Gamma–Gamma Logging with a Gamma-Ray Generator

E. M. Filippov

(Presented by the Second Inter-Institutional Conference on Electron Accelerators)

The method of radioactive logging based on measuring the intensity of scattered gamma radiation from some source of gamma rays placed in the same downhole instrument as the radiation indicator, for the purpose of determining the density of rocks penetrated by a borehole, has been called gamma–gamma logging (GGL).

Knowledge of the density of rocks helps in determining their porosity, which is necessary for calculating the reserves of oil and gas fields. The GGL method is very widely used in the study of coal and ore boreholes.

In addition, the GGL method can also be applied to solve technical problems: for locating the level of the cement sheath in the annular space of boreholes and for determining the position of coupling collars on the casing string.

The natural and artificial radiation sources used in GGL up to now have, as is known, gamma-ray energies not exceeding 3 MeV. In investigating boreholes by the GGL method, gamma-ray sources with an activity not exceeding 30 µCi may also be used; otherwise this would lead to a violation of safety regulations. The depth of investigation of rocks by the GGL method when these sources are used is small; therefore the influence of irregularities in the borehole walls leads to a decrease in the accuracy of determining rock densities. The use in GGL of gamma-ray generators (such as a betatron), which make it possible to obtain radiation of high energy and intensity, should substantially increase the effectiveness of this method.

Let us consider to what extent increasing the energy and intensity of the gamma-ray source contributes to increasing the depth of investigation and the sensitivity of the GGL method. For this purpose we shall consider three questions: the dependence of the intensity of scattered gamma radiation on the initial energy of the gamma-ray source and on the distance between the indicator and the gamma-ray source; the dependence of the depth of investigation of rocks on the initial energy of the gamma-ray source; and the dependence of the spectral composition of the scattered radiation on the density of the rocks and on the initial energy of the gamma-ray source.

1. Investigation of the dependence of the intensity of scattered gamma radiation on the initial energy of the gamma-ray source for the case of a homogeneous medium.

In examining this question we shall use the expression for the intensity of scattered radiation given in [1]:

\[ j=\frac{Q\exp(-\tau_{cp}l)}{4\pi l^\beta}. \tag{1} \]

Here \(Q\) is the output of the gamma-ray source; \(l\) is the distance between the center of the indicator and the gamma-ray source (the probe spacing); \(\beta=3.5-\sqrt{0.25+2(1-\overline{\cos\Theta})^2}\) is a coefficient depending on the mean cosine of the angle of multiply scattered gamma radiation \((\overline{\cos\Theta})\); \(\tau_{cp}=(1-\overline{\cos\Theta})\bar{\lambda}\) is the mean absorption coefficient of gamma radiation under multiple scattering. The quantity \(\bar{\lambda}\) is the mean free path of multiply scattered gamma radiation.

The quantity \(\bar{\lambda}\), taking into account the photoelectric effect or the effect of electron–positron pair production, is calculated by the following formula:

\[ \bar{\lambda}=\sqrt[3]{\frac{\displaystyle\sum_{k=1}^{n}\alpha_k\overline{R_k^2}} {\displaystyle 2\sum_{k=1}^{n}\alpha_k}} . \]

Here \(\alpha_k\) are weighting corrections that take into account the absorption of scattered gamma radiation due to the photoelectric effect or the effect of electron–positron pair production; \(\overline{R_k^2}\) is the mean square of the free path of scattered gamma radiation in rocks, which is determined from the following expression:

\[ \overline{R_k^2} = \sum_{i=0}^{n}\overline{r_i^{\,2}} + \sum_{i=0}^{n-1}\sum_{j=i+1}^{n} \overline{2r_i r_j\cos\Theta_{ij}}, \]

where \(\bar r_i\) and \(\bar r_j\) are the mean free paths of gamma quanta between individual scattering events;

\[ \overline{r_i^{\,2}} \]

is the mean square of the free path of gamma quanta between individual scattering events;

\[ \overline{\cos\Theta_{ij}} \]

is the mean cosine of the scattering angle when a gamma quantum passes from state \(i\) to state \(j\).

A detailed scheme for calculating all the quantities entering the above-mentioned formulas is given in [1]. There, as well as in [2], a comparison is given of calculated data with experimental data for a cobalt radiation source. The results of the calculation agree satisfactorily with the experimental data.

