DETERMINATION OF THE ATTITUDE ELEMENTS OF STRATA FROM ONE LINE OF AN OBLIQUE SECTION AND TWO UNORIENTED CORES

L. A. Pukhlyakov

(Presented by Professor A. V. Aksarin)

In prospecting for oil and gas, the determination of the attitude elements of strata lying at great depths is of great importance. In some cases it makes it possible to refine structures and thereby reduce the costs of exploring a deposit; in others, without it, it is altogether impossible to obtain an idea of the structure of the deep parts of certain areas. For this purpose, various methods are used under different conditions. The method considered here is applied in cases when the geologist has at his disposal a two-hole well, both shafts of which have cut a certain bed and from each of them an unoriented core has been taken. It is also possible to apply it when the geologist has two wells located relatively close to one another. Under analogous conditions the method of several oblique sections [2, 3] is also used; it is more convenient, since it does not require core sampling, but it does require that the oblique sections of different beds not be parallel to one another, which is not always the case. Thus, at times there arises an urgent need to use precisely the method of one straight line of oblique section and two unoriented cores.

The theoretical foundations of this method were set forth in one of the published works [1]; however, when it was introduced into practice, certain difficulties were encountered, caused mainly by an insufficiently detailed description of the data-processing procedure—in particular, by the absence of a description of how to find the antipodes of the points through which the circles corresponding to the attitude elements for individual cores are constructed, as well as those parts of these circles that extend beyond the contour of the stereographic projection. In connection with the foregoing, it is recommended that the work by this method be carried out in the following order.

1. The angle of inclination of the bedding plane relative to the plane perpendicular to the core axis, \(\beta\) (angle \(\angle ABC\) in Fig. 1-a), is determined. To do this it is necessary to select some characteristic interbed in the core (if necessary, it may be outlined with a pencil). Then, placing this core in the corner of some shallow box, rotate it and at the same time measure the generatrices, with the aim ultimately of finding the greatest of them, \(h_{\max}\), and the smallest, \(h_{\min}\). After this, the diameter of the given core, \(d\), is determined, the value of which, like the values

the greatest and the smallest ordinates, \(h_{\max}\) and \(h_{\min}\), is substituted into the formula for determining the angle of interest to us

\[ \operatorname{tg}\beta=\frac{h_{\max}-h_{\min}}{d}. \tag{1} \]

1. A horizontal projection of the branched part of the two-hole borehole is drawn, on which the points of intersection of both boreholes with the bed being mapped are shown, and beside them the nominal depths (depths measured along the borehole) \(l\) and their absolute elevations \(z\) are written (Fig. 1-b).

2. From the point with the higher absolute elevation to the point with the lower absolute elevation (in the present case, from point \(B\) to point \(A\)) a straight line is drawn. The angle formed by it with the “south–north” direction gives us the dip azimuth of the line of the oblique section. In the present case (Fig. 1-b) it is equal to \(26^\circ 30'\).

Fig. 1. Determination of the dip angle of the bedding plane in core (a) and of the dip direction of the line of the oblique section (b)

1. The distance between the points of intersection of the bed by the separate boreholes, \(s\), and the difference of their absolute elevations, \(z_1-z_2\), are measured; these are substituted into the formula for determining the dip angle of the line of the oblique section

\[ \operatorname{tg}\alpha_k=\frac{z_1-z_2}{s}. \tag{2} \]

In the present case (Fig. 1-b) \(s=16.5\ \text{m}\), \(z_1-z_2=21.2\ \text{m}\), \(\operatorname{tg}\alpha_k=1.28485\), \(\alpha_k=52^\circ 06'\).

1. The outline of the stereographic projection of the upper hemisphere is constructed: a circle of radius 5 cm is drawn, on which north is marked as \(0^\circ\), east as \(90^\circ\), south as \(180^\circ\), and west as \(270^\circ\). In the resulting system, the azimuth of the line of the oblique section is plotted and fixed by the radius-vector \(A_k\). In analogous fashion, the dip azimuths of the individual boreholes at the core-sampling points \(A_1\) and \(A_2\) are plotted. In the present case (Fig. 1-b) \(A_1=47^\circ\) and \(A_2=60^\circ\). On the other side of the center of the circle, the continuations of these radii are drawn as dashed straight lines (Fig. 2-a).

