Arab J Sci Eng DOI 10.1007/s13369-014-1540-6
RESEARCH ARTICLE - CIVIL ENGINEERING
A Deep Catastrophic Failure Model of Hillslope for Numerical Manifold Method and Multiple Physics Computation Hsueh-Chun Lin · Yao-Chiang Kan · Wen-Pei Sung · Yao-Ming Hong
Received: 12 March 2014 / Accepted: 30 November 2014 © King Fahd University of Petroleum and Minerals 2015
Abstract This study was aimed to create an analytical model for simulating deep catastrophic failure of hillslope (or deep-seated landslide) to help determining assessment criteria of potential risk based on numerical manifold method (NMM) and coupled multi-physics computation (MPC), in which the failure status is simulated by the NMM while the risk factors are studied by the MPC. The simulation delivers the landslide results that are compared with a laboratory test for approval. The proposed model includes a smallscale hillslope designed by two-dimensional geometry for the plane strain problem. Thus, the porous materials are considered for coupling fluid–structure interactions in hydraulic and geotechnical analyses. Meanwhile, discontinuous joints are assumed along the potential failure surfaces within the deepseated layer to simulate collapse behaviors of hillslope once the risk factors, such as effective stress and friction angle, reach the thresholds. Furthermore, the model is initially setup as laboratory scale for comparing with a hydraulic experiment that practices the failure condition caused by seepH.-C. Lin Department of Health Risk Management, China Medical University, No. 91 Hsueh-Shih Road, Taichung 40402, Taiwan Y.-C. Kan Department of Communications Engineering, Yuan Ze University, No. 135 Yuan-Tung Road, Chung-Li, Taoyuan 32003, Taiwan W.-P. Sung Department of Landscape Architecture, National Chin-Yi University of Technology, No. 57, Sec. 2, Zhongshan Rd., Taiping Dist., Taichung 41170, Taiwan e-mail:
[email protected] Y.-M. Hong (B) Department of Design for Sustainable Environment, MingDao University, No. 369 Wen-Hua Road, Peetow, Changhua 52345, Taiwan e-mail:
[email protected]
age. The simulation hence explorers the criteria of potential failure risks due to variations of slope, friction angle, and groundwater level. This study performs feasibility of the proposed model that provides a reliable procedure based on both simulation and experiment to estimate the potential risks for deep catastrophic landslides. In the future, the study can be expanded for evaluating full-scale landslide in a variety of hillslope properties. Keywords Deep catastrophic landslide · Multi-physics computation · Numerical manifold method · Potential failure · Risk assessment 1 Introduction Many landslides happened in the countries have usually been monitored by official or academic institutes for disaster prediction since the unanticipated natural hazards occurred more frequently in the past decade. The landslide is typically known to appear on the slope of hills consist of loose or soft soil–gravel materials. The hillslope with inside softening layers can be considered as a dangerous structure because unstable earth, gravel, or rock foundation become the reasons for landslides. The hillslope landslide generally includes shallow and deep catastrophic (or deep-seated) failures. The shallowseated landslide is mostly induced by precipitation so that earths and gravels covering the surface of hills will flow along the slope; however, it can be recovered and enhanced by engineering techniques. The deep catastrophic failure represents structural damage that destroys most of hillslope is probably caused by high groundwater level or seepage during a long period. The internal failures such as soil liquefaction and crack joints, which are caused by piping, seepage, groundwater, and even earthquake, will finally yield external collapses under natural loads. In the past years, many studies in
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Arab J Sci Eng Table 1 Candidates of potential energy χ for NMM formulation
χ
Form of energy
χ
Form of energy
E σ c
Strain upon block deformation Initial stress within the block Constraint spring with penalty constant to fix the block points that cannot be moved Point load on the block
I w f
Inertial force for kinetics at the time step Body force of the block Contact force due to fictitious contact springs to simulate the open–close status Friction force to stop block motions naturally
P
