Drillstring Vibrations: Classic Ideas and Modern Approaches - Wave Propagation in Drilling, Well Logging and Reservoir Applications - page 167

subscripts, represents the drillpipe in the interval L 1 < x < L 2 . Thus, our
Equation 4.2.110 is replaced by the pair of partial differential equations

2 u d,1 (x,t)/ t 2 + 1

u d,1 (x,t)/ t - E 1

2 u d,1 (x,t)/ x 2 = 0 (4.2.113a)

1

2 u d,2 (x,t)/ t 2 + 2

u d,2 (x,t)/ t - E 2

2 u d,2 (x,t)/ x 2 = 0 (4.2.113b)

2

The authors then assume that
u d,1 (x,t) = U 1 (x) e i

t

(4.2.114a)

u d,2 (x,t) = U 2 (x) e i

t

(4.2.114b)

and obtain
d 2 U 1 (x)/dx 2 + { (

2 -i 1

)/E 1 } U 1 = 0 (4.2.115a)

1

2 -i 2

d 2 U 2 (x)/dx 2 + { ( )/E 2 } U 2 = 0 (4.2.115b)

2

Their respective solutions are
U 1 (x) = C 1 sin { ( 1 2 -i 1 )/E 1 } x
+ D 1 cos { ( 1

2 -i 1

)/E 1 } x (4.2.116a)

U 2 (x) = C 2 sin { ( 2 2 -i 2 )/E 2 } x
+ D 2 cos { ( 2 2 -i 2 )/E 2 } x (4.2.116b)
The integration constants C 1 , C 2 , D 1 and D 2 are then obtained assuming that the
displacement is continuous at the pipe-to-collar position x = x p-c , that is,
u d,1 (x p-c ,t) = u d,2 (x p-c ,t) (4.2.117a)

U 1 (x p-c ) = U 2 (x p-c ) (4.2.117b)

and, that force is continuous, with
A 1 E 1 u d,1 (x p-c ,t)/ x = A 2 E 2 u d,2 (x p-c ,t)/ x (4.2.118a)

1 E 1 dU 1 (x p-c )/dx = A 2 E 2 dU 2 (x p-c )/dx (4.2.118b)
The drill bit displacement boundary condition, noting our objections, is
u 1 (0,t) = u 0 e i t (4.2.119a)
U 1 (0) = u 0 (4.2.119b)
at x = 0. Details are available in Dareing and Livesay (1968), but the limitations
of the analytical approach are severe, as discussed in Chapter 1, when more
general boundary condition models must be considered.

4.2.10.3 Jarring issues and stuck pipe problems.
“Stuck pipe,” due to “dog-legs” and/or poor “hole cleaning” in deviated
and horizontal wells, is an industry-wide problem. Drilling or “fishing jars,”
typically mounted close to the bit, are employed in the bottomhole assembly of a