Laminar Turbul Transi Founargiotakis Kelessidis Maglione

Laminar, Transitional and Turbulent Flow of Herschel–Bulkley Fluids in Concentric Annulus K. Founargiotakis,1 V. C. Kelessidis1 * and R. Maglione2 1. Department of Mineral Resources Engineering, Technical University of Crete, Polytechnic City, 73100 Chania, Greece 2. Consultant, Vercelli, Italy

An integrated approach is presented for the flow of Herschel–Bulkley fluids in a concentric annulus, modelled as a slot, covering the full range of flow types, laminar, transitional, and turbulent flows. Prior analytical solutions for laminar flow are utilized. Turbulent flow solutions are developed using the Metzner–Reed Reynolds number after determining the local power law parameters as functions of flow geometry and the Herschel–Bulkley rheological parameters. The friction factor is estimated by modifying the pipe flow equation. Transitional flow is solved introducing transitional Reynolds numbers which are functions of the local power law index. Thus, an integrated, complete and consistent set, combining analytical, semi-analytical and empirical equations, is provided which describe fully the flow of Herschel–Bulkley fluids in concentric annuli, modelled as a slot. The comparison with experimental and simulator data from various sources shows very good agreement over the entire range of flow types. On pr´esente une m´ethode int´egr´ee pour l’´ecoulement des fluides d’Herschel–Bulkley dans un espace annulaire concentrique repr´esent´e par une fente, couvrant la gamme compl`ete des types d’´ecoulement, a` savoir laminaire, en transition et turbulent. Des solutions analytiques ant´erieures sont utilis´ees pour l’´ecoulement laminaire. Des solutions d’´ecoulement turbulent sont e´ labor´ees a` l’aide du nombre de Reynolds de Metzner–Reed apr`es avoir d´etermin´e les param`etres de loi de puissance locaux en fonction de la g´eom´etrie de l’´ecoulement et des param`etres rh´eologiques d’Herschel–Bulkley. Le facteur de friction est estim´e en modifiant l’´equation d’´ecoulement dans un tube. L’´ecoulement en transition est r´esolu en introduisant les nombres de Reynolds exprim´es en fonction de l’indice de loi de puissance local. Ainsi, un ensemble int´egr´e, complet et consistant, combinant des e´ quations analytiques, semi-analytiques et empiriques, est fourni, qui d´ecrit enti`erement l’´ecoulement des fluides d’Herschel–Bulkley dans des espaces annulaires concentriques repr´esent´es par des fentes. La comparaison avec les donn´ees exp´erimentales et les donn´ees de simulateurs de diverses sources montre un tr`es bon accord pour la gamme compl`ete des types d’´ecoulement. Keywords: flow in annulus, Herschel–Bulkley fluid, laminar, transitional, turbulent flow

INTRODUCTION

N

on-Newtonian flow in annuli is encountered in many situations of importance in a variety of industries and particularly in oil-well drilling. For many non-Newtonian fluids, the two parameters models of Bingham plastic (Bingham, 1922) and of power law (Skelland, 1967; Govier and Aziz, 1972; Bourgoyne et al., 1991) are used most often because of their simplicity and the fair description of the fluid rheograms. The nonlinear three parameters model proposed by Herschel and Bulkley (1926) has not been used widely until very recently although it has been shown to fit much better rheological data of aqueous clay slurries and of drilling fluids (Fordham et al., 1991; Hemphil et al., 1993; Maglione and Ferrario, 1996; Kelessidis et al., 2005, 2007). Reasons for the non-frequent use were not

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only the complexity in derivation of the model’s three parameters (Nguyen and Boger, 1987; Hemphil et al., 1993) but also the fact that analytical solutions for laminar or turbulent flow in annuli are not possible, requiring either graphical or trial-anderror solutions (Govier and Aziz, 1972; Hanks, 1979; Fordham et al., 1991). The on-line use, however, of personal computers enabled Maglione et al. (1999, 2000), Bailey and Peden (2000) and Becker et al. (2003) to utilize Herschel–Bulkley

∗ Author to whom correspondence may be addressed. E-mail address: [email protected] Can. J. Chem. Eng. 86:676–683, 2008 © 2008 Canadian Society for Chemical Engineering DOI 10.1002/cjce.20074

