PROCEEDINGS
OF THE TOMSK ORDER OF THE RED BANNER OF LABOR POLYTECHNIC
INSTITUTE named after S. M. KIROV

Volume 151
1966

SOME METHODS OF PROFILING BOREHOLES IN PERCUSSION DRILLING

V. P. ROZHKOV

One of the problems of directional drilling is the profiling of boreholes, which involves constructing design trajectories of the holes, determining the location and initial zenith angles for setting out boreholes on the surface, finding the setting-out points of additional holes in multi-hole drilling, and certain other matters.

To solve the listed problems in directional drilling, using only the laws of natural curvature of boreholes, it is necessary to express these laws mathematically. The curvature of boreholes in a vertical plane may be considered established if the change in their curvature has been studied*). The curvature of individual sections of a borehole must be considered as a function of the magnitude of the zenith angle characterizing that section, since as it increases the forces deflecting the upper part of the tool toward the lying wall increase, and thereby the degree of use of the flattening conditioned by geological factors increases.

To determine the dependence of borehole curvature on the magnitude of the zenith angle, corresponding graphs were constructed for a number of deposits: Akhtenskoye, Bakalskoye, Inskoye, Kusinskoye, Odinochnoye, Olkhovskoye, Tashelginskoye, Temir-Tau, and Chibizhekskoye. Numerical methods of analysis applied to these dependences show that they are close to linear and may be expressed by the equations:

\[ \kappa=\frac{d\Theta}{dl}=a+b\Theta, \tag{1} \]

where \(\kappa\) is the curvature of the borehole, deg./meter;
\(l, \Theta\) are, respectively, the depth and zenith angle of the borehole;
\(a\) is the conditional initial curvature of the borehole, deg./meter;
\(b\) is the intensity of curvature gain with borehole depth, \(1/\)meter.

Similar dependences were established by A. E. Kolesnikov for the Sadonskoye and Zgidskoye deposits [1]; graphs of curvature as a function of zenith angle obtained by A. G. Kalinin for deposits of Gornaya Shoria [2] also testify to a similar dependence.

The results of processing the dependence of curvature on zenith angle for the deposits analyzed by us are presented in Table 1, the notation in which is adopted in accordance with formula (1). The values of the coefficients \(a\) and \(b\) were found by the quantile method (3).

* ) In the present case, curvature is understood as the mean increment of the zenith angle over an interval of one meter.

Table 1

For the purpose of qualitative testing and precise determination of the boundaries of a mineral deposit, the method for carrying out geological-exploration work provides that a borehole must intersect ore bodies at specified angles. Usually the angle of intersection \((\beta)\) is prescribed, formed by the tangent to the borehole trajectory at the point where it meets the ore body and by the dip direction of this body. If the dip angle is denoted by \(\alpha\), then the zenith angle of the borehole at the point of intersection will be (Fig. 1)

\[ \Theta_{\Gamma}=\left(\alpha+\beta-\frac{\pi}{2}\right). \tag{2} \]

Fig. 1. Diagram for profiling a naturally curved shot-drilling borehole.

Let us place the borehole mouth at the origin of coordinates, as shown in Fig. 1. In this case the profiling problem reduces to determining the horizontal distance between the collaring point and the point of intersection with the ore body and the zenith angle of the borehole collaring.

To solve the posed problem it is necessary to find the equation of the horizontal and vertical projection of the borehole. If the angle of inclination of the curve is taken as the argument (in our case, the zenith angle), then the parametric equations in differential form may be written as follows:

\[ dx=dl\cdot\cos\Theta, \]

\[ dy=dl\cdot\sin\Theta. \]

Substituting into these equations the value of the length element from expression (1) and integrating the right- and left-hand sides over the corresponding limits, we arrive at the initial integral equations

\[ X_i=\int_{\Theta_0}^{\Theta_i}\cos\Theta\,dl = \int_{\Theta_0}^{\Theta_i}\frac{\cos\Theta_i}{a+b\Theta}\,d\Theta, \]

\[ Y_i=\int_{\Theta_0}^{\Theta_i}\sin\Theta\,dl = \int_{\Theta_0}^{\Theta_i}\frac{\sin\Theta_i}{a+b\Theta}\,d\Theta, \]

