Power law correlations for sediment transport in pressure driven channel flows

Transcription

1 Last updated April,. Power law correlations or sediment transport in pressure driven channel lows N. A. Patankar Department o Mechanical Engineering, Northwestern University, Evanston, IL 68 D. D. Joseph, J. Wang Department o Aerospace Engineering and Mechanics, University o Minnesota, Minneapolis, MN Abstract R. D. Barree Barree & Associates LLC, Lakewood, CO 835 M. Conway, M. Asadi STIM-LAB, Duncan, OK Lit orces acting on particles play a central role in many cases, such as sediment transport, proppant transport in ractured reservoirs, removal o drill cuttings in horizontal drill holes and cleaning o particles rom suraces. We study the problem o lit using D direct numerical simulations and experimental data. 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 on log-log plots. Data rom slot experiments or ractured reservoirs is processed (or the irst time) on log-log plots. Power laws with a parameter dependent power emerge as in the case o Richardson-Zaki correlations or bed expansion by drag.. Introduction Transport o particles in a channel by luids occurs in variety o settings such as sediment transport, proppant transport in oil reservoirs, removal o drill cuttings etc. Our ocus is on the problem o sand or proppant transport in hydraulic racturing applications. Hydraulic racturing is a process oten used to increase the productivity o a hydrocarbon well. A slurry o sand or proppants in a 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 penetrating rom the well bore ar into the pay zone. When the pumping pressure is removed, the rigid sand particles act to prop the racture open. Productivity is enhanced because the proppant-illed racture oers a higher-conductivity path or oil extraction. A typical vertical racture or crack may be 3 m high, 3 m long and cm wide. The diameter o a typical sand grain is mm, so that the crack width to proppant diameter ratio is about. The proppant density is about.4 g/cc.

2 Well Bore Upper Fracture Boundary Fluid Sand Injected Early Sand Injected Late Figure. Sand (proppant) transport in a ractured reservoir. Figure shows the side view o the crack. The luid-proppant mixture is injected rom the well bore. The proppants settle to the bottom as they are dragged orward. A mound o proppant develops and grows until the gap between the top o the crack and the mound reaches an equilibrium value; this value is associated with a critical condition. The velocity in the gap between the mound and the top o the slot increases as the gap size decreases. For velocities below critical the mound gets higher and spreads laterally; or larger velocities proppant is washed out until the new equilibrium height and velocity are established (Kern, Perkins & Wyant 959). The accumulation o proppant at the bottom causes good vertical illing to be lost. This reduces well productivity and can also interere with the racture growth process by blocking downward extension. Despite many years o practice and experiments a suitable model or proppant transport or a simulation tool to predict proppant placement is yet to be ully accomplished. 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 proppant transport in slots or 5 years. The physical processes in proppant transport in ractures, described above, are settling and washout. Washout could be by sliding and slipping called the bed-load transport; however, a more eicient transport mechanism is by advection ater suspension or luidization by lit called the suspended-load transport. Lit orce plays a central role in the suspension o particles in channel lows. Joseph () proposed that ideas analogous to the Richardson-Zaki (954) correlation must come into play in problems o slurries, where the particles are luidized by lit rather than by drag. The problems o luidization by lit can be decomposed into two separate types o study: () single particle studies in which the actors that govern liting o a heavier-

3 than-liquid particle o a wall by a shear low are identiied and () many particle studies in which cooperative eects on lit-o are investigated. Direct numerical simulation (DNS) can be used to extract inormation implicit in the equations o luid-particle motion. We have investigated the lit-o o a single particle and many particles in pressure driven lows by D DNS (N. A. Patankar, Huang, Ko & Joseph a, Ko, N. A. Patankar & Joseph, Joseph, Ocando & Huang, Choi & Joseph, N. A. Patankar, Ko, Choi & Joseph b). 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 on log-log plots. Power laws emerge as in the case o Richardson-Zaki correlations or luidization by drag. Powers laws are expected rom the experimental data based on our predictions rom D numerical simulations. The primary objective o this paper is to process the data rom STIM-LAB's experiments in the same way we process data rom numerical simulations. The engineering correlations or lit-o can be used to predict proppant placement in the crack. The racturing industry makes extensive use o numerical simulation schemes based on models and programmed to run on PC s to guide ield operations. These simulations are used to predict how the racture crack opens and closes and how proppant is transported in the crack. Commercial packages dealing with these problems and propriety packages developed by oil service companies are used extensively. These numerical schemes solve the average equations or the luid and the proppant phases. The solid and the luid are considered as inter-penetrating mixtures, which are governed by conservation laws. Interaction between the inter-penetrating phases is modeled. Models or drag and lit orces on the particles must be used or luidproppant interaction. Models or the drag orce on particles in solid-liquid mixtures is a complicated issue and usually rely on the well-known Richardson-Zaki (954) correlation. Models or lit orces in mixtures are much less well developed than models or drag. Thereore, none o the packages model the all important levitation o proppants by hydrodynamic lit. The power law models we are developing rom DNS and experiments may be incorporated in the model-based simulation techniques similar to the model or drag. The experiments o Segré & Silberberg (96) have had a big inluence on studies o the luid mechanics o migration and lit. Eichhorn & Small (964) perormed experiments to study the lit and drag orces on spheres suspended in a Poiseuille low. Bagnold (974) 3

