Progressive Failure of Water-Resistant Stratum in Karst Tunnel Construction Using an Improved Meshfree Method Considering Fluid–Solid Interaction

Abstract

An improved meshfree method that considers cracking, contact behaviour and fluid–solid interaction (FSI) was developed and employed to shed light on the progressive failure of the water-resistant stratum and inrush process in a karst tunnel construction. Hydraulic fracturing tests considering different scenarios and inrush events of the field-scale Jigongling karst tunnel in three scenarios verify the feasibility of the improved meshfree method. The results indicate that the brittle fracture characteristics of the rock mass are captured accurately without grid re-meshing by improving the kernel function of the meshfree method. The complex contact behaviour of rock along the fracture surface during inrush is correctly captured through the introduction of Newton’s law-based block contact algorithms. FSI processing during inrush is accurately modelled by an improved two-phase adaptive adjacent method considering the discontinuous particles without coupling other solvers and additional artificial boundaries, which improves computational efficiency. Furthermore, the improved meshfree method simultaneously captures the fast inrush and rock failure in the Jigongling karst tunnel under varying thicknesses and strengths of water-resistant rocks and sizes of karst caves. As the thickness and strength of water-resistant rock increase, the possibility of an inrush disaster in the tunnel decreases, and a drop in the water level and an increase in the maximum flow velocity have significant delayed effects during the local inrush stage.

1. Introduction

Karst caves with high water pressure and low visibility around tunnels can be dangerous when they induce the sudden and unpredictable inrush of water in tunnel construction. Under the joint action of karst water pressure and the weak strength of the water-resistant stratum, the water-resistant rock between the tunnel and high-water pressure-hidden cavity cannot withstand the absence of pre-support and pre-reinforcement. Subsequently, cracks in water-resistant rock initiate, expand and coalesce, possibly resulting in water inrush disasters (Figure 1), severe economic losses and casualties. Therefore, understanding the internal mechanisms of progressive rock failure during water/mud inrush disasters will undoubtedly provide a critical guiding effect during the design and construction of karst tunnel structures.

The approaches used to study progressive rock failure and water/mud inrush disasters of karst tunnels are mainly divided into laboratory tests and numerical simulations. In terms of laboratory tests, Zhou et al. [1] discovered disastrous characteristics of inrush during the filling of a karst tunnel by carrying out a three-dimensional filling karst tunnel model. Then, Pan et al. [2] used the indoor FSI model to study the influence of hidden karst caves on the stability of surrounding rock under different water pressure conditions during filling. However, laboratory tests were conducted on laboratory scales and cannot replace tests of field-scale tunnels. Therefore, numerical simulation methods to investigate the progressive failure of field-scale karst tunnels were developed. Li et al. [3] employed the finite element method (FEM) to investigate the pattern of change in the stress and defamation of rock under different water pressures in karst caves. For modelling rock damage, Huang et al. [4] used a realistic failure process analytical method to study the progressive failure of water-resistant rock under karst caves with different water pressures. However, the grid method cannot reproduce a larger deformation (i.e., fictional sliding and toppling failure) of water-resistant rock. Moreover, the inrushing fluids were not physically modelled in the above studies. Comparatively, discontinuous methods, such as the discrete element method (DEM) [5] and discontinuous deformation analysis (DDA) [6], are more suitable for analysing the complex motion and contact behaviour of rock blocks during inrush disasters. Moreover, inrush mechanics in karst tunnels have been studied over the last decade with the development of computational fluid dynamics (CFD) methods [7,8] and the lattice Boltzmann method (LBM) [9,10], which made simulating progressive rock failure and inrush in tunnels feasible. Although coupled multiple solvers can reproduce the complete failure and inrush flow process in rocks, it requires additional and artificial treatment, such as the static and moving immersed boundary methods in CFD [11], which increases the complexity of the FSI programming and reduces the computational efficiency. In addition, the DEM and DDA have relatively low computational efficiency due to complex contact search algorithms and bond models, which hinders their application in field-scale karst tunnels. Among the existing methods for two-phase fluid-structure interaction, including DDA-SPH [12,13,14], DEM-SPH [15,16], MPM-SPH [17,18], DEM-CFD [19,20], MPM-CFD [21] and DEM-LBM [22,23] and other methods. In the FSI strategy, it is observed that the use of particle methods for both fluid and solid phases allows for a more natural conversion of speed and energy. In contrast, when particle methods and grid methods are used for the fluid and solid phases respectively, numerous artificial assumptions are introduced at the boundaries of particles and grids. This results in reduced robustness of the algorithm and increased complexity.

Smoothed particle hydrodynamics (SPH) [24] is a pure Lagrangian meshfree method and is then applied to reproduce the free-surface fluid [25,26,27] and solve the FSI process problem [28,29]. Based on this, coupled SPH methods to reproduce the progressive failure of rock and the inrush of the tunnel have been developed in recent years. Wang et al. [30] used coupled FEM and SPH methods to simulate the barrier wall damage and the water inrush under artificial excavation disturbance, in which FEM and SPH models represented rock wall and inrushing fluid, respectively. Peng et al. [31] adopted a coupled DDA and SPH method to simulate the water inrush process in tunnel construction, in which DDA and SPH models represent water-resistant rock and inrushing fluid, respectively. However, the coupled method requires the introduction of artificial assumptions, and the interface of two computational domains introduces additional computational errors and nonconvergence issues [32]. Additionally, it increases the programming complexity and expensive computational time. To fill the gaps mentioned above, SPH has been improved to model material behaviour and subsequent fractures by employing various local cracking strategies in recent years. For example, Chakraborty and Shaw [33] developed a pseudo-spring method for crack treatment, which successfully realised crack initiation in rocks. Yu et al. [34] used an improved kernel SPH (IKSPH) method and investigated the cracking processes in rock cells. Mu et al. [35] developed an improved SPH method to reproduce the fracture process of a rock mass. The above studies have demonstrated that by incorporating reasonable failure criteria, SPH can independently simulate the progressive failure of rocks. Meanwhile, the problem of coupled fracture and contact in SPH has been solved, making it applicable to the field of rock landslides [36]. However, complex fluid–solid interactions during inrush by the SPH method need to be rigorously defined. There has been no case study demonstrating the typical and critical processes of rock failure and water inrush in karst tunnels using the SPH framework.