whence, after a number of mathematical operations, we obtain, respectively,

\[ X_i=\frac{1}{b}\{\cos A[\operatorname{Ci}(A+\Theta_i)-\operatorname{Ci}(A+\Theta_0)]+ \]

\[ +\sin A[\operatorname{Si}(A+\Theta_i)-\operatorname{Si}(A+\Theta_0)]\}, \tag{3} \]

\[ Y_i=\frac{1}{b}\{\cos A[\operatorname{Si}(A+\Theta_i)-\operatorname{Si}(A+\Theta_0)]- \]

\[ -\sin A[\operatorname{Ci}(A+\Theta_i)-\operatorname{Ci}(A+\Theta_0)]\}, \tag{4} \]

where

\(X_i, Y_i\) are the current coordinates of the point under consideration of the curved borehole, corresponding to the zenith angle \(\Theta_i\), in m;

\(\operatorname{Si}, \operatorname{Ci}\) are, respectively, the sine integral and cosine integral;

\(\Theta_0\) is the zenith angle of the collar (or of the upper point of the section under consideration) of the borehole, in degrees;

\[ A=\frac{a}{b} \]

varies within the limits from \(-(5\div 6)\) to 10 degrees.

In profiling, the depth \(H_B\) at point \(B\), through which the borehole must pass, is always known. Thus, in essence, the magnitude of the vertical projection of the curve \(OB\) is specified. On the other hand, the depth of point \(B\) is determined by equation (3), in which the zenith angle of the collar \(\Theta_0\) still remains unknown. Consequently, by equating equation (3) to the depth \(H_B\), one can determine the value of the zenith angle of the collar. However, solving such an equation is extremely difficult, since the sine integral and cosine integral entering into it are not elementary functions. Nevertheless, the problem can be solved with a sufficient degree of accuracy if, over a certain interval of variation of the argument, the integral functions are replaced by simpler functions.

Since the zenith angles of boreholes practically vary within small limits (most often from 0 to 25 degrees), the integral functions may be replaced by simpler ones, using Table 2 as a guide.

Table 2

After replacing in equation (3) the integral functions by simpler ones and equating it to the depth \(H_{\mathrm{v}}\), we obtain the quadratic equation

\[ f\cdot \Theta_{\mathrm{op}}^{2}-q\cdot \Theta_{\mathrm{op}}+p=0, \tag{5} \]

in which

\[ \Theta_{\mathrm{op}}=\left(\frac{a}{b}+\Theta_{0}\right)=(A+\Theta_{0}), \]

\[ \rho=\left(A+\alpha+\beta-\frac{\pi}{2}\right), \tag{6} \]

and the values of the coefficients are determined from Table 3.

Table 3

When computing \(p\), the value of the integral functions may be calculated from the formulas of Table 2, but it is better to determine it from special tables (4).

Having solved the quadratic equation (5), one can determine from formula (6) the required angle of deviation, and by substituting the angle of deviation into equation (4), determine the setting point of the planned borehole.

Example. Find the point and zenith angle of deviation of a shot-drilling borehole if the assumed parameters of natural curvature are \(a=0.02\) deg/meter, \(b=0.002\) 1/meter, and the borehole must intersect, at an angle \(\beta=80^\circ\), at a depth \(H_{\mathrm{v}}=500\) m, an ore body inclined to the horizon at an angle \(\alpha=40^\circ\).

Solution. The zenith angle of deviation at the point of intersection with the ore body, determined by formula (2), is equal to \(30^\circ\). Judging from the magnitude of the zenith angle and the depth of the point of intersection, the zenith angle of deviation (calculated) will probably be within \(0.20\text{--}0.50\) radian. Calculations by the formulas of Table 3 give the following values of the coefficients of equation (5):

\[ \rho=\left(A+\alpha+\beta-\frac{\pi}{2}\right)=40^\circ, \]

\[ f=2.081\cdot \cos 10^\circ=2.0494, \]

\[ q=4.25\cdot \cos 10^\circ-0.99\cdot \sin 10^\circ=4.3573. \]

\[ p=1.757\cdot \cos 10^\circ+\sin 10^\circ\cdot \operatorname{Si}40^\circ+ \]

\[ +\cos 10^\circ\cdot \operatorname{Ci}40^\circ-0.002\cdot 500=0.9453. \]

