SIGNIFICANT PARAMETERS AFFECTING EFFICIENT CUTTINGS TRANSPORT IN HORIZONTAL AND DEVIATED WELLBORES IN COIL TUBING DRILLING: A CRITICAL REVIEW V. C. Kelessidis, G. Mpandelis, A. Koutroulis, and T. Michalakis, all Technical Univ. Of Crete, Greece. Paper presented at the 1st International Symposium of the Faculty of Mines (ITU) on Earth Sciences and Engineering, 16-18 May 2002, Maslak, Istanbul, Turkey. ABSTRACT Coiled Tubing Drilling (CTD) has grown significantly in recent years as a niche market for drilling jobs considered not good candidates for conventional rotary drilling due primarily to high cost associated with them. CTD is normally associated with high angle to horizontal and extended reach wells. It is, however, in these applications that hole problems, due to inefficient cuttings removal, become more troublesome. These problems are enhanced due to the non rotation of the inner string (the CT itself) as opposed to normal (or conventional) drilling where the inner pipe rotation imparts some motion and agitation of the drilled solids thus minimizing hole problems due to cuttings accumulation. Many parameters affect efficient cuttings transport in Coil Tubing Drilling in horizontal and deviated geometries. Among these are pump rates, well dimensions, fluid properties, solids sizes and solids loading. Drilling should be designed and carried out so that all cuttings are maintained in suspension and no cuttings bed is formed. Several attempts have been made to determine the optimum operating range of these parameters but complete and satisfactory models have yet to be developed. The purpose of this paper is to provide a critical review of the state of the art of previous approaches to efficient cuttings transport during Coil Tubing Drilling, present the critical parameters involved and establish their range according to what is observed in practice through dimensional and similarity analysis. Finally the laboratory system that has already been set up is presented. Its primary purpose is to allow the gathering of good quality data, which are really missing from the literature, which could aid into better understanding the flow of solid – liquid mixtures in annuli. INTRODUCTION Coil Tubing Drilling is the process of drilling for oil and gas wells where the traditional drill pipe is replaced with a continuous coil, the Coil Tubing. It has started for services, other than drilling, in oil and gas wells. From the introduction of Coil Tubing Drilling in early 1990s, there has been considerable increase in the number of wells drilled with CT and the number of available CT Units. The advantages of CTD are numerous and have been indicated and proved in practice by a large number of investigators. Among these are:

• ability to drill underbalanced resulting in higher ROP and less formation damage

• reduction in overall time spent and hence reduction in cost of drilling (both drilling and tripping time)

• better performance for re-entry wells (major market for CTD) and slim holes • well placed for drilling highly deviated / ultra short radius wells • smaller footprint, hence less environmental damage • requirement for less personnel • increased safety with better control of wellbore pressures (Blow Out

Preventers in operation all the times)

2

while the disadvantages are

• buckling of CT, especially in highly deviated / horizontal wells • less options for hole sizes to be drilled • no pipe rotation thus making it more difficult for efficient cuttings transport

CTD usually occurs either in the overbalanced mode (drilling fluid pressure higher than formation fluids pressure) or in the underbalanced mode. The latter is normally achieved with the injection of a gas (nitrogen, air, natural gas) or with co-production fluids resulting in three phase flow in the annulus, the drilling fluid (usually liquid), the drill cuttings (solids) and the ‘underbalanced phase’ (normally gas or foam). We will not tackle the three phase problem, at least in this paper. We are mainly concerned with the two phase problem, that of drilling fluid and drilled solids. Cuttings transport during drilling (either conventionally or with CT) has a major impact on the economics of the drilling process. Inefficient hole cleaning from the cuttings can lead to numerous problems such as stuck pipe, reduced weight on bit leading to reduced rate of penetration (ROP), transient hole blockage leading to lost circulation conditions, extra pipe wear, extra cost due to additives in the drilling fluid and wasted time by wiper tripping (the stopping of drilling and the circulation of drilling fluid while pulling out of the hole for cleaning purposes). These many problems have prompted significant research into cuttings transport during the past 50 years. Excellent reviews on the subject are given by Pilehvari et al. (1996) and Azar & Sanchez (1997). Pilehvari et al. state that fluid velocities should be maximized to achieve turbulent flow and mud rheology should be optimized to enhance turbulence in inclined / horizontal sections of the wellbore. They conclude that turbulent flow of non Newtonian fluids needs much more work and should be extended to include pipe rotation and dynamics for conventional drilling. Future work should focus on getting more experimental data, validation of fluid models, cuttings transport mechanistic models verified by comprehensive experimental data. Azar & Sanchez conclude that a combination of appropriate theoretical analyses (complete free body diagrams, accurate rheological models, accurate annular flow models), experimental studies (extensive testing concentrating on individual variables or phenomena), statistical modeling (rheological models, unstable cuttings transport conditions), and high – tech research facilities (accurate measurement of pertinent variables, analysis of video to develop flow pattern maps) will be necessary for further progress. While many problems were addressed quite successfully for conventional drilling in vertical wells in the past and also for inclined and horizontal wells in the recent past, the increase in activity of CTD has called for renewed interest into cuttings transport problems in horizontal and highly inclined annular geometries but with no rotation of the inner pipe. In the past five years there have been several theoretical, semi-theoretical and experimental investigations for assessing the important parameters for efficient cuttings transport during CTD (or conventional drilling but not taking into account the rotation of the inner pipe) in highly inclined and horizontal geometries: for CTD, Walton (1995), Leising & Walton (1998), Li & Walker (1998, 2001) and Walker & Li (2000), Cho et al. (2000, 2001) and for conventional drilling, Kamp & Rivero (1999), Nguyen & Rathman (1998), Martins & Santana (1992), Santana & Martins (1996), Masuda et al. (2000).

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Despite these efforts, there is still lack of good quality published data against which models can be compared with. Many models utilize data for validation not applicable to the situation at hand (e.g. using data with inner pipe rotation, Walton -1995, Nguyen & Rathman – 1998) or even no data at all (Martins & Santana - 1992), or compare their results with results of other models (Kamp & Rivero, 1999). Some data has been presented together with own theoretical analysis Li & Walker (1999, 2001), Walker & Li (2000) but with very limited information on experimental parameters (e.g. rheology of fluids, cuttings concentrations, etc.). It is one of the main objectives of this study to provide good quality data on cuttings transport since, as mentioned by Walker & Li (2000), ‘the highly deviated / horizontal well has placed a premium on having a reliable body of knowledge about solids transport in single and multiphase conditions’. The approach taken by many investigators on modeling cuttings transport for highly inclined and horizontal annuli (with no inner pipe rotation) is that of two or three layer model. The basic model proposed for annuli (Walton, 1995, Nguyen & Rathman, 1998, Kamp & Rivero, 1999) is adopted from the one proposed by Doron et al. (1987) for solids transport in pipes and later extended (1993, 1995, 1996) by same authors. The models are based on mass balance equations for solids and liquid plus momentum balance equations for the two or three layers resulting in a system of coupled algebraic equations. Closure relationships that describe the interaction of the two phases are needed in order to solve these equations and these are taken from published correlations. FLOW PATTERNS During the flow of solid liquid mixtures in horizontal conduits, the liquid and solid phases may distribute in a number of geometrical configurations depending on flow rates, conduit shape and size, fluid and solid properties and inclination. Natural groupings, or flow patterns, exist within which the basic characteristics of the two-phase mixture remain the same. The main parameters determining the distribution of solids in the liquid, i.e. the flow patterns, are the liquid flow velocity (hence the liquid flow rate & cross sectional area), the solids loading (volumetric or mass concentration of solids) and the properties of liquid and solids (rheology and density of liquid, density, diameter & sphericity of solids). Experimental observations for solid – liquid flow in pipes and annuli, even at low solids concentrations, suggest the following flow patterns (depicted in Figure 2), in the direction of decreasing flow rate (or velocity) (Govier & Aziz (1972), Doron et al. (1987), Doron & Barnea (1996), Cho et al. (2001), Walton (1995), Martins & Santana (1990)): • At high liquid velocities the solids may be uniformly distributed in the liquid and

normally the correct assumption is made that there is no slip between the two phases, i.e. the velocity of the solids is equal to the velocity of the liquid. This flow pattern is normally observed for fairly fine solids, less than 1mm in diameter. (This is not expected to be observed during drilling applications where diameter of solids is normally > 1mm, even in CTD where the solids produced are often called ‘fine solids’ in contrast to rotary drilling where we have coarse solids larger than 5 – 6 mm in diameter). This flow pattern is called by many the fully suspended symmetric flow pattern (Figure 2a).

4

• As the liquid flow rate is reduced there is a tendency for the solids to flow near the bottom of the pipe (or outer pipe of the annulus), but still suspended, thus creating an asymmetric solids concentration (non homogeneous). This is called the asymmetric flow pattern but the solids still move with the liquid (Figure 2b).

• A further reduction in the liquid flow rate results in the deposition of solid particles on the bottom of the pipe. The solids start forming a bed which is moving in the direction of the flow, while there may be some solids in the layer above non uniformly distributed. This is the moving bed flow pattern and the velocity below which this is happening has been given different names like, limit deposit velocity, suspension velocity, critical velocity (Figure 2c).

• Further reduction in liquid velocity (or flow rate and increase in cross sectional area) results in more and more solids deposited. We have a bed of solids that is not moving, forming a stationary bed, a moving bed of solids on top of the stationary bed and a heterogeneous liquid – solid mixture above. There is a strong interaction between the heterogeneous solid – liquid mixture and the moving bed with solids deposited on the bed and re-entrained in the heterogeneous solid – liquid mixture. There is a point of equilibrium where with the increase in height of the solids bed, the available area for flow of the heterogeneous mixture is decreased resulting in higher mixture velocities and hence an increase in the erosion of the bed by the mixture (Figure 2d).

• At even lower liquid velocities the solids pile up in the pipe (or annulus) and full blockage may occur. Experimental evidence and theoretical analysis indicate that this may occur at relatively high solids concentration, not encountered during normal drilling operations. It may occur, however, if cuttings transport is inefficient resulting in high solids concentration, especially in sections where large cross sectional areas exist (e.g. in annulus washouts).

