Analysis of the Slope Response to an Increase in Pore Water Pressure Using the Material Point Method

Home , Landslide, Pore water pressure, Slope stability analysis, Soil water (retention)

Article Analysis of the Slope Response to an Increase in Pore Water Pressure Using the Material Point Method

Antonello Troncone *, Enrico Conte and Luigi Pugliese

Department of Civil Engineering, University of Calabria, Rende, 87036 Cosenza, Italy * Correspondence: [email protected]

Received: 29 May 2019; Accepted: 9 July 2019; Published: 12 July 2019

Abstract: Traditional numerical methods, such as the finite element method or the finite difference method, are generally used to analyze the slope response in the pre-failure and failure stages. The post-failure phase is often ignored due to the unsuitability of these methods for dealing with problems involving large deformations. However, an adequate analysis of this latter stage and a reliable prediction of the landslide kinematics after failure are very useful for minimizing the risk of catastrophic damage. This is generally the case of the landslides triggered by an excess in pore water pressure, which are often characterized by high velocity and long run-out distance. In the present paper, the deformation processes occurring in an ideal slope owing to an increase in pore water pressure are analyzed using the material point method (MPM) that is a numerical technique capable of overcoming the limitations of the above-mentioned traditional methods. In particular, this study is aimed to investigate the influence of the main involved parameters on the development of a slip surface within the slope, and on the kinematics of the consequent landslide. The obtained results show that, among these parameters, the excess water pressure exerts the major influence on the slope response. A simple equation is also proposed for a preliminary evaluation of the run-out distance of the displaced soil mass.

Keywords: landslide; excess pore water pressure; post-failure stage; run-out distance; material point method

1. Introduction Excess pore water pressure, due for example to prolonged and heavy rainfall, groundwater level rising, or earthquakes, generally exerts a strong influence on the slope stability. Following the scheme originally introduced by the authors of [1,2], an increase in pore pressure induces a deformation process in the slope (pre-failure stage), which could lead to the slope collapse (failure stage) and the occurrence of large displacements of the unstable soil mass until a new condition of equilibrium is reached (post-failure stage). Some typical situations are schematized in Figure1. Specifically, Figure1a shows a cover soil overlaying a geological formation with much higher permeability and greater mechanical properties than the former soil. In these circumstances, an excess in pore water pressure owing to an increase of the hydraulic head in the upper portion of the slope (for example, due to rain infiltration) could lead to the instability of the cover soil with a slip surface located at the interface of the two materials. Figure1b shows a slope where a soil layer with low permeability (for example, a clayey layer) is interbedded in a geological formation characterized by a much higher hydraulic conductibility than that of the soil layer. An excess in pore water pressure occurring in the lower portion of the slope (for example, due to water recharge) could trigger a landslide in the portion of the slope above the soil layer. Similar considerations can be made for the dual situation shown in Figure1c, where a saturated soil layer (for example, a sandy layer) under pressure is interbedded in a

Water 2019, 11, 1446; doi:10.3390/w11071446 www.mdpi.com/journal/water Water 2019, 11, 1446 2 of 13 Water 2018, 10, x FOR PEER REVIEW 2 of 14 mucha much less less permeable permeable formation. formation. Case Case studies studies similarsimilar toto thosethose above-describedabove-described areare documenteddocumented in several published papers [3–55]]..

Figure 1. Some typical situations in which an excess of pore water pressure could trigger a landslide: (Figurea) A slope 1. Some with typical a low permeablesituations in cover which soil; an (excessb) A slope of pore with water a low pressure permeable could soil trigger layer; a ( clandslide:) A slope (a) A slope with a low permeable cover soil; (b) A slope with a low permeable soil layer; (c) A slope with a high permeable soil layer. ∆u: excess pore water pressure; k1, k2: coefficient of permeability; ∆h: increasewith a high in hydraulic permeable head. soil layer. Δu: excess pore water pressure; k1, k2: coefficient of permeability; Δh: increase in hydraulic head. Generally, stability of such slopes is assessed by analyzing only the deformation processes that occurGenerally, in the pre-failure stability and of such failure slopes stages. is as Simplifiedsessed by approaches analyzing [only6–11 ]the and deformation numerical methodsprocesses based that onoccur the inassumption the pre-failure of small and strains failure [12 – stages.21] are Simplified often used forapproaches this aim. [ On6–11 the] and contrary, numerical the post-failure methods stagebased ison usually the assumption ignored owing of small to thestrains unsuitability [12–21] are of often the above-mentioned used for this aim. methods On the contrary, to deal with the problemspost-failure involving stage is largeusually deformations. ignored owing However, to the a unsuitability prediction of of the the landslide above-mentioned kinematics aftermethods failure to isdeal very with useful problems to minimize involving the risklarge connected deformations. with theHowever, occurrence a prediction of landslides of the or landslide to establish kinematics the most suitableafter failure mitigation is very measuresuseful to forminimize land protection. the risk connected Modelling with large the deformations occurrence of requires landslides the use or ofto alternativeestablish the methods most suitable that often mitigation combine measures the performance for land protection. of different Modelling numerical large techniques deformations [22,23]. Mostrequires methods the use proposed of alternative in the methods literature that refer often to two combine distinct the approaches: performance the of discrete differe approachnt numerical and thetechniques continuum [22,23 one.]. Most The distinctmethods element proposed method in the (DEM) literature falls into refer the to former two distinct category. approaches: Application the ofdiscrete this method approach to simulateand the continuum the occurrence one. ofThe landslides distinct element and their method evolution (DEM) can falls be found into the in severalformer category. Application of this method to simulate the occurrence of landslides and their evolution can be found in several studies [24–26]. The smoothed particle hydrodynamics method (SPH) [27–28] and the material point method (MPM) [29–31] are the most used continuum approaches for modelling

