Power Law Correlations for Lift from Direct Numerical Simulation of Solid-Liquid Flow

Transcription

1 Power Law Correlations or Lit rom Direct Numerical Simulation o Solid-Liquid Flow Daniel D. Joseph Department o Aerospace Engineering and Mechanics 107 Akerman Hall, 110 Union Street SE University o Minnesota Minneapolis, MN (61) oice; (61) ax joseph@aem.umn.edu March, 001 Abstract...1 Preace... Direct numerical simulation (DNS) o solid-liquid low... Richardson and Zaki (RZ) correlations.... Single particle lit o and levitation to equilibrium Slip velocities, circulation and lit... 6 Model o slip velocity... 6 Biurcation... 8 Levitation to equilibrium o 300 circular particles Engineering correlations...1 Sand transport in ractured reservoirs Power law or bed erosion Erosion experiments; or these experiments H 1 = H and only water is moving Power it: H /W vs. R ~ in a log-log plot The inal correlation a(r G ) as R G in a log-log plot Conclusions...19 Acknowledgment...0 Reerences...0 Abstract Lit orces acting on a luidized particle plays a central role in many important applications, such as the removal o drill cuttings in horizontal drill holes, sand transport in ractured reservoirs, sediment transport and cleaning o particles rom suraces. The problem o lit is studied using direct numerical simulations. Lit ormulas which respect the act that the lit must change sign on either side o the "Segré-Silberberg" radius are discussed. An accurate analytical expression or the slip velocity o circular particles in Poiseuille low is derived. We show that the lit-o o single particles and many particles in horizontal lows ollow laws o similarity, power laws, which may be obtained by plotting simulation data in D on log-log plots. Data rom slot experiments on bed erosion or ractured reservoirs is processed (or the irst time) in log-log plots. Power laws with a parameter dependent power emerge as in the case o Richardson-Zaki correlations or bed expansion.

2 Preace My collaborators on studies o lit are H. Choi, H. Hu, P. Huang, T. Ko, D. Ocando, N. Patankar and P. Singh. This is but one aspect o a concentrated NSF supported study o direct numerical simulations o solid-liquid low. The results o such studies are collected at the project web site The whole ield is reviewed in the monograph under preparation "Interrogation o Direct Numerical Simulations o Solid-Liquid Flow, which can be downloaded rom the web site Direct numerical simulation (DNS) o solid-liquid low. The current popularity o computational luid dynamics is rooted in the perception that inormation implicit in the equations o luid motion can be extracted without approximation using direct numerical simulation (DNS). A similar potential or solid-liquid lows, and multiphase lows generally, has yet to be ully exploited, even though such lows are o crucial importance in a large number o industries. We have taken a major step toward the realization o this potential by developing two highly eicient parallel inite-element codes called particle movers or the direct numerical simulation o the motions o large numbers o solid particles in lows o Newtonian and viscoelastic luids. One o the particle movers is based on moving unstructured meshes (arbitrary Lagranian-Eulerian or ALE) and the other on a structured mesh (distributed Lagrange multiplier or DLM) using a new method involving a distribution o Lagrange multipliers to ensure that the regions o space occupied by solids are in a rigid motion. Both methods use a new combined weak ormulation in which the luid and particle equations o motion are combined into a single weak equation o motion rom which the hydrodynamic orces and torques on the particles have been eliminated. Several dierent kinds o code have been developed and tried on a variety o applications. See the project Web site, To our knowledge we are the only group to compute ully resolved particulate low at Reynolds numbers in the thousands occurring in the applications. Richardson and Zaki (RZ) correlations. The correlations o Richardson and Zaki (1954) (see also Pan, Joseph, Bai, Glowinski and Sarin 001) are an empirical oundation or luidized bed practice. They did very many experiments with dierent liquids, gases, particles and luidization velocities. They plotted their data in log-log plots; miraculously this data ell on straight lines whose slope and intercept could be determined. This showed that the variables ollow power laws; a theoretical explanation or this outstanding result has not been proposed. Ater processing the data Richardson and Zaki ound that V(φ) = V(0) (1-φ) n where V(φ) is the composite velocity which is the volume low rate divided by the cross-section area at the distributor when spheres o volume raction are luidized by drag. V(0) is the "blow out" velocity, when φ = 0; when V > V(0) all the particles are blown out o the bed. Clearly V(φ) < V(0). The RZ exponent n(r) depends on the Reynolds number R = V(0)d/ν; n =.39 when 500 < R < DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

