EULERIAN LAGRANGIAN SIMULATION OF PARTICULATE FLOW FOR SEALING FRACTURED CHANNELS

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1 IV Journeys in Multihase Flows (JEM 15) March 3-7, 15, Caminas, SP, Brazil Coyright 15 by ABCM Paer ID: JEM EULERIAN LAGRANGIAN SIMULATION OF PARTICULATE FLOW FOR SEALING FRACTURED CHANNELS Fernando C. De Lai Federal University of Technology - Paraná UTFPR, Curitiba-PR 83-91, Brazil fernandodelai@utfr.edu.br Marcos Vinicius Barbosa Federal University of Technology - Paraná UTFPR, Curitiba-PR 83-91, Brazil marcosc@gmail.com Admilson T. Franco Federal University of Technology - Paraná UTFPR, Curitiba-PR 83-91, Brazil admilson@utfr.edu.br Silvio L. M. Junqueira Federal University of Technology - Paraná UTFPR, Curitiba-PR 83-91, Brazil silvio@utfr.edu.br Abstract. The aim of this work is numerically investigate the articulate flow in a channel with a discrete fracture. Concerning the flow in the channel, the study is divided into two arts: the first one deals a single-hase fluid flow, to analyze the fluid loss along the fracture; the second art handles a two-hase (liquid-solid) flow, to investigate the fracture filling by articulate material. Both the mathematical formulation and numerical modeling are ursued via an Euler-Lagrange aroach. The couling of discrete (articles) and continuum (fluid) hases is attained by combining the Dense Discrete Phase Model (DDPM) and the Discrete Element Method (DEM). The fracture filling rocess is characterized by varying the leading arameters namely: the number of injection oints, the injection time ste, the article-fluid density ratio and the article diameter. Effects of article concentration over control arameters such as the fluid flow rate through the fracture outlet and the ressure of the mixture at the channel inlet are investigated. Results indicate the time required to comlete the injection of articles and the shae of the fixed bed formed by the articles to achieve a artial or comlete fracture sealing. Keywords: Numerical simulation, Liquid-solid flow, Fractured channel, DDPM, DEM 1. INTRODUCTION Porous formations generally found in nature consist of a fixed matrix comosed of rocks and soil embedded by fluid, such as water, gas and/or oil, and may eventually resent some discontinuities (fractures). A fracture network acts as hydraulic conductors due to the resence of referred flow, regarding the geometrical characteristics (e.g., quantity, shae, orientation) of the fractures (Dietrich et al., 5). In the drilling rocesses, different tyes of fluids are constantly interacting with the geological formation, allowing the ossible seeage of drilling fluids toward the formation. The occurrence of high ressure gradients at the wellbore and the eventual resence of fractures in the geological formation can be rather inconvenient, since a significant loss of drilling fluid may haen due to referential flow channels rovided by fractures. This henomenon imairs the well roductivity as well as the reservoirs recovery time (Schechter, 199). Therefore, different methods must be used to control the fluid flow outwards the borehole-formation system. One of these methods is the rocess of injecting solid articles to seal the fractures (Abrans et al., 1977). A better understanding of the fracture sealing rocess, by the arresting of the solid articles flowing with the fluid through the fracture, is the goal of the resent study. In this context, a mathematical model is develoed to numerically simulate the liquid-solid flow in a channel with the resence of a discrete fracture. The rocess of filling the fracture with articulate material is characterized by modeling the articulate flow via a Lagrangian aroach (Loth, 1). The resulting fracture filling rocesses are differentiated by the injection time and the resulting form of the article fixed bed along the fracture to achieve a artial or comlete fracture sealing. This effort is accomlished by observing the effect of the main injection arameters (i.e., the number of injection oints n IP, the injection time ste tip, the articlefluid density ratio / and the article diameter d ) variation over the articles concentration as well as the interaction forces between the articles and fluid hase. The article injection rocess is monitored considering several arameters, such as the leakage fluid flow of the fracture and the ressure of the mixture at the channel inlet.. PROBLEM FORMULATION