4 experimentally studied the luid orces on a body in shear low. Ye & Roco (99) experimentally measured the angular velocity o neutrally buoyant particles in a planar Couette low. Dierent analytical expressions or the lit orce on a single particle can be ound in literature. They are based on perturbing Stokes low with inertia (e.g. Rubinow & Keller 96, Saman 965, 968, Bretherton 96, Asmolov 99, McLaughlin 99, Krishnan & Leighton 995 and reerence therein) or on perturbing potential low (e.g. Auton 987, Drew & Passman 999) with a little vorticity. In particular Schonberg & Hinch (989), Hogg (994) and Asmolov (999) analytically studied the inertial migration o spherical particles in Poiseuille lows. The eect o curvature o the unperturbed velocity proile was ound to be important. The domain o parameters or which these analytic expressions are applicable is rather severely restricted. The perturbation analyses are o considerable value because they are analytic and explicit but they are not directly applicable to engineering problems like proppant transport, removal o drill cuttings, sediment transport or even lit-o o heavy single particles. Dandy & Dwyer (99) and Cherukat, McLaughlin & Dandy (999) reported computational studies o the inertial lit on a sphere in linear shear lows. Mei (99) obtained an expression or the lit orce by itting an equation to Dandy & Dwyer s (99) data or high Reynolds numbers and Saman s expression or low Reynolds numbers. The numerical results o Dandy & Dwyer (99) are said to be valid or non-rotating spheres. Hence they cannot be applied, strictly speaking, to the case o reely rotating spheres in shear lows. Kurose & Komori (999) perormed numerical simulations to determine the drag and lit orces on rotating spheres in an unbounded linear shear low. Morris & Brady (998) studied the migration o non-neutrally buoyant spheres in pressure driven lows o Newtonian luids. They perormed Stokesian dynamic simulations o a monolayer o spheres. These studies are valid in the creeping low limit. In applications such as the transport o slurries or proppants, the eect o Reynolds number on the lit orce is important. A detailed investigation o the lit-o o single and many particles in channel lows at inite Reynolds numbers is necessary. In section 3 we will review and discuss these results obtained rom our D direct numerical simulations. New discussion on the implication o these results as compared to the Richardson-Zaki correlation or drag will be presented. 4

5 Shields (936) reported one o the earliest works on modeling sediment transport. Vanoni (975) reviews the problem o sedimentation engineering. Sheild s curve (936) is used to predict the initial motion o sediment particles. For higher luid low rates the particles in the moving bed are suspended leading to transport by advection. The problem o determining the low condition at which the suspension o the sediment begins has been addressed by Bagnold (966), van Rijn (984a,b), Sumer (986) and Celik & Rodi (99), among others. There are reasons to investigate the problem urther since these studies oten provide inconsistent results in determining the condition o the initiation o suspension. In section we will discuss luidization by drag and lit. In section 3 we will discuss the results rom DNS. The experimental setup will be described in section 4. The correlations based on experimental data will be presented in section 5. Conclusions will be presented in section 6.. Analogy between luidization by drag and lit (a) (b) g g Uniorm luid low Shear low o the luid Figure. (a) Heavy particles luidized by uniorm luid low rom the bottom o a vertical column. (b) Heavy particles luidized by lit due to shear low o the luid in a horizontal channel. Gravity acts vertically downwards. Fluidization by drag and shear is depicted in the cartoons in igure. In igure a the luid enters at the bottom o a vertical column at a uniorm luidization velocity. At equilibrium, the drag exerted by the luid balances the net buoyant weight o the particles. The particle bed acquires a height corresponding to the average particle raction φ. When the luidizing velocity is increased the particle bed expands. Richardson & Zaki (954) did experiments with dierent luids, 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 & Zaki (RZ) ound that V φ = V [-φ] n(r ) () 5

6 where V φ is the luidization velocity at the column entrance or the composite velocity. V is the "blow out" velocity, when φ = ; when V φ > V all the particles are blown out o the bed. Clearly V φ < V or a luidized bed. For luidization columns with large cross section in comparison to the particle size, the RZ exponent n(r ) depends on the Reynolds number R = V d/ν only, where d denotes the particle size e.g. diameter o a spherical particle and ν is the kinematic viscosity o the luid. The power law in the RZ case is an example o what Barenblatt (996) calls "incomplete sel similarity" because the power itsel depends on the Reynolds number, a third parameter. Pan, Joseph, Bai, Glowinski & Sarin () carried out 3D DNS o the luidization o 4 spheres and obtained a correlation in agreement with equation. The Richardson-Zaki correlation gives dierent expressions or n or dierent values o R. In the appendix R. D. Barree presents a way o representing the various expressions or n by a single continuous unction. Equation describes the complicated dynamics o luidization by drag. The single particle luidization velocity plays a key role in obtaining the luidization velocity o concentrated suspensions. An expression or the drag orce F d () on a single isolated particle in an ininite ambient o the luid is given by a drag law, e.g. F () 3πηdV, =.55πρ d V d, laminar turbulent () where η is the luid dynamic viscosity, ρ is the luid density and spherical particles are considered. In a luidized bed the total orce F acting on a particle is (Foscolo & Gibilaro 984, Joseph 99) F ( ε R ) = F d ( ε ) F ( ε ),, (3) where ε is the luid raction, F d (ε) is the drag on a single particle in the luid-particle mixture and F B (ε) is the eective buoyant weight o a particle in the suspension. We have, F B (ε) = V p (ρ p - ρ c )g = εv p (ρ p - ρ )g = εf B (), where ρ p is the particle density, V p is the volume o the particle, g is the gravitational acceleration, F B () is the buoyant weight o an isolated particle and ρ c = ερ + φρ p is the eective or composite density o the luid-particle mixture. At steady conditions B 6

