On the Methodology for Quantitative Assessment of Tectonic Fracturing

Yu. N. Popov

(Presented by Professor, Doctor K. V. Radugin)

The methodology proposed below for the quantitative assessment of tectonic fracturing has found application in the study of the latter in the faces of mine workings and quarries, as well as in natural outcrops of rocks. It differs from existing methodologies by simplifying the calculations, which is highly significant in the quantitative assessment of fracturing under field conditions.

As the principal parameter characterizing a fracture system quantitatively, the averaged true distance between the fractures of the system is adopted. With an uneven face or outcrop surface and small distances between fractures, it is sometimes possible to measure the true distances between the surfaces of fractures of the system. In most cases, however, one can observe traces of fractures intersecting the surface of the face or outcrop (the exposure surface). In this case the shortest distances between the traces of fractures of an individual system are measured in meters. Determining the true distances between fractures requires recalculation in order to introduce a correction for the arbitrary position of the exposure surface relative to the fracture surface.

Most faces of mine workings are vertical or nearly vertical; vertical surfaces of rock outcrops are also often encountered. Under these conditions, the conversion of visible distances between fractures into true distances will have the following form.

In Fig. 1a, a cube with vertical faces is shown. Suppose that this cube is cut from the rock mass with the intention that the face \(AOA'O'\) should be the exposure surface, on which the shortest visible distances between the traces of fractures of the system \((\rho_{\text{vis}})\) have been measured. It is necessary to determine the true distances between the fractures of this system \((\rho)\).

Fig. 1. a — method for measuring the apparent distances \((\rho_{\mathrm{vis}})\) between fractures of a system; b — nature of the dependence of the apparent \((\rho_{\mathrm{vis}})\) and true \((\rho)\) distances between fractures of a system on the mutual orientation of the fractures and the exposure surface.

From Fig. 1, a it is evident that at the vertex \(O\) the fracture plane cuts out a triangular pyramid \(aOcd\), whose base is \(aOc\) and whose two faces \(aOd\) and \(cOd\) are right triangles. Let us rotate this pyramid about the side of the base \(ac\) by \(180^\circ\) into the position shown in Fig. 1, b. Now the face \(abc\) shows the position of the fracture in space; \(ac\) is the strike line of the fracture; \(\alpha\) is the dip angle of the fracture. The face \(abO\) is the exposure surface; \(\gamma\) is the angle of inclination of the trace of the fracture to the horizon; \(w\) is the difference between the azimuths of the strikes of the fracture and of the exposure surface (the acute angle).

From the right triangle \(mOb\),

\[ Ob=\frac{mO}{\cos\gamma}=\frac{\rho_{\text{vis}}}{\cos\gamma}; \]

from the right triangle \(jOb = Ob \cdot \cos\alpha\);

\[ \rho=\rho_{\text{vis}}\frac{\cos\alpha}{\cos\gamma}; \tag{1} \]

The angle \(\gamma\) can be determined by the known formula used to determine the dip angle in an oblique section:

\[ \operatorname{tg}\gamma=\operatorname{tg}\alpha\cdot \sin w. \tag{2} \]

Proceeding from this, let us consider the following possible special cases:

a) the strikes of the exposure surface and of the fracture coincide, i.e. \(w=0^\circ\) and \(\gamma=0^\circ\)

\[ \rho=\rho_{\text{vis}}\cdot \cos\alpha; \]

b) the strikes of the fracture and the outcrop are perpendicular, i.e. \(w=90^\circ\); then \(\operatorname{tg}\gamma=\operatorname{tg}\alpha\), \(\angle\gamma=\angle\alpha\);

\[ \rho=\rho_{\text{vis}}; \]

c) the fracture is horizontal, i.e. \(\alpha=0^\circ,\ \gamma=0^\circ\);

\[ \rho=\rho_{\text{vis}}; \]

d) the fracture is vertical, i.e. \(\alpha=90^\circ,\ \gamma=90^\circ\); in this case

\[ \rho=\rho_{\text{vis}}\cdot \sin w. \]

For a horizontal exposure surface:

\[ \rho=\rho_{\text{vis}}\cdot \sin\alpha. \]

Formula (1) is considerably simpler than those proposed by a number of authors [1, 4, 5, and others], but calculations using it require determination of the angle \(\gamma\), which can be measured directly in the field and also determined from tables or on a Wulff net during office processing of the observational results. If one takes into account that at a single observation point for fracturing the number of fracture systems rarely exceeds five to seven, then determination of the true distances between fractures by formula (1), using a slide rule, will take no more than 5–7 min.