FIGURE 2: Flow patterns for solid – liquid flow in concentric annulus Several investigators have reported that the height of the cuttings bed varies along the length of the annulus, i.e. forming dunes. This has been modeled by Iyoho &

a. SUSPENDED SYMMETRIC b. SUSPENDED ASYMMETRICa. SUSPENDED SYMMETRICa. SUSPENDED SYMMETRIC b. SUSPENDED ASYMMETRICb. SUSPENDED ASYMMETRIC

c. MOVING BED d. STATIONARY - MOVING BEDc. MOVING BEDc. MOVING BED d. STATIONARY - MOVING BEDd. STATIONARY - MOVING BED

5

Takahashi (1986) but closure relationships are needed for pressure fluctuations, as pointed out by Kamp & Rivero (1999). It should be stressed that the solids concentration in the liquid is fairly low during normal drilling operations, and for CTD rarely exceeds 2 – 4% by volume (see Appendix 1). TWO LAYER MODELING The two layer model (moving bed & clear liquid in laminar flow) proposed by Gavignet & Sobey (1989) made a basic advance but data did not conform to model predictions. The model was extended by Martins & Santana (1990), as mentioned by Martins et al. (1999) to account for distributed solids (suspended) in the liquid layer and covered the four flow patterns described above: stationary bed, moving bed, heterogeneous suspension and pseudo - homogeneous suspension. This resulted in a system of four algebraic equations and one integral equation for turbulent diffusion of solids. They state that there was negligible effect of the inclusion of solids in the liquid layer as well as the allowance for liquid movement in the solid bed. The model was further extended to non steady behavior and results were presented of computer simulations where in some cases steady state was not observed for times as large as 5500 sec! Following Martins & Santana (1992) and Walton (1995), the two layer model is as follows (Figure 3):

FIGURE 3: Schematic of two layer model There is a layer of liquid on the top side of the annulus containing distributed (suspended) solids and a layer of solids on the bottom of the outer pipe which may be moving, albeit at a very low velocity. Mass balance for the solids

MMMBBBsss CAUCAUCAU =+ (1) And mass balance for the liquid

)1()1()1( MMMBBBsss CAUCAUCAU −=−+− (2) where

θb

τs

τi

τi

FBτB

Us

UB

τsτs

τi

τi

FBτB

Us

UB

6

Us mean velocity of suspension UB mean velocity of the bed, taken as very small, i.e. 0 UM overall mixture velocity Cs mean concentration of solids in the suspension layer CB mean concentration of solids in bed CM mean feed concentration As, AB cross sectional areas occupied by the suspension layer and by the bed AM total cross sectional area, area of annulus While the momentum equations for each of the two layers are

iisss SSzpA ττ −−=

∆∆

(3)

iiBBBB SSFzpA ττ +−−=

∆∆

(4)

For the concentration of the solids in the liquid layer, the turbulent diffusion equation is solved (Appendix 3) with B.C. 2, at y = h, C = CB, thus yielding

[ ]( )∫

−−=

2/20 cos

2sinsin

exp2

π

θγγ

θγ

b

dD

duACC bp

s

Bs (5)

where τs, τb are the wall stresses on the suspension and on the bed at the walls τi is the stress at the interface between suspension and bed FB is the frictional force between the particles in the bed and the wall of the

annulus up settling velocity of solids D dispersion coefficient of solids Ss, Si, SB are the wetted perimeters of suspension, interface and bed respectively up solid settling velocity d0 annulus outer diameter θb as in figure 3 For the solution of the above equations we need closure relations for the shear stresses and the friction force, given by:

( ) 22221

21

21

BBBBBssiissss UfUUfUf ρτρτρτ =−== (6)

where fs, fi, fB are the friction factors for the suspension, interface and bed respectively and ρs, ρB are the densities of suspension and of bed respectively. At the point of slip of the bed, FB is equal in magnitude to the maximum sliding friction force between the particles and the wall, Fmax,

( )

+−==

φτ

ρρηηtanmax

iiBBpN

SAgCFF (7)

where the first term in the r.h.s. represents the submerged weight of the bed and the second term represents the Bagnold stresses (Bagnold, 1954, 1957), where he showed that when a fluid flows over a deposit of solid bed, there exist a normal stress at the interface associated with the shear stress exerted by the fluid on the solids and ρp solid particle density ρ the fluid density

7

φ the coefficient of internal friction between particles and walls η the dry friction coefficient Setting UB = 0 (hence τB = 0), FB can be regarded as the value of the particle / wall friction force required to maintain the bed at rest against the forces exerted on it (gravity, fluid stresses and fluid pressure gradient). As for the settling velocity of solid particles, up, Walton (1995) uses correlations for newtonian fluids, as Doron et al. (1987). Martins & Santana (1992) refer to a procedure proposed by Santana et al. (1991) for predicting up in non – newtonian fluids, but do not give the equations. The friction factors, as referenced by Walton (1995), can be expressed in a standard way, but no indication is given whether the ones he uses are for newtonian or non-newtonian fluids (presumed newtonian). Doron et al. (1987) use the relationships for newtonian fluids

s

hssss

ss

DUf

µρ

== ReRe046,02,0 (8), for turbulent flow

B

hBBBB

BB

DUf

µρ

== ReRe16

(9), for laminar flow

where µB = µs = µ, the liquid viscosity ρs = ρp Cs + ρ(1-Cs) ρB = ρpCB + ρ(1-CB) Dhs the hydraulic diameter of the suspension DhB the hydraulic diameter of the bed Martins & Santana (1992) use also the Fanning friction factor concept and they reference the work of Silva & Martins (1988) who studied experimentally the correlations for annular flow of drilling fluids, and for turbulent flow they use

7,0Re645,000454,0 −+=sf (10) with

( ) nnh

ns

nnK

DU

+

=−

132

8Re

2ρ (11)

As for the interfacial friction factor, fi, most researchers use the expression proposed by Televantos et al. (1979)

+−=

is

hp

i f

Dd

f 2Re51,2

7,3/

ln86,021

(12)

where he uses an interfacial roughness equal to a particle diameter, dp, and Dh is the hydraulic diameter of the pipe in the presence of the stationary bed. Televantos et al. accounting for particle collisions with the bed and for entrainement and deposition of particles, which tend to increase fi, use the factor (2*fi) instead (fI) in the above expression (12) which is the Colebrook formula for newtonian fluids flowing in rough

8

pipes. And as stated in Cheremisinoff & Gupta (1983, pg. 931) according to the study of Televantos, the magnitude of fi is not of critical importance in the model. However in their study, Martins et al. (1996), using their two layer model, they found that the value of fi can have a dramatic impact on the results of the model. They proceeded and tried, by measuring the bed thickness, to estimate and correlate the interfacial friction factor with various parameters. Finally they got the dependence of fi on Re (for non newtonian fluids, equation 11), the fluid behavior index (n), the particle diameter, dp, and the annulus hydraulic diameter, Dh. Their results, however, do not show any significant correlation with neither of these parameters, as a simple inspection of their proposed figures easily reveal! Nevertheless they proceed and propose a relation such as

( ) ( ) 34539,2360211,207116,1 /Re966368,0 −−= hpi Ddnf (13) which is also used in their next paper (Santana et al., 1996). The significance of the value of fi on the model predictions for a two (but also the three) layer model is a point for further investigation in view of these contradictory research results. Martins & Santana (1992) do not reference anything on the value and estimation of the solids dispersion coefficient, D, needed in the diffusion equation (5), presumed to use the one proposed by Doron et al. (1987), as described in Appendix 3, while Walton (1995) has a slight different approach, also described in Appendix 3. Martins & Santana (1992) predict that increasing the flow rate and the density of the liquid are the most effective ways to reduce cuttings concentration. Rheological parameters have only a moderate effect, but no comparison with experimental data is given. Santana et al. (1996) extending further this model they introduced a slip between the liquid and solid particles in the bed, as done by Doron & Barnea (1992) for pipes and found negligible effect on the final predictions. Furthermore they examined the influence of using different rheological models on their predictions. They tested five different models, Bingham plastic, Casson fluid, power law, Robertson – Stiff (3 parameter) and Herschel – Bulkley (3 parameter) and the results of the simulations were different. This demonstrated the importance of fluid rheology on the predictions of efficiency of cuttings transport, however, as stated, it was difficult to estimate which one is better, in view of the absence of own data. Based on own three criteria, fitness to rheometer data, suitability for particle sedimentation velocity predictions and consistency of bed height results, they concluded that the three parameter rheological models showed better results, i.e. satisfied the above three criteria, but there was no comparison with experimental data! Walton (1995) compares his simulation results (using water) with data of Tomren et al. (1986) with inner pipe rotation at 50 RPM, hence, as he states, no quantitative agreement is expected. The simulation predicts the trend of the data. At high flow rates data show no bed for concentric annulus which is different from his predictions. He produces a flow regime map and a map that shows how the minimum flow rate for complete suspension varies with fluid viscosity and particle concentration. Fluids of moderate viscosities are more efficient than low viscosity fluids such as water or high viscosity gels, with optimum viscosity around 20 – 30 cp at 170 s-1. Furthermore they predict that cleanouts in horizontal and highly deviated wells require pump rates an order of magnitude higher than vertical wells.

