Multiscale Modeling of Flood-Induced Piping in River Levees

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Multiscale Modeling of Flood-Induced Piping in River LeveesUsama El Shamy, P.E., M.ASCE1; and Firat Aydin2

Abstract: A three-dimensional transient fully coupled fluid-particle model is utilized to simulate flood-induced piping under river leveesand taking into account the effects of soil-fluid-structure interactions. The porous soil medium is modeled as a mixture of two phases,namely the fluid phase �water� and the particulate solid phase. The fluid is idealized as a continuum by using an averaged form ofNavier–Stokes equations that accounts for the presence of the solid particles. These particles are modeled at a microscale using thediscrete element method. The interphase momentum transfer is modeled using an established relationship that accounts for the dynamicchange in porosity and possible occurrence of nonlinear losses. The hydraulic structure �levee� is modeled as an impervious rigid blockand its motion is described by a combination of external and internal forces from the surrounding fluid and solid particles. A computa-tional simulation is conducted to investigate the response of a granular deposit when subjected to a rapidly increasing head difference. Thesimulation provided information at the microscale level for the solid phase as well as at the macroscopic level for the pore-water flow. Thesettlement and failure mechanism of the structure were captured as the hydraulic head difference gradually increased and the solid phaseunderwent subsequent deformations. The results suggest that failure of such structures may occur suddenly and at hydraulic gradients wellbelow the critical gradient. The proposed computational framework for analyzing river and flood-protection levees would provide a newdimension to the design of such vital geotechnical systems. The technique can be effectively used to investigate failure mechanisms undercomplex loading and flow conditions.

DOI: 10.1061/�ASCE�1090-0241�2008�134:9�1385�

CE Database subject headings: Seepage; Granular media; Transient flow; Discrete elements; Soil deformation; Soil-structureinteraction.

Introduction

Flooding along a river or a canal can produce hydraulic headdifferences on opposite sides of a flood-protection levee. As theexit hydraulic gradient increases and approaches the criticalvalue, seeping water begins to carry sand to the surface, formingconical sand mounds, often referred to as sand boils �Fig. 1�. Theterm piping is described as the process of the formation of apipe-like opening starting from the point of sand boils whichprogresses below the levee base toward the stream �Ojha et al.2001�. Flood-induced piping and subsequent formation of sandboils is a major cause of severe damage to river levees and earthdams. Levee failure can also result in significant damage to road-ways and nearby structures. Wolff �1997� pointed out that theprobability of levee failure due to underseepage is among thehighest causes of levee failures and increases as the floodwaterelevation increases. The investigation report of the performanceof the New Orleans flood protection systems during HurricaneKatrina made reference to a number of cases where signs of un-

1Assistant Professor, Environmental and Civil Engineering Dept.,Southern Methodist Univ., P.O. Box 750340, Dallas, TX 75275 �corre-sponding author�. E-mail: [email protected]

2Formerly, Graduate Student, Civil and Environmental EngineeringDept., Tulane Univ., 6823 St. Charles Ave., New Orleans, LA 70118.E-mail: [email protected]

Note. Discussion open until February 1, 2009. Separate discussionsmust be submitted for individual papers. The manuscript for this paperwas submitted for review and possible publication on February 13, 2007;approved on January 7, 2008. This paper is part of the Journal of Geo-technical and Geoenvironmental Engineering, Vol. 134, No. 9,September 1, 2008. ©ASCE, ISSN 1090-0241/2008/9-1385–
1398/$25.00.
JOURNAL OF GEOTECHNICAL AND GEOEN

J. Geotech. Geoenviron. Eng.

derseepage piping erosion may have occurred and contributed tothe failure of the levees �Seed et al. 2006�.

Field and experimental evidence of the occurrence of pipinghas been reported by a number of researchers �e.g., Turnbull andMansur 1961; Peter 1974; Skempton and Brogan 1994; Tomlin-son and Vaid 2000�. Many analytical expressions to predict thecritical hydraulic head at which piping is initiated have been de-veloped for sand beds. Most of the analytical work found in theliterature considers piping under steady state conditions �Sell-meijer 1988; Sellmeijer and Koenders 1991; Weijers and Sell-meijer 1993; Ojha et al. 2001, 2003�. Ozkan �2003� introduced aone-dimensional transient analytical flow model with changingwater level to study the effects of transient flow and repetitiveflood events. While these models can be used for design of leveesagainst piping, they do not account for intergranular stresses andany subsequent deformation of the soil system.

Application of computational methods in the modeling of pip-ing is widely used. Griffiths and Fenton �1993, 1997, 1998� em-ployed two- and three-dimensional finite-element models to studyseepage in spatially random soil with statistically variable soilpermeability and steady state flow. Unsteady groundwater flowmodels using the finite-element method were also presented �e.g.,Nath 1981; Koo and Leap 1998�. Lu and Zhang �2002� used afinite difference technique that accounts for heterogeneous soils.The aforementioned studies focused on solving the seepage equa-tions but did not account for the solid phase deformations.

