Analytical Justification of a Method for Determining the Attitude Elements of Rock Strata from Non-Oriented Cores on a Wulff Net
Abstract Generated abstract
The paper provides an analytical basis for determining the attitude of uniformly bedded rock strata from unoriented drill cores taken in differently oriented borehole intervals. It formulates the problem using the known orientations of borehole axes and measured apparent dip angles in cores, expressing possible bedding normals as solutions to systems of plane equations constrained to the unit sphere. The analysis shows when two or three cores yield unique or multivalued solutions, explains the role of determinant conditions and sign ambiguity, and identifies how information on normal or overturned bedding and drilling sequence can reduce formal ambiguity. It concludes that the resulting constructions can be carried out graphically on a Wulff net by locating the common intersection points corresponding to the pole of bedding.
Full Text
Analytical Justification of a Method for Determining the Attitude Elements of Rocks from Unoriented Cores on a Wulff Net
V. V. Nikolaev
(Presented by the scientific seminar of the departments of geology and exploration of mineral deposits, petrography, and mineralogy)
The weak substantiation and often low accuracy of structural constructions based on exploratory drilling materials are due, to a significant extent, to the fact that the geological documentation of boreholes is carried out mainly on unoriented core, which excludes the possibility of direct measurement of the bedding elements of stratification. Let us evaluate the possibility of solving this problem from unoriented cores taken in differently oriented intervals of one or several boreholes, for the case of uniform bedding of a bedded sequence.
The initial data for solving the problem are the orientations of the axes of the boreholes at the points where the cores were taken and the apparent dip angles of the beds in the cores (the angles between the bedding plane and the plane perpendicular to the core axis). Using the formula for determining the angle between planes [1, p. 212], we write the following series of equations:
\[ \pm \cos \omega_1 = \cos \alpha \cos \alpha_1 + \cos \beta \cos \beta_1 + \cos \gamma \cos \gamma_1, \]
\[ \pm \cos \omega_2 = \cos \alpha \cos \alpha_2 + \cos \beta \cos \beta_2 + \cos \gamma \cos \gamma_2, \]
\[ \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \tag{1} \]
\[ \pm \cos \omega_n = \cos \alpha \cos \alpha_n + \cos \beta \cos \beta_n + \cos \gamma \cos \gamma_n, \]
where $\alpha, \beta, \gamma$ are the angles formed by the normal to the stratification, and $\alpha_i, \beta_i, \gamma_i$ by the axes of the boreholes, with the positive directions of the coordinate axes; $\omega_i$ are the apparent dip angles of the beds in the cores of the corresponding boreholes. It is easy to see that these equations are the normal equations of planes perpendicular to the axes of the boreholes at the core-sampling points, with polar angles $\alpha_i, \beta_i, \gamma_i$ and polar distances $\pm \cos \omega_i$.
Considering equations (1) as elements of a certain series, we find all their combinations taken $n$ at a time. We obtain $C_{2n}^{n}$ systems of equations with three unknowns: $\cos \alpha$, $\cos \beta$, $\cos \gamma$, which determine the orientation of the sought vector (the normal to the stratification), for which, with obvious necessity, the condition must be satisfied
\[ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1 \tag{2} \]
(the equation of a sphere of unit radius). The solutions of the systems of equations composed, satisfying condition (2), will be the answer to the problem posed.
In investigating the problem, without diminishing generality, we shall restrict ourselves to the cases of solving it from three and from two unoriented cores. Having data for three boreholes (cores), one can compose six equations of planes, from which 20 systems can be formed, with three equations in each. We note that in some systems (there are 12 of them) two of the three equations contain identical coefficients at the unknowns. Obviously, these systems cannot have a solution (two parallel planes are intersected by a third) and must be excluded from further consideration.
Let us interrupt for the time being the investigation of the solutions of the eight remaining systems of equations, corresponding to the general solution of the problem, and examine the possible solutions of one of these systems, for example:
\[ \left\{ \begin{aligned} \cos \omega_1 &= \cos \alpha \cos \alpha_1 + \cos \beta \cos \beta_1 + \cos \gamma \cos \gamma_1,\\ \cos \omega_2 &= \cos \alpha \cos \alpha_2 + \cos \beta \cos \beta_2 + \cos \gamma \cos \gamma_2,\\ \cos \omega_3 &= \cos \alpha \cos \alpha_3 + \cos \beta \cos \beta_3 + \cos \gamma \cos \gamma_3,\\ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma &= 1. \end{aligned} \right. \tag{3} \]
On the basis of the investigation of a system of three equations with three unknowns [1, p. 144], we arrive at the conclusion that the solution of system (3) is exhausted by two cases.