civil engineering, hazard prevention, or soil–water conservation experienced in monitoring, detecting, analyzing, and improving stability of the hillslope to avoid disasters [1–4]. Thus, modern viewpoints participate in a variety of expertise to assess the hazardous thresholds related to structural failure by using multiple physics computation (MPC) [5] which employs the parameters that represent the identical physics coupling different governing equations of interactive physical domains to solve the finite element problems and help determining the failure conditions [6]. In addition, the risk criteria for the thresholds can be created by the simulation model to support risk assessment upon the possible interactions in diverse domains beyond actual disasters. As employing MPC to predict the critical conditions of hillslope, the hydraulic studies on simulating seepage flows in subsurface based on well-known Darcy’s law and Richard’s equation typically help awareness of porous media influenced by saturated and unsaturated pore water pressure [7–12]. The simulations enable motion and continuity of groundwater flowing in homogeneous porous media to carry out pore water pressure and stream lines. With properly controlling boundary conditions of fluxes for the reliable simulation scheme, the specific parameters can be substituted to evaluate critical infiltration or fluctuation of the nonlinear porous media behind the rapidly growing groundwater levels [13–15]. Furthermore, the risk factors behind the critical conditions can be reflected by the parameters that govern the equations in simulation. The factors usually help determining the potential failure surface on the inside of hillslope. Thus, a proper MPC model can offer the material coefficients corresponding to both of the geometrical and natural boundary conditions [16–20]. Many past studies applied the computing techniques for discovering potential failures influential factors that probably lead potential failure surfaces in which the conditions of disjoint, piping, liquefaction, or instable slope are observed [21–26]. Thus, the catastrophic damage could be preliminarily learned by laboratory-scale experiments or numerical simulations beyond the practical collapse [27–33]. The failure behavior due to the critical condition hence can be estimated by modeling a distinct (or discontinuous) block system that was built for the distinct element method and the discontinuous deformation analysis to recognize rigid body motion of particles and rocks in the early era [34,35]. In
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fr
advance, the numerical manifold method, which depends on the energy conservation theorem thus enhances finite element method, was proposed to improve computation on elastically deformable blocks [36–38]. Therefore, risk evaluation on the deep catastrophic failure of hillslope is available for analyzing the risk factors of the potential failure surface followed by simulating the failure behavior of the dangerous structure in the critical condition [39,40] (Table 1). In this study, we proposed an analytical model of hillslope for simulating the deep catastrophic failure by coupling both of multiple physics computation and numerical manifold method. The risk factors and failure criteria beyond seepage and geometry of the structure can be assessed due to the analysis procedure. Based on the laws of similitudes for dimension controlled in geotechnical and hydraulic tests [41–44], the hundredth scale-down sample of hillslope was installed in a sandbox equipment for a pilot study with controlled streamlines to approve the simulation results [45]. Thus, the qualitative comparisons of numerical computation and laboratory-scale experiment would be discussed to approve feasibility of the criteria. Dimensions and materials chosen for the proposed model are designed in the in next section in which formulas of simulation and equipments of experiment are also introduced. Then, the results are described to suggest the potential failure criteria including the thresholds of the risk factors. Consequently, the proposed model will be discussed due to the simulation and experiment results. Finally, the conclusion remarks are summarized. 2 Methods The potential failure surfaces on the inside of the hillslope could yield deep catastrophic landslide when seepage flows keep in softening the soil materials. Thus, the risk factors of the probable collapse conditions can be referred to the computational parameters given by analytical formulas that enable to quantify the critical failure condition [33]. Therefore, the simulation and risk assessment for hillslope landslide couples hydraulic computation on the seepage flows and structural analysis upon the critical conditions while the procedure can be modeled as shown in Fig. 1. The proposed model involves multi-physics computation (MPC) and numerical manifold method (NMM) to decide the required
Arab J Sci Eng Fig. 1 The procedure of multi-physics computation for evaluating potential failures of the landslide dam model