It should be noted that calculations using formula (1) are advisable if the dip angles lie within the range \(10\text{–}80^\circ\); when their values are greater or less than the indicated limits, the formulas used for the special cases are valid. The error in this case will be within the limits of the accuracy of the measurements.

The problem of determining the true distances between fractures becomes substantially more complicated if the exposure surface is inclined to the horizon at some angle.

In this case, methods are used for graphical, graphico-analytical, and analytical determination of the true distances between fractures of a system. They are based on determining the magnitude of the dihedral angle between the fracture and exposure surfaces, with subsequent intro-

with the introduction of a correction into the measured shortest distance between fracture traces \((\rho_{\text{vis}})\).

As is known, the magnitude of the dihedral angle formed by two planes is measured in a plane perpendicular to these planes, i.e., perpendicular to the line of their intersection. In our case, the line of intersection of the planes of the fracture and of the exposure is the trace of the fracture on the exposure surface; consequently, the lines of the shortest visible distances between fractures, situated along the normal to

Fig. 2. \(a\) — relationship between the measured and true distances between fractures in a plane perpendicular to the surfaces of the fracture and the exposure; \(b\) — position of a fracture in a rectangular coordinate system.

traces, lie in a plane where the magnitude of the dihedral angle is determined—\(\varphi\). Figure 2,a shows a section of a rock block by such a plane. It follows from the figure that:

\[ \rho=\rho_{\text{vis}}\cdot \sin\psi . \tag{3} \]

The magnitude of the dihedral angle \(\varphi\) can be determined graphically in projections with numerical marks or on the Wulff net (Fig. 3). The acute angle is used in the calculation.

Fig. 3. Graphical determination of the true distances between fractures in a projection with numerical marks: \(a\)—scale of inclinations; \(b\)—determination of the magnitude of the dihedral angle \(\varphi\) between the fracture plane—T and the exposure—E; \(v\)—determination of \(\rho\) from \(\rho_{\text{vis}}\) and \(\varphi\) and on the Wulff net; \(g\)—determination of the angle \(\varphi\); \(d\)—determination of \(\rho\).

To determine analytically the magnitude of the angle between two planes specified by their attitude elements, it is necessary to find the equations of these planes and then, using the known formulas of analytic geometry, determine the angle \(\varphi\). The equation of a plane (dip azimuth \(\Theta\), dip angle \(\alpha\)) can be written by means of the intercept form of the equation of a plane on the coordinate axes, under the condition that the \(XOY\) plane of the rectangular coordinate system is horizontal and the \(x\)-axis is oriented along the meridian (Fig. 2, b).

The equation of a plane in intercepts on the axes has the form [3]

\[ \frac{x}{a}+\frac{y}{b}+\frac{z}{c}=1, \]

where \(a, b, c\) are the intercepts on the corresponding coordinate axes. From Fig. 2, b it is evident that the intercepts on the axes cut off by the plane \(ACB\) can be found from the three right triangles \(AOD\), \(COD\), \(BOD\), which have the common side \(OD\), which may be taken as unity.

\[ AO=a=\frac{OD}{\cos\beta}; \qquad BO=b=\frac{OD}{\cos\gamma}; \qquad OC=c=OD\,\operatorname{tg}\alpha. \]

But \(\beta=180^\circ-\Theta,\quad \gamma=\Theta-90^\circ,\quad OD=1\); therefore

\[ a=\frac{1}{\cos\Theta}; \qquad b=\frac{1}{\sin\Theta}; \qquad c=\operatorname{tg}\alpha. \]

Substituting the obtained values into the equation of the plane, we obtain

\[ x\cdot \cos\Theta+y\cdot \sin\Theta+\frac{z}{\operatorname{tg}\alpha}=1. \]

