaster mitigation but also reduces the reliance on large-scale physical experiments, lowering the environmental footprint of the research itself and demonstrating how innovation drives sustainability [18].

This study investigated a water and mud inrush incident in the Baiyunshan Tunnel, located on an expressway section in Chongqing. Following a systematic analysis of the geological background and disaster scenario, the inrush event was identified as a karst water inrush triggered by construction blasting. A 3D visualization model of the tunnel geology was developed through the integration of multi-source data, including geological exploration, geophysical survey, and remote sensing. A 3D numerical model was subsequently established for the inrush section, with focused simulation and analysis of the connected system comprising karst caves, underground conduits, and the tunnel. The study successfully reconstructed the entire disaster process, revealing its evolutionary characteristics and underlying physical mechanisms related to destructive forces (such as the relationship between Reynolds number and drag force). This research provides a comprehensive methodological framework for high-fidelity simulation and the mechanistic analysis of water and mud inrush disasters while establishing a theoretical basis for the prevention and mitigation of similar hazards.

In summary, the study builds a complete technological chain—from geological insight and mechanistic understanding to intelligent hazard prevention—delivering key scientific and technical support for achieving safe, green, and economical tunnel construction under complex geological conditions. It represents proactive infrastructure-sector practice toward sustainable development goals.

2. Project Overview

2.1. Geological Setting

The project is located at the border between Nanchuan and Wulong District, Chongqing Municipality, China. It forms part of a dual-line tunnel system that passes through the core depression of an inverted anticline known as Tongmawan. The tunnel crosses mountainous terrain with elevations ranging from 580 m to 1480 m on both wings, as illustrated in Figure 2. On a regional scale, it is influenced by the famous fault-fold belt in Eastern Sichuan Province, primarily characterized by regional compressional tectonic movements. The maximum horizontal principal stress direction is NWW–SEE. The plate interactions in geological history caused crustal uplift, leading to the unique, layer-upon-layer folding belts and the undulating mountainous terrain typical of Eastern Sichuan Province [19].

The Tongmawan inverted anticline was formed under such compressional forces. The strata in the water inrush section of this project mainly consist of limestone layers, which are not only developed with karst caves and fissures but also intersected by faults. Two sets of X-type conjugate strike-slip faults (F4 and F6) pass through the water inrush section, making the rock structure in this area highly fractured with well-developed joints and fissures. The Tongmawan Anticline exhibits a NW–SE trending axial trend. The outcrop sequence, from oldest to youngest, comprises the Silurian Hanjiadian Formation (S₂h) at the core, followed by the Permian Liangshan Formation (P₁l) to Changxing Formation (P₃c) adjacent to the core, the Triassic Feixianguan Formation (T₁f) to Jialingjiang Formation (T₁j) on the limbs, and Quaternary deposits (Q₄) covering the entire region.

Field investigations identified a water and mud inrush event within tunnel pile K83+122 to K83+166. The surrounding rock in this zone is classified as the Middle Permian Qixia Formation (P₂q), consisting of moderately thick-bedded limestone with a measured attitude of 162°∠52°. Advanced probing indicates that the rock mass is categorized as Class III, exhibiting moderate strength, a relatively fractured condition, and well-developed joint fissures. Analysis indicates that these faults are secondary faults formed in the core area during the folding process. On-site investigations show that the surrounding rock’s strength (Rc) ranges from 25 to 30 MPa, with an integrity coefficient (Kv) ranging from 0.15 to 0.55.

2.2. Karst Development and Groundwater

Based on existing electrical exploration data, the Geode EM3D electromagnetic exploration system was deployed to conduct gridded data acquisition around the water inrush point. Six geophysical survey lines (measuring points) were arranged parallel to the tunnel axis, covering an area of approximately 0.94 km² (Figure 3). A resistivity model was developed through integrated 3D inversion and gradient analysis, with anomalous bodies extracted using isosurface slicing techniques. The integration of seismic or ground-penetrating radar (GPR) data helped reduce interpretation ambiguity. Despite limitations such as volume effects and anisotropy, the 3D spatial distribution characteristics of the karst system were effectively characterized through borehole calibration and machine learning-assisted classification, providing reliable support for engineering geological and hydrogeological investigations (Figure 3).

A geophysical survey conducted in the area aimed to delineate the spatial distribution of karst development and verify the specific location of the water and mud inrush. The survey included 2D and 2.5D transient electromagnetic method (TEM) profiles and a 3D resistivity tomography experiment. The study area is situated at the lithological boundary between limestone of the Permian Qixia Formation (P₂q) and shale of the Silurian Hanjiadian Formation (S₂h), where a set of X-shaped conjugate faults has developed. The rock formation boundary generally aligns with the NE–SW striking F4 fault. The dataset (covering chainage ZK83+156 to ZK83+176) revealed that the karst development zone exhibited a significant low-resistivity anomaly in the TEM inversion model (Figure 3b). The first water inrush point in the tunnel was precisely located at the fault intersection and within the corresponding low-resistivity karst development zone, specifically at chainage ZK83+166.

The groundwater in the working area primarily exists in the form of a karst underground river, which develops along the contact between the limestone aquifer (P₂q) on the western limb of the inverted anticline and the siltstone aquitard (S₂h) at the core. The course of the underground river generally aligns with the axial trend of the Tongmawan Anticline. Although locally offset by faults, its overall direction remains consistent and intersects the tunnel alignment at an angle of approximately 60°.

The lowest erosion base level of the regional karst underground river system is controlled by the water level of the Wujiang River (151.8 m) to the north. However, within the tunnel section, the system is governed by the local erosion base level, i.e., the water level of the Yuquan River (592.0 m) to the south. As a result, the groundwater flows in a SW215 direction. The elevation of the tunnel site is 614 m, placing it within the horizontal runoff zone of this local underground river system.

