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## 2-D versus 3-D Approach to Drag Force Analysis

The method of calculating drag force presented in the previous section is based on the analysis of a two-dimensional well profile which consists of vertical and horizontal sections only. In practice, however, the spatial factors, such as bit walk, bearing angle, dogleg, etc., will cause the hole to deviate from the normal course and result in a three-dimensional well profile (vertical, horizontal and azimuth) as shown in Fig. 4.8. These effects are particularly noticeable in the buildup portion of the hole.

The forces acting on a unit section of the casing in the buildup section of the hole are presented in Fig. 4.9. From the state of equilibrium, the differential equation for drag-associated axial force, Fa. can be expressed as follows (Maidla, 1987):

dF

-^ = Wu(l)±fbCs(l)WN(l) (4.41)

where:

 Cs(l) WN(l)

 Fa{l)

 ВД +

 (4.42)

 R(l)

 correction factor for the effect of the surface contact area between the pipe and borehole, buoyant weight projection on the principal normal direction 21 05

 = {Wb(l)2 +

 m w«(0

Hole-curvature after drilling. The results, which are a function of depth, are obtained from hole surveys, unit buoyant weight projection on the tangent direction w u(l)

 (4.43)

I dl • VF(,U (AZ sin A — f VZ cos A) ]

 Wb (I) = unit buoyant weight projection on the binormal direction Wp(1)
 (4.44)

w. b{l) = dl Wbu (AX • VY — VX • AY)

unit buoyant weight projection on the principal

normal direction

w. p(l)

Wbu (AZ cos X — VZ sin A) buoyant weight VFn ■ BF

 (к

 Л VX VY vz их uz

 (4.46) (4.47) (4.48) (4.49) (4.50) (4.51)

 h — <2 sin »2 COS в-2 sin <*2 sin 02 cos a 2 sin c*i sin 0i cos ai

 1)8

 8 =
 (4.52) (4.53) (4.54) (4.55) (4.56) (4.57) (4.58) (4.59)

arc cos (UX x VX + UY x VY + UZ x VZ)

 AX AY AZ 8 A R(l) =

UX — VX cos /3 sin/?

UY — VY cos/3 sin /3 E/Z — VZ cos /3 sin /3

bearing angle in radians, overall angle change in radians, contact angle in radians (axial).

к — h

COS [cos(0i — 02) sin Q] sin Q2 + COSQi COS Q2]

In Eq. 4.41, the positive sign implies an upward pipe movement, whereas the negative sign denotes a downward movement. Equations 4.42 to 4.45 describe projections of the distributed pipe weight on the trihedron axis (Kreyszig, 1983) associated with any given point of the well trajectory.

In Eq. 4.41, a correction factor, Cs, is introduced to take into account the effect of contact surface between the pipe and the borehole. As reported by Maidla (1987), Cs is a function of the contact surface angle, ф, and is expressed as:

 (4.60)
 + 1

C,(0 = — ф{1)

Cs varies between 1 (</>(/) = 0) and А/ж (</>(/) = ж/2) as shown in Fig. 4.10.

Initially, the circles of Fig. 4.10 are internally tangent. However, as the pipe is deformed the internal circle shifts laterally by Л<3 as illustrated by the dashed arc. The approximate pipe deformation in the direct ion of the dist ributed normal

 Fig. 4.10: Surface of the contact between borehole and the casing. (After Maidla. 1987.)

force, Wx(l), is:

 (4.61;

Ad=£ё и-(‘> t

The pipe to borehole contact surface area. o(l). is given by:

2 A

 ‘4.62!
 arctan

0(0 =

2 Y-dw + dc

where X and Y are the coordinates of the point of intersection of the two circles. The cartesian plane x-y is assumed to be normal to the pipe at point / and its origin (0,0) to lie at the borehole centreline. .V and Y are given by:

d20 ~ d2w + (dw — d3 + 2 A</)‘

 (4.63) (4.64)

Y = 0.25

duj — d0 4- 2 Ad X = 0.5 (dl — 4T2)0’5 where:

 d%n A d t E

= external diameter of the casing, in.

= diameter of the well. in.

= the approximate pipe deformation in the direction of the applied normal force, in.

= thickness of the casing, in.

= modulus of elasticity.

W)v(0 = distributed normal force from Eq. 4.42. lb/ft.

In Eq. 4.60, the following assumptions are made:

1. Pipe deformation is elastic.

2. Contact surface has the same geometry as the borehole.

3. There is a linear relationship between the contact surface correction factor. Cs and the contact angle. A.

4. Contact surface as shown in Fig. 4.10. is controlled by an arc between the intersection points of two circles.

After multiple simulation runs. Maidla (1987) found that C’S{I) was close to unity in all cases. He, therefore, concluded that the effect could be ignored in most calculations.

Generally, it is not possible to solve Eq. 4.41 analytically and instead numerical integration must be used. Equation 4.41 does not consider torsional effects which might also contribute to the normal force.