The coupled response of saturated granular soils is commonlymodeled using continuum formulations derived based on phe-nomenological considerations �Biot theory, e.g., Zienkiewicz etal. 1998� or homogenization of the micromechanical equations ofmotion �e.g., Lewis and Schrefler 1998�. In these models, the soil
skeleton is considered to be porous elastic solid and the pore fluid
VIRONMENTAL ENGINEERING © ASCE / SEPTEMBER 2008 / 1385

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is coupled to the solid through equilibrium and continuity rela-tions. Water flow is assumed to be laminar and obeys Darcy’s law.The most crucial part of such an approach lies in establishingconstitutive models for the behavior of the components of theporous medium. To the writers’ knowledge, a constitutive relationthat is capable of predicting piping and subsequent formation ofsand boils does not exist.

Apart from the considerations discussed above, there is also aneed for a computational model that accounts for the soil-structure interaction for the analysis of piping. Oner et al. �1997�used the finite-element method together with models of soil-structure interface, nonlinear soil behavior, and loading sequenceto study the soil-structure interaction mechanisms in floodwallsystems. They represented the soil by isometric quadrilateral fi-nite elements in plane strain, the sheet pile by linearly elasticone-dimensional �1D� beam-column elements, and the soil/pileinterface by special point interface elements. These interface ele-ments allow soil and pile nodes to separate under tension, andwater was allowed to intrude in these tension cracks to provideadditional hydrostatic pressures. It was concluded that a typicalfloodwall will not behave as it is assumed in conventional design-analysis methods and the soil deformation should primarily betaken into account. Following the catastrophic failure of NewOrleans levees in the aftermath of Hurricane Katrina, extensivecomputational investigation of the performance of different sec-tions of the levee systems was conducted using finite-elementsoil-structure interaction analyses �Interagency PerformanceEvaluation Task Force 2006�. These analyses provided importantunderstanding into the performance of the analyzed I-wall flood-walls which will be useful for improving future designs and as-sessments. However, as in most continuum-based techniques,some parameters that represent the soil constitutive law and theinterface elements are difficult to quantify.

Piping is an instability condition where the soil media undergoconsiderable changes and the development of an analytical modelto assess it becomes rather difficult. Terzaghi and Peck �1967�noted that a theoretical approach cannot be used to analyze sub-surface erosion. Van Zyl and Harr �1981� acknowledged that themechanics of piping are almost impossible to analyze. Flow con-ditions that would result in piping encompass several issues thathave to be accounted for when developing a computational modelfor fluid flow through deforming porous medium. In the case of

Fig. 1. Microscopic view of fluid flow th

flood-induced piping, transient analysis has to be considered since

1386 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGIN


flooding of a river results in a rapidly increasing hydraulic headthat a steady flow regime is unlikely to occur. Furthermore, undersuch extreme flow conditions, soil particles may undergo largedisplacements leading to significant changes in porosity. Varia-tions in porosity affect the soil hydraulic conductivity and defor-mation characteristics. Only micromechanical analysis may shedfull light on the intricate response mechanisms of saturated granu-lar soils when subjected to relatively high hydraulic heads.

Computational simulation of piping is achieved herein byusing a transient fully coupled continuum-discrete hydromechani-cal model to analyze the pore-fluid flow and solid phase deforma-tion of saturated granular soils when subjected to seepageconditions �El Shamy 2004; El Shamy and Zeghal 2005�. Con-ceptually, the mixture of solid particles and pore fluid is viewedas two interpenetrating media, namely the solid phase and thefluid phase �Fig. 1�. The fluid is idealized as a continuum by usinga homogenized form of Navier–Stokes equations that accounts forthe presence of the solid particles. These particles are modeled ata microscale using the discrete element method �DEM�. The in-terphase momentum transfer is modeled using an establishedrelationship.

Multiscale Coupled Fluid-Particle Model

Fully Lagrangian computational techniques �e.g., Potapov et al.2001; Cleary et al. 2006�, such as the smoothed particle hydrody-namics �SPH� and the DEM, provide the most effective tools toanalyze the coupled fluid-particle dynamics of saturated granularsoils at the level of each individual grain. As a consequence of theinvolved high computational cost, fully microscale approacheshave a reduced scope of applications, and are not practical even inanalyses of soil samples of more than few particles �Zhu et al.1999; Potapov et al. 2001�. The usefulness of these techniques iscurrently limited to investigate the local mechanisms of pore-fluidinteraction with a small number of particles.

Alternative methods may be used to quantitatively account forfluid-particle interactions when analyzing saturated granular soilsystems. Herein, the coupled response of the fluid and solidphases of saturated soil is analyzed using a micromechanical La-grangian DEM idealization of the particles �Cundall and Strack

granular soil system and proposed model
rough
1979� and a mesoscale model of the pore fluid provided by aver-

EERING © ASCE / SEPTEMBER 2008

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aged Navier–Stokes equations. This approach was pioneered byTsuji et al. �1993� for analysis of fluidized beds and since then hasbeen used by several researchers in a number of engineering ap-plications �e.g., Hoomans et al. 1996; Kawaguchi et al. 1998;Kafui et al. 2002; Zeghal and El Shamy 2004; El Shamy 2004�.This approach was recently employed in the DEM commercialsoftware package PFC3D �Itasca 2005�, which is utilized in thisstudy, as an optional feature to account for fluid-particle coupling.The following sections present an overview of this model �moredetails may be found in El Shamy and Zeghal 2005�, followed bya description of conducted numerical simulation.

