Powder Technology 239 (2013) 248 258. Contents lists available at SciVerse ScienceDirect. Powder Technology

Powder Technology 239 (2013) 248 258 Contents lists vilble t SciVerse ScienceDirect Powder Technology journl homepge: www.elsevier.com/locte/powtec Coupled CFD DEM simultion of fluid prticle interction in geomechnics Jidong Zho, Tong Shn Deprtment of Civil nd Environmentl Engineering, The Hong Kong University of Science nd Technology, Cler Wter By, Kowloon, Hong Kong rticle info bstrct Article history: Received 12 June 2012 Received in revised form 7 November 2012 Accepted 2 Februry 2013 Avilble online 8 Februry 2013 Keywords: Grnulr medi Fluid prticle interction Coupled CFD DEM Single prticle settling 1D consolidtion Sndpile This pper presents coupled Computtionl Fluid Dynmics nd Discrete Element Method (CFD DEM) pproch to simulte the behviour of fluid prticle interction for pplictions relevnt to mining nd geotechnicl engineering. DEM is employed to model the grnulr prticle system, whilst the CFD is used to simulte the fluid flow by solving the loclly verged Nvier Stokes eqution. The prticle fluid interction is considered by exchnging such interction forces s drg force nd buoyncy force between the DEM nd the CFD computtions. The coupled CFD DEM tool is first benchmrked by two clssic geomechnics problems where nlyticl solutions re vilble, nd is then employed to investigte the chrcteristics of snd hep formed in wter through hopper flow. The influence of fluid prticle interction on the behviour of grnulr medi is well cptured in ll the simulted problems. It is shown in prticulr tht snd pile formed in wter is more homogeneous in terms of void rtio, contct force nd fbric nisotropy. The centrl pressure dip of verticl stress profile t the bse of sndpile is modertely reduced, s compred to the dry cse. The effects of rolling resistnce nd polydispersity in conjunction with the presence of wter on the formtion of sndpile re lso discussed. 2013 Elsevier B.V. All rights reserved. 1. Introduction Fluid prticle interction underpins the performnce of wide rnge of key engineering pplictions relevnt to grnulr medi. Subjected to externl lods, the pore fluid in sturted grnulr mteril my fluctute or flow nd cuse prticle motion. This my work fvourbly in some cses, such s in snd production in sndstone oil reservoir, but my be n dverse fctor in other occsions, such s in the cse of internl/surfce erosion of embnkment dms nd soil slopes which my trigger instbility nd filure of these structures [21]. Conventionl pproches bsed on continuum theories of porous medi, such s the Biot theory, hve considered the interction between pore fluids nd prticles in phenomenologicl mnner. They cnnot offer microscopic informtion t the prticle level relevnt to the fluid prticle interction which my be otherwise useful in mny occsions. Indeed, s mentioned in recent review by Zhu et l. [38], quntittive understnding of the microscle phenomen reltive to fluid prticle interction could fcilitte the estblishment of generl methods for relible scle-up, design nd control of different prticulte systems nd processes. To this end, number of collective ttempts hve been mde on prticle-scle modelling of fluid prticle interction, mong which Discrete Element Corresponding uthor. Fx: +852 2358 1534. E-mil ddress: jzho@ust.hk (J. Zho). Method (DEM) plys centrl role. In prticulr, numericl pproches combining the Computtionl Fluid Dynmics nd Discrete Element Method (CFD DEM) prove to be dvntgeous over mny other options, such s the Lttice Boltzmn nd DEM coupling (LB DEM) method nd the Direct Numericl Simultion coupled DEM (DNS DEM), in terms of computtionl efficiency nd numericl convenience [37]. A typicl CFD DEM method solves the Newton's equtions governing the motion of the prticle system by DEM nd the Drcy's lw or the Nvier Stokes eqution for the fluid flow by CFD, in considertion of proper interction force exchnges between the DEM nd the CFD (see [21,27,32,33,37]). The method hs been successfully pplied to the simultion of pplictions such s fluidiztion, pneumtic conveying nd pipeline flow, blst furnce, cyclone, nd film coting (see the review by [38]). Relevnt to civil nd geotechnicl engineering, the importnt impct of fluid prticle interction on the overll behviour of soils hs long been recognized. More recently, there hs been growing interest in exploring the soil behviour using discrete modelling pproches, in sought for key mechnisms nd mitigting mesures for vrious geotechnicl hzrds (see [21] for summry). Whilst the mjority of these studies were focused on the dry soil cse bsed on DEM only, there hve been limited investigtions considering the fluid prticle interction through coupled discrete pproches s mentioned before. A hndful of exceptions include the tretment of upwrd seepge flow in soils, sinkhole process, flow under sheet pile wlls [6,7,25]. The current pper ims to develop coupled CFD DEM numericl tool to investigte vrious geomechnics problems 0032-5910/$ see front mtter 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.powtec.2013.02.003

J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 249 relevnt to mining nd geotechnicl engineering. In prticulr, two open-source DEM nd CFD pckges re employed to fcilitte the coupling between fluids nd prticles, nmely, the LAMMPS-bsed DEM code, LIGGGHTS [16], nd the OpenFOAM (www.openfom. com). The computtionl frmework hs been bsed on the CFDEM progrm developed by Goniv et l. [13], by further considering both phses of gs nd wter in the fluid simultion by OpenFOAM. The fluid prticle coupling is considered by exchnging interction forces between the two pckges during the computtion. The interction forces being considered include the drg force nd buoyncy force, which my generlly suffice for grnulr mterils in geomechnics with reltively low Reynolds number of pore flow. Such complex interction forces s unstedy forces like virtul mss force, Bsset force nd lift forces nd non-contct forces such s cpillry force, Vn der Wls force nd electrosttic force, my be importnt for certin pplictions, but will not be considered here. It is however emphsized tht the computtionl frmework is generl nd cn esily ccommodte the considertion of these forces if necessry in the future. Three problems will be employed to demonstrte the predictive cpcity of the numericl tool. They include the single-prticle settling in wter which simultes typicl sedimenttion process, the onedimensionl consolidtion nd the formtion of conicl snd pile through hopper into wter. The first two exmples re chosen due not only to their simplicity but lso the vilbility of nlyticl solutions for