Bi-power law correlations for sediment transport in pressure driven channel flows

Transcription

1 Bi-ower law correlations or sediment transort in ressure driven channel lows J. ang, D. D. Joseh Deartment o Aerosace Engineering and Mechanics, University o Minnesota, Minneaolis, MN N. A. Patankar Deartment o Mechanical Engineering, Northwestern University, Evanston, IL 6008 M. Conway STIM-LAB, Duncan, OK D. Barree Barree & Associates LLC, Lakewood, CO 8035 October, 00 Abstract Power law correlations or sediment transort in ressure driven channel low was derived in the revious work o the authors. Those correlations can be used as redictive tools in the racturing industry but they are in imlicit orms. New data rom slot exeriments or ractured reservoir were collected which enable us to derive correlations in exlicit orms and alicable over a wide range o data. In this aer we resent correlations or bed load transort o slurries as a comosition o bi-ower laws in the roant and luid eynolds number with exonents and reactors exressed as logarithmic unctions o dimensionless sedimentation numbers. The bed load transort correlations are very accurate and are aarently the only such in the ublished literature.. Introduction A good descrition o the roblem o roant lacement in ractured reservoir in which the resent work is ramed can be ound in the recent aer by Patankar, Joseh, ang, Barree, Conway & Asadi (00). Only small arts o that descrition need to be reeated here to make this document easy to understand. ydraulic racturing is a rocess oten used to increase the roductivity o oil and gas wells. A luid-roant mixture is injected through a well bore to be stimulated, at suicient ressure to oen a vertical racture enetrating rom the well bore ar into the ay zone. Figure shows the side view o the crack. A mound o roant develos and grows until the ga between the to o the crack and the mound reaches an equilibrium value; this value is associated with a critical condition. For velocities below critical the mound gets higher and sreads laterally; or larger velocities roant is washed out until the new equilibrium height and velocity are established (Kern, Perkins & yant 959).

2 ell Bore Uer Fracture Boundary Fluid Sand Injected Early Sand Injected Late Figure. (Patankar et al. 00) Sand (roant) transort in a ractured reservoir. The major objective o this aer is to rocess the exerimental data on roant transort in slots rom STIM-LAB using the method o correlations. The resultant engineering correlations or erosion and bed load transort can be used to redict roant lacement in the crack. The method o correlations alied to real or to numerical exeriments is a way to derive ormulas and analytic exressions rom the rocessing o data. enerating correlations rom exeriments is an old method which many industrial alications are based on but it has come to have a bad name, viewed as emirical and not undamental. owever, used with care and understanding o the hysics o the roblem, correlation method can generate outstanding results. A good examle is the ichardson-zaki correlation (954) which is obtained by rocessing the data o luidization exeriments. ichardson-zaki correlation describes the comlicated dynamics o luidization by drag and is widely used or modeling the drag orce on articles in solid-liquid mixtures. Our enthusiasm or correlations has to do with the surrising emergence o correlations rom the simlest kind o ost-rocessing o our numerical exeriments. e have done lit correlations or single article (Patankar, uang, Ko and Joseh 00) and or the bed exansion o many articles in slurries (Patankar, Ko, Choi and Joseh 00). The rocedure we ollow is to lot the results o our simulations in log-log lots o the relevant dimensionless variables. The surrise or us is that these lots requently come u as straight lines giving rise to ower laws which is not redictable or at least ar rom obvious. For examle, a single article will lit-o in a Poiseuille low at a certain eynolds number =Ud/ or a given settling eynolds number 3 / ( ) gd D. hen we lotted the lit-o criterion rom about 0 oints we ound that n a with an intercet a and a sloe n in the log-log lot. The straight lines are imressively straight and we generated such correlations or lit to equilibrium, or the bed STIM-LAB is a research lab in Duncan, OK, which is suorted by a consortium o oil roduction and oil service comanies. STIM-LAB has been collecting data on roant transort in slots or 5 years.

