Ind. Eng. Chem. Res. 1996, 35, 2039-2047
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Unsteady Axial Laminar Couette Flow of Power-Law Fluids in a Concentric Annulus Yuan Wang and Godwin A. Chukwu* Department of Petroleum Engineering, University of AlaskasFairbanks, 437 Duckering Building, P.O. Box 755880, Fairbanks, Alaska 99775-5880
The unsteady laminar couette flow of non-Newtonian power-law fluids in a concentric annulus is a flow phenomenon that can be applied to predict the surge or swab pressure encountered when running or pulling pipes in a liquid-filled borehole. It is similar to a moving cylinder, concentrically placed in a liquid-filled outer cylinder. This flow phenomenon has a wide variety of applications in the solution of problems in petroleum, chemical, mechanical, and ceramic industries. Several numerical models have been proposed by several authors for surge or swab pressure predictions. Though previous investigators had reported that the magnitude of the surge or swab pressure is determined by the annular geometry, the moving pipe velocity, and the drilling fluid properties, there has been no reported simple analytical presentation of the application of the unsteady couette flow phenomenon to predict either the surge or swab pressure which occurs when the casing pipes are in motion in a liquid-filled borehole. In this study, the motion equations are analytically solved for power-law fluids by the perturbation method. The solutions of the resulting pressure gradient equations are presented in both dimensionless and graphical forms and for different pipe/borehole diameter ratios and power-law index values. These allow for a more general application in annular fluid flow analysis and for predicting the surge or swab pressure during the unsteady motion of casing strings. Introduction Couette flow phenomenon occurs when two coaxial cylinders are placed such that one of which is stationary and the other is moving at a specified velocity. This flow characteristic is representative of flow in the borehole annulus where the wall of the wellbore is represented by the stationary cylinder and the drill string or casing is represented by the moving cylinder. The fluid local velocity is dependent on the velocity of the moving cylinder. The geometric representation of a concentric annulus is as shown in Figure 1. In drilling operations, the displacement of fluid encountered when running or pulling the drill string or casing in the annulus produces momentary variations in fluid pressure. This pressure change is known as the surge or swab pressure, the magnitude of which is governed by the movement of the inner pipe, the annular geometry, and the drilling fluid properties. The study of the couette flow phenomenon to predict the surge or swab pressure is interdisciplinary in nature and finds its applications in the petroleum, chemical, and mechanical industries. The effects of surge and swab pressures have been investigated and reported by numerous researchers since the early days of drilling. There has been a basic assumption in most of the previous studies of a uniform or constant pipe velocity when running an inner pipe in a liquid-filled borehole. In the studies presented by Cannon (1934), Clark (1955), Burkhardt (1961), Moore (1974), Chukwu and Blick (1991), Tao and Donovan (1955), Guckes (1975), and Yang (1993), the basic common assumption was that the pipe was in steady motion in the liquid-filled borehole. Lubinski et al. (1977) and Mitchell (1988) assumed that a compressive flow exists, which causes the pressure changes at constant pipe velocity. Yang and Chukwu (1995a,b) reported the effects of eccentricity on the * Corresponding author. Fax: 907/474-5912. E-mail: ffgac@ aurora.alaska.edu.
S0888-5885(95)00598-7 CCC: $12.00
Figure 1. Geometric representation of a concentric annulus.