both, nd consequently, they serve s benchmrks for the developed CFD DEM pckge. The snd pile formtion problem hs received much ttention in wide rnge of brnches of engineering nd science. Of prticulr interest is the phenomenon of pressure dip in sndpile observed in experiments. Vrious nlyticl pproches nd numericl studies hve been devoted to the explntion of this phenomenon, such s the fixed principl xes model [31], the rching theory bsed on limit nlysis by Michlowski nd Prk [20], s well s DEM simultions[12,17]. The occurrence of pressure dip in sndpile hs been found dependent on the construction method, prticle shpe nd other fctors [1,39].Despite the intensive studies on this topic, no widely ccepted consensus hs been reched regrding the mjor mechnism for the observed pressure dip. In prticulr, very scrce studies hve been found exploring the effect of wter on the formtion of sndpile nd on the chrcteristics of the pressure dip. Relevnt studies in this respect my hve fr wider engineering bckground closely relted to such issues s dredging nd lnd reclmtion, mining production hndling, soil erosion nd debris flow wherein the interction between soil nd wter proves to be importnt. The CFD DEM tool developed in this pper will be employed to investigte the chrcteristics of sndpile formed through hopper flow in wter, nd creful comprison will be mde ginst the dry cse. 2. Methodology nd formultion Key to the coupling between the Computtionl Fluid Dynmics method nd Discrete Element Method (CFD DEM) is proper considertion of prticle fluid interction forces. Typicl prticle fluid interction forces considered in pst studies include the buoyncy force, pressure grdient force, drg force due to the prticle motion resistnce by stgnnt fluid, s well s other unstedy forces such s virtul mss force, Bsset force nd lift forces (see, [37]). Following the pproch proposed by Tsuji et l. [27,28], we ssume tht the motion of prticles in the DEM is governed by the Newton's lws of motion nd the pore fluid is continuous which cn be described by loclly verged Nvier Stokes eqution to be solved by the CFD [3]. The interctions between the fluid nd the prticles re modelled by exchnge of drg force nd buoyncy force only. Detiled formlisms governing the three spects nd numericl solution procedures re described s follows. 2.1. Governing equtions for the pore fluid nd prticle system For prticle i treted by the DEM [9], the following equtions re ssumed to govern its trnsltionl nd rottionl motions 8 >< m i du p i dt ¼ Xn c i dω I i i dt ¼ Xn c i >: j¼1 M ij j¼1 F c ij þ F f i þ F g i where U p i nd ω i denote the trnsltionl nd ngulr velocities of prticle i, respectively. F c ij nd M ij re the contct force nd torque cting on prticle i by prticle j or the wll(s), nd n c i is the number of totl contcts for prticle i. F f i is the prticle fluid interction force cting on prticle i, which includes both buoyncy force nd drg force in the current cse. F g i is the grvittionl force. m i nd I i re the mss nd moment of inerti of prticle i. In the DEM code, either the Hooke or Hertzin contct lw is employed in conjunction with Coulomb's friction lw to describe the interprticle contct behviour. In the CFD method, the continuous fluid domin is discretized into cells. In ech cell vribles such s fluid velocity, pressure nd density re loclly verged quntities. In prticulr, specific cell cn be occupied by immiscible liquid nd ir, nd the density of cell is the weighted verge of the two phses (excluding the volume of prticles if they re present in cell). The following continuity eqution is ssumed to hold for ech cell: ðερþ þ ερu f ¼ 0 t where U f is the verge velocity of fluid cell. ε=v void /v c =1 v p /v c denoting the porosity (void frction) (v void is the totl volume of void in cell which my contin either ir or wter or both; v p is the volume occupied the prticle(s) in cell; v c is the totl volume of cell). ρ is the verged fluid density defined by: ρ=αρ w +(1 α)ρ,whereα=v w / v c =1 v /v c. α is defined in the CFD simultion by the nominl volume frction of wter phse in cell, where v w is the nominl wter phse volume in the cell nd v the nominl ir phse volume, nd v w +v =v c. Evidently, the totl void volume in cell cn be written s v void =ε(v +v w ). If α =1, the void of cell will be fully occupied by wter, nd if α =0,thevoidisfullofir.Thecseof0bαb1 normlly refers to cell with void filled by both ir nd wter. This definition of verge fluid density in conjunction with the porosity ε leds to netly expressed continuity eqution in Eq. (2), nd hs been widely followed. In ddition, s will be shown, this definition offers convenient wy to simulte the trnsition process of prticles pssing between the interfce between (pure) ir phse nd wter phse. The CFD method solves the following loclly verged Nvier Stokes eqution in conjunction with the continuity eqution in Eq. (2) ερu f t þ ερu f U f ε μ U f ¼ p f p þ ερg where p is the fluid pressure in the cell; μ is the verged viscosity; f p is the interction force verged by the cell volume the prticles inside the cell exert on the fluid. g is the grvittionl ccelertion. 2.2. Fluid prticle interction forces The motion of submerged prticles cn be significntly influenced by the fluid through either hydrosttic or hydrodynmic forces. Buoyncy force is typicl hydrosttic force, whilst hydrodynmic forces my include drg force, the virtul mss force nd the lift force, ð1þ ð2þ ð3þ

250 J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 mong others [21,37]. In this study, we consider the drg force F d nd the buoyncy force F b s the dominnt interction forces. Specificlly, the expression of drg force used by Di Felice [11] is employed: F d ¼ 1 8 C dρπd 2 p U f U p U f U p ε 1 χ where d p is the dimeter of the considered prticle. C d is the prticle fluid drg coefficient which depends on the Reynolds number of the prticle, Re p 0 1 B C d ¼ 0:63 þ 4:8 C @ qffiffiffiffiffiffiffiffia Re p 2 in which the prticle Reynolds number is determined by: ερd p U f U p Re p ¼ μ : ð6þ ε χ in Eq. (4) denotes corrective function to ccount for the presence of other prticles in the system on the drg force of the prticle under considertion, wherein χ hs the following expression 2 2 3 6 1:5 log 10 Re p 7 χ ¼ 3:7 0:65 exp4 5: ð7þ 2 As indicted by Kfui et l. [15], the Di Felice expression leds to smooth vrition in the drg force s function of porosity. The expressions in Eqs. (4) nd (5) work well for our pplictions with reltively low Reynolds numbers. Regrding the hydrosttic force, we employ the following verge density bsed expression for the buoyncy force (c.f. [15,21]): F b ¼ 1 6 πρd3 p g: 2.3. Numericl