parameters of potential failure risks. Then, the results of simulation are verified by an experiment with the sandbox upon a laboratory-scale sample to create the criteria. In this study, the COMSOL MultiphysicsTM software and self-developed NMM program were employed for analysis. 2.1 Model Design We consider a hillslope structure that is shaped in trapezoidal geometry with constant width (W = 20 units) and height (H = 25 units) while adjustable length (L) of the bottom can be increased from 30 to 62.5 units. The unit of this scalable model is initially set to 1 cm for comparing with the scale of laboratory test. Herein, we adopt length in 30, 45, and 62.5 units, respectively, with 68.2◦ , 45◦ , and 30.5◦ of inclined angles for simulation. As shown in Fig. 2, the proposed model is designed as a plane strain problem in two dimensions with both sides of a cliff waterfront and an inclined hill. The body
of hillslope structure consists of homogeneous and poroelastic materials, and thus, the water pressure head uniformly distributes on the cliff side to cause seepage. Properties of the materials chosen for simulation are listed in Table 2 in which the sand is also adopted by experiment design in laboratory for comparison. Several potential failure surfaces are candidates to yield probable deep catastrophic collapse based on pore water pressure on the inside of the hillslope. According to the model, the simulation can efficiently evaluate critical conditions and the hydraulic computation can reasonably resolve the potential failure surface. Once the failure surface is yielded, a discontinuous joint is added on geometry of the model to simulate discontinuous deformation behaviors of the structure. 2.2 Procedure of Multi-Physics Computation The proposed model implies that deep catastrophic failures probably occur as the soils inside the hillslope are gradually
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Arab J Sci Eng Fig. 2 Two-dimensional geometry of hillslope under potential failure conditions
Table 2 Materials suggested for simulating the deep catastrophic failure model of hillslope
Unit = 1 means a dimensionless parameter
Parameter
Hillslope (soil/sand)
Foundation (rock)
Unit
Porosity Permeability Density Young’s modulus Poisson’s ratio Dynamic viscosity Compressibility of fluid Internal friction angle Cohesion
0.3 10−10 1,800 80M 0.3 8.9 × 10−4 4.6 × 10−10 25◦ –40◦ 0
0.005 10−16 2,700 50G 0.2 8.9 × 10−6 4.6 × 10−10 30◦ 10,000
1 m2 kg/m3 Pa 1 Pa × sec 1/Pa 1 KPa
saturated by seepage flows. As considering the MPC procedure to conclude the critical conditions and potential failure surfaces, we employ the Richard’s equation based on Darcy’s law for necessary hydraulic computation. Meanwhile, the soil porosity and inclined angle of hillslope are studied by a tryand-error cyclic under constant water level and long-period seepage to inspect the zero effective stress of saturated soil; i.e., the critical status of failure surface can be estimated when the calculated pore water pressure eliminates the soil stress inside the hillslope. The MPC procedure herein couples hydraulic and geotechnical analyses upon the computational parameters of fluid and structure interaction. The Darcy’s law and Richard’s equation are both recalled to learn the risk factors referred by the parameters of formulas. Darcy’s law is valid for slow, viscous flow like seepage and gives a proportional equation below to yield total discharge Q due to pressure drop (b − a ) between point b and a. Q=
−k A b − a μ L
(1)
where k is the permeability of the medium, A is the crosssectional area of flow, μ is the viscosity, and L is the length over which the pressure drops. Furthermore, we consider the Richard’s equation that demonstrates the movement of water
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in unsaturated soils and gives ∂ ∂ψ ∂ = K +1 ∂t ∂t ∂z
(2)
where the left side of equation represents the movement of water content (or storage volume) by time t, the right side of equation includes the elevation z above a vertical datum, pressure head ψ, and hydraulic conductivity K that is a function of . The equation implies seepage flows by hydraulic head as dewatering the water storage in the medium. The global equilibrium of the hillslope involves internal body stresses and external hydraulic forces that lead total stress σ and displacement u under hydraulic changes in pore water pressure, i.e. −∇ · σ = Fv, ∇ =
∂ ∂ ∂ + + ∂x ∂y ∂z
(3)
where ∇ is a gradient operator, F is external hydraulic force, and v is fluid velocity. Therefore, the multiple physics can be formulated for poroelasticity and storage below. ρS
∂pf + ∇ · (ρ u) = Q ∂t
(4)
Arab J Sci Eng Fig. 3 The MPC model including domains of hillslope, foundation, water inlet and outlet for evaluating potential failure surfaces
where ρ is fluid density, pf is pressure of fluid, and Q is discharge volume. Thus, S is a storage coefficient related to porosity εp and compressibility of fluid χf , that is, S = εpχ f +
(αB − εp )(1 − αB ) K
(5)
where αB is Biot-Willis coefficient. Substituting Eq. (5) into (4), the left side can be rewritten by ∂ ρε p + ∇ · (ρu) = Q, ∂t
and
u=
−k ∇pf μ
(6)