steady couette flow of non-Newtonian power-law fluids in eccentric annuli. Letelier and Cespedes (1988) developed an analytical method for unsteady flow of power-law fluids in pipes. In their study, with the given pressure gradient profile, the velocity and shear stress profiles were obtained. Edwards, Nellist, and Wilkinson (1972) developed a numerical method for unsteady laminar flow of powerlaw fluids in pipes. In their study, velocity profiles were © 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
derived using an explicit finite difference technique with a given pressure gradient. Balmer and Florina (1979) developed a numerical method for the study of unsteady laminar flow of powerlaw fluids in pipes using an implicit finite difference method. Lal (1988) assumed an unsteady-state flow phenomenon and presented a numerical solution for his final equation. In the study of the unsteady laminar couette flow of non-Newtonian fluid presented here, it is believed that the surge or swab pressure is dependent not only on velocity but also on acceleration of the moving pipe, which is a real-life situation in drilling operations. The numerical solutions of the equations for unsteady laminar flow of non-Newtonian power-law fluid have been complex and time consuming. The method proposed in this study will attempt to solve the equations using the perturbation method. The final equations are solved analytically and presented graphically for practical application to compute the pressure associated with the pipe movements in liquid-filled boreholes. Governing Equations In the development of the equations which follow, the concentric annular geometry is represented by two cylinders positioned so that the inner cylinder moves with an acceleration, ap, and the other outer cylinder is stationary. The fluid velocity is computed relative to the velocity and acceleration of the inner pipe. The following assumptions are made in the derivation of the model equations. (1) There is unsteady-state, single-phase, incompressible fluid flow. (2) The flow regime is considered laminar. (3) The flow is isothermal with constant fluid properties. (4) The flow geometry is parallel, axial-symmetric. (5) Slip effect is not considered. (6) Gravity is neglected. (7) The pipe is considered closed (no fluid communication between the inside of the moving pipe and the annular space). The axial momentum equation for the unsteady-state, laminar flow of an incompressible fluid can be expressed in cylindrical coordinates as:
F
∂p* ∆p ∂u* τ* ∂τ* + + )) φ* ) ∂t* r* ∂r* ∂z* L
n
n
(∂u* ∂r* ) ∂u* τ* ) η(∂r* )
n
(5)
r g r0
(6)
2Frre1+nUre1-n ) ηtre
(7)
The dimensionless pressure gradient φ is defined by eq 8.
rre1+n
φ ) φ*
(8)
ηuren
and ure, tre, τre, and rre are velocity, time, shear stress, and radial reference constants, respectively. Equations 9 and 10 represent the dimensionless noslip boundary conditions which must be applied to solving the governing equations
u(rp,t) ) -vp
(9)
u(1,t) ) 0
(10)
Solution by the Perturbation Method. For relatively small values of , 1, the perturbation method can be employed by expanding the shear stress τ and velocity u in powers of . Thus:
τ ) τ0 + τ1 + τ22 + ...
(11)
u ) u0 + u12 + u22 + ...
(12)
When ) 0 and τ ) τ0, the acceleration term in eq 4 becomes zero and the equation becomes that of a steadystate case. Substitution of eqs 11 and 12 into eq 4 results in a system of linear motion (eqs 13a, 13b, and 13c).
τ0 ∂τ0 + )φ r ∂r
(1)
(13a)
-0.5
∂u0 τ1 ∂τ1 ) + ∂t r ∂r
(13b)
-0.5
∂u1 τ2 ∂τ2 ) + ∂t r ∂r
(13c)
r* e r0*
(2)
Similarily, the rheological equations are expressed as:
r* g r0*
(3)
∂u0 ) (-τ0)1/n ∂r
(14a)
∂u1 1 ) (-τ0)1/n-1(τ1) ∂r n
(14b)
The dimensionless motion equation can be expressed as:
0.5
r e r0
where the dimensionless unsteadiness number is defined by:
For a power-law fluid, the rheological equations relating the local shear stress to the local shear rate in cylindrical coordinates can be expressed as:
τ* ) -η
n
(∂u∂r ) ∂u τ ) (- ) ∂r τ)-
∂u τ ∂τ + + )φ ∂t r ∂r
(4)
and the dimensionless rheological equations for a powerlaw fluid in cylindrical coordinates are given by eqs 5 and 6.