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rheological model in fluid mechanics computations of drilling fluids. Laminar flows of Bingham plastic and power law fluids in concentric annuli have been analyzed by Fredrickson and Bird (1958) while laminar flow of Herschel–Bulkley fluids in concentric annuli has been investigated by Hanks (1979) who provided a non-analytical solution, although with some errors (Buchtelova, 1988), together with several tables covering various annulus aspect ratios. Bird et al. (1983) provided an overview of solutions for the flow of yield-pseudoplastic fluids in various conduits. Analytical solutions for power-law and yield-pseudoplastic fluids, but not for Herschel–Bulkley fluids, in concentric annuli have been presented by Gucuyener and Mehmetoglu (1992), David and Filip (1995) and Filip and David (2003). Analytical and numerical codes covering laminar flow of non-Newtonian fluids in both concentric and eccentric annuli have been presented by Haciislamoglu and Langlinais (1990), Walton and Bittleston (1991), Szabo and Hassager (1992), Yang and Chukwu (1995), Hussain and Sharif (1997), Siginer and Bakhtiyarov (1998), Meuric et al. (1998), Fang et al. (1999) and Hussain and Sharif (2000), while Round and Yu (1993) have studied numerically laminar nonNewtonian flows in the entrance of a concentric annuli. Fordham et al. (1991) provided an analysis leading to a numerical code for solving the annulus situation as well as the case of slot approximation for laminar flow, together with some experimental data. Chin (2001) has provided a general simulator covering laminar nonNewtonian flows in complex geometries. Measured and computed velocity profiles for laminar flow of yield pseudoplastic fluids in concentric and eccentric annuli have been given also by Escudier et al. (2002a,b). No analytical solutions exist for turbulent flow of Herschel–Bulkley fluids in annuli and resort is made to experimental data with the use of the friction factor. Heywood and Cheng (1984) have reported variations of predictions of different proposed correlations of as much as ±50% for turbulent flow of Herschel–Bulkley fluids in pipes. Harnett and Kostic (1990) reported that the best approach for turbulent flow of Herschel–Bulkley fluids in pipes was through the use of the Metzner and Reed (1955) graph. No such studies have been performed for the case of concentric annuli except the study by Reed and Pilehvari (1993) where a fairly complex model covering laminar, transitional and turbulent flow of Herschel–Bulkley fluids flowing in concentric annuli has been presented. Reed and Pilehvari (1993) proposed modifications to the diameters of the conduits to account for the non-Newtonian effect, using the generalized power law model of Metzner and Reed (1955) and the friction factor suggested for pipe flow by Dodge and Metzner (1959). Their model described very well their experimental data. Some confirmation of the Reed and Pilehvari model was also provided by Subramanian and Azar (2000) but there were cases for which predictions were at odds with the measurements. The approach of Reed and Pilehvari has been followed also by Merlo et al. (1995), Maglione et al. (2000) and Bailey and Peden (2000) to cover all flow regimes for flow of Herschel–Bulkley fluids and of generalized non-Newtonian fluids in concentric annuli. A consistent and accurate methodology for predicting pressure drop for laminar, transitional and turbulent flow as well as the transition points from laminar to turbulent flow for the flow of Herschel–Bulkley fluids in concentric annuli, without resorting to heavy numerical computations, does not exist and this is the scope of the present paper. And as the slot model has been used often to approximate flows in concentric annuli of large diameter

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Figure 1. Flow in an annulus and representation as a slot for a Herschel–Bulkley fluid.

ratios, normally greater than 0.3 (Guillot and Dennis, 1988; Guillot, 1990; Bourgoyne et al., 1991; Fordham et al., 1991), the slot model will be used in the present approach.

THEORY The Herschel–Bulkley model is given by:
=
y + K


n


du dy

for
>
y and

,


=
y + K − du dy

n ,

du dy

for
>
y and

>0 du dy

(1)

<0

The geometry of the system is depicted in Figure 1. There is a central core of the fluid which moves as a rigid plug if the shear stress levels are smaller than the yield stress of the fluid. The flow rate for a slot of width, w, is then given by Grinchik and Kim (1974), Fordham et al. (1991) and Kelessidis et al. (2006):

 q=

dpf /dL K

m

2w(h/2)m+2 (1 − )m+1 [ + (m + 1)] (m + 1)(m + 2)

(2)

where m = 1/n and the dimensionless shear rate, , is given by:
y .
w

=

(3)