in which the equation denotes the displacement is related to the hydraulic gradient in Darcy’s law. We can hence couple formulation above with the stress–strain (σ − ε) constitutive equation, σ − σ0 = Cε − α B pf ; ε =
1 (∇u)T + ∇u 2
(7)
where σ0 is the initial stress and C is the constitutive coefficient matrix of elasticity including Young’s modulus E and Poisson’s ratio ν. The term (αB pf ) regarding contribution of the fluid pressure is used for coupling the fluid-to-structure expression. Thus, the strain vector ε is defined by gradient displacement (∇u); i.e., the second term in the equation for computing matrix. With software COMSOL MultiphysicsTM , changes of seepage flow yielding effective stresses would be simulated
by the MPC procedure to determine when the critical condition will be reached. The proposed procedure is initialized by a MPC model as shown in Fig. 3 that illustrates domains of hillslope, foundation, water inlet and outlet, in which the seepage flows through the hillslope domain due to pressure head applied on the boundaries of water domains, while the fixed constraint is set on the interface of hillslope and foundation for evaluate stress distribution due to body loading of soils. 2.3 Procedure of Numerical Manifold Method Hydraulic computation with the MPC procedure in previous section helps determination of risk factors for potential failures. Thus, the flow lines of seepage with saturated pore water pressure imply the possible failure surfaces that are candidates to yield disconnected joints in the block system required for the NMM procedure when the discontinuous deformation analysis (DDA) of the hillslope structure is processed to simulate deep-seated collapse in landslide. This study uses the self-developed NMM program to evaluate failure behaviors of the hillslope. Herein, we briefly highlight the math formulation of NMM and contact algorithm of DDA for completing the proposed procedure. The NMM complies with the principle of energy conservation, which is obeyed by conventional finite element method (FEM), and inherits the DDA to solve both of static deformations in continuity and dynamic behaviors in discontinuity. It enables physical “cover” with arbitrary shapes which
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Arab J Sci Eng Fig. 4 The NMM model with polygonal manifold covers that contain triangle covers joining within physical domains of the rectangle block based on evaluation by MPC procedure
can replace mathematical “element” formed by interpolation function used in FEM to mesh the separable and deformable blocks of discontinuous system. Herein, polygon is considered as the shape of a cover so that the simplex integral can be applied for computing the closed-form entry of the stiffness matrix. In the NMM procedure, we therefore remodel the hillslope used in previous MPC procedure by updating joints into geometry of the structure for simulating possible failures in the critical conditions. As simulating kinematic behaviors of discontinuous blocks, the NMM formulates a variety of energy listed in Table 2 and summarizes them together to be the potential energy. The form of potential energy depends on the two-dimensional linear displacement function Ui of the ith discontinuous block which can be derived in terms of the global displacement Di associated with the transformation function Ti . Similarly, the energy in each physical cover can be formulated by local functions Ue(r) , De(r) , and Te(r) of the rth cover due to deformation within the block domain. [39] Thus, the constitutive relationship of elasticity is assumed for internal stress σ and strain ε on xx, yy, and xy planes as shown in Eq. (8). ⎫ ⎫ ⎧ ⎛ ⎞⎧ 1 ν 0 ⎨ εxx ⎬ ⎨ σxx ⎬ E ⎝ ⎠ εyy σyy = ν 1 0 ⎩ ⎭ 1 − ν2 ⎭ ⎩ σxy εxy 0 0 (1 − ν)/2
(8)
where E and ν are Young’s modulus and Poisson’s ratio, respectively. With the minimum energy theorem, the simultaneous equation can be obtained by differentiating energy by displacement D. That is well known to solve the unknown displacement vector D due to the stiffness matrix K and the external force vector F for n block domains as below. ∂E = 0 yields K(n×n) D(n×1) = F(n×1) ∂Di
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(9)
The dynamic movement of blocks can be simulated by the DDA contact algorithm that follows Coulomb’s law to calculate friction on the joint surface and assumes the springs to compute open and close effects of contact interaction. Once the inertial force can overcome the maximum static friction, the blocks start moving and the failure occurs. With the virtual contact springs, a minimum entrance distance prohibits two blocks close to penetrate each other. Meanwhile, the blocks begin to separate (or open) when tension of the spring is counted. During the simulation process, stiffness of the block and time interval of each contact both request automatic adjustment in each time step for open–close iterations. The NMM procedure hence requires cyclically repeats these iterations due to different joint and geometry conditions of the hillslope model. The results of simulation procedure will be compared with the laboratory-scale experiment in the following section. Figure 4 shows a discontinuous block system of NMM model based on the result from the MPC model for the proposed procedure. The potential failure surfaces are dependent on seepage flow lines, and the joints are assumed for weak interface among blocks in which small blocks are considered for the slope side. The physical covers are formed as arbitrary polygons while the triangular elements joining with the rectangular blocks. Herein, the blocks can move due to gravity and the foundation is fixed without movement. 