(
τ12 ∂u2 1 1/n-1 ) (-τ0) -τ2 - 2 ∂r n nτ for r e r0 and
0
)
(14c)
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2041
∂u0 1 ) -τ0 ∂r n
(15a)
∂u1 1 ) - τ01/n-1τ1 ∂r n
(15b)
(
2
∂u2 τ1 1 ) - τ01/n-1 τ2 - 2 ∂r n nτ
0
)
(15c)
for r g r0. Substituting eq 12 into the boundary eqs 9 and 10, a system of boundary equations is obtained, given by eqs 16, 17, and 18.
ui(rp,t) ) -kivp
(16)
ui(1,t) ) 0
(17)
The leading order boundary conditions are given by eqs 26 and 27.
k0 + k1 + k22 + ... ) 1
k0 ) 1 ki ) 0,
∫r
r
(19b)
Leading Order Equations and Solutions. From the series of motion eq 13a, the leading order equation for shear stress is given as
τ0 ∂τ0 + )φ r ∂r
(20)
∂(rτ0)/∂r ) φ
(21)
f0 ) -k0vp
(28)
g0 ) 0
(29)
[ (
p
u0 )
)] ∫[ ( )] 1
r
1/n
dr - k0vp r02 r
0.5φ r -
r e r0
(30)
1/n
dr
r g r0
(31)
When r ) r0, the velocities u0 in eqs 30 and 31 are equal. Hence,
[∫ (
n
(k0vp) ) 0.5φ
r0
rp
)
r02 -r r
1/n
∫r
dr -
0
) ]
(
1
r02 rr
1/n
n
dr
(32)
If a constant a0, which is only dependent on the dimensionless pipe geometry rp, dimensionless critical radius r0, and the power-law index n, is defined as:
[∫ (
1/n
a0 ) 0.5
which can be written as:
(27)
r02 -r r
0.5φ
(19a)
i>0
u0(1,t) ) 0
If the expressions for f0 and g0 are substituted into eqs 24 and 25, respectively, the expressions for the leading order velocity are obtained.
(18)
where ith order degree ) 1, 2, 3, ..., ki ) acceleration ratios for different orders, and ui ) velocities for different orders. For a steady-state fluid flow,
(26)
The integration constants f0 and g0 are obtained by substituting eqs 26 and 27 into eqs 24 and 25. Hence,
u0 )
and
u0(rp,t) ) -k0vp
r0
rp
)
r02 -r r
1/n
dr -
∫r
(
1 0
) ]
r02 rr
1/n
dr
a0 > 0 (33)
Then eq 32 becomes
The leading order solution of shear stress is obtained when eq 21 is integrated, hence,
(k0vp)n ) φa0n
1 τ0 ) (0.5φr2 + c0) r
By employing eq 8 and the reference constants and substituting them into eq 34, the expression for the pressure gradient (∆p/L) is obtained.
(22)
where c0 is the constant of integration. When r ) r0, the leading order shear stress τ0 ) 0, and eq 22 becomes
(
τ0 ) 0.5φ r -
r0 r
)
(23)
Substituting eq 23 into the rheological eqs 14a and 15a and integrating the resulting expressions, eqs 24 and 25 are obtained, respectively.
u0 )
∫r
[ (
)]
[ (
)]
r p
r02 -r 0.5φ r
u0 )
∫r
r02 0.5φ r r
(35)
( )
(36)
In field units, eq 35 becomes:
∆p 121+n ηn k0vp* ) n L 100 r *1+n a0 c
If a dimensionless quantity, Pd, is defined by eq 37
1/n
dr + f0
r e r0 (24) Pd )
and 1
( )
ηn k0vp* ∆p ) φ* ) n L r *1+n a0 c
2
(34)
()
121+n k0 n 100 a0
(37)
Then, eq 35 becomes:
1/n
dr + g0
r g r0 (25)
where f0 and g0 are constants of integration.
