PROFILING OF MEDIUM-DEPTH WELLS DURING FRACTIONAL DRILLING

V. P. ROZHKOV

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

When wells are drilled by modern methods, as a rule they become curved and therefore pass above the accepted intersection points. Such wells may partially or completely fail to fulfill the assigned geological task. To increase the efficiency of exploration, in many geological organizations deep wells are—

Table 1

Deviations of actual trajectories from planned ones for certain wells of the Tashelga deposit

...assigned geological task. To increase the efficiency of exploration, in many geological organizations deep wells are—

began to be profiled taking into account their future natural curvature. In designing wells with depths of 150–250 m, such allowance for curvature is, as a rule, not made. It is usually assumed that, because of their shallow depth, they cannot deviate significantly from straight lines. This is by no means true for all deposits.

Table 1 gives, as an example, the deviations of actual wells of the Tashelga deposit from their design (rectilinear) trajectories. As can be seen from the table, the discrepancies may reach several tens of meters. Moreover, the larger the collaring zenith angle, the more significant the discrepancy. Such relatively large deviations, as in deep wells, can be avoided by taking natural curvature into account.

Many researchers [2, 3, 4, 6] note that the zenith curvature of wells increases with their depth, and the greater the depth, the more intensive this increase. The intensity of curvature depends to a considerable extent on the magnitude of the collaring zenith angle. This is also evidenced by the data in Table 1. Differences of opinion arise only in determining the degree of influence of depth and of the collaring zenith angle on the profile of a curved well. Consequently, at small depths, when the radius of curvature does not decrease very intensively, one may assume that the well curves with a constant radius, whose magnitude depends on the collaring zenith angle.

To carry out profiling by the proposed method, it is necessary to determine the mathematical dependence of the average curvature of a well on the magnitude of the collaring zenith angle. For this purpose, statistical dependences of curvature on the collaring zenith angle were found for several deposits, and graphs were constructed from them. One such graph is shown in Fig. 1.

Fig. 1. Dependence of the average curvature \(K_{\mathrm{av}}\) for wells of the Inskoye iron-ore deposit on the collaring zenith angle \(\Theta_0\):
1 — mean point and the footage over which it was determined; 2 — empirical dependence; 3 — correlation straight line

Using the methods of mathematical statistics it was established that this dependence is in all cases close to linear. The parameters of the correlation straight lines were determined by the quantile method, as the more accurate one [1]. The processing results are presented in Table 2.

Thus, the correlation between the average curvature of the well and the zenith angle of deviation can be represented by the equation

\[ K_{\mathrm{av}}=\frac{d\Theta}{dZ}=n+m\Theta, \tag{1} \]

where \(K_{\mathrm{av}}\) is the average curvature of the entire well, degrees/m;
\(\Theta_0\) is the zenith angle of deviation, degrees.

Table 2

Parameters characterizing the curvature of medium-depth shot holes as a function of the zenith angle of deviation

The radius of curvature, as is known, is the reciprocal of curvature. Passing to the dimensions adopted in the present work, it can be determined by the formula

\[ R_{\mathrm{av}}=\frac{57,3}{n+m\Theta_0}. \tag{2} \]

The law of variation of the zenith angle as a function of well depth can be determined by integrating differential equation (1). After the indicated operation we have

\[ \Theta=\Theta_0+K_{\mathrm{cp}}\cdot L. \tag{3} \]

The laws of variation of the vertical and horizontal projections of a wellbore curving along the arc of a circle are known from the work of B. I. Spiridonov [5]. As applied to the case under consideration, they are expressed as follows:

\[ X=R_{\mathrm{cp}}(\sin\Theta-\sin\Theta_0), \tag{4} \]

\[ Y=R_{\mathrm{cp}}(\cos\Theta_0-\cos\Theta). \tag{5} \]