2 F.C De Lai, A.T. FRANCO and S.L.M. Junqueira Eulerian Lagrangian Simulation of Particulate Flow for Sealing Fractured Channels Figure 1 shows an idealized geometry of the fractured channel. The relevant zones (left) and the boundary conditions (right) are resented. The origin of the Cartesian system (right) is located at the bottom of the fracture entrance. The fracture (FR) and the free region (CH) are delimited by the x( ) coordinate. The y( ) coordinate indicates a region ustream of the fracture (UP) starting at y lup. When y( ) a transition region (TR) defined by the fracture thickness e FR and a downstream region (DW) of the fracture, ending at y ( efr ldw ), can be identified. The boundary conditions are described in terms of surfaces (1-5) of Fig. 1 (right): (1) channel inlet: uniform surface ( h CH ) and fluid velocity rofile ( U,CH,i ); () article injection: uniform surface ( h IP ) with a constant article generation ( m,ip and N,IP ) and a uniform inut velocity ( U,IP ); (3) fracture outlet: flow rate and ressure rescribed conditions with reflection condition for articles (allowing only the assage of fluid); () channel outlet: fluid and articles assage with rescribed conditions of flow rate and ressure to control the initial leakage on the fracture; (5) imermeable surfaces: fluid non-sli condition at channel and the fracture walls. hch () DW ldw hfr TR FR (5) y (3) x efr UP IP () (1) lup Figure 1. Schematic reresentation of the fractured channel: regions (left); boundary conditions (right). The fracture geometry (FR) is reresented by a thickness ( e FR ) and a length ( h FR ). A ressure gradient ( fuga ), Eq. (1), sufficient to romote a leakage through the fracture, is imosed at the fracture outlet (3). Thus, the articles are carried throughout the fluid inside the fracture due to velocity and ressure gradient rovided, allowing the comlete or artial article filling rocess. ( ) (1) fuga m,fr, o m,ch, o ref where m,fr, o and m,ch, o are the ressures values, resectively, at the fracture and channel outlets; ref is the reference ressure at the channel outlet. The invasion henomenon is considered as the initial condition for the articulate injection rocess which is erformed through the imosition of Q fuga, Eq. (), which corresonds to a art of the channel inlet flow rate ( q,ch,i ). Observe that the average fluid velocity at the inlet ( U,CH,i ) is function of the Reynolds number ( Re ). Therefore, imosing Q fuga at the fracture outlet, the values of m,fr, o and m,ch, ocan be evaluated. These values are emloyed as inut data in a second stage of the roblem, which consists of reroducing the same initial leakage, using ressure conditions for both channel ( ref ) and fracture ( fuga ) outlets. Q fuga q,fr, o ; q,ch, i hch U,CH, i Re () where q,fr,o, and 3. NUMERICAL MODEL are resectively, the fluid flow rate at the fracture outlet, density and dynamic viscosity. The model adoted to reresent a Lagrangian aroach utilizes a combination of DDPM (Pooff and Braun, 7) and DEM models (Cundall and Strack, 197). The DDPM model is resonsible for solving the couled fluid-solid equations, while the DEM model accounts for the articles collisions via the soft sheres aroach (Hoomans, ). The liquid-solid two-hase flow in the fracture-channel considers a Newtonian fluid with constant and uniform roerties. A laminar and isothermal flow (fluid and articles) is considered. A non-rotating Cartesian coordinate system is used for the fluid, with the gravitational force acting in the vertical direction. A Lagrangian framework for the trajectory