3 (a) (b) (c) Figure 1. (Pan, Joseph, Bai, Glowinski and Sarin 001). Snapshots o luidization o 104 spheres comparing experiment (right) and simulation (let) (a) V =, (b) V = 3.5, (c) V = 4.5. We carried out DNS simulations o 104 balls in a slit bed whose dimensions exactly match a real experiment. The simulation is compared with a matched real experiment and they give rise to essentially the same results (see igure 1). This simulation is presently at the rontier o DNS; it is a 3D computation o 104 spheres at Reynolds numbers based on the sphere diameter o the order o 10 3 and the agreement with experiment is excellent. The details and animation o the computation (Pan, et al 001) can be ound at The simulation o 104 spheres was carried out in the bed [depth, width, height] = [0.686, 0.30, 70.] cm. Snapshots comparing the animation with the experiment, in a rontal view are shown in igure 1. Figure shows the luidizing velocity vs. liquid raction ε=in a log-log plot; one line is or the simulation and another or the experiment. We draw a straight line with slop n =.39 through both sets o data. The it is not perect but we think rather encouraging. From the straight lines we determine the blow-out velocities V s (0) = cm/s or the simulation and V e (0) = 10.8 cm/s or the experiment, and ind the power laws V s (φ) = 8.131ε=.39 cm/s (1) Pan, et al (001) presented arguments that the discrepancy is due to the dierence in the diameter 0.635cm o the sphere in the simulation and the average diameter cm o the 104 spheres uses in the experiments. V e (φ) = 10.8ε=.39 cm/s. 3 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

4 40 Bed height (inches) Experiment Simulation V (cm/s) Figure (a). (Pan, et al 001) The bed height vs. luidizing velocity or both experiment and simulation. Fluidization velocity V (cm/s) 10 Experiment Simulation e Figure (b). Data rom Figure (a) plotted in a log-log plot. The slopes o the straight line are given by the Richardson-Zaki n =.39. The blow-out velocities V s (0) and V e (0) are deined as the intercepts at ε = 1. Single particle lit o and levitation to equilibrium. The problem o lit o and levitation to equilibrium o a single circular particle in a plane Poiseuille low was simulated using an ALE particle mover in Patankar, Huang, Ko and Joseph (001). The principal eatures o lit o and levitation to equilibrium are listed in the caption o igure 3. Heavier particles are harder to lit o. The critical lit o Reynolds number increases strongly with the density ratio. The height, velocity and angular velocity o the particle at equilibrium is given as a unction o prescribed parameters in tables and trajectories rom lit-o to equilibrium in graphs shown in Patankar, et al (001). 4 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

5 d γ Ω U p u He U p Ω P u γ (a) (b) (c) (d) Figure 3. Lit o and levitation to equilibrium. The pressure gradient in the low and on the particle is increased. The heavier than liquid particle slides and rolls on the bottom o the channel. At a critical speed the particle lits o. It rises to a height in which the lit balances the buoyant weight. It moves orward without acceleration at a steady velocity and angular velocity. The channel height is W, particle diameter d, density o luid ρ and particle ρ p, viscosity η, kinematic viscosity η/ρ. The equation o motion o luid and particles made dimensionless with [d,w,d/v, η γ w ] where γ w is the wall shear rate and p is the applied pressure gradient that drives the low, are in the orm u d R + u u = p + e + x u, () t w π ρ p du RG d 4 Solid{ R = ey + e x + ( p + D[ u] ) ndθ (3) ρ dt R w π ρ ρ p du RG R = e dt R The low is determined by our dimensionless groups, y 0 π d 4 + e x + ( p + D[ u] ) ndθ. (4) w π 0 ( ρ ρ ) ρ p d ρ γ w d ρ p gd,, R =, RG = (5) ρ w η η where R is the shear Reynolds number and R G is a Reynolds number based on the sedimentation velocity in Stokes low. The terms with the actor e x come rom the pressure gradient; the pressure gradient (de x /w in (3)) drives the particle orward and the orward motion is resisted by the integral o the shear tractions. Freely moving particles in steady low have zero acceleration. The density ratio ρ p/ ρ vanishes when the particle accelerations are zero. The critical value o the Reynolds number or lit-o increases with density o the particle rom zero or neutrally buoyant ρ p = ρ p circular particles to R = 5 or particles 1.4 times heavier than water ρ p = 1.4 ρ. Ater the particle lits o it rises to an equilibrium height in which the buoyant weight equals the hydrodynamic lit. The equilibrium height or neutrally buoyant particles is called a Segré-Silberberg radius; it is determined by a balance o wall and shear gradient eects. The equilibrium height or heavy particles is lower than the Segré-Silberberg height. The rise to equilibrium is shown in igure 4. DNS results given in Patankar, Huang, Ko and Joseph (001) show that a circular particle will rise higher when the rotation o the particle is suppressed and least when the slip angular velocity is put to zero; the reely rotating zero torque case lies between. DNS allows such a comparison, which would be diicult or impossible to carry out in an experiment. 5 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