Continuum Fluid Phase

The volumetric deformation of water is typically negligible com-pared to changes in pore volumes. The pore fluid was thereforeconsidered to be incompressible with no spatial or temporal varia-tion in its density. The averaged Navier–Stokes continuity andmomentum equations are then given by �e.g., Anderson and Jack-son 1967; Jackson 2000�

�n

�t+ � · �nv f� = 0 �1�

� f� ��nv f��t

+ � · �nv fv f�� = − n � pf� − n � · � − fi + n� ffg �2�

where n=n�x , t��porosity �in which x and t�space and timecoordinates�; v f = v f�x , t��averaged fluid velocity vector; pf

= pf�x , t��averaged fluid pressure; ��gradient operator, � f�fluid

density; fg�gravitational acceleration vector; fi= fi�x , t��aver-aged fluid-particle interaction vector; and ��viscous stress tensordefined as

� = �ed �3�

in which ��fluid viscosity; and ed�fluid deviatoric strain ratetensor. Since most energy dissipation associated with water flowthrough granular soils occurs at fluid-particle interfaces, it is suf-ficient to assume that the pore fluid is inviscid and therefore theviscous stress tensor is eliminated from Eq. �2�. The boundaryconditions associated with the above equations consist of fluidvelocity and/or pressure constraints.

Fluid-Particle Interaction

Averaged fluid-particle interactions reflect the means by whichthe fluid energy is dissipated as it moves between the voids of theparticulate system. Quantifying these interactions originated withthe experimental work and theoretical analysis published byHenry Darcy �Darcy 1856� which have been extensively used todate in a number of science and engineering applications �Trusselland Chang 1999�. Darcy provided the following relationship,commonly referred to as Darcy’s law:

�d = Ki �4�

where �d�superficial �discharge� pore-fluid velocity; K�Darcy’spermeability coefficient; and i�hydraulic gradient. For caseswhere high flow velocities and relatively coarse media were in-volved, Forchheimer �1901� proposed a two-term, nonlinear
model for the hydraulic gradient


i = a�d + b�d2 �5�

in which a�coefficient related to linear head loss; and b�similarcoefficient related to nonlinear head loss. The second term in theForchheimer equation gives rise to the contribution of the inertiaforces at relatively high fluid velocities. Eqs. �4� and �5� revealthat different laws apply to different flow regimes based on fluidvelocities. Based on a review of available data, Trussell andChang �1999� suggested that there exist four flow regimes de-pending on the amplitude of the dimensionless particle Reynoldsnumber, Rp, defined as

Rp =� fdp

� f�d �6�

in which � f�pore fluid viscosity; and dp�particle diameter. Trus-sell and Chang �1999� suggested that Darcy’s law is applicablefor values of Rp below about 1 �Darcy’s regime� where there is noinfluence of inertia forces. For values of 1�Rp�100, the fluidflow will be laminar but deviates from Darcy’s law and wouldfollow Forchheimer law as the influence of inertia forces in-creases �Forchheimer regime�. Fluid flow through porous mediacommonly encountered in geotechnical engineering applicationsfalls within these two regimes.

Under conditions applicable to Darcy’s law, the hydraulic ra-dius theory is the most effective tool for matching the character-istics of the media and the fluid to the coefficients in Darcy’s law�Trussell and Chang 1999�. Using the hydraulic radius theory�which simplifies the geometry of the pores as capillary tubes�along with a number of experimental results, Carman �1937� sug-gested the following relation between the hydraulic gradient andthe properties of the porous material for the special case of uni-form spheres:

i = 180� f

� fg� �1 − n�2

dp2n3 ��d �7�

in which n�porosity and g�gravitational acceleration. In order toobtain an equation for energy losses through porous media over awider range of flow conditions, Ergun �1952� proposed the fol-lowing empirical equation

i = 150� f

� fg� �1 − n�2

dp2n2 ��d + 1.75

�1 − n�gdpn2 �d

2 �8�

Ergun examined the performance of this equation using datafrom more than 600 experiments, mostly from various sizedspheres but also from sand and pulverized coke �Fig. 2�. The datawere obtained from experiments on granular beds of various po-rosities where the bed was subjected to a certain pressure dropand the resulting flow rate was measured. As noted by Trusselland Chang �1999�, Eq. �8� is of the same form that was suggestedearlier by Forchheimer �Eq. �5�� to account for the nonlinearlosses at high flow rates.

For small fluid velocities, the second term in Eq. �8� will benegligible and the fluid flow will mostly obey Darcy’s law. Forhigh fluid velocities resulting from flow through coarse mediumor high porosity associated with the solid phase deformation, thecontribution of the inertia forces increases and is accounted forusing the second term of Eq. �8�. Thus, Ergun’s equation caneffectively model fluid flow over a wide range of fluid velocitiesregardless whether the flow obeys Darcy’s law or not. Therefore,it is more suitable for fluid flow through a deforming porousmedium subjected to extreme conditions such as those encoun-
tered during piping.

