Current Spreading from Grounding Electrodes under Permafrost Soil Conditions

V. P. CHUMANOV, N. N. MAKSIMENKO

Presented by Professor I. D. Kutyavin

Analysis of the electrophysical characteristics of permafrost soils shows that in regions of the Far North the nonuniformity of the distribution of soil resistivity with depth is clearly pronounced. In this connection, studies are needed of the characteristics of current spreading from grounding electrodes embedded in permafrost soils, in order to investigate the distribution of potential on the soil surface within open switchgear installations.

In homogeneous soils, as is known, the absolute value of the potential at points located on the earth’s surface, all other conditions being equal, is proportional to the value of the soil resistivity and, for a vertical grounding electrode, is equal to:

\[ \varphi_x=\frac{I\cdot \rho_0}{2\pi l}\cdot \ln \frac{\sqrt{e^2+r_x^2}+e}{r_x}. \tag{1} \]

For one and the same grounding electrode, when the soil resistivity increases from \(\rho_0\) to \(\rho_1\), the magnitude of the potential at the point increases by \(\Delta \varphi \frac{\rho_1}{\rho_0}\) times.

For regions of the Far North, studies of current spreading from grounding electrodes are of interest for the following types of soil nonuniformity:

1. The grounding electrode is located in the upper layer, whose resistivity is \(\rho_1<\rho_2\) (or \(\rho_1>\rho_2\)).

2. The grounding electrode passes through the interface between layers. The first group includes ordinary surface and vertical grounding electrodes laid in the upper part of the active layer; the second includes borehole grounding electrodes, the main length of which is located in layers of high negative temperatures, where \(\rho_2<\rho_1\).

Let us consider the influence of soil nonuniformity on the magnitude of the surface potential using the example of a concentrated grounding electrode.

1. CURRENT SPREADING FROM A CONCENTRATED GROUNDING ELECTRODE UNDER THE CONDITION \(\rho_1 \ll \rho_2\)

In practice, such cases are encountered when grounding electrodes are laid in soil beneath which there are rock formations (non-ore rocks), or frozen sandy soil. Under the indicated condition it may be assumed that the entire current from the grounding electrode spreads only in the upper layer, and in any cylindrical surface of the soil at a distance from the axis of the grounding electrode it is equal to the total current:

\[ I = 2\pi x h j, \tag{2} \]

where

\(h\) — depth of the layer;
\(x\) — distance of the section of the layer \(\rho_1\) from the concentrated grounding electrode;
\(j\) — current density.

The electric-field intensity at any point on the surface of the soil is determined by:

\[ E = \rho_1 j = \frac{\rho_1 I}{2\pi h x}. \]

The expression obtained shows that, in a two-layer soil with \(\rho_1 \ll \rho_2\), the field intensity decreases from the axis of the grounding electrode considerably more slowly than in homogeneous soils, where, as is known, for a concentrated grounding electrode located at the surface of the earth,

\[ E = \frac{\rho_1 I}{2\pi x^2}. \]

If one considers the spreading of current from a grounding electrode whose equivalent radius is \(r_0\), then the voltage on the grounding electrode relative to any point of the soil is:

\[ U_x = \int_{r_0}^{x} e\,dx = \frac{\rho_1 I}{2\pi h}\int_{r_0}^{x}\frac{dx}{x} = \frac{\rho_1 I}{2\pi h}\cdot \ln\frac{x}{r_0}. \tag{3} \]

Accordingly, for a homogeneous structure,

\[ U_x = \int_{\frac{d_0}{2}}^{x} e\,dx = \frac{\rho_1 I}{\pi}\int \frac{dx}{x^2} = \frac{\rho_1 I}{2\pi}\left(\frac{1}{r_0}-\frac{1}{x}\right). \tag{4} \]

From a comparison of formulas (3, 4) it is evident that in nonhomogeneous soils, where \(\rho_1 \ll \rho_2\), the voltage decreases more slowly with distance from the grounding electrode than in homogeneous soils, which can be explained by the limited surface of current spreading.