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Water 2018, 10, x FOR PEER REVIEW 3 of 14 studies [24–26]. The smoothed particle hydrodynamics method (SPH) [27,28] and the material point methodlandslide (MPM) problems. [29–31 A] are comprehensive the most used review continuum of these approaches methods for was modelling presented landslide by [32] problems. who also Ahighlighted comprehensive the great review capability of these of methods MPM for was analysing presented the bystability [32] who of slopes also highlighted and their post the-failure great capabilityresponse.of In MPM this forlatter analysing method, the a stability continuum of slopes is discretized and their into post-failure two different response. frames: In this a latter set of method,Lagrangian a continuum points, is called discretized material into points, two diff anderent a frames: fixed acomputational set of Lagrangian mesh. points, Consequently, called material the points,method and combines a fixed computational the advantages mesh. of Consequently,the Eulerian and the method Lagrangian combines formulations the advantages and prevents of the Euleriannumerical and drawbacks Lagrangian due formulations to mesh distor and preventstion. In addition, numerical the drawbacks contact problems due to mesh between distortion. different In addition,bodies are the automatically contact problems solved between [4]. Therefore, different MPM bodies is are suitable automatically for simulating solved extreme [4]. Therefore, events MPMwhich isinvolve suitable large for simulating displacements, extreme such events as the which post-failure involve phase large of displacements, landslides [32– such34]. as the post-failure phaseIn of this landslides paper, [MPM32–34 ].is used to simulate the deformation processes occurring in the pre-failure, failureIn this, and paper, post-failure MPM isphases used of to landslides simulate the triggered deformation by a change processes in pore occurring water pressure. in the pre-failure, Referring failure,to an ideal and post-failure slope consisting phases of of cohesionless landslides triggered soils under by a drained change in conditio pore waterns, a pressure. theoretical Referring study is toperformed. an ideal slope Particularly, consisting the of influence cohesionless of the soilsmain under parameters drained involved conditions, in the a development theoretical study of a slip is performed.surface within Particularly, the slope the and influence the kinematics of the main of the parameters landslide involved after failure in the is developmentinvestigated ofin adetail. slip surfaceMoreover, within a very the slopesimple and equ theation kinematics is proposed of thefor predicting landslide afterthe run failure-out isdistance investigated of the indisplaced detail. Moreover,soil mass aas very a function simple of equation these parameters. is proposed for predicting the run-out distance of the displaced soil mass as a function of these parameters. 2. Material Point Method 2. Material Point Method The material point method was first introduced by the authors of [35]. In the standard approach, a continuumThe material is discretized point method by a was set of first subdomains introduced (Figure by the authors2). The mass of [35 of]. Ineach the subdomain standard approach, is assumed a continuumto be concentrated is discretized in a Lagrangian by a set of subdomains point (called (Figure the material2). The masspoint) of that each carries subdomain also other is assumed proprieties to beof concentrated the subdomain, in a Lagrangiansuch as velocities, point (called stresses, the stra materialins, and point) material that carries parameters also other as well proprieties as external of theloads. subdomain, Additionally, such asa background velocities, stresses, fixed Eulerian strains, andmesh material is adopted parameters to solve as the well governing as external equations loads. Additionally,and to perform a background the required fixed differentiation Eulerian mesh and is adoptedintegration to solveoperations. the governing This mesh equations covers andthe full to performdomain the of requiredthe problem differentiation under consideration. and integration Empty operations. elements Thisare also mesh required covers the outside full domain the initial of thegeometry problem in under order consideration.to capture the material Empty elements points that are will also leave required the original outside theposition. initial Therefore, geometry inthe ordersizes toof capturethe computational the material mesh points are thatlarger will than leave those the of original the initial position. geometry. Therefore, The occurrence the sizes of of large the computationaldisplacements mesh is simulated are larger updating than those the of theposition initial of geometry. the material The occurrencepoints moving of large through displacements the fixed ismesh. simulated updating the position of the material points moving through the fixed mesh.

FigureFigure 2. 2.Scheme Scheme of of the the spatial spatial discretization discretization of of the the material material point point method method (MPM). (MPM).

AA scheme scheme of of the the calculation calculation procedure procedure is is shown shown in in Figure Figure3. 3. At At a givena given time, time, any any information information storedstored in in the the material material points points is is mapped mapped to to the the computational computational mesh mesh (Figure (Figure3a). 3a). Interpolation Interpolation shape shape functionsfunctions provide provide the the relationship relationship between between material material points points and and nodes. nodes. Then, Then, the the unknown unknown variables variables areare calculated calculated by by solving solving the the governing governing equations equations (Figure (Figure3 b),3b) and, and the the resulting resulting acceleration, acceleration, velocity, velocity, andand displacement displacement of of the the material material points points are are updated updated (Figure (Figure3c). 3c). In In addition, addition, strains strains and and stresses stresses are are evaluatedevaluated using using an an appropriate appropriate constitutive constitutive model. model The. The position position of of the the material material points points is is also also updated updated after a mesh adjustment (Figure 3d). Finally, time is incremented by a time step Δt, and the solution procedure is applied at the next time step.

