Determination of Stratum Attitude Elements from Dual-Borehole Well Data

L. A. Pukhlyakov

(Presented by the scientific seminar of the Department of Fossil Fuels)

In drilling for oil and gas, wells very often deviate arbitrarily to the side. Such wells are corrected: the deviated portion of the borehole is filled with cement and a new borehole is drilled. As a rule, the data on the deviated portion of the borehole are not used. Meanwhile, on the basis of these data it is possible, without additional expense, to determine the elements of stratum attitude, which is very important in constructing structural maps of deposits. For this, the curvature elements for both boreholes are needed, as are the points at which both boreholes meet a certain stratum (their exact depth referencing), and unoriented cores taken from both boreholes. But the curvature elements for both boreholes are necessarily available, since only from them can one judge how far a given borehole has deviated from the assigned direction. Electrometric work to pick out horizons is always carried out. As for unoriented core, its recovery likewise presents no difficulty.

In order to determine from these data the elements of stratum attitude, it is first necessary to draw the horizontal projections of both boreholes and to plot on them the points of intersection with the given stratum. The technique for such constructions has been developed sufficiently [1, 2]. A straight line drawn through the projections of these points (it may be called the line of oblique section of the stratum) gives us the azimuth of this section \(A_k\), while the distance between them \(s\) and the difference of their absolute elevations \(\Delta z\) make it possible to determine the angle of dip of the stratum in the given section \(\alpha_k\)

\[ \operatorname{tg}\alpha_k=\frac{\Delta z}{s}. \tag{1} \]

From both unoriented cores the angles of inclination of the bedding plane relative to the plane of the transverse section of the core, \(\beta\), are determined. For this purpose the core is processed on a grinding machine: one side of it is ground along a plane perpendicular to the core axis, and the other along the bedding plane. Then its diameter \(d\) and the generators are measured: the greatest \(h_{\max}\) and the smallest \(h_{\min}\), the values of which are substituted into the formula for determining the quantity of interest to us,

\[ \operatorname{tg}\beta=\frac{h_{\max}-h_{\min}}{d}, \tag{2} \]

From these data, the attitude elements of the stratum can be determined on a sphere with degree divisions (on a globe). Here, first of all, the points at which both borehole shafts emerge on the upper half of the sphere are determined (meaning those parts of them from which the core was taken). For this purpose the azimuth of inclination of the first borehole, \(A_1\), is measured in the system of meridians clockwise, beginning from the 180-degree mark, and the zenith angle—in the system of parallels from the pole to the equator. The azimuth is measured precisely from 180°, and not from the zero meridian, because, if measured from zero, it would give us the direction of the lower end of the borehole. What we need, however, is the direction of the upper end, since we are working on the upper half of the sphere. The curvature elements of the second shaft are treated in the same way. As a result, both shafts give us one point each on the sphere.

Next, dividers are taken, and one leg is placed at the point corresponding to the emergence on the upper half of the sphere of the first shaft, while the other is set off at a distance corresponding to the angle of inclination of the bedding plane relative to the plane of the transverse section from the first core, \(\beta_1\). The magnitude of the angle is measured in the system of parallels along the meridian on which the given point lies. With the obtained opening of the dividers a circle is drawn. Around the point of emergence on the upper half of the sphere of the second shaft, a circle is drawn corresponding to the angle of inclination of the bedding plane from the second core, \(\beta_2\). As a result, two points of intersection of the circles are obtained on the sphere (in a special case, one point may be obtained, and even a superposition of the circles upon one another; the latter is possible when the azimuths of inclination and the zenith angles of both shafts are identical).

It is then necessary to refer to the line of oblique section. The azimuth of its inclination, \(A_k\), is measured from the zero meridian in the system of meridians clockwise, and its angle of inclination, \(\alpha_k\), in the system of parallels from the pole to the equator. Through the point obtained, a great circle is drawn, which must express the plane perpendicular to the line under consideration. This circle intersects only one of the points of intersection of the first two circles. This point will correspond to the attitude elements of the stratum. Its dip azimuth, \(A_p\), is measured from the zero meridian clockwise in the system of meridians, and the dip angle, \(\alpha_p\), in the system of parallels from the pole to the equator. Incidentally, we note that on a globe the parallels are counted from the equator to the poles, and the meridians in the eastern and sometimes [5] in the western hemispheres are counted counterclockwise.

Working on a globe is very inconvenient. This is connected not only with the fact that the counting of degrees here is carried out not in the direction required for solving the given problem, but mainly with the fact that on a globe it is very difficult to draw circles of large diameters. Finally, after solving several problems on one globe it will become completely unusable. In this connection, it is recommended that the solution of problems of this kind be carried out either on an axonometric projection or on a stereographic projection of the upper half of the sphere (upper hemisphere).