2.4 Laboratory-Scale Experiment The experiment model is designed for a laboratory-scale pilot study in which the grain size of homogeneous soil is D50 = 0.195 mm, and the sample is built in a sandbox. Herein, Darcy’s law implies that the seepage flow is laminar as is generally the case in porous media. The limit of validity
Arab J Sci Eng Fig. 5 The experiment model equipped with the sandbox in which water is pumped into the soil sample by the pumper, and practical failure is monitored by the camera
can be stated in terms of the Reynold’s number (NR ), i.e., (V D/v)prototype = (V D/v)lab where V , D, and v are fluid velocity, grain size, and kinematic viscosity, respectively. In general, the same v works in both simulation prototype and laboratory model. In the prototype, landslide usually occurs by slow seepage due to rainfalls after several days. In the model, landslide was controlled within 20 min due to fast seepage flow. Therefore, the suitable grain size in the simulation prototype can be larger than the laboratory model based on the dynamic similarity. As considering dynamic similarity of hydraulics, seepage flow is controlled by particle size for similar time rates of change motions with respect to computation in the MPC model. Thus, the sample is scaled down as 1/100 of actual dimension for simulation in the NMM model due to geometric similarity of geotechnics. The equipment of the sandbox is divided into three zones: (1) A repository zone that locates on the right side of the sandbox to store water for the test and draw water back after flow, (2) a watering zone that supplies water through a holed still plate covered by fine-grid net to produce seepage flow into the sample, and (3) a test zone that is set up at the center of the sandbox with one water pressure meter and six pore water pressure meters while the measured data can be transported to a data logger and then be transferred to a data storage, in which the data storage can immediately deliver records of groundwater level and water table level to the computer at the backend. Meanwhile, a camera is installed above the test zone to monitor failure conditions of the hillslope sample. The equipment design is shown in Fig. 5a–c that illustrates the details of components, dimensions, and installation. Herein, due to the model, the hillslope sample can be adjusted in a variety of slopes so that various groundwater levels can be studied for risk assessment as the expected seepage yields landslide. With the constant width (=20 cm) and height (=25 cm) of geometry, we further change the inclined angle of slope from 30.5◦ to 68.2◦ corresponding to length of the hillslope bottom that is decreased by 2.5 cm from 62.5 to 30 cm. Due to
proposed laboratory-scale sample, the maximum slope and the particle size approximately equal to probable collapse scale in the actual case of deep-seated landslide. The theoretical equation to calculate shear stress beyond failures can be formulated by τ = σ tan(φ) + c
(10)
where τ is shear stress, σ is normal stress, and φ is friction angle while cohesion c is zero since dry sand is the sample material. Meanwhile, the friction angle becomes zero if liquefaction occurs. The experimental procedure within the model design is expected to observe deep catastrophic failure caused by seepage, and thus, it can help approval to estimate the risk factors and modify the correlative parameters referred by the simulation procedure. 3 Results and Discussion Based on the analysis procedures coupling the MPC and NMM models, the approaches simulated a hillslope model by 68.2◦ , 45◦ , and 30.5◦ of inclined angles for three types of steep, moderate, and tender slopes while the water level is constant. In this study, the MPC model explores the microprospect with stress and flows inside the structure depending on continuity analysis. The NMM model performs the macroprospect with collapse tendency upon kinematic behaviors limited to supposed disjointed geometry. Both of the models make up limitations of the other party. Due to results from MPC analyses, the possible failure surfaces for NMM simulation are located on the contour with zero effective stresses. Thus, the critical liquefaction or piping condition with frictionless joints is considered for the deep catastrophic failure. 3.1 Risk Assessment on MPC Procedure We consider the stationary condition with the constant water level and body load for coupling the studies in hydraulics