3 exansion o many articles and in non-newtonian luids. The rediction o the ower laws or roant transort rom DNS is veriied by the engineering correlations obtained rom exerimental data in Patankar, Joseh, ang, Barree, Conway and Asadi (00). Patankar et al. rocessed the data rom slot exeriments or ractured reservoirs on log-log lots and generated ower law correlations with a arameter deendent ower. The existence o such ower laws is an exression o sel-similarity, which has not yet been redicted rom analysis or hysics. The low o disersed matter aears to obey those sel-similar rules to a large degree (Barenblatt 996). e can get ower laws when only two variables are at lay; when there are three variables or more, it would aear that we get dierent ower laws searated by transition regions. This is certainly the case or the ichardson-zaki correlation; it has one ower law relating the luidization velocity to the solids raction at low eynolds number, and another at high eynolds number with a eynolds number-deendent transition between. The ichardson-zaki correlation is an examle o what Barenblatt (996) calls incomlete sel-similarity because o the deendence o the ower on the eynolds number, a third arameter. e got such correlations between three variables or slurries, and rom numerical exeriments (Choi and Joseh 00, Patankar, Ko, Choi and Joseh 00, Pan, Joseh, Bai, lowinski and Sarin 00, Joseh and Ocando 00, Patankar, Joseh, ang, Barree, Conway and Asadi 00, Joseh 00). ere we encounter the yet more comlicated situation in which our or ive arameters enter and we ind a new tye o solution in bi-ower law correlations. The correlations aithully describe the erosion and bed load transort in a slot under a wide variety o conditions.. Exerimental setu The exerimental setu used by STIM-LAB was described thoroughly in Patankar et al. (00). ere we rovide a brie descrition o the aaratus used by STIM-LAB where the transort o roant in a horizontally oriented slot could be observed. Figure shows the aaratus schematically. Proant and water enter the 8 mm wide slot through an oen end that is 30.5 cm tall. The roant and water then move through the.44 m length o the slot where they exit via three 8 mm erorations saced 7.6 cm aart on the 30.5 cm tall end o the slot. e call attention to that new exeriments were conducted in STIM-LAB where more luids and roants were used. The correlations resented in this aer describe both old and new data very well. 3

4 Proant Metering Feeder Oen Standie Pressure egulator ose and Fluid Exit Fluid Suly ose See igure 3 Initial Proant Emlacement Channel Above Proant Perorations by 8 oot Plexiglas Slot Proant Tra Figure. (Patankar et al. 00) The exerimental setu or roant transort. Proant and luid are added at the let where they enter over the ull height o the slot. Materials exit at the right through erorations. The roant bed in the exeriments is shown in igure 3 and the ortion show in igure 3 is marked in igure. There are three distinct zones in the lat bed region. The bottom art o the bed is immobile; it is a stationary orous medium that suorts liquid throughut that might be modeled by Darcy's law. Above the immobile bed is a mobile bed in which roants move by sliding and rolling or advection ater susension 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 roant low rate Q in and out o this region are constant. At steady state, these are equal to the rate at which the luid and roant are injected in the slot. 4

5 Q Clean luid Mobile bed Q Q Q Immobile bed Figure 3. (Patankar et al. 00) Proant transort in thin luids at steady state conditions. In Case only luid is umed, Q P = 0, = ; the articles are immobile. In Case roants are also injected, Q P 0, ; there is a mobile bed o height -. The channel width = 7.94 mm. STIM-LAB carried out two tyes o exeriments. In Case only luid is umed, Q P = 0, = ; the articles are immobile. e call case erosion case. In Case roants are also injected, Q P 0, ; there is a mobile bed o height -. e call Case bed load transort case. The channel width = 7.94 mm. A simliied descrition o the exeriment is that a bed o roant is eroded by the low o water. hen roant is not injected as in Case, the aster the low o water the deeer is the channel above the roants. e are seeking to redict the height above the channel or the given luid low rate. In Case, we seek to redict both the clear luid height as well as the mobile bed height as unctions o Q and Q. In the exeriments the luid and the roant low rates are controlled and the heights and are measured. 3. Exerimental correlations or sediment transort 3. Dimensionless arameters The dimensionless arameters in this roblem are listed below: ravity eynolds number 3 [ ] gd. () D ravity eynolds number or the luid 3 D / g. ( 3/ ) () g D 5

