A„ - n = 0 ,1 ,2 ,3 ... - are coefficients of the polynomial series .
2.1.2- Inclined Holes
The longest "footage" in directional drilling consists of straight inclined holes - hence it is important to study this case. If the weight on bit is zero the only force to act between the tangency point (i.e., the point where the "suspended" drill string contacts the bore hole wall) and the bit is the drill string's own weight. This restoring force, often caUed pendulum force, tends to bring the bit to the vertical. On the other hand as load is added to the bit a reaction force tends to deviate the trajectory away from the vertical. The balance between these two forces determines if the bore hole angle will tend to increase, decrease or remain constant. Lubinski and W oods (1953) acknowledged that the problems of deviation in slightly inclined holes and stability in vertical holes are governed by the same equation (a proper set of boundary conditions should be applied to each case). They pointed out, however, that solution by series presents poor convergence for large weights on bit and an iterative method may be necessary - both formulations are presented in Appendix 2. At the end of Lubinski and Woods' paper Szego
sh ow s how solution can be extended to large inclinations if minor modifications in the iteration method are introduced (Appendix 2). Lubinski and Woods' experimental work showed that helical buckling of drill collars is unlikely to occur for most practical applications.
For a certain range of tilt angle, the bit drills in the direction of the resultant force if the formation is isotropic. In anisotropic formations, however, the bit teeth do not remove soil equally and consequently a deviation force (Figure 2.1.2.1) appears in the direction of more material removal (Bradley, 1975). Lubinski and Woods also developed a method to account for anisotropic formations. Since anisotropic formation properties are not identical in all directions, it follows that when formation drillability is lower along the bedding planes than perpendicular to them, the bit does not proceed in the direction of the resultant force (Figure 2.1.2.2). This difference in drillability is quantified by the drilling anisotropy index h w hich is an empirical constant obtained from experiment - for isotropic formations h is zero. Table 2.1.2.1 lists formulae for equilibrium in anisotropic formations.
Particular Case Formula
General case
—1 a (l - hcos^ x ) - ^ h s i n x c o sx ^ ~ f (j)\
— \ a h s i n x + sin^x)
\CC /
Slightly inclined formations
s i n x = X c osx = l h a Horizontal formations
:
a 1- h Vertical formations — - 1- h aEquilibrium in Anisotropic Formations Table 2.1.2.1
where :
a is the bore hole angle,
(j) is the angle of the resultant force on bit and
X is the formation angle (or dip of formation).
Lubinski and Woods also explained how a dog-leg - defined as the rate of change of hole angle - may originate if weight on bit or formation dip vary
abruptly. They advocated that instantaneous drops in weight on bit can cause larger dog-legs in dipped formations than in horizontal beds. Continuation of a dog-leg is illustrated in Figure 2.1.2.3: when bit meets the formation interface there is an instantaneous change of hole angle proportional to the formation dip - ^ - a = h x ~ where W is the angle of actual drilling. This formula applies to small formation dip, in general the change of hole angle depends also on the weight on bit, drill collar size, radial clearance and hole angle. Drilling proceeds at this new angle until the drill collar touches the discontinuity point, then the bit follow s a path parallel to the previous hole until the drill collar lays again in the new hole whose angle is defined by the new set of parameters. Lubinski and Woods recommended that abrupt drops in weight on bit should be avoided.
W oods and Lubinski (1954) published tw o papers containing direct applications of the previous theory. In their first work, it was shown how a new equilibrium configuration can be assessed once one of the following parameters is changed: hole angle, hole size, drill collar outside diameter, weight on bit or formation dip. It is assumed that the mud density is ten pounds per gallon (1198.2 kg/m^) and the inside diameter is 37.5% of the outside diameter but it should be stressed that the results are not very sensitive to these two variables. In their second paper, the weight on bit per hole diameter is introduced as a performance criterion. It is shown that proper combinations of radial clearance and collar diameter can optimise penetration rate for certain formations and hole inclinations. Results presented show that drilling efficiency improves when heavy drill collars are used in small radial clearances and some deviation is tolerated.
2.1.3 - Use of Stabilisers
Essentially stabilisers are curved blades placed on drill collars to stiffen bottom hole assemblies, increase the ability to drill straight and reduce undesirable bit movements. In addition they centralise drill collars making sure they rotate around their ow n axis - thus m inim ising lateral displacements. In short stabilisers improve drilling performance by reducing bit wear and allowing more weight on bit to be carried (and consequently increasing the rate of penetration). Stabiliser efficiency increases as the outside diameter of the blades approaches the bit diameter (Short, 1982).
Stabilised drill strings - often called packed hole assemblies - may contain up to eight elements but normally three or four are common place.
Woods and Lubinski (1955) presented a solution for single stabilised drill strings in slightly inclined holes. The solution technique was omitted from the paper but it is likely that they used a power series method, as reviewed in Appendices 1 and 2, to solve the governing equations. Figure 2.1.3.1 shows the system of co-ordinates and the assumed boundary conditions. The solution requires that :
- ^ 4 T 2 - T 4 U 2 - U 4 0 0 0 ( X 2 - X 4 )