solution schemes for coupled CFD DEM computtion In the coupled CFD DEM scheme, the fluid phse is discretized with typicl cell size severl times of the verge prticle dimeter. At ech time step, the DEM pckge provides such informtion s the position nd velocity of ech individul prticle. The positions of ll prticles re then mtched with the fluid cells to clculte relevnt informtion of ech cell such s the porosity. By following the corse-grid pproximtion method proposed by Tsuji et l. [27], the loclly-verged Nvier Stokes eqution in Eq. (3) is solved by the CFD progrm for the verged velocity nd pressure for ech cell. The obtined verged vlues for the velocity nd pressure of cell re then used to determine the drg force nd buoyncy force cting on the prticles in tht cell. Itertive schemes my hve to be invoked to ensure the convergence of relevnt quntities such s the fluid velocity nd pressure. When converged solution is obtined, the informtion of fluid prticle interction forces will be pssed to the DEM for the next step clcultion. LIGGGHTS hs been dopted s the DEM pckge nd the finite-volume-method bsed OpenFOAM code is employed s the CFD solver. A customized OpenFOAM librry, CFDEM, developed by Goniv et l. [13], hs been modified to wrp the OpenFOAM fluid solver into the LIGGGHTS solution procedure to solve the coupled problem. The InterDyMFom solver is modified in the OpenFOAM to solve the loclly verged Nvier Stokes eqution. ð4þ ð5þ ð8þ Idelly, informtion on interction forces should be exchnged immeditely fter ech step of clcultion for the DEM or the CFD. This, however, my request excessive computtionl effort in prctice. For the problems to be treted in this pper, numericl experience shows tht for ech CFD computing step, exchnging informtion fter 100 steps of DEM clcultion will ensure sufficient ccurcy nd efficiency. If the time steps for DEM nd CFD re sufficiently smll, more steps for DEM re lso cceptble. 2.4. Two pproches clculting the void frction of fluid cell The CFD DEM method employed here generlly considers fluid cell with size severl times of the men prticle dimeter. It is interesting to compre two different methods in clculting the void frction for fluid cell which re shown in Fig. 1 in demonstrtive 2D view (our code is 3D). Fig. 1 illustrtes the centre void frction method. In this method, if the centre of prticle i is found locted in fluid cell j, the totl volume of the prticle will be counted into the clcultion of the void frction for tht cell. For exmple, Prticles A, B, C nd D in Fig. 1 will ll be counted into the clcultion of void frction for Cell 2. Whilst for the cse of Prticle E, it cn be considered either entirely to Cell 2 or Cell 4, but not both. Apprently, this pproch will overestimte the void frction for some cells whilst underestimting it for others in the neighbourhood. An improved method is shown in Fig. 1b, where the exct volume frction of prticle i in fluid cell j, ϖ ij, is ccurtely determined (ϖ ij =v ij p /v i p, where v i p is the totl volume of prticle i nd v ij p is the exct portion of volume of prticle i in cell j). Evidently, ϖ ij [0,1]. When prticle is entirely locted in Cell j (such s the cse of Prticles B nd C with respect to Cell 2), ϖ ij =1; when it is totlly outside tht cell, ϖ ij =0. Otherwise, its vlue is in between 0 nd 1. ϖ ij is then used to clculte the void frction of the concerned cells. The ltter pproch is termed s the divided void frction method. Evidently, the first pproch cn be regrded s specil cse of the second, with ϖ ij either equl to 1 or 0, depending on its centroid loction with respect to the cell. The two pproches ffect how the fluid prticle interction forces re clculted. In clculting the interction force pplied to DEM prticle i, simplified centre-position pproch (similr to the centre void frction method mentioned bove) hs been followed for ll cses. Specificlly, the verge fluid velocity U f in Eq. (4) nd the verge fluid density ρ in both Eqs. (4) nd (8) re chosen entirely ccording to the cell the prticle centre is locted in. As such, the totl interction force pplied to prticle i is F f i ¼ F d i þ F b i : Fig. 1. Schemtic of two different pproches to clculte the void frction for fluid cell. () The centre void frction method; (b) the divided void frction method. b ð9þ

J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 251 This expression is followed in both the centre nd the divided void frction methods. However, in clculting the interction forces for fluid cell j, the contributing weight of ech prticle relevnt to the cell hs been considered s follows p f p j ¼ Xn j ϖ ij i¼1 F d i þ F b i =v j c ð10þ where ϖ ij is the weight of volume frction of prticle i in cell j. n j p is the totl number of prticles relevnt to fluid cell j, nd v c j is the cell volume. For the divided void frction method ϖ ij cn be ccurtely determined, whilst for the centre void frction method we my simply set ϖ ij =1 for prticle whose centre is locted in cell j, nd ϖ ij =0 otherwise. velocity (m / s) Prticle velocity (m / s) Fluid 0.7 0.5 0.3 0.1 Air Cells Fluid Cells Anlyticl Centre Divided Trnsit Cells 3. Benchmrking exmples It is instructive to benchmrk the coupled CFD DEM progrm presented bove first. Two simple problems with nlyticl solutions vilble re chosen for this purpose. The first is the single sphericl prticle settling from ir into wter, nd the second is the clssicl one-dimensionl consolidtion problem in soil mechnics. 3.1. Single sphericl prticle settling from ir to wter Sedimenttion, or the settling of prticle(s) into wter, hs been problem of interest for hundred yers. Stokes [24] ws mong the erliest who hs ttempted to describe the sedimenttion of sphere nlyticlly. He hs found tht the settling velocity of sphere in fluid is directly proportionl to the squre of the prticle rdius, the grvittionl force nd the density difference between solid nd fluid nd is inversely proportionl to the fluid viscosity, s follows (see lso, [8]) u p ðþ¼ t 1 ρ p ρ f d 3 "!