6 Note that is constant in the exeriments, hence, can be viewed as the dimensionless orm or the kinematic viscosity o the luid D /. Fluid eynolds number based on channel width ~ V Q ~ Q, where V D D. (3) Proant eynolds number based on channel width V Q Q, where V. D D (4) Particle diameter/channel width d/. eight o bed/channel width /. e are seeking correlations between the height o the bed /, / and the rescribed arameters. The deendence o / and / on the three eynolds numbers, and arises rom the undamental mechanics, as shown in our revious DNS works. In the rocess o itting the bed load transort data, we ound that the resence o the gravity eynolds number or the luid in the correlations is necessary or satisactory itting. ence, the correlations are in the orm / (,,, ) where i =,. i (i =, ) deend on eight dimensional i variables namely,, d,d,, Q, Q and g. The undamental dimensions are M (mass), L (length), and T (time). The Pi theorem indicates that the total number o dimensionless arameters involved in develoing correlations or i is six. The number o dimensionless arameters in / (,,, ) is ive; one less than the number indicated by the Pi i theorem. The reason is that d and are constant in the exeriments, hence, d/ does not aear in the correlations. 3. Power law correlations or the erosion case The erosion case: = = inds the critical condition o the initial motion o the roant. Only luid is injected in the channel and the article bed is immobile. There is an equilibrium value o corresonding to a given luid low rate. hen the luid low rate is increased beyond the critical value or a given initial height, the roants are eroded rom the bed and washed out until a new equilibrium height o the clear luid region above an immobile bed is achieved or the new low rate. 6

7 In the erosion case, three dimensionless arameters, /, and enter the ower law correlation : m( ) / a( ) (5) The values o a and m are listed in table as unctions o. Details o the derivation o the ower law correlations or the erosion case can be ound in Patankar et al. (00) a m Table. The reactor a( ) and exonent m( ) in the ower law correlations or the erosion case. 3.3 Bi-ower law correlations or the bed load transort case Bed load transort is another name or the transort o sediments. In bed load transort, both luids and roants lay imortant roles in determining and. Thereore we seek correlations or / and / in terms o and with the coeicients as unctions o and. To create correlations, we need data and a data structure. An examle o the way the data is structured or rocessing correlations is given or 0/40 Ottawa in water in table. : 0 3 kg/m 3, : kg/m 3, d: m, D: 0-3 Pa.s, : m, : , : Q (m 3 /s) Q (m 3 /s) (0 - m) (0 - m) / / Shield s (936) curve also gives the critical condition or the initiation o sediment motion. The Shields arameter J S is deined as: S [ ] gd, where J is a measure o the shear stress on the article bed. I we take J D V~, then S [ ~ DV ] gd [ d ]. From the Shield s (936) curve one obtains S = s ( [ d ] ). Equation 5, alicable or roant transort in narrow channels has / as another arameter. Nothing close to the bi-ower law correlations has been ut orward or sediment transort. 7

8 Table : Exerimental data or the bed load transort case with 0/40 Ottawa and water. e look or correlations in the bi-ower law orm with ive dimensionless arameters involved: m (, ) n ( ) c ( ) ; (6) m (, ) n ( ) c( ). (7) Following are the rocedures we used to achieve the bi-ower correlations: () Dierent kinds o roant and luid are used in bed load transort exeriments and lead to dierent values o and. For each single case, we develo bi-ower law correlations o and. () The reactors and exonents in these correlations are unctions o and. e imlement curve-itting to ind analytical exression or these coeicients. (3) Curve itting imlies that c, n, c and n can be reasonably aroximated by logarithmic unctions o. hile the trend o m, m is less obvious. (4) e use the redicted c, n, c, and n by the logarithmic unctions o and vary m, m in the bi-ower law correlations to match the measured and consistently. The new m and m turn out to be also logarithmic unctions o, but with sloes and intercets as unctions o. (5) ith the exlicit and analytical exressions or all the coeicients in the bi-ower law known: c ( ), c ( ), n ( ), n ( ), m (, ), m (, ), we redict and and comare them with the exerimentally measured values. The analytical exressions c ( ), c ( ), n ( ), n ( ), m (, ) and m (, ) can be adjusted obtain the best it or and. ence, we obtain the inal orm or the analytical exressions: (8) (3). Then they are inserted to (6) and (7), giving rise to (4) and (5) as the inal orm or the biower law correlations. Next, we resent the analytical exressions or the reactors and exonents in the bi-ower law correlations. These exressions are lotted in igures