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Kamp & Rivero (1999), who made an excellent review of layer models, use also a two layer model with eccentric annuli. They utilize mass flux of cuttings that are deposited, φs,dep, or resuspended, φs,susp, per unit interface, mass balance equations for solid, liquid and mixture (3 equations) and momentum equations for heterogeneous layer and the bed (2 equations). They thus have a system of five equations with five unknowns, Cs, Us, UB, p, h, where p is pressure and h is the height of the bed (they have assumed that CB = 0,52). For the solution they need closure relationships for the fluxes, φs,susp and φs,dep and for the shear stresses τs, τι, τB . They use the diffusion equation for the concentration in the suspension layer, but instead of using CB in the solution, they use the average concentration, Cs. The deposition of particles, φs,dep, is taken as proportional to Cs and up. The resuspension flux, φs,susp is taken as a function of interfacial shear stress, τi. The settling velocity up is derived for newtonian fluids through standard procedures, while some reference is given on the effect of non newtonian fluids but they do not state which ones are used. There is no equation given for the solid dispersion coefficient, D, except referencing appropriate modeling (Taylor, 1954), which is followed by Doron et al. (1987, Appendix 3) as well as the work of Mols and Oliemans (1998), where, though, no specific equation for the prediction of D is given! Their results, using as a base case rather large annulus for CTD, d0 = 17,5 in. (44,4, cm) and di = 9,0 in. (22,8 cm) and a flow rate Q = 50 GPM (189,25 lpm) resulting in very low velocity in the annulus (Vann = 2,8 cm/s!) and ROP = 50 ft/hr (hence volumetric concentration ~ 17%) showed that starting from well distributed solids in the entrance (h = 0), there is almost immediately bed formation in the annulus (h = 0,125d0), at an axial distance of only one outer annulus diameter. The height of the bed then remains constant along the flow channel. These results are also derived for the other pertinent parameters. The bed height decreases as the flow rate is increased but the rate of decrease is much smaller at higher flow rates, contrary to what is expected. Results are not very sensitive to mud viscosity, in contrast to experimental and field evidence which show that turbulence promotes cuttings transport. They do not compare their results with experimental data, rather with predictions of correlation based models (Larsen et al., 1993 and 1997 and Jalukar et al., 1996), which of course are based on their own experimental results. Kamp & Rivero predictions show similar trends as the correlation based models, but they are far from being quantitatively close! It is recognized that the model overpredicts cuttings transport at a given liquid flow rate. They state the need for good closure terms with respect to particle resuspension and particle settling velocities. Li & Walker (1999) presented results of their experimental study utilizing various empirical correlations, devised from their own data as well as from other investigators. One of the main correlations used is that cuttings erosion follows logarithmic expression with time. They devised a computer model based on dimensional analysis and using these correlations. They then study the sensitivity of predictions on various important parameters (including underbalanced conditions). Most important variable is the liquid velocity, but in general, no quantitative data is given in the paper. They extended their study (Walker & Li, 2000) to cover the effects of particle diameter, dp, fluid rheology and eccentricity. They conclude that for particles with diameters dp < 0,76mm, smaller particles are easier to clean, while for dp>0,76 mm, smaller particles are harder to clean, however they tested only three particle sizes (0,15 mm, 0,76 mm and 7mm). Furthermore the critical velocity of water for full suspension was only 12% lower for the 7mm size compared to the 0,76mm particles, while for 20lb Xanvis, the critical velocity was only 10% higher for the 7mm sizes compared to the 0,76mm sizes. These results indicate that their conclusions, at best, are not fully supported by their own data. Fluid rheology plays a

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significant role with low viscosity fluids in turbulent flow giving optimum results for hole cleaning, similar to the results of Walton (1995). They also found that the critical velocity for full suspension is higher for fully eccentric annulus compared to the concentric annulus. Similar two layer model has also been used by Doan et al. (2000) for underbalanced drilling under transient conditions. Here the two phase (gas – liquid) mixture in the suspension layer is considered as pseudo – homogeneous with fixed properties. They also consider deposition fluxes, derived from hindered settling of solids and erosion (or resuspension) fluxes as a function of interfacial shear velocity, in a manner similar to Kamp & Rivero (1999). They allowed slip between the particles and the fluid both in the suspension layer and the bed and accounted for the non newtonian behavior of the liquid. They use a correlation for solids settling velocity for non newtonian fluids (see Appendix 2) but use hindered settling expression proposed for newtonian fluids, as of Doron et al. (1987)! Their results (run for water and mud only, without a gas phase), showed that for the suspension flow pattern, cuttings velocity matches that of liquid, i.e. there is no slip between the two phases. However, when a bed is formed, then the solids move in the suspension at velocities much lower (about ½) of the liquid velocity. Their predictions compare favorably with own experimental results, except in the cases of very dilute solid concentrations (less than 0,05% by volume) due to the fact that a smooth interface (assumed in the model) did not really existed. Effective transport is achieved when cuttings velocity is greater than zero and they establish a range of parameters for which they can determine the minimum inlet cuttings concentration for this to be achieved. It should be noted that what differs between all above mentioned approaches with the approach for layer modeling in pipes (as of Doron et al., 1987) is mainly the geometry of the cross section, with all the above simulating eccentric annulus, i.e. 1. the contact areas for estimating the frictional forces due to the shear stresses for

the two layers 2. the friction factors, taken as the ones proposed for pipe flow but using the

hydraulic diameter of the annulus 3. the concentration of solids in the suspension layer, solved for the case of annulus THREE LAYER MODELING The ‘inadequacies’ in the predictions from the two layer models when compared to limited data and the extension of the two layer to the three layer model by Doron & Barnea (1993) for flow of solid – liquid mixtures in pipes led to the extension of the two layer model in annulus to the three layer model (Nguyen & Rathman, 1998 and Cho et al., 2000). The development follows that of the two layer model approach, with the inclusion of a moving bed on top of a stationary bed of solids. We thus have two mass balance equations (for solids and liquid) and three momentum equations, one for each layer, the suspension heterogeneous solid - liquid layer, the moving bed and the stationary bed. For a concentric annulus, the three layer model is depicted in Figure 4.

The mass balance equations are

MMMsbsbsbmbmbmbsss CUACUACUACUA =++ (14), for solids )1()1()1()1( MMMsbsbsbmbmbmbsss CUACUACUACUA −=−+−+− (15)

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for liquid. For the momentum equations we have:

smbsmbsss SSzpA ττ −−=

∆∆

(16), for the suspension layer

smbsmbmbmbmbsbmbsbmbmbsbmb SSSFFzpA τττ +−−−−=

∆∆

(17),

for moving bed layer, and

bmbsbmbsbmbsbsb FSFzpA ≤++

∆∆ τ (18),

for the stationary bed.

FIGURE 4: Schematic representation of 3 – layer pattern

where the various stresses and frictional forces are shown in Figure 4 and Sk are the wetted perimeters for each layer, as before. The equation for the stationary bed (18) serves as a condition to be satisfied whenever a stationary bed is predicted but it is not part of the solution (Doron and Barnea, 1993). The concentration in the suspension layer is derived from the diffusion equation, as for the two layer model, with the interface now being (mb), rather than (sb – one layer). The solution to the above equations requires, as for two layer modeling, closure relationships for the shear stresses and the frictional forces, which are taken as for the two layer modeling, through the use of Re and friction factor (f) relationships. Nguyen & Rathman (1998) use the above formulation but avoid using the diffusion equation. Instead, they predict the thickness of the moving bed, hmb, based on analysis of Bagnold forces, as derived by Wilson & Tse (1987)

φρρτ

tan)( mbp

smb gUh

−= (19)

where τs is the intergranular shear stress and the moving bed velocity, Umb, is predicted from turbulent boundary layer theory as

s.l.

m.b.s.b.

τs

τsmb

τsmb Fmb

Fmbsb

τmbsb

τmbsb

τb Fb

Fmbsb

s.l.

m.b.s.b.

τs

τsmb

τsmb Fmb

Fmbsb

τmbsb

τmbsb

τb Fb

Fmbsb

Us

Umb

Ub

12

v

smbmb K

Uρτ /

34

= (20)

with Kv the von Karman constant. Nguyen predicts full erosion of bed at annular velocity of Us ~ 3,8 ft/s (1,2 m/s) for flow of water in a 5 by 1,9 in. annulus (12,7 by 4,8 cm), with particle diameters of 6,4 mm, ROP = 50 ft / hr (15,2 m/hr) corresponding to 0,42% solids volumetric concentration at this velocity. This compares favorably with the results of Ford et al. (1990) and Peden et al. (1990) who measured minimum transport velocity for these conditions of 3,3 ft /s. Various runs of the simulator showed that liquid density and annulus eccentricity had significant effect. The use of the assumed value of CB = 0,62 (instead of the standard used of 0,52) had minor effect, while a low viscosity bentonite mud showed better cuttings transport compared to that of water. However, the use of the correlations, e.g. that of interfacial friction factor fi, was not studied, nor the type of non newtonian fluid, albeit they use correlations for friction factors for non newtonian fluids (through the use of the generalized Re for non newtonian fluids), nor the use of correlations for settling velocities in non newtonian fluids. Cho et al. (2000) use above formulation with the diffusion equation, with an experimental expression for the settling velocities of solids through non newtonian fluids of Chien (1994)

( ) 0)/)()(5exp(45,19/)5exp(45,02 =∆−+ ρρφρµφ pppp dudu (21) with φ the sphericity of particles (taken as 0,8 for drill cuttings). They further take into account the effect of hindered settling, taken from the correlation developed by Thomas (1963), as )9,5exp( Cuu ph −= . For the particle dispersion they use the approach of Walton (1995, and Appendix 3). They use as a base case a low viscosity bentonite mud, with n = 0,7 and K = 0,29 Pasn, in a 5 by 1,9 in. annulus, ρp = 2,6 g/cm3, ρ = 1,1 g/cm3 and dp = 2,3mm. Their results show that for liquid annulus velocities in the range of 0,3 – 1,5 m/s (typical velocities encountered in CTD applications), a bed of cuttings is formed. At about 0,6 – 0,9 m/s annular velocities, suspension starts to occur, similar to Nguyen and Rathman results, and the critical velocity for full suspension is around 1,2 – 1,5 m/s. The most significant parameter is the annulus liquid velocity and should be of the order of 1,1 – 1,4 m/s (compared to conventional drilling of 0,6 – 0,9 m/s) because the minimum pressure drop is obtained at these flow rates and a minimum height of stationary bed is also obtained. There is a slight effect of rheological parameters. It should be noted that when the moving bed vanishes, the model does not reduce to a two layer model (as it should do as a limiting case), instead, a new model covering the two layers is solved. No comparison is made to the predictions versus that of two layer model to justify the added complexity of the three layer model.