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Comparing Eq. �8� to Eq. �4� one may define an equivalentDarcy’s coefficient of permeability as

K =� fg

150� f�1 − n�2

n2dp2

+ 1.75�1 − n�� f�d

n2dp

�9�

Noting that the discharge velocity is related to actual relative fluidvelocity as �d=n�v f − vp�, where vp�average particle velocity, Eq.�8� may then be rewritten in vector form as

fi = 150� f�1 − n�2

ndp2

�v f − vp� + 1.75�1 − n�� f�v f − vp�

dp

�v f − vp�

�10�

in which vp�average particle velocity vector; and dp�equivalentparticle diameter. Note that for the coupled transient analysis pre-sented in this study, Eq. �10� would capture the impact of thedeformation of the solid phase and subsequent changes in poros-ity on the momentum transfer.

Discrete Solid Phase

The discrete element method �Cundall and Strack 1979� was usedto idealize the assemblage of soil particles using distinct spheres.DEM is essentially a Lagrangian technique �mesh free� whereeach individual particle is treated separately. The motion of aparticle p is governed by the momentum equations �Itasca 2005�

mpvp = mpfg + �c

fc + fd �11�

Ip�p = �c

rc � fc �12�

where vp and �p�translational and rotational velocity vectors �asuperposed dot indicates time derivative�; mp�particle mass;Ip�particle moment of inertia; fc refers to interparticle force atcontact c �c=1,2 , . . . �; rc�vector connecting the center of the

Fig. 2. Performance of Ergun’s equation compared to results of alarge number of experiments on particles of different materials, sizes,and shapes �Ergun 1952�

particle to the location of the contact c; and fd�drag force exerted



by the fluid on the particle p. This force includes buoyancy andfluid-particle interaction terms

fd = �− �pf +f f − p

1 − n�Vp �13�

in which Vp�volume of particle p. A contact force fc between twoparticles consists of normal fn and shear fs components. The in-terparticle forces are dictated by contact laws which are directfunctions of grain stiffness properties and relative movements atthe contacts. The normal component was idealized using a linearspring stiffness which is connected in parallel to a viscous dash-pot. The shear contact force was modeled using an elastic springin series with a frictional slider. The shear and normal forces arerelated by a slip Coulomb model �Itasca 2005�.

Coupled Response

The coupled fluid-particle response was obtained using the com-mercial software PFC3D v3.1 �including the optional fluid cou-pling feature �Itasca 2005��. In this software, particle velocitiesand locations are obtained by integrating the equations of motion�Eqs. �11� and �12�� using an explicit central finite differencealgorithm. The fluid domain is discretized into parallelepipedcells and the averaged Navier–Stokes equations �Eqs. �1� and �2��are solved using the finite volume technique. A space staggeredgrid scheme, as shown schematically in Fig. 1, is employed toensure stability �Harlow and Welch 1965�. The pore fluid pressure

pf as well as the averaged solid phase parameters n, vp, and dp areevaluated at grid nodes, while the fluid velocity v f is computed onstaggered nodes located at the center of the finite volume �cell�sides. Average per-unit-volume drag forces are computed for in-dividual fluid cells based on mean values of solid and fluid phaseparameters and state variables. These drag forces are then appliedto individual solid particles proportionally to their volumes �Eq.�13��. Displacements of these particles subjected to drag forces,external loads, and contact forces are computed subsequentlyusing Eqs. �11� and �12�.

Simulation

The classical problem of seepage beneath a hydraulic structurewas selected to examine the potential of the aforementioned com-putational model to simulate the response of the solid particles toflow-induced seepage forces. A hydraulic structure with a lengthof 5.0 m is constructed over a 9.7 m deep deposit of cohesionlesssoil �Fig. 3�. Use was made of the high g-level concept commonlyimplemented in centrifuge modeling �Kutter 1992� in order toreduce the total number of particles that can be used in a DEMsimulation to a manageable size. This approach was found to bevery effective in DEM simulations to model boundary value prob-lems and has been adopted in a number of applications �e.g.,Zeghal and El Shamy 2004; El Shamy 2004; Jiang et al. 2006�.Uniformly graded spherical soil particles with a uniformity coef-ficient of 1.4 �Fig. 4� were generated and settled under 1 g untilthere was no further movement of particles. A 100 g gravitationalfield was then applied until a submerged state condition is main-tained. The particles properties are shown in Table 1. These prop-erties were chosen to resemble a synthetic soil with an overallresponse that is similar to real soils �see the Appendix�. Thechoice of the model length �0.55 m� was based on a 2D finite-
element analysis using the popular software SEEP/W �Geo-Slope

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Int. Ltd. 2004� to asses the impact of the lateral boundaries onflow characteristics �Aydin 2007�. The 0.05 m dimension in the ydirection was found to have insignificant influence on the ampli-tude of the initial stress distribution after generating the deposit,as can be seen in Fig. 5. In this figure, the computed verticaleffective stress is compared to that using the geostatic stressequation

Fig. 3. Three-dimensional view of particle deposit and employedfluid mesh in conducted simulation

Fig. 4. Grain size distribution of soil particles used in simulation



�zz� = �subz �14�

where �sub�average submerged unit weight of the soil �about9.87 kN /m3� and z�depth below ground level. The computedlateral normal stresses reflect a coefficient of earth pressure at restof about 0.5.