9 4 3 c.30 0 ln( ).9 0 ; (8) 4-3 c.5 0 ln( ).33 0 ; (9) c = -.30 " ln( ) c and c c c c = ln( ) Figure 4. Preactors c and c as logarithmic unctions o. n 0.07ln( ) 0.0; (0) n ln( ) () n = -0.07ln( ) - 0. n and n n n n = ln( ) Figure 5. Exonents n and n as logarithmic unctions o. m m ln( ) ; () ln( ). (3) 9

10 m c glycol 5 c glycol water 50 F water E+07 Figure 6. Exonent m as a logarithmic unction o with the sloes and intercets as unctions o. m c glycol 5 c glycol water 50 F water Figure 7. Exonent m as a logarithmic unction o with the sloes and intercets as unctions o. From igures 4-7, we can see that c, n, c and n can be reresented by logarithmic unctions o, while m and m are logarithmic unctions o with sloes and intercets as unctions o. Equation () imlies that or any, the logarithmic curve m (, constant) asses through the oint (m =., ln( )=5.). Equation (3) shows that such a oint or m (, ) is (m =.0, ln( )=.67). In igure 6 and 7, we can see the two oints. 0

11 To obtain satisactory itting or the data o bed load transort, we ind that c, n, c and n deend on, whereas m and m deend on both and. Following is a ossible hysical exlanation or such deendence. m and m are exonents or the luid eynolds number and could be sensitive to the kinematic viscosity o the luid, D /. Note that is a dimensionless orm o the kinematic viscosity o the luid. In contrast, the reactors c and c and the exonents o the roant eynolds number n and n are less sensitive to the luid roerties. ence, dose not aear in the exressions or c, n, c and n. In table 3, the bed load transort exeriments with the corresonding and are listed. Note that the roant and luid used in these exeriments and their roerties can be ound in table A.. c, c, m, m, n and n listed in table 3 are redicted by (8) (3) corresonding to and listed in the irst and second columns. These c, c, m, m, n and n have been lotted in igures 4 7, indicated by oints. c c n n m m Table 3: and or bed load transort exeriments and the corresonding c, c, m, m, n and n redicted by (8) (3). Inserting the analytical exressions (8) (3) into (6) and (7), we get the inal orm or the biower law correlations: [ ln( ) ] [5. ln( )] [ 0.07ln( ) 0.0], (4)

12 [ ln( ) ] [.67 ln( )] [ ln( ) 0.304] (5) e emhasize that equations (4) and (5) are exlicit and redictive correlations or roant transort. By (4) and (5), and can be redicted rom the rescribed arameters:,, d,d,, Q, Q. e redict / and / by (4) and (5). In igure 8 and 9, we lot the redicted values against the exerimentally measured data. Ideally, all the oints should be on the straight-line y=x. It can be seen that the redicted values are in good agreement with the exerimental data.. redicted / measured / =3.50e3 =.59e4 =.8e3 =8.90e3 =648 =7.5 =07 =.8e4 =7.35e4 =6 =73 =644 =5.8 Figure 8. The redicted values o / by equation (4) versus the exerimentally measured values or the cases listed in table 3.

13 redicted / =5.8 =7.5 =07 =6 =73 =644 =648 =.8e3 =3.50e3 =8.90e3 =.8e measured / =.59e4 =7.35e4 Figure 9. The redicted values o / by equation (5) versus the exerimentally measured values or the cases listed in table 3. To test this correlation, exeriments were conducted in the slot which is 4.88 m long and. m high. (Note that the correlations are extracted rom exeriments conducted in the slot which is.44 m long and 30.5 cm high.) Following are the rescribed arameters or the exeriments in the 4.88-m-long slot: 0/40 Ottawa with water, : kg/m 3, : kg/m 3, d: m, D: Pa.s, : m, : , : redicted / redicted / redicted (m) redicted (m) measured (m) measured (m) Table 4. The redicted and in comarison with the measured and in exeriments conducted in the slot which is 4.88 m long and. m high. e can see that the agreement between redicted values o and and measured values is encouraging. It is also noted that table 4 does not show good agreement between redicted and measured values o -. Such inconsistency can be generated in two ways. First, the measurement o the mobile bed height - is not quite accurate. The total height is airly 3