The ground surface elevation of this section ranges between 900 and 1100 m, forming a synclinal negative topographic depression characterized by karst features such as sinkholes, solution grooves, and solution spikes. These features are prone to forming localized zones of strong vertical permeability and even hydraulic drops. The distinctive geomorphology of the area indirectly enhances recharge to the underground river system, particularly during the rainy season. According to calculations [20], the water inflow at the Baiyunshan Tunnel portal section during the normal water period is 27,711 m³/d, with a maximum value of 138,557 m³/d. However, field observations recorded a peak inflow of 390,000 m³/d, which exceeded the calculated maximum by three to four times. This discrepancy is likely attributable to the intense infiltration of surface rainfall through sinkholes and fractures. For example, in the tunnel segment corresponding to pile ZK83+156 ~ ZK83+226, an identified sinkhole (BY14) was present on the surface, along with a fault-associated infiltration zone inferred through geophysical electrical surveys.

3. Water and Mud Inrush in Tunnel Construction

3.1. In Site Water-Mud Inrush Incident Scenarios

Baiyunshan Tunnel is constructed as part of Section 6 of the Banan–Pengshui segment on the Chongqing–Hunan highway duplicate line. It is designed as a fully separated twin-bore tunnel, with the left bore extending 6442 m and the right bore 6404 m in total length. The tunnel has a maximum overburden of 803.6 m, qualifying it as a deep-buried, extra-long mountain tunnel.

The water inrush section is located on the western limb of an inverted anticline within a well-developed karst groundwater system. The conjugate fault system (F4 and F6), formed under local compressive stress during folding, exhibits significant hydraulic conductivity, and is interpreted to have hydraulically connected to the underground river system, thereby inducing the water and mud inrush event.

The first water inrush occurred at the upper bench of the right wall of the left tunnel face at pile ZK83+166, triggered by excavation blasting from February 27 to March 29, 2022. The blast instantly released approximately 150,000 m³ of water along with 5000 m³ of sediment. The inrushing water, as shown in Figure 4, resembled a river gushing out of the tunnel portal. The flow was turbid and sediment-laden, extending several hundred meters downstream with a thickness ranging from 0.1 to 0.3 m, partially accumulating and blocking the tunnel portal. Intermittent water inrushes continued for nearly two months, ceasing only around the time when the second water inrush incident occurred.

The second water inrush occurred at the face of the left tunnel at pile ZK83+212 from 30 March to 15 June 2022. This incident was caused by the tunnel excavation directly exposing a karst cavity, allowing groundwater to gush out from the underground river system connected to the cavity. The initial water inrush lasted for half an hour, with a discharge of 15,600 m³ and a small amount of sediment. Subsequently, similar intermittent water inrushes continued, gradually decreasing in volume. The event persisted for two and a half months until the initial support was installed at the location, and anti-inrush sealing measures were implemented.

3.2. Analysis of Water and Mud Inrush Discharge

Based on textual descriptions of two water and mud inrush phenomena documented on-site [20], this study visually represents the two inrush processes using bar charts (Figure 4). It is important to note that as the data were derived from descriptive records of geological events rather than actual monitoring measurements, and because field conditions during the inrush prevented precise quantification, the water volume estimates are subject to significant uncertainty. To quantitatively characterize this uncertainty, Gaussian noise, i.e., Gauss (0, 1000), was introduced into the flow data. Furthermore, while the volume of mud inflow was quantitatively recorded only during the initial event at the first water inrush point on 27 February 2022, descriptions of mud inflow at other times were primarily qualitative.

As can be seen from Figure 5, the two water inrush events exhibited characteristics of intermittency and variability. Specifically, (1) the water inrushes typically occurred within a few hours, rather than persisting throughout the entire day; (2) the volume of water inrush could vary by more than tenfold. For example, at station ZK83+166, the initial water inrush event discharged approximately 150,000 cubic meters of water within 85 min, equivalent to an average flow rate of 1764.7 m³/min. In contrast, subsequent water inrush events involved a water volume of only about 7000 cubic meters, accompanied by approximately 5000 cubic meters of mud inrush, resulting in a combined average flow rate of about 58.8 m³/min; and (3) if the intensity of the inrush is expressed as flow rate per unit time, it is evident that the initial inrush events at both points essentially represent the maximum intensity and were released within a short period. As previously mentioned, the first inrush point had a flow rate of 1771.1 m³/min. Similarly, the second inrush point discharged 15,600 m³ of water in 29 min, resulting in a flow rate of 537.9 m³/min, which was the second highest intensity after the first event.

3.3. Analysis on the Evolution Mechanism of Water and Mud Inrush Disaster

The water and mud inrush event at Baiyunshan Tunnel occurred on 27 February 2022 and represents a typical case of karst cave + conduit + tunnel-type water inrush triggered by dynamic disturbance of excavation. According to the study by Li et al. [11], the water inrush disaster in tunnels consists of three components: the disaster source, the water inrush channel, and the impermeable barrier structure (as shown in Figure 6).

① Disaster Source: The source of energy, which is a mixture of water, deposits, and cavities within a specific spatial area, exhibits distinct energy storage characteristics. The disaster source is the primary factor triggering the water inrush event.

② Water Inrush Channel: The dominant migration pathway of the disaster source, which is the location where the underground water, mud, sand, and other mixed materials couple and evolve in transit. The water inrush channel is an essential condition for the occurrence of the water inrush disaster.

③ Impermeable Barrier Structure: The final barrier preventing the disaster source from entering the tunnel, which is the structure where the final rupture and water inrush occur. The rupture leading to the water inrush is a dynamic failure process induced by both the migration of the disaster source and the disturbances caused by tunneling activities at the working face.