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after a mesh adjustment (Figure3d). Finally, time is incremented by a time step ∆t, and the solution WaterprocedureWater 20182018,, 1010 is,, xx applied FORFOR PEERPEER at REVIEWREVIEW the next time step. 44 ofof 1414

FigureFigure 3. 3. ComputatiComputatiComputationalonalonal schemescheme scheme ofof of MPM:MPM: (( aaa))) MM Mapapap information information fromfrom from material material popo pointsintsints toto to gridgrid grid nodes;nodes; nodes; ((bb)) SolutionSolution ofof of thethe the governinggoverning governing equationsequations equations atat at thethe the nodes;nodes; nodes; ((cc)) ( UcU)pdatepdate Update ofof thethe of the materialmaterial material pointpoint point information;information; information; ((dd)) U(Udpdatepdate) Update ofof thethe of material thematerial material pointpoint point positionposition position... Two diﬀerent approaches have been developed in the last years: the single-layer approach [36–39] TwoTwo differentdifferent approachesapproaches havehave beenbeen developeddeveloped inin thethe lastlast years:years: thethe singlesingle--layerlayer approachapproach [[3366–– and the multi-layer one [40–43]. A scheme of these approaches is shown in Figure4. 3939]] andand thethe multimulti--layerlayer oneone [[4400––4433]].. AA schemescheme ofof thesethese approachesapproaches isis shownshown inin FigureFigure 44..

Figure 4. Scheme of the different approaches of the MPM (adapted from [44]). FigureFigure 4.4. SchemeScheme ofof thethe differentdifferent approachesapproaches ofof thethe MPMMPM (adapted(adapted fromfrom [[4444]).]). In the former approach, a saturated soil is discretized by a set of material points. Each material point representsInIn thethe bothformerformer solid approach,approach, phase and aa saturatedsaturated water phase. soilsoil isis The discretizeddiscretized material pointsbyby aa setset move ofof materialmaterial attached points.points. to the EachEach solid materialmaterial skeleton pointgivingpoint representsrepresents a Lagrangian bothboth description solidsolid phasephase of andand the solidwaterwater phase phase.phase. motion. TheThe materialmaterial Water motion pointspoints move occursmove attachedattached simultaneously toto thethe solidsolid with skeletonthatskeleton of the giving giving solid phase. a a Lagrangian Lagrangian In the multi-layer description description approach, of of the the solid solid solid and phasephase fluid phases motion. motion. are Water Water represented motion motion separately occurs occurs simultaneouslybysimultaneously means of diff witherentwith thatthat sets of of materialthethe solidsolid points. phase.phase. EachInIn thethe material multimulti--layerlayer point approach,approach, carries only solidsolid information andand fluidfluid of phasesphases the phase areare representedthatrepresented it represents. separatelyseparately Therefore, byby meansmeans this latter ofof differentdifferent approach setssets is ofof suitable materialmaterial to modelpoints.points. problems EachEach materialmaterial where pointpoint pore carriescarries fluids move onlyonly informationseparatelyinformation from of of the the solidphasephase phase. that that However,it it represents. represents. the number Therefore, Therefore, of material this this latterlatter points approachapproach required is is to suitable suitablediscretize to to a porousmodel model problemsdomainproblems considerably wherewhere porepore fluidsfluids increases movemove compared separatelyseparately to fromfrom the single-layerthethe solidsolid phase.phase. formulation, However,However, making thethe numbernumber the multi-layerofof materialmaterial pointsformulationpoints requiredrequired computationally toto discretizediscretize muchaa porousporous more domaindomain expensive considerablyconsiderably than the multi-layer increasesincreases compared one.compared The approach ttoo thethe singlesingle considered--layerlayer formulation,informulation, the present makingmaking paper isthethe the multimulti single-layer--layerlayer formulationformulation one. In particular, computationallycomputationally the two-phase muchmuch single-point moremore expensiveexpensive formulation thanthan thethe is multiused,multi-- inlayerlayer which one.one. all TheThe information approachapproach about consideredconsidered a saturated inin thethe porous presentpresent medium paperpaper isisis carried thethe singlesingle by- one-layerlayer set one.one. of material InIn particular,particular, points, thewhereasthe twotwo--phasephase pore water singlesingle pressure--pointpoint formulationformulation is an additional isis used,used, variable inin whichwhich [44 ]. allall The informationinformation reason for aboutabout using aa this saturatedsaturated formulation porousporous is mediumthatmedium the e isisff ects carriedcarried of the byby relativeoneone setset ofof motion materialmaterial between points,points, solid whereaswhereas and water porepore waterwater phases pressurepressure should is notis anan be additionaladditional significant variablevariable for the [problem[4444]].. TheThe reason consideredreason forfor usingusing in the thisthis present formulationformulation study [ 4 isis,45 thatthat]. In thethe addition, effectseffects ofof as thethe already relativerelative said, motionmotion this choice betweenbetween considerably sosolidlid andand waterreduceswater phasesphases the computational shouldshould notnot bebe costs significantsignificant with respect forfor the tothe the problemproblem multi-layer consideredconsidered formulation. inin thethe The presentpresent analyses studystudy presented [4[4,,4455]].. InIn in addition,addition, asas alreadyalready said,said, thisthis choicechoice considerablyconsiderably reducesreduces thethe computationalcomputational costscosts withwith respectrespect toto thethe multimulti--layerlayer formulation.formulation. TheThe analysesanalyses presentedpresented inin thethe followingfollowing sectionssections havehave beenbeen carriedcarried outout usingusing thethe ANURA3DANURA3D codecode,, developeddeveloped byby thethe ANURA3DANURA3D MPMMPM ResearchResearch CommunityCommunity [[4466]]..