6 Y = y/d W s = g/-w p W s =g / 4 R = 16. W =g/-w W s =g/ s p Freely rotating 0 < W s < g/ W s = 0 ρ p = 1.01 ρ 3 Y = y/d R = 5.4 Freely rotating Neutrally buoyant: r /r p = < W s < / g W s = 0 1 r /r p = Time (second) Time (second) Figure 4. (Patankar, Huang, Ko and Joseph 001.) Rise vs. time or R w = 16. and 5.4. Compare rise o reely rotating and nonrotating particles. Nonrotating ones rise more. A neutrally buoyant, reely rotating particle rises closer to the center line than the Segré-Silberberg experiment; the nonrotating one rises even more. Models which ignore particle rotation overestimate lit. A yet smaller lit is obtained when the slip velocity is entirely suppressed (Ω s = 0), but the particle does rise. The greater the slip angular velocity, the higher the particle will rise. Slip velocities, circulation and lit. In commercial packages or slurry low in pipes, conduits and ractured oil and gas reservoirs, lit orces are not modeled, and in academic studies they are not modeled well. Possibly the best known and most used ormula or lit is the Rayleigh Formula L = ρ U=Γ=or aerodynamic lit. Here U is the orward velocity in still air that is produced by an external agent like a rocket engine, and Γ is the circulation, which is a complicated quantity determined by boundary layer separation. The lit on a ree body in a shear low is analogous and the lit ormulas that have been proposed are in the orm o U s, the slip velocity, times ρ Γ,=where=Γ=is a dierent quantity or dierent modelers. The slip velocity is the luid velocity at the particle center when there is no particle minus the particle velocity. Since it is the luid motion rather than an external agent which drives the motion o the particle, it might be expected that U s > 0. Since ree particles in shear low migrate to an equilibrium radius, the associated =Γ=ought to change sign at this radius; in act none o the lit ormulas that have been proposed do change sign; i they are right at one side o the equilibrium they are wrong on the other. The slip angular velocity discrepancy deined as the dierence between the slip angular velocity o a migrating particle and the slip angular velocity at its equilibrium position is positive below the position o equilibrium and negative above it. This discrepancy is the quantity that changes sign above and below the equilibrium position or neutrally buoyant particles, and also above and below the lower equilibrium position or heavy particles. On the other hand the slip velocity discrepancy U s - U se does not change sign Joseph, Ocando and Huang (001). Model o slip velocity. A long particle model was proposed in Joseph, Ocando and Huang (001), which leads to an explicit expression or the particle velocity U p o a circular particle in a Poiseuille low. Reerring to Figure 5 we ind that U A = φ + ψ=h A d, U B = φ + ψ=h B d (6) 6 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

7 where b φ = η ψ = γ ( d + h + h ) h h ( h + h ) ( h + d / ) ( h + h ) and the particle velocity U p is given by U p = γ (U A + U B ) /. B A B A A B B A B The slip velocity U s is given by p U s h η d + h d + U = B A p > 0 and U s = 0 when d = 0. (7) y p 1 > p p 1 u A (y ) p d U A h A τ A τb U B u B (y) y h B x Figure 5. (Joseph, Ocando and Huang 001.) The circular particle is replaced with a long rectangle where short side is d. The rectangle is so long that we may neglect the eects o the ends o the rectangle at sections near the rectangle's center. The rectangle is sheared at the shear rate o the circular particle Ω p γ / (see Figure 4). The velocity proile is Poiseuille low on either side o the particle and U A and U B determined by requiring that the pressure gradient p balance the shear stress. A comparison o the velocity proile o the long particle model with the velocity proile through the center o the circular particle computed by direct numerical simulation is given in Figure 6. Such a good agreement is surprising. 7 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

