ON THE RELATIONSHIP BETWEEN THE ANISOTROPY COEFFICIENTS OF THE ELASTIC AND THERMAL-CONDUCTIVITY PROPERTIES OF ROCKS

Yu. L. Boyarko

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

Rocks, for the most part, possess anisotropy (inequality of properties in different directions within the body under study), which is due to a certain orientation of the minerals composing them.

The anisotropy of the mechanical properties of rocks has been noted by a number of researchers who have studied the physicomechanical properties of rocks, and it is also taken into account and used in the mining and stone-working industries. For example, it is known that anisotropic rocks split easily in one direction—along schistosity—whereas they are drilled much more easily in another direction—perpendicular to schistosity. The latter quality, namely the ease of drilling in the direction perpendicular to schistosity or bedding, as compared with the direction parallel to schistosity or bedding, is presently taken into account by the Unified Time Standards for drilling (UTS for drilling operations, 1963) through the use of coefficients. The authors of the UTS take into account the decrease in drilling speed of boreholes with increasing dip angle of the rocks and increase the time standards by introducing a correction coefficient, with numerical values from 1.05 to 1.20, which also depends on the drillability category of the rock. However, from observations and studies by a number of authors there follows the proposition that the anisotropy of hardness is strongly dependent on the degree of dynamometamorphism [5]. The more metamorphosed the rocks, the more distinctly expressed is the anisotropy of a number of properties of these rocks.

Table 1, for example, gives data obtained by the author in studies to determine the drillability of rock in different directions relative to bedding or schistosity, which were carried out under laboratory conditions.

An analysis of the data in Table 1 shows that the difference in drillability in different directions relative to the plane of schistosity is very variable and depends on the degree of metamorphism of the rocks. Thus, mica schists, granitic gneiss, and amphibolite, which contain high-temperature anhydrous minerals, belong to strongly metamorphosed rocks. Accordingly, their drilling anisotropy index is very high. Argillites, clay shales, and hard coal belong to weakly or almost unmetamorphosed rocks. Their drilling anisotropy index is correspondingly low.

In connection with this, the introduction of a general correction coefficient into drilling-time norms, taking into account the magnitude of the angle at which the borehole axis meets the plane of schistosity, is only the first step toward improving output and time standards in this direction.

Table 1

The anisotropy of the hardness of rocks, moreover, has an enormous influence on borehole deviation, for which published materials are available [2, 3].

All this indicates that, in prospecting deposits, it is necessary to study carefully the mechanical properties of rocks from the standpoint of their anisotropy, so that these data may subsequently be used in combating borehole deviation, in directional drilling, and in determining the actual drillability of rocks.

The anisotropy of the mechanical properties of rocks is characterized by differences in such indices as Young’s modulus, Poisson’s ratio, and others, when these are determined in different directions in specimens relative to the plane of schistosity. Most often, the ratio of Young’s modulus determined in the rock along the plane of schistosity, $E_{\parallel}$, to Young’s modulus determined in the direction perpendicular to schistosity, $E_{\perp}$, is taken as the index of anisotropy of mechanical properties [7].

However, determining Young’s modulus and other physicomechanical indices of rocks is associated with considerable difficulties, since it requires a rather well-equipped laboratory facility and is quite laborious in terms of time expenditure. Suffice it to say that, with any method of determining the modulus of elasticity—static or dynamic—it is necessary to prepare specimens of special shape

and dimensions, which is often difficult. It is especially difficult to carry out such determinations under field conditions.

However, when it is a question of determining only the index of anisotropy of the mechanical properties of rocks, or, as it may also be called, the coefficient of elastic anisotropy \(A_m = \dfrac{E_{\parallel}}{E_{\perp}}\), one can use other vector properties of the rock, whose method of determination is many times simpler, and then establish between them a correlational or analytical dependence.