In order to compensate for the employed high g level, viscousfluid was used in the simulation �Kutter 1992�. The use of viscousfluid in the simulations results in a permeability of the model that

Table 1. Characteristics of Conducted Numerical Simulation in ModelUnits

Particles

Diameter 1.7–8.5 mm

Normal/shear stiffness 1e5 N /m

Critical damping ratio 0.10

Friction coefficient 0.5

Density 2,650 kg /m3

Number of particles 22,303

Initial average porosity of soil 0.39

Walls

Normal/shear stiffness 1e5 N /m

Friction coefficient 0.5

Structure

Width �y-direction� 0.05 m

Height �z-direction� 0.065 m

Length �x-direction� 0.05 m

Number of clumped particles �ncl� 5,764

Diameter of the particles composing the structure 2.0 mm

Density of a clump particle ��cl� 12,922.4 kg /m3

Fluid

Density 1,000 kg /m3

Viscosity 0.1 Pa s

Boundaries

Width �y-direction� 0.05 m

Depth �z-direction� 0.097 m

Length �x-direction� 0.55 m

Computation parameters

Time step for DEM 4�10−6 s

Time step for fluid 2�10−5 s

Number of fluid cells �x ,y ,z� 44�3�8

Applied g level 100

Fig. 5. Initial distribution of normal stresses along depth of deposit


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is the same as that of the prototype �see Eq. �9�� in accordancewith centrifuge scaling laws. Fig. 3 shows the setup that wasanalyzed in the simulation along with the employed three-dimensional fluid mesh.

Modeling Hydraulic Structure

The hydraulic structure was generated using clumped sphericalparticles which behave as a rigid body. That is, regardless of theforces acting upon it, the structure will remain intact. The contactforces between the clump particles are not computed during thesimulation. However, the interactions between the clumped par-ticles �the structure� and the soil particles are accounted for. Thespherical clumped particles had a radius of 1 mm and were as-sembled to resemble near-realistic conditions between the hydrau-lic structure and the soil particles. The structure is composed of abase and four lateral walls. The base of the hydraulic structurewas composed of clumped particles spaced at one particle radiuscenter-to-center in both the x and y directions to provide a roughinterface surface between the structure and the deposit particles.The lateral walls were made of particles arranged at one particlediameter center-to-center in both the lateral and vertical direc-tions. The stress applied by the structure on the underlying soilrepresented a structure that is 6.5 m high �in z direction�, 5 mwide �in x direction�, and has a density of 2,000 kg /m3. Only a5 m strip was considered in the normal direction to analyze thepresumably infinite structure. Since soil-structure interaction wastaken into consideration, any deformation of the solid phase mayresult in displacement and/or rotation of the structure.

The forces that act on the structure were calculated analyti-cally and were taken into account during the simulation �Fig. 6�.In this figure, L ,H ,B represent, respectively, the length, height,and width of the structure and Hw is the height of the water level.W is the self-weight of the structure, p is the water pressure forcorresponding Hw, U is the uplift force, Fx is the horizontal forcedue to water pressure, Fr is the frictional force between the hy-draulic structure and the soil particles, and M is the momentcreated on the centroid of the structure as a result of these forces.During the course of the simulation, forces acting on the structure�Fx ,W, and U� were calculated as the water level increased andcarried to the centroid of the structure along with their turningmoment. This centroid lies at a height of 1.96 m from the base of

Fig. 6. Free body diagram of forces acting on hydraulic structure

the structure. The frictional force Fr as well as the soil reaction



were computed automatically by PFC3D as a result of the contactforces of deposit particles in contact with the base of thestructure.

Note that there is partial coupling between the fluid and thestructure in the sense that changes in the fluid phase affect thestructure response. On the other hand, the structure movementaffects the fluid response only through the deformation of theunderlying particles and changes in porosity in the fluid cellsdirectly underneath the structure. Since the clump in PFC3D be-haves as a rigid body, translational and rotational motion equa-tions are sufficient to describe its motion. The translationalmotion of the center of mass is described in terms of its position,velocity, and acceleration. The rotational motion of the clump isdescribed in terms of its angular velocity and acceleration �Itasca2005�.

After generating the particles composing the structure, the de-posit was allowed to come to equilibrium. An impermeableboundary condition �zero pressure gradient across the boundary�was imposed at the base as well as at all four lateral walls for thesolution of the fluid equations. An impermeable boundary condi-tion was also imposed at the interface between the hydraulicstructure and the deposit. The lateral walls as well as the basewall had contact properties equal to that of the deposit particles�Table 1� for the solution of the particles equations of motion. Theinitial conditions were chosen to correspond to a practical sce-nario where the initial head difference is approximately 1.0 m�Fig. 7�. The fluid flow was then allowed to reach steady stateunder these conditions. The spatial variation in porosity corre-sponding to this initial stage is shown in Fig. 8. The rapid increasein the upstream water level was simulated by defining prescribedpressure on the upstream cells while maintaining a value of zeroat the downstream. The deposit was then subjected to the headincrease with the specified rate �100 Pa /s�. This rate of increasein water level in the upstream is extremely high compared to realflooding scenarios and is mainly due to computational timelimitations.