Water 2019, 11, 1446 5 of 13 the following sections have been carried out using the ANURA3D code, developed by the ANURA3D Water 2018, 10, x FOR PEER REVIEW 5 of 14 MPM Research Community [46]. 3. Parametric Study 3. Parametric Study Figure 5 shows an ideal slope the geometry of which is defined by two parameters: inclination Figure5 shows an ideal slope the geometry of which is defined by two parameters: inclination angle, α, and slope height, H. The slope consists of a saturated soil with a linear elastic perfectly angle,Water 2018α,, 10 and, x FOR slope PEER height, REVIEW H. The slope consists of a saturated soil with a linear elastic perfectly5 of 14 plastic Mohr-Coulomb constitutive model and non-associated flow rule. The soil is purely frictional. plastic Mohr-Coulomb constitutive model and non-associated flow rule. The soil is purely frictional. In addition, it is assumed that no change in the shear strength parameters occurs with increasing In3. Parametric addition, it Study is assumed that no change in the shear strength parameters occurs with increasing strain. Therefore, the case of the soils with strain-softening behavior is not considered in the present strain. Therefore, the case of the soils with strain-softening behavior is not considered in the present studyFigure. The soil5 shows parameters an ideal used slope in thethe geometryanalysis are of whichindicated is defined in Table by 1, two where parameters: γ is the unit inclination weight, study. The soil parameters used in the analysis are indicated in Table1, where γ is the unit weight, anglE’ ise, Young’s α, and modulus,slope height, ν’ is H. Poisson’s The slope ratio, consists c’ is the of effectivea saturated cohesion, soil with φ’ isa thelinear shearing elastic resistanceperfectly E’ is Young’s modulus, ν’ is Poisson’s ratio, c’ is the effective cohesion, φ’ is the shearing resistance plasticangle, Mohr ψ is the-Coulomb angle ofconstitutive dilation, modeland k and is the non coefficient-associated of flow permeability. rule. The soil Considering is purely frictional. that the angle, ψ is the angle of dilation, and k is the coefficient of permeability. Considering that the numerical Innumerical addition, simulations it is assumed are that carried no change out under in the drained shear st loadinrengthg, parameter a high values occurs of the with coefficient increasing of simulations are carried out under drained loading, a high value of the coefficient of permeability is strainpermeability. Therefore, is assumed. the case Asof theexplained soils with below, strain this-softening choice of behavior k also satisfies is not considered the stability in condition the present of assumed. As explained below, this choice of k also satisfies the stability condition of the solution and studythe solution. The soil and parameters leads to a used reduction in the analysisof the calculation are indicated time. in TheTable authors 1, where have γ isalso the foundunit weight, that a leads to a reduction of the calculation time. The authors have also found that a different choice of E’different is Young’s choice modulus, of E’ andν’ is ν’Poisson’s (which ratio, satisfies c’ is the the aboveeffective-mentioned cohesion, stability φ’ is the condition) shearing resistance does not E’ and ν’ (which satisfies the above-mentioned stability condition) does not significantly affect the angle,significantly ψ is theaffect angle the conclusions of dilation, ofand the kpresent is the study. coefficient Finally, of different permeability. values Considering of φ’ are assumed that the in conclusions of the present study. Finally, different values of φ’ are assumed in the calculations, as numericalthe calculations simulations, as indicated are carried in Table out 1. under drained loading, a high value of the coefficient of indicatedpermeability in Table is assumed.1. As explained below, this choice of k also satisfies the stability condition of the solution and leads to a reduction of the calculation time. The authors have also found that a different choice of E’ and ν’ (which satisfies the above-mentioned stability condition) does not significantly affect the conclusions of the present study. Finally, different values of φ’ are assumed in the calculations, as indicated in Table 1.

Figure 5. Slope model considered in the analysis. α: inclination angle; H: slope height. Figure 5. Slope model considered in the analysis. α: inclination angle; H: slope height. Table 1. Soil properties used in the analysis. γ: unit weight, E’: Young’s modulus; ν’: Poisson’s ratio; φThe’: shearing analysis resistance is carried angle; outψ under: angle plain of dilation;-strain k: conditions coefficient ofby permeability; using a computational c’: effective cohesion. mesh made up of triangular elements with an average size of 1 m (Figure 6). Initially, three material points are Material γ (kN/m3) E’ (kPa) ν’ φ’( ) c’ (kPa) ψ ( ) k (m /s) distributed within each element. Boundary conditions are◦ simulated by rollers located◦ on the left and right sidesSoil ofFigure the model5. Slope 20 to model constrain co 25,000nsidered the horizontal in the 0.3 analysis. dis 30–40placement, α: inclination and 0 angle; by hinges H: slope 0 at height. the bottom 0.01 of the mesh where both vertical and horizontal displacements are prevented. A boundary condition is also The analysis is carried out under plain-strain conditions by using a computational mesh made enforcedThe analysisat the top is of carried the mesh out to under fix the plain-strain vertical displacement. conditions by using a computational mesh made up of triangular elements with an average size of 1 m (Figure 6). Initially, three material points are up of triangular elements with an average size of 1 m (Figure6). Initially, three material points are distributed within each element. Boundary conditions are simulated by rollers located on the left and distributed within each element. Boundary conditions are simulated by rollers located on the left and right sides of the model to constrain the horizontal displacement, and by hinges at the bottom of the right sides of the model to constrain the horizontal displacement, and by hinges at the bottom of the mesh where both vertical and horizontal displacements are prevented. A boundary condition is also mesh where both vertical and horizontal displacements are prevented. A boundary condition is also enforced at the top of the mesh to fix the vertical displacement. enforced at the top of the mesh to fix the vertical displacement.