7 F i. e i. e ( ε, R ) =,. Fd ( ε ) = FB ( ε ) = εfb ( ). F ( ε ) = εf ( ). d d, (4) For spherical particles, equations, and 4 give 3 [ πd ][ ρ p ρ ] or R G 6 8ε = n( R ) n( R ) 3πηdVφ ε, g = n.55πρ d Vφ ε R, laminar n( R ) [ ε R ], turbulent φ φ ( R ), laminar turbulent, (5) where R G 3 = ρ [ ρ p ρ ] gd η represents the Reynolds number based on the sedimentation velocity scale V G = [ ρ ρ ] gd η and ρ V d η. Equation 5 is another orm o the p R = φ φ correlation or luidization by drag and can be written as R G p( R ) q( R ) ( R ) R ε = a. (6) φ Figure b shows the luidization o particles by shear low observed in experiments and numerical simulations. At equilibrium the average lit exerted by the luid should balance the net buoyant weight o the particles. When the applied shear rate is increased the particle bed expands. This is similar to the luidization by drag where the mechanism or bed expansion is dierent. Correlations analogous to equation 6 may be expected or luidization by shear. In that case a Reynolds number based on the applied shear rate should be deined instead o R φ. The preactor and the exponents may be determined rom experimental or numerical data. In the ollowing sections we will irst show that our previous DNS results are in agreement with the above expectation. New discussion based on some o our previous DNS results will be presented. We will then show that the experimental data obtained rom STIM-LAB also give power law correlations. 3. Direct numerical simulation (DNS) o solid-liquid lows We used a numerical method, described in detail by Hu (996), Hu & N. A. Patankar () and Hu, N. A. Patankar & Zhu (), to study the lit-o o a single particle in Newtonian and viscoelastic luids (N. A. Patankar et. al a, Joseph et. al, Ko et. al ). It is an 7

8 Arbitrary-Lagrangian-Eulerian (ALE) numerical method using body-itted unstructured inite element grids to simulate particulate lows. A closely related numerical method or particulate lows, based on a Chorin (968) type ractional step scheme, was introduced by Choi (). Choi & Joseph () and N. A. Patankar et. al (b) use this scheme to study the luidization by lit o 3 circular particles in a plane Poiseuille low by direct numerical simulation. 3. Single particle lit-o The principal eatures o lit-o and levitation to equilibrium in D simulations are shown in igure 3. A heavy particle reely translating and rotating in contact with a plane wall in Poiseuille low is lited rom the wall and suspended in the luid i the shear Reynolds number R is greater than a critical value. Beyond the critical shear Reynolds number the particle rises rom the wall to an equilibrium height at which the buoyant weight F B just balances the hydrodynamic lit L rom the luid. In igure 3, γ w is the shear rate at the wall in the absence o the particle, U p and Ω p are the translational and angular velocities o the particle, respectively, at steady state and h e is the equilibrium height. L Ω p y x γ w h e F B U p Figure 3. Lit-o and levitation to equilibrium o a single particle. At steady state the lit orce on a reely rotating and translating circular particle in a Poiseuille low o a Newtonian luid depends on various parameters, (, h, ρ, η ) L = γ w e d,h, (7) where d is the particle diameter and H is the channel height. At equilibrium L = (πd /4)(ρ p - ρ )g. Using this relation and non-dimensionalizing (Buckingham s Pi theorem) we get R G = H he R,,, (8) d d 8

9 ρ γ w d where the shear Reynolds number R = and the gravity Reynolds number or non- η dimensional lit R G ( ρ ρ ) 3 ρ p gd =. Freely moving particles in steady low have zero η acceleration. The density ratio ρ p/ ρ vanishes as a parameter when the particle accelerations are zero. In practical applications the particle acquires some inite separation distance rom the wall due to the presence o surace roughness. N. A. Patankar et. al (a) and Ko et al. () deined the critical Reynolds number as the minimum shear Reynolds number required to lit a particle to an equilibrium height greater than.5d. They reported that the eect o the channel height on the critical shear Reynolds number or H /d > was not signiicant. At the critical condition, equation 8 implies that R G should be a unction o R only. The correlation that N. A. Patankar et al. (a) ound or the lit-o o a single circular particle in a Newtonian luid is o the orm R G = ar n, a =.36, n =.39. (9) This is similar to the orm expected rom equation 6 and shows that sel-similarity lies at the oundation o solid-liquid lows. Similar correlations were ound or lit-o in Oldroyd-B luids by Ko et. al (). There the luid elasticity was an additional parameter. We ound that obtaining a general expression or R G in terms o the R, h e /d and H /d is not straightorward. This is primarily because o the presence o multiple equilibrium position o a heavy particle at the same Reynolds number, irst detected by Choi & Joseph (). It was explained by N. A. Patankar et al. (a) to be due to the presence o a turning point biurcation o the equilibrium solution (igure 4). Turning point biurcations have also been ound in computations o levitation to equilibrium o particles in viscoelastic luids o Oldroyd-B type (Ko et al. ). This implies that the correlation unction in equation 8 will not be a single valued monotonous orm. We have not observed multiple equilibrium heights o the particle bed in our many particle simulations at the parameters we tested. Hence, many particle correlations or lit in terms o a single particle variable analogous to equation is not preerred. In the next subsection we see that it is more convenient to obtain many particle correlations similar in orm to equation 6. 9