The total run time to conduct the simulation from the particlegeneration process until the end of the simulation using a Xeon

Fig. 7. Initial and boundary conditions in simulation

Fig. 8. Snapshot of initial porosity of deposit at initial conditions�t=0.0 s�


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dual-core processor with a speed of 3.00 GHz is about 10 days.Multiple solid and pore-fluid state variables were monitored dur-ing the course of the simulation. Table 1 summarizes the compu-tational data for the solid and fluid phases as well as othercomputational details. Two separate sets of output are retrievedfrom the simulation. The first set provides the information asso-ciated with the averaged Navier–Stokes equations �porosity, pres-sure, and average fluid and particle velocities� and is provided atthe center of the fluid cell �see Fig. 3�. The porosity for a particu-lar fluid cell is calculated from the volume of particles that theircenters lie within this cell. Note that this is a simplified compu-tation of the porosity that avoids the calculation of the volume ofthe complex geometric shapes of particles intersected by celledges and corners. The second set of output is that computed fromthe particulate phase, such as the stress tensor, and is computedfrom averaging over a spherical control volume �measurementsphere� as illustrated in Fig. 9. The results presented in the fol-lowing sections are in prototype units exclusively.

Fluid Flow Characteristics

The total head contours at the initial state, under steady statecondition, were compared with a widely used finite-elementmethod software in geotechnical engineering �SEEP/W, Geo-Slope Int. Ltd. 2004�, and the results obtained were in goodagreement �Fig. 10�. The progressive increase in the uplift pres-

Fig. 9. Spherical control volumes used to evaluate stresses in vicinityof hydraulic structure

Fig. 10. Total head contours from two-dimensional finite elementanalysis �top� using SEEP/W compared with simulation results �bot-tom� at initial conditions �H=1.0 m�



sure underneath the hydraulic structure is shown in Fig. 11 atselected instants of head increase.

Investigation of the evolution of water velocity as the up-stream water level kept rising indicates progressive increase in theamplitude of water velocity vectors �Fig. 12�. High water veloci-ties were observed in the zone surrounding the structure. Thehighest velocity was always next to the toe of the structure in thedownstream side and next to the heel of the structure in the up-stream side �Fig. 13�. The amplitude of the water velocity de-

Fig. 11. Progressive increase in head difference with time in thevicinity of the hydraulic structure

Fig. 12. Average seepage velocity vectors at selected head differenceinstants �maximum amplitude=0.9 m /s�


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creases significantly as the distance from the structure increases inboth directions. As shown below, the structure experienced sig-nificant settlement and tilting causing failure �Fig. 14�. The am-plitude of seepage velocity decreased significantly in the upperlayers of the deposit directly underneath the downstream side andthe toe of the structure as a result of this movement as can be seenfrom the seepage velocity at a head difference of 4.1 m �Figs. 12and 13�.

Hydraulic Structure Response

Due to the soil-fluid-structure interactions, any deformation of thesolid phase caused the structure to experience displacement androtation �Fig. 14�. Soil particles in this simulation were not onlyunder the effect of their own weight and fluid flow, but also the

Fig. 13. Average seepage velocities versus distance at 0.61 m belowground level

Fig. 14. Snapshots of particles and hydraulic structure at selectedtime head difference instants



stresses induced by the weight of the hydraulic structure. The firstsignificant settlement, which may be considered as the failure ofthe structure, was observed at about 1.6 s �Fig. 15� correspondingto a water height of 2.6 m. Complete failure occurred just after3 s at a water level of about 4.1 m. These findings show the factthat failure can happen at hydraulic gradients that are significantlylower than the critical value as discussed below.

Solid Phase Response

The most commonly used criterion in geotechnical engineering toassess a soil deposit susceptibility to piping is based on evaluatingthe safety factor against piping defined as the critical gradientdivided by the hydraulic gradient. The critical gradient estimatedfrom Terzaghi’s classical formula �Terzaghi and Peck 1967� isgiven by

ic =��s − �w�

�w�1 − n� �15�

where ic�critical gradient; �s�specific weight of soil particles;and �w�specific weight of water. Thus, the critical gradient asdefined by Eq. �15� is a macroscale material property that dependsmainly on porosity. In the present simulation, the critical locationfor piping to take place is at the toe of the structure on the down-stream side. In order to assess the factor of safety against pipingin that location, the changes in porosity and the correspondingcritical gradient as well as the hydraulic gradient were producedas the head difference increased �Fig. 16�. Several discontinuitiesare observed in both plots. These discontinuities follow the sud-den changes in porosity that resulted mainly from the way theporosity was calculated in the computational model. As statedearlier, the porosity within a cell is calculated from particles thattheir centers fall within the cell. This will result in a jump inporosity as a particle center becomes outside the cell and its vol-ume is not included in the porosity calculation.

The initial value of the critical gradient at the toe of the struc-ture was about 0.919. As indicated by Fig. 16a, the soil at the toeof the structure on the downstream side experienced a volumeincrease particularly around the time of first significant settlementof the structure �H=2.6 m�. During the course of the simulation,

Fig. 15. Settlement of hydraulic structure in vertical direction withincrease in head

the smallest value of the safety factor was about 1.9 �at H


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=4.0 m�. At the time of significant settlement of the structure�H=2.6 m� the factor of safety was about 2.5, implying that thesoil near the toe of the structure is safe against piping. It may thenbe concluded that the settlement of structure, and its subsequentfailure, cannot be solely attributed to the formation of piping.