# pg 1 exp 1 μ f t 18 μ f 27 ρ p d 3 p ð11þ where u p (t) denotes the settling velocity of the sphericl prticle. d p is the dimeter of the prticle. g is the stndrd grvity. The term outside the brcket of the RHS of Eq. (11) is the so-clled terminl velocity. Notbly, the finding by Stokes [24] pplies to the slow prticle motion cse with low Reynolds numbers. In the benchmrking simultion of the problem by the CFD DEM method, sphericl prticle of d p =1mm is dropped from 45 mm high from the centre of continer (see inset of Fig. 3) with dimension L W H=20mm 10mm 50mm. The continer is divided into homogeneous mesh of 20 10 30. The cells re decomposed into three regions. The upper (20 10 14) cells re pure ir cells where α=0, nd the bottom (20 10 15) cells re pure wter cells where α=1. There is one lyer (20 10 1) of trnsitionl cells where α is specified s 0.5. The viscosity of wter nd ir used in the clcultion re specified to be: μ f =9.982 10 4 P s nd μ = 1.78 10 5 P s. The densities use the following vlues: ρ p =3 10 3 kg/m 3, ρ f =998.2kg/m 3, nd ρ =1.2kg/m 3. Hertzin contct lw is used nd the continer wll is ssumed to hve the sme contct prmeters s the prticle: Young's modulus E=5 10 6 P, Poisson's rtio ν=5, nd the coefficient of restitution ζ=0.3. The predictions re compred in Fig. 2 ginst the nlyticl solution. Also compred in the figure re the two methods on clculting the void frction of the fluid cell. It is evident from Fig. 3 tht the predicted settling velocities of the prticle by both methods gree well with the nlyticl solution. The numericl predictions cpture well the shrp reduction of velocity when the prticle hits the wter nd bounces bck when it hits the continer bottom. The settling process b -0.1 0.1 0.3 0.5 t (s) 0 Centre-round prticle Centre-interfce 0.15 Centre-bottom Divided-round prticle Divided-interfce 0.10 Divided-bottom 5 0-5 0.1 0.3 0.5 t (s) Fig. 2. Comprison of the CFD DEM prediction nd the nlyticl solution for singleprticle settling in wter with the centre nd divided void frction methods. () Prticle velocity (inset: the settling problem nd CFD mesh); (b) Fluid cell velocity t different loctions: round the prticle, the centre interfce (trnsition) cell nd the bottom centre cell. in the wter lso compres well with the nlyticl solution. Interestingly, the centre void frction method ppers to perform slightly better thn the divided method. This my hve been cused by the use of identicl cell size with the prticle dimeter. However, rther different scenrio is observed in the next exmple. Fig. 2b presents the fluid cell velocity t three loctions: round the prticle, t the centre of the trnsition zone nd t the bottom of the continer (ll long the centre line). As cn be seen, the velocity of the fluid cell round the prticle bers close correltion with the motion of the prticle. The prticle motion impcts the cell t the trnsitionl interfce only temporrily, nd its interction with the bottom cell is clerly observed before the prticle hits the bottom. 3.2. One-dimensionl consolidtion The proposed method hs lso been benchmrked by the clssicl one-dimensionl (1D) consolidtion problem in soil mechnics. A similr problem hs been discussed by Suzuki et l. nd Chen et l. [7,25]. According to the 1D consolidtion theory by Terzghi [26], the dissiption of excess pore pressure in one-wy drined soil

252 J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 Top prticle settlement (mm) b z (m) 0-2 -4-6 -8 lyer subjected to surfce surchrge cn be described by the following eqution: p t ¼ C 2 p v z 2 ð12þ where p denotes the excess pore pressure during the consolidtion, z is the verticl coordinte in the dringe direction, nd C v is the coefficient of consolidtion given by k p C v ¼ ρ w gm v where k p is the permebility, k p ¼ d2 ε 3 ρ w g 150μð1 ε ð13þ [30]. m Þ 2 v is the coefficient of volume chnge, m v =Δε v /Δσ v (Δε v nd Δσ v re the vritions of verticl strin nd verticl stress, respectively) which cn be determined from the mteril properties nd problem specifiction. In ddition, non-dimensionl time cn be defined to conveniently describe the normlized time process [41] T v ¼ C vt H 2 Anlyticl Centre Divided -0 0.5 1.5 2.0 T 0.10 8 6 4 2 Tv=1.6 Tv=0.8 Tv= Tv= Tv=5 Tv=0.175 Anlyticl Centre Divided Tv=0 0 p / p 0 0.8 Fig. 3. Benchmrking of CFD DEM simultions of the consolidtion settlement nd the dissiption of excess pore pressure with the clssic Terzghi's nlyticl solution to the 1D consolidtion problem. Centre: centre void rtio method; Divided: divided void rtio method. (For interprettion of the references to colour in this figure legend, the reder is referred to the web version of this rticle.) ð14þ where H is the height of the soil lyer (its initil vlue being H 0 ). The initil nd boundry conditions for the one-wy dringe problem re: pz; ð 0 Þ ¼ p 0 ; pð0; tþ ¼ 0; p z ¼ 0: z¼h ð15þ The nlyticl solution to Eqs. (12) (15) for the excess pore wter pressure during the consolidtion process is (see [10]) p ¼ Xn¼ n¼1 2p 0 nπ! ð nπz 1 cosnπþsin H exp n2 π 2 T v 4 ð16þ where n denotes n integer number. In simulting the one-dimensionl consolidtion problem, we consider soil column comprised of 100 equl rdius spheres (r=0.5mm) which re supposed to be sturted in wter. The dimension of the column is 1 mm wide nd 100 mm high, the sme s tht treted by Suzuki et l. nd Chen et l. [7,25]. The column is discretized into fluid cells of 2 mm high ech. Hooke contct lw is dopted for the DEM computtion, nd the vlues of relevnt prmeters re dopted s the sme in Suzuki et l. [25] (ρ p =2650kg/m 3,contctstiffnessk n =100N/m, ρ f =998.2kg/m 3, fluid viscosity μ f =0.9982 10 3 P-s, Grvity constnt g=9.81m/s 2 ). All prticles re initilly plced t the centre line of the column without ny overlp nd re emerged in wter. The grvittionl force nd buoyncy force re then switched on to llow the prticles to settle to hydrosttic stte (see lso [7]). Once the initil consolidtion is finished, surchrge lod p 0 =100P is then pplied t the top of the column. The simulted settlement of the top prticle nd the dissiption of excess pore wter with time re compred in Fig. 3 ginst the nlyticl solutions. The performnces of the two void frction clcultion methods re lso compred. As shown in Fig. 3, the predicted settlements of the top prticle by both methods compre well with the nlyticl solution, except t T v =0. Whilst the nlyticl solution ssumes n instntneous buildup of excess pore pressure throughout the column once the surchrge is pplied, the CFD DEM clcultion needs certin time to build up the whole excess pore wter. The numericl nd nlyticl solutions hence re not totlly comprble t the instnt of T v =0. Following similr strtegy s suggested by Chen et l. [7], we hve shifted the time mesure of the numericl computtion to certin smll time to mtch the initil excess pore wter pressure field for the nlyticl cse, from which instnt of time the two solutions re then compred. Nevertheless, it is suggested tht the predicted quntities, including both the settlement nd the excess pore wter pressure, t the erly stge of the consolidtion remin less relible due to the sme reson. This explins the discrepncy between the numericl methods nd the nlyticl solution for the dissiption of excess pore pressure in the cse of T v =0.175 in Fig. 3b (note tht the T v =0 cse hs been imposed by the initil conditions for p). As shown in Fig. 3b, except in the erly stge nd the cse of T v = 0.8, the predicted dissiptions of excess pore pressure using both methods of void frction clcultion re in good greement with the nlyticl solution. It is of prticulr interest to discuss the cse of T v =0.8 in Fig. 3b. The curve in red circle (see online) represents the predictions by the centre void frction method. As compred to the nlyticl solution, the divided void frction method evidently provides significntly better predictions thn the centre void frction method, which exemplifies the potentil pitfll ssocited with the ltter. A further inspection of the results revels tht the initil consolidtion (driven by grvittionl force nd buoyncy force) hs resulted in settlement round mm for some prticles on the top. At T v =0.8 of the norml consolidtion stge, there is n extr settlement round 9 mm occurring for these prticles. The totl