DISCUSSION ON THE LAYER MODELS Careful analysis of the published results by previous investigators shows that a two or three layer pattern will form almost immediately, even when starting from a homogeneous distribution of solids at the entrance. Kamp & Rivero (1999) show that the solids concentration, or equivalently the height of the bed are constant at two to three hydraulic diameters (d0 – di) from the entrance! Similar results have been obtained by Masuda et al. (2000).

13

From the proposed tentative solutions from the various investigators it is evident that the framework for the problem solution is similar, with main differences whether a two or three layer model used and the closure relationships which include, among others:

• Solid distribution in the heterogeneous liquid - solid layer • Interfacial friction factor between the heterogeneous liquid layer and moving

bed of solids • Whether Bagnold stresses are taken into account • Terminal velocity of solids in newtonian or non-newtonian fluids, taking into

account the effect of hindered settling, the effect of walls (normally not taken into account)

• Fluid friction factors for fluid & walls of annulus, normally taken for newtonian fluids and using correlations developed for pipe flow.

The results of Cho et al. (2000) and Nguyen & Rathman (1998) show that full suspension occurs at annulus velocities of 1,2 – 1,5 m/s, which are at the upper limit of the flow rates encountered in CTD. Hence, consideration should be given for the determination of minimum suspension velocities of the cuttings which are not derived from bed erosion models. MINIMUM SUSPENSION VELOCITY FOR HORIZONTAL FLOW Most of the work done and reported in the literature for the minimum suspension velocity of solids in conduits is for the pipe geometry, since this is the geometry used for solids transport using liquids (mainly water) (Govier & Aziz, 1972, Cheremisinoff & Gupta, 1983, and the many references quoted there). In addition, much of the reported work is for high solids loading, up to 40 – 50%, with different particle sizes, from fines to coarse particles (up to 7mm). The differences of the studied situations with the one we encounter during drilling applications, especially for drilling with coil tubing are: • the geometry is annulus (most often eccentric) • the inlet concentration of solids is fairly low, rarely exceeding 4% by volume • the fluid used could be non-newtonian, shear thinning, many times modeled as

power law fluid • there is a limitation on the maximum flow rate to be achieved with the upper limit

defined by the capacity of downhole motor and maximum surface pressure sustained (Leising & Walton, 1998). Typical flow rates range between 2 – 3 bpm (318 – 477 lpm) which for typical annulus sizes results in the velocity of the fluid in the annulus in the range of 0,3 – 1,5 m/s (see Table 1).

Not much recent work has been done on the minimum suspension velocity, especially in annulus geometry. Most models deal, as mentioned and analyzed above, with the two or three layer model. The minimum suspension velocity is derived from the bottom end of the spectrum, i.e. when the bed disappears. It appears though that the critical flow rate for bed erosion is seldom sufficient to maintain the particles in suspension (Leising & Walton, 1998) who finally conclude that • hole cleaning in bed transport is inefficient • a suspension condition that relies only on bed erosion will under predict the flow

rate necessary for efficient suspension, and hence cuttings removal. If the flow in the annulus is laminar, the cuttings will inevitably form a bed, given sufficient length. Since in CTD there is no pipe rotation (no mechanical stirring as in rotary drilling), the role of turbulence must be examined for maintaining particles in suspension (Leising & Walton, 1998, Cho et al., 2000, Nguyen & Rathman, 1998). The question then posed is whether the flow rate necessary to suspend the particles in turbulent flow is practical in view of the limitations on maximum Q and ∆p imposed

14

for CTD. Leising & Walton (1998) predicted that fluids of low to moderate viscosity (5 – 15 cp at nominal shear rate of 170s-1) were optimal for hole cleaning. A full model, however, for predicting these critical velocities in terms of the flow parameters for full suspension of particles is not given. Based on work on solids transport in pipes, as reported by Govier & Aziz (1972) and Cheremisinoff & Gupta (1983), the critical velocity for solids suspension can be derived from consideration of forces acting on the particles and the requirement that the forces balance so that particles remain in suspension. Very relevant is the work of Davies (1987) where for full suspension, but not necessarily for a symmetric concentration profile, the balance of forces is between: • the downward sedimentation force, Fds • and the upward eddy fluctuation force, Fue = eddy pressure * area These forces are estimated as:

npds CgdF )1(

63 −∆= ρπ

(22)

( ) ( )4/' 22pue duF πρ= (23)

where dp spherical particle diameter ρ liquid density ρp solid density ∆ρ density difference = ρp – ρ C solid volumetric concentration g acceleration of gravity u’ fluctuating component of transverse velocity Note also that for the sedimentation force, hindered settling has been taken into account through the term (1-C)n. When these forces balance, we get

( ) ρρ /182,0' ∆−= gdCu pn (24)

The fluctuating velocity (u’) for liquid only is related to energy dissipation based on turbulence theory and the fact that we are concerned with eddies of the size of the particle diameter, hence,

( ) pM dPu =3' (25) where PM is the power dissipated per unit mass of fluid, i.e.

dfUP M

M32

= (26)

with f the friction factor UM the mixture velocity d the pipe diameter and using Blasius relationship,

µρ dUf M== Re,

Re079,025,0 (27)

we get finally

15

( ) ( ) ( ) ( ) ( ) 42,03/192,012/13/1 /16,0' −= ddUu pMρµ (28) the equation being dimensional and valid for pipe flow and only liquid flowing. Davies claims that the presence of solids dampens (u’) and he takes this into account by letting

Cuu p α+

=1

'' (29)

with α a constant, from work on agitated vessels, in conformity with other investigators (Cheremisinoff & Gupta, 1983, pg. 914, Hsu et al., 1989, Pechenkin, 1972) but contradictory to others. For example, Julian & Dukler (1965), as mentioned by Govier & Aziz (1972, pg. 483) who studied vertical gas solid flow in pipes (with results equally applicable to horizontal flows of solid – liquid mixtures), state that for dilute gas – liquid systems ‘the solids make their influence felt by modifying local turbulence in the gas phase, increasing turbulence fluctuations, mixing length, eddy viscosity and hence frictional pressure drop, ∆pf’. This is also evident from measured curves of ∆pf vs Q, where ∆pf in the presence of solids is higher than for pure gas or liquid flowing.

Utilizing the above dampened equation for u’p, we get, after some algebra,

( )( ) ( ) ( ) ( ) ( )54,0

046,009,018,054,009,1 2/1108,1

∆−+= −

ρρρµα gddCCU p

nM (30)

with the equation being dimensional. Utilizing the observed maximum versus concentration, Davies estimates the value of α = 3,64. This final equation is then similar to Durand’s correlation (1953), as stated by Davies, apart from the concentration dependence. The value of (n) is set from the equations for hindered settling, taken as n = 4 for 1 < Rep < 10 falling to ~ 3 when Rep approaches 100. His final predictions were close to published experimental data but about 1,35 times higher. The above mentioned approach has not been used neither for the flow in an annulus nor for non – newtonian fluids. The main challenges are then

1. to relate (u’) to main flow velocity (UM) for an annulus and for non newtonian fluids

2. to examine the dampening or not of the non-newtonian liquid velocity fluctuations by the presence of solids and quantify it

3. to examine the effect of hindered settling for solids 4. to compare predictions with good quality data 5. to verify whether values of UM so predicted conform to the maximum flow rate

and pressure drop imposed for CTD. EXPERIMENTAL SYSTEM In order to shed more light into the efficient cuttings transport for CTD and aiming at providing good quality data, we established an experimental system for studying the solids carrying capacity of liquids in horizontal and inclined annulus, with no inner pipe rotation. Before embarking into the setting up of the experimental system, a dimensional analysis was done to determine the significant parameters affecting the process. If we visualize the process for the two layer model (liquid – solid suspension and a stationary bed), the height of the bed, h, will depend on the following parameters: dp particle diameter

16

d0 diameter of outer pipe of annulus dι diameter of inner pipe of annulus V annular liquid (or suspension) velocity Vp solid settling velocity µ liquid viscosity ρp, ρ particle and liquid density g acceleration of gravity Dimensional analysis then suggests equation (31) below,

hence, the main parameters are geometrical parameters d0, di, dp physical properties ρp, ρ, µ velocity ratio (C)

Reynolds number µρhVd

=Re , dh the hydraulic diameter ( = d0 – di)

with the last term in the r.h.s. (A) easily shown to be

))((Re2 FrA= (32)

hgdVFr /2= (33) 223 / µρ gdA h= (34)

Hence, for full simulation of field conditions, we should have between our system and field conditions not only geometric similarity, i.e. similar values for )/( 00 iddd − ,

pdd /0 , idd −0 , but also dynamic similarity, i.e. similar values for volumetric

concentration, µρhVd

=Re and hgdVFr /2= .

Literature review of SPE papers describing case studies with CTD, since its first application in 1991 to date (Ramos et al., 1992, Leising & Newman, 1992, Leising & Rike, 1994, Walton, 1995, Goodrich et al., 1996, Weighill et al., 1996, Elsdborg et al., 1996, McGregor et al., 1997, Svendsen et al., 1998, Kara et al., 1999, Kirk & Simbiring, 1999, Stiles et al., 1999, Portman, 2000, and McCarty & Stanley, 2001) gave us the results of Table 1.

