CALCULATION OF THE TIME TO ELECTRICAL DISCHARGE IN ROCK UNDER THE ACTION OF A LOCAL HIGH-INTENSITY ELECTRIC FIELD

A. A. VOROB'EV, V. S. DMITREVSKII

(Presented by the scientific seminar of the Department of Electrical Insulation and Cable Engineering)

In [1] the hypothesis was put forward that electric charges can accumulate in dielectric rocks of the Earth’s crust, creating high electric fields. When a local electric field reaches a value equal to the electric strength of the medium, an electrical discharge will occur in the rocks, accompanied by the release of the accumulated energy over short time intervals.

The fluctuation theory of the fracture of solids [2] makes it possible to obtain the dependence of time (durability) on the magnitude of the stress at which the fracture process develops. According to this theory, the cause of the fracture of solid materials is thermal fluctuations, i.e., local short-term increases in internal energy that cause ruptures of some chemical bonds in molecules. In the absence of external loading, with the equilibrium state maintained, the ruptured chemical bonds recombine. An applied tensile stress disturbs the thermodynamic equilibrium in the material and creates the possibility for the number of ruptured bonds to accumulate. All-round compression, by reducing the distance between atoms, promotes the restoration of bonds or, in other words, increases the potential barrier.

On the basis of the fluctuation theory of fracture of solid dielectrics, a dependence of the time on breakdown of dielectric materials in general and dielectric rocks in particular [3] was obtained in the form

\[ \tau=\tau_{0}\, \frac{ \exp \dfrac{D}{2kT} }{ \left[ \exp \dfrac{\sqrt{(A\beta E)^{2}\pm \gamma^{2}\sigma^{2}}}{2kT} \ln \dfrac{2De}{\sqrt{(A\beta E)^{2}\pm \gamma^{2}\sigma^{2}}} \right]-1 }, \tag{1} \]

where \(\tau\) is the time to fracture of the material;

\(\tau_{0}\)—the relaxation time of bonds under thermodynamic equilibrium;

\(D\)—the energy of bond dissociation;

\(T\)—absolute temperature;

\(k\)—Boltzmann’s constant;

\(A,\gamma\)—constants depending on the structure of the dielectric material;

\(\beta\)—a coefficient taking into account the concentration of electric-field intensity at sites of inhomogeneities;

\(E\)—the mean electric-field intensity;

\(\sigma\)—mechanical stress.

In equation (1), the sign before the product \(\gamma^2\sigma^2\) is taken as plus in the presence of tensile stresses and as minus under all-around compression.

The change of temperature with depth in the earth’s interior can be written approximately as

\[ T = T_0 + ah, \tag{2} \]

where \(T_0\) is the temperature at the Earth’s surface;
\(a\) is the temperature gradient in the Earth’s crust;
\(h\) is the depth of occurrence of the layer under consideration.

The mechanical stress due to all-around compression in the earth’s interior may approximately be expressed in the form

\[ \sigma = bh. \tag{3} \]

Substituting the values of \(T\) and \(\sigma\) into equation (1), we obtain

\[ \tau = \tau_0 \frac{ \exp \dfrac{D}{2\kappa (T_0 + ah)} }{ \left[ \exp \dfrac{\sqrt{(A\beta E)^2 - \gamma^2 b^2 h^2}}{2\kappa (T_0 + ah)} \ln \dfrac{2De}{\sqrt{(A\beta E)^2 - \gamma^2 b^2 h^2}} \right] - 1 }. \tag{4} \]

Analysis of equation (4) shows that the time to breakdown of the rock at a given electric-field strength varies with depth along a curve with a minimum. Taking the derivative of \(\tau\) with respect to \(h\) and setting it equal to zero, we found an equation for determining the value \(h_{\mathrm{m}}\) corresponding to the minimum time to breakdown of the rock at depth, in the form

\[ \sqrt{(A\beta E)^2 - \gamma^2 b^2 h_{\mathrm{m}}^2} = \frac{T_0 + ah_{\mathrm{m}}}{ah_{\mathrm{m}}} \left(\frac{\gamma b h_{\mathrm{m}}}{D}\right)^2 + \tag{5} \]

\[ + \left[ \frac{(A\beta E)^2}{D} + \frac{T_0}{ah_{\mathrm{m}}} \frac{(\gamma b h_{\mathrm{m}})^2}{D} \right] \ln \frac{2De}{\sqrt{(A\beta E)^2 - (\gamma b h_{\mathrm{m}})^2}}. \]

For rocks, approximate values of the coefficients appearing in equation (5) are given in Table 1.

Table 1

Using equation (5), the value of the depth \(h_{\mathrm{m}}\) was found and, for different electric-field strengths, the minimum time to the appearance of a discharge in the rock was determined from expression (4).

The values of \(h_{\mathrm{m}}\) and of the time to discharge as functions of the electric-field strength, for different values of the coefficients \(a\) and \(b\), calculated from equations (4) and (5), are given in Table 2.

