Gamma-Neutron Logging

V. M. Zaporozhets, V. V. Sulin

(Presented at the Second Inter-Institute Conference on Electron Accelerators)

The development of charged-particle accelerators for the investigation of boreholes makes it possible to raise the question of creating a new method of radioactive logging—gamma-neutron logging (GNL), in which rocks are irradiated by a flux of gamma rays of high energy and intensity, and the resulting photoneutron radiation of the rocks is recorded.¹)

In view of the small cross section and the high threshold energy of \(\gamma, n\) reactions for most rock-forming elements, the realization of GNL presents great difficulties. It is therefore of interest to consider possible methods for carrying out GNL, to estimate quantitatively the effectiveness of the method, and, on this basis, to determine the requirements that must be satisfied by a borehole gamma-quantum generator for GNL. The feasibility of these requirements characterizes the practicability of the method.

Among the theoretically possible methods for detecting differences in the photoneutron properties of rocks (recording changes in the intensity of gamma radiation occurring as a result of the photocapture of gamma quanta by the nuclei of atoms in the rock; recording the flux density of fast photoneutrons; recording the flux density of slow neutrons), we shall orient ourselves toward the method of recording the flux density of thermal neutrons, which is practically the most feasible, since industrial geophysics has at its disposal scintillation counters of slow neutrons with an efficiency of the order of 10–15%.

Let us consider a source of gamma quanta located at point \(O\) of an infinite homogeneous medium and emitting \(Q(E)\) gamma quanta per unit time with energy \(E\) (Fig. 1). At the same point there is an indicator of slow neutrons. The medium is characterized by the gamma-quantum absorption coefficient \(\mu(E)\), by the content of a certain element under investigation, \(N\) nuclei in \(1 \text{ cm}^3\), in region II and by its absence in region I, bounded by radius \(r_0\). The cross section of the \(\gamma, n\) reaction for the element under investigation is \(\sigma_{\gamma,n}(E)\). The slowing-down parameter of the fast photoneutrons formed in the medium is \(L_f(E)\).

We shall calculate the flux density of slowed-down photoneutrons at point \(O\), where the indicator is located. In region II of the medium let us select an ele—

¹) That is, neutrons emitted by the nuclei of atoms of rocks as a result of photonuclear reactions.

an elementary volume \(dv\), located at a distance \(r\) from the point \(O\) (Fig. 1). Let us determine the flux density of thermal neutrons \(dn(E)\) at the point \(O\), arriving at it from the volume \(dv\), as a result of the production in the rock of fast neutrons by the \(\gamma,n\) reaction and their subsequent slowing down. For this purpose we shall use the formula given in [1], which determines the density \(df(E)\) of thermal neutrons at the point \(O\) as a function of \(Q(n)\), the number of thermal neutrons produced per unit time in the elementary volume \(dv\) as a result of the slowing down of fast neutrons in the medium:

Fig. 1. Illustration for the derivation of formula (4)

\[ df(E)=\frac{Q(n)dE\,\tau}{8\pi L_f(E)x(E)}\,e^{-\frac{r}{L_f(E)}}\,dv, \tag{1} \]

where \(\tau\) is the lifetime of a thermal neutron in the medium,

\[ x(E)=1-\frac{L^2}{L_f^2(E)}, \]

\(L\) is the diffusion length of thermal neutrons.

As is known, the diffusion of thermal neutrons obeys the law of the kinetic theory of gases. From this theory it follows that the flux density \(dn(E)\) of diffusing particles is related to the density \(df(E)\) of these particles by the following relation:

\[ dn(E)=df(E)\frac{v_T}{4}, \tag{2} \]

where \(v_T\) is the velocity of a thermal neutron.

Replacing in formula (1) the function \(Q(n)\) by the function

\[ Q_1(E)=\frac{Q_n}{N\sigma_{\gamma,n}(E)}. \tag{3} \]

Let us relate the function \(Q_1(E)\) introduced by us to the function \(Q(E)\), characterizing the radiation of the gamma-quantum source, by the following relation:

\[ Q_1(E)=Q(E)[1+A(E)]\,\frac{e^{-\mu(E)r}}{4\pi r^2}, \tag{4} \]

where \(A(E)\) is a coefficient that takes into account the increase in the number of gamma quanta entering the \(\gamma,n\) reaction as a result of scattering of primary gamma quanta that have not yet undergone scattering on electrons

atoms to react with nuclei via the \(\gamma,n\) reaction; \(\dfrac{e^{-\mu(E)r}}{4\pi r^2}\) is a coefficient accounting for the absorption of gamma quanta by the rock on their path to the volume \(dv\). After substituting (1), (3), and (4), formula (2) takes the following form:

\[ dn(E)=Q(E)[1+A(E)]\times \]

\[ \times \frac{e^{-\mu(E)r}}{4\pi r^2}\,dE\,N\sigma_{\gamma,n}(E)\, \frac{v_T\tau}{3u^3(E)\alpha(E)}\, e^{-\frac{n}{L_f(E)}}\,dv . \tag{5} \]