8 u [cm/s] 30 0 Undisturbed DNS Shear slip model u [cm/s] 30 0 Undisturbed DNS Shear slip model WALL y [cm] WALL WALL y [cm] WALL u [cm/s] 30 0 Undisturbed DNS Shear slip model u [cm/s] 30 0 Undisturbed DNS Shear slip model WALL y [cm] WALL WALL y [cm] WALL Figure 6. (Joseph, Ocando and Huang 001.) Comparison o the velocity proiles (Figure 5.) or the long particle model with the velocity proile on a line through the circular particle center computed by DNS or constrained motion at R = 0. In a constrained motion the y position o the particle is ixed, lateral motion is suppressed, but the particle is otherwise ree to translate and rotate under the action o the hydrodynamic orces and torques. Biurcation. A turning point biurcation o steady orward low o a single particle at equilibrium was ound in direct simulations o rise trajectories reported in Choi and Joseph (001); the height and particle velocity change strongly at such a point. A computational method advanced in Patankar, Huang, Ko and Joseph (001) looks or the points on lit vs. height curve at which lit balances buoyant weight. This gives both stable and unstable solutions and leads to the "biurcation" diagram shown in Figure 7, which shows there are two turning points, hysteresis, but no new branch points. 8 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

9 5 4 Equilibrium height Equilibrium height as a unction o shear Reynolds number or a particle o density 1.01 g/cm 3 Stable branch Unstable branch Shear Reynolds number Figure 7. (Patankar, Huang, Ko and Joseph 001.) Turning point "biurcations" shown in the height vs. Reynolds number curve. There are two stable branches separated by an unstable branch. Similar turning point biurcations have been ound also in computations o levitation to equilibrium o viscoelastic luids o Oldroyd-B type. Similar instabilities have been ound at yet higher Reynolds numbers. Biurcations o sedimenting particles, including Hop biurcations to periodic motions, have been reported in the literature. It is probable that all the phenomena known or general dynamic systems occur also or particulate lows. Levitation to equilibrium o 300 circular particles. The transport o a slurry o 300 heavier than liquid particles in a plane pressure driven low was studied using DNS in Choi and Joseph (001). Time histories o luidization o the particles or three viscous luids with viscosities η = 1, 0. and 0.01 (water) were computed at dierent pressure gradients. The study leads to the concept o luidization by lit in which all the particles are suspended by lit orces against gravity perpendicular to the low. The time history o the rise o the mean height o particles at a given pressure gradient is monitored and the rise eventually levels o when the bed is ully inlated. The time taken or ull inlation decreases as the pressure gradient (or shear Reynolds number) increases (see Figure 8). At early times, particles are wedged out o the top layer by high pressure at the ront and low pressure at the back o the particle in the top row (t = 1 in Figure 8a, t = 0.9 in Figure 8b). The dynamic pressure at early times basically balances the weight o the particles in the rows deining the initial cubic array. This vertical stratiication evolves into a horizontally stratiied propagating wave o pressure, which tracks waves o volume raction. The pressure wave is strongly involved in the liting o particles. For low viscosity luids like water where R G is large the particle-laden region supports an "interacial" wave corresponding to the wave o pressure. I R /R G is large the interace collapses since the stronger lit orces push wave crests into the top o the channel, but the pressure waves persist (Figure 9). 9 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

10 (a) Figure 8. (Choi and Joseph 001.) (a) Snapshots o the luidization o lit o 300 circular particles ρ p = 1.01 g/cm 3 when η = 1 poise (R = 5.4, R /R G = 1.8). The low is rom let to right. The gray scale gives the pressure intensity and dark is or low pressure. At early times particles are wedged out o the top layer by high pressure at the ront and low pressure at the back o each and every circle in the top row. The vertical stratiication o pressure at early times develops into a "periodic" horizontal stratiication, a propagating pressure wave. The inal inlated bed has eroded, rather tightly packed at the bottom with luidized particles at the top. (b) Fluidization o 300 particles (R = 10, R /R G = 0.08). The conditions are the same as in 9(a) but the ratio o lit to buoyant weight is greater and the luidization is aster and the particle mass center rises higher than in the previous Figures. (b) (a) Figure 9. (Choi and Joseph 001.) (a) Fluidization o 300 particles (η = 0. poise, R = 150, R /R G = 1.63). The inal state o the luidization at t = 5 sec has not ully eroded. The particles that lit out o the bed can be described as saltating. A propagating "interacial" wave is associated with the propagating pressure wave at t = 5. (b). Fluidization o 300 particles (η = 0. poise, R = 450, R /R G =0.54). The low is rom let to right. The particles can be lited to the top o the channel. We did correlations in numerical experiments in D. Correlations work. We studied the levitation o 300 particles in a Poiseuille low, Patankar, Ko, Choi and Joseph (000), Choi and Joseph (001), and created a data bank which when plotted on a log-log plot give rise to straight lines; this is to say that lit results or luidized slurries are power laws in appropriate dimensionless parameters. This shows that luidization o slurries by lit also alls into enabling correlations o the RZ type. The method o correlations is a link between direct simulation and engineering application. (b) 10 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

