Carbon dioxide is one of the most common greenhouse gases, countries all over the world are eager to find a way to eliminate its harm to the environment^[1,2]. Capturing industrial CO₂ and injecting it into deep formation for long-term storage is a promising method to mitigate the greenhouse effect^[3]. CO₂-EOR technology not only meets this requirement, but also realizes secondary exploitation of the oil field, which can increase oil and gas production to increase economic benefits. The “In Salah” project in Algeria has been successfully injected over 3×10⁴ t of CO₂ into fractured sandstone formation in five years, proving the feasibility and considerable economic value of this method^[4].

At present, many methods monitoring CO₂ storage status have been successfully applied all over the world^[5], such as seismic^[6,7,8], CT^[9] and well temperature logging^[10,11], etc. In CO₂-EOR technology, CO₂ saturation is a key parameter to characterize the distribution and migration of CO₂ gas. As an important method of gas saturation evaluation in nuclear geophysical exploration, pulsed neutron logging technology plays an indispensable role in the evaluation of CO₂ saturation in the CO₂ flooding reservoirs^{[12,13,14,15,16,17]}. In recent years, the Schlumberger company has designed a multifunctional pulsed neutron logging tool, which uses a long source distance YAP (yttrium aluminate) crystal detector combined with a pulsed neutron yield monitor to measure fast neutron scattering cross-section (σ_f) to quantitatively monitor the gas saturation of the reservoirs^[18]. However, characterizing the σ_f of 14 MeV neutron with inelastic gamma information directly will be affected by the formation lithology to a certain extent, leading to errors in the interpretation of gas saturation.

Aiming at the difference in the deceleration of fast neutrons by CO₂, water and heavy oil, based on the theory of fast neutron scattering and secondary gamma distribution and the three-detector pulsed neutron logging technology, this paper proposes a method characterizing σ_f by the combination of secondary inelastic and capture gamma information to overcome the influence of formation lithology on σ_f. In view of the presence of shale in the actual reservoirs, the corresponding CO₂ saturation interpretation model has been given, and the influences of various factors on the relationship between σ_f and the formation porosity have been explained. Thus, a more complete quantitative evaluation method for gas saturation of CO₂ injection heavy oil reservoirs has been established.

The pores of heavy oil reservoirs are generally filled with high-density and high-viscosity heavy oil, and some pores also contain water. In the process of CO₂ flooding, the pore fluid is gradually displaced by CO₂ gas to form an oil-gas two-phase or oil-gas-water three-phase fluid. CO₂ has hydrogen index of zero, and the density, thermal neutron capture cross-section, fast neutron scattering cross-section etc. quite different from formation water and heavy oil. The CO₂ injected into the formation will cause great changes in the thermal neutron capture cross-section and fast neutron scattering cross-section of the formation. As a parameter that can provide information independent of the other neutron measurements for formation evaluation^[18], the fast neutron scattering cross-section has good performance on monitoring gas saturation of CO₂ flooding heavy oil reservoirs quantitatively.

According to the ENDF/B-VII.0 nuclear database^[19], the microscopic cross-section of formation elements can be obtained, and then the macroscopic elastic scattering cross-section of the medium can be obtained by the calculation formula. Fig. 1 shows the relationships between the elastic scattering cross-sections of different formation matrix minerals and pore fluids with neutron energy. It can be seen that the elastic scattering cross-section of gas is significantly smaller than that of formation matrix mineral and pore fluids, and this feature of CO₂ is particularly obvious. In this paper, the elastic scattering cross-section is taken as σ_f because of its difference in formation media to evaluate the CO₂ saturation quantitatively.

The numerical integration method is used to calculate the σ_f values of formation matrix minerals and pore fluids, and the results are shown in Table 1. It can be seen that the gases, especially CO₂, have lower σ_f, the formation matrix minerals have generally larger σ_f, and fresh water and heavy oil have even larger and similar σ_f. Therefore, based on the characteristic of ultra-low elastic scattering cross-section of CO₂, the oil-water-gas three- phase fluid in the CO₂ flooding heavy oil reservoir can be regarded as gas-liquid two-phase fluid.

When entering formation, fast neutrons would collide with the formation nuclei and slow down. In this process, secondary inelastic gamma-ray is generated by the inelastic scattering of fast neutrons and formation elements. In the spherical model shown in Fig. 2, the deuterium- tritium neutron source (D-T) located at point O, emitted fast neutrons with an energy of 14 MeV uniformly around, and the spherical surface with a radius of r was used to record the inelastic gamma-ray from the formation.