TABLE 1: Field Data of CTD parameters SOURCE Dhole OD ID Dhydr Aann Q Vann ROP dens Pbh

(in) (in) (in) (in) (cm2) (bpm) (m/s) (ft/hr) (g/cm3) (psi) SPE 23875 2,000 1,688 1,90 9 to 12 2.500SPE 24594 4,750 1,750 1,438 3,000 98,8 2,00 0,54 30 1,0

" " 4,750 2,000 1,688 2,750 94,0 3,20 0,90 1,0 SPE 27433 3,880 1,750 2,130 60,8 4,750 2,000 2,750 94,1 6,250 2,000 4,250 177,7 SPE 29491 5,000 1,900 3,100 108,3 2,38 0,58

" " 5,000 1,900 3,100 108,3 4,76 1,16 SPE 35128 3,750 2,375 1,375 10 to 70 1,1

( )f

ie

pk

pd

ib

i

a

pi

gddVVddV

ddd

dd

ddh

−

−

−

−

∝

− 2

2300

0

00

0

)(µ

ρρ

ρρµ

ρ

17

SPE 35544 3,500 2,875 0,625 20,2 7 to 25 SPE 37074 2,875 75 to 200 SPE 38397 4,750 2,375 2,375 3 to 50 SPE 50405 3,750 2,375 1,375 40 to 50 1,5 SPE 54496 3,750 2,000 1,750 51,0 1,50 0,78 1,1 SPE 54502 6,125 2,375 3,750 161,4 2,50 0,41 20 1,1 SPE 57459 4,750 2,375 2,375 SPE 62744 2,750 2,000 20 SPE 67824 2,750 12 to 21

" " 4,125 12 to 21 SPE 68436 5,000 1,900 3,100 108,3 1,25 0,31 1,1 5.900

" " 5,000 1,900 3,100 108,3 6,23 1,52 1,1 7.700where Q is listed in barrels / min (1 bpm = 159 lpm), and pbh is the bottom hole pressure in psi (1 psi = 6,9 kPa). It is unfortunate that some information is missing, because it was not stated explicitly. In particular, the viscosity of the fluids used, while density was mentioned and ranged from that of water to a maximum of 1,1 g/cm3. Based on the above values, we have constructed a flow loop, shown in Figure 6.

FIGURE 6. Schematic of annulus flow loop: 1 – annulus section, 2 – measuring section, 3 – tank, 4 – agitator, 5 – pump, 6 – Coriolis flow meter, 7 – pressure

transducer, 8 – P/C for data acquisition The annulus is 5m long, with outer diameter d0 = 7cm and inner diameter di = 4 cm, and currently is concentric. It is supported by a metal structure that can be tilted from horizontal, hence various inclinations from horizontal can be studied (maximum inclination ~ 35 degrees). There is a plastic tank, holding about 500 liters of liquid, equipped with a variable speed agitator (1,5 KW). The flow is achieved with a 7,5 KW slurry pump capable of delivering 700 lpm at 441 Kpa (4,35 atm). The flow rate as well as density and temperature are monitored with a Coriolis mass flow meter (Rheonik, RHM 30). Viscosity of liquids is measured with a continuously variable shear rate coaxial cylinder viscometer (Grace, type M3500a) with shear rate rates from 0,01 to 600 RPM. A comparison of the pertinent variables between the experimental system and field operations is given in Table 2 below.

18

TABLE 2: Comparison of field conditions with experimental simulation

CONCLUSIONS A critical review has been presented for the effect of the various parameters on efficient cuttings transport in concentric and eccentric annuli. The modeling and experimental results show that: • the most significant parameter is the annulus mixture velocity (flow rate and cross

sectional area) • flow should be turbulent in the annulus • maximum flow rates should conform to maximum rates imposed by downhole

motor and maximum pressure allowable for Coil Tubing • liquid density is an important parameter (increase in buoyancy for higher ρ) • eccentricity plays a significant role with a dramatic decrease in cuttings transport

efficiency for fully eccentric annulus. From the published results and models to date, the main issues are the following: • need for good quality data, for concentric and eccentric annulus, with conditions

similar (dynamic similarity) to the ones observed in the field • establish theoretically the observed link (in the field and experiments) between

fluid rheology and efficient cuttings transport • establish the best rheological models for the liquids, two versus three parameter

models, since added complexity of three parameter models is still questionable • resolve the contradiction for using low viscosity fluids versus moderate viscosity

fluids, where results show better suspension characteristics for the latter but the former promote turbulence

• better understand the solids distribution in the suspension layer. The dispersion coefficient of solids is very critical and better predictions are needed

• should the approach be the determination of minimum suspension velocity or the modeling of layers for efficient cuttings transport ?

550 0 -- 18001800

%CMC, non newtonian%CMC, non newtonian

0,8 0,8 –– 44

1,5 1,5 –– 2 2 -- 44

2,52,5

1,051,05

32 32 –– 450450

20 20 –– 700700

0,430,43

0,570,57

7,07,0

4,04,0

Experimental conditionsExperimental conditions

60 60 -- 180180γγNwNw (12V/d(12V/d00--ddii) (s) (s--11))

water & polymerswater & polymersµµ

0,8 0,8 –– 44CCvv, % , % κ.ο.κ.ο.

2 2 toto 77ddp p (mm)(mm)

2,52,5ρρpp (g/cm(g/cm33))

1,02 1,02 –– 1,141,14ρ (ρ (g/cmg/cm33))

30 30 –– 150150V (cm/s)V (cm/s)

240 240 –– 475475Q (Q (lpmlpm))

0,370,37--0,470,47--0,500,50--0,610,61((ddii--dd00) / d) / d00

0,400,40––0,530,53––0,630,63––0,820,82ddiiii/ d/ d0 0

7,0 7,0 –– 9,5 9,5 –– 12,112,1dd0 0 (cm)(cm)

5,08 5,08 -- 6,03 6,03 -- 7,37,3ddii (cm)(cm)

Real conditionsReal conditionsParameterParameter

550 0 -- 18001800

%CMC, non newtonian%CMC, non newtonian

0,8 0,8 –– 44

1,5 1,5 –– 2 2 -- 44

2,52,5

1,051,05

32 32 –– 450450

20 20 –– 700700

0,430,43

0,570,57

7,07,0

4,04,0

Experimental conditionsExperimental conditions

60 60 -- 180180γγNwNw (12V/d(12V/d00--ddii) (s) (s--11))

water & polymerswater & polymersµµ

0,8 0,8 –– 44CCvv, % , % κ.ο.κ.ο.

2 2 toto 77ddp p (mm)(mm)

2,52,5ρρpp (g/cm(g/cm33))

1,02 1,02 –– 1,141,14ρ (ρ (g/cmg/cm33))

30 30 –– 150150V (cm/s)V (cm/s)

240 240 –– 475475Q (Q (lpmlpm))

0,370,37--0,470,47--0,500,50--0,610,61((ddii--dd00) / d) / d00

0,400,40––0,530,53––0,630,63––0,820,82ddiiii/ d/ d0 0

7,0 7,0 –– 9,5 9,5 –– 12,112,1dd0 0 (cm)(cm)

5,08 5,08 -- 6,03 6,03 -- 7,37,3ddii (cm)(cm)

Real conditionsReal conditionsParameterParameter

19

• for the modeling of minimum suspension velocity, the needs are: o to find relationships to link the turbulent fluctuating velocity component to

the main flow parameters for annulus and non newtonian fluids o to get better relationships for hindered settling of solids in non newtonian

fluids o to examine the effect of the presence of solids on the turbulent fluctuating

velocity for non Newtonian liquids o to compare predictions with good quality data

• for the layer modeling, the needs are for o justification of the use of three layers (more complex) vs the two layer

model with good quality data o better prediction of the solids dispersion coefficient, D, hence

determination of Pe o validation of the ‘diffusion’ equation for the annulus o better closure relationships

for interfacial friction factor, between heterogeneous layer and moving (or stationary bed)

for the wall friction factors for the heterogeneous layer and for the moving bed, valid for non newtonian fluids

REFERENCES Azar, J. J., Sanchez, R. A., 1997, ‘Important Issues in Cuttings Transport for Drilling Directional Wells’, SPE 39020, presented at the 5th Latin American and Caribbean Petroleum Engineering Conference and Exhibition, Rio de Janeiro, Brazil, 30 august – 3 September. Bagnold, R., A., 1954, ‘Experiments of a gravity free dispersion of large solid spheres in a Newtonian fluid under shear’, Proc. R. soc., vol. A225, 49 – 63 Bagnold, R., A., 1957, ‘The flow of cohensionless grains in fluids’, Phil. Trans. R. Soc., vol. A249, 235 - 297 Ceylan, K., Herdem, S., and Abbasov, T., 1999, ‘A Theoretical Model for Estimation of Drag Force in the Flow of Non – Newtonian Fluids Around Spherical Solid Particles’, Powder Techn., Vol. 103, 286 – 291. Cheremisinoff, N., P. & Gupta, R., 1983, ‘Handbook of Fluids in Motion’, Ann Arbor Science, Michigan Chhabra, R. P., and Peri, S. S., 1991, ‘Simple Method for the Estimation of Free – Fall Velocity of Spherical Particles in Power Law Liquids’, Powder Techn., Vol. 67, 287 – 290. Chien, S. F., 1994, ‘Settling Velocity of Irregularly shaped Particles’, SPE 26121, presented at the 69th Annual Technical Conference and Exhibition, New Orleans, Louisiana, Sept. 25 – 28. Cho, H., Shah, S. N., Osisanya S. O., 2000, ‘A Three Layer Modeling for Cuttings Transport with Coil Tubing Horizontal Drilling’, SPE 63269, presented at the 2000 SPE Annual Technical Conference and Exhibition, Dallas, Texas, USA, 1 – 4 October. Cho, H., Shah, S. N., Osisanya S. O., 2001, ‘Selection of Optimum Coil tubing Parameters Through the Cuttings Bed Characterization’, SPE 68436, presented at the 2001 SPE / IcoTA Coil Tubing round Table, Houston, Texas, USA, 7 – 8 March.