Table 2 makes it possible to trace the influence of temperature and pressure gradients on the minimum time to breakdown of a dielectric rock.

With increasing temperature gradient and decreasing pressure gradient, the value of \(h_{\mathrm{m}}\) increases, i.e., the depth at which discharges occur increases, and the time to breakdown decreases. Of greatest interest for estimating the conditions for the appearance of discharges in the Earth’s crust are depths of 5–15 km.

At these depths the rocks have the lowest conductivity. Apparently, it is precisely at these depths that one may expect the occurrence of an electric field of the greatest strength.

It is seen from Table 2 that at these depths the time to discharges in rocks, at temperature gradients of 30 deg/km and field strengths of \(10^5\) V/m, may be measured in several tens of seconds.

In the calculations presented, only the petrostatic pressure was taken into account; this pressure increases the height of the potential barrier and hinders the development of the discharge. In rocks, owing to wave processes—for example, because of the influence of the attraction of the Moon and the Sun—tensile stresses appear. Tensile stresses may also arise during tectonic processes.

Table 2

| \(E\), V/m | \multicolumn{2}{c}{\(a = 10\ \dfrac{\text{deg}}{\text{km}}\)
\(b = 2.4 \cdot 10^7\ \dfrac{\text{N}}{\text{m}^2\text{ km}}\)} | \multicolumn{2}{c}{\(a = 30\ \dfrac{\text{deg}}{\text{km}}\)
\(b = 2.4 \cdot 10^7\ \dfrac{\text{N}}{\text{m}^2\text{ km}}\)} | \multicolumn{2}{c}{\(a = 10\ \dfrac{\text{deg}}{\text{km}}\)
\(b = 1.2 \cdot 10^7\ \dfrac{\text{N}}{\text{m}^2\text{ km}}\)} | \multicolumn{2}{c}{\(a = 30\ \dfrac{\text{deg}}{\text{km}}\)
\(b = 1.2 \cdot 10^7\ \dfrac{\text{N}}{\text{m}^2\text{ km}}\)} |
|---:|---:|---:|---:|---:|---:|---:|---:|---:|
| | \(h_m\), km | \(\tau\), sec | \(h_m\), km | \(\tau\), sec | \(h_m\), km | \(\tau\), sec | \(h_m\), km | \(\tau\), sec |
| \(10^4\) | 0.645 | \(10^{24}\) | 0.74 | \(4 \cdot 10^{22}\) | 1.45 | \(2.5 \cdot 10^{23}\) | 1.527 | \(4 \cdot 10^{20}\) |
| \(10^5\) | 6.65 | \(8 \cdot 10^{15}\) | 7.4 | \(5 \cdot 10^6\) | 14.4 | \(2.2 \cdot 10^{11}\) | 15 | 57.5 |
| \(10^6\) | 45 | \(6 \cdot 10^{-8}\) | 52.5 | \(4 \cdot 10^{-11}\) | — | — | — | — |

Tensile stresses lower the potential barrier and cause a decrease in the time to breakdown of the rock. In this case, discharges in rocks may arise at considerably lower electric-field strengths than those given in Table 2.

The process of rock breakdown in the subsurface will be accelerated when its temperature increases, which may occur, for example, when magma rises.

The hypothesis put forward makes it possible to offer certain proposals for forecasting incipient earthquakes associated with tectonic processes in the Earth’s crust.

Before the onset of an earthquake, one should expect a local increase in pressure exerted by magma on the Earth’s crust, which will entail the appearance of tensile stresses in the Earth’s crust and a rise in temperature at the sites of the future earthquake.

The appearance of mechanical tensile stresses and an increase in temperature will lead to a sharp increase in the number of local discharges in the Earth’s crust.

Each local discharge is accompanied by the release of electromagnetic energy. By installing sufficiently sensitive electromagnetic-wave sensors, it is possible to detect the initial stages of discharges in the Earth’s crust.

Since discharges may appear long before the beginning of an earthquake, an increase in their intensity will indicate a developing electrical process in the subsurface, which may culminate in a tectonic one.

Observations carried out in Central Asia and Kamchatka have shown that seismic shocks were often preceded by a disturbance of the electric field in the atmosphere and the Earth’s crust.

References

1. A. A. Vorob'ev. On the question of the initiation of earthquakes by underground thunderstorm phenomena. In: Electrophysical Apparatus and Electrical Insulation. Moscow, “Energiya,” 1970.

2. A. A. Tager. Physicochemistry of Polymers. Moscow, “Khimiya,” 1968.

3. V. I. Trafov, A. A. Vorob'ev, V. S. Dmitrievskii, V. I. Merkulov, V. A. Lkhamazhapov. Aging of solid dielectrics and the reliability of electrical insulation. Report at the IV Section of the Scientific Council of the Academy of Sciences of the USSR on electrophysical problems of electric power engineering. Novosibirsk, December 1–3, 1970.