2. The distance from the center of the projection is determined which must express the magnitude of the dip angle of the line of the oblique section, \(\rho_k\):

\[ \rho_k=5\operatorname{tg}\frac{\alpha_k}{2}, \tag{3} \]

and also its antipode, \(\lambda_k\),

\[ \lambda_k = 5 \operatorname{Ctg} \frac{\alpha_k}{2}. \tag{4} \]

The first of these distances, \(\rho\), is laid off from the center to the periphery in the direction of the azimuth of the line of the oblique section (on the solid line). The point obtained is denoted by the letter “\(K\).” The second, \(\lambda_k\), is laid off in the opposite direction (from the center to the periphery along the dashed line). Through the points obtained one must draw a circle in such a way that its center lies exactly midway between them; more precisely, one must draw the part of this circle that lies inside the outline of the stereographic projection (the arc NKP in Fig. 2-a). However, since the distance between the given points may turn out to be too large, it is recommended that the position of the center of this circle be determined relative to the point “\(K\)” through the arithmetic mean of the obtained quantities

\[ r_k = \frac{\rho_k + \lambda_k}{2}. \tag{5} \]

In the present case \((\alpha_k = 52^\circ 06')\), \(\rho_k = 2.444\ \text{cm}\) and \(\lambda_k = 10.229\ \text{cm}\), \(r_k = 6.336\ \text{cm}\). This value is laid off from the point “\(K\)” in the direction of the center and then farther along the dashed straight line. This gives the second point \(O_k\) (Fig. 2-a). In the process of photographing Fig. 2, all linear quantities,

Fig. 2. Determination of the attitude elements of a bed from the line of an oblique section and two unoriented cores by the graphical-analytical method on a stereographic projection, reduced by a factor of two

including the value \(r\), proved to be reduced by a factor of two. If this value turns out to be too large, one may use a Wulff net. For this purpose, the construction described must be carried out on tracing paper and, in addition, through the center of the projection (the intersection) a perpendicular to the straight line under consideration must be drawn. Then the tracing paper must be placed on the Wulff net so that one of the meridians passes through the points where this perpendicular meets the outline of the stereographic projection and through the point “\(K\).” The part of this meridian lying inside the projection outline is traced in ink. This will be the arc NKP.

1. On the radius-vectors corresponding to the dip azimuths of the individual boreholes, at the core-sampling points and on their dotted extensions, circles are constructed expressing the relationship between the dip angles of the beds in the cores, \(\beta_1\) and \(\beta_2\), and the zenith angles of the borehole at the points where they were sampled, \(\Theta_1\) and \(\Theta_2\). The distances between the extreme points of these circles and the center of the projection should be determined by the formulas

\[ \rho'_1 = 5 \operatorname{tg} \frac{\Theta_1-\beta_1}{2}, \tag{6-a} \]

\[ \rho''_1 = 5 \operatorname{tg} \frac{\Theta_1+\beta_1}{2}, \tag{6-b} \]

\[ \rho'_2 = 5 \operatorname{tg} \frac{\Theta_2-\beta_2}{2}, \tag{6-c} \]

\[ \rho''_2 = 5 \operatorname{tg} \frac{\Theta_2+\beta_2}{2}. \tag{6-d} \]

The obtained quantities are laid off in the corresponding directions, namely \(\rho'_1\) and \(\rho''_1\) along the straight line corresponding to the dip azimuth of the first borehole, and \(\rho'_2\) and \(\rho''_2\) along the straight line corresponding to the dip azimuth of the second borehole. In this case the quantities \(\rho''\), which are always positive, are laid off from the center along the dotted straight lines, while \(\rho'\)—when positive—are laid off along the dotted straight lines, and when negative along the solid ones. Then the midpoints between the given points are determined and, accordingly, circles are drawn through them.

Let us assume that in the case under consideration the dip angle in the first core, \(\beta_1\), has proved to be \(80^\circ 30'\), and in the second, \(\beta_2\), \(72^\circ 00'\), while the zenith angles are \(20^\circ 30'\) at the first point and \(11^\circ 00'\) at the second. In this case the quantities of interest to us should be as follows: \(\rho'_1=-2.887\) cm, \(\rho''_1=+6.066\) cm, \(\rho'_2=-2.945\) cm, \(\rho''_2=+4.424\) cm and, plotting them on the drawing (Fig. 2-b), we obtain: on the straight line corresponding to the first borehole, the points \(B\) (on the solid line), \(C\)—(on the dotted line) and \(O_1\)—the center between them; and on the straight line corresponding to the second borehole, the points \(L\) (on the solid line), \(M\) (on the dotted line) and \(O_2\)—the center between them.