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Davies, J. T., 1987, ‘Calculation of Critical Velocities to Maintain Solids in Suspension in Horizontal Pipes’, Chem. Engr. Science, Vol. 42 (7), 1667 - 1670 Doan, Q. T., Oguztoreli, M., Masuda, Y., Yonezawa, T., Kobayashi, A., and Kamp, A., 2000, ‘Modeling of Transient cuttings Transport in Underbalanced Drilling’, IADC / SPE 62742, paper presented at the 2000 IADC / SPE Asia Pacific Drilling Technology, Kuala Lumpur, Malaysia, 11 – 13 Sept. Doron, P., Garnica, D. and Barnea, D., 1987, ‘Slurry Flow in Horizontal Pipes: Experimental and Modeling’, Int. J. Mult. Flow, Vol. 13 (4), 535 - 547 Doron, P. and Barnea, D., 1992, ‘Effect of the No – Slip Assumption on the Prediction of Solid – Liquid Flow Characteristics’, Int. J. Mult. Flow, Vol. 18 (4), 617 - 622 Doron, P. and Barnea, D., 1993, ‘A Three Layer Model for Solid – Liquid flow in Horizontal Pipes’, Int. J. Mult. Flow, Vol. 19 (6), 1029 - 1043 Doron, P. and Barnea, D., 1995, ‘Pressure Drop and Limit Deposit Velocity for Solid – Liquid Flow in Pipes’, Chem. Eng. Science 50 (10), 1595 – 1604. Doron, P. and Barnea, D., 1996, ‘Flow Pattern Maps for Solid – Liquid Flow in Pipes’, Int. J. of Multiphase Flow, 22 (2), 273 – 283 Durand R., 1953, Basic Relationships of the Transportation of Solids in Pipes – Experimental and Modeling’, Proc. 5th Minneapolis Int. Hydraulics Conv., Minneapolis, MN, 89 – 103. Elsborg, C., Catter, J., and Cox, R., 1996, ‘High Penetration Rate Drilling with Coiled Tubing’, SPE 37074, paper presented at the 1996 SPE International Conference on Horizontal Well Technology held in Calgary, Canada, 18-20 November 1996.

Felice, R. Di., 1999, ‘The Sedimentation Velocity of Dilute Suspensions of Nearly Monosized Spheres’, Int. J. Mult. Flow, Vol. 25, 559 – 574. Ford, J. T., Peden, J. M., Oyeneyin, M. B., Gao E., Zarrough, R., 1990, ‘Experimental Investigation of Drilled Cuttings Transport in Inclined Boreholes’, SPE 20421, presented at the 65th SPE Annual Technical Conference, New Orleans, LA, USA, September 23 – 26. Gavignet A. A., and Sobey, I. J., 1989, ‘Model Aids cuttings Transport Prediction’, J. of Petrol. Techn., Sept., 916 – 921. Goodrich, G.T. , Smith, B E., and Larson, E.B., 1996, ‘ Coiled Tubing Drilling Practices at Prudhoe Bay’, SPE 35128, paper presented at 1996 lADC / SPE Drilling Conference held in New Orleans Louisiana, 12-15 March 1996 Govier, G. W., and Aziz, K., 1972, ‘The Flow of Complex Mixtures in Pipes’, R.E. Krieger Pub. Co., Florida Hsu, F-L., Turian, R., M., and Ma, T-W., 1989, ‘Flow of Noncolloidal Slurries in Pipelines’, AIChE Journal, Vol. 35 (3), 429 - 442 Iyoho, A. W., and Takahashi, H., 1993, ‘Modeling Cuttings Transport in Horizontal, Eccentric Wellbores’, unsolicited paper SPE 27416.

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Jalukar, L. S., Azar, J. J., Pihlevari, A. A. and Shirazi, S. A., 1996, ‘Extensive Experimental Investigation of the Hole size Effect on Cuttings Transport in directional Well Drilling’, paper presented at the ASME Fluids Engineering Division Annual Summer Meeting, San Diego, California, July 7 – 12. Julian, F. M., and Dukler, A. E., 1965, AIChE J., Vol. 11, 853. Kamp A. M., Rivero S., 1999, ‘Layer Modeling for Cuttings Transport in Highly Inclined Wellbores’, SPE 53942, presented at the 1999 Latin American and Caribbean Petroleum Engineering Conference and Exhibition, Caracas, Venezuela, 21 – 23 April. Kara, D.T., Gantt, L.L., Blount, C.G., and Hearn, D.D., 1999, ‘Dynamically Overbalanced Coiled Tubing Drilling on the North Slope of Alaska’, SPE 54496, paper presented at the 1999 SPE / ICoTA Coiled Tubing Roundtable held in Houston, Texas, 25–26 May 1999. Kirk, A., and Sembiring, T., 1999, ‘Application of C.T.D. Offshore, Indonesia Phase One Pilot Project’, SPE 54502, paper presented at the 1999 SPE/ICoTA Coiled Tubing Roundtable held in Houston, Texas, 25–26 May 1999. Larsen, T. I., Pilehvari, A. A. and Azar, J.J., 1993, ‘Development of a New Cuttings Transport Model for High – Angle Wellbores Including Horizontal Wells’, SPE 25872, presented at the 1993 SPE Rocky Mountain Regional / Low Permeability Reservoir Symposium, Denver, Colorado, Apr. 12 – 14. Larsen, T. I., Pilehvari, A. A. and Azar, J.J., 1997, ‘Development of a New Cuttings Transport Model for High – Angle Wellbores Including Horizontal Wells’, SPE Drill. and Completion, Vol. 12 (2), 129 – 135. Leising, L. J., and Newman, K. R., 1992, ‘Coil Tubing Drilling’, SPE 24594, presented at the 67th SPE Annual Technical Conference and Exhibition, Washington, DC, Oct. 5 – 7. Leising L. J., and Walton, I. C., 1998, ‘Cuttings Transport Problems and Solutions in Coiled Tubing Drilling’, IADC / SPE 39300, presented at the 1998 IADC / SPE Drilling Conference, Dallas, Texas, USA, 3 – 6 March. Leising L. J., and Rike, E. A., 1994, ‘Coil Tubing Case Histories’, IADC / SPE 27433, paper presented at the IADC / SPE Drilling Conference, Dallas, TX, Feb. 15 – 18. Li, J. and Walker S., 1999, ‘Sensitivity analysis of Hole Cleaning Parameters in Directional Wells’, SPE 54498, presented at the 1999 SPE / ICoTA Coil Tubing round Table, Houston, Texas, USA, 25 – 26 May. Li, J. and Walker S.,, 2001, ‘Coiled – Tubing Wiper Trip Hole Cleaning in Highly Deviated Wellbores’, SPE 68435, presented at the 2001 SPE / ICoTA Coil Tubing round Table, Houston, Texas, USA, 7 – 8 March. McCarty, T., M., and Stanley, M. J., 2001, ‘Coiled Tubing Drilling: Continued Performance Improvement in Alaska’, SPE / IADC 67824, Paper presented at the SPE/IADC Drilling Conference, Amsterdam, 27 Feb. – 1 March, 2001

22

McGregor, B., Cox, R. and Best, J., 1997, ‘Application of Coil Tubing Drilling Technology on Deep Under Pressured Gas Reservoir’, SPE 38397, paper presented at the 2nd North American Coil Tubing Round Table, Montgomery, Texas, 1 – 3 April, 1997. Machac, I., Ulbrichova, I., Elson, T. P., and Cheesman, D. J., 1995, ‘Fall of Spherical Particles through Non – Newtonian Suspensions’, Chem. Engr. Science (Shorter Communications), Vol. 50 (20), 3323 – 3327. Martins, A. L., and Santana, C. C., 1990, ‘Modeling and Simulation of Annular Axial Flow of Solids and Non – Newtonian Mixtures’, paper presented at the 1990 III Encontro Nacional de Ciencias Termicas, Itapema, SC, Brazil Martins, A. L., and Santana, C. C., 1992, ‘Evaluation of Cuttings Transport in Horizontal and Near Horizontal Wells – A Dimensionless Approach’, SPE 23643, presented at the Latin American Petroleum Engineering Conference, II LACPEC, Caracas, Venezuela, March 8 – 11, 1992. Martins, A. L., Sa, C. H. M. and Lourenco, A. M. F., 1996, ‘Experimental Determination of Interfacial Friction Factor in Horizontal Drilling with a Bed of Cuttings’, SPE 36075, paper presented at the Fourth SPE Latin American and Caribbean Petroleum Engineering Conf., Port of Spain, Trinidad & Tobago, 23 – 26 April, 1996. Martins, A. L., Santana, M. L., and Gaspari, E. F., 1999, ‘Evaluating the Transport of Solids Generated by Shale Instabilities in ERW Drilling’, SPE Drill. and Completion, Vol. 14 (4), 254 - 259. Masuda, Y., Doan, Q., Oguztoreli, M., Naganawa, S., Yonezawa, T., Kobayashi, A., Kamp, A., 2000, ‘Critical Cuttings Transport Velocity in Inclined Annulus: Experimental Studies and Numerical Simulation’, SPE / Petroleum Society of CIM 65502, paper presented at the 2000 SPE / Petroleum Society of CIM on Horizontal Well Technology, Calgary, Alberta, 6 – 8 November, 2000 Miura, H., Takahashi, T., Ichikawa, J., and Kawase, Y., 2001, ’Bed Expansion in Liquid–Solid Two-Phase Fluidized Beds with Newtonian and Non-Newtonian Fluids over the Wide Range of Reynolds Numbers’, Powder Techn., Vol. 117, 239 – 246. Mols, B. and Oliemans, V. A., 1998, ‘A turbulent Diffusion Model for Particle Dispersion and Deposition in Horizontal Tube Flow’, Int. J. Mult. Flow, Vo. 24 (1), 55 – 75. Nguyen, D., and Rathman, S. S., 1998, ‘A three layer Hydraulic Program for Effective Cuttings Transport and Hole Cleaning in Highly Deviated and Horizontal Wells’, SPE Drilling & Completion, Vol. 13 (3), 182 – 189. Pechenkin, M., V., 1972, ‘Experimental Studies of Flows with High Solid Particle Concentration’, Proc. Cong. A.I.R.H., Tokyo. Peden, J. M., Ford, J. T., and Oyeneyin, M. B., 1990, ‘Comprehensive Experimental Investigation of Drilled Cuttings Transport in Inclined Wells Including the Effects of Rotation and Eccentricity’, SPE 20925, paper presented at Europec, The Hague, Netherlands, Oct. 22 – 24.