Based on the disaster chain model theory (Figure 1), the water and mud inrush at Baiyunshan Tunnel exemplifies a classic composite disaster involving karst caves, subterranean rivers, and tunnel excavation. Its development conforms to a four-stage mechanism:

Predisposing Geological Environment (Upstream): The upstream area had unfavorable conditions, including karst conduits, water-bearing faults, weathered troughs, and sinkholes. The sinkhole at labelled BY14, along with faults F4 and F6 interconnected with an underground river, created a highly permeable flow path, with argillaceous infillings prone to water softening increasing instability.

Triggering Mechanism (Midstream): Tunnel excavation altered the in situ stress, potentially opening pre-existing fractures and allowing water ingress. Blast-induced vibrations may have raised the pore pressure, causing fractures to propagate and breach the tunnel.

Disaster Evolution (Downstream): A negative feedback loop developed between solid transport and conduit erosion. The mud–water mixture exhibited a shear-thinning Herschel–Bulkley rheological behavior, with flow velocity increasing as the conduit diameter expanded, i.e., v ∝ d 2, under same hydraulic gradient.

Aftermath: The initial inrush involved high-volume discharge that partially submerged the TBM and working platforms, lasting from 27 February to 1 March 2022. The event transitioned from continuous to intermittent flow. A subsequent inrush occurred between 30 March and 15 June 2022, with episodic water discharges carrying sediments, rock fragments, and pebbles, indicating a direct hydrologic connection between the tunnel and the surface-water–groundwater system via rainfall recharge.

4. Numerical Simulation Model Development

4.1. 3D Numerical Model of a Karst Cave+Conduit+Tunnel System

The phenomenon of mud and water inburst in tunnels involves an immiscible multiphase flow process comprising air, water, and mud. Accurately tracking the evolving interfaces among these three phases, as well as simulating the associated mass transport and flow regime evolution, constitute two fundamental challenges in the numerical modeling of such flows [11]. Various numerical approaches have been developed to capture fluid interfaces, including the Lattice Boltzmann Method (LBM) [21], the Level Set Method (LSM) [22,23], the Phase-Field Method (PFM) [22,24], and the Volume of Fluid (VOF) method [25]. Among these, the Phase-Field Method has demonstrated distinct advantages for capturing gas–liquid interfaces due to its inherent mass conservation characteristics, computational efficiency, and robust numerical stability. Additionally, particle-based tracking methods provide an effective means to simulate the motion of the solid-rich mud phase, thereby facilitating the realistic representation of mud surge dynamics. These computational techniques are essential for addressing the critical scientific questions posed by Li et al. [11], namely:

(1)

The mechanisms of displacement and energy release during the interaction of solid, liquid, and gas phases from hazard sources; and

(2)

The governing principles of multiphase mass transport and flow pattern transformation within water-mud inrush channels.

This study investigated a water and mud inrush-prone section of the Baiyunshan Tunnel. Leveraging an existing large-scale 3D model, we first extracted geological data from the portal to the disaster site. Through the systematic integration of multi-source data (high-precision topographic point clouds, geophysical electrical profiles, regional stratigraphy, tunnel design, and borehole records), we applied data fusion and intelligent extraction to identify reliable “hard data” defining the karst cave–conduit system. Augmented by geological expert knowledge for scientific interpretation, we successfully constructed a 3D spatial model of the karst system, as outlined in the technical workflow of Figure 7.

This methodology not only ensures the scientific validity and reliability of the model but also provides precise 3D geological model support for research on the mechanisms of water inrush and mud outburst disasters. The 3D karst conduit–tunnel system was numerically simulated using the finite element method implemented in COMSOL Multiphysics 6.3. This methodology necessitates the synthesis of multi-source datasets—including geological, geophysical, and hydraulic interpretations—to characterize dominant structural features, coupled with structural geology expertise for identifying principal groundwater flow paths and directions in the karst system. Through integrated data assimilation, the spatial configuration of hydraulically interconnected caves, subsurface conduits, and tunnels was delineated. These heterogeneities—including karst caves, conduits, and tunnels—were discretized as explicit structural elements, with model accuracy rigorously constrained through iterative calibration against field observations and in situ monitoring data. Consequently, the implemented model incorporates characteristic features identified from interdisciplinary datasets, encompassing karst cavities, conduit networks, major fault zones, and the engineered tunnel structure.

In COMSOL Multiphysics 6.3 software, the specific parameters used for finite element mesh (FEM) generation are presented in

Table 1. Specific parameters for FEM in COMSOL Multiphysics 6.3 software.

Item	Value
Number of Tetrahedral Elements	4,999,400
Curvature Factor	0.6
Element Volume Ratio	1.029 × 10−6
Average Element Quality	0.6631
Minimum Element Quality	0.1622
Maximum Element Growth Rate	1.15
Smallest Element/m	12.6
Largest Element/m	42.1

.

4.2. Numerical Simulation Strategy

A true three-dimensional model was developed, comprising three primary structural components representing karst caves, conduits, and tunnels. By employing a pairwise coupling strategy between solid-water and water-gas phases, these three hydraulically interconnected components simulate the complex three-phase (solid–water–gas) fluid dynamics in realistic subsurface geometries. This integrated configuration enables an accurate geometric and hydraulic representation of the karst cave–conduit–tunnel system at true spatial scale and depth. The cave modules function as natural storage reservoirs, accounting for significant groundwater storage capacity. An array of sensors is embedded within corresponding components to monitor hydraulic pressure and flow velocity. Furthermore, traceable particle phases are incorporated within the connecting conduits to simulate the transport dynamics of mud suspensions. The technical strategy for the simulation of tunnel water–mud inrush is illustrated in Figure 8.