The solid phase deformation is a result of the stresses inducedby the hydraulic structure as well as the flowing water. Thestresses due to the structure are not constant with time as thestructure is subjected to increasing hydrostatic and uplift pres-sures. While the increase in uplift pressure would tend to reducethe structure-induced stresses within the soil, the turning momentsdue to the hydrostatic and uplift pressures tend to overturn thestructure around its toe causing stresses to increase at that loca-tion and the particles directly under the toe of the structure tomove downward. On the other hand, water flow applies mostlyupward forces at the exit face near the toe of the structure on thedownstream side producing a tendency to lift the particles at theexit face. This combined effect of downward pressure from thestructure and upward drag forces from water flow at the toe

Fig. 16. Computation of safety factor against piping at toe of struc-ture on downstream side with increase in head difference: �a� changesin porosity and critical gradient; �b� hydraulic gradient and safetyfactor against piping

causes the particles next to the toe on the downstream side to



move upward in a failure mode similar to that of a bearing capac-ity failure mechanism �Fig. 17�. In this figure, it can be seen thatthere were two instants of failure on two different failure surfaces.The first instant occurred as the upstream water level approacheda height of 2.6 m. The structure tilted and settled but remained inthis new position until the water level reached a height of 4.0 m.Upon reaching this height, another failure plane can be observedand the structure experienced excessive titling and displacement.

The above argument is supported by investigating the evolu-tion of stresses in the vicinity of the structure as the hydraulichead rapidly increases in the upstream. The six components of thestress tensor were monitored within the spherical control volumesshown in Fig. 9. The normal and shear stresses are shown in Figs.18 and 19, respectively, along with plots of changes in their am-plitude relative to initial conditions. The normal stresses representthe effective stresses in the three orthogonal directions x, y, and z.The vertical effective stress distribution under the structure showsthe familiar bell shaped trend due to high stresses under the hy-draulic structure �Fig. 18�. The changes in the vertical effectivestress �zz� were consistent with the forces acting on the structure.There is a general decrease in the stress values due to the effect ofincreasing uplift force with increasing water level and are pro-nounced under the heel of the structure. On the other hand, thereis an increase in the amplitude of vertical effective stress near thetoe of the structure due to the effect of the overturning momentinduced by the lateral and uplift water pressures around the toe ofthe structure. The shear stress values in the xy and yz directionsdid not change significantly, since the water flow in the y direc-tion is negligible, compared to the shear stress in the xz direction�Fig. 19�.

The normal and shear stresses described above were used toplot the effective stress Mohr’s circles at different instants ofwater level increase near the toe of the structure �Fig. 20� for thestresses in the xz plane �the longitudinal middle strip of the de-posit�. In all of these plots, starting from the initial conditions, thecircles moved closer to the Mohr–Coulomb failure envelope astime passes. The failure envelope corresponds to an angle of in-ternal shearing resistance equal to 27.4°, based on the results ofdirect shear test simulations �Appendix�. While the failure modeunder the structure may not be adequately represented by a directshear test failure mode, the simulated test gives an indication of

Fig. 17. Snapshots of accumulated soil particle and structure dis-placement vectors at selected head increase instants �maximumamplitude=1.0 m�

the shear strength of the synthetic soil employed.


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Fig. 18. Effective stresses ��ii� , i=x ,y ,z� and changes in effective stresses ��ii� , i=x ,y ,z� versus distance at 1.25 m below ground level

Fig. 19. Shear stresses ��ij , i , j=x ,y ,z� and changes in shear stresses ��ij , i , j=x ,y ,z� versus distance at 1.25 m below ground level

1394 / JOURNAL OF GEOTECHNICAL AND GEOENVIRONMENTAL ENGINEERING © ASCE / SEPTEMBER 2008

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Mohr’s circles are drawn only until a head difference of 2.5 mjust before the instant of first significant settlement of the struc-ture �H=2.6 m�. Initially, the stress state at zones located adjacentto the toe on the downstream side and just under the toe of thestructure �locations b and c, respectively� was relatively close tothe failure envelope. The stress state at a zone located adjacent tothe toe on the structure side �location a� is far from the failureenvelope because of the high confining stress. These plots showedthat just after water reached a height of 2.5 m, the particles lo-cated under the corner and adjacent to the toe failed and werefollowed by the structural failure. After this first failure instant,the new position of the structure and the particles piled adjacentto its toe and produced higher vertical effective stresses at thecenters of the three measurement spheres as the structure becamecloser to their centers �see Fig. 18�. That caused Mohr’s circlesdrawn for heads more than 2.6 m to move away from the failureenvelope. As noted earlier, the second instant of significant defor-mation resulted from particle movement along a different failuresurface �see Fig. 17� that was not captured by the monitoringmeasurement spheres.