10 Multiple stable equilibrium positions are not observed or neutrally buoyant particles. A neutrally buoyant particle has a stable equilibrium position between the channel axis and the wall. This is known since the well-known experiments o Segré & Silberberg (96). They ound that at low Reynolds number Poiseuille lows, the equilibrium position o a sphere rom the pipe axis was ound to be.6 times the pipe radius. A general expression or single particle lit orce analogous to the drag ormula, valid at all Reynolds numbers, has not been obtained yet. Joseph et al. () have suggested that 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 may be an important variable to model lit. It 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 translational slip velocity discrepancy does not change sign. The translational slip velocity is the luid velocity at the particle center when there is no particle minus the particle velocity. The angular slip velocity is similarly deined. 5 Equilibrium height 4 3 Equilibrium height as a unction o the shear Reynolds number or a particle o density. g/cm 3 Stable branch Unstable branch Shear Reynolds number Figure 4. (Patankar, Huang, Ko & Joseph ) Turning point "biurcations" shown in the equilibrium height (cm) vs. Reynolds number curve. There are two stable branches separated by an unstable branch. The channel height is cm. 3. DNS o levitation to equilibrium o 3 circular particles. The transport o a slurry o 3 heavier than liquid particles in a plane pressure driven low o Newtonian luids was studied using D DNS by Choi & Joseph () and N. A. Patankar et al. (b). Particles are initially placed at the bottom o a periodic channel o height H in a close packed ordered coniguration. The low is driven by an external pressure gradient. At steady condition, the particle bed reaches a constant height (igure 5). The height o the clear

11 luid region above the particle bed is H and the average luid raction in the particle bed is ε. Similar to equation 8 we expect that H R G = R,ε, ε, 3 max d where ε = - 4 Nπd [ H H ], ε l, max Nπd = -, 4H l () where N is the number o particles, l is the channel length and ε max is the luid raction in the particle bed i the particles occupy the entire height o the channel i.e. i H =. Figure 5. (N. A. Patankar, Ko, Choi & Joseph b) Lit-o o 3 heavy particles in a plane pressure driven low o a Newtonian luid, Re = 8. Contour plot o the horizontal velocity component is shown. During the simulations ε max and H /d were constant (N. A. Patankar et al. b, Choi & Joseph ). Thereore, R G should be a unction o R and ε only. N. A. Patankar et al. (b) obtained the ollowing correlation

12 R or R G G = 3.7 = ε 9.5 R.49 ε max H H H H 9.5 R.49. () The correlation above is o the same orm as that expected rom equation 6. This shows that luidization o slurries by lit also alls into enabling correlations o the RZ type and the above correlation by N. A. Patankar et al. (b) could be called a Richardson-Zaki type o correlation or luidization by lit. Lit results or luidized slurries are power laws in appropriate dimensionless parameters. These power laws are in the orm o engineering type 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 DNS results are in agreement with the expected power law orm in equation 6 rom the analogy between luidization by drag and shear. In the next sections we analyze the experimental data or proppant transport to veriy the prediction o power laws rom DNS. 4. Experimental setup Kerns, Perkins, and Wyant (959) reported the earliest experimental investigation o proppant transport in narrow slots. STIM-LAB did more experiments to better understand the processes involved in proppant transport by water and other thin luids. We have analyzed the data obtained rom their experiments. The apparatus used by STIM-LAB was constructed so that the transport o proppant in a horizontally oriented slot could be observed. A schematic o the apparatus is shown in igure 6.

13 Proppant Metering Feeder Open Standpipe Pressure Regulator Hose and Fluid Exit Fluid Supply Hose See igure 7 Initial Proppant Emplacement Channel Above Proppant Perorations by 8 oot Plexiglas Slot Proppant Trap Figure 6. The experimental setup or proppant transport. Proppant and luid are added at the let where they enter over the ull height o the slot. Materials exit at the right through perorations. Proppant can be added at a constant rate and water low rate is also constant. Proppant and water enter the 8 mm wide slot through an open end that is 3.5 cm tall. The proppant and water then move through the.44 m length o the slot where they exit via three 8 mm perorations spaced 7.6 cm apart on the 3.5 cm tall end o the slot. The proppant and water low rates were varied, proppants o varying size and density were added and water at dierent temperatures was used. Observations were recorded and portions o the experiments were video taped. The evolution o the proppant bed in the experiments is well described in igure 7. The portion shown in igure 7 is marked in igure 6. In the steady state there is an initial development length (see igure 6) ollowed by a lat bed region shown in igure 7 and marked in igure 6. There are three distinct zones in the lat bed region. The bottom part o the bed is immobile; it is a stationary porous medium that supports liquid throughput that might be modeled by Darcy's law. Above the immobile bed is a mobile bed in which particles move by sliding and rolling or advection ater suspension or a combination o these modes. Above the mobile bed is the clear luid zone. At steady state the volumetric luid low rate Q and the volumetric proppant low rate 3