11 Correlations allow generalizations rom 0 or 30 data points into a continuum o points reaching even beyond where we can compute. Because you get so much rom correlations even expensive calculations are cheap. The correlation we ound or lit-o o a single particle is in the orm R G = ar n, a =.36, n = 1.39 (8) where R and R G are deined by (5); a and n are obtained by plotting about 5 data points in a log R vs. log R G plane (Patankar, Huang, Ko and Joseph (001), Figure 10). The straight lines that come out are amazing; they show that sel-similarity lies at the oundation o solid-liquid lows. Similar correlations were ound or lit-o in viscoelastic luids Ko, Patankar and Joseph (001), Patankar, et al (001) (Figure 11). For 300 particles in Poiseuille low we processed simulation data or the rise o the center o gravity o particles in the slurry; rom the height rise we can compute the solid raction φ. Processing data in loglog plots (Figure 1) we got R G = (1-φ) R 1.49 (9) This could be called a Richardson-Zaki type o correlation or luidization by lit Patankar, Ko, Choi and Joseph (001) R G = ar n W/d = 1 & 48 : a =.3648, n = W/d = 6 : a = 1.810, n = W/d = 4 : a = 1.596, n = RG W/d = 48 W/d = 1 W/d = 6 W/d = 4 Data rom dynamic simulations (W/d = 1) R Figure 10. (Patankar, Huang, Ko and Joseph 001.) The plot o R G vs. the critical shear Reynolds number R or lito on a logarithmic scale at dierent values o the channel width/diameter ratio W/d. This has evidently reached its asymptotic W/d value when W/d = DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

12 Newtonian luid : Oldroyd-B luid : Oldroyd-B luid : Oldroyd-B luid : λ / λ 1 = 1.0 λ / λ 1 = 0.5 λ / λ 1 = 0.15 λ / λ 1 = 0.0 R G R G = ar n λ / λ 1 = 1.0 : a =.3648, n = λ / λ 1 = 0.5 : a = , n =.0805 λ / λ 1 = 0.15 : a = , n =.1356 λ / λ 1 = 0.0 : a = , n = Figure 11. (Ko, Patankar and Joseph 001.) RG vs. R or lit-o o an Oldroyd B luid with dierent relaxation/ retardation time ratios in a log-log plot (W/d = 1, elasticity E = l1h/rd.) 10 R (b) ε R G K = R G K K = R R Figure 1. (Patankar, Huang, Ko and Joseph 001.) An engineering correlation (9) or lit-o rom numerical simulations o 300 circular particles in plane Poiseuille lows o Newtonian luids (W/d = 1). Engineering correlations We have already demonstrated that two-dimensional simulations o solid liquid lows give rise to power laws. These power laws are in the orm o engineering correlations; to use them in applications we need rules or converting two- to three-dimensional results. The goal o our uture work is to generate power laws or engineering applications by processing results o simulations in 3D just as we have done in D. The processing o data or the luidization o 104 spheres rom a simulation and experiment which leads to comparison in Figure is an example o what can be done. 1 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