23

Pilehvari, A. A., Azar, J. J., Shirazi, S. A., 1996, ‘State of the Art Cuttings Transport in Horizontal Wellbores’, SPE 37079, presented at the 1996 SPE International Conference On Horizontal Well Technology, Calgary, Canada, 18 – 20 November. Pilehvari, A. A., Azar, J. J., 1999, ‘State of the Art Cuttings Transport in Horizontal Wellbores’ SPE Dril. & Compl, Vol 14 (3), 196 – 200. Portman, L., 2000, ‘Reducing the Risk, Complexity and Cost of Coiled Tubing Drilling ‘, IADC / SPE 62744, paper presented at the 2000 IADC/SPE Asia Pacific Drilling Technology held in Kuala Lumpur, Malaysia, 11–13 September 2000.

Ramos Jr., A. B., Fahel, R.A,,Chaffin, M., and Pulis, K., H., 1992, ‘Horizontal Slim-Hole Drilling With Coiled Tubing: An Operator’s Experience’, SPE 23875, paper presented at the 1992 lADC/SPE Drilling Conference held In New Orleans, Louisiana, February 10-21, 1992. Santana, C. C., Massarani, G., and Sa, C.H.M., 1991, ‘Dynamics of Solid Particles in Non – Newtonian Fluids: The Wall and concentration Effects’, presented at the 22nd Annual Meeting of the Fine Particle Society, USA, July 31 – Aug. 2. Santana, M. and Martins, A. L., 1996, ‘Advances in the Modeling of the Stratified Flow of Drilled Cuttings in High Angle and Horizontal Wells’, SPE 39890, paper presented at the International Petroleum Conference and Exhibition, Mexico, March 3 – 5, 1996. Silva, M. G. P., and Martins, A. L., 1988, ‘Evaluation of the Rheological Behavior, Equivalent Diameter and Friction Loss Equations of Drilling Fluids in Annular Flow Conditions’, paper presented at the 1st World Conference on Experimental Heat Transfer, Fluid Mechanics and Thermodynamics, Dubrovnik, Yugoslavia. Stiles, E. K. , DeRoeun M. W. , Terry, I. J., Cornell, S. P. and DuPuy, S. J., 1999, ‘Coiled Tubing Ultrashort-Radius Horizontal Drilling in a Gas Storage Reservoir: A Case Study’, SPE 57459, paper presented at the 1999 SPE Eastern Regional Meeting held in Charleston, West Virginia, 21–22 October 1999. Svendsen, Ø. , Saasen, A., Vassøy, B.,Skogen E., Mackin, F. and Normann, S. H., 1998, ‘Optimum Fluid Design for Drilling and Cementing a Well Drilled with Coil Tubing Technology’, SPE 50405, paper presented at the SPE International Conference on Horizontal Well Technology held in Calgary, Alberta, Canada, 1–4 November 1998. Taylor, G., 1954, ‘The Dispersion of Matter in Turbulent Flow through a Pipe’, Proc. R. Soc., Vol. A223, 446 – 468. Televantos, Y., Shook, C. A., Carleton A. and Streat, M., 1979, ‘Flow of Slurries of Coarse Particles at High Solids Concentrations’, Can. J. Chem. Engr., vol. 57, 255 – 262. Thomas, D. G., 1963, ‘Transport characteristics of Suspensions: Relation of Hindered Settling Floc Characteristics to Rheological Parameters’, AiChE J., Vol. 9, 310 – 316. Tomren, P. H., Iyoho, A. W., and Azar, J. J., 1986, ‘Experimental Study of Cuttings Transport in directional Wells’, SPE Drill. Engr., Febr. 1986, 43 – 56.

24

Walker S., and Li, J., 2000, ‘The Effects of Particle Size, Fluid Rheology, and Pipe Eccentricity on Cuttings Transport’, SPE 60755, paper presented at the 2000 SPE / ICOTA Coil Tubing Round Table, held in Houston, TX, 5 – 6 April. Walton, I. C., 1995, ‘Computer Simulator of Coiled Tubing Wellbore Cleanouts in Deviated Wells Recommends Optimum Pump Rate and fluid Viscosity’, SPE 29491, presented at the Productions Operations Symposium, Oklahoma City, OK, USA, 2 – 4 April. Weighill, G., Thereby, H., and Myrholt, L., 1996, ’Underbalanced Coiled Tubing Drilling Experience on the Ula Field’, SPE 35544, paper presented at the European Production Operations Conference and Exhibition, Stavanger, Norway, April. 16 – 17. Wilson, K. C., and Tse, J. K. P., 1987, Deposition Limit for Coarse Particles Transport in Inclined Pipes’, Proc. 9th Intl. Conf. on Hydraulic Transport of Solids in Pipes, BHRA Fluid Engr, Cranfield, UK, 149.

25

APPENDIX 1 – VOLUMETRIC CONCENTRATION OF SOLIDS DURING CTD The solids concentration in the liquid is fairly low during normal drilling operations, and for CTD rarely exceeds 2 – 4% by volume. It can easily be estimated from the following

)1(*)*4/(*)(

)1(*)*4/(*)(% 2

2

φπ

φπ

−+

−=

holem

hole

DROPQ

DROPC

where %C concentration of solids, % volume / volume ROP rate of penetration Dhole open hole diameter (bit diameter) φ formation porosity (here taken as 20%) Qm mud flow rate It should be noted that in the many studies referenced herein, the bed porosity is rarely taken into account. For the various field parameters as shown in Table 1, taking two values of Dhole (5 in. and 3 ¾ in., we can easily compute the volumetric concentration of solids (%C) for various flow rates and these calculations are shown in the figures below. It is seen that %C can vary from about 0,1% to max 3,5% for the observed parameters in practice.

Solids Volumetric Concentration for Dhole=5in

0,00

0,50

1,00

1,50

2,00

2,50

3,00

3,50

0 20 40 60 80 100

ROP (ft/hr)

Solid

s C

once

ntra

tion

(%v/

v)

Qm=1 bpm Qm=2 bpm Qm=3 bpm Qm=5 bpm

Solids Volumetric Concentration for Dhole=3 3/4in

0,00

0,50

1,00

1,50

2,00

0 20 40 60 80 100

ROP (ft/hr)

Solid

s C

once

ntra

tion

(%v/

v)

Qm=1 bpm Qm=2 bpm Qm=3 bpm Qm=5 bpm

26

APPENDIX 2 – TERMINAL SETTLING VELOCITY of SOLIDS There are many correlations published for predicting solids terminal settling velocity for newtonian and non-newtonian liquids. A literature review shows the following, with the following nomenclature: CD drag coefficient (dimensionless) g acceleration of gravity (m/s2) dp particle diameter (m) ∆ρ density difference, (ρp – ρ) / ρ (-) ρp solid density (kg/m3) ρ liquid density (kg/m3) V solid terminal velocity (m/s) Rep Reynolds number based on particle diameter Ar Archimedes number µ liquid viscosity (kg/(ms)) K consistency index of fluid (kg/(msn)) n flow behavior index (-) From the definition of drag coefficient

234

V

gdC pD

ρ

ρ∆= , and the proposed relationships here below, one can get V

normally by a trial and error procedure. NON – NEWTONIAN FLUIDS (power law) CHHABRA & Peri (1991) from

KdV np

n−=

2Re

ρ, through

)2/2()2/2(

)2/2()2/2(

3

/4Re

−−

−+−

Κ

∆== nn

nnpn

Dgd

CArρ

ρρ

they correlated available experimental data in the form of

baAr=Re , for Re < 104 and got

−= nn

a 73,051,0exp , 16,0954,0−=

nb

For given values of dp, ∆ρ, ρ, Κ, n, (Ar) is calculated and (Re) is predicted, from which we can directly determine settling velocity, hence CD. DOAN et al. (2000) with

K

dVn

np

n

1

2

8Re

−

−

=ρ

they use

27

{ }

989Re44,0

989ReReRe15,0124 687,0

>=

<+

=

D

D

C

C

MACHAC et al. (1999)

from KdV np

n−=

2Re

ρ and

[ ] 1000ReRe36,0Re25,2 06,031,0 <+= −DC

MIURA et al. (2001)

with KdV np

n−=

2Re

ρ and

510Re

4,0Re4

Re24

<+

+=DC

which was proposed for newtonian fluids, but used the same equation with the above definition of non-newtonian Re. CEYLAN et al. (1999)

with KdV nn−

=2

Re ρ and

Re24

*n

DX

C = , where,

( )33

)3/3(

4*

3532

*

53

232*

10Re10Re244

10Re10Re10log13

1

10Re33

<<+=

<<+

−+=

<+−

==

−−

−−

−−

nnn

nn

nn

nn

nXX

nnXX

nnnXX

NEWTONIAN LIQUIDS

we have, for µ

ρ pVd=Re :

FELICE (1999)

28

2

5,0Re8,463,0

+=DC

CHEREMISINOFF & GUPTA (1983 )

500Re44,0

500Re3Re4

Re24

3ReRe24

3/1

>=

<<+=

<=

D

D

D

C

C

C

DORON ET AL. (1987)

5

6,0

10*2Re50044,0

500Re1,0Re5,18

<<=

<<= −

D

D

C

C

Using the above equations, and for a variety of fluid viscosity, solid diameters, but ∆ρ/ρ = 1,5, we computed the terminal velocity and these are shown in the following figure. We see that apart for the correlation of Chhabra et al. (1991), not shown because it consistently underpredicted settling velocity compared to the other correlations, the proposed correlations for non-newtonian fluids coincide with the data for newtonian

Drag Coefficient

0,1

1

10

100

0,1 1 10 100 1000 10000Re

Cd

Doron-newt Chere-newt Felice-newt Doan Ceylan Machac Miura

29

fluids and predictions are fairly well overlapped. We can therefore conclude that using one of the above correlations we can fairly accurately estimate the terminal settling velocity of solid spheres in non-newtonian or newtonian fluids. EXPERIMENTAL DETERMINATION OF SOLID SETTLING VELOCITIES IN NEWTONIAN & NON – NEWTONIAN LIQUIDS Some data has been derived for the settling velocities of single spheres in liquids. Data was obtained in a cylinder of length of 46 cm, diameter of 6 cm, using solid glass beads of ρp = 2,5 g/cm3, of three diameters, dp1 = 1 – 1,25 mm (took a mean of 1,125 mm), dp2 = 2 – 2,3mm (took a mean of 2,15 mm) and dp3 = 2,85 – 3,3 mm (took a mean of 3,075 mm). The time of travel of the sphere over a fixed distance (L varied between 7 – 45 cm) was monitored with a stop watch, and values of time were averaged. Standard deviations were of the order of 2 – 7% of the mean values. The liquids used were CMC solutions in water at 0,5%, 1% and 2% by volume, with values for the 1% K = 0,04 Pasn, n = 0,85 2% K = 0,294 Pasn, n = 0,77 For the 0,5%, the value obtained was almost newtonian, with a viscosity of 5,44 cp. The measured velocities for each liquid and for each solid diameter (total six conditions) are compared with the predictions from the above presented correlations in the following figure.