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Fig. 6 Seepage and stress distribution of various hillslopes simulated by MPC procedure: a flow lines with pore water pressure contour for the slope inclined in 68.2◦ , b von Mises stress contour for the slope inclined in 68.2◦ , c flow lines with pore water pressure contour for the
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slope inclined in 45◦ , d von Mises stress contour for the slope inclined in 45◦ , e flow lines with pore water pressure contour for the slope inclined in 30.5◦ , f von Mises stress contour for the slope inclined in 30.5◦
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Fig. 7 Failure simulation of discontinuous block system for various slopes of hillslope in critical conditions computed by NMM procedure: a bottom length = 30, slope = 68.2◦ , no friction, b bottom length = 45, slope = 45◦ , no friction, c bottom length = 62.5, slope = 30.5◦ , no friction
and solid mechanics. Seepage flow lines with contours of pore water pressures and soil stresses for the proposed MPC procedure are resulted on Fig. 6, in which three figure sets denoted by (a)(b), (c)(d), and (e)(f) illustrate contour conditions for the steep, moderate, and tender slopes, respectively. For the example of Fig. 6c, d, the seepage lines are highlighted inside the 45◦ hillslope, and thus, the flows distributed within the foundation are ignored in this study. We are hence interested in finding the critical portion of both contours combining pore water pressures and soil stresses. The zero effective stress below the ground, which is yielded by soil stress subtracted from pore water pressure (both are about 2,000 Pa), can be found in the zone around the lower flow lines approximately 5–10 units from the bottom of hillslope. Within this area, the flow velocity raises upward at the slope nearby to yield probable piping conditions. It means the candidate failure surface happens for the critical condition when pore water pressure becomes positive in the saturated soils to eliminate soil stress and lead further liquefaction potentials for deep-seated collapse. As considering the critical status, the upper seepage flow line beginning at about 17.5-18 units from the bottom is another potential failure joint for loosening soil layers near
the top of hillslope and probably yielding shallow-seated collapse. Similar risk assessment on potential failures can be applied for other cases, i.e., the steep and tender slopes. Due to the potential risk assessment above, we can determine three possible failure joints for the discontinuous block system required by the NMM model in the next step. 3.2 Failure Simulation by NMM Procedure The block system formed by NMM is shown on Fig. 4 that follows the seepage lines resulted from the previous MPC procedure to produce the cracks and joints among the blocks. As the critical condition of liquefaction occurs, the cohesion and friction would not be counted in computation while 500 time steps were used in simulation. By repeating the procedure with trial and error, the typical failures for three types of hillslopes are concluded and shown in Fig. 7. Figure 7a presents a steep slope for totally collapsing along the failure surface. In fact, the steep case herein implies an unstable slope that is even damaged under the condition with friction and without failure surface. Figure 7b performs a moderate slope that includes both of the shallow and deep catastrophic failures. The small blocks in the weak zone of slope slide
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Arab J Sci Eng Fig. 8 Deep catastrophic failure of 45◦ hillslope in the sandbox: a initial failure at water level = 15.05 cm due to seepage and piping, b complete failure at water level = 17.66 cm due to seepage and piping
along the upper failure surface, while the large blocks near the lower failure surface have been apparently destructive to push out the small blocks at the toe of hillslope. Herein, the potential piping zone can be learned. In the case of the tender slope, the structure is steadier with respect to other two types. Only slight separation is initially found in the blocks upon slope surface during a limited time-step period. The obvious damage as shown in Fig. 7c, which is similar to the previous cases, will be carried out by more time steps. Herein, the simulation in systematic study presents a prospect of macro-analysis for evaluating possible risk tendency. Collapse of the distinct elements is mostly limited to the geometry of blocks and joints that present individual solids, but not particles, within the potential failure zones. It might be improper to express actual behavior of cohesionless sands in this study.