Since the interception point is known before profiling begins, the distance \(H\) along the vertical from the wellhead to this point, i.e., the vertical projection, is known with a certain degree of accuracy. On the other hand, the vertical projection, taking account of dependence (2), is determined by the expression

\[ X=\frac{57.3(\sin\Theta_{\mathrm{г}}-\sin\Theta_0)}{n+m\Theta_0}. \tag{6} \]

Consequently, from this expression, if the sine of the zenith angle of drilling is replaced by this same angle expressed in radians, it is possible to determine the setting zenith angle of the well. The limiting relative error of such a replacement is equal to \(\delta_{\max}=(17\cdot\Theta_0^2)\%\) [6]. For drilling zenith angles equal to 10, 20, and \(30^\circ\), the error will be, respectively, 0.5; 1.9; and 4.6%. Such errors will have practically no effect on the final result. The replacement is also justified by the fact that the well-setting angle can practically be specified no more accurately than \(0.5^\circ\), and dependence (6) itself is statistical in character.

After replacement we obtain

\[ \Theta_0=\frac{57.3\cdot\sin\Theta_{\mathrm{г}}-Hn}{1+Hm},\ \text{degrees}, \tag{7} \]

where \(\Theta_0,\ \Theta_2\) are respectively the drilling zenith angle and the zenith angle at the interception point, in degrees; the zenith angle at the interception point is measured on the geological section along the well or is calculated;

\(H\) is the depth of the interception point along the vertical, m; in equation (4) it corresponds to the quantity \(X\);

\(n,\ m\) are the parameters of well curvature; the values of these parameters are given in Table 2.

The further solution of the problem presents no difficulties. From equation (2) the radius of curvature of the well is determined, and from equality (5) the horizontal displacement, i.e., the location of drilling the well.

Example. Design a well if the interception point is located at a depth of 185 m, and the zenith angle of the well at the interception point must be \(13^\circ\). The curvature parameters have the following values:

\[ n=0.0477\ \text{degrees/m},\quad m=0.000608\ 1/\text{m}. \]

Solution. From formula (7) we have

\[ \Theta_0=\frac{57.3\cdot0.225-185\cdot0.0477}{1+185\cdot0.000608}=3.66=3^\circ40'. \]

The horizontal displacement is equal to

\[ H_{\mathrm{г}}=Y=\frac{57.3(0.9979-0.9744)}{0.0477+3.66\cdot0.000608}=27\ \text{m}. \]

Thus, the well must be drilled in at 27 meters (horizontally) from the point of interception, with an inclination angle equal to 3.5° (after rounding to 0.5°).

When drilling in wooded or sharply dissected terrain, much time is spent preparing the site for the drilling unit. In such cases it is useful to use natural sites or to drill several wells from one specially prepared site. In this case profiling is reduced only to determining the inclination angle of the kickoff and constructing the well profile, since the points of interception and kickoff are known in advance.

Fig. 2. Scheme of profiling a well curved along an arc of a circle: \(\Theta_0, \Theta_{\mathrm{g}}\)—inclination angles of the well respectively at the kickoff point and at the interception point; \(H_{\mathrm{v}}, H_{\mathrm{g}}\)—distance from the kickoff point to the interception point respectively vertically and horizontally

To solve the stated problem one may use equations (2)—(5). After jointly solving equations (4) and (5) and carrying out some trigonometric transformations, we obtain

\[ \frac{X}{Y}=\operatorname{tg}\left(\frac{\Theta+\Theta_0}{2}\right) \]

or, as applied to the case under consideration (Fig. 2),

\[ \frac{H_{\mathrm{g}}}{H_{\mathrm{v}}}=\operatorname{tg}\left(\frac{\Theta_{\mathrm{g}}+\Theta_0}{2}\right). \]

Hence it is easy to determine the sum of the kickoff inclination angle and the inclination angle of the well at the point of interception. This sum can also be determined in a simpler way. From relation (8) it is clear that the argument of the tangent is equal to the inclination angle \(\Theta_{\mathrm{p}}\) of the straight line connecting the kickoff point with the interception point. This angle can be measured on the geological section along the well, and twice its value gives the sum of the kickoff inclination angle and the inclination angle at the point of interception. Thus,

\[ \Theta_{\mathrm{g}}+\Theta_0=2\operatorname{arctg}\left(\frac{H_{\mathrm{g}}}{H_{\mathrm{v}}}\right)=2\Theta_{\mathrm{p}}=C. \tag{9} \]