The data shows that the predictions work well, within about +/- 25% at low velocities, to less than +/- 10% at high velocities.

Measured vs Predicted Settling Velocities

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30Measured Velocities (cm/s)

Pre

dict

ed V

eloc

ities

(cm

/s)

'Doan' 'Ceylan' 'Machac' 'Miura' Felice'Chere'misinoff' Doron '+25%' '-25%'

30

APPENDIX 3: MODELING THE CONCENTRATION OF SOLIDS IN THE HETEROGENEOUS MIXTURE Many researchers use the ‘solid – diffusion’ equation, which is derived on the assumption that turbulent eddies counterbalance the settling of solids due to gravity. Using 1-D modeling, with ‘y’ the distance from the bottom of the pipe (in the presence or absence of a bed), a balance between rate of settling and upward transport by turbulent forces results in (Doron et al., 1987, Walton, 1995, Kamp & Rivero, 1999)

02

2=+

dyCdD

dydCu p (A3.1)

where C the volumetric concentration of solids = C(y) only up the terminal settling velocity of solids D the dispersion (or diffusion) coefficient of solids the above equation is solved using the two boundary conditions B.C. 1 no flux of solids at the top wall, i.e.

[ ] 0=

−+−

−

−

==

dydyp dy

dCDCu (A3.2)

B.C. 2 the mean concentration over the cross sectional area is Cm, i.e.

∫∫=Am dAyC

AC )(1

(A3.3)

The solution to the equation, for the case of pipe and annulus is

−+=Dyu

BAyC pexp)( (A3.4)

and using the first boundary condition we get, A = 0, while for the 2nd boundary condition we proceed as follows, with reference to the figure below (following Walton, 1995):

If we let x, y the horizontal and vertical coordinates and γ the angle above the horizontal axis of the general points on the circumference of the outer cylinder. Then

)cos(

)cos()sin(

0

00

γ

γγγ

Rx

dRdyRy

=

=→=

the area of the annulus is given by

x

y

γ

R0

Ri

x

y

γ

R0

Ri

31

( )∫ ∫∫

∫∫

∫ ∫∫ ∫ ∫ ∫

− −−

−−

− −− −

−=−=

→−=

→

−=

−=

2/

2/

2/

2/

2220

22/

2/

2220

2/

2/

2/

2/00

0 0

)][cos()][cos()][cos(2

))cos())(cos(())cos())(cos((2

220

0

0

0

π

π

π

π

π

π

π

π

π

π

γγγγγγ

γγγγγγ

dRRdRdRA

dRRdRRA

xdyxdydxdydxdyA

ii

ii

R

R

R

R

R

R

x R

R

x i

i

i

i

which of course finally gives ( )220 iRRA −= π .

For the mean concentration of solids in the annulus, we have

∫∫ ==A

m dAyCAC )(

∫−

−2/

2/00 ))cos())(cos()((*2

π

πγγγ dRRyC ∫

−

2/

2/))cos())(cos()((

π

πγγγ dRRyC ii

and since

−=

−=

00expexp)(dyPeB

Dyu

ByC p

with Ddu

Pe p 00

*= (A3.6)

we get

[ ] ∫−

−−=2/

2/

20220 )][cos(

2)sin(

exp**2π

πγγ

γd

PeRRBAC im

and since ( )220 iRRA −= π , we get finally

IntC

B m2π

= (A3.5)

with ∫−

−=2/

2/

20 )][cos(2

)sin(exp

π

πγγ

γd

PeInt (A3.7)

Hence, the concentration profile for solids in the concentric annulus is given by

IntdyPe

CyC

m

−

= 00exp

2)( π

(A3.8)

valid for the values of

32

00 & dyddyd ii <<−<<− while for pipe, with

Ddu

Pe p= (A3.9)

with d the pipe diameter, we get,

IntdyPe

CyCm

−

=exp

2)( π

(A3.10)

with the value of Int given by (A3.7) NOTE: Many investigators solve the diffusion equation in the presence of stationary or moving bed and hence use as a second boundary condition the following at y = h C = Cb where h = height of bed Cb = concentration of solids in bed, normally taken as Cb = 0,52 for

cubing packing of spheres Under these conditions the final result is, for both pipe and annulus (with the definition for Pe for pipe and annulus as in (A3.9 and A3.6),

−−=

dh

dyPe

CyCb

exp)( for pipe, (A3.11)

−−=

000exp)(

dh

dyPe

CyC

b, for annulus (A3.12)

It should be noted that to get the solutions (A3.8 and A3.10) or (A3.11 and A3.12) it has been implicitly assumed that up ≠ f(y), hence up is the terminal, not the hindered terminal velocity of solids and D ≠ f(y). Doron et al. (1987) and Walton (1995), although they assumed this, they later use the hindered settling velocity which is obviously not correct, as already pointed out by Kamp & Rivero (1999). It is apparent that neither (A3.8) nor (A3.12) give the absolute value of C(y) for the annulus. They give C(y) as a function of either the bed concentration Cb (assumed 0,52) or the mean concentration Cm in the suspension layer. The major difficulty in these solutions is the determination of Pe. ESTIMATION OF PE NUMBER It has been shown that the concentration of solids in the heterogeneous layer depends on Peclet number, Pe,

DduPe P= , for pipe and

Ddu

Pe P 00 = , for annulus

where uP the solids terminal settling velocity d the pipe diameter d0 the annulus outer pipe diameter D the solids dispersion coefficient

33

and for Pe <<1, the process is diffusion controlled, while for Pe>>1 the process is gravity controlled. To get the minimum value of Pe that could be encountered during CTD, uP & d0 should be minimum, while D should be maximum. We take (dp)min ~ 1mm. From Appendix 2, we can easily estimate that minimum uP is of the order of 0,008m/s., while for d0 the minimum is ~ 2 ¾ in. (0,07m) (Table 1). For the dispersion coefficient D, there are two approaches suggested so far in the literature: 1. that of Doron et al. (1987) where they suggest (following Talyor, 1954) that

))((052,0 * rUD = , or for annulus, )2/)((052,0 * hDUD =

where (U*) is the friction velocity = 2/fU , and r is the hydraulic radius, which for a full pipe is equal to the pipe radius = d/2, and for annulus, Dh is the hydraulic diameter = d0 – di U is the mean liquid velocity and f is the friction factor.

For typical values of annulus velocity (Table 1), U varies from 0,4 to 1,5 m/s, while for newtonian fluids it has been suggested that optimum µ ~ 10 – 20 cp (Leising & Walton, 1998).

For turbulent flow, 25,0Re079,0

=f

For D to be maximum, U should be maximum and f maximum or Re minimum. Re is minimum at the onset of turbulent flow (~ 2100) and hence, fmax = 0,0116. Hence,

smDfUD h /10*7,2))(2/(052,0 24maxmaxmaxmax

−== or Dmax ~ O (10-4) 2. that of Walton (1995), where he suggests that

3/10 Re***014,0 pp udDD = , where

D0 is given as 05,012,0

24,15,0

0 <

= CifCD , hence (D0)max ~ 0,8 m2/s.

dp is the particle diameter, up is the particle terminal velocity and Re is the Re of flow. For this situation, for D to be maximum, D0 should be maximum (0,8), (dp)max ~ 6mm, (up)max ~ 0,1 – 0,2 m/s and Remax ~ 10.000, hence (D)max ~ 10-4

m2/s, same order of magnitude as the approach of Doron et al. Hence the minimum values of Pe for annulus for typical CTD is of the order of

5~10

)07,0)(008,0(~4−

Pe

34

SOLID CONCENTRATION PROFILES It is worthwhile to examine the behavior of the function C(y) for various values of Pe and for 0 < y < d0. It should be noted that for Pe <<1 the process is diffusion (dispersion) controlled and expect a more uniform distribution of solids in the heterogeneous layer and for Pe>>1, the process is gravity controlled and expect to see the solids accumulating near the bottom of the pipe. It has been shown above that for drilling applications the minimum values that Pe should be of the order of ~ 5. Using equation A3.8 together with equation A3.7 we calculated the values of C(y)/Cm for various values of Pe. The results are shown in Figure A3.1 below. We include calculations for Pe 1, 2 and 5.

FIGURE A3.1 Concentration distribution in annulus (1-D modeling) for various Pe

numbers We see from the above figure that the solids may be distributed very close to the pipe bottom (hence the liquid above is free of solids) at the minimum estimated value of Pe (~5), while for Pe ~ 1, there is a good distribution of solids throughout the channel. Hence, the determination of the proper value that Pe takes for the situation at hand is very critical. And the main difficulty in estimating Pe is the value of the dispersion (or diffusion) coefficient of solids, D.

Variation of C/Cm for various Pe

0,00,10,20,30,40,50,60,70,80,91,0

0,0 0,5 1,0 1,5 2,0C/Cm

y/d 0

Pe=1 Pe=2 Pe=5