ELECCIÓN DE LA TECNOLOGÍA DE ACCESO REMOTO

The displacement equation derived from the method of virtual work is :

dX* d X ( W ^ - p X c o s a ) ^ =p sina (2.1.5.12)

which is identical to the two-dimensional equation (2.1.5.6b) if the applied torque is set to zero. Integration of the above expression leads to equation (2.1.4.1). When forces acting on nodes and axial forces are considered constant, the following equilibrium equation can be used :

d^Y d^Y

where P is the axial force. Solution of equation (2.1.5.13) for each element approximates the solution of equation (2.1.5.12) for the whole structure. For large hole angles and small clearances the bending moments due to axial forces vanish and the second term in equation (2.1.5.13) can be neglected, producing :

d^Y

E I ^ = 0 (2.1.5.14)

dX^

Equation (2.1.5.14) can be solved directly. Equation (2.1.5.13), on the other hand, requires iteration but it should only be used when the hole angle is small and radial clearance is significant. Three situations were verified: slick assembly; two-stabiliser assembly (one close to the bit and the other in varying positions); and two field problems. Results agree with established solutions but it is felt that linear analysis is inadequate for large weights on bit. It also seems that the position of the second stabiliser rules the hole angle trend.

Alternative approaches were proposed by MiUheim, Jordan and Ritter (1978). In the first model the drill string is divided in to straight beam elements with circular cross-sections and six degrees of freedom - three translations and three rotations in each node. Contact is represented by gap elements consisting of straight trusses with constant cross-sections: Figure 2.1.5.3 illustrates how the Young's modulus changes when a pre-specified strain is

reached. A more refined model employs curved beam elements and non­ linear elastic foundations composed of five springs whose stiffnesses are very small until contact occurs. This model may represent initial curvature effects and continuous contact, hence curved elements might be better suited to curved bore holes while straight beam elements can be used for large displacem ents in straight holes. Investigations were carried out on assemblies with up to four stabilisers - comparisons with actual responses produced excellent results (within 3% of each other!) except for four stabilisers.

2.2- Dynamic Analysis

There was a boom in dynamic analysis in the eighties as a natural step following an understanding of the static phenomenon. The rotary speed and friction coefficients play important roles in the assessment of bit trajectory, especially in azimuth control while static analysis is basically limited to the prediction of inclination. The dynamic analysis is dealt with below in two parts: harmonic and transient responses.

2.2.1 - Harmonic Response

Millheim and Apostal (1981) extended the finite element static version (M illheim and al, 1978) by including inertial and frictional forces proportional to acceleration and velocity, respectively. The external force vector Fj^ acting in each node N is assumed to be harmonic :

^ Fi + F^ s i n Q t + F^ c o s Q t (2.2.1.1)

where :

Q is the rotational sp eed ,

FjJ is the steady component and

F^ and F^ are the force vectors respectively proportional to s i n O t

and cos Q t.

Using similar notation, the response vector has the following pattern :

The program does not account for bit tooth/formation interaction, as a result it may overestimate the right-hand tendency of the bit. Only friction due to pipe rubbing on the wall is taken into account - the wall is considered to be infinitely hard, side cutting is neglected and the well bore is assumed to be circular. In addition, torque on bit is disregarded. Experiments conducted in five directional wells - in which four had a near bit stabiliser and the other one w as a slick assembly - drilled under controlled conditions show agreement with the numerical results. Four different vibratory paths are observed and classified according to their energy levels (Figure 2.2.1.1). For low rotary speeds (low energy) friction forces predominate and the pipe keeps in contact with the wall. On increasing rotary speed, the pipe tends to lift and contact the wall at two points. At even higher energy levels the pipe travels in four quadrants touching the wall intermittently, in what is called pipe whip. In this unstable region the inclination and direction of forces may change dramatically. Finally above this whip speed, the pipe turns around the well bore in stable paths.

2.2.2 - Transient Response

The "steady state" dynamic analysis does not account for intermittent drül string/ formation interaction. The formulations necessary for computing transient responses are quite comprehensive and take a long time to simulate a few seconds of drilling. This suggests that they are more suited to research rather than practical in-field application. Eronini, Somerton and Auslander (1982) proposed an integrated dynamic model for rock-drilling rigs. The drill string (as a transmission line), the rotary drilling bit (considered as a "non-ideal" transformer), the rock-bit mechanics (fracture model) and a performance criterion (penetration rate, for instance) are combined altogether in one model whose set of differential equations is solved by an implicit trapezoidal scheme. Bottom hole cleaning and tooth wear are also taken into account. Baird, Caskey, Wormley and Stone (1985) implemented a three-dimensional transient finite element program which includes :