14 Q p in and out o this region are constant. At steady state, these are equal to the rate at which the luid and proppant are injected in the slot. Q Clean luid H H Mobile bed Q Q p Q p Immobile bed Figure 7. Proppant transport in thin luids at steady state conditions. In Case only luid is pumped, Q P =, H = H ; the particles are immobile. In Case proppants are also injected, Q P, H H ; there is a mobile bed o height H - H. The channel width W = 8 mm. STIM-LAB carried out two types o experiments looking at the transport o proppants in thin luids. In Case only luid is pumped, Q P =, H = H ; the particles are immobile. In Case proppants are also injected, Q P, H H ; there is a mobile bed o height H - H. The channel width W = 8 mm. A simpliied description o the experiment is that a bed o proppant is eroded by the low o water. When proppant is not injected as in Case, the aster the low o water the deeper is the channel above the proppants. We are seeking to predict the height above the channel or the given luid low rate. In Case, we seek to predict both the clear luid height as well as the mobile bed height as unctions o Q and Q p. In the experiments the luid and the proppant low rates are controlled and the heights H, H are measured. In the DNS o 3 particles reported by Choi & Joseph () and N. A. Patankar et al. (b) (igure 5), we have a set up similar to that in igure 7. The value o H in igure 7 is equivalent to the height o the channel in the simulations. In the simulations, data is obtained or a ixed value o H /d. This is not the case with the experimental data. 5. Experimental correlations or sediment/proppant transport We develop combined correlations or the two cases studied experimentally by STIM-LAB. 5. Case : H = H = H 4

15 This case inds the critical condition o the initial motion o the proppant. Only luid is injected in the channel. The particle bed is immobile. There is an equilibrium value o H corresponding to a given luid low rate. When the luid low rate is increased beyond the critical value or a given initial height H, the proppants are eroded rom the bed and washed out until a new equilibrium height H o the clear luid region above an immobile bed is achieved or the new low rate. The data or Case was independently analyzed by D.D. Joseph & J. Wang and N.A. Patankar. We obtained similar correlations, presented below, rom both analyses. Table gives the data rom these experiments. The parameters o this problem are listed below: ~ ρ VW ρ Q ~ Q Fluid Reynolds number based on channel width Rq = =, wherev =. η Wη W 3 ρ [ ρ p ρ ] gd Gravity Reynolds number RG =. η Particle diameter/channel width Height H/channel width d/w. H/W. Figure 8 shows a plot o R q vs. H/W at dierent values o R G. R q = 558.(H/W).394 R G = R G = 45.8 Rq R q = 3.9(H/W).68 R G = R q = 53.4(H/W).937. H/W Figure 8. Plot o R q vs. H/W at dierent values o R G on a logarithmic scale. 5

16 Proppants d (cm) H (cm) η ρ Q ρ p R G V ~ (gm/cm-s) (gm/cc) (cc/s) (gm/cc) (cm/s) / Brady R q H/W / Ottawa /4 Light Beads / Carbolite / Carbolite /3 Banrite / Badger Table : Data rom experiments on the initiation o sediment motion (Case ). From igure 8 we get the ollowing orm or the correlation o the initiation o sediment motion R q = a[h/w] n, () 6

17 where a and n are unction o R G. We see that a and n may be regarded as constants or 5.37 R G The values o a and n listed in igure 8 are plotted in igures 9a and 9b, respectively. a E+6 R G Figure 9(a). Plot o a vs. R G on a logarithmic scale. n R G Figure 9(b). Plot o n vs. R G on a logarithmic scale. We deine an eective Reynolds number as 7

18 R e = R q [W/H] n, (3) where, the values o n are given in igure 8 and plotted in igure 9b. This leads to the ollowing expression or the critical eective Reynolds number R cr e or the initiation o bed motion R cr e = a(r G ) (4) The value o a is given in igure 8 and plotted in igure 9a. R e may be regarded as a Reynolds number that accounts or the eect o the side walls o the channel. Shield s (936) curve also gives the critical condition or the initiation o sediment motion. τ The Shields parameter S is deined as: S =, where τ is a measure o the shear stress [ ρ ρ gd p ] on the particle bed. I we take τ = η V~ W, then ~ ηv S = [ ρ ρ ] gd p = Rq[ d ] W. From the Shield s R G (936) curve one obtains (see also, Vanoni 975) S = s ( R [ d q ] ). Equation 4, applicable or W proppant transport in narrow channels has W/H as another parameter. 5. Case : H H There are two sets o data or Case :. /4 Ottawa sand: The parameters are given in Table. ρ p.65 gm/cc d.6 cm ρ gm/cc η. gm/cm-s W.8 cm R G 3496 Table. Parameters or experiments with /4 Ottawa sand in Case.. 6/3 Carbolite sand: The parameters are given in Table 3. 8

19 ρ p.7 gm/cc d.9 cm ρ gm/cc η. gm/cm-s W.8 cm R G 9 Table 3. Parameters or experiments with 6/3 Carbolite sand in Case. In this case the luid and the proppant are both injected at a speciied volumetric low rate. Since the proppants are injected, there is a moving particle bed in the channel. At equilibrium there is a clear luid region o height H above a moving bed o height H -H. The experimental data are given in tables 4 & 5. First we wish to obtain a correlation or the luid Reynolds number R q in terms o R G and ε. It is more convenient to obtain experimental correlations in terms o H and H instead o ε. Power laws in terms o H /H were also obtained rom numerical simulations (equation ). H H Q p Q Q=Q p +Q Q p /Q Q /Q cm cm cc/s cc/s cc/s Table 4. Experimental data or Case with Ottawa sand. 9