This paper aims to propose an improved meshfree method that considers cracking, contact behaviour and FSI to reproduce the progressive rock failure and water inrush of a karst tunnel. This method combines the kernel broken cracking strategy and failure criterion to simulate the brittle fracturing of rock. Damaged and intact particles form new contacts, upon which frictional slip, compression/separation, and rotation of blocks occur after rock damage. δ-Weakly compressible SPH (δ-WCSPH) is employed to model the fluid domain. Moreover, a two-phase adaptive adjacent method considering the discontinuous particles is developed to rigorously define FSI processing during inrush. A verified hydraulic fracturing test considering the different initial flaws, water pressure, initial particle spacing and smoothing kernel length is conducted. The progressive rock failure and water inrush of the Jigongling karst tunnel are calculated and analysed by considering the influence of karst cave radii, the thickness and strength of water-resistant rock, and corresponding physical vulnerabilities are investigated. Finally, the advantages and disadvantages of the improved SPH are discussed.

2. Method

2.1. Constitutive Model of Rock

Cauchy stress tensor of rock $\mathit{\sigma }$ consists of two parts: isotropic pressure P and deviatory stress τ, which is calculated as follows [37]:

$${\mathit{\sigma }}^{\alpha \beta }=-P{\delta }^{\alpha \beta }+{\mathit{\tau }}^{\alpha \beta },$$

(1)

The isotropic pressure $P=Kb({\displaystyle \frac{\rho }{{\rho }_0}}-1)$ of the material obeys Hooke’s law; $\dot{\mathit{\tau }}$ denotes the stress rate; under small deformation conditions, the stress rate is proportional to the strain rate,

$$\dot{\mathit{\tau }}=G{\overline{\mathit{\epsilon }}}^{\alpha \beta }=G({\dot{\mathit{\epsilon }}}^{\alpha \beta }-{\displaystyle \frac{1}{3}}{\delta }^{\alpha \beta }{\dot{\mathit{\epsilon }}}^{\gamma \gamma }),$$

(2)

where ${\overline{\mathit{\epsilon }}}^{\alpha \beta }$ denotes the shear deformation, $G$ denotes the shear modulus, and ${\dot{\mathit{\epsilon }}}^{\alpha \beta }$ denotes the strain rate tensor, which is formulated as follows:

$${\mathit{\epsilon }}^{\alpha \beta }={\displaystyle \frac{1}{2}}({\displaystyle \frac{\partial {\mathit{\nu }}^{\alpha }}{\partial {\mathit{x}}^{\beta }}}+{\displaystyle \frac{\partial {\mathit{\nu }}^{\beta }}{\partial {\mathit{x}}^{\alpha }}}),$$

(3)

By introducing the SPH kernel approximation, the equations of mass and momentum conservation can be obtained as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\rho }_i}{\mathrm{d}t}}={\displaystyle \sum _{j=1}^N{m}_j({\mathit{v}}_i^{\alpha }-}{\mathit{v}}_j^{\alpha }){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_i^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{j=1}^N{m}_j(}{\displaystyle \frac{{\mathit{\sigma }}_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\mathit{\sigma }}_j^{\alpha \beta }}{{\rho }_j^2}}+{\mathit{\Pi }}_{ij}-{\mathit{T}}_{ij}){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_i^{\alpha }}{\mathrm{d}t}}={\mathit{v}}_i^{\alpha }\end{array}\right.+{\displaystyle \frac{{\mathit{f}}_i^{\alpha }}{m_i}}$$

(4)

where ${\mathit{v}}_i^{\alpha }$ and ${\mathit{v}}_j^{\alpha }$ is the velocity of base particles i and adjacent particles j, respectively. Π_ij is the artificial viscosity, which is unutilised to balance nonphysical vibrations [38]. The B-spline kernel $W_{ij}$ is used [39] in this study. m and $\rho$ represents mass and density, respectively; ${\mathit{x}}^{\alpha }$ and ${\mathit{v}}^{\alpha }$ is the coordinate position and velocity component, respectively; t is calculation time; ${\mathit{f}}^{\alpha }$ is an external force.

2.2. Constitutive Model of Fluid

For a fluid like water, a weak compression SPH method is adopted to model free-surface flow. The discretisation governing equations, which consist of the mass and momentum conservation equations for fluid, are calculated as follows [40]:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\rho }_a}{\mathrm{d}t}}={\displaystyle \sum _{b=1}^N{m}_b({\mathit{v}}_a^{\alpha }-{\mathit{v}}_b^{\alpha })}{\displaystyle \frac{\partial W_{ab}}{\partial {\mathit{x}}_a^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_a^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{b=1}^N{m}_b(}{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\displaystyle \frac{P_b}{{\rho }_b^2}}+{\mathit{\Pi }}_{ab}){\displaystyle \frac{\partial W_{ab}}{\partial {\mathit{x}}_a^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_a^{\alpha }}{\mathrm{d}t}}={\mathit{v}}_a^{\alpha }\end{array}\right.+{\displaystyle \frac{{\mathit{f}}_a^{\alpha }}{m_a}}$$