J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 253 much ttention in clssic grnulr mechnics. The occurrence of pressure dip cn be cused by mny fctors, e.g., the bse deflection [36], the prticle shpe [34,40] nd the construction methods [4,29]. Apprecible pressure dip hs been observed in snd pile prepred by loclized flow source such s hopper, whilst using rining sieve produces snd pile with centrl pek norml stress. Whilst dominnt body of existing studies on sndpiling hs been focused on the cse of dry grnulr mterils, reserch on sndpile formtion in n environment of wter is scrce. The ltter cse my find rther interesting pplictions in prctice, rnging from silos to rod nd dm constructions, lnd reclmtion nd dredging, mine product nd tiling hndling s well s soil erosion nd debris flow. A deeper understnding towrds the fundmentl principles governing the stress trnsmission in sttic grnulr solids submerged in wter my led to not only considerble dvnces in the theory of grnulr mechnics but lso improved technologies for relevnt prcticl pplictions mentioned bove. The CFD DEM method developed bove will be employed in this pper to exmine the behviour of sndpiling in wter. Prticles re poured from hopper through continer filled with wter to form conicl snd piles on circulr flt pnel plced t the bottom of the continer (see Fig. 4). To ensure proper conicl sndpiles cn be formed, the circulr pnel is limited by round bffle with 2 mm high (similr to wy employed in the tests by [17]). The prticles flowing beyond the bffle will drop off nd will be removed from considertion. Menwhile the corresponding cses without the presence of wter (herefter referred to s the dry cses) will lso be simulted for comprison. Whilst the prticle shpe is found ffecting the chrcteristics of the pressure dip in sndpile (see [2,34,35,40]), it is considered pproximtely here by considering the rolling resistnce mong sphericl prticles. Following the model by Zhou et l. [35], one hs Fig. 4. Illustrtion of the formtion of snd pile through hopper flow in wter: () during the formtion of the sndpile; (b) the finl stte of the sndpile. settlement of such prticles thus reches round 0.5 mm, which my exctly led to sitution tht the centre of the concerned prticle comes cross the boundry of two neighbouring cells (similr to the cse of Prticle E in Fig. 1). According to the centre void frction method, there will be sudden jump of void frction for neighbouring cells nd hence the drg forces s well, which my led to the observed erroneous results shown in Fig. 3b. Whilst the divided void frction method offers improved ccurcy, the centre void frction method is dvntgeous in terms of efficiency, especilly when the simulted system is extremely lrge. Evidently, if the prticle size is very smll reltive to the fluid cell, the difference in predictions between the two void frction methods is expected to become smll. The overll performnce of the CFD DEM progrm hs been found stisfctory with the bove two benchmrking exmples. The fluid prticle interction ppers to be resonbly cptured. The numericl lgorithms solving the governing equtions of both the CFD nd the DEM prts re generlly stble nd robust. 4. Appliction to conicl sndpiling in wter Hndling nd processing of grnulr mterils re commonplce in mny engineering brnches nd industries. The piling of grnulr medi, for exmple, hs been common in open stockpiles in griculture, chemicl engineering nd mining industries. The ngle of repose nd the stress distribution in snd pile hve been the focus of DEM studies on snd pile formtion (see [38] on review of this topic). In prticulr, the pressure minimum in the verticl stress profile of the bse of snd pile hs been n interesting phenomenon ttrcting ω M r ¼ μ r F n R rel r j j ω rel ð17þ where M r is the torque between two contcted prticles. F n is the contct norml force. R r is the rolling rdius defined by R r =r i r j /(r i +r j ) where r i nd r j re the rdii of the two sphericl prticles in contct. μ r is the coefficient of rolling resistnce. Zhou et l. nd Zhou nd Ooi [34,35] hve emphsized the importnce of rolling friction in chieving physiclly/numericlly stble sndpiles. To highlight its role in the wet cse, comprtive study of two rolling resistnce cses, μ r =0 nd μ r =0.1, is conducted. Note tht smll bffle used for the ground pnel is especilly useful to ensure the forming of proper sndpiles in the cse of free rolling (μ r =0). In ddition, we hve exmined both monosized nd polydisperse grin size distribution. The polydisperse pcking follows typicl grin size distribution of snd. To render the two cses comprble, the men grin size of the polydisperse pcking is chosen to be equl to the prticle size of the monosized cse. Tble 1 summrizes the relevnt prmeters used in the subsequent computtion. We simulte rel cse of forming snd pile in 10 s, mong which round 6 s is spent in pouring ll prticles through the hopper into the wter nd onto the circulr pnel nd round 4 s for the relxtion of ll prticles (some my fll off the receiving pnel) until they finlly settle down (with n overll kinetic energy reching mgnitude round 10 14 J). Becuse very smll time steps hve been used in both the DEM nd CFD computtions to solve the problem, dequte ccurcy cn be chieved by stepping 1000 DEM clcultions fter one step of CFD computtion. The totl computing time for ech reliztion of sndpile in wter, on 4-core Intel CPU (3.0 GHz) desktop computer, is round 2 dys. The finl stble-stte sndpile will be used to extrct such informtion s stress distribution, repose ngle, void rtio distribution nd contct force chins for the subsequent nlysis. In prticulr, both the centre nd divided void frction methods hve been used for the problem. Only mrginl