14 reliable measurement but the traction caret thickness is subject to interretation esecially at higher rates where a clear demarcation between the clear layer and the traction caret is more diicult to see since some roant is still transorted by viscous orces. The measurements were made by dierent observers and variations in the recorded heights could be in the order o one centimeter. That s in the same order o -. Due to such errors, the redicted values o - are not in good agreement o the measured values. Second, our correlations are derived or the total heights and the coeicients in the correlations are adjusted to obtain the best it or and. Thereore, the agreement between redicted and measured values o - is not as good the agreement or and. The irst reason has to do with the exerimental uncertainty; the second with the method o rocessing. The corrlations or / and / are bi-ower laws in and with the coeicients as logarithmic unctions o and. In the inal orm o the correlation, equation (4) or (5), eight itting arameters are used. These arameters emerge as the intercets and sloes o the logarithmic unction and are not arbitrary. Considering the wide range o exerimental data covered by the correlations, we believe that the number o itting arameters is reasonable. Our bi-ower law correlations reveal the hidden sel-similarity in the low o disersed matter. They are also in a convenient orm and alicable to a wide range o data. e believe that our correlations rovide a romising way to redict transort o roant. 3.4 Logistic dose curve itting or / and / The bi-ower law correlation gives good rediction o / and / or the bed load transort case. owever, it is not comatible with the erosion case. hen aroaches zero, / and / tend to ininity. Thereore we need a dierent correlation to account or the transition region rom the erosion case to the bed load transort case. e it the data or / and / to a logistic dose curve (see the Aendix in Patankar et al. 00 or details) to determine a unction valid in the transition region; this itting eectively combines the ower law or the erosion case and the bi-ower law or the bed load transort case. e seek to determine the unction: M C (6) ( T ) N 4

15 and M C. (7) ( T hen =0 (the erosion case), (6) and (7) reduce to: ) N M, (8) C M C. (9) For the erosion case, = = ; hence, C =C, M =M and we recover the ower law correlation reduce to: m( ) / a( ) or the erosion case. hen >>T and >>T, (6) and (7) ( N N ) M C T, (0) ( N N ) M C T. () Thereore, we recover the bi-ower law correlations (6) and (7) or the bed load transort case. Comaring (0) and () to (6) and (7), we observe that M and M should be unctions o both and ; hence, the exonent M =M in the ower law correlations or the erosion case should be unctions o and. owever, most o the erosion exeriments were conducted using water at dierent temeratures and lead to a small range o (See table A.); we do not have enough data to ind an analytical exression M (, )=M (, ). owever, we ind that C, M, T, N, C, M, T and N can be reasonably aroximated by unctions o a single variable ; hence, the coeicients in (6) and (7) are unctions o only. e do not have data or erosion and bed load transort with the same (See table A. and A.) so that we use (6) and (7) to it data rom erosion and bed load transort with dierent but close : : Erosion case with =86.8 and bed load transort case with =09, : Erosion case with =5 and bed load transort case with =648, 3: Erosion case with = and bed load transort case with = , 4: Erosion case with = , and bed load transort case with =

16 The results o itting erosion with =86 and bed load transort with =09 are resented below: () 0. (I = 0.95) (I = 0.946) ( ) 68.3 ( ) 59.4 By () & (3), / and / are comuted and tabulated in table 5. / / comuted / 0.56 error comuted / Table 5: Data structure or the combination o erosion case with =86 and bed load transort case with =09. / and / calculated by equation () & (3) are listed in comarison with the exerimentally measured values. The error is comuted by (calculated value - measured value). error (3) 6