From Figure 8, it can be seen that a simulation employs a hybrid approach, integrating the Phase-Field and Particle Tracking Methods in a parallel computational framework. The simulation strategy was proposed to simulate the mixed processes of water–gas drive and mud–water migration, with the aim of addressing the two aforementioned challenges, proposed by Li et al. [11], associated with water inrush and mud outburst. Among the strategies, the Phase-Field Method is employed to simulate the water–air interface with distinct phase separation, while the Particle-Tracking Method addresses the coupled flow problem involving mud–water interactions. This approach implicitly assumes that the two physical processes operate independently without mutual interference. This configuration aligns with the common real-world scenario where highly viscous and dense mud is largely unaffected by water flow. Due to the functional limitations of the current version of COMSOL Multiphysics, the particle tracking module is auxiliary in nature and cannot effectively transfer parameters to the laminar flow or phase-field modules; therefore, the two models are treated separately in the numerical solution. Meanwhile, there exists a clear order-of-magnitude difference between the dynamic viscosities of water and mud, and the mutual influence between their flow processes is relatively small. Based on these numerical constraints and physical characteristics, the real-time feedback effect of high-viscosity mud flow on the evolution of the water–gas interface was not considered in this study. Furthermore, given that the water inrush dominated this incident, with the mud inrush accounting for only a minimal portion, the constitutive model adopted for the fluid was primarily characterized as Newtonian.

4.3. Governing Equations

4.3.1. Fluid Control Equations

The flow of mud phase and water phase adopts the laminar flow model, and the fluid satisfies the Navier–Stokes equation for incompressible fluids and the continuity equation:

$$\left\{\begin{array}{c}\rho {\displaystyle \frac{\partial \mathit{u}}{\partial t}}+\rho \left(\mathit{u}·\nabla \right)=\nabla ·\left[-pI+\mu \left(\nabla \mathit{u}+{\left(\nabla \mathit{u}\right)}^T\right)\right]+\rho g+{\mathit{F}}_{\mathit{s}\mathit{t}}\\ \nabla \left(\rho \mathit{u}\right)=0\end{array}\right.$$

(1)

where u is the fluid velocity; ρ is the volume average density; p is the pressure; μ is the dynamic viscosity of the fluid; g is the gravitational acceleration; F_st is the surface tension.

However, to accurately represent water and mud as fluids with distinct flow behaviors, the phase-field and particle-tracking models employ separate sets of fluid flow parameters. This implementation establishes two dedicated simulation pathways within our parallel strategy, each performing its designated function. Crucially, the Navier–Stokes equations are solved using two different parameter sets to separately model the flow processes of water and mud.

4.3.2. Phase-Field Model

The phase-field method assumes that the interface between two phases is not a discontinuous form like a cliff, but a smooth transition region with a specific thickness, and there are diffusion effects and interfacial stresses between fluids in this transition region. Specifically, within a single medium fluid, all physical quantities remain constant; while in the transition region of the two-phase interface, these physical quantities change continuously without mutation.

The phase-field method is a robust numerical simulation technique whose core concept lies in representing sharp interfaces as smooth transition regions. This formulation allows for natural and efficient handling of complex interfacial topology changes. As a result, it has been extensively adopted in various frontier fields, including carbon storage with enhanced gas recovery (CSEGR) in petroleum engineering [26], CO₂ geological storage in environmental science [27], as well as multiphase flow interface dynamics and pore-scale flow studies in fluid mechanics [28]. It has thus emerged as a pivotal tool for bridging microscopic mechanisms with macroscopic phenomena.

Based on the above assumptions, the phase-field method identifies the transition regions of multiphase fluids and different phases by introducing phase-field variables ϕ. Among them, in different phase regions, the phase-field variables take constant values, usually defined as 1 and −,1 respectively; in the transition region of the two-phase contact, the phase-field variables change continuously in the interval (−1, 1).

Using the phase-field method as the interface capture algorithm, the movement process of the two-phase interface is described by the Cahn–Hilliard equation, and the specific expression, which is a mathematical statement of the evolution of the parameter ϕ under the transport by the fluid velocity u, comes from the calculation of the Navier–Stokes equation as follows:

$$\left\{\begin{array}{c}{\displaystyle \frac{\partial \varphi }{\partial t}}+\mathit{u}·\nabla \varphi =\nabla ·{\displaystyle \frac{\gamma \lambda }{{ϵ}^2}}\nabla \mu \\ \mu ={\displaystyle \frac{\delta \mathcal{F}}{\delta \varphi }}=\lambda \left[-{\nabla }^2\varphi +{\displaystyle \frac{\left({\varphi }^2-1\right)\varphi }{{ϵ}^2}}\right]\\ F={\int }_V\left[{\displaystyle \frac{\lambda }{2}}{\left|\nabla \varphi \right|}^2-{\displaystyle \frac{\lambda }{4{ϵ}^2}}{\left({\varphi }^2-1\right)}^2\right]dV\end{array}\right.$$

(2)

where μ is the chemical potential; γ represents the mobility; λ represents the mixing energy density; ϵ represents the interface thickness control parameter; ϕ is the phase-field variable;

Additionally, in the phase-field method, the surface tension F_st aforementioned in Equation (1) is the product of the chemical potential (μ) and the gradient of the phase-field variable (∇ϕ):

$${\mathit{F}}_{\mathit{s}\mathit{t}}=\mu \nabla \varphi$$

(3)

This expression establishes the connection between the two distinct fluids—water and air—in the phase-field model. The specific parameters used for the model are listed in

Table 2. Simulation Parameters for Phase-Field Model.