Figure 6. Mesh employed in the analysis.

The initial stress state of the slope is reproduced using the well-known gravity loading procedure, with the soil that is assumed to be fully saturated. Afterwards, an excess of pore water pressure, Δu, is imposed at the slopeFigure base 6. Mesh (Figure employed 5) owing in the to analysis. which motion of the unstable soil mass starts or accelerates if motion isFigure in progress 6. Mesh. A employed similar scheme in the analysis. was considered by the authors of [32] to simulateThe initial successfully stress state the of Selborne the slope slope is reproduced failure. Excess using pore the well-knownwater pressure gravity could loading be due procedure, to a rapid withwaterThe the recharge soilinitial that or stress is a assumedphreatic state level to of be the rise, fully slope as saturated. occurs is reproduced in Afterwards, the situations using an schematized excessthe well of- poreknown in waterFigure gravity pressure, 1. Although loading∆u, procedurein these situations, with the, an soil increase that is assumedin pore pressure to be fully is favoured saturated by. Afterwards, the presence an of excess soils withof pore different water pressure,permeability, Δu, isslope imposed failure at involvesthe slope a base homogeneous (Figure 5) owingsoil like to in which the slope motion schem of theatized unstable in Figure soil mass5. startsSince or accelerates ANURA3D if motion uses an is explicitin progress dynamic. A similar formulation scheme towas solve considered the governing by the authorsequations, of [32]the tosolution simulate is successfully conditionally the stable, Selborne i.e. slope, a time failure.-step E lessxcess than pore awater criti calpressure value could must be be due used to a inrapid the water recharge or a phreatic level rise, as occurs in the situations schematized in Figure 1. Although in these situations, an increase in pore pressure is favoured by the presence of soils with different permeability, slope failure involves a homogeneous soil like in the slope schematized in Figure 5. Since ANURA3D uses an explicit dynamic formulation to solve the governing equations, the solution is conditionally stable, i.e., a time-step less than a critical value must be used in the

Water 2019, 11, 1446 6 of 13 is imposed at the slope base (Figure5) owing to which motion of the unstable soil mass starts or accelerates if motion is in progress. A similar scheme was considered by the authors of [32] to simulate successfully the Selborne slope failure. Excess pore water pressure could be due to a rapid water recharge or a phreatic level rise, as occurs in the situations schematized in Figure1. Although in these situations, an increase in pore pressure is favoured by the presence of soils with diﬀerent permeability, slope failure involves a homogeneous soil like in the slope schematized in Figure5. Since ANURA3D uses an explicit dynamic formulation to solve the governing equations, the solution is conditionally stable, i.e., a time-step less than a critical value must be used in the calculations. This critical value can be evaluated according to the Courant–Friedrichs–Levy condition [47]. For the case considered in the present study, it was found that a value of ∆t = 0.01 s is suﬃcient to satisfy this condition. In addition, this value reduces the calculation time.

Results and Discussion In this section, a study is carried out to analyze the influence of the main involved parameters on the slope response. These parameters are slope height H, inclination angle α, soil shearing resistance angle φ’, and excess pore water pressure ∆u. The values considered for H range between 5 m and 10 m, and those for α between 10◦ and 20◦. In addition, φ’ varies from 30◦ to 40◦ and ∆u from 20 kPa to 80 kPa. The results presented in Figure7 refer to the case with H = 5 m, ∆u = 40 kPa, φ’ = 35◦, and α = 15◦, and show the soil displacements calculated at four different times after imposing an excess in pore pressure at the depth indicated in Figure5. The safety factor of this slope, SF, is equal to 2.70 or 1.13 when the soil is assumed dry or fully saturated, respectively. Therefore, the slope is initially stable. These values of SF have been obtained using the phi/c reduction procedure implemented in the finite elementWater 2018 code, 10 PLAXIS, x FOR PEER 2D REVIEW [48]. 7 of 14

FigureFigure 7. Total7. Total displacement displacement ofof the material material points points at atdifferent diﬀerent times, times, for the for case the with: case φ’ with: = 35°;φ ’α= = 35◦; α = 15°;15◦ ;HH == 5 5m m,, and and Δu∆ u= 40= 40kPa. kPa. (a) (t=0.5a) t = s;0.5 (b) s;t=1.0 (b) ts;= (c)1.0 t=2.5 s; ( cs;) t(d)= 2.5t=6.5 s; s. (d R:) t run= 6.5-out s. distance. R: run-out distance.

