Single-layer soil-water coupled SPH method and its application to sinkhole simulation

Abstract

In this study, a single-layer SPH approach that takes into account full soil-water interactions is proposed. The approach updates the propagation of pore pressure through combination of volumetric strain and Darcy's law, accounting for the momentum equation, soil constitutive behavior, and the development of pore pressure at each timestep of the simulation. The proposed method is validated by analytical solutions of consolidation problems. To showcase its capability in simulating large-deformation problems with hydro-mechanical interactions, a physical test of a seepage-induced sinkhole was simulated using the proposed SPH method. The good agreements suggest that the proposed method can capture the key features of sinkhole developments and serve as a promising tool to explore the associated failure mechanism. A series of parametric studies are then conducted to reveal the influences of material properties and hydraulic conditions on the failure behavior of sinkholes, including failure patterns, influence zone, and surface settlement.

Appendices

Appendix 1

The theory, implementation and validation of the Mohr-Coulomb model in SPH method were elaborated by Mori [42]. The main concepts and example validations would be briefly presented here.

1.1 Elastic response

Prior to the plastic regime, an isotropic linear elastic model is adopted:

$$\begin{array}{*{20}c} {{\text {d}} \varvec{\varepsilon}^{{e}} = {\varvec{D}}^{ - 1} {\rm{d}}\varvec{\sigma} } \\ \end{array}$$

(37)

$$\begin{array}{*{20}c} {D = \left[ { \begin{array}{*{20}c} {a_{1} } \\ {a_{2} } \\ {a_{2} } \\ 0 \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} {a_{2} } \\ {a_{1} } \\ {a_{2} } \\ 0 \\ 0 \\ 0 \\ \end{array} \begin{array}{*{20}c} {a_{2} } \\ {a_{2} } \\ {a_{1} } \\ 0 \\ 0 \\ { 0 } \\ \end{array} \begin{array}{*{20}c} {0 } \\ 0 \\ 0 \\ G \\ 0 \\ { 0 } \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ G \\ { 0 } \\ \end{array} \begin{array}{*{20}c} 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ { G } \\ \end{array} } \right]} \\ \end{array}$$

(38)

where $a_{1} = K + 4G/3$, $a_{2} = K - 2G/3$.

1.2 Yield function

This study adopts the Mohr-Coulomb yield function to define the onset of plastic straining, with the center position $\sigma_{o}$ and radius $r{ }$ of the Mohr-circle formulated as:

$$\begin{array}{*{20}c} {\sigma_{o} = \frac{1}{2}\left( {\sigma^{\alpha } + \sigma^{\beta } } \right)} \\ \end{array}$$

(39)

$$\begin{array}{*{20}c} {r = \frac{1}{2}\sqrt {\left( {\sigma^{\alpha } - \sigma^{\beta } } \right)^{2} + 4\left( {\sigma^{\alpha \beta } } \right)^{2} } } \\ \end{array}$$

(40)

where $\sigma^{\alpha }$, $\sigma^{\beta }$, $\sigma^{\alpha \beta }$ are the effective stresses in $x$, $y,$ and $xy$ directions when $\sigma^{\gamma }$ is the intermediate effective stress. The principal effective stresses are then formulated as:

$$\begin{array}{*{20}c} {\sigma_{1} = \sigma_{o} + r} \\ \end{array}$$

(41)

$$\begin{array}{*{20}c} {\sigma_{3} = \sigma_{o} - r} \\ \end{array}$$

(42)

The Mohr-Coulomb failure criterion is defined as:

$$\begin{array}{*{20}c} {f^{s} = \sigma_{1} - \sigma_{3} N_{\phi } - 2c\sqrt {N_{\phi } } } \\ \end{array}$$

(43)

$$\begin{array}{*{20}c} {N_{\phi } = \frac{1 + \sin \phi }{{1 - \sin \phi }}} \\ \end{array}$$

(44)

where $\sigma_{1}$ and $\sigma_{3}$ are the major and minor principal effective stresses, and $c, \phi$ represent the cohesion and friction angle of the soil, respectively.

1.3 Flow rule

This study adopts the non-associated flow rule, with the plastic potential function defined as follows:

$$\begin{array}{*{20}c} {g^{s} = \sigma_{1} - \sigma_{3} N_{\psi } } \\ \end{array}$$

(45)

$$\begin{array}{*{20}c} {N_{\psi } = \frac{1 + \sin \psi }{{1 - \sin \psi }}} \\ \end{array}$$

(46)

and $\psi$ is the dilatation angle. The flow rule determines the magnitude and direction of plastic strain increments. For the elastic behavior, the directions of elastic strains are governed by direction of stress increment $d{\varvec{\sigma}}$; while for plastic behavior, the direction of plastic strains is governed by the current stress state σ and the plastic potential $g^{s}$:

$$\begin{array}{*{20}c} {\text{d} \varvec{\varepsilon}^{p} = \lambda^{s} \frac{{\partial g^{s} }}{{\partial {\varvec{\sigma}}}}} \\ \end{array}$$

(47)

where $\lambda^{s}$ is the plastic multiplier to be elaborated in the next section. The incremental plastic strains in the principal directions are derived as follows:

$$\begin{array}{*{20}c} {\text{d} \varepsilon_{1}^{p} = \lambda_{s} } \\ \end{array}$$

