ON THE MAGNITUDE OF THE STATIC FORCE REQUIRED TO EXTRACT A PIPE STRING FROM A CURVED WELL

V. P. ROZHKOV

(Presented by the scientific seminar of the Department of Exploration Engineering)

Determining the static force required to extract a pipe string from a curved well*) is of great importance in the design of hoisting mechanisms and structures, and in calculating the strength of elements of a drill-pipe string. Closely connected with the problem of determining the static force is the problem of finding such a rational profile of a directional well that, when it is drilled, a minimum of energy and time would be expended on hoisting the tool. This question acquires especially great importance in the design of deep directional wells and has not only technical but also economic significance.

The static force is readily determined if the well is straight. However, such wells are encountered comparatively rarely. If a well (especially one of considerable depth) does not follow a previously designed profile, then, as a rule, it is curved arbitrarily. In the case of well curvature, the static force depends not only on the proper weight of the string, but also on the profile of the well.

In the technical literature [1, 2], the problem of calculating friction forces and the static force is considered either for a constant intensity of zenith and azimuthal curvature, or for a constant zenith or azimuthal angle. Despite such conditions, the final dependences prove to be very cumbersome and applicable only to individual intervals of the well. Moreover, for naturally curved shot-drilled wells, which are characterized by an increase in the intensity of zenith curvature with depth, these dependences cannot be applied at all.

In the present article, the case of well curvature only in a vertical plane will be considered. In practice, plane-curved wells are encountered comparatively rarely. However, if wells are drilled across the strike of the main structures, then their azimuthal curvature is usually insignificant, and the wells may, with a sufficient degree of accuracy, be regarded as plane-curved.

Wells drilled in hard rocks may, in separate intervals, either decrease or increase the zenith angle; on the whole, such

*) For brevity of expression, hereafter it will be called simply the static force.

wells, as a rule, are made to flatten out. One of such wells is shown in Fig. 1. Suppose that a string of drill pipes has been lowered into it. It is required to determine the static force necessary for extracting them.

In solving the stated problem, let us assume that the well is curved so smoothly that, when the tool is lifted, no additional resistance arises because of a sharp bend. This assumption is justified by the fact that drilling of geological-exploration wells in most cases is carried out with long string assemblies. At the same time, this assumption proves to be so essential that it permits a considerable simplification of the final relationships.

Consider a separate element of the string of length \(dl\) with its center of gravity at point \(O\). Let the average weight of one meter of pipes in the well (allowing for corrections for the Archimedean force and for weighting due to the joints) be equal to \(q\). Then the average weight of the selected pipe element is equal to

\[ dG = qdl. \]

For solving the stated problem, the weight of an element in the inclined part of the well is replaced by the components \(M\) and \(K\). In addition, to the element we apply the force \(T_1\), acting from the side of the lower-lying part, and the second force \(T_2\)—from the side of the overlying part. The element will be in a state of rest if the sum of all forces acting along its axis is equal to zero. In analytic form this condition may be written as

\[ T_2 = T_1 + M + fK \]

Fig. 1. Diagram of a flattening well (a) and diagram of the resolution of the gravitational forces of an elementary section of a pipe string in a curved well (b)

or, after substituting the value of each term, as the differential equation

\[ dT = T_2 - T_1 = q\left(dl \cdot \cos \Theta_i + fdl \cdot \sin \Theta_i\right), \tag{1} \]

where \(dT\) is the increment of the static force on the elementary section of the pipe string under consideration;
\(\Theta_i\) is the zenith angle of the well at the point under consideration;
\(f\) is the coefficient of friction of the drilling tool against the walls of the well.

The total static force is equal to

\[ T = q \int_{(l)} dl \cos \Theta_i + \int_{(l)} dl \cdot \sin \Theta_i = q \int_0^x dx + qf \int_0^y dy. \tag{2} \]

Since the expressions under the integral signs are nothing other than the projections of the element \(dl\) onto the vertical and horizontal axes, the definite integrals are the projections of the string

pipes lowered into the well, or, in other words, by the projections of the curved well. The static force in this case can be expressed by the simple formula

\[ T = q(X + fY), \tag{3} \]

where \(X, Y\) are, respectively, the vertical and horizontal projections of the curved well.

According to the data of B. I. Vozdvizhenskii and M. G. Vasil'ev [3], “because of the small clearance between the walls of the well and the casing pipe, the latter offers considerable resistance to lifting, especially if it carries a sludge pipe open at the top.” Therefore, it is customary to introduce into the analytical relation a correction coefficient \(\eta\), whose value (according to B. I. Vozdvizhenskii and M. G. Vasil'ev) is equal to \(1.3\)—\(1.6\). Then the static force is finally expressed as

\[ T = q\eta(X + fY). \tag{4} \]

A straight inclined well may be regarded as a special case of a curved well. Therefore, applying formula (4) to it, we obtain an expression known from drilling mechanics [3]:

\[ T = \eta ql(\cos \Theta + f\sin \Theta). \]

For curved wells, similar final relationships are very complex. In such a case one should resort to a preliminary calculation of the projections, which can be determined by any of the known analytical or graphical methods.

Fig. 2. Possible variants of flattening-out (\(a\)) and “straightening-out” (\(b\)) wells, whose bottoms arrive at one and the same point

It follows from formulas (3) and (4) that there is an interesting consequence: if the well as a whole flattens out and its maximum projections correspond to the bottomhole, then the static force does not depend on the particular path by which the well reached the bottomhole, but depends only on the magnitude of its vertical and horizontal projections. Thus, for all variants of curved wells which at the final depth arrive at one and the same point (Fig. 2, \(a\)), the static force will be one and the same. This proposition, of course, must under no circumstances be extended to wells in which the zenith angle is constantly decreasing, since, because of the tension of the string, an additional force arises that presses the string against the lower wall. Wells of this type are shown in Fig. 2, \(b\). Nevertheless, relation (4) may be applied to wells having separate intervals of straightening-out, but in general having the character of flattening-out. This is all the more permissible because a correction coefficient has been introduced into the formula.

In designing directional wells it is necessary to strive to make the static force when lifting pipes from the final depth as small as possible. This will make it possible to lift inst-

tools at higher speeds and, consequently, will lead to a saving of time in hoisting operations.

Reducing the vertical projection most often is not possible. Reducing the horizontal projection (for a given curvature intensity) will lead to a decrease in the zenith angle at which the well meets the ore body and, at the same time, to a decrease in the well footage. Thus, in order to shorten the time required for drilling a directional well, it must intersect the ore body at the minimum permissible angle.

Conclusion

If the borehole of a flattening well is so smooth that, when a pipe string moves along it, the latter does not experience sharp bends, then a sufficiently rigorous mathematical relationship can be obtained between the hoisting force and the mutual spatial arrangement of the wellhead and bottom. Such a relationship is described by formulas (3) and (4).

REFERENCES

1. V. A. Kharkov. On the static force required for hoisting a pipe string from a curved well. NKH, No. 5, 1959.

2. M. M. Aleksandrov. Quantitative assessment of the friction force during the movement of a tool in a curved well. Izv. vyssh. uchebn. zaved., Neft’ i gaz, No. 4, 1959.

3. B. I. Vozdvizhenskii, M. G. Vasil’ev. Drilling Mechanics. 3rd ed. Gostoptekhizdat, Moscow, 1954.