2. ACCOUNTING FOR THE CONDUCTIVITY OF THE LOWER LAYER

In accordance with the theory of the electric field in nonhomogeneous soils, given by Prof. V. V. Burgsdorf, the potential at any point of the first layer from a point source located in this layer (with the origin of coordinates located on the soil surface) is determined by:

\[ \varphi_{11}=\frac{i\rho_1}{4\pi}\left\{ \frac{1}{\sqrt{r^2+(y-\eta)^2}}+ \frac{1}{\sqrt{r^2+(y+\eta)^2}}+ \sum_{n=0}^{n=\infty}\left[ \frac{k^n}{\sqrt{r^2+(y-2nh-\eta)^2}}+ \frac{k^n}{\sqrt{r^2+(y-2nh+\eta)^2}}+ \frac{k^n}{\sqrt{r^2+(y+2nh-\eta)^2}}+ \frac{k^n}{\sqrt{r^2+(y+2nh+\eta)^2}} \right]\right\}. \tag{5} \]

where

\[ k\frac{\rho_2-\rho_1}{\rho_2+\rho_1} \]

is the coefficient of soil nonuniformity;

\(\eta\) is the depth of the point source from the surface;

\(y, r\) are the coordinates of the point of interest;

\(h\) is the thickness of the upper soil layer.

When the point source is located in the second layer, for the potential at any point of the first layer we shall correspondingly have:

\[ \varphi_{12}=\frac{i_2\rho_2}{4\pi}(1+k)\sum_{n=0}^{n=\infty}\left[ \frac{k^n}{\sqrt{r^2+(y-2nh-\eta)^2}}+ \frac{k^n}{\sqrt{r^2+(y+2nh+\eta)^2}} \right]. \tag{6} \]

With point current sources present simultaneously in the first and second layers, the potential of the first layer is

\[ \varphi_1=\varphi_{11}+\varphi_{12}. \]

For points located on the soil surface, where \(y=0\), and \(r=\sqrt{x^2+z^2}=x\) (for \(z=\mathrm{const}\)), the potential due to a source of the first layer is

\[ \varphi'_{11}=\frac{i_1\rho_1}{4\pi}\left\{ \frac{2}{\sqrt{x^2+\eta^2}}+ \sum_{n=0}^{n=\infty}2k^n\left[ \frac{1}{\sqrt{x^2+(2nh-\eta)^2}}+ \frac{1}{\sqrt{x^2+(2nh+\eta)^2}} \right]\right\}. \tag{7} \]

The first term of the expression obtained, \(\varphi'_{11}=\dfrac{i\rho_1}{2\pi}\cdot\dfrac{1}{\sqrt{x^2+\eta^2}}\), determines the potential on the soil surface from a point source \(i\) located in homogeneous soil at depth \(\eta\). The second term

\[ \Delta\varphi_{11}=\frac{i_1\rho_1}{4\pi}\sum_{n=0}^{n=\infty}2k^n\left( \frac{1}{\sqrt{x^2+(2nh-\eta)^2}}+ \frac{1}{\sqrt{x^2+(2nh+\eta)^2}} \right) \]

takes into account the influence of soil nonuniformity on the surface potential. Quantitatively, this influence depends on the coefficient of nonuniformity and on the position of the current source. The greatest influence of nonuniformity is manifested at \(\eta=0\), that is, when the current source is on the soil surface. In this case

\[ \Delta\varphi_{11}=\frac{i_1\rho_1}{4\pi}\sum_{n=0}^{n=\infty}4k^n\frac{1}{\sqrt{x^2+(2nh)^2}}. \tag{8} \]

For a nonuniformity where \(\rho_1<\rho_2\), \(k>0\), the quantity \(\Delta\varphi_{11}\) tends to a finite positive number, which results in a slowing of the decay of the soil-surface potential with increasing distance from the source—

...of the source. Accordingly, for $\rho_1 > \rho_2$, $\Lambda\varphi_{11}$ tends to a finite negative number, which affects the acceleration of the attenuation of the surface potential.

From formula (7) one can find the potential on the ground surface from a source $i_2$ located in the lower layer:

\[ \varphi_{12}= \frac{i_2\rho_2}{4\pi}(1+k) \sum_{n=1}^{n=\infty} 2k^n \frac{1}{\sqrt{x^2+(2nh+r_2)^2}}, \tag{9} \]

where $r_2$ is the depth of the source $i_2$ below the surface. Analysis of formulas (8) and (9) shows that in inhomogeneous soils the distribution of potential on the earth’s surface depends not only on the degree of inhomogeneity and the location of the current source, but also on the ratio of the currents $i_1$ and $i_2$ flowing from the grounding electrodes.

Let us determine the relative change in the potential on the surface of an inhomogeneous soil with respect to the potential of a homogeneous soil in the presence of one point source $i_1$.