The dependence, obtained from formula (1), of the intensity of scattered gamma radiation on the initial energy of the gamma-ray source for rocks (ores) with densities of 1, 2, 3, 4, and 5 g/cm\(^3\) and the corresponding effective atomic numbers\(^1\) 7.42; 11.4; 15.67; 19.3; 22.6 is shown in Fig. 1. The curves presented in this figure correspond to the case of using an integral indicator of gamma quanta. From Fig. 1, in addition to the known dependence of the intensity of scattered—

\(^1\) In studying the interaction of gamma radiation with media of complex chemical composition, instead of the individual atomic numbers of the elements composing these media, effective atomic numbers are introduced [1].

From the dependence of the intensity of scattered gamma radiation on the density of the medium it is also evident that the initial energy of the gamma-quantum source (when investigating boreholes by the gamma-gamma logging method), depending on the density of the medium being studied and the size of the probe, should be increased only up to a certain value; with further increase the curve \(j=f(E)\) tends toward an asymptote and does not change the character of the dependence being studied.

Fig. 1. Dependence of the intensity of scattered gamma radiation on the energy of the gamma-quantum source for three probe sizes: \(20\ \text{cm}\) (upper solid curves, \(1\text{--}5\)), \(50\ \text{cm}\) (lower solid curves, \(1\text{--}5\)), and \(80\ \text{cm}\) (dashed curves). The code of the curves corresponds to the rock density in \(\text{g}/\text{cm}^3\).

Thus, for example, for a probe size of \(20\ \text{cm}\) it is inadvisable to use gamma-quantum sources with initial energy above \(8\ \text{MeV}\), and for a probe of \(80\ \text{cm}\)—above \(15\ \text{MeV}\). This is due to the fact that, as their initial energy increases, gamma quanta change the direction of their motion to a lesser degree during scattering. Therefore, when investigating boreholes by the gamma-gamma logging method with small probes, scattered radiation from high-energy sources will reach the radiation indicator to a lesser extent than when investigating with large probes.

The use of large probes in the investigation of boreholes by the GGL method is advantageous because increasing the size of the probe leads to an increase in the sensitivity of the GGL method to changes in rock density. This is evident, in particular, from the curves obtained from formula (1) for probes of 20, 50, and 80 cm and a source with an initial energy of 5 MeV (Fig. 2). The dependences have a similar character for other energies of gamma-ray sources as well. However, the use of large

Fig. 2. Dependences of the intensities of scattered gamma radiation (in relative units \(j'\)) on rock density for three probe sizes (20, 50, and 80 cm) and a source with an initial energy of 5 MeV

probe sizes also requires an increase in the strength of the gamma-ray source. It was noted above that, according to safety regulations, when investigating boreholes by the GGL method one may not use natural or artificial gamma-ray sources with strengths greater than 30 μCi. Therefore, at present it is practically impossible to use probes larger than 50 cm without violating safety regulations.

In this connection, the use of surface-controlled gamma-ray generators, such as the betatron, as high-power sources will make it possible to increase substantially the sensitivity of the GGL method to changes in the density of rocks and ores by increasing the probe size.

The use of large probe sizes is also advantageous because it leads to an increase in the depth of investigation of rocks and, consequently, to an increase in the path length of gamma radiation in these rocks. This circumstance, in turn, will lead to a more accurate determination of the density of rocks and ores, as we shall show below.

2. Calculation of the depth of investigation of rocks and of their volume scattering the radiation.

By the depth (or penetration depth) of investigation of rocks intersected by a cylindrical borehole we shall mean the distance, in the direction of the borehole radius, from its wall to a certain cylindrical surface, the scattered radiation from which still reaches the gamma-ray detector located on the borehole axis.

We shall determine the depth of investigation of rocks from the expression for the intensity of scattered radiation. The intensity of gamma radiation \(j\), scattered in a rock volume \(v\), depending on the initial energy of the gamma-ray source, the rock density, and the coordinates of the positions of the detector and of the point source of gamma rays on the borehole axis, is determined by the following expression:

\[ J=\frac{Q\rho A_0}{32\pi^2}\int_V \frac{d\sigma}{d\Omega}\, \frac{\exp\left[-\frac{r-R_0}{r}\left(\tau R_1+\tau_{\mathrm{cp}}R_2\right)\right]\,d v} {R_1^2\left(R_{21}^{b}+bR_{21}R_{22}+R_{22}^{2}\right)}, \tag{2} \]

where \(Q\) is the intensity of the gamma-ray source;