n ∆p ηvp* Pd ) L r *1+n c
(38)
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Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
If k0 ) 1 (steady state), then:
Pd ) Pd0 )
b0 ) 121+n 100a0n
5(A + C)r02 - 5(Arp2 + C) - 30r04(2r0 - rp - 1)] (49)
(39)
n ∆p ηvp* Pd0 ) L r *1+n
and
(40)
c
[
a0 ) (2r03 - rp3 - 1) - 6r02(2r0 - rp - 1) -
For k0 * 1, then:
(41)
In order to obtain the dimensionless pressure gradient Pd for pipe acceleration ap in different cases (ap < 0, ap > 0, ap ) 0), the following assumptions are made: (1) The critical radius r0 is the same for both steadyand unsteady-state flow. (2) The critical radius r0 is also considered the same for different power-law index values, n. (3) The acceleration ratio k0 does not change for different power-law index values, n, and different pipe velocities, vp. If, for example, the value of n is 0.5, the leading order velocities of eqs 30 and 31 can be expressed as
k0vp 3 (r - 6r02r + A - 3r04/r) 12a0
r e r0
(42)
and
u0 ) -
k0vp 3 (r - 6r02r + C - 3r04/r) 12a0
r g r0
(43)
where A and C are constants.
A ) -rp3 + 6r02rp +
3r04 - 12a0 rp
C ) -1 + 6r02 + 3r04
(44) (45)
When r ) r0, the shear stress τ0 ) 0 and the shear rate ∂u0/∂r ) 0. In this case, the leading order velocity u0 is equal to the maximum velocity u0max. Substitution of r ) r0 into eq 42 gives
u0max )
k0vp (1 - 6r02 + 8r03 - 3r04) 12a0
(46)
2 uavg0 ) 1 - rp2
∫r u0r dr p
(47)
Integration of eq 47 results in eq 48
∫r1u0(r,t) r dr
2 uavg0 )
k0vpb0
p
2
)
1 - rp
1 - rp2
uavg )
rp2vp
(51)
1 - rp2
For a steady-state condition, i.e., k0 ) 1 and uavg0 ) uavg, eq 52 is obtained after substitution of uavg0 ) uavg into eqs 48 and 51.
b0 ) rp2
(52)
From eq 49, it is found that b0 is a function of r0 and rp. When eq 52 is solved to obtain the critical radius r0, it is found that r0 is only a function of the pipe geometry rp, as shown in Figure 2. This r0 can also be used for unsteady-state fluid flow with different power-law index values, n. First-Order Equations and Solutions. When the leading order velocity eqs 42 and 43 are substituted into eq 13b and the resulting expression is integrated, the first-order shear stress τ1 is obtained for the respective boundary condition
k0vp1 τ1 ) (2r4 - 20r02r2 + 5Ar - 30r04 + B/r) 240a0 r e r0 (53) and
k0vp1 τ1 ) (2r4 - 20r02r2 + 5Cr - 30r04 + D/r) 240a0 r g r0 (54) where the constants B and D are defined by eqs 55 and 56, respectively.
B ) 48r05 - 5Ar02
(55)
D ) 48r05 - 5Cr12
(56)
If we assume that the power-law index factor n is equal to 0.5 (the same assumption made for the leading order solution) and we substitute the leading order and firstorder shear stress eqs 23, 55, and 56 into the first-order rheological eqs 14b and 15b, the first-order average velocity given by eq 57 is obtained.
(48) uavg1 ) -
where the constant b0 is defined as
1 2 - - 1 /12 (50) r0 rp
From the geometry of the moving inner pipe and stationary outer pipe, the average fluid velocity uavg can be expressed as
When the velocity eqs 42 and 43 are integrated, the leading order average velocity uavg0 in the annulus is obtained. 1
)]
(
3r04 Pd ) k0nPd0
u0 )
1 [2(2r05 - rp5 - 1) - 20r02(2r03 - rp3 - 1) + 60a0
k01.5vp0.5vp1b1
k1vp(r02 - rp2) 1 - rp2
-
1 - rp2
(57)
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2043
Figure 2. Dimensionless critical radius, r0, for different diameter ratios, do/Di.