Parameter	Phase-Field Model
	Water	Air
Density (ρ)/kg·m−3	1000	2.5
Dynamic Viscosity (μ)/Pa·s	0.001	1.8 × 10−5
Surface Tension (σ)/N·m−1	0.07
Wetted Wall (θ)/rad	π/2
Interface Thickness Control (ϵ)/m	0.6
Inlet Pressure (p)/Pa	0
Outlet Pressure (p)/Pa	0

.

The initial and boundary conditions for the phase-field model were set as follows. A monitoring surface was set up at the interface between the water–mud inrush and the tunnel to measure the flow rate at this location. According to field records, the water inrush reached 15,000 m³ over 90 min. By converting this into an average flow rate and adjusting the water column height to ensure that the simulated flow matched the field measurements, the initial water column height was finally determined to be 18 m after multiple calculations and validations (Figure 7), with the tunnel air environment as the initial condition. A free water surface in the karst cave and a zero-pressure boundary condition at the tunnel outlet were adopted. Since this sudden water inrush occurred in an area with intense karst development, and after prolonged rainfall infiltration, it can be assumed that a certain amount of water already existed in the karst cavities before excavation. Therefore, the inlet in the boundary conditions can be regarded as a free water surface. In this case, the influence of rainfall on groundwater runoff can be neglected. This configuration establishes the prerequisite environment for subsequent transient calculations using the phase-field method.

4.3.3. Particle-Tracking Model

The particle-tracking model is a Eulerian–Lagrangian approach. It treats the mud as a fluid phase, which is a continuous medium described within a Eulerian framework, while the discrete particle phase is treated as individual particles tracked in a Lagrangian framework. This method resolves particle trajectories by computing the net force acting on each particle and integrating its equation of motion according to Newton’s second law.

$$m_p{\displaystyle \frac{d{\mathit{u}}_p}{dt}}=\sum \mathit{F}$$

(4)

where u_p is the vector of particle velocity; m_p is the particle mass; ΣF is the resultant force on the particle.

The movement behavior of particles in mud fluid is relatively complex, and the external forces they are subjected to mainly include flow resistance (viscous resistance), pressure gradient force, rotational lift force, Saffman lift force, virtual mass force, Basset force, thermogravitation, and gravity. For the convenience of calculation, this simulation assumes that the particle concentration in the fluid system is low and the particle size is small, and the interaction between particles can be ignored [29,30]. Therefore, the forces on the particles mainly consider the drag force and gravity of the liquid.

(1)

The resultant gravity can be expressed as:

$${\mathit{F}}_g=m_{p}g{\displaystyle \frac{{\rho }_p-\mathit{\rho }}{{\rho }_p}}$$

(5)

where F_g is the resultant force of gravity and buoyancy on the particle; ρ_p is the particle density; ρ is the fluid density.

(2)

The drag force can be expressed as:

$$\left\{\begin{array}{c}{\mathit{F}}_D={\displaystyle \frac{1}{{\tau }_p}}{m}_p\left|\mathit{u}-\mathit{v}\right|(\mathit{u}-\mathit{v})\\ {\tau }_p={\displaystyle \frac{{\rho }_p{d}_p^2}{18\mu }}\end{array}\right.$$

(6)

where F_D is the drag force of the mud on the particle; τ_p is the particle velocity response time; u is the mud velocity calculated by Navier–Stokes; v is the particle velocity (m/s); d_p is the particle diameter (m); μ is the dynamic viscosity (Pa·s).

The specific parameters used for the particle-tracking model are listed in

Table 3. Particle-Tracking Model Simulation Parameters.

Parameter	Particle-Tracking Model
	Mud	Particle
Density (ρ)/kg·m−3	1700	2700
Dynamic viscosity (μ)/Pa·s	0.5
Particle diameter (dp)/m	0.07
Inlet pressure (p)/Pa	0
Outlet pressure (p)/Pa	0

.

The initial fluid distribution for the simulated mudflow was set as a 10.4-m-high mud column (Figure 7), with the tunnel remaining air-filled as previously described. The initial particle release domain was located within the karst conduit, with the entrance set at the junction between the conduit and the tunnel—specifically at the breach formed during actual construction. This configuration replicates the initial state of the mudflow prior to its mobilization. Based on this setup, subsequent calculations for mud outburst with particle-tracking under unsteady flow conditions can be performed.

5. Results and Analysis

5.1. Simulation Results of Flow Velocity in Tunnel

In the simulation results presented in Figure 9a–f, three horizontal elevation planes are defined: Level 1 (606 m above reference) intersects both the karst cave and the tunnel; Level 2 (616 m) passes through the mid-height of the karst cave; and Level 3 (623.6 m) corresponds to the top of the karst cavity.

Notable discrepancies were observed in the magnitude ranges of the velocity fields across these elevation levels, as indicated by the respective scale legends. The velocity at Level 1 spanned from 0 to 10 m/s, representing the region of highest flow velocity. In contrast, Level 3 exhibited the most constrained velocity range, between 0 and 0.072 m/s, reflecting markedly lower flow intensities at this elevation.

As shown in Figure 9, distinct annular zones of velocity variation were visible at different elevations within the karst cave, indicating the formation of a depression funnel. During this process, groundwater is channeled into the tunnel through subterranean conduits, establishing the tunnel as the primary drainage pathway. This mechanism ultimately triggers a water inrush disaster. Velocity readings at the junction between the tunnel and the conduit showed a maximum flow velocity of up to 10 m/s.

Simultaneously, the simulation results show that at the initial stage (t = 10 s), the flow velocity at the tunnel heading was relatively high, reaching 6~7 m/s. By t = 60 s, however, the velocity at the same location had decreased significantly to only 1~2 m/s. This temporal evolution of velocity is fully consistent with the characteristic flow behavior of a real fluid.