The vector, or, as they are also called, tensor properties of rocks include, for example, thermal conductivity. In that case, if it is necessary to determine only the coefficient of anisotropy of the thermal conductivity of a rock, it is sufficient to carry out the following operations. A rock specimen taken from a core obtained in a borehole or from the face of a working is ground on a surface perpendicular to the schistosity, bedding, or some other \(S\)-surfaces. Then this surface is heated to a temperature of \(70\text{--}90^\circ\mathrm{C}\) and covered with a thin layer of melted paraffin. After the specimen has cooled to normal temperature, a heated metal rod (best of all an electric soldering iron) with a spherical tip is pressed tightly against the ground surface. It is held for 3–5 minutes, as a result of which the rock is heated at the point of contact with the metal, and the heat spreading through it melts the paraffin. If the rock is anisotropic, then the propagation of heat takes place in all directions at different rates. Heat spreads more rapidly in the direction of greater density. As a result, an ellipse of melted paraffin is formed on the surface of the specimen (Fig. 1). The lengths of the axes of the ellipse are measured. The ratio of the length of one axis to the other gives the coefficient of anisotropy of the rock’s thermal conductivity. Its relation to the coefficient of elastic deformation is as follows.

Fig. 1. Ellipse of thermal conductivity on the surface of the investigated specimen of anisotropic rock

All thermal properties of solid nonmetallic bodies, according to Debye’s theory, can be related to a system of acoustic standing waves, whose propagation velocity depends on the modulus of elasticity of the medium under consideration [6].

Having established an analogy between molecular heat transfer in gases and thermal conductivity in solid nonmetallic bodies by means of phonons, Debye derived the equation

\[ \lambda = \frac{1}{4} c \cdot u \cdot \bar{l}, \tag{1} \]

where \(\lambda\) is the coefficient of thermal conductivity,
\(c\) is the heat capacity,
\(u\) is the speed of sound,
\(\bar{l}\) is the mean free path of phonons.

This equation is applicable both to crystalline and to amorphous nonmetallic materials.

On the other hand [2], the coefficient of thermal conductivity is determined by the equation

\[ \lambda = a \cdot \rho \cdot c, \tag{2} \]

where
\(a\) is the coefficient of thermal diffusivity,
\(\rho\) is density,
\(c\) is heat capacity,

For the case of an anisotropic material we have

\[ \left. \begin{aligned} \lambda_{\parallel} &= 1/4 \cdot c \cdot u_{\parallel} \cdot \bar{l}_{\parallel} \\ \lambda_{\perp} &= 1/4 \cdot c \cdot u_{\perp} \cdot \bar{l}_{\perp} \end{aligned} \right\} \tag{3} \]

\[ \left. \begin{aligned} \lambda_{\parallel} &= a_{\parallel} \cdot \rho \cdot c \\ \lambda_{\perp} &= a_{\perp} \cdot \rho \cdot c \end{aligned} \right\}, \tag{4} \]

where the quantities \(c\) and \(\rho\) are scalar, while the quantities \(\lambda\), \(u\), \(c\), and \(a\) are vectorial, i.e., dependent on the direction in which they are determined.

The velocity of propagation of sound oscillations in a rock is determined from the equation [4]

\[ u = \sqrt{\frac{E}{\rho}}, \tag{5} \]

where
\(E\) is Young’s modulus,
\(\rho\) is the density of the rock.

For an anisotropic medium we have

\[ \left. \begin{aligned} u_{\parallel} &= \sqrt{\frac{E_{\parallel}}{\rho}} \\ u_{\perp} &= \sqrt{\frac{E_{\perp}}{\rho}} \end{aligned} \right\}. \tag{6} \]

Using the initial equations (1), (2), and (5), we can write

\[ \lambda = 1/4 \cdot \sqrt{\frac{E}{\rho}} \cdot c \cdot l, \]

whence

\[ E = \frac{16 \cdot \lambda^{2} \cdot \rho}{c^{2} \cdot \bar{l}^{2}}, \]

or, replacing \(\lambda\) by expression (2),

\[ E = \frac{16 \cdot a^{2} \cdot \rho^{3}}{\bar{l}^{2}}. \tag{7} \]

For an anisotropic medium, using equations (3), (4), and (6), we can write the relation

\[ \frac{E_{\perp}}{E_{\parallel}} = \frac{\bar{l}_{\parallel}^{2} \cdot a_{\perp}^{2}} {\bar{l}_{\perp}^{2} \cdot a_{\parallel}^{2}}. \tag{8} \]

The velocity of propagation of a thermal wave depends on a number of factors, such as thermal conductivity, heat capacity, thermal diffusivity, etc.

In the present case we are interested only in those factors that possess vector properties, i.e., have different magnitudes in one and the same anisotropic specimen when they are determined along different directions. In equation (7), such properties are possessed by the thermal diffusivity \(a\) and the mean free path of phonons \(l\).