According to the fast neutron scattering theory^[20], the fast neutron flux distribution where inelastic scattering occurs is:

The intensity of inelastic gamma-ray depends on the type of the formation element and its microscopic inelastic scattering cross-section. The number of gamma photons produced by element k in one collision is i_k, the corresponding microscopic inelastic scattering cross-section is σ_k, then the number of gamma rays produced in a spherical shell with a radius of r and a thickness of dr is:

The number of inelastic gamma photons in the sphere model is equal to the integral of dI from r=0 to r=+. With the influence of gamma attenuation considered, assuming that the σ_f are equal everywhere of the formation, the mass attenuation coefficient (μ_m) of the formation and the average number of gamma photons (i) generated by the interaction of one fast neutron with the formation elements are constant, ignoring the gamma photon contribution outside the detectors, according to the theory of neutron gamma coupling field^[21], the inelastic gamma flux can be expressed as:

Eq. (3) is simplified by Lagrange Mean Value Theorem:

In Eq. (4), α is a proportionality factor related to the source distance, which is a fixed value for a given instrument. It can be seen from Eq. (4) that the inelastic gamma flux is not only related to the value of σ_f, but also related to the formation density. Therefore, σ_f can be regarded as a function of inelastic gamma flux and formation density. Based on the multi-detector pulsed neutron logging tool, the formation density can be characterized by the combination of inelastic and capture gamma count ratio of dual-detector^[22]:

In Eq. (5), A, B, and C are constants, which are obtained by fitting simulation data or measured data of calibration well.

Combining Eqs. (4) and (5), the quantitative characterization of σ_f can be realized by the inelastic and capture gamma of the near and far detectors:

where $K=\frac{1}{R\left( \alpha -1 \right)}$ $L=\frac{A\alpha {{\mu }_{\text{m}}}}{\alpha -1}$ $M=\frac{B\alpha {{\mu }_{\text{m}}}}{\alpha -1}$ $N=\frac{C\alpha {{\mu }_{\text{m}}}}{\alpha -1}+\ln \frac{4\pi R}{i{{\sigma }_{\text{in}}}{{\phi }_{\text{0}}}}$

Eq. (6) shows that the σ_f can be characterized by combining inelastic and capture gamma information of the near and far detectors, and the coefficients K, L, M, and N can be obtained by fitting simulation data or calibrating well measured data.

σ_f represents a macroscopic physical characteristic of the reservoir, which follows the physical model of bulk-volume rock. Using Eq. (6) to obtain the value of σ_f; for sandstone reservoirs filled with CO₂ and heavy oil in pores, the σ_f satisfies the following equation:

Then the gas saturation interpretation model of the reservoir is:

The formation of pure sandstone matrix is an ideal condition, but the actual formation matrix always contains a certain proportion of shale or other solid minerals. Taking the shale-bearing formation as research object, based on the difference in the value of σ_f of shale and sandstone matrix, σ_f of the formation satisfies the following equation:

It can be seen from Eq. (9) that the presence of shale will cause changes in the physical model of bulk-volume rock. When the formation contains shale, the shale content at the measurement point and the porosity of the formation need to be combined to correct the sandstone formation model shown in Eq. (7). The interpretation model of gas saturation in shale-bearing reservoir is:

Compared with the gas saturation interpretation model for pure sandstone formation shown in Eq. (8), Eq. (10) takes into account shale-bearing formation conditions. When the formation contains other solid minerals, it is also necessary to modify the gas saturation interpretation model to meet the actual formation matrix conditions.

Monte Carlo simulation (MCNP) is a random sampling or statistical experiment method. It can be used to simulate neutrons, photons, electrons and their coupled transport processes, which has been widely applied in radioactive logging simulation, response and instrument manufacturing^[23]. In this paper, MCNP was used to establish the instrument and formation model to simulate all the processes of fast neutron transport, including fast neutron slowing down to produce inelastic gamma, capture of thermal neutron to produce capture gamma and gamma attenuation. Based on the simulation results, the characterization of σ_f and gas saturation response were analyzed to realize the quantitative monitoring of gas saturation of CO₂ flooding heavy oil reservoir.

The numerical calculation model built in this study is shown in Fig. 3. The formation model is set up as a casing well with a height of 150 cm and a diameter of 120 cm. The borehole is 20 cm in diameter and filled with fresh water. The formation matrix is sandstone, limestone or dolomite. The casing is 0.7 cm thick stainless steel, and a 3 cm thick CaSiO₃ cement filled between the casing and the formation. The reservoir has a porosity changing from 0 to 40% with an interval of 5%, the CO₂ saturation changed from 0 to 100%. The formation pressures were set at 35, 50 and 65 MPa, and the formation temperatures were set at 363.15, 393.15 and 423.15 K. The densities of CO₂ gas, fresh water and heavy oil were 0.50, 1.00 and 0.95 g/cm³ respectively at the formation pressure of 35 MPa and formation temperature of 363.15 K.

