Introduction

In recent years, liquefaction resulting from seismic events has become a major concern due to its impact on structures, buildings and other infrastructure during and after an earthquake. Almost all major earthquakes are accompanied by soil liquefaction, as evidenced by the 1976 Tangshan earthquake in China (Fu and Tatsuoka 1984), the 1989 Loma Prieta earthquake in the USA (Bartlett and Youd 1995), the 1995 Hyogo-ken-Nambu earthquake in Japan (Shibata et al. 1996), and the 2008 Wenchuan earthquake in China (Huang and Jiang 2010). Liquefaction-induced ground failure has become one of the leading causes of infrastructure damage during an earthquake. Under seismic loading, the rapid increase in pore water pressure quickly decreases the shear strength of the unconsolidated sediment, possibly triggering large shear deformation. As flow failure of the ground during an earthquake may be caused by either the dynamic force due to the seismic acceleration or the static gravity force due to the topography of the ground (Tamate and Towhata 1999), flow deformation of post-liquefaction soil may continue under a gravitational load after seismic shaking ceases. In this case, the ground surface is prone to large-scale flow deformation regardless of whether it is flat or gently inclined. Uzuoak et al. (1998) suggests the shear strain may sometimes exceed 100%. This can develop into a regional seismic liquefaction event that can cause extensive damage. During the Tajik earthquake in 1989, the sliding mass traveled a distance of 2 km due to the liquefaction of the saturated soil, despite the fact that the ground surface was nearly flat (Ishihara 2009). More recently, during the 2003 Tokachi-oki earthquake in Japan, fluidized subsurface soil was expelled onto the surface and flowed about 1 km along a water channel (Kokusho 2009).

Liquefaction hazards are associated with substantial losses. Compared with the liquefaction itself, the damage resulting from the large scale deformation of the liquefied soil can be even more serious. Previous studies of soil liquefaction have mainly focused on influencing factors, initial conditions and liquefaction predictions, and only recently has attention turned to the importance of the large deformation associated with liquefaction. These studies generally use conventional solid mechanics-based methods and assume a relatively limited maximum shear (Yang and Elgamal 2002; Yoshimine et al. 2006). However, with liquefaction-induced ground flow, the shear strain of the soil may exceed 100% in the flow condition (Fig. 1), i.e, the state of the soils has been changed from solid to fluid.

Fig. 1
figure 1

The mechanical state of the liquefied soil under loading

Full size image

In the past few years, a series of laboratory tests conducted by different researchers has shown that post-liquefaction soil behaves in a similar manner to a viscous fluid (Sasaki et al. 1992; Towhata et al. 1999; Hwang et al. 2006). Thus, due to the phase transition, it is difficult to apply solid mechanics based on small or finite deformation theory to the deformation properties of the post-liquefaction soil; a new analytical method is required.

Computational fluid dynamics (CFD) is a computer-based method which makes use of a discretization of the algebraic equations governing flow and their subsequent mathematical manipulation and solution (Versteeg and Malalasekera 1995). With the development of computer technology, numerical simulation methods have been rapidly developed, expediting the use of such applications as the CFD method to the study of liquefaction-induced flow deformation. In recent years, a few preliminary attempts have been made to promote the application of the CFD method to deformation problems related to liquefaction (e.g., Uzuoka et al. 1998; Hadush et al. 2001; Hwang et al. 2006). However, the research is very limited compared with that using solid mechanics and hence there is no generally accepted theory concerning its use in describing non-linear and deformation characteristics. In addition, little material is available concerning the study of large deformation flows induced by liquefaction.

The present study is a further development in the ongoing efforts to assess the flow features of post-liquefaction soils. In this paper, the post-liquefaction soil is assumed to be a viscous fluid, and a numerical model based on fluid dynamics is proposed for the flow behavior. The Pressure Implicit with Splitting of Operators (PISO) algorithm is used for solving the Navier–Stokes equations, and the volume of fluid (VOF) model is used to track interfaces in the numerical scheme. To verify the validity of the method, modeling of the liquefaction-induced soil flow was undertaken.

Numerical model description

The numerical modeling of ground flow involves the solution of the Navier–Stokes equations, which are based on the assumption of the conservation of mass and momentum, and the tracking of phase interface kinematics. The conservation of mass is described by the following equation:

$$ \frac{\partial \rho }{\partial t} + \nabla \left( {\rho {\varvec{u}}} \right) = 0, $$
(1)

and the conservation of momentum is described as:

$$ \frac{\partial \rho {\varvec{u}}}{\partial t} + \nabla (\rho {\varvec{uu}}) = \rho {\varvec{f}}_{v} + \nabla {\varvec{T}}.$$
(2)