To determine the angle \(\Theta_{\mathrm{op}}\), it is necessary to solve the quadratic equation (5)

\[ 2.0494\,\Theta_{\mathrm{op}}^{2}-4.3573\,\Theta_{\mathrm{op}}+0.9453=0. \]

The roots of this equation are \(\Theta_{\mathrm{op}_1}=1.881\) radian and \(\Theta_{\mathrm{op}_2}=0.246\) radian. The first root cannot satisfy the conditions of our

Fig. 2. Nomogram for determining the calculated zenith drilling angle \(\Theta_{\mathrm{op}}\) (or the depth \(H_{\mathrm{в}}\)) of naturally deviated shot-drilling boreholes.

\[ \Theta_{\mathrm{op}} = f(A + B);\ \text{degrees} \]

Diagram for use

Visible labels in the nomogram:

• scale of marks
• Coefficient \(k\), in thousandths
• \(H,\ \mathrm{m}\)
• \(\Theta_{\mathrm{op}} = A + B\Theta_{\mathrm{в}},\ \text{degrees}\)
• \(A = 5^\circ\)
• \(A = 10^\circ\)
• \(\Theta_{\mathrm{op}} - A + B\Theta_{\mathrm{в}},\ \text{degrees}\)
• \(B\)

Usage scheme

Fig. 3. Nomogram for determining the length of the horizontal projection \(H_{\mathrm{г}}\) of a naturally curved shot-drilling borehole.

Visible labels on the nomogram:

• \(H_{\mathrm{г}},\ \mathrm{m}\)
• mark scale
• quadrant \(\beta\) in the plane [[unclear: abbreviated word]], \(x\)
• \(\theta_{0\mathrm{п}}=\mu+\theta_{\mathrm{в}},\) degrees
• \(\theta_{\mathrm{г}0}=\mu+\theta_{\mathrm{г}},\) degrees
• \(A\)

example, since in this case the zenith angle of drilling should be obtuse. The solution of the quadratic equation is the second root. Passing from radian measure to degree measure, we obtain

\[ \Theta_{\mathrm{op\,deg}}=57.3\,\Theta_{\mathrm{op\,rad}}=14^\circ 06'. \]

Using equation (6), we find the value of the zenith angle of drilling the borehole (actual)

\[ \Theta_0=\Theta_{\mathrm{op}}-\frac{a}{b}=4^\circ 06'. \]

To find the location for setting out the borehole, we shall use equation (4)

\[ Y_B=\frac{1}{0.002}\{\cos 10^\circ[\operatorname{Si}(40^\circ)-\operatorname{Si}(14^\circ06')]- \]

\[ -\sin 10^\circ[\operatorname{Ci}(40^\circ)-\operatorname{Ci}(14^\circ06')]\}=133.6\ \text{m}. \]

Thus, the borehole must be drilled 133.6 m (in plan) from the point of undercutting of the ore body, at a zenith angle of \(4^\circ06'\).

Analytical profiling with the aid of equations (3) and (4), although possible, is nevertheless cumbersome and requires appreciable expenditures of time. The indicated inconveniences can be overcome if nomograms are constructed from equations (3) and (4), and profiling is then carried out with their aid.

Reducing the indicated dependences to the form

\[ X_i b=[\cos A\cdot \operatorname{Ci}(A+\Theta_i)+\sin A\cdot \operatorname{Si}(A+\Theta_i)]- \]

\[ -[\cos A\cdot \operatorname{Ci}(A+\Theta_0)+\sin A\cdot \operatorname{Si}(A+\Theta_0)], \]

\[ Y_i b=[\cos A\cdot \operatorname{Si}(A+\Theta_i)-\sin A\cdot \operatorname{Ci}(A+\Theta_i)]- \]

\[ -[\cos A\cdot \operatorname{Si}(A+\Theta_0)-\sin A\cdot \operatorname{Ci}(A+\Theta_0)], \]

it is easy to see that nomograms with binary scales (5) can be constructed from them; moreover, with such a representation of the parametric equations, the amount of computational work in constructing the nomograms is substantially reduced, since the expressions in square brackets in each equation are identical in their structure. Such nomograms are presented in Figs. 2 and 3.

We shall illustrate the use of the nomograms with the example of solving the previous problem.

