For the SPH method, the entire domain is represented as a set of randomly distributed particles with no connection between them. Two main steps are required to obtain the fluid governing equations in SPH form, namely, the kernel approximation and the particle approximation. Firstly, approximation of the field function and its derivative are derived from an integral representation method by using a smoothing kernel function. Secondly, these approximations are replaced by the sum of all neighboring particles in the support domain. It is the kernel approximation and particle discretization that make the SPH method work well for free surface flows and large deformations [17,38].

The integral representation of a function f(x) may be given as the following kernel approximation using the smoothing technique:

(1)f(x)≅∫Ωf(x′)W(x−x′,h)dx′where f(x) is a field function of the three-dimensional position vector x, and W(x−x′,h) is the smooth function. The smoothing length h that determines support domain Ω. The discrete form of a function at particle i can eventually be described as:

(2)f(xi)=∑j=1Nmjf(xj)ρjWijwhere mj and ρj are the mass and density of particle j, respectively, and Wij=W(xi−xj,h) indicates the smoothing function. In the same manner, the particle approximate form for the spatial derivative of the function can be formulated as:

(3)df(xi)dx=∑j=1Nmjf(xj)ρj∂W∂xi

There are several kernel functions that are widely utilized. Typically, higher-order kernel functions often enhance the computing accuracy, and substantial time is required. The cubic spline kernel, which belongs to the B-spline family, is one of the kernels that is commonly employed in SPH method [17] and is given by:(4)W(x−x′,h)=αd×{23−q2+12q3,0≤q<116(2−q)3,1≤q<20,q≥2where q=|x−x′|h and αd=157πh2 in 2D and αd=32πh3 in 3D, respectively. For hydrodynamics of fluids represented in the Lagrangian formulation, the SPH formulations of Navier-Stokes equations can be written as follows:

(5)dρidt=∑j=1Nmjνij∂Wij∂xi

(6)dνidt=∑j=1Nmj(σiρi2+σjρj2)∂Wij∂xi+g

νij=(νi−νj) indicates the velocity difference between particle i and particle j; σ represents the total stress tensor and g is the gravity acceleration, respectively.

Due to the density approximation, the pressure field in the SPH formulation for weakly compressible fluids generally exhibits instability and numerical noise. Particularly, in the present case, the noise of the pressure field may be critical. In this study, the delta-SPH method [30] was adopted, which was implemented by adding a diffusive term in the continuity Eq. (5) and it can be written as:

(7)dρidt=∑j=1Nmjνij∂Wij∂xi+ξhc0∑j=1Nψijmjρj∂Wij∂xiwhere ξ is the diffusive coefficient, which is usually adopted as 0.1; c0 is the speed of sound at reference density; and ψij is written as:

(8)ψij=2(ρjρi−1)(xi−xj)|xi−xj|2+εhh2where εh is a small constant and is usually adopted as 0.01.

Regarding the momentum Eq. (6), the total stress tensor σ can be written as the summation of an isotropic component pressure p and a viscous shear stress component τ. To remove the numerical oscillations, the Monaghan type artificial viscosity Πij was adopted in this study, which is most widely used in the current SPH simulation. After considering the artificial viscosity, the momentum equation can be written as:

(9)dνidt=−∑j=1Nmj(piρi2+pjρj2+Πij)∂Wij∂xi+∑j=1Nmj(τiρi2+τjρj2)∂Wij∂xi+g

(10)Πij={−αΠc¯ijϕij+βΠϕij2ρ¯ij,vij.xij<00,vij.xij>0where αΠ and βΠ are two constants and are commonly set as 0.1 and 0.01, respectively; Detailed parameter definitions can be found in reference [39].

In this work, the temperature of landslides is considered as a constant. Therefore, the energy equation is not considered. In the typical SPH method for resolving the compressible flows, the particle motion is governed by the pressure gradient, whereas the particle pressure is determined using an appropriate Equation of State (EoS).

(11)pi=c02ρ0γ[(ρiρ0)γ−1]where γ is commonly set to be 7.0; c0 is the speed of sound and is commonly set at least 10 times the maximum fluid velocity for the sake of keeping the density fluctuation within 1% which can be calculated by β(ghmax)1/2. β is a constant and is set as 10 in this study.

