ANALISIS ESTADISTICO:

the upper end and to the bit. Eliminating g from the last two equations and using simplified notation (i.e., S(xi) = Si; S W = S2,' etc) the following

system of equations is obtained :

a P i + b Q i + c Ri = 0 (A1.20a)

a P2 + b Q2 + c R2 = 0 (A1.20b)

a (Si - S2) + b (Ti - T2) + c (Ui - U2) = 0 (A1.20c)

To avoid trivial solution (i.e.,a = b = c = 0 ) the determinant given below must be zero :

Pi Qi Ri

Pz Qz ^ 2

Sj - S2 Tj - T2 Hi - H;

= 0 (A1.20d)

Assum ing several values for Xi in equation (A1.20d) corresponding values of Xz can be calculated by trial and error with results shown in Figure A1.3. For small values of Xj more weight on bit - i.e., larger Xz - is proportionally required to buckle the pipe: in other words, short drill strings tend to be more stable. Furthermore, it is noted that above Xi = - 6

the critical Xz tends to a constant level around 1.9. In fact it is later shown that the bending moment and deflection, up to the second m ode, are alm ost negligible for the portion of drill string close to the kelly's bushing, hence it should not matter if the top end is assumed at %i = - 6 or

Xi = - 7 . It is possible, however, that another limit should be considered if buckling of third and higher orders are calculated. The author's ow n calculation of equation (A1.20d) resulted in the pair %% = - 6, %2 = 1.939

which virtually coincides with Lubinski's solution, but another pair (xi = - 7, Xz = 1.9046) was also investigated. It is later seen that "extending" the upper end to %i = - 7 does not substantially improve the results, especially when analysing the lower string portion between the neutral point and the bit.

P o i n t o f t a n g e n c y f o r c r i t ic a l c o n d i t i o n o f f i r s t o r d e r

In the point of contact, defined as at distance the inclination vanishes so that :

a F3 + b G3 + c Hs = 0

ày dx

(A1.21)

which combined with equations (A1.20a) and (A1.20b) gives

'^3 A H i a 0

Pi Qi Ri ■b 0>

A Qz Rz. c 0

(A1.22)

The determinant must be zero for a non-trivial solution and by trial and error it is found that = 0.143 (for Xi = -6, Xi = 1.939 ) and X3 = 0.0724 (for

Xi = -7, X2 = 1.9046 ).

E q u a t i o n c o e f f i c i e n t s f o r c r i t ic a l c o n d i t i o n o f f i r s t o r d e r

So far the coefficients a, b, c and g are unknown and consequently the displacements, bending moments and so on cannot be determined. The system com posed of equations (A1.20a), (A1.20b) and (A1.20c) is indeterminate. However, at the contact point the deflection is equal to the apparent radius, i.e., yix^) = c (C = m e ) s o that :

a S3 + b T3 + c U3 + g = c (A1.23)

w here C = (Dh - Dj/2, Dh is the hole diameter and D is the pipe outside diameter. Elim inating g from equations (A1.23) and (A1.19c) and rewriting expressions (A1.20a) and (A1.20b) gives :

^ 3 ^ 3 “ ^ ^ 3 “ a c

Qi b 0^

Pz Qz Pz _ c 0

(A1.24)

The equation coefficients may be obtained if values of %2 and X3 are

P o i n t s o f t a n g e n c y f o r 'W O B ' a b o v e c r i t ic a l c o n d i t i o n o f f i r s t o r d e r

After buckling, equation (A1.8) is introduced to account for the contact force, hence two different equations are valid for below and above the point of contact. The subscript i refers to above { i = 1 ) and below {i = 2 )

the tangency point. Thus :

y ( x ) = Ui S (x) + bi T ( x ) + q U (x ) + g, (A1.25)

The following boundary conditions must be satisfied :

At upper string : = 0 y(xi) = 0 = 0 y fe ) = c (A1.26a)

At lower string : = 0 yixi) = 0 = 0 y (x^ = c (A1.26b)

Furthermore the bending moment should be continuous at % = x^, or :

F3 + b i Q3 + Cj Rs = « 2 P3 + bz Qs + C2 R3 (A1.26c)

The follow ing set of equations is formed from boundary conditions (A1.26a), (A1.26b) and (A1.26c) :

Pi + bi Qi + jR] = 0 (A1.27a) Ui Fs + biG3 + c , H s = 0 (A1.27b) ui (S3 - S i) + bi (T3 - T i) + c, (U3 - U i ) = c (A1.27C) Uz P2 + bzQz + C2 R2 = 0 (A1.27d) az F3 + bz G3 + C2 H3 = 0 (A1.27e) az (S3- Sz) + bz (T3- Tz) + C2 (U 3- Uz) = c (A1.27f) Ui P3 + bi Q3 + Cj R3 - (uz P3 + bz Q3 + C2 R3) = 0 (A1.27g)

Note that gi and gz were eliminated, as indicated in equations (A1.27c) and (A1.27f). There w ill be a solution if the follow ing determinant vanishes : p, Qz Ri 0 0 0 0 Ps Ga Hs 0 0 0 0 $3 — Sj T s - T , U s - U , 0 0 0 c 0 0 0 _{Pi Qi Ri} 0 0 0 0 _{Ps Gs H}3 0 0 0 0 Ss —S2 T3- T2 U3- U 2 c Ps Qs Rs - P s - Q s - R s 0 = 0 (A1.28)

Table A l.l contains results for the tangency point.

X2 ^ 3 Com m ent

1.940 0.145 Critical condition of first order

2.600 0.942

3.200 1 . 6 6 8

3.753 2.346 Critical condition of second order

4.000 2.672

4.218 3.098 Second buckle contacts the hole wall

Tangency Point Table A l.l

In the critical condition of second order (% 2 = 3.753, Xs = 2.346 ) the reaction

force and the angle of the deflected drill string are zero at the kelly's bushing.

Equation coefficients for W O B above critical condition of first order

Table A1.2 was produced by substituting values of Xz and x^ in equations (A1.27a) to (A1.27c) and (A1.27d) to (A1.27f). With these coefficients the buckled shape and bending moments, for instance, can be calculated.