\(A_0\) is Avogadro’s number;

\(\rho\) is the density of the rock;

\(R_0\) is the borehole radius;

\(r\) is the distance from the borehole axis to the scattering volume in the direction of the borehole radius;

\(R_1\) and \(R_2\) are, respectively, the distances from the point source to the scattering volume and from the scattering volume to the detector;

\(R_{21}\) is the path of the gamma quantum along the direction \(R_2\) in the rock;

\(R_{22}\) is the same in the air of the borehole \((R_2=R_{21}+R_{22})\);

\(\tau\) is the total linear coefficient of absorption and scattering of the primary radiation in the rock;

\(\tau_{\mathrm{cp}}\) is the average coefficient accounting for absorption of gamma radiation under multiple scattering in the rock along the direction \(R_2\);

\(\dfrac{d\sigma}{d\Omega}\) is the differential scattering cross section according to Klein–Nishina–Tamm;

\(b\) is the same as in formula (1); the quantity \(r-R_0\) in this formula corresponds to the depth of investigation of the rocks.

Let us carry out a numerical calculation for two different media: coal \((\rho=1.3\ \text{g/cm}^3\) and \(Z_{\mathrm{eff}}=6.92)\) and sedimentary rock \((\rho=2.6\ \text{g/cm}^3\) and \(Z_{\mathrm{eff}}=13)\), for gamma-ray sources with energies \(1.25\ \text{MeV}\) \((K=2.45)\) and \(14.75\ \text{MeV}\) \((K=28.9)\), for probe sizes of 5, 20, 50, and 80 cm.

Fig. 3. Dependence of the intensity of scattered gamma radiation reaching the detector (in relative units) on the depth of investigation of rocks.

The results of the calculation are shown in Fig. 3. As can be seen, an increase in the energy of the gamma-ray source leads to an increase in the depth of investigation of the rocks; an increase in the density of the rock and in its effective atomic number leads to a decrease in the depth of their investigation.

In addition, from these curves it is also evident that, with a change in the size of the probe, the depth of investigation changes. This is especially clearly seen from the curves of Fig. 4, constructed for the case of recording 90% of the radiation reaching the detector. In the same figure, for the same conditions, the dependence is given of the volume of rock scattering the gamma radiation that reaches the gamma-quantum detector. From Fig. 4 it is seen that:

a) the maximum values of the depth of investigation of rocks correspond to a probe 20 cm in size, which is due to the most optimal conditions for scattering gamma radiation;

b) the volume of the rock scattering the radiation, as the probe size increases, tends to an asymptote, and the more slowly, the greater the initial energy of the gamma-quantum source.

Fig. 4. Dependence of the investigation depth of sedimentary rock \(r-R_0\) (solid curves) and its volume \(v\) (dashed curves) on probe size

Thus, the increase in probe size, leading to an increase in the sensitivity of the GGL method to changes in medium density, is due exclusively to the magnitude of the volume of rock scattering the gamma radiation. All this is caused by the fact that the increase in the volume of rock occurring through an increase in probe size leads to an increase in the path length of gamma radiation in the rock and, consequently, to a greater dependence of the intensity of the recorded radiation on the parameters of the medium (density and effective atomic number). In this connection, the optimal probe should be taken to be one that corresponds to the emergence of the curve \(v=f(l)\) onto the asymptote.

Thus, in investigating boreholes by the GGL method it is necessary to use gamma-quantum sources having, as far as possible, the highest energy within limits up to 15 MeV. At the same time, for a source of a particular energy it is necessary to select a definite optimal probe.

It should be noted that increasing the initial energy of the gamma-quantum source does not lead to a change in the equilibrium spectrum of scattered radiation, as we shall show below.

1. Investigation of the dependence of the spectral composition on the density of rocks and on the initial energy of the gamma-quantum source.

The expression determining the change in the radiation spectrum from its initial energy in some medium has the following form:

\[ \tau(K)\varphi(K)=\int_{K}^{K_0}\varphi(K')\,\Gamma(K,K')\,dK' + Q\delta(K_0-K), \tag{3} \]

where \(\varphi(K)\) is the distribution function of the scattered gamma radiation with respect to energy \(K\);

\(\tau(K)\) is the total linear coefficient of absorption and scattering of gamma radiation in the substance;

\(\Gamma(K,K')\) is the probability of transition of a gamma quantum from a state with energy \(K'\) to a state with energy \(K\);

\(\delta(K_0-K)\) is the Dirac delta function;

\(Q\) is the strength of the source of gamma quanta;

\(K_0\) is the initial energy of the source of gamma quanta, in units of the rest energy of the electron;

\(K\) is the energy of gamma quanta of interest to us (in the same units as \(K_0\)).