254 J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 Tble 1 Physicl nd geometric prmeters used in the sndpiling simultions. Chrcteristics of the pckings Monosized 15,000 prticles, 2 mm in dimeter. Polydisperse 15,000 prticles, dimeter rnged from 1 to 3 mm (men=2 mm, the cumultive grin size distribution shown in Fig. 5) Prticle nd contct prmeters Prticle density 2700 kg/m 3 Interprticle friction coefficient μ=0.7 Young's modulus (Hertz model) 70 GP (prticle prticle contct) 700 GP (prticle wll contct) Poisson's rtio 0.3 Restitution coefficient 0.7 Rolling friction (using the torque model of [35]) μ r =0&0.1 Geometry of the hopper & Hopper dimeter 14 mm the circulr pnel Hopper height (from the hopper bottom to the receiving pnel) 40 mm Rdius of the receiving pnel 5 cm Bffle height 2 mm Simultion control Time step (DEM) 5 10 7 s Time step (CFD) 5 10 4 s Simulted rel time 10 s (20,000,000 steps in DEM) Note tht in prctice, different vlues for both the coefficient of interprticle friction nd the coefficient of restitution should be used for the in ir nd in wter cses, i.e., ccording to Mlone nd Xu [19]. For simplicity, they re kept the sme in both cses in this study to highlight the pure effect of wter presence (e.g., interction forces). difference hs found between the predictions by the two methods. Hence only the simultions by the divided void frction method will be presented in the subsequent sections. 4.1. Repose ngle The repose ngle ϕ of sndpile formed in wter (referred in the sequel s wet cse ) hs been compred to tht for the dry cse, for both monosized nd polydisperse pckings. In mesuring the repose ngle, the position of ech prticle is projected onto the plne of r z where r denotes its horizontl distnce to the xis of the pile (ssumed to be identicl to the xis of the hopper). The pek of ll sndpiles obtined in our study hs been found rther flt with verticl height H slightly less thn the conicl pex H s shown in Fig. 6. The pex height H will be used to normlize the verticl pressure profile. The obtined results re summrized in Tble 2 for the cse of μ=0.7. It is observed from Tble 2 tht, for the monosized cse without considertion of rolling resistnce, the repose ngle for sndpile formed in wter is firly close to tht in the dry cse. However, if the rolling resistnce is considered, it becomes considerbly smller thn tht in the dry cse. From our simultion the difference is found to be round 9. However, the observtion is quite different for the polydisperse cse, where the obtined repose ngle in the dry cse Cumultive Probbility 100 80 60 40 differs only round 1 from the wet cse. Its vlue in dry cse is slightly greter thn the wet cse for the free rolling cse (μ r =0), but is mrginlly smller thn the ltter in considertion of rolling resistnce. In ny of these cses, considering rolling resistnce leds to pprecibly incresed repose ngle for sndpile thn otherwise. This is consistent with the observtion by Zhou nd Ooi [34]. Menwhile, our study indictes tht there is mixed effect of the polydispersity of pcking on the obtined sndpile. Without considertion of rolling resistnce, the monosized nd polydisperse pckings produce roughly the sme repose ngle. When the rolling resistnce is considered, much smller repose ngle is found for dry polydisperse pcking thn dry monosized cse, wheres it is greter in the wet polydisperse cse thn in the wet monosized cse. 4.2. Pressure dip Fig. 7 depicts the verticl pressure profiles t the bse of sndpiles obtined from our simultions. Cler pressure dip t the centre is found for ll cses. Tble 2 lso presents the specific vlues of the normlized dip nd pek pressures. In prticulr, the effects of the following fctors on the observed pressure dip cn be identified () Wter. A sndpile formed in wter generlly hs fltter dip ( smller reltive pressure dip) thn the dry cse. The difference in the reltive pressure dip cn be two times s much. (b) Rolling resistnce. Under otherwise identicl conditions, the considertion of rolling resistnce my led to n increse in the reltive pressure dip for monosized pckings, but moderte decrese for the polydisperse cse. (c) Polydispersity. A polydisperse smple generlly leds to smller reltive pressure dip thn monosized one. 4.3. Void rtio It is interesting to explore the fetures of both verge void rtio nd the locl void rtio in ech sndpile. We employ the Voronoi 20 0 1.5 2.0 2.5 3.0 Dimeter d (mm) Fig. 5. Cumultive grin size distribution of the polydisperse pcking used for sndpiling simultion. Fig. 6. Determintion of the repose ngle for snd pile.

bprobbilityprobbility J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 255 Tble 2 Comprison of the chrcteristics of sndpiles obtined for the dry/wet cses nd monosized nd polydisperse pckings (interprticle friction coefficient μ=0.7). Sndpile chrcteristics Monosized pcking Polydisperse pcking μ r =0 μ r =0.1 μ r =0 μ r =0.1 Dry Wet Dry Wet Dry Wet Dry Wet Repose ngle ϕ ( o ) 22.4 2 31.5 22.6 23.3 2 27.4 26.0 Finl prticle number in the sndpile 13,348 13,138 11,604 9476 14,681 15,000 14,328 14,989 Normlized dip stress 0.3865 61 0.345 68 0.572 0.81 3 0.719 Normlized pek stress (by gh ) 54 0.724 0.721 0.776 0.869 0.93 0.87 0.878 Reltive pressure dip (%) 40.90 8.70 52.15 13.92 34.18 12.90 27.59 18.11 Averge void rtio 1.243 1.27 1.376 1.397 1.459 1.481 1.545 1.571 Fbric Anisotropy 717 368 988 0.3919 516 839 0.1825 0.1853 Reltive pressure dip=(pek stress dip stress)/pek stress. tesselltion of sndpile to clculte these quntities. Shown in Fig. 8 re the Voronoi tesselltion cells for typicl sndpile. Since ech Voronoi cell is occupied by single prticle, the locl void rtio cn be conveniently determined. Bsed on the locl void rtio, the verge void rtio cn lso be obtined. In clculting the void rtios, prticles/cells in the bottom lyers which re below the height of the bffle hve been excluded for considertion. Normlized verticl stress 0.8 r =0, dry r =0.1, dry r =0, wet r =0.1, wet 3.5 3.0 2.5 2.0 1.5 0.5 dry- r = 0 wet- r = 0 dry- r = 0.1 wet- r = 0.1 b Normlized verticl stress 0.8 rtn ( ) / z, = 0.7 0.8 r =0, dry r =0.1, dry r =0, wet r =0.1, wet c 0.8 1.2 1.4 1.6 1.8 2.0 void rtio,e dry- r = 0 wet- r = 0 dry- r = 0.1 0.8 wet- r = 0.1 rtn ( ) / z, = 0.7 Fig. 7. Profile of verticl pressure t the bse of snd piles for () monosized pckings nd (b) polydisperse pckings. 0.8 1.5 2.0 2.5 3.0 void rtio,e Fig. 8. () Voronoi tesselltion of sndpile; (b) comprison of the locl void rtio distribution for monosized pckings; (c) locl void rtio distribution for polydisperse pckings. In both (b) nd (c), the symbols re numericl dt, nd the dsh or solid lines re fittings by Gmm distribution.