Every circle constructed on a sphere is also represented in stereographic projection, or on a Wulff net, in the form of a circle [3, pp. 60–63]. Hence the solution of the problem under consideration on a Wulff net reduces to the following.

1. Tracing paper is placed over the net and fastened immovably to it. Then the net with the tracing paper is positioned in the usual way, namely, one of its poles is placed at the top and the other at the bottom. The net is numbered. The upper pole is denoted by the number \(0^\circ\), the lower pole by \(180^\circ\), the right end of the equator by \(90^\circ\), and the left end of the equator by \(270^\circ\).

1. From the 180-degree division of the limb, clockwise, the azimuths of inclination of both borehole axes are read off; these are fixed by straight lines (radii).

2. The tracing paper is detached from the net and brought into such a position that the radius corresponding to the azimuth of inclination of the first axis coincides with one half of one of the diameters of the net, for example with the right half of the horizontal diameter.

3. In the system of meridians of the net, from the center toward its periphery—in the present case from left to right—the value of the zenith angle of the first axis is read off. The point obtained will express the elements of curvature of the first axis.

4. To the right and to the left of the point obtained, the value of the angle of inclination of the bedding plane in the first core, \(\beta_1\), is laid off. On the chosen diameter two more points are obtained; through them it is necessary to draw a circle with its center located exactly midway between them. This circle will express the relation between the elements of curvature of the first axis and the angle of inclination of the bedding plane in the first core. If, in the course of the construction described, it turns out that the sum of the angles \(\Theta\) and \(\beta\) exceeds \(90^\circ\), and one of the points lying on the circle must fall on the lower hemisphere, then its distance from the center of the net \((r)\) should be determined by the method developed for analogous problems by D. Fisher [6], that is, by the formula

\[ r = R \operatorname{tg} \frac{\Theta + \beta}{2}, \tag{3} \]

where \(R\) is the radius of the net, usually equal to 10 cm.

1. In an analogous way, a second circle is constructed, which must express the relation between the elements of curvature of the second axis and the angle of inclination of the bedding plane in the second core.

2. The tracing paper is brought back to its initial position, after which, from the zero division of the limb of the net, clockwise, the azimuth of inclination of the line of the oblique section \(A_{\ldots}\) is read off. Through the point obtained a third radius is drawn, which is then made to coincide with one half of the horizontal diameter. The center of the tracing paper, as before, remains coincident with the center of the net.

3. In the system of meridians, from the center toward the periphery, the angle of inclination of the bed in the investigated section, \(\alpha_k\), is read off. One more point is obtained. From this point, along the meridian on which it lies, an arc is drawn, which will express the plane perpendicular to the line of the oblique section of the bed. It will intersect only one of the intersection points of the first two circles. Let us mark this point with a small circle.

4. Through the point marked with a small circle, a fourth radius is drawn, which is made to coincide with one half of the horizontal diameter. Then, along this radius, from the center of the net to the point marked with a small circle, the angle is read off in the system of meridians; this angle will be the true dip angle of the bed, \(\alpha\).

5. The tracing paper is brought back to its initial position. In doing so, the fourth radius will substantially change its position (in the particular case it may remain in place). The division of the limb of the net that lies opposite one of its ends (its other end must lie at the center) will express the azimuth of dip of the bed.

Solving the problem under consideration on the Wulff net is very convenient, but not always possible. The point is that sometimes one of the points of the first or second circle may lie near the nadir of the lower hemisphere. In such a case, it is not possible to plot it on the sheet of paper on which the solution is being carried out. For solving problems of this kind,

it is recommended to use an axonometric projection of the upper hemisphere. All circles drawn on the sphere are represented in this projection as ellipses, whose major semiaxes will be equal to the radii of these circles \(r\), while the minor semiaxes are related to them by the relation

\[ b = r \cdot \cos \gamma, \tag{4} \]

where \(\gamma\) is the angle of inclination of the plane of the circle to the plane of projection. It is recommended that the work be carried out in the following order.

1. A circle of radius 10 cm is drawn, and in it two principal diameters are drawn—the vertical and the horizontal. The upper end of the vertical diameter is taken as zero; the right end of the horizontal is denoted by the number \(90^\circ\), etc.

2. In the resulting system, from the 180th division of the limb, the azimuths of inclination of both wellbores are laid off; these are fixed by straight lines (radii).