5.2. Simulation Results of Water and Mud Inrush Scenarios

As previously described, the phase-field method is a thermodynamics-based mesoscale simulation approach that employs a continuous phase-field variable and a diffuse-interface model to describe the evolution of phase interfaces. Its core concept lies in replacing the traditional sharp interface with a smoothly varying order parameter (e.g., ϕ = ±1), forming a nanoscale transition region at the interface. The system evolution is governed by the Cahn–Hilliard or Allen–Cahn equations, which minimize the free energy functional comprising bulk and gradient energy terms, enabling the interface to spontaneously undergo complex topological changes such as nucleation, coalescence, and splitting.

In phase-field simulations conducted with COMSOL, the volume fraction of a fluid phase can be obtained by performing volume integration over regions where the order parameter corresponds to that phase (e.g., ϕ = 1). This allows precise identification and delineation of the interface in water–air two-phase flows. By visualizing the resulting phase distribution, it becomes possible to reconstruct realistic dynamic scenarios of fluid motion, thereby providing an intuitive and reliable numerical basis for the analysis of interface-dominated multiphysics processes.

Figure 10a–f presents the simulation results of the water and mud inrush process in an underground pipeline–tunnel system at different time steps. The mud inrush process is reproduced using particle tracking technology, which models the mud as solid particles with volume, mass, and viscosity. Within the first 30 s, the simulated mud flow, driven by the fluid, initially gushes out and accumulates in the tunnel, indicating the occurrence of the mud inrush phenomenon (as shown in Figure 10a–c).

Subsequently, the mud inrush process essentially concludes, and a particle-laden water flow is generated and gradually develops, eventually filling the 60-m tunnel section in the simulation domain with an average flow velocity of approximately 1.2 m/s (as shown in Figure 10d–f). The legend represents the velocity range of the water flow, showing that the flow front exhibited a higher velocity and greater impact force, around 6~7 m/s, while the middle and rear sections of the flow slowed down to about 1~2 m/s, with localized vortex formation observed.

5.3. Hazard Mechanism Analysis in Relation to Water and Mud Gushing

When water–mud-bearing structures are exposed during excavation, the drainage of these structures into underground spaces can trigger water and mud inrushes [31]. Such inrushes are typically sudden, posing immediate threats to the safety of personnel and equipment within underground chambers. A notable example is the Lingjiao tunnel, where direct exposure of karst caves led to sudden mud inrushes. The ejected mud destroyed all construction facilities and equipment at the working face, with two excavators, one loader, and one concrete pump being swept out of the tunnel [32]. The occurrence of these phenomena signifies the entry into the third phase of the previously mentioned disaster chain for tunnel water–mud inrush—the phase of disaster evolution.

The numerical simulation results provide a relatively clear visualization of the water and mud inrush process. By adjusting the parameters, a certain degree of consistency with actual field disaster scenarios can be achieved. Given the challenges in obtaining on-site observational data for such disasters, a strategy of repeated simulations was adopted to summarize the motion characteristics and behavioral patterns of the water–mud–air three-phase fluid system under various conditions, which will contribute to a deeper understanding of the mechanisms behind water and mud inrush.

The limitations of the numerical approach mainly lie in the following aspects. First, the model simplifies the mud fluid as a homogeneous Newtonian fluid and treats solid particles as a discrete phase with interactions through drag force. However, in actual mud inrush events, the slurry often exhibits strong non-Newtonian fluid characteristics (such as Bingham plasticity or shear-thinning behavior), where the internal friction angle and cohesion play a controlling role in the initiation and transport processes. Second, the model does not account for the erosive effect of the fluid on the pipe wall or the subsequent dynamic changes in channel morphology, which may constitute a key positive feedback mechanism in the evolution of real disasters. Field investigations indicate that the main evolution of the water and mud inrush lasted approximately 90 min, whereas significant erosion-induced changes in karst conduits and the tunnel surrounding rock typically require much longer timescales. During the event, erosion was limited to the millimeter–centimeter scale and was negligible compared with the meter-scale dimensions of the conduits and tunnel; therefore, fluid-induced erosion and the associated evolution of conduit morphology can be neglected.

To gain a deeper understanding of the physical picture revealed by the simulation results despite these limitations, we theoretically compared the mud inrush process in tunnels with the movement of surface debris flows [33,34]. This analogy is reasonable, as both are essentially shear-driven flow processes of solid–liquid mixtures under gravity. Based on this, a key dimensionless number—the Reynolds number (Re)—can be introduced to bridge the microscopic fluid behavior with macroscopic transport characteristics. Based on dimensional analysis, the impact effect of debris flows—governed by kinetic energy (ρv²) or potential energy (ρgh)—is primarily determined by both the Froude number (Fr) and the Reynolds number (Re), making it a function of these two dimensionless parameters.

$${\displaystyle \frac{P}{\rho v^2}}=a\times {Re}^b+c\times {Fr}^d$$

(7)

where P is the impact pressure of debris flows; a, b, c, d are empirical constants.

However, there are notable differences in the flow characteristics between tunnel mud inrushes and typical debris flows. First, mud inrushes generally occur in channels with relatively gentle slopes. Since their movement is not primarily driven by the conversion of gravitational potential or kinetic energy, the resulting impact effect is relatively limited. Second, the material composition of mud inrushes is dominated by clay particles, with a very low content of coarse grains such as gravel or cobbles, resulting in significantly higher viscosity. In such high-viscosity, low-gradient flows, viscous forces—in addition to inertia—play a key role in governing the flow behavior. Therefore, for this type of mud inrush, studying the impact effect alone offers limited practical value. Instead, in-depth analysis of the relationship among flow velocity, drag force, transport distance, and the Reynolds number holds greater significance for engineering guidance.