PySPH is an open-source framework for the simulation of the SPH method [40]. The original PySPH code (version 1.0b1) is able to handle with the free-surface flows, but it is incapable of dealing with non-Newtonian flows. A relevant constitutive model should be developed in the original code to allow for the landslide simulation. In Newtonian fluid dynamics, the shear stress of the fluid can be expressed by the following equation:

(12)τ=με˙n+τywhere ε˙ is the shear strain rate. The Newtonian fluid is represented by Eq. (12), when n=1 and τy=0. For a Newtonian fluid, μ is the dynamic viscosity that is given as a constant. The Bingham flow model, as a typical non-Newtonian fluids, can describe the relationship between the shear strain rate and the shear stress of soil [41]. When the shear stress τ is greater than the yield strength τy, the soil begins to deform. For non-Newtonian fluids, the dynamic viscosity depends on the shear rate or the shear rate history. The relation between a non-Newtonian fluid's shear stress tensor and shear rate tensor can be expressed as:

(13)τ=μeffε˙where μeff is the effective viscosity that is changed over time. The effective viscosity of the Bingham model can be formulated as:

(14)μeff=μB+τyγ˙where μB is the Bingham viscosity and τy is the yield stress, respectively.

Assuming that under the condition of shear rate, γ˙→0, μeff in Bingham model approaches an infinite value, therefore, the direct application is not available. To solve this problem, existing studies employed a variety of strategies to avoid numerical divergence, i.e., the simple regularization [42] and the threshold regularization [43]. In this study, the general Cross model [44] was chosen to characterize the non-Newtonian behavior of flow-type landslides.

(15)μeff=μ0+(Kγ˙)mμ∞1+(Kγ˙)mwhere μ0 and μ∞ represent the fluid's viscosity at extremely low and high shear rates, respectively; K and m are the constant parameters of the Cross model. For the sake of simplicity, the methodology in which these values are derived by incorporating the more commonly utilized Bingham fluid parameters was adopted [29]. Assuming m=1 in Eq. (15), the effective viscosity in the Cross model can be formulated as:(16)μeff=μ0+Kγ˙μ∞1+Kγ˙

Typically, Kγ˙>>1, by Eq. (14) with Eq. (16), the remaining two parameters in the Cross model can be given as follows:

(17)μ∞=μB

(18)K=μ0τy

The Mohr-Coulomb yield criterion with the cohesion c and frictional angle φ, which is adopted to describe the soil yielding process, is introduced into Eq. (12) [3], which can be written as:

(19)τ=μγ˙n+c+σtan⁡φwhere σ is the pressure in this status. Related literature [45] pointed out that the numerical solution's accuracy is satisfactory as the viscosity at a low shear rate is 1000 times greater than that at a high shear rate.

(20)μ0=1000μ∞

Notice that the advantage of the Cross model over the Bingham model is that the effective viscosity is a continuous variable, and the numerical instability is avoided, as shown in Fig. 1.

The boundary treatment has been a main challenge in SPH simulation and affects the calculation accuracy as well as the efficiency. It is vital to select an adequate boundary condition that represents the effect of solid boundary in order to closely describe the dynamic mechanism of landslides, as shown in Fig. 2.

A simple and convenient boundary condition was proposed that can consider the boundary friction. In this work, two types of boundary conditions need to be set. Firstly, when a particle approaches to a solid boundary, the kernel function will be truncated, and an error in the solution result will occur. To address this issue, a generalized wall boundary condition by Adami et al. [46] (see Eq. (23)), was introduced. By extrapolating from the neighboring fluid particles, the pressure on the solid boundary is obtained as:

(21)Pi=∑j(Pj+(gj.rij)ρj)Wij∑jWij

Secondly, the discrete particles of landslides may slip at the boundary and thus suffer from the slippage resistance (see Eq. (24)). Referring to the concept of pressure interpolation on the solid boundary, the value of the viscosity force on the solid boundary was given by:

(22)τi=∑jτjWij∑jWij

(23)dvipdt=−∑jmj(piρi2+pjρj2)∂W∂xi⏟dvidt←fluid−∑jmj(piρi2+pbρb2)∂W∂xi⏟dvidt←boundary

(24)dviτdt=∑jmj(τiρi2+τjρj2)∂W∂xi⏟dvidt←fluid+∑kmj(τiρi2+τbρb2)∂W∂xi⏟dvidt←boundary

Through the above two steps, the no-slip boundary conditions considering friction on the solid boundary was successfully set. When calculating the viscous forces on the boundary, both the dynamic viscosity coefficient can be obtained by interpolation from the fluid, or a fixed viscosity coefficient can be set.

Prediction-correction algorithm was adopted to execute the time integration. For continuity equations and momentum equations of the form, all the values of time-variant quantities are predicted at the time step n+1, based on the time step n+1/2, as follows:(25)ρin+12=ρin+12Δtdρindt,vin+12=vin+12Δtdvindt

Afterwards, these values are updated at another half time step:

(26)ρin+12=ρin+12Δtdρin+12dt,vin+12=vin+12Δtdvin+12dt

At the end of the time step, the values are calculated as follows:

(27)ρin+1=2ρin+12−ρin,vin+1=2vin+12−vin

The selection of the magnitude of time step Δt is limited due to the stability reasons based on several criteria: the Courant Friedrichs Lewy (CFL) condition, the force terms, and the viscous diffusion terms, as follows:

(28)Δt=min(0.25hcmax+|vmax|,0.125h2v,0.25(h|g|)1/2)

The flow chart of the SPH with proposed boundary condition is shown in Fig. 3.

The Shuicheng flow-like landslide event was triggered by intense rainfall in July 2019 (Fig. 7a). A total of three heavy rainfall events occurred in the week before the occurrence of landslides, with a cumulative rainfall of 288.9 mm. Under the influence of continuous heavy rainfall, the landslide began to lose stability and rapidly slid along the slope at high speed. After roughly 700 m of landslide movement, two diversions were formed after hitting a small ridge and subsequently impacting and burying the residential buildings on the lower part of the slope. The mass flow of landslide spread along the slopes for about 1490 m and was deposited in the gully depressions. Due to the suddenness of the landslide and the lack of protective measures such as blocking, the disaster destroyed 21 houses and 43 people were killed.

The Guizhou Provincial Geological Disaster Emergency Technical Guidance Center immediately carried out the field investigations following the occurrence of landslide. The primary characteristics of the flow-like landslides are depicted in Fig. 7b as well as a typical cross-section of the Shuicheng gully in Fig. 5c. According to the field investigations, the entire landslide is approximately 1300 m long, with a large distance of around 460 m between the front and rear edges, and a volume of about 191.2 × 10⁴ m³. According to Zhou et al. [48], the average density of the Shuicheng landslide mass was about 2000 kg/m³. Depending on the test results, the angle of internal friction and the cohesion of the flow-like landslide can be set approximately as 30° and 30 kPa, respectively. In the previous simulations, the Bingham model was widely applied to simulate the landslides or debris flows considering a range of dynamic viscosities from 20 to 500 Pa⋅s [49]. In this work, the empirical physical experimental equation is used to obtain the dynamic viscosities [50], which is given below:

(29)μB=0.621⋅exp⁡(0.173⋅CV)where Cv is the volume concentration. According to the investigation report, the best-fitted Cv is 44%, resulting in a dynamic viscosity of 255 Pa⋅s.

According to the simulation results, the whole process duration was about 90 s. Fig. 8 presents the slope configurations at different time points, reflecting the movement of landslides. The particles slipped down from the top of the landslide and then moved along the steep slope by gravity. Finally, these particles reached an equilibrium state and were deposited in the lowlands. The overall performance of the sliding was accelerated motion in the period 0–30 s and decelerated motion after 30 s.