- Three planes have one common point lying on the sphere. This is possible if the determinant of the system of equations of the three planes is different from zero, that is,
\[ \Delta = \begin{vmatrix} \cos \alpha_1 & \cos \beta_1 & \cos \gamma_1\\ \cos \alpha_2 & \cos \beta_2 & \cos \gamma_2\\ \cos \alpha_3 & \cos \beta_3 & \cos \gamma_3 \end{vmatrix} \ne 0. \tag{4} \]
The inequality \(\Delta \ne 0\) means that the normal vectors of the planes are not coplanar. The coordinates of the common point of the three planes (they, of course, also satisfy the equation of the sphere, since \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\)) are determined by the formulas
\[ \cos \alpha = \frac{\Delta_1}{\Delta}, \qquad \cos \beta = \frac{\Delta_2}{\Delta}, \qquad \cos \gamma = \frac{\Delta_3}{\Delta}, \]
where
\[ \Delta_1 = \begin{vmatrix} \cos \omega_1 & \cos \beta_1 & \cos \gamma_1\\ \cos \omega_2 & \cos \beta_2 & \cos \gamma_2\\ \cos \omega_3 & \cos \beta_3 & \cos \gamma_3 \end{vmatrix}, \qquad \Delta_2 = \begin{vmatrix} \cos \alpha_1 & \cos \omega_1 & \cos \gamma_1\\ \cos \alpha_2 & \cos \omega_2 & \cos \gamma_2\\ \cos \alpha_3 & \cos \omega_3 & \cos \gamma_3 \end{vmatrix}, \]
\[ \Delta_3 = \begin{vmatrix} \cos \alpha_1 & \cos \beta_1 & \cos \omega_1\\ \cos \alpha_2 & \cos \beta_2 & \cos \omega_2\\ \cos \alpha_3 & \cos \beta_3 & \cos \omega_3 \end{vmatrix}. \tag{5} \]
Rectangular and spherical coordinates are related by the relations
\[ \cos \alpha = \sin \vartheta \cos \lambda,\qquad \cos \beta = \sin \vartheta \sin \lambda,\qquad \cos \gamma = \cos \vartheta, \]
where \(\vartheta\) is the dip angle, and \(\lambda\) is the dip azimuth of the rocks. Hence,
\[ \vartheta = \gamma,\qquad \sin \lambda = \frac{\cos \beta}{\sin \vartheta}. \]
- Three planes intersect along one straight line, having one or two common points with the sphere.
Such cases are possible if all third-order determinants of the matrix
\[ \begin{pmatrix} \cos\alpha_1 & \cos\beta_1 & \cos\gamma_1 & \cos\omega_1\\ \cos\alpha_2 & \cos\beta_2 & \cos\gamma_2 & \cos\omega_2\\ \cos\alpha_3 & \cos\beta_3 & \cos\gamma_3 & \cos\omega_3 \end{pmatrix} \]
are equal to zero, but at least one of the minors of determinant (4) is nonzero.
Since two of the three planes intersecting along one straight line determine the spatial position of the latter, system (3) in the present case may be represented as a system of equations of a sphere and two planes. Expressing \(\cos\alpha\) and \(\cos\beta\) from the equations of the planes and substituting their values into the equation of the sphere, we obtain a quadratic equation with one unknown, \(\cos\gamma\). If, in solving this equation, the discriminant proves greater than zero, the equation will have two distinct real roots; if the discriminant proves equal to zero, the equation will have one real root. Cases in which the discriminant is less than zero are excluded, since the rectangular coordinates of the points are expressed by real numbers. Thus, system (3) may have a two-valued or a unique solution.
We do not give the solution of the system of equations of the sphere and two planes in general form, in view of the obvious simplicity and the unwieldiness of the computations. We note only that, in the case under consideration, the problem has a unique solution if the borehole axes and the normal to the bedding are coplanar (Fig. 1), and a two-valued solution if the latter are not coplanar, while the borehole axes are coplanar.
The case in which three planes have no common point (the equations are inconsistent) would be possible under the condition that determinant (4) is equal to zero and at least one of its minors is nonzero, if at the same time one of the third-order determinants (5) were nonzero. The latter is excluded:* if determinant (4) is equal to zero, and
\[ \cos\omega_i=\cos\alpha \cos\alpha_i+\cos\beta \cos\beta_i+\cos\gamma \cos\gamma_i, \]
then all third-order determinants (5) are equal to zero. The case in which three planes coincide or are parallel (the determinant and all its minors are equal to zero) is also impossible, since, by the condition of the problem, the cores were taken from differently oriented intervals of the boreholes.