The flow rate can then be given in terms of pressure drop and the annulus geometric parameters as: q=

(r22 − r12 )(r2 − r1 )1+m ((1/K)(dpf /dL))m (1 − (
y /[(r2 − r1 )/2](dpf /dL)))1+m 2m (m + 1)(2m + 4)




×


y +m+1 [(r2 − r1 )/2](dpf /dL)



(4)

while the mean velocity, designated as, V, is given by:

 V=

dpf /dL K

m

(h/2)m+1 (1 − )m+1 ( + m + 1) . (m + 1)(m + 2)

(5)

Using Equation (5), the velocity profiles can be recast in terms of the dimensionless shear rate as follows: u = (m + 2) V 0≤y≤



 ;

h (1 − ) 2

m+2 u = V +m+1

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(1 − )m+1 − (1 −  − (y/(h/2)))m+1 (1 − )m+1 ( + m + 1)

(6)

h h (1 − ) ≤ y ≤ (1 + ) 2 2

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(7)

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u = (m + 2) V



(1 − )m+1 − (−1 +  + (y/(h/2)))m+1 (1 − )m+1 ( + m + 1)

 ;

h (1 + ) ≤ y ≤ h. 2

(8)

Laminar flow is amenable to analytical solution but for transition and turbulent flows, empirical equations must be used. Following Metzner and Reed (1955), use is made of the local power law parameters. The relevant Rabinowitsch (1929) equation for flow in an annulus, modelled as a slot, can be shown to be: ˙ Nw


w d 12V = = f (
w ) = (d2 − d1 ) 3 d
w



12V d2 − d1



2 + 3



12V d2 − d1

 . (9)

This equation shows that the shear stress at the wall,
w , depends only on the Newtonian shear rate at the wall ˙ Nw . The local power law parameters, K  , n  , are then defined by: n =

d ln (
w ) d ln (˙ Nw )



2n + 1 ˙ Nw . 3n





(13)

The relationship between (n  ) and the Herschel–Bulkley parameters can be derived by combining the laminar flow solution (Equation (2)) with the definition of (n  ), Equation (10), as it is done in the case of pipe flow, giving finally: n =

n(1 − )(n + n + 1) 1 + n + 2n + 2n2  2

(14)

while the K  parameter can be shown to be: K =


y + K((2n + 1/3n )˙ Nw )n (˙ Nw )n

(15)

Equations (14) and (15) fully define the generalized power law parameters of Herschel–Bulkley fluids flowing in an annulus, modelled as a slot, and are functions of the Herschel–Bulkley rheological parameters,
y , K, n, and of the particular flow situations, through
w . The friction factor, f, is estimated, after adjusting the pipe flow equation proposed by Dodge and Metzner (1959) to flow in annuli, as: 1

 =

0.395 4  log [ReMRa f 1−n /2 ] −  1.2 . (n )0.75 (n )

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f

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(16)

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Re1 = 3250 − 1150(n )

(18)

Re2 = 4150 − 1150(n ).

(19)

fl1 =

(12)

V(d2 − d1 ) V 2−n (d2 − d1 )n = . e K  (12)n −1

A transition region, instead of a single transition point, is defined for a range of Reynolds numbers, with Re 1 denoting the beginning of transition and Re 2 the end of transition. Guillot (1990) has suggested to take these limits from the Dodge and Metzner (1959) graph, thus giving:

(11)

The generalized Reynolds number for the flow of Herschel–Bulkley fluid in an annulus then becomes, with e =
/˙ w : ReMRa =

(17)

(10)

The Herschel–Bulkley wall shear rate, ˙ w is then related to the Newtonian shear rate, ˙ Nw , by: ˙ w =

dpf 2fV 2 = . dL d2 − d1

The friction factor for the transition region, f tr , is estimated at the particular Reynolds number, Re MRa , with linear interpolation between the laminar friction factor at Re 1 , f l1 , and the turbulent friction factor, f t2 , at Re 2 . Other interpolation approaches have not proven to provide better estimates. Hence, f l1 can easily be shown to be:

and
w = K  (˙ Nw )n

Guillot and Dennis (1988) have reported good agreement of this equation with their data for flow of drilling fluids and of cement slurries in concentric annuli. The pressure drop is then computed with:

24 Re1

(20)

while f t2 is taken from the solution of Equation (16) with Re MRa = Re 2 . Finally, the transitional friction factor is estimated as: ftr = fl1 +