(5)

where a and b denote the base and adjacent fluid particles, respectively, $P_a$ and $P_b$ are the pressure of fluid particles a and b, respectively. Additionally, the pressure $P={\displaystyle \frac{c_0^2{\rho }_0}{\gamma }}[{({\displaystyle \frac{\rho }{{\rho }_0}})}^{\gamma }-1]$ is determined by the equation of state [41]. To reduce the susceptibility of the pressure field to numerical noise, a density diffusion term is incorporated into the continuity density equation and is calculated as follows:

$${\displaystyle \frac{\mathrm{d}{\rho }_a}{\mathrm{d}t}}={\displaystyle \sum _{b=1}^N{m}_b({\mathit{v}}_a^{\alpha }-{\mathit{v}}_b^{\alpha })}{\displaystyle \frac{\partial W_{ab}}{\partial {\mathit{x}}_a^{\alpha }}}+0.2h{c}_0{\displaystyle \sum _{b=1}^N(\frac{{\rho }_a}{{\rho }_b}}-1){\displaystyle \frac{1}{({\mathit{x}}_a^{\alpha }-{\mathit{x}}_b^{\alpha })+0.01h^2}}{m}_j({\mathit{x}}_a^{\alpha }-{\mathit{x}}_b^{\alpha }){\displaystyle \frac{\partial W_{ab}}{\partial {\mathit{x}}_a^{\alpha }}}$$

(6)

2.3. Cracking Strategy of Rock

To determine crack initiation in solids like rock, the Mohr–Coulomb criterion with tensile truncation is applied, yielding reasonable results. In intact rocks, two adjacent particles are initially connected by a virtual bridge (kernel), as defined by the kernel function of base particles, indicating ‘full interaction’ (Figure 2a). Based on this, once the rock particle reaches a failure state, the kernel function does not control the particle (i.e., kernel broken) [42,43,44,45,46]. ‘No interaction’ is formed between the damaged and intact base particles (Figure 2b). To model damage growth and crack propagation, a linear elastic brittle law is employed, as illustrated in Figure 2c. A parameter ψ is introduced to enhance the kernel function. When the failure criterion is met by the rock particle, ψ is set to 0; otherwise, ψ equals 1. This approach defines the path for crack propagation.

$${\displaystyle \frac{\partial {\Re }_{ij}}{\partial {\mathit{x}}_i}}=\psi {\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i}}$$

(7)

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\rho }_i}{\mathrm{d}t}}={\displaystyle \sum _{j\in U}{m}_j({\mathit{v}}_i^{\alpha }-}{\mathit{v}}_j^{\alpha }){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}+{\displaystyle \sum _{j\in \Re }{m}_j({\mathit{v}}_i^{\alpha }-}{\mathit{v}}_j^{\alpha }){\displaystyle \frac{\partial {\Re }_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_i^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{j\in U}{m}_j(}{\displaystyle \frac{{\mathit{\sigma }}_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\mathit{\sigma }}_j^{\alpha \beta }}{{\rho }_j^2}}+{\mathit{\Pi }}_{ij}-{\mathit{T}}_{ij}){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}+{\displaystyle \sum _{j\in \Re }{m}_j(}{\displaystyle \frac{{\mathit{\sigma }}_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\mathit{\sigma }}_j^{\alpha \beta }}{{\rho }_j^2}}+{\mathit{\Pi }}_{ij}-{\mathit{T}}_{ij}){\displaystyle \frac{\partial {\Re }_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_i^{\alpha }}{\mathrm{d}t}}={\mathit{v}}_i^{\alpha }\end{array}\right.+{\displaystyle \frac{{\mathit{f}}_i^{\alpha }}{m_i}}$$

(8)

where U and $\mathbf{\Re }$ represents damage and intact particles, respectively.

2.4. Contact Algorithm of Rock

In this section, a contact algorithm for rocks is described: when rock particles are damaged, they automatically become discontinuous particles with cracks (Figure 3a). A block contact based on Newton’s law is established between intact particles (IP) i and discontinuous particles (DP) j. Once penetration between two rock particles is detected, the contact will be activated immediately (Figure 3b). The penetration between IP i and DP j is determined as follows (Figure 3c):

$$\left\{\begin{array}{l}{U}^n=R_i+R_j-d_{ij}\\ ∆U^s=∆{\mathit{v}}_i-∆{\mathit{v}}_i\cdot \mathit{n}\end{array}\right.,$$

(9)

where R_i and R_j are the diameters of the two particles, and the tangential contact force $F_i^{Cs}$ acting on the intact particle is calculated by $F_i^{Cs}={\left\{F_i^{Cs}\right\}}_{\mathrm{update}}+k_s\Lambda U^s$. Once the fraction limit is exceeded (i.e., $‖{\mathit{F}}_i^{Cs}‖\ge u‖{\mathit{F}}_i^{Cn}‖$), sliding is permitted. To address this, the tangential contact force can be finally derived as follows:

$$\overline{{\mathit{F}}_i^{Cs}}=\left\{\begin{array}{l}u‖{\mathit{F}}_i^{Cn}‖{\displaystyle \frac{{\mathit{F}}_i^{Cs}}{‖{\mathit{F}}_i^{Cs}‖}}\begin{array}{cc} & \end{array}‖{\mathit{F}}_i^{Cs}‖\ge u‖{\mathit{F}}_i^{Cn}‖\\ {\mathit{F}}_i^{Cs}\begin{array}{ccccc} & & & & \end{array}‖{\mathit{F}}_i^{Cs}‖<u‖{\mathit{F}}_i^{Cn}‖\end{array}\right.,$$

(10)

where the frictional coefficient, μ, is incorporated into the force, which is then added to the momentum Equation (8) to update the particle’s velocity and position. As a result, the governing equations for both intact and discontinuous particles can be reformulated as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\rho }_i}{\mathrm{d}t}}={\displaystyle \sum _{j\in U}{m}_j{\mathit{v}}_{ij}^{\alpha }}{\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\beta }}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_i^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{j\in U}{m}_j(}{\displaystyle \frac{{\mathit{\sigma }}_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\mathit{\sigma }}_j^{\alpha \beta }}{{\rho }_j^2}}+{\mathit{\Pi }}_{ij}-{\mathit{T}}_{ij}){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\beta }}}+{\displaystyle \frac{{\mathit{F}}_i^E}{m_i}}+{\displaystyle \frac{{\mathit{F}}_i^{Cn}}{m_i}}+{\displaystyle \frac{\overline{{\mathit{F}}_i^{Cs}}}{m_i}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_i^{\alpha }}{\mathrm{d}t}}={\mathit{v}}_i^{\alpha }\end{array}\right.$$

