STUDY OF ROCK DENSITY AT THE ITATSKY LIGNITE DEPOSIT USING A RADIATION METHOD

V. E. Olkhovatenko, V. K. Kuleshov, V. V. Uchaikin

(Presented by the scientific seminar of the Department of Hydrogeology and Engineering Geology)

In recent years, radiometric methods have been increasingly widely used in engineering-geological investigations. By these methods it is possible, comparatively quickly and with sufficient accuracy, to determine the density and moisture content of rocks.

The methods and apparatus developed at the present time make it possible to conduct investigations both under field and under laboratory conditions [1, 2, 3, 5, 6, 7].

The present paper reports the results of studies of the density of sedimentary rocks of the Itatsky lignite deposit by the method of gamma-radiation attenuation.

The Itatsky deposit is one of the largest in the Kansk-Achinsk basin, and it is proposed to mine it by the open-pit method using powerful rotary excavators. During exploration of the deposit, the Department of Hydrogeology and Engineering Geology of the Tomsk Polytechnic Institute is carrying out detailed engineering-geological investigations. A large part of these investigations is occupied by determination of the unit weight of the rocks, which is a design characteristic in calculations of the stability of the walls of the planned open pits. The existing standard methods for determining the unit weight of rocks—the hydrostatic weighing method, cutting rings, paraffin coating—are rather cumbersome and require considerable time expenditures; therefore, recently, works have appeared in the literature devoted to determining the density of rocks and soils by irradiating them with a beam of gamma quanta [2, 3, 5].

When the rocks under study are irradiated with collimated monoenergetic gamma radiation, the intensity of the latter decreases. The attenuation of the gamma-radiation intensity as a result of the interaction of gamma quanta with rocks is caused by the partial removal of gamma quanta from the beam due to Compton scattering, photoelectric absorption, and pair production.

The law of attenuation of the intensity of collimated gamma radiation when passing through rock is described by the equation

\[ I = I_0 e^{-\mu_{\text{lin}} x}, \tag{1} \]

where \(I\) and \(I_0\) are the intensity of the radiation recorded by the detector in the presence and absence, respectively, of the rock sample being irradiated;

\(\mu_{\text{lin}}\) is the linear coefficient of attenuation of the intensity of the gamma-radiation beam of the given energy for the given rock, cm\(^{-1}\);

\(x\) is the thickness of the irradiated layer of material, cm.

The quantity \(\mu_{\text{lin}}\) characterizes the relative decrease in the intensity of collimated radiation by a layer of the irradiated substance 1 cm thick.

The latter is proportional to the density of the substance and depends on the radiation energy, the atomic number, and the atomic weight of the substance. In our case, the isotope Cs\(^ {137}\) (662 MeV), 0.5 mg-eq Ra, was used as the radiation source. The thickness of the samples studied was constant and equal to 20 mm.

The linear attenuation coefficient is related to the mass (volume) attenuation coefficient by the relation

\[ \mu_{\text{vol}}=\frac{\mu_{\text{lin}}}{\rho}, \qquad \text{where} \tag{2} \]

\(\mu_{\text{vol}}\) is the volume attenuation coefficient, cm\(^2\)/g;
\(\rho\) is the density of the sample under study, g/cm\(^3\).

The latter is determined, on the basis of the above dependences, by the formula

\[ \rho=\frac{\ln I_0-\ln I}{\mu_{\text{vol}}\cdot x}, \ \text{g/cm}^3 . \tag{3} \]

The symbols are the same as in the formulas given above.

To study rock density in our case, an apparatus was used whose block diagram is shown in Fig. 1. The flux of gamma—

Fig. 1. Block diagram of the apparatus for determining rock density in samples by the radiation method.
1 — lead container; 2 — Cs\(^ {137}\) radiation source; 3 — sample under study; 4 — collimator; 5 — radiation detector; 6 — FEU-52 photomultiplier; 7 — lead shielding; 8 — cathode follower assembled according to White’s circuit; 9 — BD-5 integral discriminator; 10 — PP-8 counting device; 11 — VS-22 stabilized rectifier; 12 — BN-40 power-supply unit; 13 — BN-41 power-supply unit.

quanta from source 2, passing through sample 3 and collimator 4, fall on the radiation detector, which is a scintillation counter consisting of a scintillation crystal 4 NaI (Te) 70×70 and a photomultiplier 6, FEU-52 of the venetian-blind type. The pulses of the FEU unit, arising as a result of scintillations from gamma quanta in the crystal and converted into voltage pulses, are transmitted through cathode follower 8, assembled according to White’s circuit [9] on a 6N3P tube, to the input of integral discriminator 9 of type BD-5 (10).