17 calculated / I = 0.95 y = x exerimental / Figure 0(a): Exerimental / vs. calculated / using (). calculated / y = x I = exerimental / Figure 0(b): Exerimental / vs. calculated / using (3). e can see that the comuted / and / are in good agreement with the exerimentally observed values. Cases, 3 and 4 are rocessed in the same manner and resultant coeicients C, M, T, N, C, M, T, N and I-squared values are listed in table 6. Note that in table 6, we use the o the bed load transort case or the combination o data. C M T N I C M T N I

18 Table 6: The coeicients in the logistic dose curve itting o / and / or the data rom erosion and bed load transort with close. e only list the o the bed load transort case in the irst column. Next we obtain the unctions C ( ), M ( ), T ( ), N ( ), C ( ), M ( ), T ( ), and N ( ). 0. C =C C =C = I = Figure : The coeicient C = C as a unction o. C. (4) C (I = 0.979).40 0 M =M M =M = I = Figure : The coeicient M = M as a unction o. M 0.5 M (5) (I = 0.994) 8

19 00 T = T and T 0 T = I = I = T T Figure 3: The coeicient T and T as unctions o. 4.7 T.47 0 ; (6) (I = 0.930) T. (7) (I = 0.976) N and N N N Figure 4: The coeicient N and N lotted against. e use a natural cubic sline to interolate N ( ) and N ( ) and the results are lotted in igure 5. Because we do not have enough data, the sline interolation is not reliable and could be signiicantly changed when more data become available. Such a itting is at best tentative and is meant to show that exressions or N ( ) and N ( ) could be obtained i we had enough oints. 9

20 Figure 5. Use natural cubic sline to interolate N ( ) and N ( ). The resultant exressions o the sline interolation are: 3 N A B D E (8) where the values o A, B, D, E are listed in the ollowing table: ange A B D E Table 7: The coeicients in the sline intoolation or N ( ). N 3 A B D E (9) where the values o A, B, D, E are listed in the ollowing table: The inal correlations: ange A B D E Table 8: The coeicients in the sline intoolation or N ( ) , (30) N ( ) ( )

21 (3) N ( ) ( ) where N ( ) and N ( ) are exressed in (8) and (9). Correlations (30) and (3) can be used to redict / and / with the rescribed arameters,,, d D, Q and Q. These correlations are comatible with both the ower law or the erosion case and the bi-ower law or the bed load transort case. e develoed the logistic dose curve by itting the data or erosion and bed load transort using equation (6) and (7). The curve should deend on and but we do not have enough data to determine how it deends on. This itting using the dose curve is indeendent o the revious ower law and bi-ower law correlations and the resultant equation (30) and (3) do not reduce recisely to equation (5), (4) and (5) at the resective limits. The number o itting arameters utilized in (30) and (3) is more than in (4) and (5). The reason is that we are not able to ind elegant exressions or N ( ) and N ( ) rom sarse data (see igure 4). e emhasize that the logistic dose curve reresents a correlation or transition situations between erosion and bed load transort; it is not as accurate as the ower law (5) or erosion or the biower law (4) and (5) or bed load transort. 4. Discussion In a sense our results here realize the oortunity which is resented by digital technology or imlementing the old tried and true method o correlations to big data sets. Our mantra is the secrets are in the data. The same method works well or data rom numerical and rom real exeriments. e have used the ollowing rocedure not only or this but also or other multihase low rocesses. First we roose candidates or controlling dimensionless arameters and list the data required to orm these numbers in a sreadsheet. Then we identiy two arameters and lot the results o the exeriments or those two in log-log lots under conditions in which other arameters are ixed. e have a good choice when the lots come u as straight lines in the log-log lot. ere and elsewhere we have had excellent results in this search using the arameters suggested by making the governing sets o PDE s dimensionless. The results o this kind o ower law rocessing is that the sloes and intercets o the straight lines in log-log lots, or the reactors and exonents o the ower laws these lines imly, deend on the arameters we have ixed. hen we look at the variation o these arameters, the reactors and