(11)

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\rho }_d}{\mathrm{d}t}}=0\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_d^{\alpha }}{\mathrm{d}t}}={\displaystyle \frac{{\mathit{F}}_d^E}{m_d}}+{\displaystyle \frac{{\mathit{F}}_d^{Cn}}{m_d}}+{\displaystyle \frac{\overline{{\mathit{F}}_d^{Cs}}}{m_d}}\\ {\displaystyle \frac{\mathrm{d}{\mathit{x}}_d^{\alpha }}{\mathrm{d}t}}={\mathit{v}}_d^{\alpha }\end{array}\right.$$

(12)

where ${\mathit{F}}_i^E$ and ${\mathit{F}}_d^E$ denote the external force of intact and discontinuous particles, respectively. Therefore, Equation (12) is used to describe the mass and the governing equation for the motion of DP particles [47].

2.5. Fluid–Solid Interaction Scheme

An SPH method incorporates a two-phase adaptive adjacent algorithm that accounts for DPs, allowing for the simulation of FSI processes like hydraulic fracturing during inrush. When fluid particles get into the kernel support of solid particles, the solid particles are regarded as the deformable boundary to impose velocity and displacement boundary conditions on the fluid particles. Consequently, the force ${\mathit{f}}_i^S$ exerted on the solid particle i by the neighbouring fluid particles can be derived as follows:

$${\mathit{f}}_i^S=-m_i{\displaystyle \sum _a{m}_a(}{\displaystyle \frac{P_i}{{\rho }_i^2}}+{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\mathit{\Pi }}_{ia}){\nabla }_{i}W({\mathit{x}}_{ia}),$$

(13)

According to Newton’s third law, the equal and opposite reaction due to neighbouring solid particles exerted on the solid particle is obtained from the following:

$${\mathit{f}}_a^F=-m_a{\displaystyle \sum _i{m}_i(}{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\displaystyle \frac{P_i}{{\rho }_i^2}}+{\mathit{\Pi }}_{ai}){\nabla }_{a}W({\mathit{x}}_{ai}),$$

(14)

When hydraulic fracturing of rock occurs, referring to the DEM–SPH coupling treatment [15], the discontinuous particles d are regarded as repulsive particles [48]. In addition, interaction forces ${\mathit{f}}_d^F$ and ${\mathit{f}}_a^S$ are treated as the external force (i.e., pressure), which are an equal and opposite reaction from the interacting discontinuous and water particles and are described using the following formula:

$${\mathit{f}}_a^S=-{\mathit{f}}_d^F=\left\{\begin{array}{c}\chi \left[{(r_0/r_{ad})}^{q_1}-{(r_0/r_{ad})}^{q_2}\right](x_{ad}/r_{ad}), \mathrm{if} r_0/r_{ad}<1\\ 0, \mathrm{if} r_0/r_{ad}\ge 1\end{array}\right.$$

(15)

These interaction forces are then incorporated into both the fluid and solid momentum equations. Consequently, the momentum equations for intact rock particles, discontinuous particles, and fluid particles can be rewritten as follows:

$$\left\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}{\mathit{v}}_i^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{j\in U}{m}_j(}{\displaystyle \frac{{\mathit{\sigma }}_i^{\alpha \beta }}{{\rho }_i^2}}+{\displaystyle \frac{{\mathit{\sigma }}_j^{\alpha \beta }}{{\rho }_j^2}}+{\mathit{\Pi }}_{ij}-{\mathit{T}}_{ij}){\displaystyle \frac{\partial W_{ij}}{\partial {\mathit{x}}_i^{\alpha }}}+{\displaystyle \frac{{\mathit{F}}_i^E}{m_i}}+{\displaystyle \frac{{\mathit{F}}_d^{Cn}}{m_i}}+{\displaystyle \frac{\overline{{\mathit{F}}_i^{Cs}}}{m_i}}-m_i{\displaystyle \sum _a{m}_a(}{\displaystyle \frac{P_i}{{\rho }_i^2}}+{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\mathit{\Pi }}_{ia}){\nabla }_{i}W({\mathit{x}}_{ia})\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_a^{\alpha }}{\mathrm{d}t}}={\displaystyle \sum _{b=1}^N{m}_b(}{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\displaystyle \frac{P_b}{{\rho }_b^2}}+{\mathit{\Pi }}_{ab}){\displaystyle \frac{\partial W_{ab}}{\partial {\mathit{x}}_a^{\alpha }}}+{\displaystyle \frac{{\mathit{f}}_a^{\alpha }}{m_a}}-m_a{\displaystyle \sum _i{m}_i(}{\displaystyle \frac{P_a}{{\rho }_a^2}}+{\displaystyle \frac{P_i}{{\rho }_i^2}}+{\mathit{\Pi }}_{ai}){\nabla }_{a}W({\mathit{x}}_{ai})+\chi \left[{(r_0/r_{ad})}^{q_1}-{(r_0/r_{ad})}^{q_2}\right]({\mathit{x}}_{ad}/r_{ad})\\ {\displaystyle \frac{\mathrm{d}{\mathit{v}}_d^{\alpha }}{\mathrm{d}t}}={\displaystyle \frac{{\mathit{F}}_d^E}{m_d}}+{\displaystyle \frac{{\mathit{F}}_d^{Cn}}{m_d}}+{\displaystyle \frac{\overline{{\mathit{F}}_d^{Cs}}}{m_d}}-\chi \left[{(r_0/r_{ad})}^{q_1}-{(r_0/r_{ad})}^{q_2}\right]({\mathit{x}}_{ad}/r_{ad})\end{array}\right.$$

(16)

where ${\displaystyle \frac{\mathrm{d}{\mathit{v}}_i^{\alpha }}{\mathrm{d}t}}$, ${\displaystyle \frac{\mathrm{d}{\mathit{v}}_a^{\alpha }}{\mathrm{d}t}}$ and ${\displaystyle \frac{\mathrm{d}{\mathit{v}}_d^{\alpha }}{\mathrm{d}t}}$ denote the momentum of intact rock i, fluid a and discontinuous d particles, respectively. An appropriate minimum time step and the XSPH method [49] were employed to prevent mutual penetration between fluid particles and solid particles.