In this paper the fluid is assumed to follow the generalized Newton’s Law; the constitutive model can be described by the following equation:

$${\varvec{T}} = - p{\varvec{I}} + 2\eta \left({\varvec{S}} - \frac{1}{3}\nabla {\varvec{u}}\right) .$$
(3)

Applying the constitutive model above to Eq. 2 results in the Navier–Stokes equation:

$$ \frac{\partial \rho {\varvec{u}}}{\partial t} + \nabla (\rho {\varvec{uu}}) = \rho {\varvec{f}}_{v} - \nabla p + \nabla {\varvec{\tau}}$$
(4)

where:

$$ \nabla {\varvec{\tau}} = \eta \Updelta {\varvec{u}} + \frac{1}{3}\eta \nabla (\nabla {\varvec{u}}) $$
(5)

Note, in the equations above, the absence of sources of mass and momentum, where ρ is the fluid density, S is the strain rate tensor, f v is the volume force, u is the velocity tensor, η is the viscosity, T is the stress tensor, p is the static pressure and τ is the deviatoric stress tensor.

To solve the Navier–Stokes equations, the PISO algorithm (Issa 1986; Oliveira and Issa 2001) is used. This is a pressure–velocity calculation procedure based on a finite volume discretization on a staggered grid of the governing equations. The purpose of a staggered grid is to evaluate scalar variables, such as pressure and density, at ordinary nodal points, while velocities are defined at the cell faces between the nodes. The arrangement for a two-dimensional flow calculation is shown in Fig. 2. The PISO algorithm provides a higher degree of accuracy for pressure and velocity corrections than the SIMPLE algorithm (Barton 1998), which was used by Uzuoka et al. (1998) in the analysis of liquefaction-induced lateral spreading.

Fig. 2
figure 2

Locations of variables in staggered grid

Full size image

Due to the existence of a free surface with good mobility during the flow, a method for interface tracking can be used to describe the interface distribution and motion characteristics. In the past, a variety of methods, such as Level Set, Particle-In-Cell (PIC), Marker-and-Cell (MAC) and Volume of Fluid (VOF) have been developed for phase interface tracking (Oliveira and Issa 2001; Andrews et al. 1996; Tomé et al. 2006; Hirt and Nichols 1981). The VOF method, which was initially introduced by Hirt and Nichols (1981), is known for its capacity for interface tracking, including stratified flows, free-surface flows, filling, sloshing and so on. As there is a typical phase interface existing between the air and the liquefied soil, the VOF method was considered appropriate for the description of flow configurations in post-liquefaction soil. The most important feature of the VOF method is the introduction of the phase function F(x, y, t) for the description of the interface. F represents the areal (two-dimensional) or volumetric (three-dimensional) fraction of a cell being occupied by a fluid in the computational domain. If F = 1, the cell is full of a fluid phase; if F = 0, the grid is full of another fluid phase; if 0 < F < 1, the cell is intersected by the phase interface. The position of the interface is determined by the F values of the neighboring cells. For two immiscible flows, the conservation form of the transport equation is:

$$ \frac{\partial F}{\partial t} + u\frac{\partial F}{\partial x} + v\frac{\partial F}{\partial y} = 0. $$
(6)

A detailed description of the VOF method is available in Hirt and Nichols (1981).

The Geometric Reconstruction scheme, which represents the interface between fluids using a piecewise linear approach, is applied to reconstruct the free surface between the liquefied soil and the air. As is shown in Fig. 3, the interface is described by a linear slope within each cell and the linear slope is used for the calculation of fluid flow through the cell faces. This approach is based on the work of Youngs (1982).

Fig. 3
figure 3

Comparison of actual and reconstructed interface shapes represented by geometric reconstruction

Full size image

Applications and analysis

Model test description

Figure 4 shows the layout of the model box designed to reproduce and test the flow behavior of post-liquefaction soil. The soil container is made of transparent synthetic glass lined with an inner box. The external box is 950 mm long, 330 mm side and 350 mm high while the inner box is 320 mm long and 320 mm wide with a maximum height of 250 mm. The inclination of the model box may be changed if necessary. The properties of the sample sand used in the test are shown in Table 1. Liquefaction of the saturated soil sample in the inner box is generated by an electric motor which imparts a sinusoidal acceleration up to a maximum of 0.15 g at a frequency of 1 Hz. Excess pore water pressure is measured by pore pressure transducers installed at depths of 50 and 100 mm in the model. During the test, when the excess pore water pressure ratio (the ratio of excess pore water pressure to initial effective vertical stress) reaches 0.90 and remains stable (Fig. 5), the soil sample is considered to have reached the liquefaction state. The electric motor is then turned off, the baffle is quickly pulled out and the process of the post-liquefaction soil flow along the model box is recorded with a high-speed camera.