The three-detector pulsed neutron logging tool was used to monitor the CO₂ gas saturation with shielding added between the source and each detector. The neutron pulse emission time was designed at 40 μs, and the measurement time of inelastic and capture gamma-ray was 0-40 μs and 50-1000 μs, respectively. The near, middle and far detectors of the logging tool coded 1, 2, and 3, recorded the count of inelastic and capture gamma-ray at the same time. 17-4PH steel was used as the material of the instrument housing, with a thickness of 0.5 cm. The source was a D-T pulsed neutron source, which uniformly emitted high-energy fast neutrons (14 MeV) to the formation. The detector material was LaBr₃ crystal, and the source distances of the three detectors were 27.5, 40.0, and 60.0 cm respectively. The crystal of three detectors were 5 cm in diameter and 5, 10 and 10 cm long, respectively.

By simulating different reservoirs containing heavy oil and CO₂, we obtained the three-detector inelastic and capture gamma counts of different lithologic reservoirs (sandstone, limestone and dolomite) of different porosities (0-30%, interval of 5%) saturated with heavy oil or CO₂. The purpose of the combination of inelastic and capture gamma count ratio shown in Eq. (5) is to eliminate the influence of formation density. Considering the counting statistics and accuracy of σ_f, the gamma information of near and far detectors was selected to quantitatively characterize σ_f. By Eq. (6) and multiple linear regression method, a equation converting inelastic and capture gamma information to σ_f has been established:

To verify the applicability of the Eq. (11), the calculated σ_f values and the theoretical σ_f values set in the formation model were compared. The error between the calculated and the theoretical values is expressed as:

Table 2 shows the error analysis of the calculated σ_f and theoretical σ_f values of the heavy oil saturated and CO₂ saturated sandstone reservoirs with different porosities. Table 3 shows the error analysis of the calculated σ_f and theoretical σ_f values of the CO₂ saturated sandstone, limestone and dolomite reservoirs with different porosities. These calculations were used to verify the accuracy of the characterization equation under different oil-gas- bearing and lithologic conditions. It can be seen that Eq. (11) can characterize the formation σ_f accurately, and the error between the calculated value and the theoretical value can be controlled within ±2%. Combining inelastic and capture gamma of the pulsed neutron logging can quantitatively characterize the σ_f of CO₂ flooding heavy oil reservoir.

In actual heavy oil reservoirs, the use of Eq. (11) to characterize the reservoir σ_f may cause errors because of the influence of factors such as shale content, heavy oil density, formation temperature and pressure, and gas type, etc. The σ_f obtained by Eq. (11) should be corrected first for the influence of different wellbore-formation factors to ensure the accuracy and realize the universal application of the equation in CO₂ flooding effect evaluation.

The instrument and formation model shown in Fig. 3 was established based on Monte Carlo numerical calculation method. Under the condition of a sandstone reservoir with the mixed fluid of heavy oil and CO₂ in pores, the formation porosity is set to 0-40% with an interval of 5%, and the CO₂ saturation is set to 0, 25%, 50%, 75% and 100% respectively. By the characterization equation of σ_f, the relationship between σ_f and formation porosity under different CO₂ saturations is obtained, as shown in Fig. 4. It can be seen that the CO₂ saturation of the reservoir has a great influence on the relationship between σ_f and porosity. At low CO₂ saturation, heavy oil is the main component of pore fluid, except for the formation matrix, the heavy oil dominates the deceleration to neutrons. As the CO₂ saturation increases, CO₂ in the pores gradually increases, the deceleration of CO₂ on neutrons becomes gradually significant and finally dominant. Compared with the formation matrix, heavy oil has higher σ_f, while CO₂ has much lower σ_f. Therefore, at low CO₂ saturations (0 and 25%), σ_f shows an upward trend with the increase of formation porosity; at high CO₂ saturations (50%, 75% and 100%), σ_f shows a clear downward trend with the increase of formation porosity. It can be seen that the macroscopic σ_f is determined by porosity and gas saturation in CO₂ flooding heavy oil reservoir. The macroscopic σ_f of formation can be characterized with gamma count, then combined with the physical model of bulk-volume rock, the gas saturation of the CO₂ flooding heavy oil reservoir can be quantitatively monitored.