3. Validation Example

Hydraulic Fracturing of Two Pre-Existing Flawed Rocks

To verify the feasibility of the improved SPH framework for simulating FSI and cracking behaviour, a numerical simulation was performed, following the work of Mu et al. [50], to analyse how flaw water pressure affects crack propagation and coalescence in rock samples with two parallel prefabricated flaws. Three samples, each subjected to different flaw water pressures (P = 0, 4, 8 MPa), were selected for the simulation. The rock sample, as shown in Figure 4, has dimensions of 76.2 mm in length and 152.4 mm in width and is composed of 18,000 particles. Prefabricated flaw 1 has an inclination angle of β = 45°, while the angle between the line connecting the inner tips of prefabricated flaws 1 and 2 and the extension line of flaw 2 is α = 45°. The initial particle spacing ΔP is set to 0.8 mm, with a smoothing kernel length of h = 0.7 mm to prevent SPH kernel interaction between particles across the DPs. A loading rate of 0.5 m/s was applied to the upper and lower boundary particles, with a time step of ∆t = 5 × 10⁻⁸ s. The mechanical parameters for this test are listed in

Table 1. Parameters of SPH particles for hydraulic fracturing of two pre-existing flawed rocks.

Parameters	Rock Matrix	Water
Density (kg/m3)	2650	1000
Normal stiffness (GPa/m)	0.2	–
Friction coefficient	tan (35°)	–
Particle spacing (m)	8 × 10−3	8 × 10−3
Boundary particle spacing (m)	8 × 10−3	8 × 10−3
Number of IPs	18,000	50
Kernel function	B-spline	B-spline
Kernel smoothing length (m)	0.7 × 10−3	0.7 × 10−3
DP radius (m)	0.8 × 10−3	–
SPH time step (s)	5 × 10−8	5 × 10−8
δ–SPH coefficient	–	0.1
Shape parameter, mshape	20	–
Artificial viscosity parameter ${\beta }_1$	2	2
Artificial viscosity parameter ${\beta }_2$	4	4
Artificial stress parameter e	0.3	0.3
Artificial stress parameter n	4	4

. While the density, elastic modulus, Poisson’s ratio, and internal friction angle represent the actual physical properties of the material, the cohesion and tensile strength values reflect the bond strength between particles.

The failure modes with initial flaw water pressures of 4 MPa calculated by the improved SPH method are depicted in Figure 5. The apparent elastic modulus (6.12 GPa) and uniaxial compressive strength (UCS) (17.6 MPa) obtained from the stress–strain curve of the improved SPH model in this paper are highly consistent with the results of Total Lagrangian SPH (TLSPH) with an initial flaw water pressure of 4 MPa, and the error is within 1%. Moreover, the coalescence and failure modes of double cracks are highly consistent with those calculated by the TLSPH method. In summary, the results proved the accuracy and feasibility of the improved SPH method for simulating fracture and FSI processes.

Compared with TLSPH based on the implicit seepage control equation [50], in improved SPH, the actual fluid particle is directly modelled based on δ-WCSPH and can flow based on the water pressure along cracks, which leads to a larger variation in the final UCS, with initial different water pressures. For example, when the water pressure in fractures reaches 0 MPa and 8 MPa, the UCS value (6.5 MPa) calculated by the improved SPH method is greater than that (2.8 MPa) of TLSPH [50].

The crack propagation and coalescence modes with initial flaw water pressures of 4 MPa are depicted in Figure 6. The tensile wing crack outside the tip and anti-plane crack calculated by the improved SPH method is verified by TLSPH [50], indicating that the improved SPH method can effectively simulate the generation and propagation of hydraulic cracks. This is an important basis for the subsequent simulation of hydraulic fracturing in karst tunnels.

Figure 7 presents a direct comparison of crack distributions and stress-strain curves for SPH models with varying initial particle spacings (ΔP), demonstrating the robustness of the approach. The results show that particle spacing has a minor impact on the final cracking pattern. As seen in Figure 7b, the peak stress for ΔP = 1 × 10⁻³ m is noticeably lower than the TLSPH result, primarily due to reduced accuracy [50]. As ΔP decreases, the simulated crack paths become more accurate, improving the precision of the stress-strain curves. However, while reducing particle spacing enhances the computational accuracy of SPH models, it also substantially increases the computational cost [42]. To balance precision and computational efficiency, ΔP of 8 × 10⁻⁴ m was chosen for this case. This principle was applied to determine the particle spacing in all SPH models.

In Figure 8, a direct comparison of crack distributions and stress–strain curves are presented for SPH models with different smoothing kernel lengths (h). In fluid and soil simulations, larger kernel length could very likely induce high accuracy in SPH [51]. However, if the kernel length becomes overly large, it can lead to some abnormal phenomena or instability in computation. For instance, in the case of a rock crack, assuming that there are two groups of IPs separated by a layer of DPs, it is crucial to prevent the SPH kernel interaction of the IPs on the two sides of DPs. This is because there is no physical link between the intact parts on each side of the crack. To address this issue, a small support domain approach inspired by previous studies was employed to avoid such complications [52,53]. Through our comparative analysis in Figure 8, it is observed that when the kernel length h is overly large in the cases of h = 7.5 × 10⁻⁴ m and 8 × 10⁻⁴ m, they lead to abnormal crack distributions, especially in the bottom section. Conversely, if the kernel length is overly small, the accuracy of the computational results is compromised, as evidenced by the case of h = 0.65 mm, which failed to accurately reproduce the quasicoplanar shear crack. In terms of stress–strain curves, increasing the kernel length results in higher peak stress values. This phenomenon is primarily caused by the interaction between the IPs on either side of the DPs, leading to a false cohesive force effect. Therefore, a kernel length of h = 7 × 10⁻⁴ m provides the most accurate results, with a difference of only 1% compared to the TLSPH result [50]. Through this investigation, we have highlighted the importance of selecting an appropriate kernel length in simulating rock fracture to ensure both accuracy and stability in the computational results. The chosen kernel length strikes a balance between capturing the discontinuous nature of rock fractures and maintaining the desired level of accuracy in stress distribution. The smoothing kernel lengths employed in all SPH models were selected following this principle.