Figure 3. Dimensionless critical radius, r0, for different dimensionless flow rates, Qd.
Figure 4. (a) Acceleration ratio, K0, for different diameter ratios, do/Di, and corrected acceleration, a (ft/s2), when a < 0. (b) Acceleration ratio, K0, for different diameter ratios, do/Di, and corrected acceleration, a (ft/s2), when a > 0.
where the constant b1 is given by eq 58
first orders.
b1 )
1 {(1 - rp8) - 22r02(1 - rp5) + 1.5 2880a0
uavg0 + uavg1 ) uavg
8A(r05 - rp5) + 8C(1 - r05) - 30r04(1 - rp4) + 2
3
3
2
3
8(B - 5r0 A)(r0 - rp ) + 8(D - 5r0 C)(1 - r0 ) 180r06[2rp2 ln rp + 1 - rp2] + 2E(r02 - rp2) + 2F(1 - r02) + 24Br02(r0 - rp) + 24Dr02(1 - r0) (58) Acceleration Ratio. It is assumed in this study that the total average fluid velocity in the annulus is equal to the sum of the average velocities of the leading and
(59)
Substitution of eqs 48, 57, and 51 for the average velocities into eq 59 gives:
[
2
2
k0b0 - k1(r0 - rp ) +
]
k01.5v′b1 0.5
v
) rp2
(60)
Since the acceleration ratio k0 is assumed to be the same for different pipe velocities, it is assumed that vp ) 1 and vp′ ) ap; hence, eq 60 becomes
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Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
k0b0 - [k1(r02 - rp2) + k01.5apb1] ) rp2
(61)
Since only the leading and first-order equations are considered, the first-order acceleration ratio becomes
k1 ) (1 - k0)/
(62)
When eq 62 is substituted into eq 61, the resulting expression is given by eq 6
(1 - k0)r02 + k01.5apb1 ) 0
(63)
In eq 63, as b1 > 0, then
k0 > 1 when ap > 0
(64a)
k0 < 1 when ap < 0
(64b)
k0 ) 1 when ap ) 0
(64c)
When eq 7 is used to determine the unsteadiness number, the initital fluid density F0 is assumed for calculation. For different fluid densities F, corrected acceleration a is used.
a ) Fap/F0
(65)
Equation 63 then becomes:
(1 - k0)r02 + k01.5ab1 ) 0
(66)
Application of the Solution To Determine the Surge or Swab Pressure. The application of the solution of the derived equations can be useful in the early prediction and control of surge or swab pressure which is encountered when running or pulling tubular pipes in the liquid-filled borehole. In order to use the final equations, families of curves are generated from which the dimensionless pressure gradient, Fd, is obtained. The following step-by-step procedure is followed to determine the surge or swab pressure. (1) From eqs 67 and 65, the pipe acceleration and corrected pipe acceleration are obtained, respectively.
ap ) dvp*/dt
(67)
(2) From eq 7, the unsteadiness number is determined. Here, F and η are given data; ure and tre are selected as 1 ft/s and 1 s, respectively. (3) From eq 52, the parameter b0 is determined, which is substituted into eq 49 to obtain the critical radius r0 for different diameter ratios, do/Di, where the dimensionless radius rp is equal to do/Di. A plot of r0 vs do/Di is made as shown in Figure 2. Similarly, a plot of r0 vs the dimensionless flow rate Qd is made as shown in Figure 3, where Qd is obtained from eq 68.
Qd )
1 1 - (do/Di)2
(68)
(4) The solution of eq 66, at assumed values of a and do/Di and calculated values of , gives the acceleration ratio k0. Plots of k0 vs do/Di and k0 vs Qd for different values of a are made as shown in Figures 4a,b and 5a,b, respectively. (5) From eq 37, at assumed values of a, n, and do/Di, different dimensionless pressure gradients, Pd, are obtained.