13 We are presently doing 3D simulations or lit-o and levitation to equilibrium o single spheres and enhancing our simulators or eicient computation o migration and lit o slurries o many spherical particles. These simulators will be used to generate power laws or practical application to sand transport in ractured reservoir, among others. Sand transport in ractured reservoirs. Hydraulic racturing is a process oten used to increase the productivity o a hydrocarbon well. A slurry o sand in a highly viscous, usually elastic, luid is pumped into the well to be stimulated, at suicient pressure to exceed the horizontal stresses in the rock at reservoir depth. This opens a vertical racture, some hundreds o eet long, tens o eet high, and perhaps an inch in width, penetrating rom the well bore ar into the pay zone. When the pumping pressure is removed, the sand acts to prop the racture open. Productivity is enhanced because the sand-illed racture oers a higher-conductivity path or luids to enter the well than through the bulk reservoir rock, and because the area o contact or low out rom the productive ormation is increased. It ollows that a successul stimulation job requires that there be a continuous sand-illed path rom great distances in the reservoir to the well, and that the sand is placed within productive, rather than non-productive, ormations. Well Bore Sand Injected Early Upper Fracture Boundary Fluid Sand Injected Late Figure 13. (Kern, Perkins and Wyant 1959) Sand transport in a ractured reservoir is dierent than the eroded bed o 300 particles in Figure 9(a) and 10(a) because particles are injected. The creation o algorithms to simulate continuous injection is one o our simulation projects. In a slot problem a particle laden (say 0% solids) luid is driven by a pressure gradient and the particles settle to the bottom as they are dragged orward. Sand deposits on the bottom o the slot; a mound o sand develops and grows until the gap between the top o the slot and the mound o sand reaches an equilibrium value; this value is associated with a critical velocity. The velocity in the gap between the mound and the top o the slot increases as the gap above the mound decreases. For velocities below critical the mound gets higher and spreads laterally; or larger velocities sand will be washed out until the equilibrium height and velocity are reestablished (see Figure 13). The physical processes mentioned here are settling and washout. Washout could be by sliding and slipping; however, a more eicient transport mechanism is by advection ater suspension which we studied by direct simulation. Despite many years o practice and experiments many o the most essential luid dynamic properties o proppant transport, other than luidization by lit, are not well understood. To help our studies o these properties be ocused and practical, we have partnered with STIM-LAB, a research laboratory in Duncan, OK, which is supported by a consortium o oil production and oil service companies. STIM-LAB has been collecting data on sand transport in slots or 15 years. We have begun to process the data rom STIM-LAB's experiments or power laws in the same way we process data orm numerical simulations. An example is given just below. Power law or bed erosion. STIM-LAB carried out two types o experiments looking at the transport o proppants in thin luids. The irst experiment can be described as a lit-o or erosion experiment. A somewhat simpliied description o the experiment is that a bed o proppant is eroded by the low o water. Proppant is not injected. The aster the low o water the deeper is the channel above. We are seeking to predict the height above the channel. The evolution o the proppant bed in the experiments is well described in the diagram o Figure 13 with the caveat that experimental cell has a inite length. In the steady state there is an initial development 13 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

14 length ollowed by a lat bed that is divided into zones shown in Figure 14. The bottom o the bed is immobile, it is a stationary porous media that supports liquid throughput that might be modeled by Darcy's law. Above the immobile bed is a mobile bed in which particle slide and roll but do not lit. The traction carpet is a ully luidized bed in which particles move orward in ree motion under a balance o buoyant weight and lit. The lit o region lies between the stationary and luidized bed. Q p Q L Traction carpet moves orward Fluidized bed the height o 1- grains; rolls and bounces Mobile bed; moving at a very low velocity H 1 H Clean Fluid (open channel layer) Immobile bed. Figure 14. Proppant transport in thin luid at steady state conditions. In the erosion experiments only luid is pumped, Q P = 0, H 1 = H, the particles don't move and H is determined by Q P. The slot used in the erosion experiments was 8' long, ' high and 3/16" wide; riction rom the close side walls is most important in the slot and in real ractures. The open channel zone is deined by the presence o a channel, or proppant-ree conduit, above the proppant pack. This zone comprises most o the length o the model and probably is also the dominant zone in a ractured well. Proppant moves in response to the shear stress generated by the moving luid in the channel. In summary, the channel base is eroded until an equilibrium height is reached or a given velocity. I velocity decreases, the channel is stable. I velocity increases, the channel depth increases. Most sand erosion and transport is rom this zone. In the experiment, low was established through the slot, causing proppant to erode rom the top o the pack and a channel to orm. Once a channel ormed above the proppant it was allowed to equilibrate at least until the lower plane bedorms dominated the length o the slot. I necessary, the low rate was then reduced until no there was no movement o proppant in the channel. When no proppant movement was detected along the channel base, the low rate, channel height, dierential pressures, and temperatures were recorded. The low rate was then increased and the procedure repeated to generate the data in Table 1; the nomenclature or this table is given just below. The processing or power laws in our D simulations were ramed in terms o a Reynolds number or the orward low R = Vd/η (10) and a sedimentation Reynolds number based on the settling velocity R G = ρ ( ρ ρ ) The luid velocity V is related to the pressure drop across the low cell. p η gd 3 ρ ( ρ ρ ) gd p η in Stokes low. (11) 14 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