20 H H Q p Q Q=Q p +Q Q p /Q Q /Q cm cm cc/s cc/s cc/s Table 5. Experimental data or Case with Carbolite sand. A correlation obtained or Case should reduce to the correlation or Case when H = H. From Case we see that when R G = 3496 (Ottawa) and 9 (Carbolite) the critical eective Reynolds number R cr e = 53.4 and n =.937. We deine an eective Reynolds number at the luid proppant-bed interace as R e = R q (W/H ).937 or the Carbolite and Ottawa data. This is consistent with the deinition o R e in equation 3 since H is the height o the clear luid region. Figure shows a plot o R e vs. ln(h /H ) or the combined Carbolite and Ottawa data.

21 8 R e = 877(H /H ).648 Re 6 4 R e = 6838ln(H /H ) ln(h /H ) Figure. Plot o R e vs. ln(h /H ) or the combined Carbolite and Ottawa data. Two regimes are observed or the given values o R G : Regime : R e R cr e = 6838ln(H /H ). (5) Regime : R e = 877(H /H ).648. (6) Note that the y-intercept is taken to be equal to R cr e. Regime shows logarithmic behavior whereas regime shows power law behavior. When H /H =, we recover the correlation in equation 4. Thus equations 5 & 6 represent a combined correlation or the data rom Cases &. Regimes and have an overlap region; we may estimate the transition to begin at R e =. The ollowing general orm o the correlation may be proposed: ρ Q (W/H ) n- a(r G ) = 6838ln(H /H ); regime, (7) H η and ρ Q (W/H ) n- = 877(H /H ).648 ; regime. (8) H η Equations 5 & 6 can also be represented by a single equation using a technique described in the appendix by R. D. Barree. The single equation is given by

22 877 H.648 R e = (9) 9 5 H + H H.9 Equation 9 implies that correlations in the entire range can be represented by power laws connected by transition regions. The logarithmic regime above lies in the transition type region. The above correlation suggests the ollowing model or proppant transport. Consider some experiment with a channel o given height H and width W. Begin the luid low in the channel. Ater a critical value Q, obtained rom equation 4, the luid begins to move or erode the proppant in the bed. This results in a proppant low rate corresponding to the given values o Q and H. The height H remains almost the same but the depth H -H increases as the luid low rate is increased. The depth H to which the bed moves can be ound rom equation 5. This is Regime or the bed erosion regime. This regime may also be regarded as the bed-load transport regime. The correlation suggests a logarithmic behavior in the bed erosion regime. Further increase in the luid low rate induces suspension o the particles, hence the bed begins to inlate or expand. This situation is identical to our numerical simulations. Bed expansion now causes the value o H to decrease. This is Regime or the bed expansion regime. As expected rom our numerical simulation results, the experimental data also shows power law relation between the luid Reynolds number and H /H. The orms o numerical and experimental correlations are not identical obviously due to additional complexities such as three dimensionality and narrow slot in the case o experiments. Nevertheless, we observe that a power law behavior lies at the oundation o these lows. In order to obtain a ully predictive model another correlation or the proppant low rate is required. We ound that the data correlates best with a non-dimensional variable R p or proppant ρ Q p low rate given by R p =. The data is plotted in igure. η H

23 Re - Re cr R e - R e cr =.356R p.85 R e - R e cr = 57.7R p.397 R p Figure. Plot o R e R e cr vs. R p. Once again we observe two regimes consistent with those in igure. The correlations are Bed erosion regime: R e R e cr = 57.7R p.397 Bed expansion regime: R e R e cr =.356 R p.85 () () The transition point is around R e =. Using a technique presented in the appendix by R. D. Barree, equations & can be given by a single equation representing power laws connected by transition regions.397 p 57.7R cr R e Re = () 5.9 R p Equations 5- represent correlations or proppant transport in a narrow channel. Power laws are observed in agreement with the expectation rom numerical results and the analogy between luidization by drag and lit. The correlations in equations 5- can be used as a predictive tool e.g. to obtain the values o H and H or given values o luid and proppant low rates. A correlation in terms o ε, giving an expression identical to the orm in equation 6, may 3

24 be desirable to develop models or sediment transport or use in simulators. We are currently investigating this aspect urther. Prediction o the transition point between the bed erosion and bed expansion regimes may require urther experimental and numerical investigation. 6. 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 basis or this belie are the excellent correlations o experiments on luidization and sedimentation done by Richardson & Zaki and the correlations or sediment transport in horizontal channels obtained rom our numerical simulations and the analysis o the experimental data rom STIM-LAB. Results o two dimensional simulations o solid-liquid lows give rise to straight lines in loglog plots o the relevant dimensionless Reynolds numbers. Power laws are also obtained rom the analysis o experimental data. 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, experiments and the world o engineering correlations. The correlations obtained can be used as predictive tools or as a basis or models or sediment transport in simulators used or design purposes. Acknowledgments We acknowledge support by the National Science Foundation KDI/New Computational Challenge grant (NSF/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. NAP acknowledges the support rom Northwestern University through startup unds. 4