3 IV Journeys in Multihase Flows (JEM 15) of the articles is accounted. The articles are considered solid with sherical shaes, regarding a homogeneous articulate flow (articles with the same shae, dimensions and roerties). The rotational motion of the articles is neglected, i.e., no angular velocity. Furthermore, erfectly elastic collisions (without change of form or coalescence due to imact) are assumed. The mathematical formulation consists of two sets of equations: one for the fluid and one for the articles. The DDPM conservation equations of mass and momentum for the continuous (fluid) hase are exressed resectively by: t ( u ) ( u ) ( u u ) ( u ) g FDPM SDPM t (3) () where t is the time, is the volume fraction of continuum hase, u is the fluid velocity vector, g is the gravity acceleration vector, F DPM is the couling term to exchange momentum due to the interaction of discrete hase forces and S DPM is the source term due to dislacement of fluid in relation to entry of articles in a given control volume. Newton s second law rovides the trajectory of the discrete hase articles (velocity and osition) by solving the set of ordinary differential equations exressed in terms of the forces that influence the articles acceleration: du m F F F F F F (5) dt d gb g vm ls DEM dx dt u (6) where m reresents the article mass; x, the article osition vector; u the article velocity vector; F d, the drag force; F gb, the summation of gravity and buoyancy forces; F g, the ressure gradient force; F vm, the virtual mass force; F ls, the Saffman lift force and F DEM the collision forces of the articles. Table 1 summarizes the equations forces related to the discrete hase model. Table 1. Equations for the forces acting on the discrete hase of the articles. Forces Equations 18 CD Re Drag force Fd m ( u u ) d Gravitational and buoyance forces F gb m ( ) g Pressure gradient force F f g u u m m D Virtual mass force Fvm Cvmm ( u u ) Dt Saffman lift force ( ) d F ( u u ) 1/ ij ls, i Cm ls, j, j d 1/ ( d lk d kl ) Collision forces FDEM Fn Ft ( knn n( u1 n1 )) n1 a Fn t1 where is the article density, C D is the article fluid drag coefficient, calculated by the Morsi and Alexander (197) model and Re u u d / is the article Reynolds number. C vm is the virtual mass force coefficient. C ls is the Saffman lift constant (Li and Ahmadi, 199), where dij ( ui, j u j, i ) / reresents the deformation rate tensor and the term dlkd kl can be exressed as a function of the mean shear rate for a simle shear flow (Ounis and Ahmadi, 1991). The collision forces are a combination of normal ( F n ) and tangential ( F t ) forces. The normal force is calculated through a sring-dashot model (Luding, 1998), where k,,, u 1 and n1 are resectively, the sring coefficient, the overla between the articles, the daming coefficient, the relative velocity and the unit normal vector at contact. The tangential force is based on the Coulomb friction equation, where a is the friction coefficient and t 1 is the unit tangential vector at contact.

4 F.C De Lai, A.T. FRANCO and S.L.M. Junqueira Eulerian Lagrangian Simulation of Particulate Flow for Sealing Fractured Channels De Lai (13) brings details of the numerical modeling of the DEM arameters (e.g., stiffness and daming constants, restitution and friction coefficients, scales of article time ste). The discretization of the equations for the fluid hase uses the Finite Volume Method (Patankar, 198). The solution of the couled set of equations is based on the Pressure-Based Solver with a segregated method (Chorin, 1968), through the ressure-velocity PC-SIMPLE algorithm couling (Vasquez e Ivanov, ). An imlicitly first order time discretization is emloyed. The interolation scheme for the advective terms is the uwind first order (Versteeg and Malalasekera, 1995). The evaluation of the gradients is taken by the Least Squares Cell-Based method (Anderson and Bonhus, 199). The Node Based Averaging technique (Ate et al., 8) via Gaussian kernel function, for accumulation or distribution variables, is alied.. RESULTS AND DISCUSSION Preliminary results with verification roblems for the terminal velocity were erformed and have shown good agreement with the literature (Mordant and Pinton, ). This roblem consists in the settling of a article initially at rest on a quiescent fluid that accelerates under gravity influence and attains its maximum velocity named as terminal or settling velocity. The comarison between the exerimental and numerical data available in Mordant and Pinton () are dislayed in Fig.. Two cases were simulated for different falling articles (a-glass and b-steel) droed in a water 988. kg m kg m s ). reservoir ( and.1 u [m s -1 ].8 (a). u [m s -1 ].3 (b).6.. Mordant e Pinton () - Ex. Mordant e Pinton () - Num. Present..1 Mordant e Pinton () - Ex. Mordant e Pinton () - Num. Present t [s].1..3 t [s] Figure. Verification roblem for the terminal velocity of a article on water: 3 3 (a) glass ( d,5[mm] and 56[ kg m ] ); (b) steel ( d,8[mm] and 771[ kg m ] ). Table shows the configurations simulated to characterize the fracture filling rocess, resented in Fig.1. Variation of the main arameters injection ( n IP, tip, / and d ) values are exected to affect the concentration of the injected articles on the channel. For all cases simulated, the channel and fracture geometric arameters are constant ( hch.5 [m] ; lup.9 [m] ; ldw.5 [m] ; lz d ; hfr.9 [m] ; efr.1[m] ). An initial configuration, to achieve the invasion roblem, is set for Re 5 and Qfuga 1%. A mixture of water and glycerin (73.7%) is emloyed: kg m and kg m s. The case is considered as a standard injection configuration. The injection arameters are varied (according to the case ) to increase the injected articles concentration (values highlighted in Tab. ). case n IP Table. Main injection arameters for the simulated cases. tip [s] / [mm] d m N k,ip [kg s ],IP [s ] n[nm ] Preliminary tests required to simulate the cases of Tab. to determine the arameters named as secondary are erformed. Table 3 summarizes the values of the secondary arameters used to characterize the main arameters injection of the filling rocess. The articles distribution, colored as a function of u (left) and the fluid velocity field u (right) for different injection times t i are shown in Figure 3.

5 IV Journeys in Multihase Flows (JEM 15) Table 3. Values of secondary arameters used for the resented simulation cases. Parameters Symbol Value Unit Comutational mesh VC 8 VC Size of unit volume Outut length of the channel Time ste of the discrete hase Particle restitution coefficient Surface osition of the injection Surface length of the injection l Z.1 [m] l DW.5 [m] t.e- [s] e i.9 [-] l IP. [m] h IP.35 [m],53,15,158,1 t i [s] -1-1 u [ms ] u [ms ] Figure 3. Filling rocess for the case : trajectory of the articles (left); velocity field of the fluid (right).

6 F.C De Lai, A.T. FRANCO and S.L.M. Junqueira Eulerian Lagrangian Simulation of Particulate Flow for Sealing Fractured Channels Figure shows the trajectory of the articles for two injection times ti 3; 6 [s], considering different views of the fractured channel. One observes the change direction of the articles (see vectors) at the fracture inlet region. According to the filling rocess and the fixed bed, the intensity of articles in the fracture inlet is changed. The three-dimensional acking (3D view) of the Lagrangian framework used for the articles trajectory is observed for a unit volume. -1 u [ms ],53,15,158,1 Fracture inlet region Fracture view 3D unit volume view Figure. Results of the articles trajectory, colored according to the article velocity fractured channel: t 3s (left) and t 6s (right). i i u, for different views of the Figure 5 shows the monitoring arameters for the fluid flow rate ( Qfuga q,fr, o / q,ch, i ) at the fracture outlet (left) and the ressure ( Pm,CH, i m,ch, i / fuga ) of the mixture (fluid and articles) at the inlet channel (right), to comare the cases of filling rocess resented in Table. Configurations with a higher solid concentration show a more efficient sealing with resect to the time elased for the injection rocess. Moreover, this increase in concentration roduces a significant raise in the ressure gradient to reserve the mixture flow, requiring a higher ressure P m,ch, i at the channel inlet. The stabilization value of Q fuga determines the end of the injection rocess. When Qfuga indicates that the fracture is fully sealed and the flow in the channel is re-established. On the other hand, a value of Q fuga different from zero indicates that a failure or an inefficient sealing rocess occurred. In this situation, the ressure gradient at the fracture outlet is higher than the ressure loss generated by the ack of articles that forms a fixed bed in the fracture. 6

7 IV Journeys in Multihase Flows (JEM 15). Q fuga,fr,o Figure 5. Monitoring arameters: fluid flow at the fracture outlet (left); ressure mixture at the channel inlet (right). 5. ACKNOWLEDGEMENTS The authors acknowledge the financial suort rovided by PETROBRAS S.A., ANP by means of the Human Resource Program (PRH) and the National Research Council (CNPq). The authors also exress their gratitude to PETROBRAS ersonnel for roviding technical assistance. 6. REFERENCES 1 3 t i [s] Abrans, A., et al., Mud design to minimize rock imairment due to articles invasion. J. Pet. Technol. Vol. 9, No. 5, Anderson, W. and Bonhus, D.L., 199. An Imlicit Uwind Algorithm for Comuting Turbulent Flows on Unstructured Grids. Comuters Fluids, Vol. 3, No. 1, Ate, S.V., Mahesh, K. and Lundgren, L., 8. Accounting for finite-size effects in simulations of disersed articleladen flows. Int. J. Multihase Flow, Vol. 3, No. 3, Chorin, A.J., Numerical solution of navier-stokes equations. Mathematics of Comutation, Vol, Cundall, P.A., Strack, O.D.L., A discrete numerical model for granular assemblies. Géotchnique, Vol. 9, No. 1, De Lai, F.C., 13. Numerical Simulation of Particulated Flow for Filling of Fractured Channel. M.Sc. dissertation, Universidade Tecnológica Federal do Paraná, Curitiba, PR, Brazil. Dietrich, P., Helmig, R., Sauter, M., Hötzl, H., Köngeter, J. and Teutsch, G., 5. Flow and Transort in Fractured Porous Media. Sringer, Berlin. Hoomans, B.P.B.,. Granular Dynamics of Gas-Solid Two-Phase Flows. Ph.D. thesis, Twente Univ., Netherlands. Li, A. and Ahmadi, G., 199. Disersion and Deosition of Sherical Particles from Point Sources in a Turbulent Channel Flow. Aerosol Science and Technology, Vol. 16, Loth, E., 1. Particles, Dros and Bubbles: Fluid Dynamics and Numerical Methods. Cambridge University Press. Luding, S., Collisions & Contacts between two articles. In: Physics of dry granular media. Kluwer Academic Publishers, Dordrecht, 1st edition. Mordant, N. and Pinton, J.F.,. Velocity measurement of a settling shere. Eur. Phys. J. B, Vol. 18, Morsi, S.A. and Alexander, A.J., 197. An Investigation of Particle Trajectories in Two-Phase Flow Systems. Journal of Fluid Mechanics, Vol. 55, No., Ounis, H. and Ahmadi, G., Motions of small articles in a turbulent simle shear flow field under microgravity condition. Physics of Fluids A: Fluid Dynamics, Vol. 3, No. 11, Patankar, S.V., 198. Numerical Heat Transfer and Fluid Flow. Hemishere/McGraw-Hill, New York. Pooff, B., Braun, M., 7. A Lagrangian Aroach to Dense Particulate Flows. In: International Conference on Multihase Flow, Leizig, Germany. Schechter, R.S., 199. Oil Well Stimulation. Prentice Hall, Englewood Cliffs, New Jersey, EUA. Vasquez, S.A. and Ivanov, V.A., A hase couled method for solving multihase roblems on unstructured meshes. In: Proceedings of the ASME Fluids engineering Division Summer Meeting, Boston. Versteeg, H.K. and Malalasekera, W., An introduction to comutational fluid dynamics. Harlow, UK: Longman Grou Ltd. 7. RESPONSIBILITY NOTICE case P m,ch,i 8 The authors are the only resonsible for the rinted material included in this aer. 6 case t i [s]