From formula (7) we find:

\[ m=\frac{\varphi_{11}}{\varphi'_{11}} =1+\sqrt{x^2+r_1^2}\sum_{n=1}^{n=\infty} k^n \left[ \frac{1}{\sqrt{x^2+(2nh-r_1)^2}} + \frac{1}{\sqrt{x^2+(2nh+r_1)^2}} \right]. \]

If one takes into account that the field of a grounding electrode of any shape in an unbounded medium, as the distance from the grounding electrode increases, approaches the field produced by a point current source, then the expression obtained can be used to estimate approximately the degree of distortion of the potential curve for any grounding-electrode design.

In the regions of the Far North in summertime, owing to thawing of the active layer, $\rho_1 < \rho_2$. In accordance with formulas (8) and (9), this causes a slowing of the fall of the potential curve when current spreads from grounding electrodes located both in the upper and in the lower layer. In wintertime, when $\rho_1 \gg \rho_2$; $|k| < 0$, the inhomogeneity of the soil contributes to attenuation of the potential curve; however, owing to the high values of $\rho_1$, the resulting potential-drop curve will still remain flat.

The enlargement of the zone of current spreading over the soil surface imposes special requirements on the choice of a method for measuring the resistance to current spreading of grounding electrodes in regions of the Far North.

Existing experience in measuring grounding resistances, based on the use of the MS-08 grounding tester and the A—V method, shows that when the potential probe is moved away from the grounding electrode being measured to distances of 20 m (or to a distance of $5d$ for grid grounding devices), the measurement errors caused by the slow attenuation of the potential curve reach 50% and more. The foregoing makes it possible to conclude that the required accuracy in measuring the resistance of grounding devices in regions of the Far North can be achieved only by recording potential curves. In doing so, it is necessary to install a number of potential probes driven into the ground to a depth of 0.5–0.6 m. The first probe should preferably be placed at a distance of not less than 20–30 m from the grounding electrode being measured. The subsequent probes, respectively, should be placed at distances of 10–20 m from one another. To record the complete potential curve,

in this case it is necessary to install 8–11 probes. The current probe must be installed at a distance of 150–200 m from the grounding electrode being measured. The actual value of the grounding-electrode resistance is determined from the stabilized resistance values on the potential curve.

The recommended method for measuring the resistances of grounding devices when measurements are carried out in summer presents no difficulties, since, owing to the high conductivity of the surface soil layers, the maximum permissible resistances of the potential and current probes are readily achieved. In winter the resistances of probes driven into the soil to a depth of 0.5–0.8 m reach 3000–5000 ohms, which limits the possibilities of using the MS-08 meter. The required sensitivity of the latter in a number of cases can be achieved by treating the soil at the place where the probes are installed with a strong saline solution and by using, as the current probe, one of the operating grounding electrodes of the system. A more universal method when measurements are carried out in winter is—

Fig. 1. Calculated potential curves of point grounding electrodes:

\(1 — \rho_1 = 0.5 \cdot 10^2\ \text{ohm·m},\ \rho_2 = 10 \cdot 10^2\ \text{ohm·m};\ \alpha_{\text{pr}} = 0.55.\)

\(2 — \rho_1 = 0.5 \cdot 10^2\ \text{ohm·m},\ \rho_2 = \infty,\ \alpha_{\text{pr}} = 0.25\)

\(3 — \rho_1 = \rho_2;\ \alpha_{\text{pr}} = 0.86\)

…the \(A\)—\(V\) method is used. Raising the working voltage on the grounding electrode and the current probe to 2000–3000 V makes it possible, for the above-mentioned values of probe resistances, to carry out measurements by means of \(A\)—\(V\) with the required accuracy. In this case it is necessary to use an electrostatic or tube voltmeter and shielded leads used in the measurement circuit.

The nonuniformity of the distribution of specific resistance with depth affects not only the choice of methods for measuring the spreading resistance of grounding electrodes, but also, to a considerable extent, the safety conditions for personnel servicing outdoor switchgear.

As is known, the magnitude of touch and step voltage is determined by the steepness of the drop of the potential curve. Under permafrost-soil conditions, owing to the limited space for current spreading, the zero-potential zone from a single grounding electrode is removed to a distance greatly exceeding 20 m. At the same time, the potential curve is correspondingly raised at a distance of 0–1.0 m. Owing to the rise of the potential curve, the touch and step voltages are substantially reduced.