By introducing the mean cosine of the angle of singly scattered gamma radiation, the kernel \(\Gamma(K,K')\) can be reduced to the degenerate form \(\Gamma(K')\), and the integral expression (3) to a differential one; its solution presents no particular difficulties and is analyzed in detail in [3]\(^1\).

The energy spectra of a cobalt source of gamma quanta, obtained as a result of calculation for rocks and ores with densities

Fig. 5. Dependence of the distribution of the differential spectrum of scattered gamma radiation on the energy \(K\) for three media with densities 1, 2, 3, 4, 5 g/cm\(^3\) (the index of the curves corresponds to the density of the rock)

1, 2, 3, 4, and 5 g/cm\(^3\) and with corresponding effective atomic numbers 7.42; 12.74; 15.67; 19.3; 22.6 are shown in Fig. 5. It is of interest to compare these data with the results of a numerical calculation of equation (3), obtained by P. R. Karr and I. K. Lamkin [4]

\(^1\) In [3], instead of the function \(\varphi(K)\), its value multiplied by the ratio of the total coefficient of absorption and scattering of gamma quanta to the scattering coefficient was used throughout.

for aluminum \((Z = 13,\ \rho = 2.6\ \text{g}/\text{cm}^3)\) at different energies of the gamma-quantum source, shown in Fig. 6. Comparison of these data leads to the following conclusions.

1. In the energy spectrum of scattered gamma radiation, depending on the density of the medium and its material composition, an accumulation of soft gamma radiation occurs with a sharply pronounced maximum, the position of which depends on the properties of the medium.

2. A change in the energy of the gamma-quantum source does not lead to a displacement of the maximum of the soft component of the spectrum of scattered gamma radiation. Therefore, the presence in the betatron radiation of a complex spectrum (a white spectrum) will not lead to a displacement of the maximum of the spectrum of scattered radiation.

Fig. 6. Dependence of the distribution of the differential spectrum of scattered gamma radiation in aluminum on the energy \(K'\) for three gamma-quantum sources: 1 — \(0.5\) MeV, 2 — \(2.55\) MeV, and 3 — \(5.1\) MeV

Thus, on the basis of these calculations it may be asserted that, by recording the maximum energy values of the soft component of scattered gamma radiation with the aid of a differential gamma spectrometer, it will be possible to distinguish the rocks penetrated by a borehole according to their density and material composition, provided that the body of the instrument is made of a material that allows the soft gamma radiation to pass through to the spectrometer, including the maximum of the energies of interest to us.

As a result of all the foregoing, the following procedure may be outlined for investigating boreholes by the gamma–gamma logging method with the aim of increasing the accuracy of determining the density of rocks and their material composition:

1. Use gamma-quantum sources, insofar as possible, of higher power and energy.

2. Carry out the investigation with an optimal probe, ensuring, as far as possible, the study of the largest volume of rock scattering the gamma radiation.

3. Use a differential gamma spectrometer as the gamma-quantum indicator and record the soft component of the spectrum of scattered gamma radiation in order to ensure the greatest sensitivity of the gamma–gamma logging method to changes in the density of rocks and their material composition.

At present, the use of natural and artificial gamma-quantum sources in borehole investigations by the gamma–gamma logging method makes it possible to determine the density of rocks with an accuracy of up to \(0.1 \div 0.05\ \text{g}/\text{cm}^3\). When gamma–gamma logging is performed with a betatron, it will be possible substantially to increase the accuracy of determining the density of the rocks and ores penetrated by the borehole.

References

1. Filippov E. M. On the theory of gamma–gamma logging (GGL). Applied Geophysics. Gostoptekhizdat, issue 17, 1957.

2. Filippov E. M. Gamma–gamma logging. The Use of Radioactive Isotopes in the Petroleum Industry (collection of reports). Gostoptekhizdat, 1957.

3. Filippov E. M. Investigation of the energy spectrum of scattered gamma radiation in rocks of different mineralogical composition and different density. Applied Geophysics. Gostoptekhizdat, issue 19, 1958.

4. P. R. Karr and J. O. Lamkiu. Phys. Rev., 76, 12, 1949.