When the circle is drawn, part of it may prove to lie outside the contour of the stereographic projection (the primitive circle). This will mean that the upper end of the perpendicular to the bedding plane has fallen in the lower hemisphere. This part of the circle is drawn with a dotted line (Fig. 2-b), and at the same time that circle is constructed which will be traced by the lower end of the perpendicular to the bedding plane, more precisely that part of this circle which will prove to lie in the upper hemisphere. For this purpose, first of all the distances from the center of the projection to the antipodes of the extreme points of the circle that has gone beyond the contour of the stereographic projection are determined, in the present case the points \(B\) and \(C\). The calculation is made by the formulas

\[ \lambda'_1 = 5 \operatorname{Ctg} \frac{(\Theta_1-\beta_1)}{2}, \tag{7-a} \]

\[ \lambda''_1 = 5 \operatorname{Ctg} \frac{(\Theta_1+\beta_1)}{2}. \tag{7-b} \]

In the present case (with \(\Theta_1=20^\circ 30'\) and \(\beta_1=80^\circ 30'\)) these quantities prove to be \(\lambda'_1=-8.660\) cm and \(\lambda''_1=4.1217\) cm. These distances are laid off in such a way that the antipodes are opposed to the original points—are located on the other side of the center of the projection. In the present case point \(B\) is located in the upper part of the drawing, and point \(C\)—

in the lower part, while their antipodes are, conversely: point \(B'\) in the lower part, and point \(C'\) in the upper (Fig. 2-b). After this, the midpoint between the antipodes is determined—the point \(O'\) in Fig. 2-b—which will be the center of the circle of interest to us. Placing one leg of the compass at this point, with the other we draw the required part of the circle, in the present case (Fig. 2-b) the arc \(D'C'E'\).

If the radius of the resulting circle proves too large, the Wulff net may be used to construct it. In this case, in addition to the antipode of the point located on the straight line corresponding to the direction of dip of the borehole, it is necessary to have the antipodes of the points of intersection of the given circle with the contour of the stereographic projection. They are situated exactly opposite one another with respect to the center of the projection. In the present case, one of them (point \(D\)) corresponds to point \(D'\), and the other (point \(E\)) to point \(E'\). Then some meridian or parallel of the Wulff net is superposed on these three points (more precisely, brought under them), and then traced. In the present case (Fig. 2-b) there was no need to resort to the Wulff net.

The obtained circles, or more precisely their parts lying within the contour of the stereographic projection, must intersect with the arc constructed from the data along the straight line of the oblique section (in Fig. 2-b, the arc \(NKP\)), thereby forming two triangles, one of which should be many times smaller than the other. In the ideal case the smaller triangle should shrink to a point. The center of gravity of this triangle will reflect the attitude elements of the bed. Namely, the radius vector drawn through it (in Fig. 2 it is denoted by the symbol \(A_{\text{п}}\)) will give on the limb the azimuth of dip of the bed (in the present case it is equal to \(75^\circ\)), and its distance from the center \(\rho_\Delta\), equal in the present case to 3.0 cm, when substituted into the formula

\[ \operatorname{tg}\frac{\alpha_{\text{п}}}{2}=\frac{\rho_\Delta}{5} \tag{8} \]

will give the dip angle of the bed \(\alpha_{\text{п}}\). If the sizes of the triangles prove equal or nearly equal, this will mean that the attitude elements of the beds at the core-sampling points are different and that the problem will have no solution. In the present case it has a solution, namely
\(\rho_\Delta = 3.0\) cm, \(\alpha_{\text{п}} = 62^\circ\), and \(A_{\text{п}} = 75^\circ\).

REFERENCES

1. L. A. Pukhlyakov. Determination of the attitude elements of a bed from data of a two-hole borehole. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 167, 1967.

2. L. A. Pukhlyakov. Two-face boreholes as a means of determining the attitude elements of beds. Collection “Results and prospects of geometrization of mineral raw-material deposits. Proceedings of the seminar held January 29–31, 1969.” Moscow, 1969.

3. L. A. Pukhlyakov, I. Z. Karamaev, B. A. Nikulin. Use of an emergency borehole shaft to determine the attitude elements of beds in the undermining zone of the Astashevskaya area. Izvestiya Tomskogo politekhnicheskogo instituta, vol. 216, 1971.

ERRATA