(ReMRa − Re1 ) × (ft2 − fl1 ) . (Re2 − Re1 )

(21)

Derivation of pressure drop from the flow rate requires iterative solution. Similarly, derivation of the flow rate for a given pressure drop, normally used for design purposes, although gives a direct solution for laminar flow, it still requires iteration because the type of flow, laminar, turbulent or transition, is not known a priori and is known only after computing the pressure drop which gives the wall shear stress,
w , on which the local power law parameter, n  , depends. The solution procedure for the former case will be given here below, as the latter case can be easily resolved. The current approach uses a bisection method and the flow diagram is given in Figure 2. Two values for
w are assumed, one close to the yield stress and the second at very high value. The third value is then the arithmetic mean of the two. The Newtonian shear rate at the wall is computed together with the dimensionless shear stress  from Equations (3) and (9). The local power law parameters are then computed, n  from Equation (14) and K  from Equation (15). Then the generalized Reynolds number is estimated from Equation (13). The limits for laminar and turbulent flow can be computed and accordingly, the friction factor is computed. The flow rate is then estimated from:

 q=A×

2 ×
w f ×

(22)

The procedure converges when |q–q given | ≤ ε. Following the above procedure, any flow problem can be solved without

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Figure 2. Flow diagram for the solution process.

Figure 3. Flow diagram for the solution process for transitional points.

prior knowledge of the type of flow (laminar, transitional, or turbulent) and the procedure, based on theoretical and semitheoretical grounds, has proven to work as it will be demonstrated later. Many times is of great interest to know the start and end points of laminar to transition and transition to turbulent flow, that is, Re 1 and Re 2 . For accomplishing this, the procedure shown in Figure 3 is followed. Input data for the procedure are the geometry (d 2 –d 1 ), fluid rheology and density (). The average velocity, V, is considered as variable parameter. The first step consists in assuming that the local power law index equals the Herschel–Bulkley index, n = n. Then, the Newtonian shear rate at the annulus wall is calculated from Equation (9). Following this, the Herschel–Bulkley shear rate at the wall, the shear stress at the wall, the dimensionless shear stress,  1 , and the local power-law flow-behaviour index n1 , are calculated. The local consistency index K and the generalized Reynolds number ReMRa are then calculated. Subsequently, the local wall shear rate and the shear stress, the dimensionless shear stress,  2 , and the local power-law flow-behaviour index n2 , are again calculated. The procedure is repeated until the following conditions are satisfied, for the beginning of transition, V1 and ReMRa1 and for the end of transition for V2 and ReMRa2 :

RESULTS

ReMRa1 − 3250 + 1150 · nj ≤ ε,

(23)

ReMRa2 − 4150 + 1150 · nj ≤ ε

(24)

The error in the calculation ε, is chosen to be smaller than 10−4 . It has been seen that differences between computed (˙ wj−1 and ˙ wj ) and (
wj –1 and
wj ) values become trivial after j = 4.

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The accuracy of the predictions of the proposed model has been determined by comparing with experimental data and simulator results from works reported previously. More specifically, experimental data for laminar flow in concentric annulus, presented by Fordham et al. (1991), Okafor and Evers (1992) and Hansen et al. (1999) will be used for laminar flow. Similarly, controlled field data provided by Langlinais et al. (1983) will be used to cover all three flow regime types, laminar, transitional and turbulent flows. Comparison of the predictions of this work with predictions from full simulators (Hanks, 1979; Fordham et al., 1991) will also be performed. Comparison of predictions from the approach presented in this work with experimental results of Okafor and Evers (1992) is shown in Figure 4 for an annulus aspect ratio of 0.62. There is excellent match of the predictions of this work with the experimental results. For a different fluid but the same annuli, the comparison with experimental results of Okafor and Evers (1992) is shown in Figure 5. There is excellent match with the data up to the velocity of 0.65 m/s, which corresponds to Reynolds number of 541. There is deviation for the last four points (maximum deviation is of the order of ∼17%), having velocities between 1.0 and 1.25 m/s which correspond to Reynolds numbers 992 and 1275, respectively, thus the flow is still in the laminar regime. The slope is similar but there is constant under-prediction the cause of which is unknown. In Figure 6, the comparison of predictions with experimental laminar flow results of Hansen et al. (1999) is shown. Very good match is seen for the laminar flow data with slight underprediction for the points close to transitional flow. In the graph, predictions in the transition region are also shown and the comparison with the data shows that probably the transition occurs earlier than is predicted by current methodology.