It is worth investigating the changes in the coefficient of per-meability with time as an important property of the porous me-dium. Spatial and temporal changes in permeability coefficientwere monitored during the course of the simulation �Fig. 21�. Theequivalent permeability coefficient �Eq. �9�� was used to calculatethe K values. Three different K values are plotted �at the upstreamside, under the structure, and at the downstream side� to assess the

Fig. 20. State of stress underneath structure as given by three mea-surement spheres shown in Fig. 17: �a� adjacent to toe on structureside; �b� adjacent to toe on downstream side; and �c� under toe

effect of velocity terms on the permeability. As can be seen from



Fig. 21, including the velocity terms resulted in about 25% de-crease in the amplitude of the permeability coefficient. This is dueto the relatively large particles used in this study which resulted ina flow that does not obey Darcy’s law as the value of particleReynolds number �Rp� in the vicinity of the structure ranged be-tween 1 and 66, indicating that the flow was within Forchheimerregime as discussed before. During piping, permeability increasedas expected as a result of increased porosity. At the upstream sideof the structure, K increased in a stepwise fashion while the trendunder the structure was quite smooth. At the downstream sidepermeability changes were more complex where sand boils wereobserved. In a related study, fluid with 100 times the viscosityreported for the current simulation was used. The obtained fluidvelocities were much smaller and including the velocity term inEq. �9� did not alter the amplitude of the coefficient of permeabil-ity. However, the overall response of the system was similar to thepresent simulation �Aydin 2007�.

Discussion

The previous sections illustrated the capabilities of the modelemployed to obtain the response of the hydraulic structure torapidly increasing upstream water level. Due to computationallimitations, simplifications were introduced and their impact isdiscussed in this section. The particle diameters used in the simu-lations are large compared to real soils. If realistic values of soilparticle diameters were used, the total number of particles wouldbe much higher. This would lead to a significant increase in com-putational time and conducting the simulation would be com-pletely impractical. Accordingly, the fluid cell dimensions werealso large as their size depends on the size of the particles �inorder to obtain reasonable averaged quantities�. A sensitivityanalysis shows that a size of about 2.5 times the mean diameter issufficient �El Shamy 2004�. Another drawback of using large par-ticle diameters is the sensitivity of the computed cell porosity tothe volume of one particle leaving or entering the cell. As men-tioned earlier, porosity is calculated from the volume of particleswith centers inside the cell. This computational scheme led todiscontinues in the time histories of porosity and porosity-based

Fig. 21. Changes in coefficient of permeability with head increase

variables as can be seen in Fig. 16.


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The effects of the high g level employed are not fully under-stood yet. Utilizing high gravitational field reduced the resolutionof the results. The computed fluid flow state variables �pressureand velocities� were obtained at the center of the first row of thecells near the surface which after employing the scaling lawswould be about 0.61 m below the surface. The minimum radiusfor the measurement spheres, which is again controlled by theparticle sizes that could be used to compute the stresses, wasabout 1.25 m. Therefore, the closest depth to obtain stresses was1.25 m below the surface as a result of the high g level employed.Such low resolution of stress measurement resulted in inability tocapture the precise shape of the failure surface based on soilstrength.

Conclusions

This paper examines the potential of a three-dimensional fullycoupled fluid-particle model to simulate large soil deformations

Fig. 22. Results of direct shear test on synthetic sample representing a�c� shear stresses versus horizontal displacement; �d� volume change



resulting from extreme flow conditions that lead to the occurrenceof piping and failure of the hydraulic structure. The approachaccounts for transient flow conditions, spatial and time variationsin porosity, and subsequent changes in the permeability of thesoil. The fluid flow patterns agree with solutions using more com-mon finite-element-based formulations. The mesh-free nature ofDEM allows particle movements to be tracked as they respond tothe seepage forces. The simulation captured a failure mechanismof the hydraulic structure due to the rapid rise of upstream waterlevel similar to a bearing capacity failure. The combined action ofstresses induced by the structure and water flow may lead to asudden failure at hydraulic gradients less than the critical gradi-ent. This approach appears to be a very effective tool to modelsaturated granular deposits and geotechnical systems when sub-jected to high seepage forces such as those encountered duringflooding of a river. With the current rate of advancement in com-putational power and the development of codes that utilize paral-lel computing, implementation of this method would become

ed deposit: �a� initial �undeformed� sample; �b� sample after shearing;horizontal displacement; and �e� Mohr–Coulomb failure envelope

nalyzversus


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more computationally attractive and would facilitate modeling ofsmaller particle sizes and finer fluid mesh. Such implementationwould provide a higher resolution of particle movement and theintricate pore-fluid flow patterns.

Acknowledgments

This research was supported by the Louisiana Board of RegentsSupport Fund, Grant No. LEQSF�2005-07�-RD-A-32. This sup-port is gratefully acknowledged. The writers would also like tothank anonymous reviewers for helpful suggestions and valuableremarks that helped improve the manuscript for this paper.

Appendix. Direct Shear Test Simulations

Direct shear test numerical simulations were conducted usinggranular soil samples that represent the analyzed deposit. Thesamples were sheared between two cylinders having a diameter of75 mm. The height of the sample inside the cylinders after theconsolidation phase was 44 mm �Fig. 22�. The particles werepacked at a porosity of 39%, similar to that of the average poros-ity of the analyzed deposit. The samples were tested under threedifferent normal stresses �25, 50, and 100 kPa� and the stresstensor within the soil sample was monitored during the course ofthe simulation. During a specific test, the normal stress ��N� wasmaintained constant during shearing through a servo mechanismsuch that the velocity of the top wall is constantly adjusted tomaintain the desired stress �Itasca 2005�. As shown in Fig. 22, theemployed simulation parameters �Table 1� resulted in responsepatterns similar to that of granular soils. Interpretation of the testresults suggests that the angle of internal friction of the deposit isabout 27.4°.

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