3.3 Approval of Laboratory-Scale Experiment The laboratory-scale experiment was proceeded to water the hillslope sample slowly in the sandbox. According to the experiment design, the hillslope samples were categorized in a variety of the steep levels while the high risk usually occurs as the slope is larger than 45◦ . In the test, the samples with various slopes were therefore corresponding to the steep levels which imply a variety of risks. In general, the major failures were observed at two steps: (1) scouring sands inside the sample to cause partial collapse and (2) piping sands at the bottom of the sample to yield large failure surface. Herein, we adopted one of the samples for approving the simulation. In this pilot study, the case of 45◦ hillslope sample was practiced to compare with previous simulation. Due to the camera photographs as shown in Fig. 8a, b, the seepage began to flow through bottom of the hillslope sample when water was continuously pumped into the watering zone. The partial collapse initially occurred since inside sands were scoured out by piping as the water level was arising to 15.08 cm. However, there is no sliding happened due to balance of water pressures on both sides of the sample. Thus, the advanced failure surface appeared because of piping at the inside of hill-
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slope sample as the water level was 17.66 cm that approaches to the beginning of the upper seepage flow line in previous simulation. Finally, the complete collapse was experienced when the water in the repository zone was pumped out. The test results approve the simulation according to the proposed model of deep catastrophic failure. Due to results and discussion above, we would conclude the comparison for the simulation with respect to the experiment. In the aspect of the experiment, the water level increased quickly since discharge of the pumper in the test zone did not reach in equilibrium with watering volume. It speeded up the process to yield the deep catastrophic failure but probably lost control in balancing the water level with respect to the actual condition. On the other hand, the MPC and NMM procedures simulated multiple physics coupling hydraulic computation and structural analysis helped prediction of potential failures and determination of risk criteria beyond critical conditions. The simulation performs theoretical failure tendency with virtual prediction based on micro- and macro-prospects while the experiment supports the actual conditions. The proposed model enables the risk assessment of deep catastrophic failure of hillslope.
4 Conclusion Remarks This study contributed a prototype of hillslope model for evaluating deep catastrophic failures by coupling multiple physics computation and numerical manifold method for simulating possible failure conditions as well as approval of the laboratory-scale experiment. The analyses estimate risk factors due to structural, hydraulic, and geotechnical parameters behind finding the potential failure surfaces. The simulation procedure begins the preliminary study by using a MPC model to compute the contour of zero effective stress that help presenting the possible failure conditions according to seepage flow lines. Then, a discontinuous block system in a NMM model is created to simulate failure behaviors of hillslope structure. Within the model, the poroelastic materials are assumed for comparing with the practical results from the
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experiment in the sandbox equipment. Finally, the failures of hillslope are learned by the proposed model while the similar collapses are practically approved by the laboratory-scale test. In the future, the proposed model can be further applied with reasonable potential failure conditions for practical risk assessment upon the full-scale hillslope by adjusting the scalable unit to meter in the computation procedure. Acknowledgments The authors would like to appreciate the research support from National Science Council of Taiwan, the Republic of China, with the Project No. NSC101-2625-M-039-001, 101-2625M167-001, and 102-2313-B-451-001.
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