HYBRID FEM-DEM APPROACH APPLIED TO BEDLOAD TRANSPORT

XI Simósio de Mecânica Comutacional II Encontro Mineiro de Modelagem Comutacional Juiz De Fora, MG, 28-30 de Maio De 2014 HYBRID FEM-DEM APPROACH APPLIED TO BEDLOAD TRANSPORT José L. D. Alves, Carlos E. Silva, Marcus V. S. Casagrande, Pietro G. Cosentino, Fábio T. Alves, Leandro C. Gazoni alves@lamce.coe.ur.br LAMCE/COPPE/UFRJ - BRAZIL Av. Athos da Silveira Ramos, 149, Centro de Tecnologia - Bloco I - Sala 214, Cidade Universitária, Rio de Janeiro, 21941-909, RJ, Brasil Alvaro L. G. A. Coutinho, José Camata, Renato Elias NACAD/COPPE/UFRJ - BRAZIL Paulo Paraizo PETROBRAS Abstract. Among the many geological rocesses o interest to oil industry, understanding turbidity currents can hel exlain where and how organic matter was deosited and erhas transormed by other rocesses o geological scale into hydrocarbons. The intent o the work, is to assess local, small scale arameters and their uscaling. The hybrid model is based on a Lagrangian-Eulerian aroach under a class o the named Unresolved Discrete Particle Method (UDPM). In this aroach, a Lagrangian descrition is used or the article system emloying the Discrete Element Method (DEM) while a ixed Eulerian mesh is used or the luid hase modeled by inite element method (FEM). Fluid motion equations are solved by an aroriate FEM imlementation. Closure equations are used to comute drag and lit orces over the articles in the DEM ramework. Volume averaged momentum sink terms are included in the luid equations. The resulting couled DEM-FEM model is integrated in time with a subcycling scheme. The aorementioned scheme was alied in the simulation o a seabed current to analyze which mechanisms leads to the emergence o bedload transort and sediment susension, and also quantiy the eective viscosity o the seabed in comarison with the ideal no-sli wall condition. To comare the behavior o the articles alling in a luid medium, a simulation o a salt lume in ree all was erormed, and the main characteristics o the system are discussed in comarison with a qualitative exeriment. Keywords: DEM, FEM, bedload, UDPM CILAMCE 2013 Proceedings o the XXXIV Iberian Latin-American Congress on Comutational Methods in Engineering Z.J.G.N Del Prado (Editor), ABMEC, Pirenóolis, GO, Brazil, November 10-13, 2013

Hybrid FEM-DEM aroach alied to bedload transort 1 INTRODUCTION Among the many geological rocesses o interest to oil industry, understanding turbidity currents (tyically a gravity or density current) can hel exlain where and how organic matter was deosited and erhas transormed by other rocesses o geological scale into hydrocarbons. In this work we describe a contribution to the study o turbidity transort in scales smaller than TFM (two-luid models). The intent o the work, art o a large scale simulation roect, is to assess local, small scale arameters and their uscaling. The hybrid model is based on a Lagrangian-Eulerian aroach under a class o the named Unresolved Discrete Particle Method (UDPM) according to the classiication resented in Hoe (2008). In this aroach, a Lagrangian descrition is used or the article system emloying the Discrete Element Method (DEM) while a ixed Eulerian mesh is used or the luid hase modeled by inite element method (FEM). This technique has been successully alied to the study o luidized bed in catalytic reactors (Hoomans, 1996). Fluid motion or the incomressible and viscous luid is governed by Navier-Stokes equations which are solved by an aroriate FEM imlementation (Elias, 2005). Closure equations are used to comute drag and lit orces over the articles in the DEM ramework (Cho, 2005). Volume averaged momentum sink terms are included in the luid equations. The resulting couled DEM-FEM model is integrated in time with a subcycling scheme. 2 MOTION EQUATIONS In this aer a three-dimensional Lagrangean aroach is used with the Discrete Element Method (DEM) to describe the articles, and a three-dimensional Eulerian aroach with the Finite Element Method (FEM) is used to describe the luid behavior. 2.1 Particle motion According to Hoe (2008), there are two main classiications to the simulation o article s interaction: the hard-shere model and the sot-shere model. In the hard-shere model the articles are modeled as rigid bodies and interact through instantaneous collisions. However, in the sot-shere model, the equations o motion o each article are solved numerically, requiring a contact orce model. The sring-damer model is the most widely used, showing a good comromise between accuracy and eiciency. Due to the large number o ossible simultaneous contacts, directly related to the high concentration o articles, the sot-shere model is the most suitable and, thereore, the imlemented. The equation o motion o each article can be written as: m x= where x G E C A is the articles accelerations, m the articles mass, G is the gravity orce, E is the is the contact orce and is the drag orce. Other ossible ways o interaction A buoyancy, C between articles (e.g. adhesion, aggregation, disaggregation) can be naturally treated within the ramework o DEM technique. Once the articles have been maed in the Eulerian domain discretized by a tetrahedra mesh, it is ossible to determine by interolation the luid velocity at the oint occuied by the XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional (5)