Figure 1 shows calculated curves of the change in potential of a single point grounding electrode for different degrees of soil nonuniformity; analysis of these curves shows that the role of the nonuniformity of permafrost soils in the distribution of potential on the ground surface is significant. The latter circumstance indicates the

Fig. 2. Potential curves of surface grounding electrodes
(\(l=4\) m, \(\varnothing=40\) mm, \(h=0.35\)), 1, 2 — in the area of the Talnakh settlement; 3, 4 — in the area of point “A,” Norilsk; 5 — for a pipe laid along the ground surface; 6 — for a vertical pipe (\(l=3.5\) m, \(\varnothing=100\) mm).

Fig. 3. Potential curves for two parallel surface tubular grounding electrodes \((l = 4\ \text{m},\ \varnothing = 40\ \text{mm},\ h = 0.35\ \text{mm})\).
1 — point “A,” Norilsk, 8.8.64; 2 — settlement “Talnakh”; 3 — point “A,” 9.9.64; 4 — point “A,” 18.10.64; 5 — the same, 4.12.64.

the need to take into account soil inhomogeneity when considering the complex of questions connected with ensuring personnel safety from currents flowing into the earth from various grounding devices. In addition, the physical conditions of current spreading in permafrost soils must be made the basis for designing grounding devices and for selecting the permissible value of grounding resistance in high- and low-voltage installations operated in regions of the Far North.

Figure 2 shows the potential curves of horizontal and vertical grounding electrodes measured in the region of Norilsk and the settlement “Talnakh.” The indicated curves were obtained with the aid of an MS-08 grounding meter. The potential probes were made of crowbars \(\varnothing 22\ \text{mm}\), driven into the ground to a depth of \(0.6\text{–}0.8\ \text{m}\).

Analysis of the data (Fig. 2) shows that in summer, under permafrost-soil conditions, when current spreads from single horizontal grounding electrodes, the touch voltage, in comparison with homogeneous soils, decreases by a factor of 3; correspondingly, for the case of vertical grounding electrodes, by a factor of 2.8.

Figure 3 presents potential curves for two horizontal strips connected to one another by an insulated conductor along the earth’s surface (the region of Norilsk and the settlement “Talnakh”). In summer, owing to current spreading mainly through the thawed active layer, the resulting potential curve between horizontal grounding electrodes laid 6.0 m apart is considerably leveled out, which correspondingly leads to a reduction in the touch and step voltages \(\alpha_{\text{t}} = 0.07;\ \alpha_{\text{st}} = 0.17\).

Fig. 4. Lithological section of the soil at point “A,” Norilsk.

Fig. 5. Potential curves of surface grids:
1 — \(4 \times 4\) m, 2 — \(6 \times 6\) m, 3 — \(10 \times 10\) m.

We have also investigated the effect of freezing of the active and accumulation layers of the soil on changes in the distribution of potential over the soil surface. The results of these studies show that in winter the character of the potential curves on the soil surface practically does not change (see Fig. 3).

The observed features of the change in the potential curves indicate the need to study current spreading from a system of surface grounding electrodes in order to justify the selection of the most economical cell of the surface grounding electrode from the condition of equalizing the potential within the cell and of the optimal value of the utilization coefficient of grounding electrodes.

A qualitative analysis of the questions posed above shows that, under conditions of the Far North, the potential curve in the zone of the grounding electrode will be straightened significantly better than under conditions of homogeneous soils.

We measured the distribution of potential within cells of surface grounding electrodes of the following dimensions: \(4 \times 4\); \(6 \times 6\); \(10 \times 10\) m. The cells were made of pipes \(\varnothing 4\) cm laid in the soil to a depth of \(0.3\) m. The lithological section of the soil at the study site is shown in Fig. 4. The test was carried out on 10.9.64, i.e., during the period of maximum thawing of the active layer. The potential curves for the indicated grounding electrodes are shown in Fig. 5; their analysis shows that increasing the cell of surface grounding electrodes from 4 to 10 m does not lead to a substantial change in the degree of equalization of the potential curve within the grounding-electrode cell.

CONCLUSIONS

1. Temperature changes of the soil within the active layer cause a slowed attenuation of the potential curve over the soil surface within the grounding device of an electrical installation. This leads to a substantial reduction in the touch and step coefficients in grounding devices operated in regions of the Far North.

However, the magnitude of the utilization coefficient of surface grounding electrodes in summer is significantly lower than under conditions of homogeneous soils.

1. In regions of permafrost soils, satisfactory results when measuring the resistance to spreading of grounding electrodes can be obtained only by taking potential curves.

2. The physical conditions of current spreading from individual grounding electrodes in inhomogeneous soils indicate the need for studies to determine the optimal cell dimensions of surface grounding electrodes operated in regions of the Far North.