25 Appendix Fitting power-law data with transition regions by a continuous unction: Application to the Richardson-Zaki correlation R. D. Barree Barree & Associates LLC Many data sets representing naturally occurring phenomena can be described using a Sigmoidal distribution unction. One such unction that is particularly useul in itting physical data is the logistic dose response curve given by: ( b a) y = a + c x + t d (A) In this equation each o the constant terms or coeicients (a, b, c, d and t) have readily apparent physical signiicance, which allows data modeling to be accomplished almost by inspection. y y = + + ( ) ( x )... x Figure A. A typical logistic dose response curve. Figure A shows the dose-response unction or a =, b =, c =, d =, and t =. As can be seen, the coeicients a and b represent the values o the lower and upper plateaus o the unction, respectively, or its range. The coeicient t deines the value o the independent variable 5

26 (x) where the unction deviates rom the constant irst plateau value. The sharpness o curvature during the deviation rom the irst plateau is determined by the coeicient c. The slope o the power-law straight line in transition rom the irst plateau to the second plateau is determined by the product o coeicients c and d. The slope in this example is negative because both exponents are positive in the denominator o the rational raction... Curve B : Curve A : y =. ( ) + x y =. ( ). + x.. x y Curve C : y = + ( x ) Figure A. Eects o changing signs o coeicients c and d in the logistic dose response curve. The eects o changing the signs o the exponents can be examined (igure A). I the sign o coeicient c is changed, the plot is essentially rotated about a line parallel to the y-axis through the transition value t (compare curves A and B in igure A). I the sign o the coeicient d is changed, the plot is rotated about a line parallel to the x-axis through the upper bound b (compare curves A and C in igure A). These relationships allow construction o a transition unction in any general orm. Another useul property o this unction is that the bound corresponding to the coeicient a can be eliminated by setting its value to zero. With a = the unction yields a horizontal line at the upper bound value b and a power-law line o slope c d which extends to ininity (Curve A, igure A). Various unctions can then be modeled by products o unctions with speciied power-law slopes and transition points (igure A3). For curve A in igure A3 c =, d = and or curve B c =.6, d =. The inal power-law slope in the product is then c d + c d. 6

27 y Curve B: y =.6 [ + x ] Curve A : y = + ( x )... x Product o curves A & B Figure A3. Obtaining a curve rom the product o two dierent logistic dose response curves. Combinations o these unctions can be used in various orms to model many commonly observed phenomena. The logistic dose response curve can also be multiplied by a linear power law unction to impose an overall slope to the unction. Quite complex systems can be modeled by combining rational ractions or products o multiple unctions. This method has been used to model the Richardson-Zaki correlation that relates bed luidization velocity to the solids volume-raction o particles in suspension. The Richardson- Zaki correlation is given by equation. Speciically, the various unctions representing the exponent n are (Richardson & Zaki, 954) d n = D n = n = n = 4.45R n =.39. d D when d D R R when. when.3 5 < R when < R R when <.;. < R < R < 5; < < ; (A) 7

28 In these relations d is particle diameter, and D is the diameter o the luidization column. These unctions can be replaced with a single continuous orm o the logistic dose response curve where R is the independent variable and n is the dependent result (igure A4). n n(.<r<) n(<r<) n(<r<5) n(r<.) n(r>5) n(general). R Figure A4. A continuous logistic dose response curve or the Richardson-Zaki exponent n or d/d =. Figure A4 shows a continuous curve or the Richardson-Zaki exponent n. The continuous orm o the unction is ormed assuming that n should not decrease below a value o.39 or any value o Re. The continuous orm is generated by the equation n =.39 + ( d ) R + T This unction sets a minimum value o n =.39 and a maximum that is a unction o the ratio o particle size to vessel diameter (irst o equations A). The transition value T is also a weak unction o the diameter ratio d/d, and is given by e.7 D. (A3) T. = + d + D. (A4) The inal calculation o n is then given by a combination o equations A3 and A4. 8

29 The same technique has been applied to obtain equations 9 and or proppant transport in horizontal channels. Reerences E.S. Asmolov, 99 Dynamics o a spherical particle in a laminar boundary layer. Fluid Dyn. 5, E.S. Asmolov, 999 The inertial lit on a spherical particle in a plane Poiseuille low at large channel Reynolds number. J. Fluid Mech. 38, T.R. Auton, 987 The lit orce on a spherical body in a rotational low. J. Fluid Mech. 83, R.A. Bagnold, 966 An approach to the sediment transport problem or general physics. Geological Survey Pro. Paper 4-I, Washington, D.C. R.A. Bagnold, 974 Fluid orces on a body in shear-low; experimental use o stationary low. Proc. R. Soc. Lond. A. 34, G.I. Barenblatt, 996. Scaling, Sel Similarity and Intermediate Asymptotics. Cambridge Univ. Press. F.P. Bretherton, 96 Slow viscous motion round a cylinder in a simple shear. J. Fluid Mech., I. Celik and W. Rodi, 99 Suspended sediment-transport capacity or open channel low. J. Hydr. Engrg. ASCE, 7, 9-4. P. Cherukat, J.B. McLaughlin, and D.S. Dandy, 999 A computational study o the inertial lit on a sphere in a linear shear low ield. Int. J. Multiphase Flow 5, H.G. Choi, Splitting method or the combined ormulation o luid-particle problem, to appear in Comput. Meth. Appl. Mech. Engrg. H.G. Choi and D.D. Joseph,. Fluidization by lit o 3 circular particles in plane Poiseuille low by direct numerical simulation, accepted, J. Fluid Mech. A.J. Chorin, 968 Numerical solution o the Navier-Stokes equations. Math. Comput., D.S. Dandy, and H.A. Dwyer, 99 A sphere in shear low at inite Reynolds number: eect o shear on particle lit, drag and heat transer. J. Fluid Mech. 6, D.A. Drew, and S.L. Passman, 999 Theory o multicomponent luids, Springer-Verlag, New York. 9