Figure 8 compares the final profile of the landslide for a slope initially with α = 15°, φ’ = 35° and different values of H and Δu. The coordinates of the profile are plotted in dimensionless form with respect to H. For the sake of completeness, the initial configuration of the slope is also indicated in Figure 8 by a dashed line. The results show that, for a given value of Δu, the dimensionless post- failure profile and the associated run-out distance are slightly affected by the initial height of the slope. On the contrary, substantial differences in the final geometry of the landslide can be observed when a different value of Δu is considered. Specifically, the greater is Δu, the larger are the accumulation and depletion zones of the displaced material. Similar considerations can be made about the shape and position of the slip surface, which are essentially unchanged when a different value of H is considered (Figure 9), but strongly depend on the imposed Δu (Figure 10).

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As can be observed, a slip surface develops within the slope at depths less than that where ∆u is imposed (Figure7a). When this slip surface is completely developed (Figure7b), the slope collapses and the unstable soil mass moves downstream. Displacements undergone by the material points define the kinematics of the landslide. With increasing time, displacement increases (Figure7c) until all material points attain a new condition of equilibrium (Figure7d). At this time, it is possible to detect the final post-failure profile of the landslide and the run-out distance R, which is defined as the distance between the tip of the displaced material and the toe of the slope before failure (Figure7d). In conclusion, using MPM all phases (pre-failure, failure, and post-failure) of a landslide can be effectively simulated. Figure8 compares the final profile of the landslide for a slope initially with α = 15◦, φ’ = 35◦ and different values of H and ∆u. The coordinates of the profile are plotted in dimensionless form with respect to H. For the sake of completeness, the initial configuration of the slope is also indicated in Figure8 by a dashed line. The results show that, for a given value of ∆u, the dimensionless post-failure profile and the associated run-out distance are slightly affected by the initial height of the slope. On the contrary, substantial differences in the final geometry of the landslide can be observed when a different value of ∆u is considered. Specifically, the greater is ∆u, the larger are the accumulation and depletion zones of the displaced material. Similar considerations can be made about the shape and position of the slip surface, which are essentially unchanged when a different value of H is considered (Figure9), Water 2018, 10, x FOR PEER REVIEW 8 of 14 but strongly depend on the imposed ∆u (Figure 10).

Figure 8. Dimensionless post-failure proﬁles for the case with α = 15◦, φ’ = 35◦ and diﬀerent values of ∆Figureu and 8. H. Dimensionless post-failure profiles for the case with α = 15°, φ’ = 35° and different values of Δu and H. In particular, Figure 10 shows that the slip surface enlarges upstream with increasing ∆u. In other words, the greater is ∆u, the greater is the volume of the unstable soil mass. Numerical simulations are

Figure 9. Location of the slip surface in dimensionless coordinates, for the case with α = 15°, φ’ = 35°, Δu = 40 kPa, and different values of H.

In particular, Figure 10 shows that the slip surface enlarges upstream with increasing Δu. In other words, the greater is Δu, the greater is the volume of the unstable soil mass. Numerical simulations are also performed using different values of α and φ’, in order to investigate their influence on the location of the slip surface and the final configuration of the landslide. Figure 11 shows the location of the slip surface and the post-failure profile obtained for the case with H = 5 m, φ’ = 35°, Δu = 40 kPa, and for two values of α. The slip surface moves upstream when a greater value of α is considered. In addition, the greater is α, the larger is the volume of the displaced material (Figure 11). A similar trend can be observed when a lower value of the angle of shearing resistance is assumed (Figure 12).

Water 2018, 10, x FOR PEER REVIEW 8 of 14

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also performedFigure 8. Dimensionless using different post values-failure of profilesα and forφ’, the in ordercase with to investigate α = 15°, φ’ = their 35° and influence different on values the location of of theΔu slip and surface H. and the final configuration of the landslide.

FigureFigure 9.9. Location of the slip surfacesurface inin dimensionlessdimensionless coordinates,coordinates, forfor thethe casecase with withα α= = 1515°,◦, φφ’’ == 3535°,◦, Water ∆2018Δuu =, 1040, xkPa kPa, FOR, and PEER different di ﬀREVIEWerent values of H. 9 of 14

In particular, Figure 10 shows that the slip surface enlarges upstream with increasing Δu. In other words, the greater is Δu, the greater is the volume of the unstable soil mass. Numerical simulations are also performed using different values of α and φ’, in order to investigate their influence on the location of the slip surface and the final configuration of the landslide. Figure 11 shows the location of the slip surface and the post-failure profile obtained for the case with H = 5 m, φ’ = 35°, Δu = 40 kPa, and for two values of α. The slip surface moves upstream when a greater value of α is considered. In addition, the greater is α, the larger is the volume of the displaced Water 2018, 10, x FOR PEER REVIEW 9 of 14 material (Figure 11). A similar trend can be observed when a lower value of the angle of shearing resistance is assumed (Figure 12). Figure 10. LocationLocation of the slip surface in dimensionless coordinates,coordinates, forfor thethe casecase withwithα α= = 1515°,◦, φφ’’ = 35°35◦,, and different diﬀerent values of ∆Δu.u.

Figure 11 shows the location of the slip surface and the post-failure proﬁle obtained for the case with H = 5 m, φ’ = 35◦, ∆u = 40 kPa, and for two values of α. The slip surface moves upstream when a greater value of α is considered. In addition, the greater is α, the larger is the volume of the displaced material (Figure 11). A similar trend can be observed when a lower value of the angle of shearing resistance is assumed (Figure 12). Finally, the inﬂuence of α, φ’ and ∆u on the dimensionless run-out distance, R/ H, is documented in FiguresFigure 13 10. and Location 14. The of resultingthe slip surface values in ofdimensionless R/H are also coordinates, shown in Tablefor the2. case As expected,with α = 15°, R /φ’H = decreases 35°, withand increasing differentφ values’, and of increases Δu. with an increase in α and ∆u.