Figure 5. (a) Acceleration ratio, K0, for different dimensionless flow rates, Qd, and corrected acceleration, a (ft/s2), when a < 0. (b) Acceleration ratio, K0, for different dimensionless flow rates, Qd, and corrected acceleration, a (ft/s2), when a > 0.
(6) Plots of dimensionless pressure gradients Pd for different diameter ratios do/Di and corrected pipe accelerations a and different power-law index values n are prepared. Figures 6 and 7 illustrate such plots for a ) -1 ft/s2 and a ) 1 ft/s2, respectively. Figure 8 is a special case for the steady-state condition, that is, a ) 0. Based on these plots, the corresponding values of Pd can be obtained for given do/Di, n, and a values. (7) Plots of dimensionless pressure gradients Pd for different dimensionless flow rates Qd and corrected accelerations a, at selected power-law index values n, are prepared. Figures 9 and 10 show such plots for a ) -1 ft/s2 and a ) 1 ft/s2, respectively. Figure 11 is a
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2045
Figure 6. Dimensionless pressure gradient, Pd, for different diameter ratios, do/Di, and corrected acceleration, a ) -1 ft/s2, at selected power-law index values, n.
Figure 8. Dimensionless pressure gradient, Pd, for different diameter ratios, do/Di, and corrected acceleration, a ) 0 ft/s2, at selected power-law index values, n.
Figure 7. Dimensionless pressure gradient, Pd, for different diameter ratios, do/Di, and corrected acceleration, a ) 1 ft/s2, at selected power-law index values, n.
Figure 9. Dimensionless pressure gradient, Pd, for different dimensionless flow rates, Qd, and corrected acceleration, a ) -1 ft/s2, at selected power-law index values, n.
special case for the steady-state condition, that is, a ) 0. Based on these plots, the corresponding values of Pd are obtained for given Qd, n, and a values. (8) Using eq 38, the surge or swab pressure gradient can be calculated. Example Calculation. In a well, a 7-in. casing string is lowered at a velocity of 0.5 ft/s into a 10-in. hole. The mud in the hole has a power-law index n equal to 0.8 with a fluid property η equal to 0.626 58(bf - sn)/100 ft2 and density equal to 10 ppg. (a) With no pipe acceleration, what is the surge pressure? (b) What will be the expected surge pressure
when the corrected pipe acceleration is 1, 2, 3, 4, or 5 ft/s2? Solution: When a ) 0 ft/s2, it is a steady fluid flow. (a) Two methods can be used to calculate the pressure gradients. (1) Chukwu and Blick’s method (1991):
∆p/L ) 2.401 psf/ft ) 0.0168 psi/ft (2) The perturbation method: For do/Di ) 0.7 and n ) 0.8, Pd is 120 using Figure 8. From eq 40, the surge pressure gradient is:
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Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996
Figure 10. Dimensionless pressure gradient, Pd, for different dimensionless flow rates, Qd, and corrected acceleration, a ) 1 ft/s2, at selected power-law index values, n. n ∆p ηvp* Pd0 ) L r *1+n
Figure 11. Dimensionless pressure gradient, Pd, for different dimensionless flow rates, Qd, and corrected acceleration, a ) 0 ft/s2, at selected power-law index values, n. Table 1. Calculated Results of the Surge Pressure Gradient
c
0.8
(0.626 58)(0.5 )(120)
corrected pipe acceleration (ft/s2)
surge pressure gradient [psf/ft (psi/ft)]
Chukwu and Blick (1991) Steady-State Method [psf/f (psi/ft)]
0 1 2 3 4 5
2.383 (0.0165) 3.972 (0.0276) 5.958 (0.0414) 8.341 (0.0579) 11.122 (0.0772) 14.101 (0.0979)
2.401 (0.0168)
)
51.8 ) 2.383 psf/ft ) 0.0165 psi/ft
(b) When a ) 1 ft/s2: For do/Di ) 0.7 and n ) 0.8, Pd ) 200 is obtained from Figure 7. The surge pressure gradient from eq 38 is: 0.8 ∆p (0.626 58)(0.5 )(200) ) L 51.8