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Figure 4. Comparison of predictions with measurements from data of Okafor and Evers (1992). Computations are for a fluid with properties
= 1036.3 kg/m3 , K = 0.29647 Pasn , n = 0.5818 and  y = 9.431 Pa for an annulus with d 1 = 0.04826 m, d 2 = 0.07727 m.

Figure 7. Comparison of predictions with experimental results of Langlinais et al. (1983). Fluid data:  y = 0.7625 Pa, K = 0.0424 Pasn , n = 0.8476. Density is
= 1056.4 kg/m3 . Annulus dimensions d 2 = 0.062 m and d 1 = 0.0334 m.

Figure 5. Comparison of predictions with data of Okafor and Evers (1992) for fluid with
= 1066.2 kg/m3 , K = 0.11934 Pasn , n = 0.75534,  y = 0.622 Pa with d 1 = 0.04826 m, d 2 = 0.07727 m. Figure 8. Comparison of predictions of current model with the full numerical solution of Hanks (1979). Data for laminar flow:  y = 10 Pa, K = 0.1 Pas, n = 0.8 and different aspect ratio  = r 1 /r 2 .

Figure 6. Comparison of predictions with experimental results of Hansen et al. (1999). Fluid properties are  y = 2.45 Pa, K = 0.14 Pasn , n = 0.60. Annulus dimensions d 2 = 0.1016 m and d 1 = 0.089 m. Density (not reported, water and xantham gum solution) was taken as 1000 kg/m3 for computing the transition region.

Langlinais et al. (1983) presented an extensive range of pressure drop measurements for single phase flow of different drilling fluids in an 1829 m vertical well. Predictions of this work with their results for the full range of flow regimes, laminar, transitional and turbulent flow is given in Figure 7. Very good match with the data at all flow types is seen. Similarly, the correct prediction of the points for laminar-transition and transition-turbulent flow regimes is observed. Hanks (1979) provided the full solution to laminar flow of Herschel–Bulkley fluids in concentric annuli and presented his results in tabular and graphical form. In Figure 8, the predictions from this work are compared to the solution of Hanks, in the usual p–Q manner, where the curves of Hanks are taken from graphs of the original paper which were provided in dimensionless forms

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and for a range of annuli diameter ratio. One observes very close match of the predictions of this work with the predictions of the full numerical solution of Hanks (1979) for all aspect ratios, even down to an aspect ratio of 0.1. Fordham et al. (1991) presented experimental results and compared with the predictions of their numerical solution for the flow of a 0.5% xanthan gum in water, in a 3 m long, 4 cm by 5 cm annuli for a range of flow rates. Comparison of the predictions of this work with their experimental data as well as their predictions

Figure 9. Comparison of predictions with measurements and predictions of Fordham et al. (1991). All laminar data. Conditions are:  y = 1.59 Pa, K = 0.143 Pa sn , n = 0.54, for an annulus with d 2 = 0.05 m and d 1 = 0.04 m. Predictions of Fordham et al. (1991) coincided with their measured values.

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(which matched their data) is shown in Figure 9. The data cover laminar flow only. There is excellent match of model predictions both with the experimental data and the predictions of the simulator of Fordham et al. (1991) which uses the full laminar flow solution in an annulus.

A consistent model which predicts the pressure drop for laminar, transitional and turbulent flow of Herschel–Bulkley fluids in concentric annuli, modelled as a slot, has been presented. For laminar flow, the model provides analytical solution for the pressure drop giving also full velocity profiles. For transitional and turbulent flows, semi-analytical solutions have been provided using local power law approach giving equations which relate the local power parameters, K , n , to the rheological parameters of Herschel–Bulkley model and to the annulus flow geometry. A transition region has also been defined which depends on (n ). The predictions for laminar flow have been compared to literature data containing experimental data and predictions of numerical simulators which model fully flow in annuli. The comparison ranges from good to very good. The predictions for transition and turbulent flow compare favourably well with the scarce data from literature for drilling fluids flowing in a model 1829 m well. More experimental data is needed which will span the transition and turbulent flows to further test the transition equations.