The integral discriminator unit includes an inverter, a preamplifier, and a discriminating stage. The amplifier, built on a triode-pentode according to the “binary” circuit, provides an amplification of \(\sim 20\). The discriminator, made as a monostable multivibrator stage with both tubes conducting and a diode that quenches self-oscillations in the absence of external signals, has a sensitivity of \(\sim 100\) mV and produces an output

standard pulse. The input sensitivity of the discriminator is varied by an attenuator inserted before the inverter. High stability is ensured by negative feedback on the signal in the preamplifier, by stabilization of the DC operating point of the single-shot tubes, and by supplying the diode-tube channel in the discriminator with voltage. The amplitude of the output standard pulse from the discriminator is sufficient to trigger the PP-8 counting device.

Power for the FEU was supplied from a high-voltage stabilized rectifier 11 of type VS-22. Power for the cathode follower and the integral discriminator was supplied from power units 12, 13, BN-40 and BN-41. The scintillation counter, together with the cathode follower, was made as a single remote unit placed in lead shielding. The activity of the source was determined on the basis of safety in working with it when changing samples. The dimensions of the collimator are determined by the irradiated surface, by the occurrence of edge effects in the crystal, and, if possible, by a higher counting rate in order to reduce the measurement error. The latter consists of errors of the measuring apparatus and errors caused by the discrete nature of radioactive radiation. In our case, through the use of highly stable power sources and the selection of the optimum operation of the cathode follower, the measurement error of the apparatus did not exceed 1.5%.

From the standpoint of reducing the statistical error of measurement, the choice of the measurement method is important.

In the case of the integral method of measurement, the error can be estimated as follows:

\[ \delta u = \frac{1}{\overline{u}} \sqrt{\frac{D(U)}{2RC}}, \tag{4} \]

where \(\delta u\) is the fluctuation of the voltage \(u\) at the output of the integrating circuit with parameters \(R\) and \(C\), relative to the mean value \(\overline{u}\) of the output voltage;

\(D(U)\) is the voltage variance.

The random variable \(U\) can be represented as the product of the random variables \(\xi_1\) and \(\xi_2\)

\[ U = \xi_1 \cdot \xi_2, \tag{5} \]

where \(\xi_1\) takes the value 1 if the photon has been recorded by the device, and 0 otherwise, while \(\xi_2\) obeys the distribution \(P(U)\,dU\), which describes the amplitude spectrum of the pulses coming from the photomultiplier. Then, in accordance with probability theory,

\[ \overline{U} = M(\xi_1) M(\xi_2), \tag{6} \]

and

\[ D(U) = D(\xi_1)D(\xi_2) + [M(\xi_1)]^2 D(\xi_2) + [M(\xi_2)]^2 D(\xi_1), \tag{7} \]

where \(M(\xi)\) is the mathematical expectation of the quantity \(\xi\).

We approximate the photopeak in the amplitude distribution (Fig. 2) by a Gaussian with variance \(\sigma^2\) and mean value \(U_0\), and the Compton distribution by a uniform distribution from 0 to \(U_2\), in such a way that the area under the entire distribution is equal to \(\varepsilon \overline{N}_0\), where \(\varepsilon\) is the efficiency of photon registration and \(\overline{N}_0\) is the mean number of photons incident on the crystal per unit time. The ratio of the area under the photopeak to the area under the entire distribution is the photo-fraction. We denote it by \(P\). The presence of a discrimination threshold \(U_1\) will cause these quantities to change. Denoting the new quantities by \(\varepsilon^*\) and \(P^*\), one can write:

\[ \varepsilon^{*}=\varepsilon\left[1-\frac{U_{1}}{U_{2}}(1-p)\right] \tag{8} \]

and

\[ P^{*}=P\left[1-\frac{U_{1}}{U_{2}}(1-P)^{-1}\right]. \tag{9} \]