José L. D. Alves, Carlos E. Silva, Marcus Casagrande article. The drag orces acting in each article can be rocessed through emirical laws involving the relative velocity between the article and the luid. The contact orce comes rom the sot-shere model, consisting o linear elements o stiness and daming (O Sullivan, 2011) in the normal and tangential directions o the shere contact. Additionally, or the tangential direction it is considered the Coulomb riction law (Figure 1). F =K C c n t (6) c c c c, t Ft = min( coulombfn, Ft ( t, t )) t (7) where F n is the normal, enetration at the shere, n K n the stiness coeicient, the temoral enetration rate, F, ) C n F t the daming coeicient, the tangential orce, n the coulomb the riction coeicient and the tangential orce beore sliing, been identical to the equation o normal orce with the resective coeicients. t ( t t (a) (b) Figure 1: Contact model between two sheres: (a) normal direction, (b) tangential direction. The gravity and buoyancy orces are calculated trivially. According to Hoomans (2000), the drag orce acting on a article can be modeled as: A 1 2 ' = d Cd v v 8 where d is the article s diameter, suericial relative velocity, deined as: v is the luid s seciic mass and v (8) is the luid s 2 v v = u v u (9) being the orosity, u the luid s velocity and v the article s velocity. The eective drag ' coeicient C d consists o a correction to the drag coeicient C d due to the orosity. According to Wen and Yu (1966), this correction can be alied ollowing the equations 10, 11 and 12. C = (10) ' 4. 7 d C d XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

Hybrid FEM-DEM aroach alied to bedload transort The drag coeicient o an isolated article is modeled by Rowe and Henwood (1961) as: 24 0. 687 (1 0.15(Re ) ' C d = Re 0.44 ), Re Re 1000 1000 (11) Where the Reynolds number is deined as: Re u v = d (12) 2.2 Fluid motion While the article motion is described through discrete elements, the luid is described as a continuum medium, thus the luid domain is divided in tetrahedric cells, cells suiciently small to reresent the luid motion, and big enough to accommodate inside several articles, homogenizing locally the orosity. In this work, it was suosed that the variation o orosity in time and sace is negligible or the luid, so that the set o equations o Navier-Stokes or the incomressible Newtonian luid low does not need to be corrected by orosity. Thus, according to Hoomans et al. (1996), the resulting equations are listed in equations 13, 14 and 15. u x = 0 ui t u u x 2 ui = x x x i F (13) (14) F u _ v (15) where is the coeicient o moment transer, the luid s seciic mass, the ressure, dynamic viscosity and v _ the mean velocity o the articles contained in an element. The last term, F, reresents the moment transer rom the article to the luid, acting as a source or a sink. According to Kuiers (1992), or orosities bellow 0.80, is deined by the Ergun equation in the ollowing orm. 150 1 2 d 2 = 1 1.75 u v d while or orosities above 0.80, the ollowing correlation was resented by Wen and Yu (1966): 1 2. 65 3 = C d u v (17) 4 d (16) XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

José L. D. Alves, Carlos E. Silva, Marcus Casagrande 2.3 General asects o the couled imlementation The overall diagram o the couling is resented in Fig. 6. Ater the solution o the set o luid s equation (FEM/CFD cycle), the drag orces or each article are calculated, which allows the rogress o the solution or article motion (DEM cycle). Tyically, article dynamics requires small time stes comared to luid low, demanding a subcycle aroach, i.e., the comutation o various DEM cycles or each FEM/CFD cycle. Ater the inalization o DEM cycle, the source/sink terms are evaluated or each inite element in the mesh and F roceeding to a new ste o the Navier-Stokes equations integration or the FEM/CFD cycle. Figure 2: DEM/FEM couling structure. 3 STUDIED CASES The oerationality o the code is veriied in a qualitative way, using two cases or which two-way couling between DEM and FEM is intrinsically imortant. Channel low; Salt lume. In the ollowing sections, it is resented the descrition and results o both cases. 3.1 Channel low The aim o this case is to comare the velocity roile o the luid near the bottom o the channel using the no-sli boundary condition at the bottom wall and using a bed o articles to reresent the soil (Figure 3). XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