Therefore, the hazardous effects of high-viscosity tunnel mud inrushes—such as flow velocity and transport distance—are fundamentally governed by their flow regime. Accurately quantifying these effects requires a clear characterization of the intrinsic relationship between the Reynolds number and the drag force, that is, the resistance exerted by the fluid on particles. It serves as follows:

$$\left\{\begin{array}{c}{F}_d={\displaystyle \frac{1}{2}}{C}_d{\rho }_{d}A{v}^2\\ Re={\displaystyle \frac{\rho vL}{\mu }}\end{array}\right.$$

(8)

where L is the characteristic length, such as particle size; ρ_d is the particle density; ρ is the density of mud flow; A is the reference area (typically the frontal projected area); C_d is the drag coefficient. It is the function of Reynolds (Re), i.e., $C_d=f\left(Re\right)$.

5.4. Predicting Mud Conditions: Effects of Parameter Variations

This study investigated the influence of key parameters—mud density, particle size, and dynamic viscosity—on mudflow dynamics. Numerical simulation schemes were performed to quantify the runout distance, flow velocity, and drag force, and to establish their correlation with the Reynolds number. The parameter settings of different simulation designs are listed in

Table 4. Parameter settings across different simulation designs.

Design A
Mudflow Density/kg·m−3	1500
Dynamic Viscosity of Mudflow/Pa·s	0.1	0.5	1.0	5.0	10.0
Particle Velocity/m·s−1	0.94	1.00	1.27	3.45	5.7
Design B
Mudflow Density/kg·m−3	1700
Dynamic Viscosity/Pa·s	0.1	0.5	1.0	5.0	10.0
Particle Velocity/m·s−1	0.93	0.97	1.28	3.90	5.71
Design C
Mudflow Density/kg·m−3	1900
Dynamic Viscosity/Pa·s	0.1	0.5	1.0	5.0	10.0
Particle Velocity/m·s−1	0.92	0.98	1.28	3.5	5.72

.

Based on the disaster characteristics described previously and the subsequent geological investigation results [16], the material involved in the mud influx primarily originated from sediments stored within the karst cave–underground conduit system. The composition was dominated by muddy fluid, mixed with a certain amount of well-rounded gravel and cobbles. For parameter configuration, the fluid viscosity was set to represent a transition from the muddy flow type (dynamic viscosity: 1~10 Pa·s) to the water-stone flow type (dynamic viscosity: 0.1~5 Pa·s). Accordingly, three simulation schemes were designed based on the debris flow density parameters (dilute: 1500 kg/m³, transitional: 1700 kg/m³, dense: 1900 kg/m³). Due to the limited flow distance of muddy fluids, particularly the propensity for dense flows to deposit at tunnel exits, a simulation monitoring point (labelled T-13) was positioned near the tunnel entrance of the underground conduit (Figure 11d) to acquire data on flow velocity and drag force.

Based on the current simulation results, the model effectively reproduced the main phenomena of the water and mud inrush disaster, with the fitting results showing a high degree of agreement with the scattered data. The coefficient of determination (R²) reached above 0.90. As shown in Figure 11a,b, both the travel distance and flow velocity of the mud flow exhibited a decelerating increasing trend with rising Re number. Figure 11c presents a semi-logarithmic plot that demonstrates that the drag force on mud particles decays exponentially with increasing Re number, which is generally consistent with the theoretical exponential decay of the drag coefficient (C_d) with Re number. Furthermore, when analyzing density as a parameter representing inertial force, it was observed that under the same Re number conditions, higher density corresponded to shorter travel distance and lower flow velocity. In contrast, the drag force on particles showed little variation across different density conditions, indicating that it is less influenced by inertial forces and is primarily governed by viscous effects, particularly in the high-viscosity (low Re < 2000) regime. In the low-viscosity (high Re > 4000) regime, the drag force remained consistently at a low constant value. In the low Reynolds number range shown in Figure 11c (Re < 10²), the flow was in a viscosity-dominated Stokes flow regime. At this stage, the drag force on the particles is primarily contributed by the fluid’s viscosity and is weakly related to particle density. As a result, the simulation results under different density conditions almost overlapped, indicating that particle motion is relatively insensitive to density under low Reynolds number and high-viscosity conditions.

5.5. Predicting Mud Flow: Results of Parameter Variations

Through systematic parameter-variation simulations, this study clearly reconstructed the dynamic evolution of mudflow throughout its entire process, i.e., from initiation, transport, to deposition, visually demonstrating how its motion state is governed by key hydraulic and physical parameters.

Figure 12 presents the transport distance, flow velocity, and drag force of the mudflow under different simulation schemes. To facilitate a direct comparison, all data were extracted at a specific time instant of 30 s after the simulation start and are comprehensively presented in the form of a Poincaré plot of drag force, which integrates the results from combinations of three density values, corresponding to three simulation designs, with varied five dynamic viscosity coefficients.

Based on the graphical analysis, when the density is held constant, the transport distance of mudflow generally exhibits a decreasing trend as the dynamic viscosity coefficient increases. This behavior aligns with the general principle that “higher fluid viscosity leads to poorer flowability”, thereby validating the reliability of the simulation. Under a constant dynamic viscosity coefficient, an increase in density similarly results in a reduction in transport distance; however, the magnitude and sensitivity of this effect are less pronounced than those induced by variations in dynamic viscosity. Based on the results shown in Figure 11, Figure 12 can serve as a reference for fitting the field mudflow monitoring data. It should be noted that the relationship curve established in this study was based on a 60-m-long local model and was primarily intended to reveal the relative variations and trend characteristics of mudflow dynamics. Considering that the actual tunnel engineering scale is much larger and geological conditions are heterogeneous, directly extrapolating this dimensionless Reynolds number relationship has limitations. Therefore, when applying this relationship curve in engineering practice, it should be calibrated with field monitoring data and, if necessary, the prediction accuracy can be improved through multi-segment local simulations or on-site calibration. This enables the quantitative inversion of key physical parameters of the mudflow, thereby providing a theoretical basis for risk assessment and the design of preventive measures against water and mud inrushes during tunnel construction.