ACKNOWLEDGEMENT The paper is dedicated to the memory of the first author, Mr. Konstantinos Founargiotakis, who tragically has left us in December 2006.

NOMENCLATURE

L m n n q q given r2 r1 Re 1 Re 2 Re MRa u

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flow area (m2 ) diameter of inner tube of annulus (m) diameter of outer tube of annulus (m) pressure drop (Pa/m) friction factor friction factor at Re 1 friction factor at Re 2 slot height (m) flow consistency index (Pa sn ) flow consistency index for local power law parameters (Pa sn ) length (m) inverse of flow behaviour index (=1/n) flow behaviour index flow behaviour index for local power law parameters flow rate (m3 /s) given value of flow rate (m3 /s) outer cylinder radius of annulus (m) inner cylinder radius of annulus (m) Reynolds number for beginning transition laminar to transitional flow Reynolds number for beginning transition transitional to turbulent flow Metzner–Reed Reynolds number for annulus velocity (m/s)

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mean velocity (m/s) slot width (m) distance from bottom wall of slot (m)

Greek Symbols

CONCLUSIONS

A d1 d2 dp f /dL f f l1 f t2 h K K

V w y

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˙ ˙ Nw ˙ w e  

y
w

shear rate (s−1 ) Newtonian shear rate on the wall (s−1 ) wall shear rate (s−1 ) effective viscosity, (Pa s) dimensionless shear stress for annulus fluid density (kg/m3 ) shear stress (Pa) yield stress for Herschel–Bulkley fluid (Pa) wall shear stress (Pa)

REFERENCES Bailey, W. J. and J. M. Peden, “A Generalized and Consistent Pressure Drop and Flow Regime Transition Model for Drilling Hydraulics,” SPE Drill. Completion 15, 44–56 (2000). Becker, T. E., R. G. Morgan, W. C. Chin and J. E. Griffith, “Improved Rheology Model and Hydraulic Analysis for Tomorrow’s Wellbore Fluid Applications,” Paper SPE 84215 presented at the SPE Productions and Operations Symposium, Oklahoma City, OK (2003). Bingham, E. C., “Fluidity and Plasticity,” McGraw-Hill, New York (1922). Bird, R. B., G. C. Dai and B. J. Yarusso, “The Rheology and Flow of Viscoplastic Materials,” Rev. Chem. Eng. 1, 1–70 (1983). Bourgoyne, A. T., M. E. Chenevert, K. K. Millheim and F. S. Young, Jr., “Applied Drilling Engineering,” SPE Textbook Series, Richardson TX (1991). Buchtelova, M., Comments on “The Axial Laminar Flow of Yield-Pseudoplastic Fluids in a Concentric Annulus,” Ind. Eng. Chem. Res. 27, 1557–1558 (1988). Chin, W. C., “Computational Rheology for Pipeline and Annular Flow,” Gulf. Pub. Co., Boston (2001). David, J. and P. Filip, “Relationship of Annular and Parallel-Plate Poiseuille Flows for Power Law Fluids,” Polym. Plast. Technol. Eng. J. 34, 947–960 (1995). Dodge, D. W. and A. B. Metzner, “Turbulent Flow in Non-Newtonian Systems,” AIChE J. 5, 189–204 (1959). Escudier, M. P., P. J. Oliveira and F. T. Pinho, “Fully Developed Laminar Flow of Purely Viscous Non-Newtonian Liquids Through Annuli, Including the Effects of Eccentricity and Inner-Cylinder Rotation,” Int. J. Heat Fluid Flow 23, 52–73 (2002a). Escudier, M. P., P. J. Oliveira and F. T. Pinho, “Fully Developed Laminar Flow of Non-Newtonian Liquids Through Annuli: Comparison of Numerical Calculations with Experiments,” Exp. Fluids 33, 101–111 (2002b). Fang, P., R. M. Manglik and M. A. Jog, “Characteristics of Laminar Viscous Shear-Thinning Fluid Flows in Eccentric Annular Channels,” J. Non-Newtonian Fluid Mech. 84, 1–17 (1999). Filip, P. and J. David, “Axial Couette-Poiseuille Flow of Power-Law Viscoplastic Fluids in Concentric Annuli,” J. Petr. Sci. Eng. 40, 111–119 (2003).

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Manuscript received November 17, 2006; revised manuscript received October 18, 2007; accepted for publication December 16, 2007.

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