Landslides usually have a long distance, and the measurement and selection of various parameters may vary greatly in different locations. Therefore, it is meaningful to conduct a sensitivity analysis to determine how various rheological parameters affect the simulation results. Table 2 lists the eight calculation cases with different rheological parameters. To explicitly quantify the differences of the sensitivity of different parameters εv, was evaluated using the following equation:

(30)εv=(∑iNΔvj2/∑iNv12)1/2where Δvj is the velocity difference between case j and case 1, and N = 18 represents the number of velocity at an interval of 5 s. The results of all the eight cases are presented in Table 2 and Fig. 9, which show that the internal friction angle has more influence on the computation accuracy in comparison with the cohesion of landslides.

Since the 2D simulation cannot reflect the diffusion and lateral contraction, 3D simulation on the real complex terrain is necessary. In this section, the 3D terrain is generated from the original topographic map at a scale of 1/2000 and homogenizes the mesh. The previously created mesh is then transformed into a sequence of boundary particle particles. Similarly, the landslide is discretized into a succession of particles each with its own set of properties. In this simulation, the particle distance is set to 7.5 m, resulting in 12825 boundary particles and 7010 fluid particles. The strength parameters adopted in the 3D simulation are the same as in the 2D simulation (Table 3). Based on this model, Shuicheng landslide was numerically simulated in 3D terrain, and the results are shown in Fig. 10. The color of the particles in the figures represents the sliding velocity at various instants.

The results suggest that the front flow takes about 28 s to travel nearly 520 m to reach the isolated island. The front flow velocity reaches a maximum with an average velocity of 29.8 m/s. Due to the good stability of the soil near the isolated island, it diverts the debris to the sides and forms a divergence. The diversion of the stable slope protects the lower residential houses and creates an “island of safety”. The existence of isolated islands has dramatically slowed down the landslides. Therefore, the front flow remains stable at about 90 s and is deposited in the lowlands. Eventually, the estimated total volume of deposition is 1.1 × 10⁶ m³, which is less than the field investigation value 1.91 × 10⁶ m³. The reason might be that the simulation process does not take into account the entrapment process. In addition, the simulated landslide paths and deposited areas matched well with the results of field investigations. The topographic data, in this case, were obtained from the 1/2000 scale topographic map, and it is considered that more accurate topographic data would help to obtain better simulation results.

The displacement and velocity time histories of the landslide front and rear were investigated to gain a better understanding of the landslide characteristics in the present simulation. Fig. 11 depicts the velocity time series of the landslide's front and rear. At around 20 s and around 45.5 s after failure, the flow-like landslide front and rear reach a maximum velocity of 29.2 and 26.5 m/s, respectively. Afterwards, an isolated island was encountered, the landslide diverged and the path was changed, and the speed gradually decreased. After moving for 70 s, the landslide came to a halt and was deposited in a lower area. It should be noted that the changes in velocity at various moments represent the complexity of the 3D terrain. Fig. 12 shows the maximum displacements of the front and rear during the motion were 1099 and 859 m, respectively, at the time of 90 s after failure. Ma et al. [51] conducted extensive field investigations and calculated the maximum velocity of 25 m/s², and a landslide duration of 60 s using the empirical equations. Both values turned out to be slightly smaller than the simulated results. Quantitative validation of the landslide simulation based on the field measured data and empirical equations were challenging. The error may be triggered by a combination of natural and artificial reasons, which makes the simulation more challenging. In addition, it can be seen that the flow's pressure field is formed smoothly. Fig. 13 depicts the evolution of the flow-like landslide pressure field over time. As previously stated, delta-SPH successfully corrects pressure field noise, particularly for the high viscosity fluids.

The landslide material is discretized into a succession of SPH particles of almost the same diameters in this model. The greater the number of particles and the narrower the particle spacing, the greater the computing time required. It should be mentioned that, in the current code, the SPH module is accelerated based on Open Multi-Processing (OpenMP). In theory, the higher the number of particles in the SPH, the more accurate the simulation will be, but a balance needs to be reached with the needed calculation time. Fig. 14 depicts the relationship between the program runtime and thread count. The results show that the computational efficiency of the proposed SPH model improves as the number of threads increases. Although the OpenMP code is more computationally efficient than the serial code on the CPU, the computation time consumption is large when more particles exist.