30 R. Eichhorn, and S. Small 964 Experiments on the lit and drag o spheres suspended in a Poiseuille low. J. Fluid Mech., P.V. Foscolo, and L.G. Gibilaro, 984 A ully predictive criterion or transition between particulate and aggregate luidization. Chem. Engng. Sci. 39, A.J. Hogg, 994 The inertial migration o non-neutrally buoyant spherical particles in twodimensional lows. J. Fluid Mech. 7, H.H. Hu, 996 Direct simulation o lows o solid-liquid mixtures. Int. J. Multiphase Flow, H.H. Hu, and N.A. Patankar, Simulation o particulate lows in Newtonian and viscoelastic luids, to appear, Int. J. Multiphase Flow. H.H. Hu, N.A. Patankar, and M.Y. Zhu, Direct numerical simulations o luid-solid systems using the Arbitrary-Lagrangian-Eulerian technique, to appear, J. Comp. Phys. D.D. Joseph, D. D. 99 Generalization o the Foscolo-Gibilaro analysis o dynamic waves. Chem. Engng. Sci. 45, D.D. Joseph, Interrogations o direct numerical simulation o solid-liquid low. submitted or publication. D.D. Joseph, D. Ocando and P.Y. Huang,. Slip velocity and lit, accepted, J. Fluid Mech. T.K. Kern, T.K. Perkins and R.E. Wyant, 959. The mechanics o sand movement in racturing, Petroleum Transactions, AIME 6, T. Ko, N.A. Patankar and D.D. Joseph,. A note on the lit-o o a single particle in viscoelastic luids, submitted, Phys. Fluids. G.P. Krishnan, and D.T. Leighton, 995 Inertial lit on a moving sphere in contact with a plane wall in a shear low. Phys. Fluids 7, R. Kurose, and S. Komori, 999 Drag and lit orces on a rotating sphere in a linear shear low. J. Fluid Mech. 384, J.B. McLaughlin, 99 Inertial migration o a small sphere in linear shear lows. J. Fluid Mech. 4, R. Mei, 99 An approximate expression or the shear lit orce on a spherical particle at inite Reynolds number. Int. J. Multiphase Flow 8, J.F. Morris, and J.F. Brady, 998 Pressure-driven low o a suspension: buoyancy eects. Int. J. Multiphase Flow. 4,

31 T.-W. Pan, D.D. Joseph, R. Bai, R. Glowinski and V. Sarin,. Fluidization o 4 spheres: simulation and experiment, accepted, J. o Fluid Mech. N.A. Patankar, P.Y. Huang, T. Ko and D.D. Joseph, a. Lit-o o a single particle in Newtonian and viscoelastic luids by direct numerical simulation, accepted, J. Fluid Mech. N.A. Patankar, T. Ko, H.G. Choi and D.D. Joseph, b. A correlation or the lit-o o many particles in plan Poiseuille o Newtonian luids, accepted, J. Fluid Mech. J.F. Richardson and W.N. Zaki, 954. Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs. 3, S.I. Rubinow, and J.B. Keller, 96 The transverse orce on a spinning sphere moving in a viscous luid. J. Fluid Mech., P.G. Saman, 965 The lit on a small sphere in a slow shear low. J. Fluid Mech., 385-4; and Corrigendum 968 J. Fluid Mech. 3, 64. A. Shields, 936 Anwendung der aenlichkeitsmechanik und der turbulenzorschung au die geschiebebewegung. Mitteilungen der Preussischen Versuchsanstalt ur Wasserbau und Schibau, Berlin, Germany, translated to English by W.P. Ott and J.C. Uchelen, Caliornia Institute o Technology, Caliornia. J.A. Schonberg and E.J. Hinch 989 Inertial migration o a sphere in Poiseuille low. J. Fluid Mech. 3, B.M. Sumer, 986 Recent developments on the mechanics o sediment suspension. Transport o suspended solids in open channels, Euromech 9, Neubiberg, W. Bechteler, ed., Balkema, Rotterdam, The Netherlands, 3-3. G. Segré, and A. Silberberg, 96 Behavior o macroscopic rigid spheres in Poiseuille low, Part. Experimental results and interpretation. J. Fluid Mech. 4, L.C. van Rijn, 984a Sediment transport, Part I: Bed load transport. J. Hydr. Engrg. ASCE, L.C. van Rijn, 984b Sediment transport, Part II: Suspended load transport. J. Hydr. Engrg. ASCE, J. Ye, and M.C. Roco, 99 Particle rotation in a Couette low. Phys. Fluids A 4, -4. 3