Since the number of pulses registered per unit time obeys the Poisson distribution, one may write

\[ M(\xi_{1})=\varepsilon^{*}\overline{N}_{0} \tag{10} \]

and

\[ D(\xi_{1})=\varepsilon^{*}\overline{N}_{0}. \tag{11} \]

Then

\[ \overline{U}=\varepsilon^{*}\overline{N}_{0}M(\xi_{2}), \tag{12} \]

and

\[ D(U)=\varepsilon^{*}\overline{N}_{0}D(\xi_{2})+(\varepsilon^{*}\overline{N}_{0})^{2}D(\xi_{2}) +\left[M(\xi_{2})\right]^{2}(\varepsilon^{*}\overline{N}_{0})= \]

\[ =(\varepsilon^{*}\overline{N}_{0})^{2} \left\{\left[1+\frac{1}{\varepsilon^{*}\overline{N}_{0}}\right]M(\xi_{2}) -\left[M(\xi_{2})\right]^{2}\right\}. \tag{13} \]

The mathematical expectations of the quantity \(\xi_{2}\) and of its square are found as follows:

\[ M(\xi_{2})=\int_{0}^{\infty}UP(U)\,dU =\frac{1-P^{*}}{U_{2}-U_{1}}\int_{U_{1}}^{U_{2}}U\,dU +\int_{U_{2}}^{\infty}UP(U)\,dU \simeq \]

\[ \simeq \frac{(1-P^{*})(U_{2}+U_{1})}{2}+P^{*}U_{0}, \tag{14} \]

\[ M(\xi_{2}^{2})=\int_{0}^{\infty}U^{2}P(U)\,dU \simeq \frac{(1-P^{*})^{2}}{3} \left(\frac{U_{2}^{3}-U_{1}^{3}}{U_{2}-U_{1}}\right) +(P^{*})^{2}(\sigma^{2}+U_{0}^{2}). \tag{15} \]

Hence

\[ D(U)=(\varepsilon^{*}\overline{N}_{0})^{2} \left\{ \left[1+\frac{1}{\varepsilon^{*}\overline{N}_{0}}\right] \left[ \frac{(1-P^{*})^{2}}{U_{2}-U_{1}}\frac{(U_{2}^{3}-U_{1}^{3})}{3} +(P^{*})^{2}(\sigma^{2}+U_{0}^{2}) \right] \right. \]

\[ \left. -\left[ \frac{(1-P^{*})(U_{2}+U_{1})}{2}+P^{*}U_{0} \right]^{2} \right\}. \tag{16} \]

Substituting expressions (12), (14), and (16) into formula (4), one can obtain the relative fluctuations for an arbitrary discrimination threshold. Consider the case in which discrimination is carried out near the photopeak (Fig. 2). Then \(U_{1}=U_{2}\) and

\[ \delta U= \sqrt{ \frac{1}{2RC} \left[ \frac{\sigma^{2}}{U_{0}^{2}} \left(1+\frac{1}{\varepsilon^{*}P^{*}\overline{N}_{0}}\right) +\frac{1}{\varepsilon^{*}P^{*}\overline{N}_{0}} \right] }. \tag{17} \]

Here the quantity \(\sigma^{2}\) is related to the energy resolution of the crystal \(a\) by the relation

\[ \sigma^{2}=\frac{a^{2}}{8\ln 2}. \tag{18} \]

At a comparatively low counting rate it is advisable to apply pulse normalization. In this case \(\sigma^2 \simeq 0\), and expression (17) takes the form

\[ \delta U=\frac{1}{\sqrt{2RC\varepsilon^{*}P^{*}\overline{N}_{0}}}. \tag{19} \]

When operating in the counting mode (using a counting instrument), the relative fluctuations in the number of photons recorded during \(t\) sec are determined by the formula

\[ \delta N=\frac{1}{\sqrt{\varepsilon^{*}P^{*}\overline{N}_{0}t}}. \tag{20} \]

Comparison of expressions (17), (19), and (20) shows that the optimal measurement method is the counting method, since the principal requirement in our case is high accuracy of the results.