256 J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 As shown in Tble 2, the presence of wter leds to slightly incresed void rtio s compred to the dry cse. The considertion of rolling resistnce nd polydispersity, however, my result in significntly looser sndpile thn otherwise. Shown in Fig. 8 re the locl void rtio distributions nd fittings for both the monosized nd polydisperse cses. As cn be seen, Gmm distribution fits much better for the monosized cses (regrdless dry or wet, considering rolling resistnce or not) thn for the polydisperse cses. In ech of the monosized cses, best fit Gmm distribution slightly underestimtes the pek probbility of the locl void rtio. In the polydisperse cses, the optiml Gmm distribution provides systemticlly overestimtion in the smll void rtio region nd underestimtion in the til prt, but nevertheless cptures the pek well. The polydisperse cse reches pek probbility t slightly smller void rtio thn the corresponding monosized cse. In ll cses, the presence of wter or considering rolling resistnce my led to the pek void rtio shifted rightwrds to bigger vlue. Such n effect cused by the considertion of rolling resistnce is more obvious thn by the presence of wter. Menwhile, we hve further visulized the distribution of locl void rtio in sndpile in Fig. 9. In the monosized cse s shown in Fig. 9, two cler dense res re observed in the dry sndpile which serve s the nchoring points for n rch to be formed round the sndpile centre nd induce the observed pressure dip. In the wet cse the distribution of void rtio t the bottom hs been much smoothed nd no prticulr denser res re present. In the polydisperse cse, the presence of wter ppers to hve rther limited impct on the locl void distribution where similr locl void rtio distributions re found in both the dry nd the wet cses. 4.4. Fbric structure nd fbric nisotropy Indictive informtion of pressure dip cn be obtined from the contct force network (or fbric) of sndpile [5,18,23], which is shown in Fig. 10 for the μ r =0 cse in the present study. In the dry monosized cse (Fig. 10 upper pnel), the strong force chins show n pprecible orienttion with n inwrd inclintion ngle of round 70. This indictes tht the weights of the upper prticles of the sndpile re trnsferred to the bottom long these inclined chins rther thn long the verticl direction. The bottom centre prt of the sndpile is shielded from supporting the weights, which explins the strong pressure dip observed in this cse. In contrst, in the wet monosized cse (Fig. 10 bottom pnel), the contct force chins re more preferbly oriented to the verticl direction, nd no effective shield cn be formed to deflect the upper weights. A much reduced pressure dip is nturlly found for this wet cse. The observtion differs for the polydisperse cses shown in Fig. 10b. The polydispersity ppers to totlly chnge the force trnsmission pttern, s hs been noticed by Luding [18] s well. In both the dry nd wet cses, the strong force chins re more verticlly oriented, nd result in reduced pressure dip in these cses. Moreover, for both monosized nd polydisperse cses, the presence of wter renders the entire contct force network more homogeneous thn in the dry cse, nd the Fig. 9. Comprison of the locl void rtio contour (μ r =0). () The monosized cse, (b) the polydisperse cse. In ech cse, the upper figure corresponds to the dry pcking nd the bottom the wet pcking.

J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 257 Fig. 10. Comprison of contct force networks in the sndpile for () the monosized cse nd (b) the polydisperse cse. Free rolling (μ r =0) is considered for both cses. gretest contct forces re lrger in the dry cse thn in the wet cse. Not presented here, the considertion of rolling resistnce renders the force chins more verticlly oriented, in similr mnner s the effect of polydispersity. Evidently, the fbric structure in Fig. 10 is not isotropic, nd the degree of fbric nisotropy in these contct networks cn indeed be quntified. To this end, we employ interprticle-contct-bsed fbric tensor proposed by Stke [22] nd use its second invrint to quntify the degree of fbric nisotropy in sndpile (see [14] on similr wy of using the fbric tensor nd its invrint). The results re summrized in Tble 2. As is shown, the fbric nisotropy is modertely reduced in the presence of wter for monosized smples, wheres the opposite trend is observed in the polydisperse cse. The considertion of rolling resistnce generlly leds to significnt increse of fbric nisotropy, wheres the polydispersity results in reduced fbric nisotropy. A positive correltion is observed between the fbric nisotropy nd pressure dip rtio for the monosized cses. No pprent correltion cn be found for the polydisperse cses. 5. Concluding remrks A coupled CFD DEM method hs been presented to simulte the interction between fluid nd prticles in grnulr medi. In the method, we employ the DEM to simulte the motion nd interctions of prticles for grnulr prticle system, nd use the CFD to solve the loclly verged Nvier Stokes eqution for fluid flow. The interction between fluid nd prticle is considered by exchnging such interction forces s drg force nd buoyncy force between the DEM nd the CFD. Through two benchmrking exmples nd nother ppliction to the formtion of sndpile in wter, the following conclusions cn be mde: The proposed method is dequtely robust nd efficient to be pplied to the simultion of fluid prticle interction for wide vriety of problems in geomechnics. The behviour of fluid prticle interction in grnulr medi cn be resonbly cptured by the proposed method, s hs been demonstrted by benchmrking with the single prticle settling in wter problem nd the one-dimensionl consolidtion problem. Bsed on the CFD DEM simultion of the conicl sndpile problem, it is observed tht, () the presence of wter my help to form sndpile with more homogeneous internl structures in terms of locl void rtio, contct force network nd fbric nisotropy. It my lso help to reduce the reltive pressure dip; (b) considering the rolling resistnce mong prticles my led to greter reltive pressure dip for the monosized cse nd smller one for the polydisperse cse. The observtion holds for both dry nd wet cses; (c) sndpile formed by using polydisperse grnulr mteril my hve smller