Figure 11. Dimensionless post-failure profile and slip surface location for the case with H = 5 m, φ’ = 35°, Δu = 40 kPa, α = 10° (a), and α = 20° (b).

Figure 11. DimensionlessDimensionless post post-failure-failure profile proﬁle and and slip slip surface surface location location for forthe thecase case with with H = H5 m,= 5φ’ m, = φ α α 35°,’ = Δu35◦ =, ∆40u kPa,= 40 α kPa, = 10° =(a)10, and◦ (a ),α and= 20° (=b).20 ◦ (b).

Figure 12. Dimensionless post-failure profile (a) and slip surface location (b) for the case with H = 5 m, α = 15°, Δu = 40 kPa, and different values of φ’.

Finally, the influence of α, φ’ and Δu on the dimensionless run-out distance, R/H, is documented in Figures 13 and 14. The resulting values of R/H are also shown in Table 2. As expected, R/H decreases with increasing φ’, and increases with an increase in α and Δu.

Figure 12. Dimensionless post-failure profile (a) and slip surface location (b) for the case with H = 5 m, α = 15°, Δu = 40 kPa, and different values of φ’.

Water 2018, 10, x FOR PEER REVIEW 9 of 14

Figure 10. Location of the slip surface in dimensionless coordinates, for the case with α = 15°, φ’ = 35°, and different values of Δu.

Figure 11. Dimensionless post-failure profile and slip surface location for the case with H = 5 m, φ’ = Water 2019, 11, 1446 9 of 13 35°, Δu = 40 kPa, α = 10° (a), and α = 20° (b).

Water 2018, 10, x FOR PEER REVIEW 10 of 14 Figure 12. Dimensionless post-failure proﬁle (a) and slip surface location (b) for the case with H = 5 m, Water 2018Figure, 10, 12.x FOR Dim PEERensionless REVIEW post -failure profile (a) and slip surface location (b) for the case with H 10= 5 of 14 α = 15◦, ∆u = 40 kPa, and diﬀerent values of φ’. m, α = 15°, Δu = 40 kPa, and different values of φ’.

Figure 13. Dimensionless run-out distance, R/H, versus φ’, for α = 20° and some values of Δu. Figure 13. Dimensionless run-out distance, R/H, versus φ’, for α = 20 and some values of ∆u. Figure 13. Dimensionless run-out distance, R/H, versus φ’, for α = 20° and◦ some values of Δu.

Figure 14. Dimensionless run-out distance, R/H, versus φ’, for ∆u = 60 kPa and some values of α. Figure 14. Dimensionless run-out distance, R/H, versus φ’, for Δu = 60 kPa and some values of α. TableFigure 2. Values 14. Dimensionless of the dimensionless run-out distance, run-out, R/H, R/H, versus calculated φ’, for using Δu = the 60 kPa MPM. andφ some’: shearing values resistance of α. angle;Tableα 2.: inclination Values of the angle; dimensionless∆u: excess run pore-out, water R/H, pressure. calculated using the MPM. φ’: shearing resistance Tableangle; 2.α: Values inclination of the angle; dimensionless Δu: excess run pore-out, water R/H, pressure. calculated using the MPM. φ’: shearing resistance angle; α: inclinationα = 10 angle;◦ Δu: excess pore water pressure.α = 15◦ α = 20◦ φ’ ∆u α = 10°∆u ∆u ∆u ∆u ∆uα = 15°∆ u ∆u ∆u ∆u ∆uα = 20°∆ u φ’ Δu (◦) 20Δu kPaα = 10°40 kPa Δu 60 kPaΔu 80 kPa Δu20 kPa 40Δu kPaα = 15°60 kPaΔu 80 kPaΔu 20 kPa Δu40 kPa Δu60 kPaα = 20°80 Δu kPa Δu φ’(°) 20Δu kPa 30 40Δu 0.12 kPa 60 0.18Δu kPa 0.2180Δu kPa 0.36 20Δu kPa 0.19 40Δu 0.79kPa 60 0.91Δu kPa 0.9780Δu kPa 0.9220Δu kPa 1.43 40Δu kPa 1.82 60 2.15Δu kPa 80Δu kPa (°) 20 kPa35 40 0.06kPa 60 0.06 kPa 0.1480 kPa 0.15 20 kPa 0.05 40 0.60kPa 60 0.73 kPa 0.7580 kPa 0.4420 kPa 0.88 40 kPa 1.17 60 1.25 kPa 80 kPa 30 0.1240 0.18 0.03 0.040.21 0.100.36 0.110.19 0.04 0.79 0.31 0.570.91 0.610.97 0.260.92 0.63 1.43 1.10 1.131.82 2.15 30 0.91 0.97 0.92 1.43 1.82 2.15 35 0.120.06 0.180.06 0.210.14 0.360.15 0.190.05 0.790.60 0.73 0.75 0.44 0.88 1.17 1.25 3540 0.060.03 0.060.04 0.140.10 0.150.11 0.050.04 0.600.31 0.730.57 0.750.61 0.440.26 0.880.63 1.171.10 1.251.13 40 0.03 0.04 0.10 0.11 0.04 0.31 0.57 0.61 0.26 0.63 1.10 1.13 On the basis of all these results, the following equation is proposed to relate R/H to α, φ’, and Δu: On the basis of all these results, the following equation is proposed to relate R/H to α, φ’, and Δu: R ∆u = 푒−b tan φ′ (1) RH ∆au = 푒−b tan φ′ (1) H a where Δu is expressed in kPa. The coefficients a and b appearing in Equation (1) were evaluated whereusing aΔu trial is -expressedand-error in procedure. kPa. The Specifically,coefficients a for and any b appearing assumed valuein Equation of the ( slope1) were angle, evaluated these usingcoefficients a trial were-and -determinederror procedure. as those Specifically, providing for the anybest assumed agreement value between of the Equation slope angle, (1) and these the coefficientsnumerical simulations. were determined Finally, as thethose expressions providing relatingthe best a agreement and b to αbetween were obtained Equation by ( 1fitting) and thethe numericalavailable results. simulations. These Finally,expressions the are:expressions relating a and b to α were obtained by fitting the available results. These expressions are: a = 350e−7 tan α (2) −7 tan α a = 350e (2)