Integrating the expression over the entire volume \(v\), we obtain the value of the flux density of slowed photoneutrons \(n(E)\) at the point \(O\):

\[ n(E)=Q(E)[1+A(E)]\, \frac{N\sigma_{\gamma,n}(E)v_T\tau\,dE}{32\pi L_f^3(E)\alpha(E)} \times \]

\[ \times \frac{e^{-r_0\left[\mu(E)+\frac{1}{L_f(E)}\right]}} {\mu(E)+\frac{1}{L_f(E)}} . \tag{6} \]

As an example of the use of the obtained formula (6), let us carry out a numerical calculation of the number of slowed photoneutrons emitted by deuterium nuclei. Determining the deuterium content in rocks is of interest because, according to available information, its content in oil is 2–3 times higher than in the formation waters of oil fields [2]. For selective identification of deuterium, which has a low threshold energy \((E_\gamma \simeq 2.24\ \text{MeV})\) for the \(\gamma,n\) reaction, it is inadvisable to use high-energy gamma radiation. Therefore we shall assume that the energy of the gamma quanta does not exceed \(10\ \text{MeV}\). This greatly simplifies the calculation, since it makes it possible to take into account only single scattering of gamma quanta by atomic electrons. The influence of subsequent scattering of gamma quanta in the case under consideration was not taken into account because, after the second collision, each gamma quantum loses such a large part of its energy that thereafter it will have only a small probability of entering into the \(\gamma,n\) reaction with deuterium nuclei.

Furthermore, at a small initial energy of the gamma quanta, the radiation reaching region II is characterized by an approximately uniform distribution of gamma quanta in energy within the limits from the initial energy to the threshold energy \((2.24\ \text{MeV})\). This is evident, for example, in Fig. 2, constructed from the data of work [3] for an initial gamma-quantum energy of \(5\ \text{MeV}\). Therefore, in the case under consideration one may assume that, along with gamma quanta of the initial energy \(E_0\), singly scattered gamma quanta with energy enter into the \(\gamma,n\) reaction

\[ E_{\mathrm{cp}}=\frac{E_{\mathrm{por}}+E_0}{2}, \tag{7} \]

where \(E_{\mathrm{por}}=2.24\ \text{MeV}\). The function \(A(E)\), characterizing the probability of formation, after scattering, of gamma quanta capable of entering into the \(\gamma,n\) reaction, is calculated in the present case from the formula

\[ A(E)=\frac{\mu(E_{cp})}{\mu(E_0)}\cdot\frac{E_0-E_{por}}{q}, \tag{8} \]

where \(q\) is the energy loss of gamma quanta in one scattering event.

Taking formulas (7) and (8) into account, expression (6) can be transformed as follows:

\[ \begin{aligned} n(E)=&\ \frac{Q(E)\sigma_T \tau N\,dE}{32\pi} \Bigg[ \frac{\sigma_{\gamma,n}(E_0)}{L_f^3(E_0)\alpha(E_0)} \cdot \frac{ e^{-r_0\left[\mu(E_0)+\frac{1}{L_f(E_0)}\right]} }{ \mu(E_0)+\frac{1}{L_f(E_0)} } \\ &\quad+ \frac{\mu(E_{cp})}{\mu(E_0)} \frac{E_0-E_{por}}{q} \cdot \frac{\sigma_{\gamma,n}(E_{cp})}{L_f^3(E_{cp})\alpha(E_{cp})} \frac{ e^{-r_0\left[\mu(E_{cp})+\frac{1}{L_f(E_{cp})}\right]} }{ \mu(E_{cp})+\frac{1}{L_f(E_{cp})} } \Bigg]. \end{aligned} \tag{9} \]

As a source of gamma quanta in gamma-neutron logging, one may use a gamma-quantum generator with an accelerating tube employing the \(p,\gamma\) reaction, or a small-sized betatron.