(48)

$$\begin{array}{*{20}c} {\text{d} \epsilon_{2}^{p} = 0} \\ \end{array}$$

(49)

$$\begin{array}{*{20}c} {\text{d} \varepsilon_3^{p} = - \lambda_{s} N_{\psi } } \\ \end{array}$$

(50)

1.4 Hardening law

In the elasto-plastic analysis process, an initial assumption is made with all the strain components being elastic, in order to obtain the ‘trial stresses’ (${\varvec{\sigma}}^{*}$). A plastic correction is subsequently made if ${\varvec{\sigma}}^{*}$ is found to exceed the yield criterion, in which case the corrected principal stresses are given by:

$$\begin{array}{*{20}c} {\sigma_{1} = \sigma_{1}^{*} - \lambda_{s} \left( {a_{1} - a_{2} N_{\psi } } \right)} \\ \end{array}$$

(51)

$$\begin{array}{*{20}c} {\sigma_{2} = \sigma_{2}^{*} - \lambda_{s} \left( {a_{2} - a_{2} N_{\psi } } \right)} \\ \end{array}$$

(52)

$$\begin{array}{*{20}c} {\sigma_{3} = \sigma_{3}^{*} - \lambda_{s} \left( {a_{2} - a_{1} N_{\psi } } \right)} \\ \end{array}$$

(53)

To satisfy the consistency condition, Eqs. (51) to (53) can be substituted into Eqs. (43) and (44) with $f^{s} = 0$, in which case the plastic multiplier $\lambda^{s}$ is formulated as:

$$\begin{array}{*{20}c} {\lambda^{s} = \frac{{\sigma_{1}^{*} - \sigma_{3}^{*} N_{\phi } - 2c\sqrt {N_{\phi } } }}{{a_{1} - a_{2} N_{\psi } - \left( {a_{2} - a_{1} N_{\psi } } \right)N_{\phi } }}} \\ \end{array}$$

(54)

With the corrected principal stresses, the normal and shear stresses in Cartesian directions can be obtained. These updated stress values are used in Step 2.e of Fig. 2 and are updated at each step.

1.5 Verification

A 1D compression test in plane strain condition is simulated to test the implementation of Mohr-Coulomb models and boundary treatments in the SPH method, with the parameters adopted shown in Table 4, and geometry and boundary conditions shown in Fig. 26a. The model consists of 153 soil particles with uniform initial interval of 0.01 m.

Fig. 26

a Geometry of the compression test; b Comparison of SPH simulation results with theoretical values

Table 4 Parameters adopted in SPH model for compression test

The relationship between the axial strain and axial stress of the particle located at the model center are illustrated in Fig. 26b. Upon yielding, the axial strains develop further without an increase in axial stress, which follows the plastic behavior of the model under drained conditions. The results of SPH simulation in the response of volumetric strain of the particle located at the model center are plotted in Fig. 26c. The volumetric strains stay constant in the plastic region with zero dilation angle. The simulations results by the SPH method are almost identical to the theoretical values.

Appendix 2

The Poisson’s ratio was set as zero in the simulations of 1D consolidation in Sects. 4.1 and 4.2. To investigate the influence of the Poisson’s ratio on simulation results, a case with Poisson’s ratio of 0.25 is conducted with other parameters being the same as the base case. The comparisons of pore water pressure and strain profiles with analytical solutions are illustrated in Fig. 27. Compared with Fig. 6a, changing the Poisson’s ratio of material would affect material response and the SPH simulation results, but the response from SPH simulations still match very well with the analytical solutions.

To further investigate the influence of timestep on simulation results, a series of cases with various timesteps ($\text{d}t = 5 \times 10^{ - 6} \, \text{s}, 1 \times 10^{ - 5} \, \text{s}, 5 \times 10^{ - 5} \, \text{s}, 5 \times 10^{ - 4} \, \text{s}, 1 \times 10^{ - 3} \, \text{s}, 5 \times 10^{ - 3} \, \text{s}$) are simulated with other parameters same as the base case, and the results are illustrated in Fig. 28. The simulation results are not significantly affected by the timestep when the SPH model is stable, but model collapse may occur when the timestep is too large. As expected, the computation time increases with smaller timesteps. As a result, the timesteps adopted in this study are in some cases larger than the values defined by the $t_{cr}$ (Eq. 23), as the CFL [33] and Anderson and Wendt [5] criteria only provide reference values to enhance numerical stability. The values adopted in this work are adjusted to balance the model stability and accuracy with demands on the computational time.

A series of tests with different particle intervals ($\text{d}x = 0.01\, {\text{m}}, 0.02\, {\text{m}} , 0.04\, {\text{m}} , 0.05\, {\text{m}}$) are conducted to investigate their influences on consolidation simulations, with other parameters identical to the base case. The response of pore pressure and strain at the bottom of the model at time $t = 10$ s are shown in Fig. 29, with the corresponding computation time. The results indicate that the accuracy of the simulation results improves slightly with reduction in the initial particle spacing, but the computation time increases significantly in the meantime. To strike a balance between the model accuracy and computational efficiency, the small-strain consolidation cases involve initial particle spacing of 0.02 m.

Fig. 27

Comparisons of pore pressure and strain profiles between SPH and analytical solutions with Poisson’s ratio of 0.25

Fig. 28

Influences of the timestep on surface settlement at the finial state of the consolidation

Fig. 29

Influences of the initial particle spacing