The zenith angle for drilling the borehole is determined from the nomogram in Fig. 2. For this purpose we compute the following quantities

\[ A=\frac{a}{b}=10^\circ, \]

\[ \Theta_{\mathrm{rp}}=A+\Theta_r=40^\circ. \]

For \(\Theta_{\mathrm{rp}}=40^\circ\) and \(A=10^\circ\), on the left-hand graph we find point \(B\) on the scale of marks. Similarly, on the middle graph, from the quantities \(H_B\) and \(b\), point \(C\) is found. Drawing a straight line through points \(B\) and \(C\) to its intersection with the scale of marks of the right-hand graph, we obtain point \(D\). From this point and the curve corresponding to \(A=10^\circ\), we find that \(\Theta_{\mathrm{op}}=14^\circ10'\); then the zenith angle of drilling is

\[ \Theta_0=\Theta_{\mathrm{op}}-A=4^\circ10'. \]

The magnitude of the horizontal displacement can be determined from the nomogram in Fig. 3. To do this, from \(\Theta_{\mathrm{op}}\) and \(A\) on the left-hand graph the point \(M\) is determined, and from \(\Theta_{\mathrm{gr}}\) and \(A\) on the right-hand graph—the point \(N\). By connecting these points with a straight line, we obtain, on the mark scale of the middle graph, the point \(K\). This point and the ray corresponding to the value \(v = 0.002\) determine the horizontal displacement \(H_{\mathrm{g}} = 137\) m.

Thus, the discrepancies between the results of the analytical and nomographic profiling methods are quite insignificant.

The nomograms may prove useful in constructing projected deviated wells on a section. For this it is necessary to know the magnitudes of the initial and final zenith angles and, in addition, to specify several intermediate values so that approximately equal distances are obtained between the points. Thus, we have a series of zenith angles \(\Theta_0, \Theta_1, \Theta_2, \ldots \Theta_i\). Adding to this series the quantity \(A\), we obtain \(\Theta_{\mathrm{op}}, \Theta_{\mathrm{gr1}}, \Theta_{\mathrm{gr2}}, \ldots \Theta_{\mathrm{gri}}\). Using the latter series and the quantities \(A\) and \(v\) as initial data, on the middle graphs of the nomograms we shall obtain the horizontal and vertical displacements of the borehole points corresponding to the specified zenith angles, i.e., the coordinates of individual points of the borehole.

When profiling wells by the proposed methods, the following must be borne in mind:

1. The curvature of wells is influenced by causes that distort regular curvature; the proposed methods describe only probable design trajectories, and therefore deviations of actual boreholes from the design ones within certain limits are entirely admissible.

2. The nomograms and formulas should be used interval by interval if, in the section of the well, rocks are encountered that differ sharply in degree of anisotropy, or if the well crosses a complex geological structure; but if the well crosses monoclinically bedded strata with similar physico-mechanical properties, then the entire well can be profiled at once.

3. In developing these methods of profiling, it was assumed that the wells should be set across the strike of the structures and therefore would have comparatively small azimuthal curvatures. An azimuthal deviation of up to \(15\)—\(20^\circ\) from the original direction has a comparatively small effect on the profile of the well in the vertical plane. With considerable well depth, even with such azimuthal deviations the actual point of intersection may deviate appreciably horizontally from the design point of intersection, which, of course, must be taken into additional account in profiling.

4. To increase the service life of the nomograms, before use it is desirable to place them under a thin sheet of organic glass and to carry out all constructions on the glass.

LITERATURE

1. A. E. Kolesnikov. Investigation of the operation of flexible assemblies in the process of artificial curvature of wells and the development of their design. Author’s abstract of Candidate’s dissertation, Moscow, 1963.

2. A. G. Kalinin. Regularities of well curvature in the iron-ore deposits of Mountain Shoria. “Izv. vyssh. uchebn. zaved.,” geology and exploration, No. 11, 1959.

3. G. A. Alekseev. Finding the form of the relationship between random quantities by the quantile method. Appendix II to M. A. Velikanov’s book Measurement Errors and Empirical Dependencies, Gidrometeoizdat, Leningrad, 1962.

4. E. Jahnke and F. Emde. Tables of Functions with Formulas and Curves. Third edition. Fizmatgiz, Moscow, 1959.

5. L. L. Gavra. Fundamentals of Nomography with Applications from Mechanical Engineering. Second edition. Mashgiz, Moscow—Leningrad, 1962.