3. The distance from the center of the first ellipse to the center of the projection of the hemisphere is determined; it is expressed by the relation

\[ r_1 = 10 \cdot \cos \beta_1 \cdot \sin \Theta_1, \tag{5} \]

where \(\beta_1\) is the angle of inclination of the bedding plane in the first core relative to the plane perpendicular to its axis, determined by relation (2), and \(\Theta_1\) is the zenith angle of the borehole at the point of core sampling. The value \(r_1\) may be determined from the table for processing curved and multibore wells when measuring curvature elements at 10 m intervals [2], using the values \(\Delta x\) for this purpose. In doing so, the values \(\beta_1\) must be found in the column of azimuths \(A\), and the values \(\Theta_1\) in the column \(\Theta\). It is of interest to note that the distance from the center of the projection of the hemisphere to the projection of the point of intersection with it of the borehole axis will be somewhat greater and is determined by the relation

\[ r_{\mathrm{sk}} = 10 \sin \Theta_1. \tag{6} \]

1. The value \(r_1\) is laid off on the first straight line (radius). Through the point obtained, a straight line is drawn perpendicular to this radius, and along it are laid off the major semiaxes of the ellipse, determined by the relation

\[ a_1 = 10 \cdot \sin \beta_1. \tag{7} \]

1. Along the initial radius, the minor semiaxes of the first ellipse are laid off, their values being determined by the relation

\[ b_1 = 10 \cdot \sin \beta_1 \cdot \cos \Theta_1. \tag{8} \]

For their determination, the tables of the ten-meter interval [2] may also be used; only in this case the values \(\Theta_1\) must be found in the columns of azimuths \(A\), and the values \(\beta_1\) in the columns \(\Theta\). The answers are still found in the columns \(\Delta x\).

1. The first ellipse is constructed from the points obtained.

2. In an analogous manner, the second ellipse is constructed; it must represent the projection of the circle drawn on the sphere around the point where the second wellbore emerges onto its surface.

3. On the same projection of the hemisphere, counting from the zero division of the limb clockwise, a third radius is plotted—the radius corresponding to the azimuth of inclination of the line of the oblique section \(A_k\), on which, in centimeters, a segment corresponding to the angle of inclination of this line is laid off,

\[ r_k = 10 \sin \alpha_k \tag{9} \]

The measurement is made from the center toward the periphery. The point obtained is marked with an asterisk.

1. Perpendicular to the third radius, a third diameter is drawn through the center of the entire system. The points at which it meets the circle are marked with triangles.

2. Through the points obtained (the asterisk and the triangles), a third ellipse is drawn; its major semiaxis must be equal to 10 cm, and its minor semiaxis must be expressed by relation (9).

3. The point of intersection of the three ellipses will give us the attitude elements of the bed. Namely, the dip angle of the bed, \(\alpha\), can be determined from the distance of this point from the center of the circle \((r_{\mathrm{п}})\), in centimeters,

\[ \sin \alpha_{\mathrm{п}} = \frac{r_{\mathrm{п}}}{10}. \tag{10} \]

To determine the dip azimuth, one must draw a fourth radius through the point of intersection of the three ellipses and read the value of the division at its end on the limb.

Note. If the third ellipse does not intersect either of the two points of intersection of the first two ellipses, this will mean that the attitude elements of the bedding planes in the cores are not entirely consistent with their attitude elements if the determination is carried out from three points spaced apart from one another. In such a case, as the initial point one should take that point of intersection of the first two ellipses which is least distant from the third, and draw through it a normal to the latter. The point of intersection of the normal with the third ellipse will most accurately express the attitude elements of the bed of interest to us.

If the first two ellipses do not intersect at all, or if both points of their intersection are equally distant from the third ellipse, this will mean that the attitude elements of the bed at the core-sampling points are completely different from one another; therefore they should not be used to determine the general direction of dip of the bed. Incidentally, this is not a shortcoming of the method, but rather its advantage, since it saves the operator from erroneous conclusions.

D. Fisher [6], N. F. Frolov and E. F. Frolov [4], and many other authors have dealt with the problem of unoriented core; however, for this purpose they used only two unoriented cores, as a result of which they generally obtained a multivalued solution and only exceptionally a unique one. In the problem considered here, a third element of the bed marker in both boreholes was used. This ensures uniqueness of the solution and much greater accuracy.

REFERENCES

1. L. A. Pukhlyakov. Multi-hole wells as a means of identifying deep structures. Izv. TPI, vol. 127, issue 2, 1965.

2. L. A. Pukhlyakov. Tables for processing deviated and multi-hole wells when measuring curvature elements at intervals of 10 and 25 meters. Izd. TGU, 1966.

3. I. N. Ushakov. Mining Geometry (Subsurface Geometry). Gosgortekhizdat, 1962.

4. N. F. Frolov, E. F. Frolov. Geological Observations and Constructions in Drilling Deviated Wells. Gostoptekhizdat, 1957.

5. A. N. Khramov. Paleomagnetic Correlation of Sedimentary Sequences. Gostoptekhizdat, 1958.

6. D. J. Fisher. Drillhole Problems in the Stereographic Projection. Economic Geology, vol. XXXVI, No. 5, 1941.