Water 2019, 11, 1446 10 of 13

On the basis of all these results, the following equation is proposed to relate R/H to α, φ’, and ∆u:

R ∆u b tan φ = e− 0 (1) H a where ∆u is expressed in kPa. The coefficients a and b appearing in Equation (1) were evaluated using a trial-and-error procedure. Specifically, for any assumed value of the slope angle, these coefficients were determined as those providing the best agreement between Equation (1) and the numerical simulations.Water 2018, 10, x Finally, FOR PEER the REVIEW expressions relating a and b to α were obtained by fitting the available results.11 of 14 These expressions are: = 7 tan α a 350e− 3 (2) 3 − b = (tan α) 2 (3) 203 3 b = (tan α)− 2 (3) 20 AA comparisoncomparison ofof thethe valuesvalues of of R R/H/H calculated calculated using using Equation Equation (1) (1 to) to those those obtained obtained using using MPM MPM is is shown in Figure 15. Considering that almost all the results are arranged close to a 45° steep straight shown in Figure 15. Considering that almost all the results are arranged close to a 45◦ steep straight line,line, itit cancan bebe concludedconcluded thatthat thethe valuesvalues ofof RR/H/H calculatedcalculated usingusing thethe proposedproposed equationsequations andand thosethose derivedderived fromfrom thethe numericalnumerical simulationssimulations are inin fairlyfairly goodgood agreement,agreement, inin spitespite ofof thethe complexitycomplexity ofof thethe problemproblem underunder consideration.consideration. The predictive capacity of the proposed equation is better for low valuesvalues ofof R R/H./H. On On the the contrary, contrary, some some discrepancies discrepancies between between simplified simplified and numericaland numer predictionical prediction occur atoccur high at values high values of R/H. of AlthoughR/H. Although affected affected by these by these limitations, limitations, Equation Equation (1) could (1) could be useful be useful for for an approximatean approximate prediction prediction of the of run-out the run distance-out distance of landslides of landslides in situations in situations similar to those similar considered to those inconsidered the present in study.the present study.

FigureFigure 15.15. Comparison of the dimensionlessdimensionless run-outrun-out distance evaluated using Equation (1)(1) toto thatthat providedprovided byby MPM.MPM. RR/H:/H: dimensionlessdimensionless run-outrun-out distance.distance. 4. Conclusions 4. Conclusions In the present paper, the material point method is used to analyze the response of slopes to an In the present paper, the material point method is used to analyze the response of slopes to an excess in pore water pressure due for example to a water recharge or a phreatic level rise, as often occurs excess in pore water pressure due for example to a water recharge or a phreatic level rise, as often in several real situations. Referring to an ideal slope consisting of cohesionless soils under drained occurs in several real situations. Referring to an ideal slope consisting of cohesionless soils under conditions, at the base of which an excess in pore water pressure occurs, attention is mainly focused drained conditions, at the base of which an excess in pore water pressure occurs, attention is mainly focused on the development of a slip surface within the slope in the pre-failure phase, and the kinematics of the landslide in the post-failure stage. A parametric study is carried out to analyze the influence of the main involved parameters on the shape and position of the slip surface as well as on the final configuration of the displaced material and its run-out distance. The results have shown that, although all these parameters affect the slope response, the excess pore pressure exerts the greatest influence on the results. Finally, a simple equation is proposed for an approximate prediction of the run-out distance of the unstable soil mass in situations similar to those considered in the present study.

Water 2019, 11, 1446 11 of 13 on the development of a slip surface within the slope in the pre-failure phase, and the kinematics of the landslide in the post-failure stage. A parametric study is carried out to analyze the influence of the main involved parameters on the shape and position of the slip surface as well as on the final configuration of the displaced material and its run-out distance. The results have shown that, although all these parameters affect the slope response, the excess pore pressure exerts the greatest influence on the results. Finally, a simple equation is proposed for an approximate prediction of the run-out distance of the unstable soil mass in situations similar to those considered in the present study.

Author Contributions: A.T. designed the analyses and wrote the paper; E.C. directed the preparation of the article and revised it; L.P. performed the numerical analyses using the ANURA3D code. All authors contributed equally to the preparation of this work. Funding: This research received no external funding Conﬂicts of Interest: The authors declare no conﬂict of interest.

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