On the other hand, from relations (1) and (3) we have

\[ \Theta_{\mathrm{g}}-\Theta_0=L\left(n+m\Theta_0\right). \tag{10} \]

The joint solution of equations (9) and (10) leads to the final result

\[ \Theta_0=\frac{C-Ln}{2+Lm},\ \text{deg.}, \tag{11} \]

where \(L\) is the depth of the well to the point of interception, m.

In view of the small depth, instead of it one may substitute into formula (11) the distance between the kickoff point and the interception point \((S)\).

At depths of up to 250 m and radii of curvature of 1000–3000 m, the discrepancy between \(S\) and \(L\) will be 0.6–0.3 m. The distance between the collar and the point of intersection (in a straight line) can be measured on the geological section along the borehole or calculated by the formula

\[ S=\sqrt{H_{\mathrm{г}}^{2}+H_{\mathrm{в}}^{2}}. \tag{12} \]

Consequently, the more practical relation has the form

\[ \Theta_{0}=\frac{C-Sn}{2+Sm},\ \text{grad}, \tag{13} \]

where \(C\) is in grad, \(S\) is in m, \(n\) is in grad/m, and \(m\) is in \(1/\text{m}\).

Example. To check the accuracy of the proposed methods, solve the preceding example under the condition that the projections of the borehole are known (27 and 185 m).

Solution.

\[ C=2\operatorname{arctg}\left(\frac{27}{185}\right)=16^\circ37'=16{,}62^\circ. \]

\[ S=\sqrt{27^{2}+185^{2}}=187\ \text{m}. \]

\[ \Theta_{0}=\frac{16{,}62-187\cdot0{,}0477}{2+187\cdot0{,}000608} =3{,}60^\circ=3^\circ36'. \]

In the preceding example, \(\Theta_{0}=3^\circ40'\) was obtained; i.e., the convergence of the results of calculation by the proposed relations is very high.

When constructing the profile of a shallow fractional-drilling borehole (because of the small depth), on the design section it may be represented as a straight line connecting the collar and the point of intersection. Such a borehole, however, must be drilled at the calculated angle, rounded to \(0.5^\circ\). The actual profile must without fail be constructed from the results of measurements of the borehole position in space.

LITERATURE

1. G. A. Alekseev. Determination of the form of relationship between random variables by the quantile method. Appendix to the book by M. A. Velikanov, Measurement Errors and Empirical Relations, Gidrometeoizdat, Leningrad, 1962.

2. A. M. Kurmashev. Investigation of natural profiles of geological-prospecting boreholes by probabilistic-statistical methods. In: Methods and Techniques of Prospecting, no. 44, ONTI VITR, 1964.

3. V. P. Rozhkov, Yu. L. Boyarko. Some causes and regularities of zenith curvature of boreholes in fractional drilling. Proceedings of Higher Educational Institutions—Geology and Prospecting, no. 3, 1965.

4. A. A. Sorokin. Regularities of borehole curvature and methods for detecting them. Information collection of the Geological Prospecting Service of Rudny Altai, 1, Ust-Kamenogorsk, 1960.

5. B. I. Spiridonov. Analytical calculation of prospecting-borehole profiles. Materials on the Geology and Mineral Resources of Western Siberia. Publishing House of Tomsk University, Tomsk, 1964.

6. Machine-Builder’s Handbook. In six volumes, ed. N. S. Acherkan, vol. 1, 3rd ed., Mashgiz, Moscow, 1961.

7. S. S. Sulakshin. Basic questions in the methodology for detecting regularities of borehole curvature. In: Methods and Techniques of Prospecting, no. 48, ONTI VITR, 1964.