4. Numerical Simulation of the Karst Tunnel

4.1. Engineering Background

To evaluate the improved SPH method’s capability in simulating cracking, contacts, and fluid–structure interaction (FSI) performance, numerical simulations were carried out using the Jigongling karst tunnel in Hubei, China, as a case study (Figure 9a). This tunnel, which is approximately 4500 m in length, has a strike of about 126° and is buried to a depth of up to 338.5 m. The surrounding rock, comprised of water-soluble limestone and shale, along with numerous karst features such as depressions and valleys, makes the tunnel susceptible to water inrush and instability. A significant event, as reported by Gao et al. [54], involved an inrush at the ZK 19+509 drill hole section (Figure 9a) with a water pressure of 0.2 MPa and a flow rate of 35–45 L/s, which continued for over 10 days and caused severe flooding, halting construction.

4.2. Computation Model and Parameters

Karst caves under water-filling conditions greatly influence the stability of the tunnel, which is prone to water/mud inrushes [4]. In this regard, the assumption of the tunnel passing through a homogeneous stratum is set, and a water-bearing karst cave is located above the tunnel face [55]. A two-dimensional tunnel model with dimensions of 70 m by 70 m was established, given that the radius of the stress influence from the excavation is roughly three times the tunnel face width. The model features a tunnel face with a height of approximately 9 m and a maximum width of 10 m (Figure 9b). Additionally, a karst cave with a radius D of 6 m is located on the tunnel face, positioned about R = 1 m from the top of the tunnel face.

The SPH mechanical parameters are directly derived from partial differential equations (PDEs), providing clear physical significance and eliminating the need for complex parameter calibration. For instance, macro-mechanical parameters such as the elastic modulus (E = 3 GPa), density (ρ = 2500 kg/m³), Poisson’s ratio (ν = 0.3), and the internal friction angle (ϕ = 35°) are directly obtained from these equations [54]. However, similar to the DEM, the cohesion and tensile strength of SPH particles do not have direct macroscopic physical significance and are typically set higher than their actual values [56]. These parameters are calibrated through a trial-and-error process by fitting the simulated stress-strain curve to the experimental data from unconfined compression tests on the rock. After calibration, the cohesion and tensile strength of SPH particles were set to c = 25 MPa and R_t = 12.5 MPa, respectively, as per [54,55]. The final parameters used for SPH particles for the QYS karst tunnel simulation are listed in

Table 2. Parameters of SPH particles used in Jigongling tunnel.

Parameters	Rock Matrix	Water
Density (kg/m3)	2500	1000
Normal stiffness (MPa/m)	9.6	–
Friction coefficient	tan (30°)	–
Particle spacing (m)	0.2	0.2
Boundary particle spacing (m)	0.2	0.2
Number of IPs	70,500	1512
Kernel function	B–spline	B–spline
Kernel smoothing length (m)	0.2	0.3
DP radius (m)	0.2	–
SPH time step (s)	1 × 10−4	1 × 10−4
δ–SPH coefficient	–	0.1
Shape parameter, mshape	20	–
Artificial viscosity parameter ${\beta }_1$	2	2
Artificial viscosity parameter ${\beta }_2$	4	4
Artificial stress parameter e	0.3	0.3
Artificial stress parameter n	4	4

. To accurately simulate free-surface water flow and rock falls, gravity acceleration (g = 9.8 m/s²) was applied. The influence of factors such as the size of the karst cave, the thickness of water-resistant rock, and the rock mass strength was studied in the Jigongling tunnel, with different calculation scenarios detailed in

Table 3. Design of calculated cases under different influential factors.

Scenario	Thickness of Water-Resistant Rock D (m)	Strength Reduction Factor of Water-Resistant Rock K	Karst Cave Radius R (m)
I	1	1.4	2/4/6/8
II	1/1.5/2/2.5	1.4	6
III	1	0.6/0.8/1/1.2/1.4/1.6	6

K = 1 indicates the actual strength of the rock surrounding the tunnel, and K >= 1 indicates the weakened strength of the characterised surrounding rock; otherwise, the strength of the characterised surrounding rock increases.

.

4.3. Calculation Results and Analysis

General Inrush Behaviour

To illustrate the simulated inrush process, the case study involving the karst tunnel (with R = 6 m, D = 1 m, and K = 1) was selected to show the evolution of rock damage and water inrush, as depicted in Figure 10a. The detailed brittle failure in the rock wall is highlighted within the rectangular area outlined by the dashed line. Before 0.05 s, the cave and the tunnel remain relatively stable in the elastic stage. During 0.05–0.75 s, hydraulic cracks initiate at the top of the tunnel near the cave due to stress concentration. At 0.5 s, hydraulic cracks expand at the tunnel vault near the cave with the joint action of water pressure and the gravity of the overlying rock mass. The increase in the velocity of inrush flow enhances the width of the hydraulic crack pathway. At 1.0 s, the rock blocks slide along the crack pathway under the joint effect of friction resistance, hydrodynamic pressure and self-gravity. Subsequently, rock blocks drop from the tunnel roof with local inrush flow, which is verified by the field observation (Figure 10(b1,b2)). Meanwhile, fluid from the water-bearing structure continuously flows into the tunnel. The local water inrush channel is further expanded and gradually turns into an overall water inrush. Subsequently, the inrush water fills the tunnel face at t = 1.25 s, which is consistent with the field observation [57] (Figure 10(b3)). The distribution of the inrush water velocity is depicted in Figure 10a. It demonstrates that the improved SPH method accurately captures the free-surface flow.