Fig. 4
figure 4

Configuration of the model box equipped with transducers for liquefaction tests

Full size image
Table 1 Parameters for the sand used in the model tests
Full size table
Fig. 5
figure 5

Experimental time history of the excess pore water pressure ratio

Full size image

Model application

A finite volume method (FVM) with an Eulerian–Eulerian approach was used to study the air–liquid interactions. The phase-coupled PISO algorithm was used to couple the pressure and velocity. The diffusion phase of the control equation used the central difference method and the momentum equations used the first order upwind scheme.

The CFD code FLUENT 6.3, a well-known computer program for modeling fluid flow, was employed to verify the applicability of the proposed model (Fluent Inc 2006). Gambit 2.3.16 was used to generate the 2D geometries and their grids. As shown in Fig. 6, the length of the model box is 0.95 m and the height 0.20 m; by using a quadrilateral mesh, the overall number of nodes was 2,016 and the number of cells was 1,900. The left, right and bottom sides were set as WALL, which is used to bound fluid regions. The upper edge was set as PRESSURE-OUTLET to define a static pressure at the outlet boundary; here set to standard atmospheric pressure with a value of 1.013 × 105 Pa (Fig. 7). The model parameters used for the simulation of flow liquefaction are listed in Table 2.

Fig. 6
figure 6

Computational mesh

Full size image
Fig. 7
figure 7

Boundary conditions and position of output points

Full size image
Table 2 Parameters used for the simulation of flow liquefaction
Full size table

Results and analysis

Figure 8 shows the configuration of the post-liquefaction sand flow obtained from the model test. The flow of the liquefied sand gradually extended after the baffle was removed and lasted for 5 s. Figure 9 shows the flow configuration comparison of the model tests and simulation results for the post-liquefaction sand. It can be seen that, in general, the numerical approach reproduces the flow trend of the post-liquefaction sand and is in good agreement with the flow configuration. However, the simulated flow of sand was found to be a little faster than the flow recorded in the test. In the numerical simulation, the left side of the sand was observed to flow more substantially than in the model tests. Figure 10 shows the computed velocity vectors of the flow at a range of times from 1 to 5 s. It is clear that the velocity on the left side of the model had a very low magnitude. The vector directions indicate the flow orientation of the post-liquefaction sand.

Fig. 8
figure 8

Configuration of the model test

Full size image
Fig. 9
figure 9

Configuration comparison of model tests and simulation

Full size image
Fig. 10
figure 10

Simulation results of velocity vectors

Full size image

Figure 11 shows the time histories of flow velocity at the selected output points depicted in Fig. 7. The flow of the post-liquefaction soil reaches its maximum velocity (from 8 to 9 cm/s) about 1 s after the instigation of flow. Thereafter, the flow begins to slow down. However, the degree of deceleration differs depending on the output point; the velocity at Point 3 decreased most quickly, followed by the velocities at Points 2 and 1. This is because Point 3 is closer to the free surface and is therefore more significantly affected by gravity, the main trigger of flow.

Fig. 11
figure 11

Simulated time histories of velocities

Full size image

To evaluate laterally spreading features of the post-liquefaction sand flow, a simple calculation of engineering strain based on the one-dimensional movement of a specified point was carried out. Point A, in the lower right corner of the soil sample, was selected for this calculation (Fig. 12). Point A′ indicates the new location of Point A at different times during flow. L is the initial length of the soil sample, dL is the length increase in the flow direction; the engineering strain at Point A is defined as ε = dL/L. Figure 13 shows a comparison of engineering strain for the model tests and the numerical simulation results. It can be seen that in the first 3 s the simulated engineering strain was smaller than for the model test; however, this discrepancy decreases as the flow deformation progresses, with the values the same after 4 s of flow.

Fig. 12
figure 12

Specified point selected for engineering strain calculation

Full size image
Fig. 13
figure 13

Comparison of the engineering strain between model test and simulation results

Full size image

The strain-hardening behavior which occurs during flow is not considered here as the post-liquefaction soil is considered to be a viscous fluid. However, as evidenced by the model test, the excess pore water pressure dissipated rapidly and the liquefied soil recovered its strength, returning to the solid phase while still flowing. Thus, the simulation over-estimates the final flow deformation.

Conclusions

To overcome the limitations of traditional solid mechanics for the study of flow behavior in post-liquefaction soil, this paper presents a method based on fluid dynamics. 2D CFD simulations were conducted with a VOF model to predict the flow behavior of post-liquefaction soil, which is assumed to be a viscous fluid. The proposed model is capable of predicting flow liquefaction of soils in which the shear strain might exceed 100%.

A comparison of numerical and experimental results shows that the applied CFD method performs well, with the simulation showing a realistic lateral displacement and a velocity profile. The numerical approach reproduces the flow trend of the post-liquefaction sand and is in agreement with the model test flow configuration.