15 Proppants d (cm) H (gm/cm s) (gm/cc) (cm) η ρ ρ p υ Q (gm/cc) (cm /s) (cc/s) V ~ R G (cm/s) R ~ H /W / Brady / Ottawa /40 Light Beads / Carbolite /0 Carbolite /30 Banrite /0 Badger Table 1: Data set or lit-o experiments rom Table in Appendix Nomenclature and scaling parameters ρ p density o the particles, gm/cm 3 d mean diameter o the particles, cm ρ density o the luid, gm/cm 3 η dynamic viscosity o the luid, gm/cm-s p average pressure gradient applied in the low direction (i available), gm/(cm-s) Q volumetric low rate o the luid, cm 3 /s (continues on next page) 15 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

16 Nomenclature and scaling parameters (continued) M p mass low rate o the proppants, gm/s Q p volumetric low rate o the proppants = M p /ρ p, cm3/s Q T total volumetric low rate (luid + proppant) = Q + Q p, cm 3 /s h open channel height cm; (H ) H 1 see Figure 14, cm H see Figure 14, cm W channel width cm Α area cc; (A = W * H ) υ V kinematic viscosity c /s; υ = η ρ Q Q luid velocity cm/s; V = = ; A WH ~ Q V = ; W V ~ luid velocity cm/s; R Reynolds number (based on V); Vd R = ; υ ~ Vd ~ R ~ Reynolds number (based on V ~ RH ); R = = ; υ W ( ρ p ρ ) gd gravity parameter; G = ; ηv 3 ρ ( ρ p ρ ) gd R = η τ Shields parameter is deined as: S = ; ( ρ ρ gd G R G gravity Reynolds number; GR ; S G = p ) ηv 1 I we take τ = ηγ w and V = γ wd, then S = =. ( ρ p ρ ) gd G It is important to do correlations in terms o dimensionless parameters; this leads to maximum generality. To see this consider power laws, which we ound, that are o the orm R G = ar n (1) or lit-o. For R larger than (R G ) 1/n /a, the particle o radius d and density ρ p will levitate. This equation (1) may be written as 3 n ( ρ ρ ) gd p ρ η Vd = a η (13) Suppose we did experiments or lit-o in a certain luid with a given proppant o dierent size. We would ind a correlation o the orm V = bd m (14) We wouldn't know that m = 3 / n 1 or how all the other parameters enter into the correlation. 16 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

17 Erosion experiments; or these experiments H 1 = H and only water is moving. STIM-LAB did experiments on bed erosion. These are essentially lit-o experiments since the low rate is dropped to a critical lit-o value below which particles are not eroded rom the bed. In Table 1 we reormulate the data rom the STIM-LAB experiments or processing in terms o R G and Vd R ~ ~ = (15) ν where ~ Q V = (16) W and Q is the volume low rate. We use V ~ because Q and W are prescribed data. There are seven groups in the data shown in Table 1; each one corresponds to a value o R G. The data corresponding to a given R G is ordered by increasing H ; the larger R corresponds to a larger H more or less, but there are exceptions. Power it: H /W vs. R ~ in a log-log plot. It is rather obvious that the height H will increase with Q at a ixed R G (or ixed proppant and luid). With R G ixed we can hope or a two-parameter correlation. In Figure 15 we show seven more or less straight lines or the seven values o R G. The power law or these is given by H = a m( RG ) ( R G ) R (17) W where a(r G ) and m(r G ) are listed in Table. The value o the exponent m(r G ) 0.87 or all cases except R G = corresponding to nearly neutrally buoyant particles (ρ p = 1.05 gm/cc). R G a(r G) m(r G ) E E E E E E E Table. The coeicients (as unctions o R G ) used in power it or H /W. 17 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

18 10 R G =86.84 R G =51.16 R G =90.8 R G = R G = X R G =034.9 R 10 1 G = X X H /W X X 10 0 Power it or H /W H /W = a(r )*R G m(rg) R Figure 15. H /W vs. R ~ in log-log plot or dierent values o R G (see Table ). Inspection o Table shows that when R G 51 m(r G ) 0.87 (18) whereas when R G = (ρ p = 1.05) we get m(r G ) (19) This shows that the exponent m does depend on and we may hope to describe this dependence in intervals (as is true or the Richardson-Zaki n(r), which depends in intervals on the Reynolds number R). For example, we could suppose that there is a certain value o R G = R GC or which the m(r G ) take on the two values, (18) i R G > R GC and (19) is R G < R GC where R GC Summarizing we may propose where m(r G ) are given approximately by (18) and (19). H = a m( RG ) ( R G ) R (0) W The inal correlation a(r G ) as R G in a log-log plot. We plotted a(r G ) as given in Table against R G in a log-log plot. This plot is shown in Figure 16 and gives rise to the ormula a = R G. (1) For the coeicient a(r G ) a(r G ) = b R k G. 18 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