Additionally, Figure 10c shows the evolution of the maximum velocity of the inrush water and the water level. The inrush process can be categorised into five distinct stages. Stage I is mainly the equilibrium stage of fluid in the karst cave after FSI activation. The maximum flow velocities in the karst cave increase sharply and simultaneously to a threshold. Following stage II, a slow rise in the inrush velocity is observed, and the hydraulic fracturing of the water-blocking rock is responsible for this maximum velocity increase. Moreover, bending and cracking lead to an increase in the volume of the local karst cave and eventually slow the water level decline in the karst cave. Subsequently, water and mud surge into the tunnel, causing a rapid increase in flow velocity and a drop in the water level (stage III), which is accompanied by the complete failure of the water-resistant wall (Figure 10a). The surge reaches its maximum velocity as the fluid flow’s suddenly released energy dissipates. Then, overall water/mud inrush is achieved at stage IV, and the maximum inrush velocity gradually decreases due to energy release. Both the water levels and maximum velocities are stable in stage V.

4.4. Influence of the Thickness of Water-Resistant Rock

Figure 11a,b illustrates the evolution of water level, maximum flow velocity, and crack count induced by shear and tensile stresses for different thicknesses of the water-resistant rock (D = 1, 1.5, 2, 2.5 m). The improved SPH results show that the thickness of the water-resistant rock has a significant impact on the inrush process. A greater thickness improves rock mass stability and reduces the risk of water inrush. For example, with a thickness of D = 2.5 m, the rock remains stable, resulting in a steady water level and flow velocity. As the thickness of the water-resistant wall increases, the water level decreases while the increase in flow velocity is delayed. Additionally, the duration of stages II–III of the inrush varies with thicknesses, reflecting the rock wall’s failure process. Thinner walls exhibit more pronounced jet-like flows due to the failure process affecting the flow channel.

Cracks, primarily caused by tensile stress when the major principal stress exceeds the tensile strength, align with the observed bending and breaking failure patterns (Figure 11(c1–c4)). During stages II–III, the number of cracks increases significantly and stabilises in later stages. With a thickness of D = 2.5 m, tensile damage is characterised by a greater number of radial cracks as the major principal stress exceeds the tensile strength.

4.5. Influence of the Different Strengths of the Rock Mass

In this section, the effect of the strength of water-resistant rock (K = 0.6, 0.8, 1, 1.2, 1.4, 1.6) on the water level, inrush flow and crack count are investigated quantitatively in Figure 12a,b. With increasing shear and tensile strength of water-resistant rock (K < 1), the water level drops significantly, the drop in the water level and increase in the flow velocity occur earlier, and the corresponding stability of water-resistant rock increases during stages III–IV. With decreasing shear and tensile strength (K > 1), intuitively, crack initiation occurs earlier, and the time from fracturing to the complete failure of water-resistant rock is shorter. As the strength of the rock decreases, the peak water velocity increases in stage II due to hydraulic fracturing. Furthermore, the volume of failed rock blocks is smaller during stage III. For example, Figure 12c reveals that the volume of the failed rock block is the largest at K = 0.8 and the smallest at K = 1.6. We infer that a lower strength of water-resistant rocks favours hydraulic pressure release. In contrast, a higher strength of rock maintains a higher hydraulic pressure in the karst cave, which can push forward larger individual rock blocks. Tensile cracks dominate and exhibit a significant increase as the strength of rocks decreases.

4.6. Influence of Different Radii of Karst Caves

In this section, the influence of the different radii of karst caves (R = 2, 4, 6, 8) is investigated quantitatively in Figure 13. Obviously, most maximum velocity curves increase significantly, and the corresponding water level decreases in stages II–III. However, the peak flow velocity decreases as the karst cave radii increase in all stages. As R = 2, although the water level drops, the maximum water velocity is basically stable. This result indicates that the phase characteristics of inrush weaken as the karst cave radii are smaller than a certain threshold. As R increases, the same arrival time of the peak flow velocity is calculated by SPH for different cases. Therefore, compared with the thickness and strength of the rock, the final time and arrival time of stages II–III are less affected by karst cave radii. Moreover, it is evident that as R increases, tensile and shear cracks significantly increase. The main reason is that the longer FSI coupling boundary of large-scale caves leads to severe hydraulic fracturing.

5. Discussion

5.1. Physical Vulnerability Induced by Inrush in Karst Areas

Before a water or mud inrush event, it is crucial to quantitatively assess the extent of damage and associated risks to operations below the tunnel crown (Figure 14a). The impact pressure, P_t, is commonly used to evaluate physical vulnerability and potential harm to individuals exposed to flooding [58,59]. Based on field observations indicating that water or mud inrush typically flows vertically [60], the maximum impact pressure P_t affecting individuals can be determined as follows [59]:

$$P_t={\displaystyle \frac{1}{2}}{\rho }_{a}gH+{\rho }_a{\mathit{\nu }}_a^2$$

(17)

where ${\rho }_a$ is the density of inrush water. H is the average width of the water/mud flow channel. Assuming that water/mud inrush continuously flows in a vertical direction according to field observations [60], a schematic illustration is shown in Figure 14b. Due to the vertical flow, the static water pressure equals gravity, and Equation (17) is modified as follows:

$$P_t={\rho }_{a}gH+{\rho }_a{\mathit{\nu }}_a^2$$

(18)