Fig. 2. Spectrum of scattered gamma radiation
\(E\) — energy of gamma radiation,
\(N\) — number of gamma quanta in arbitrary units

The energy of the gamma radiation of a gamma-quantum generator depends on the target material and the energy of the accelerated protons. The energy of the bremsstrahlung gamma radiation of a betatron depends on the type of target and the energy of the accelerated electrons. The intensity of gamma radiation in both cases depends on the strength of the current of the accelerated particles (protons or electrons) onto the target. Since the spectrum of gamma radiation in these two cases is different, we shall consider each of them separately. As the final goal of the calculation, we shall set the estimation of the required current of charged particles onto the target in order to obtain a sufficient counting rate of photoneutrons by an indicator placed at point \(O\).

Let us first consider the simpler case in which an accelerator tube is used as the gamma-quantum generator, giving monochromatic gamma radiation of energy \(E_0\).

In this case, taking \(Q(E)dE = 1\) in formula (9), we obtain the flux density of slowed-down photoneutrons per one gamma quantum of the generator radiation; the corresponding counting rate of slowed-down photoneutrons may be obtained from the formula

\[ J = n(E)S\varepsilon . \tag{10} \]

Figure 3 (curve 1) shows the dependence, obtained by calculation for this condition from formulas (9) and (10), of the counting rate of slowed-down photoneutrons on the energy of gamma quanta for an oil-bearing sandstone with a porosity of 30%, containing \(0.618 \cdot 10^{19}\) deuterium atoms in \(1\text{ cm}^3\), i.e., twice its content in formation waters. The parameters of the slow-neutron indicator were taken to be those of the scintillator now widely used \((S = 20\text{ cm}^2, E = 15'' )\) [4].

As can be seen from Fig. 3, with an increase in the energy of the gamma quanta the counting rate at first increases sharply, and then less intensively. It may be assumed that with a further increase in the energy of the gamma quanta the counting rate will not increase without limit. Indeed, although at high energies, as a result of multiple scattering, a considerable number of gamma quanta capable of entering the \(\gamma, n\) reaction with deuterium nuclei may be formed, this will be compensated by an increase in \(L\), the third power of which enters into the denominator of formula (8), and by a decrease in the cross section of the \(\gamma, n\) reaction for gamma quanta that have not undergone scattering. Thus, for reasons of efficiency in the use of the gamma-quantum generator and in order to reduce interference from the occurrence of the \(\gamma, n\) reaction with other elements in GNL aimed at detecting deuterium, it is not advisable to raise the gamma-quantum energy above 5–7 MeV.

Fig. 3. Dependence on the energy of the gamma-radiation source of the counting rate \(J\) of photoneutrons for oil-bearing sandstone with a porosity of 30%. 1 — case of using a gamma-quantum generator with an accelerator tube emitting gamma quanta with energy \(E_0\); 2 — case of using a betatron with maximum energy of bremsstrahlung gamma radiation \(E_{\max}\).

Let us now consider the case of using a betatron as the gamma-quantum generator for photoneutron logging.

It is known that the bremsstrahlung gamma radiation of a betatron has a continuous spectrum, which can be described by the well-known Schiff formula [5], giving good agreement with experimental data [6]. In this case, to solve equation (9) one would have to integrate it with respect to \(E\) over the interval of its values from the threshold energy of the \(\gamma, n\) reaction to the maximum energy of the betatron \(E_{\max}\). In view of the complexity of this calculation, we shall confine ourselves to an approximate computation.

We characterize the energy of the betatron bremsstrahlung gamma radiation by its mean value \(E_{\text{av}}\), the definition of which is explained in Fig. 4, which shows part of the spectrum of the betatron gamma radiation for one of the values of the maximum energy of the bremsstrahlung gamma radiation, calculated by Schiff’s formula.

Fig. 4. Spectrum of the betatron bremsstrahlung gamma radiation with maximum gamma-quantum energy \(E_{\max}\), calculated by Schiff’s formula.
\(E\) is the energy of the betatron bremsstrahlung gamma radiation in arbitrary units.

The distribution of gamma quanta by energy is characterized by the quantity \(K\), which is the ratio of the areas \(S_2\) and \(S_1+S_2\) (Fig. 4) and shows what part of the source gamma radiation is capable of reacting with the nuclei of deuterium atoms by the \(\gamma,n\) reaction.