Let us continue the study of the general solution of the problem. Comparing the solutions of the eight remaining systems of equations, we observe that the solutions of the corresponding pairs of systems differ only in signs. Thus, if system (3) has, for example,
\[ \Delta_3= \begin{vmatrix} \cos\alpha_1 & \cos\beta_1 & \cos\omega_1\\ \cos\alpha_2 & \cos\beta_2 & \cos\omega_2\\ \cos\alpha_3 & \cos\beta_3 & \cos\omega_3 \end{vmatrix}, \]
then the system
\[ \begin{cases} -\cos\omega_1=\cos\alpha' \cos\alpha_1+\cos\beta' \cos\beta_1+\cos\gamma' \cos\gamma_1,\\ -\cos\omega_2=\cos\alpha' \cos\alpha_2+\cos\beta' \cos\beta_2+\cos\gamma' \cos\gamma_2,\\ -\cos\omega_3=\cos\alpha' \cos\alpha_3+\cos\beta' \cos\beta_3+\cos\gamma' \cos\gamma_3,\\ \cos^2\alpha'+\cos^2\beta'+\cos^2\gamma'=1 \end{cases} \]
has
\[ \Delta'_3= \begin{vmatrix} \cos\alpha_1 & \cos\beta_1 & -\cos\omega_1\\ \cos\alpha_2 & \cos\beta_2 & -\cos\omega_2\\ \cos\alpha_3 & \cos\beta_3 & -\cos\omega_3 \end{vmatrix}. \]
Consequently, \(\cos \gamma = -\cos \gamma'\). Similarly, \(\cos \beta = -\cos \beta'\), \(\cos \alpha = -\cos \alpha'\). The solutions of such pairs of systems of equations geometrically represent opposite vectors that uniquely determine the orientation of the bedding. On this basis we conclude that, for certain relationships among the angles appearing in equations (3), the problem may have a one-, two-, three-, or four-valued solution.
Without repeating the entire logical chain of the preceding arguments, let us determine the possible number of solutions of the problem from two cores. In the present case, obviously, four equations of planes perpendicular to the axes of the boreholes can be written, and from them four pairs of equations can be formed. Each such pair of equations, taken with equation (2), forms a system of three equations with three unknowns. The solutions of two of the four systems will differ from the solutions of the other two only in sign. Each of these two systems, as was proved (case 2), may have a one- or two-valued solution. Consequently, the problem of determining the attitude elements of rocks from two and three cores (the second group of cases) may have a one-, two-, three-, or four-valued solution.
Fig. 1. Geometric form of the relation between the orientation of boreholes, the apparent dip angles of beds in cores, and the attitude elements of rocks.
The axes of the boreholes (cores) as vectors are reduced to a common origin (shown by dash-dotted lines); \(C_1, C_2, C_3\) are the points of their intersection with the upper hemisphere; \(\omega_1, \omega_2, \omega_3\) are the acute apparent dip angles of the beds in the cores of the corresponding boreholes; \(p_1, p_2, p_3\) are the polar distances of the planes perpendicular to the borehole axes \((p_i = \pm \cos \omega_i)\). Point \(H\) is the pole of bedding.
The multivaluedness of the solution of the problem in a number of cases may prove to be partly or wholly formal. The formal multivaluedness of the solution is due to the fact that in cores the acute apparent dip angles of beds are always measured. We often do not know whether we are dealing with normal or overturned bedding of rocks and in what sequence ...
...each borehole intersects the beds (in the direction from the roof to the floor or in the opposite direction). If it proves possible to obtain answers to these questions, the formal multivaluedness of the solution of the problem (but not multivaluedness in general, in the general case) can be eliminated.
Let us consider the following example (Fig. 1). It is known that the bedding of the rocks is normal and that borehole \(C_1\) intersects the beds in the direction from the roof to the floor. Borehole \(C_3\) intersects the beds in the reverse sequence, which can be established from indications of normal and overturned bedding of the beds in the cores or on the basis of correlating sections along the boreholes. In the cores of boreholes \(C_1\) and \(C_3\), the acute apparent dip angles of the beds, equal respectively to \(\omega_1\) and \(\omega_3\), were measured. From Fig. 1 it is seen that the problem formally has a three-valued solution (points \(H\), \(H_1\), and \(H_2\) on the sphere), while its actual solution is the point \(H\), that is, the solution of the system
\[ \left\{ \begin{aligned} + \cos \omega_1 &= \cos \alpha \cos \alpha_1 + \cos \beta \cos \beta_1 + \cos \gamma \cos \gamma_1,\\ - \cos \omega_3 &= \cos \alpha \cos \alpha_3 + \cos \beta \cos \beta_3 + \cos \gamma \cos \gamma_3,\\ \cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma &= 1. \end{aligned} \right. \]
Obviously, in order to eliminate the formal multivaluedness of the solution of the problem, it is necessary to set up and solve such a system in which, in the equations of the planes, the constant terms would be taken with the corresponding signs. The rule for determining these signs may be formulated as follows: planes perpendicular to the axes of boreholes that intersect the bed in the direction from the roof to the floor must have a positive polar distance, while planes perpendicular to the axes of boreholes that intersect the beds in the reverse sequence must have a negative polar distance.
The solution of the system (or systems) of equations set up from specific initial data can be obtained graphically. The geometrical meaning of the equations is clear from the preceding discussion. The graphical solution of this system reduces to finding the pole of the bedding of the rocks as the common point of circles that are the lines of intersection of planes with polar angles \(\alpha_i, \beta_i, \gamma_i\) and polar distances \(\pm \cos \omega_i\) with the upper hemisphere of a sphere of unit radius. The necessary constructions and the determination of the bedding elements of the rocks can readily be performed in stereographic projection with the aid of a Wulff net.
REFERENCES
- Privalov I. I. Analytical Geometry. 29th ed., stereotyped. Nauka, Moscow, 1964.