6. Discussion

This study aimed to simulate the initiation and progression of water inrush disasters in karst tunnels. The actual engineering case underpinning our research clearly indicates that the dominant material involved in the disaster process was water. Specific data show that a brief mud outburst occurred only during the initial onset, with a peak flow rate (58.8 m³/min) constituting less than 3.3% of the peak water inrush flow rate (1764.7 m³/min), and no subsequent mud outburst of comparable magnitude was observed. The intermittent water discharges over the following month resulted in only slightly turbid water. Therefore, the core physical process under investigation is a water-dominated hydrodynamic problem. Within this context, adopting an incompressible Newtonian fluid constitutive relation in the current model serves the primary purpose of revealing the flow mechanisms, energy evolution, and interaction with surrounding rock during large-scale water influx within a conduit structure. This simplification is reasonable and effective for focusing on the main research objective and establishing a foundational analytical framework.

Nevertheless, we fully understand and concur with the reviewers’ insightful points. Actual mud outburst slurries, especially high-concentration mixtures rich in debris, often exhibit significant non-Newtonian rheological properties such as yield stress and shear-thinning behavior, which are more accurately described by constitutive models like Herschel–Bulkley or Bingham. We fully acknowledge the limitations of the current model:

(1)

Limited applicability to pure mud flows or inflows with high mud content: The Newtonian assumption neglects the “plug flow” effect caused by yield stress, potentially overestimating near-wall velocities and affecting predictions of flow stoppage mechanisms and final deposition morphology.

(2)

Current state of parameter basis: Implementing precise non-Newtonian fluid simulation relies on obtaining reliable rheological parameters (e.g., yield stress τ_y, consistency coefficient K, flow index n) through field sampling or experimental measurement. Detailed rheological data specific to this case is currently not fully available.

Furthermore, the issue of model scale extrapolation warrants discussion here. To reveal the underlying mechanisms, this study utilized a 60-m-long typical conduit segment for localized, detailed simulation, from which dimensionless relationships for analysis were derived. However, actual tunnel engineering operates on a much larger scale within highly heterogeneous geological environments. Consequently, inherent limitations exist regarding geometric similarity and boundary condition representativeness when directly extrapolating relationships derived from a small-scale local model to full-scale engineering applications. For engineering reference, the quantitative relationships established herein require calibration and correction against specific field monitoring data, or necessitate adjustments for scale effects through methods like segmental simulation or inverse analysis.

In summary, the conclusions of this study regarding the evolution of water inrush dynamics, key influencing factors, and relative trends are robust within the specific disaster scenario of a “water-dominated” event under the Newtonian fluid assumption. Simultaneously, we explicitly state that: first, this model is not suitable for precise prediction centering on mud slurry rheology; second, quantitative extrapolation of the model requires careful consideration of scale effects. In future work, we will focus on collecting rheological data for typical slurries, incorporating more universal non-Newtonian constitutive models, and developing cross-scale simulation methodologies to construct a more comprehensive framework for simulating the entire process of water and mud inrush disasters.

7. Conclusions

Based on the comprehensive investigation and numerical simulation of the water and mud inrush disaster in the Baiyunshan Tunnel, the following conclusions can be drawn:

• The water and mud inrush event at Baiyunshan Tunnel represents a typical case of karst cave–conduit–tunnel disaster triggered by excavation activities. The integration of multi-source data enabled the construction of a high-fidelity 3D geological model that accurately characterizes the spatial distribution of the fault–karst conduit–groundwater system, providing a solid foundation for mechanism analysis and numerical simulation.
• The developed numerical model, employing a coupled Phase-Field and Particle-Tracking approach, successfully reconstructed the complete disaster process and captured the essential physics of the three-phase (air–water–mud) flow behavior. The simulation results demonstrate good agreement with field observations in terms of flow velocity evolution and disaster progression patterns.
• Parameter sensitivity analysis revealed that both dynamic viscosity and density significantly influence mudflow behavior, with viscosity playing a dominant role in controlling transport characteristics. The established relationships between Reynolds number and key parameters (flow velocity, travel distance, and drag force) provide quantitative insights into mudflow dynamics across different flow regimes.
• The exponential decay of drag force with increasing Reynolds number, consistent with theoretical predictions of drag coefficient behavior, highlights the crucial role of viscous effects in high-viscosity mudflows. This understanding is particularly valuable for predicting the mobility and impact of similar mud inrush events.
• The comprehensive methodology presented in this study, from geological characterization to numerical simulation and parameter analysis, establishes a framework for the high-fidelity simulation of water and mud inrush disasters. The findings contribute to improved risk assessment and the development of more effective prevention strategies for tunneling projects in karst terrains.

For this water-dominated disaster case, adopting a Newtonian fluid constitutive model effectively captured the key flow characteristics and evolution patterns of the water-inrush process. The established flow parameter relationships provide a mechanistic reference for similar engineering scenarios. Meanwhile, the study explicitly identified the limitations of the current model in describing the non-Newtonian characteristics of high-concentration slurry and in scale extrapolation. Future work should further integrate non-Newtonian rheological models and multi-scale simulation methods to develop a more universal framework for disaster simulation and risk assessment, thereby offering more precise theoretical support for disaster prevention in karst tunnel engineering.