Hybrid FEM-DEM aroach alied to bedload transort Figure 3: Particles coniguration ater settling. The simulation addresses the channel low coniguration shown in Figure 5. Reresenting the soil, a olydiserse distribution comrising 4800 articles with diameter ranging rom 0.2mm to 0.3mm were settled in a channel o rectangular cross section with 12mm height, 1.8mm width and 12mm length. The entire channel was discretized with 6000 tetrahedra as deicted in Figure 4. Figure 4: Fluid mesh utilized or simulations. The emloyed boundary conditions are the velocity o 30mm/s at the let wall or the nodes above the seabed and 0mm/s or the nodes bellow seabed (i alied), no-sli at the bottom wall, ree sli at the uer wall and no enetration at side walls (Figure 5). At time t = 0s the articles are rozen and the sink term is nulled to the luid develo a steady state roile. This methodology was alied to acilitate the convergence. Ater 0.05s the articles were released and the sink term comuted. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

José L. D. Alves, Carlos E. Silva, Marcus Casagrande Figure 5: Channel low boundary conditions or no-sli wall (let) and article bed (right). The solution was achieved ater the simulation reaches the steady state. The Figure 6 illustrates the obtained velocity roiles. Each marker reresents a measure realized at the resective node o the mesh. Figure 6: Fluid velocity roiles. It can be observed rom the grah that the sloes o the roiles are slight dierent. This dierence is due to the interaction o the luid and the article that has a dierent eect than the no-sli wall condition. To assess this dierence, both sloes were calculated resulting in a XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

Hybrid FEM-DEM aroach alied to bedload transort 1.86º or the no-sli cond. seabed and a 2.82º or the articulate seabed. The ratio between both sloes is 152%, thereore the eective viscosity o the luid in the articulate seabed is 50% greater than the eective viscosity o the luid in the no-sli condition seabed. Table 1: Physical roerties attributed to the simulation o channel low. Proerty Value Unit (SI) Fluid time ste 0.005 s Simulation duration 5 s Dynamic viscosity 0.001 Pa.s Fluid s seciic mass 1000 Kg/m 3 Fluid domain size 12 x 12 x 1.8 mm Mesh Structured, 6000 tetrahedra with 0.6 mm side - DEM time ste 0.00001 (500 subcycles) s Number o articles 4800 Unit Particle seciic mass 2170 Kg/m 3 Contact rigidity 1.0E+5 N/m Particle diameter < 0.2, 0.3 > mm 3.2 Salt lume in ree all Due to the lack o hysical exeriments involving articles with controlled arameters, this work resents a comarison between a numerical simulated and a qualitative exeriment o a salt lume. This exeriment consists in a batch o salt released in a reciient illed with water. The main obective o this case is to comare the numerical and exerimental eatures o the salt lume. A minimal characterization o the salt batch was erormed over a samle o 300 salt grains, measuring grain sizes using digital imaging and a calier as a reerence scale. The results were interolated assuming a Gaussian distribution o the equivalent diameter o the salt set. The robability density unction and the numerical measurements are resented in Figure 7. The additional roerties utilized in the numerical simulation are listed in Table 2. The salt lume was reresented numerically by a set o 28224 sherical articles organized in a volume 20x20x40mm in a luid domain with 100x100x250mm. The salt lume is centered in the X-Y lane and is distant 200mm rom the bottom. The diameter o the articles was generated randomly according to the adusted Gaussian distribution. The boundary condition or the luid is no-sli in every wall. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

José L. D. Alves, Carlos E. Silva, Marcus Casagrande Figure 7: Salt grain size distribution. Time evolution o the numerical exeriment are resented in Figure 8 and Figure 9. The ormation o a dome shae is the main eature observed. The dome shae ormation mechanism is due to the recirculation o the luid. Initially, the luid and articles are in reose. Ater the articles start to move downward driven by the gravity orce, a drag orce is generated at the article and, in reaction, the same orce is exercised over the luid trough the source moment. This source moment imels the luid in center o domain downward that, because o incomressibility, imels the surrounding luid uward, causing the circulation o the luid. The circulation creates an axial comonent o the luid velocity that transorts the articles axially, creating the dome roile. Figure 8: Evolution o the numerical salt lume. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