22 exonents sometimes are exressible as ower laws or logarithmic unctions, and sometimes they are not. In any case we may and do imlement curve-itting rocedures or the reactors and exonents to resent exlicit ormulas in analytic orm or the rediction o uture events. Processing o data when the reactors and exonents are not exressible by ower laws or logarithmic unctions, as in the case o logistic dose curve itting (see igure 5), can be carried out by sline or other tye o itting algorithm, but accurate its are obtained only when there are suicient data. e might add that the search or the governing dimensionless numbers in multihase low is a way to achieve a dee understanding o the underlying hysics. The method o correlations is an excellent rocedure to guide the research because the data doesn t lie. Acknowledgments e acknowledge suort by the National Science Foundation under grant oortunities or academic liaison with industry (OALI), by the DOE, Deartment o Basic Energy Sciences, by a grant rom the Schlumberger oundation, rom STIM-LAB Inc. and by the Minnesota Suercomuter Institute. NAP acknowledges the suort rom Northwestern University through startu unds. Aendix Following tables give descrition to the roant and luid used in the erosion and bed load transort exeriments. roant luid (0 3 kg/ m 3 ) (0 3 kg/ m 3 ) d (0-4 m) D (0-3 Pa.s) (0-3 m) 60/4 Brady water /4 Ottawa water /4 Light Beads water /30Banrite water /0Badger water /0 Carbolite water /0 Carbolite 80 F water Table A.: Proant and luid arameters in erosion exeriments. roant luid (0 3 kg/ m 3 ) (0 3 kg/ m 3 ) d (0-4 m) D (0-3 Pa.s) (0-3 m) 0/40 sand 0c glycol

23 6/30 ceramic 0c glycol /0 sand 0c glycol /60 Sand 5c glycol /40 sand 5c glycol /40 sand 5c glycol /60 sand water /40 sand water /40 sand water /40 Bauxite water /30 Ceramic water /30 Ceramic water /40 Sand 50 o F water /30 Ceramic 50 o F water Table A.: Proant and luid arameters in bed load transort exeriments. eerences.i. Barenblatt, 996. Scaling, Sel Similarity and Intermediate Asymtotics. Cambridge Univ. Press... Choi and D.D. Joseh, 00. Fluidization by lit o 300 circular articles in lane Poiseuille low by direct numerical simulation, J. Fluid Mech. 438, u, N.A. Patankar, and M.Y. Zhu, 00 Direct numerical simulations o luid-solid systems using the Arbitrary-Lagrangian-Eulerian technique, J. Com. Phys. 69, D.D. Joseh, 00 Interrogations o direct numerical simulation o solid-liquid low. htt:// submitted or ublication. D.D. Joseh, and D. Ocando 00. Sli velocity and lit, J. Fluid Mech., 454, T.K. Kern, T.K. Perkins and.e. yant, 959. The mechanics o sand movement in racturing, Petroleum Transactions, AIME 6, T.-. Pan, D.D. Joseh,. Bai,. lowinski and V. Sarin, 00. Fluidization o 04 sheres: simulation and exeriment, to aear J. Fluid Mech. N.A. Patankar, P.Y. uang, T. Ko and D.D. Joseh, 00. Lit-o o a single article in Newtonian and viscoelastic luids by direct numerical simulation, J. Fluid Mech. 438, N.A. Patankar, D.D. Joseh, J. ang,.d. Barree, M. Conway and M. Asadi, 00 Power law correlations or sediment transort in ressure driven channel lows, Int. J. Multihase Flow, 8(8)

24 N.A. Patankar, T. Ko,.. Choi and D.D. Joseh, 00. A correlation or the lit-o o many articles in lan Poiseuille o Newtonian luids, J. Fluid Mech. 445, J.F. ichardson and.n. Zaki, 954. Sedimentation and Fluidization: Part I, Trans. Instn. Chem. Engrs. 3, P.N. owe, 987. A convenient emirical equation or estimation o the ichardson-zaki exonent, Chem. Engng. Sci. 4, A. Shields, 936 Anwendung der aenlichkeitsmechanik und der turbulenzorschung au die geschiebebewegung. Mitteilungen der Preussischen Versuchsanstalt ur asserbau und Schibau, Berlin, ermany, translated to English by.p. Ott and J.C. Uchelen, Caliornia Institute o Technology, Caliornia.. Segré, and A. Silberberg, 96 Behavior o macroscoic rigid sheres in Poiseuille low, Part. Exerimental results and interretation. J. Fluid Mech. 4,