) 3.972 psf/ft ) 0.0276 psi/ft Or, for do/Di ) 0.7 and n ) 0.8, Pd0 ) 120 is obtained from Figure 8, and k0 ) 1.9 is obtained from Figure 4b. The surge pressure gradient obtained by substituting eq 41 into eq 38 is: 0.8 ∆p 0.626 58(1.9 × 0.5) (120) ) L 51.8
) 3.983 psf/ft ) 0.027 65 psi/ft The calculated results of the surge pressure gradient at different corrected pipe accelerations are shown in Table 1. Conclusion A perturbation technique is employed for the generalized solution of the unsteady annular couette flow of non-Newtonian power-law fluids. The analytical solution is presented as a family of curves which can be used in field operations to determine the surge or swab pressure encountered during the unsteady motion of the inner pipe in a liquid-filled borehole. It is concluded in this study that the increase in pipe acceleration should
be minimized especially toward the bottom of the hole where the surge pressure effects are more detrimental to the formation stability. Nomenclature a ) corrected acceleration of the moving (inner) pipe aavg ) average acceleration of fluid flow ap ) acceleration of the moving inner pipe do ) outside diameter of moving inner pipe Di ) inside diameter of stationary outer pipe ki ) different orders of acceleration ratios n ) power-law index Pd ) dimensionless pressure gradient Pd0 ) dimensionless pressure gradient for steady-state flow ∆p/L ) pressure gradient Qd ) dimensionless flowrate defined by eq 68 r ) dimensionless radial coordinates r* ) radial coordinate rc ) dimensionless inner radius of the stationary outer pipe rc* ) inner radius of the stationary outer pipe rp ) dimensionless outer radius of the moving inner pipe rp* ) outer radius of the moving inner pipe r0 ) dimensionless critical radius r0* ) critical radius t ) dimensionless time coordinate t* ) time coordinate u ) dimensionless axial fluid velocity u* ) axial fluid velocity uavg ) dimensionless average velocity u0 ) dimensionless leading order of the axial fluid velocity
Ind. Eng. Chem. Res., Vol. 35, No. 6, 1996 2047 u0mzx ) dimensionless leading order of the maximum velocity ui ) dimensionless different orders of the axial fluid velocities vp ) dimensionless velocity of the moving inner pipe vp* ) velocity of the moving inner pipe z* ) axial coordinate Greek Letters ) unsteadiness number φ ) dimensionless pressure gradient φ* ) pressure gradient F ) fluid density F0 ) initial fluid density τ ) dimensionless axial shear stress τ* ) axial shear stress τi ) dimensionless different orders of the axial shear stress η ) consistency index value of power-law fluid
Literature Cited Balmer, R. T.; Florina, M. A. Unsteady flow of an inelastic powerlaw fluid in a circular tube. J. Non-Newtonian Fluid Mech. 1980, 7, 189-198. Burkhardt, J. A. Wellbore pressure surges produced by pipe movement. J. Pet. Technol. 1961, June, 595-605. Cannon, G. E. Changes in hydrostatic pressure due to withdrawing drill pipe from the hole. API Drill. Prod. Pract. 1934, 42-47. Chukwu, G. A. Surge and swab pressure competed for couette flow of power-law fluids. Ph.D. Dissertation. University of Oklahoma, Norman, OK, 1989. Chukwu, G. A.; Blick, E. F. Surge and swab pressure models, Part 2: Surge and swab pressure from couette flow of power-law fluids through a concentric annulus. Pet. Eng. Int. 1991, Nov, 61-62. Clark, E. H., Jr. Bottom-hole pressure surges while running pipes. Pet. Eng. Int. 1955, Jan, B.68-B.96.
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Received for review September 26, 1995 Accepted March 29, 1996X IE950598X X Abstract published in Advance ACS Abstracts, May 15, 1996.