In view of the fact that the method involves determining only the anisotropy coefficient of the mechanical properties of rocks, using the above-described and theoretically substantiated relationship between thermal conductivity and Young’s modulus, we may replace the product \(l_{\parallel}^{2}\cdot a_{\perp}^{2}\) by \(\Theta_{\perp}\) and the product \(l_{\perp}^{2}\cdot a_{\parallel}^{2}\) by \(\Theta_{\parallel}\), calling \(\Theta\) the vector index of the thermal conductivity of the rock.

Thus, we obtain the equation

\[ \frac{E_{\perp}}{E_{\parallel}}=\frac{\Theta_{\perp}}{\Theta_{\parallel}} . \tag{9} \]

The quantities \(\Theta_{\perp}\) and \(\Theta_{\parallel}\) can be determined in dimensionless units by measuring the lengths of the axes or semi-axes of the ellipse obtained on the specimen when paraffin is melted, and by squaring these quantities.

The results of the experimental determination of the anisotropy coefficients are given in Table 2.

Table 2

The rock specimens under study must necessarily be massive, since the method is based on the use of a point source of heat that forms, in the specimen under investigation, a volumetric radial heat flow. In isotropic rocks such a flow propagates in the form of a hemisphere of radius \(R\), and on the polished surface, when paraffin is melted, a circle of radius \(R\) is formed.

In anisotropic rocks the isothermal surface of the heat flow has the form of a semi-ellipsoid. For the majority of anisotropic rocks, the proposition is valid that two mutually perpendicular vectors characterizing thermal conductivity in the plane parallel to the schistosity are equal to one another, i.e., \(\lambda_{\parallel}'=\lambda_{\parallel}''\). However, research practice shows that some rocks having a so-called trachytic texture are anisotropic in all directions, and for them the following expression is valid:

\[ \lambda_{\perp}\ne \lambda_{\parallel}'\ne \lambda_{\parallel}'' . \]

For the study of anisotropic rocks characterized by the equality \(\lambda_{\parallel}'=\lambda_{\parallel}''\) and called transversely isotropic, it is sufficient to polish

one plane perpendicular to the plane of schistosity. In this case the thickness of the specimen must be somewhat greater than the minor semiaxis of the ellipse obtained on the polished surface when the paraffin melts. Otherwise (in the case of using a thin plate), the correct geometric form of the isometric surface will be disturbed and, owing to the distorted heat distribution, errors will appear in the ellipse on the polished surface.

The heater must have a spherical surface with the minimum radius of the sphere (1–1.5 mm), in order to reduce errors arising from additional heating of the surface under investigation by convection and radiation from the heater.

To study orthotropic rocks \((\lambda_{\perp} \ne \lambda_{\parallel} \ne \lambda_{II})\), it is necessary to polish three mutually perpendicular faces, one of the planes being perpendicular both to the schistosity and to the lineation of the rock.

The errors obtained in measuring the axes of the ellipses, as well as in determining Young’s modulus on a press, give some difference in the anisotropy coefficients, as is evident from Table 2. However, for the field method such agreement is quite sufficient. Moreover, the accuracy of measurement can be increased by further improving the research procedure.

REFERENCES

1. L. I. Baron et al. Determination of the Properties of Rocks. Gosgortekhizdat, 1962.

2. Yu. L. Boyarko. “Mechanics of Failure of the Face in Anisotropic Rocks and the Process of Borehole Curvature in Them.” Izv. TPI, vol. 120, 1962.

3. A. G. Kalinin. Curvature of Boreholes. Gosgeoltekhizdat, 1963.

4. Edited by Yu. A. Nelender. Ultrasound in Construction Engineering. Gosstroiizdat, 1962.

5. Yu. A. Rozanov. “The Influence of Macrostructural Features of Rocks on the Anisotropy of Their Mechanical Properties.” Transactions of the Institute of Geological Sciences, Academy of Sciences of the USSR, issue 146, petrographic series (No. 42), 1952.

6. V. S. Chirkin. Thermal Conductivity of Industrial Materials. Mashgiz, 1962.

7. N. I. Shatsov et al. Drilling of Oil and Gas Wells. Gostoptekhizdat, 1961.

8. S. N. Shorin. Heat Transfer. Vysshaya shkola, 1964.