The average inrush velocity can be determined using the improved SPH method, and g denotes the gravity acceleration (9.8 m/s²). The calculated flow width of all scenarios is H = 1.5 m–6 m. Therefore, the average impact pressure P_t is calculated, and the influence of three different scenarios on the impact pressure is depicted in Figure 14. It is evident that under all scenarios without water inrush, the value of P_t remains below the upper limit (100 kPa) for human safety [59,61]. For instance, D = 2.5 in scenario I and K = 0.6 in scenario II in Figure 15. Once water/mud inrush occurs, P_t during stages II–IV significantly exceeds the upper limit for human safety during stages III–IV, and P_t reaches its peak within 1 s, indicating that the inrush hazards are instantaneous. In stage V, after the inrush flow stabilises, P_t may decrease and then be below the upper limit for stability, as observed at R = 4 m in scenario III, D = 1 m in scenario I and K = 1, 1.4–1.6 in scenario II. In addition, as R decreases, the peak P_t increases. However, at R = 2 m, although inrush occurs, the value of P_t remains below the loading limit that the human body can withstand. Notably, the influence of falling rock impacts during inrush is not considered, which may lead to an underestimation of the assessment of damage and risk.

5.2. Advantages and Limitations of Method

Compared with previous SPH damage frameworks, such as the “pseudo-spring” method [52], general particle dynamics (GPD) [62], the improved kernel of SPH (IKSPH) method [63], and improved bond-based SPH (IBB–SPH) [64], which have made great contributions in simulating material behaviour and subsequent fractures, the improved SPH framework in this paper has an advantage in rigorously modelling the contact behaviour after a fracture. More importantly, coupling the fracture model and δ-WCSPH model in a single SPH framework based on an improved two-phase adaptive adjacent integral coupling algorithm basically realises the complete failure process of the inrush disaster chain in karst tunnels.

To celebrate SPH mechanical parameters, which are strictly derived from PDEs, the physical meaning is definitive without complex parameter calibration. However, similar to the DEM, the cohesion and tensile strength of particles do not have macroscopic physical significance and are typically more significant than the actual values, which need to be calibrated by a trial–and–error process by fitting the simulated stress–strain curve to that of the unconfined compression experiment of the studied rock. The previous SPH damage framework also has a similar advantage. In addition, the contact and slip after crack propagation can be captured by the improved SPH method, which is an effective supplement to the discontinuous methods (DDA and DEM) to solve the contact problem of the rock mass. For the calculated time of the improved SPH method, due to the linked list search method [65] and leapfrog time integration algorithm [66], the time of the inrush disaster chain calculated by improved SPH is less than that of discontinuous methods, such as DDA and DEM. As an example, we performed a water and mud inrush simulation for the Jigongling Tunnel using an R9000P computer equipped with an AMD Ryzen 7 5800H CPU, with a base frequency of 3.2 GHz and an overclocked speed of 3.8 GHz. The proposed SPH method was employed to simulate the full 2 s process of water and mud inrush. The single-threaded runtime was 0.4 h, indicating high computational efficiency and strong engineering applicability.

Both the improved meshless method and FDEM can handle the continuous and discrete behaviours of materials and are very suitable for simulating the cracking and fragmentation of continuous materials. FDEM has a more powerful elastic–brittle constitutive model, while the improved meshless method has a more natural transformation from continuity to discontinuity. The accuracy of FDEM depends on the quality of the mesh and may not be as natural as the improved meshless method under large deformation conditions. In addition, the simulation time of FDEM is longer than that of the improved meshless method. However, the improved meshless method has the problems of tensile instability and boundary truncation that occur when SPH simulates solids. These problems will reduce the solution accuracy of the improved meshless method.

The scope and conditions of the improved SPH method are respectively applicable to rock mechanics problems that consider both fluid and solid phases at the same time. The solution is based on the explicit equation. It has a weak solution ability for the seepage process and a strong ability for solving non-free surface flow.

Since this study mainly focuses on dynamic simulation, the transient seepage of fluid in the rock mass before cracking is not considered. Therefore, to realise a more realistic water inrush process, the seepage equation of porous media and the fracture surface are introduced into the improved SPH model to restore the water seepage when cracks develop [50]. The three-dimensional karst tunnel SPH model and the corresponding parallel algorithm will be developed in our future works. In addition, many other factors affect the process of water/mud inrush, such as the lateral pressure coefficient of the tunnel and the location and shape of karst caves. Furthermore, discontinuous properties (such as joints and bedding) of water-resistant rock should be considered in future studies. In addition, regarding the processing of acoustic wave detection in tunnel sites [67,68], SPH needs further development of the wave equation to simulate the response of elastic waves in rock formations.

6. Conclusions

A meshfree method that considers cracking, contact and fluid–solid interaction (FSI) is improved to simulate the progressive failure of a water-resistant stratum and the inrush process in a karst tunnel. After being validated, the progressive failure and physical vulnerability of a karst tunnel in three scenarios were investigated. The main conclusions are as follows:

(1)

The improved SPH method explicitly simulates progressive failure involving the loss of material continuity and brittle fracture characteristics without re-meshing. Moreover, a two-phase adaptive adjacent algorithm considering the DPs can faithfully capture the FSI and hydraulic fracturing;

(2)

The evolution of water inrush in the Jigongling karst tunnel calculated by the SPH method is divided into the equilibrium stage, hydraulic fracturing, local inrush accompanied by the complete failure of the water-resistant wall, overall inrush and stable inrush stages;

(3)

With increasing thickness and strength of the water-resistant wall, the possibility of water/mud inrush in the tunnel decreases, and a drop in the water level and increase in the flow velocity have a delayed effect during stages III–IV; however, such a delayed effect is less affected by karst cave radii. As the strength of the rock decreases, the peak inrush velocity increases in stage II due to hydraulic fracturing. In addition, as karst cave radii increase and the thickness and strength of rock decrease, tensile and shear cracks in rock significantly increase in stages II–III;

(4)

Once inrush occurs in the Jigongling karst tunnel, the impact pressure during stages III–IV significantly exceeds the upper limit for human safety. In stage V (i.e., stable inrush), the impact pressure significantly decreases and then may be below the upper limit for human safety, which is consistent with the field observations.