In this way, taking into account the signs of the intercepts on the axes, one can write the equation of any plane. If the value of the intercept \(z\) is always positive, then the signs of the intercepts \(x\) and \(y\) vary depending on the dip azimuths of the plane (Table 1). After reducing the equation of the plane to normal form, we obtain

Table 1

Relationship between the orientation of planes and the signs of the intercepts cut off by them on the coordinate axes

\[ \cos\Theta\cdot \operatorname{tg}\alpha x+\sin\Theta\cdot \operatorname{tg}\alpha y+z-\operatorname{tg}\alpha=0. \]

The angle between two intersecting planes is determined by the formula [3]

\[ \cos\varphi=\pm \frac{A\cdot A+B\cdot B+C\cdot C} {\sqrt{A_1^2+B_1^2+C_1^2}\sqrt{A_2^2+B_2^2+C_2^2}}. \]

In our case \(A=\cos\Theta\cdot \operatorname{tg}\alpha,\quad B=\sin\Theta\,\operatorname{tg}\alpha,\quad C=1\). Then the angle between two planes specified by the attitude elements \(\Theta_1,\ \alpha_1\) and \(\Theta_2,\ \alpha_2\), after a series of transformations, will be determined by the formula

\[ \cos\varphi=\pm(\cos\Theta_1\cdot \cos\Theta_2+\sin\Theta_1\cdot \sin\Theta_2)\sin\alpha_1\cdot \sin\alpha_2+\cos\alpha_1\cdot \cos\alpha_2. \tag{4} \]

True distances between fractures \((\rho)\) can be determined by substituting the value \(\varphi\) into formula (3),

\[ \rho=\rho_{\mathrm{vis}}\cdot \sin \left[\arccos \pm \left(\cos \Theta_{1}\cdot \cos \Theta_{2}+\sin \Theta_{1}\cdot \sin \Theta_{2}\right)\sin \alpha_{1}\cdot \sin \alpha_{2}+\right. \tag{5} \]

\[ \left. +\cos \alpha_{1}\cdot \cos \alpha_{2}\right], \]

or, knowing the value of \(\cos \varphi\),

\[ \rho=\rho_{\mathrm{vis}}\sqrt{1-\cos^{2}\varphi}. \tag{6} \]

The analytical solution of the problem, as we see, is rather laborious, but it may be used to check graphical determinations of the true distances between fractures of a system.

In the literature, in the quantitative assessment of fracturing, use is usually made of fracture frequency—the number of fractures of a definite system falling on an oriented segment of definite length. We have taken as the frequency \((n)\) the number of fractures of a system intersecting a straight-line segment, perpendicular to them, of length \(1\ m\).

\[ n=\frac{1}{\rho}. \]

The aggregate of systems studied at a point of observation for fracturing is quantitatively characterized by the specific density of fracturing, which shows how many fractures of different systems are intersected by a sphere \(1\ m\) in diameter. The specific density is equal to the sum of the frequencies of all fracture systems represented at the observation point [2].

\[ N=\sum_{i=1}^{i=n} n_i \qquad \text{or} \qquad N=\sum_{i=1}^{i=n}\frac{1}{\rho_i}, \]

where \(N\) is the specific density of fractures, \(n_i\) is the frequency of fractures of a definite system, and \(\rho_i\) is the true distance between fractures of an individual system.

LITERATURE

1. V. A. Bukrinsky, V. V. Mikhailova. Study of the relation of fracturing to tectonic structures of rocks. Moscow, 1963.

2. V. M. Vorob’ev. On the methodology for the quantitative assessment of fracturing of rocks. Proceedings of the Second All-Union Conference on the Problem of Fractured Oil Reservoirs. 1965.

3. M. Ya. Vygodsky. Handbook of Higher Mathematics. Fizmatgiz, 1963.

4. M. N. Polyansky. Mathematical substantiation of a method for studying macrofracturing of sedimentary rocks. Izv. vuzov, Geol. i razv., 1963, No. 12.

5. E. M. Smekhov. Fractured rocks and their reservoir properties. Proceedings of VNIGRI, issue 132, 1959.