19 10 0 a=0.8007r G Daniel D. Joseph, "PowerLaw Correlations or Lit rom Direct Numerical Simulation o Solid-Liquid Flow" 10 1 a Rg Figure 16. Power it or a(r G ) vs. R G. Combining now (0) and (1) H W ~ m( R G = R G R ) () where m( R G ) or R G > RGC, where 86 < R GC < 500 and m or R G < R GC. We are proposing () as a widely applicable correlation or lit-o valid beyond where data has already been taken. More experiments, testing () validity or dierent values o W under more extreme conditions ought to be undertaken; o particular interest are light particles or which R G < R GC. When R G > R GC we can write () as H ρ = W ( ρ ρ ) gd ρ Qd p η W η (3) Equation (3) gives the racture height in terms o given quantities. Formula () can be expressed in ~ RG H terms o the Shield s parameter 1/G by writing R =. GW Conclusions We believe that research leading to optimal techniques o processing data or correlations rom real and numerical experiments is ounded on the ar rom obvious property o sel similarity (power laws) in the low o dispersions. The bases or this belie are the excellent correlations o experiments on luidization and sedimentation done by Richardson and Zaki and the correlations or liting o slurries in horizontal conduits obtained orm numerical experiments described here. The method o correlations is a new link between DNS and engineering practice. 19 DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc

20 Results o two dimensional simulations o solid-liquid lows give rise to straight lines in log-log plots o the relevant dimensionless Reynolds numbers. The extent and apparent universality o this property is remarkable and shows that the low o these dispersions are governed by a hidden property o sel similarity leading to power laws. These power laws make a powerul connection between sophisticated high perormance computation and the practical world o engineering correlations. The same methods or processing data are applied to numerical and real experiments. The Richardson-Zaki correlation is o great relevance in seeing the image o the uture. The power law in the RZ case is an example o what Barenblatt (1996) calls "incomplete sel similarity" because the power itsel depends on the Reynolds number, a third parameter. The three parameter bed erosion correlation () is an example o the kind o correlation we expect to emerge rom 3D simulations. More generally we expect that the processing o real data rom both real and numerical experiments will lead to amilies o straight lines in log-log plots connected by transition regions. Acknowledgment This work was partially supported by the National Science Foundation DKI/New Computational Challenge grant (SNF/CTS ), by the DOE, Department o Basic Energy Sciences, by a grant rom the Schlumberger oundation, rom STIM-LAB Inc. and by the Minnesota Supercomputer Institute. Reerences Papers marked with have been written in the last year and have not yet been published; they can be ound and downloaded rom our web site, G.I. Barenblatt Scaling, Sel Similarity and Intermediate Asymptotics. Cambridge Univ. Press. H.G. Choi and Daniel D. Joseph, 001. Fluidization by lit o 300 circular particles in plane Poiseuille low by direct numerical simulation, accepted by J. Fluid Mech. D.D. Joseph, D. Ocando and P.Y. Huang, 001. Slip velocity and lit, accepted by J. Fluid Mech. T.K. Kern, T.K. Perkins and R.E. Wyant The mechanics o sand movement in racturing, Petroleum Transactions, AIME 16, T. Ko, N.A. Patankar and D.D. Joseph 001. Lit-o o a single particle in an Oldroyd-B luid. Submitted to Phys. Fluids. N.A. Patankar, T. Ko, H.G. Choi and D.D. Joseph, 001. A correlation or the lit-o o many particles in plan Poiseuille o Newtonian luids, accepted by J. Fluid Mech. N.A. Patankar, P.Y. Huang, T. Ko and D.D. Joseph, 001. Lit-o o a single particle in Newtonian and viscoelastic luids by direct numerical simulation, accepted by J. Fluid Mech.. T.-W. Pan, D.D. Joseph, R. Bai, R. Glowinski and V. Sarin, 001. Fluidization o 104 spheres: simulation and experiment, J. o Fluid Mech. to appear. J.F. Richardson and W.N. Zaki, Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs. 3, DDJ/001/papers/LitCorrelations/4thCon-MultiphFlow.doc