258 J. Zho, T. Shn / Powder Technology 239 (2013) 248 258 reltive pressure dip thn using monosized mteril; nd (d) the locl void rtio in sndpile with monosized prticles yields Gmm distribution. The chrcteristic is not so obvious for sndpile formed by polydisperse prticles. The observtions mde bove still need rigorous verifictions by experiments in the future. The study nevertheless constitutes first step towrds effective modelling the complex interction between fluids nd prticles in porous medi such s snd. Further improvements my be mde by considering more relistic prticle shpe nd more resonble interction forces in the coupling nlysis. Whilst it hs been developed for pplictions relevnt to geotechnicl engineering, the proposed pproch cn be eqully useful for problems in other fields such s mining nd chemicl engineering where the fluid prticle interction is considered importnt. Acknowledgement The study ws supported by the Reserch Grnts Council of Hong Kong (RGC/GRF 623609). References [1] J. Ai, Prticle scle nd bulk scle investigtion of grnulr piles nd silos, Ph.D. thesis, The University of Edinburgh, 2010. [2] J. Ai, J.F. Chen, J.M. Rotter, J.Y. Ooi, Assessment of rolling resistnce models in discrete element simultions, Powder Technology 206 (2011) 269 282. [3] T.B. Anderson, R. Jckson, Fluid mechnicl description of fluidized beds. Equtions of motion, Industril nd Engineering Chemistry Fundmentls 6 (1967) 527 539. [4] B. Brockbnk, J.M. Huntley, R.C. Bll, Contct force distribution beneth three dimensionl grnulr pile, Journl de Physique II Frnce 7 (1997) 1521 1532. [5] M.E. Ctes, J.P. Wittmer, J.-P. Bouchud, P. Cludin, Jmming, force chins, nd frgile mtter, Physicl Review Letters 81 (1998) 1841 1844. [6] F. Chen, Coupled flow discrete element method ppliction in grnulr porous medi using open source codes, Ph.D. thesis. University of Tennessee, 2009. [7] F. Chen, Eric C. Drummb, G. Guiochon, Coupled discrete element nd finite volume solution of two clssicl soil mechnics problems, Computers nd Geotechnics 38 (5) (2011) 638 647. [8] F. Conch, Settling velocities of prticulte systems, KONA Powder nd Prticle Journl 27 (2009) 18 37. [9] P.A. Cundll, O. Strck, A discrete numericl model for grnulr ssemblies, Geotechnique 29 (1979) 47 65. [10] B.M. Ds, Advnced Soil Mechnics, Hemisphere Publishing Corportion nd McGrw-Hill, 1983. [11] R. Di Felice, The voidge function for fluid prticle interction systems, Interntionl Journl of Multiphse Flow 20 (1994) 153 159. [12] C. Goldenberg, I. Goldhirsch, Friction enhnces elsticity in grnulr solids, Nture 435 (2005) 188 191. [13] C. Goniv, C. Kloss, A. Hger, S. Pirker, An open source CFD DEM perspective, Proc. of the 5th OpenFOAM Workshop, June 22-24 2010, (Göteborg). [14] N. Guo, J. Zho, The signture of sher-induced nisotropy in grnulr medi, Computers nd Geotechnics 47 (2013) 1 15. [15] K.D. Kfui, C. Thornton, M.J. Adms, Discrete prticle continuum fluid modelling of gs solid fluidised beds, Chemicl Engineering Science 57 (2002) 2395 2410. [16] C. Kloss, C. Goniv, LIGGGHTS: new open source discrete element simultion softwre, Proc. 5th Int. Conf. on Discrete Element Methods, London, UK, August 2010, (website: www.liggghts.com). [17] Y.J. Li, Y. Xu, C. Thornton, A comprison of discrete element simultions nd experiments for sndpiles composed of sphericl prticles, Powder Technology 160 (2005) 219 228. [18] S. Luding, Stress distribution in sttic two-dimensionl grnulr model medi in the bsence of friction, Physicl Review E 55 (4) (1997) 4720. [19] K.F. Mlone, B.H. Xu, Prticle-scle simultion of het trnsfer in liquid-fluidised beds, Powder Technology 184 (2008) 189 204. [20] R.L. Michlowski, N. Prk, Admissible stress fields nd rching in piles of snd, Geotechnique 54 (2004) 529 538. [21] C. O'Sullivn, Prticulte Discrete Element Modelling: A Geomechnics Perspective, Spon Press (n imprint of Tylor & Frncis), London, 2011. [22] M. Stke, Fbric tensor in grnulr mterils, in: P.A. Vermeer, H.J. Luger (Eds.), Deformtion nd Filure of Grnulr Mterils, Blkem, Rotterdm, 1982, pp. 63 68. [23] J.H. Snoeijer, T.J.H. Vlugt, M. vn Hecke, W. vn Srloos, Force network ensemble: new pproch to sttic grnulr mtter, Physicl Review Letters 92 (5) (2004) 054302. [24] G.G. Stokes, On the theories of internl friction of fluids in motion nd of the equilibrium nd motion of elstic solids, Trnsctions Cmbridge Philologicl Society 8 (9) (1844) 287 319. [25] K. Suzuki, et l., Simultion of upwrd seepge flow in single column of spheres using discrete-element method with fluid prticle interction, Journl of Geotechnicl nd Geoenvironmentl Engineering 133 (1) (2007) 104 110. [26] K. Terzghi, Theoreticl Soil Mechnics, Wiley, New York, 1943. [27] Y. Tsuji, T. Kwguchi, T. Tnk, Discrete prticle simultion of two-dimensionl fluidized bed, Powder Technology 77 (1993) 79 87. [28] Y. Tsuji, T. Tnk, T. Ishid, Lgrngin numericl simultion of plug flow of cohesionless prticles in horizontl pipe, Powder Technology 71 (1992) 239 250. [29] L. Vnel, Howell D., J. Clrk, R.P. Behringer, E. Clément, Effects of the construction history on the stress distribution under snd-pile. lnl eprint, cond-mt/9906321, 1999. [30] L.M. Wrren, C.S. Julin, H. Peter, Unit Opertions of Chemicl Engineering, 7th edition, McGrw-Hill, New York, 2005, pp. 163 165. [31] J.P. Wittmer, P. Cludin, M.E. Ctes, J.P. Bouchud, An explntion for the centrl stress minimum in snd piles, Nture 382 (1996) 336 338. [32] B.H. Xu, Y.Q. Feng, A.B. Yu, S.J. Chew, P. Zulli, A numericl nd experimentl study of gs solid flow in fluid-bed rector, Powder Hndling nd Processing 13 (2001) 71 76. [33] B.H. Xu, A.B. Yu, Numericl simultion of the gs solid flow in fluidized bed by combining discrete prticle method with computtionl fluid dynmics, Chemicl Engineering Science 52 (1997) 2785 2809. [34] C. Zhou, J.Y. Ooi, Numericl investigtion of progressive development of grnulr pile with sphericl nd non-sphericl prticles, Mechnics of Mterils 41 (2009) 707 714. [35] Y.C. Zhou, B.D. Wright, R.Y. Yng, B.H. Xu, A.B. Yu, Rolling friction in the dynmic simultion of sndpile formtion, Physic A 269 (1999) 536 553. [36] Y.C. Zhou, B.H. Xu, R.P. Zou, A.B. Yu, P. Zulli, Stress distribution in sndpile formed on deflected bse, Advnced Powder Technology 14 (2003) 401 410. [37] H.P. Zhu, Z.Y. Zhou, R.Y. Yng, A.B. Yu, Discrete prticle simultion of prticulte systems: theoreticl developments, Chemicl Engineering Science 62 (2007) 3378 3396. [38] H.P. Zhu, Z.Y. Zhou, R.Y. Yng, A.B. Yu, Discrete prticle simultion of prticulte systems: review of mjor pplictions nd findings, Chemicl Engineering Science 63 (2008) 5728 5770. [39] I. Zuriguel, T. Mullin, J.M. Rotter, The effect of prticle shpe on the stress dip under sndpile, Physicl Review Letters 98 (2007) 028001. [40] I. Zuriguel, T. Mullin, The role of prticle shpe on the stress distribution in sndpile, Proceedings of the Royl Society of London, Series A 464 (2008). [41] T.W. Lmbe, R.V. Whitmn, Soil Mechnics, Wiley, New York, 1969.