Hybrid FEM-DEM aroach alied to bedload transort Figure 9: Evolution o luid velocity ield o the numerical salt lume. The exerimental result is illustrated at Figure 10. First, a similar dome shae is observed. Desite the salt lume is not centered, the set o images shows an initial small conglomerate o salt grains that gradually transorms in a dome shae with a tail o grains, the same eatures that aeared in the numerical simulation. Secondly, the numerical and exerimental results have inherent dierences because o several oints, as cubic shae o salt grains, chemical interaction between salt and water, and initial conditions o salt grains. Figure 10: Evolution o the exerimental salt lume. Finally, it is imortant to highlight that the eature resented in the salt lume case can be only observed in a two-way simulation, since the articles motion due to gravity generates the luid low that causes the dome shae o the articles. Although quantitative data could not be recovery or comarison uroses, the numerical simulation rovided a deeer knowledge o the henomenon. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

José L. D. Alves, Carlos E. Silva, Marcus Casagrande Table 2: Proerties o the salt lume numerical exeriment Proerty Value Unit (SI) Fluid time ste 0.001 s Simulation duration 1.0 s Dynamic viscosity 0.001 Pa.s Fluid s seciic mass 1000 Kg/m 3 Fluid domain size 0.1 x 0.1 x 0.25 m Mesh Structured, 12500 tetrahedra with 0.01 m side - DEM time ste 0.00001 (100 subcycles) s Number o articles 28224 Unit Particle seciic mass 2170 Kg/m 3 Contact rigidity 1.0E+2 N/m Particle diameter Gaussian Distribution (μ = 309 ; σ = 114) μm FINAL REMARKS This work resented the main asects o a successul imlementation o a DEM-FEM hybrid aroach to a roblem o bedload transort. The case studies showed excellent qualitative results, reresenting the main hysical eatures exected. For uture work we intend to extend the veriication o the imlementation with other cases and ossible validation by comaring the exerimental results. ACKNOWLEDGMENTS The authors recognize and acknowledge the invaluable suort given to this work by Petrobras S.A., CNPq and ANP, through the Programa de Formação de Recursos Humanos, PRH-02. REFERÊNCIAS Cho, S. H., Choi, H. G., Yoo, J. Y., 2005. Direct numerical simulation o luid low laden with many articles. International Journal o Multihase Flow, Volume 31, Issue 4, Pages 435-451. Elias, R. N., Martins, M. A. D., Coutinho, A. L. G. A, 2005. Parallel Edge-Based Inexact Newton Solution o Steady Incomressible 3D Navier-Stokes Equations. Lecture Notes in Comuter Science, v. 3648,. 1237-1245. van der Hoe, M. A., van Sint Annaland, M., Deen, N. G., Kuiers, J. A. M., 2008. Numerical simulation o dense gas-solid luidized beds: A multiscale modeling strategy. Annual Review o Fluid Mechanics, 01/2008; 40:47-70. Hoomans, B.P.B., Kuiers, J.A.M., Briels, W.J., van Swaai, W.P.M., 1996. Discrete article simulation o bubble and slug ormation in a two-dimensional gas-luidised bed: A hardshere aroach. Chemical Engineering Science, Volume 51, Issue 1, Pages 99-118. Hoomans, B.P.B., Kuiers, J.A.M., Swaai, van W.P.M, 2000. Granular dynamics Simulation o segregation henomena in bubbling gas-luidised beds. Powder Technology, Volume 109, Issues 1 3, Pages 41-48. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional

Hybrid FEM-DEM aroach alied to bedload transort Kuiers, J.A.M., van Duin, K.J., van Beckum, F.P.H., van Swaai, W.P.M., 1992. A numerical model o gas-luidized beds. Chemical Engineering Science, Volume 47, Issue 8, Pages 1913-1924. O Sullivan, C., 2011. Particulate Discrete Element Modelling: A Geomechanics Persective. London, Son Press/Taylor & Francis. Rowe, P. N., Henwood, G. A., 1961. Drag orces in a hydraulic model o a luidized bed-art I. Trans. Instn Chem. Engrs, 39, 43. Wen, C. Y., Yu, Y. H., 1966. Mechanics o luidization. Chem. Engng Prog. Sym. Ser. 62 (62), 100. XI Simósio de Mecânica Comutacional e II Encontro Mineiro de Modelagem Comutacional