In expression (20), \(\overline{N}_{0}\) is determined by the formula

\[ \overline{N}_{0}=3.7\cdot 10^{10}\cdot A\cdot \gamma \cdot G \cdot e^{-\mu x}\cdot B, \]

where

\(A\) is the activity of the source in curies,
\(\gamma\) is the number of quanta per decay;
\(G\) is the geometrical attenuation factor;
\(e^{-\mu x}\) is the attenuation of the beam of gamma quanta by the material of the specimen under study of thickness \(x\);
\(B\) is the buildup factor.

The geometrical attenuation factor of radiation in the cone bounded by the edge of the collimator is calculated by the formula

\[ G=0.5\left[1-\frac{1}{\left(1+\frac{4r^{2}}{R^{2}}\right)^{\frac{1}{2}}}\right], \]

where

\(r\) is the radius of the collimator aperture;
\(R\) is the distance from the source to the detector crystal.

The error due to the discrete nature of radioactive radiation did not exceed 0.3%, and the total error was no more than 1.8%, which was achieved by choosing the optimal dimensions of the scintillator. From the standpoint of efficiency, its dimensions should be as large as possible. However, increasing the size of the crystal makes the installation more expensive. In our case, the combination of the FEU-52 and an NaJ (Te) \(70\times50\) crystal proved very successful. The diameter of the multiplier photocathode completely coincides with the diameter of the crystal. At the same time, the photomultiplier has an effective light-collection system, and the efficiency of recording quanta by the scintillation crystal is 0.75 (13). The high efficiency of the scintillator makes it possible to reject (discriminate) noise pulses from the FEU and pulses caused by scattering of gamma quanta. Figure 2 presents the spectral distribution of the amplitudes of pulses entering the discriminator. As can be seen from Fig. 2, the area under the curve characterizing the distribution of noise pulses and pulses caused by scattering of gamma quanta is rather large. These pulses do not carry useful information and constitute interference that increases the measurement error. Setting the discrimination threshold in the region of the dip makes it possible to reduce the errors to a minimum.

Density was determined on sandstones, siltstones, and argillites, which are most widespread within the Itatsky lignite deposit. These rocks have a similar mineralogical composition, differing somewhat in granulometric composition and, to a greater extent, in structural and textural

Fig. 2. Spectral distribution of voltage-pulse amplitudes at the input of the integral discriminator:
1 — pulse spectrum in the absence of a rock sample; 2 — pulse spectrum in the presence of the investigated sample; 3 — region of the spectrum consisting of PMT noise and radiation scattered in the collimator; 4 — region characterizing Compton scattering in the crystal and in the investigated sample; 5 — region characterizing photoabsorption.

features. The material composition has only a slight effect on the results of determining rock density by the radiation method.

As a result of the studies carried out, a dependence was obtained between the intensity of the registered radiation and rock density, presented in Fig. 3. From Fig. 3 it is seen that this dependence is linear in the density interval from 1.70 to 2.36 g/cm³ and for the given accuracy of the measurements. The calculated and experimental straight lines are also shown there; they almost completely coincide. In this connection, when determining the density of rocks close in composition and state to the investigated samples, either the calculated or the experimental straight line may be used.

Processing of the obtained materials by methods of mathematical statistics showed that the individual values of bulk density and radiation intensity obey the normal distribution law and have a close correlation. The correlation coefficient was 0.83. Along with determining the density of the noted rocks by the radiation method, their density for all samples was also determined by the ordinary stan-

Fig. 3. Dependence of counting rate on rock density.
I—experimental straight line; II—calculated straight line; 1—sandstones; 2—argillites; 3—siltstones.

dard method. Comparison of the results showed that the discrepancy is no more than ±0.05 g/cm³. This is within the accepted accuracy limits for determining this characteristic.

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