Thus, on the basis of Schiff’s formula, the dependence on the maximum energy of the betatron gamma radiation of the quantities \(E_{\text{av}}\) and \(K\), shown in Fig. 5, was obtained. From these data, using formulas (9) and (10), the dependence of the counting rate of photoneutrons by an indicator placed at point \(O\) on the maximum energy of the betatron, shown in Fig. 3 (curve 2), was obtained. In this calculation, in formula (8) the quantity \(Q(E)\) was replaced by the quantity \(KQ(E_{\max})\), the quantity \(E_0\) by the quantity \(E^{0}_{\text{av}}\), the difference \(E_0-E_{\text{thr}}\) by the difference \(E^{0}_{\text{av}}-E_{\text{thr}}\), and \(E_{\text{av}}\) by the quantity

\[ E_{1\text{av}}=\frac{E_{\text{thr}}-E^{0}_{\text{av}}}{2}. \]

The remaining conditions adopted in calculating curve 2 are the same as in calculating curve 1. The less steep rise of curve 2 with increasing \(E\), as compared with curve 1, is explained by the considerable content in the betatron radiation of gamma quanta with energies below the threshold energy of the \(\gamma,n\) reaction for deuterium.

Using curves 1 and 2 (Fig. 3), one can estimate the necessary magnitude of the charged-particle current on the target of the gamma-quantum generator for photoneutron logging. It is necessary that the charged-particle current provide such a yield of gamma quanta at which the co-

the counting rate of slowed-down photoneutrons in the borehole would ensure the required measurement accuracy and would substantially exceed the counting rate of the natural neutron radiation of rocks.

Fig. 5. Dependence of the parameters \(E_{\mathrm{av}}\) and \(K\) on the maximum energy \(E_{\max}\) of the betatron

Let us consider the case when measurements by the GNL method are performed at stops in the movement of the downhole instrument in the borehole. The relative error of the measurements \(\delta\) is determined in this case by the formula

\[ \delta = \frac{1}{\sqrt{Jt}}, \tag{11} \]

where \(J\) is the counting rate of slowed-down photoneutrons in pulses/sec,
\(t\) is the measurement time in seconds.

Taking as the maximum permissible values \(\delta = 0.1\) and \(t = 600\) sec, we obtain that \(J\) must not be less than 0.2 pulses/sec. This value of \(J\) exceeds the counting rate of neutrons spontaneously emitted by rocks, which, when the same indicator is used as in GNL, will on average be \(\sim 0.03\) pulses/sec [7, 8].

The calculation shows that, in order to obtain a counting rate of slowed-down photoneutrons of 0.2 pulses/sec when using as the gamma-radiation source a betatron with a maximum gamma-quantum energy of 5 MeV, an electron current of the order of 0.02 μA is required. The gamma-radiation dose rate in air at a distance of 1 m from the target of such a betatron will be approximately 1 roentgen per minute.

When using gamma-quantum generators with an accelerator tube employing reactions of the \(p,\gamma\) type on targets of \(Li^7\), \(F^{19}\), or \(B^{11}\), proton currents on the target are required equal, respectively, to 35, \(3 \cdot 10^4\), and \(16 \cdot 10^5\) μA (see Table 1, compiled from literature data).

The electron current on the target in existing betatrons is measured in hundredths of a microampere; therefore, in principle, it is possible to use a betatron as a gamma-quantum generator for gamma-neutron logging. The possibility of using for this purpose a gamma-quantum generator with an accelerator tube is less obvious, since the manufac-

Table 1

Nuclear reactions \(p,\gamma\)

…tion of targets from \(Li^{6}\) is associated with large expenditures and difficulties. Also presenting a difficulty is the creation of a voltage source of \(0.4\)–\(0.5\) MeV, necessary for carrying out the \(p,\gamma\) reaction on \(Li^{6}\).

Fewer difficulties arise in creating a gamma-quantum generator with an accelerating tube intended for investigating the lithology of rocks. In this case a more accessible target of \(Li^{7}\) may be used, ensuring the production of radiation with energy up to 17.6 MeV. Since the cross sections of the \(\gamma,n\) reaction for sandstone and clay are approximately the same as for deuterium (respectively 2.5; 4.2 and 2.3 millibarns), and the content of atoms entering into the reaction is \(\sim 3 \cdot 10^{3}\) times greater than in the investigation of deuterium, the following current values will prove sufficient in investigating the lithological character of rocks: betatron \(\sim 1 \cdot 10^{-5}\) μA, accelerating tube with a \(Li^{7}\) target \(\sim 10\) μA.

The generally favorable data obtained here make it possible to regard as expedient the further development of the theory of the GNL method, of borehole accelerators for its implementation, and also the conducting of experimental irradiation of rock samples using a laboratory betatron.

REFERENCES

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