Actual reservoirs often contain different types of shale which affect the deceleration ability of the reservoir to fast neutrons. The CO₂ saturation of shale- bearing reservoir should be calculated by Eq. (10). Taking chlorite as the shale component for research, based on the instrument and formation model shown in Fig. 3, the sandstone formation was set at shale contents of 10%, 20%, 30%, and 40%, porosities of 0-40% with an interval of 5%, and CO₂ saturations of 0, 50%, and 80%, respectively. The relationships between σ_f and the porosity of the formation at different shale contents were simulated, and the results are shown in Fig. 5. In general, when the shale is chlorite, the presence of shale makes the calculated values of σ_f significantly smaller than those of the pure sandstone formation. At high CO₂ saturation, the curves of σ_f with formation porosity at different shale contents are approximately parallel. At low CO₂ saturation, with the increase of porosity, the influence of shale content on σ_f gradually decreases. This means that the change of σ_f caused by shale is dependent upon the formation porosity and shale content, and in actual reservoirs, it is more reasonable to interpret CO₂ saturation of shale-bearing formations by using σ_f and the physical model of bulk-volume rock jointly.

This paper proposes a quantitative interpretation method for evaluating the gas saturation of CO₂-injected heavy oil reservoirs by using σ_f. However, in actual CO₂-injected heavy oil reservoirs, the gas saturation evaluation method may fail sometimes due to the uncertainty of logging environment. Therefore, it is necessary to analyze and correct the formation influencing factors (such as temperature, pressure and wellbore environment, etc.), which is the basis for ensuring the applicability of monitoring CO₂ gas saturation with σ_f quantitatively.

Changes in formation temperature and pressure are the main factors that cause the dissolution of CO₂ in heavy oil to form miscible oil for CO₂-injected reservoirs. Due to differences in CO₂ solubility, the miscible oil varies in density under different temperature and pressure conditions. According to the research of Marra et al.^[24], the change in the density of the miscible oil caused by the changes of temperature and pressure can be calculated according to the following equations:

Based on the instrument and formation model shown in Fig. 3, under the condition of a sandstone formation, the formation porosity is set to 0-40% with an interval of 5%, and the CO₂ gas saturation is set at 0, 50% and 80%. The formation temperature is set to 363.15 K, and the formation pressure is changed to 35, 50, 65 MPa to study the influence of formation pressure. Similarly, the formation pressure is set to 35 MPa, and the formation temperature is changed to 363.15, 393.15 and 423.15 K to study the influence of formation temperature. It can be seen from Fig. 6 that the formation temperature and pressure conditions have little effect on the CO₂ gas saturation calculated with σ_f. According to calculation by Eqs. (13) and (14), when the formation temperature is 363.15 K and the formation pressure of heavy oil reservoir increased from 35 MPa to 65 MPa, the density of the miscible oil would increase from 0.950 to 0.963 g/cm³. Similarly, at the formation pressure of 35 MPa, when the formation temperature increased from 363.15 K to 423.15 K, the density of the miscible oil would increase from 0.950 g/cm³ to 0.983 g/cm³. Clearly, the variation range of miscible oil density caused by the change of formation temperature and pressure is relatively small, so the deceleration capacity of formation to fast neutrons hardly changes. Together with the formation matrix factor, the deceleration capacity of the reservoir to fast neutrons remains unchanged. Within the studied formation temperature and pressure ranges, the miscible oil density will not change drastically, and the change in heavy oil density caused by changes in reservoir temperature and pressure is not enough to affect the deceleration ability of the formation to fast neutron. Therefore, when evaluating CO₂ gas saturation in CO₂ flooding heavy oil reservoir with σ_f, the influences of formation temperature and pressure can be ignored.

In heavy oil reservoirs developed by CO₂ flooding, heavy oil would change in density with certain temperature and pressure range due to the changes of CO₂ solubility with temperature and pressure^[25]. At the temperature of 333.15 K, when the pressure increases from 35 MPa to 65 MPa, calculation shows the density of heavy oil is 0.94 to 1.00 g/cm³. Based on the instrument and formation model shown in Fig. 3, the relationships between σ_f and porosity at the heavy oil densities of 0.95, 0.97, and 0.99 g/cm³ were simulated. It can be seen from Fig. 7 that at the low CO₂ saturation (0), the density of heavy oil has some influence on σ_f; with the increase of heavy oil density, the calculated σ_f shows an increasing trend. But in the reservoir with high CO₂ saturation (80%), the influence of heavy oil density is not obvious, the reason is that most of the reservoir fluid has been displaced by CO₂. For actual heavy oil reservoirs developed by CO₂ flooding, the heavy oil density after multiple cycles of CO₂ injection would change in much smaller range than that studied in this work. The changes in heavy oil density after CO₂ flooding has a negligible effect on the measurement of CO₂ saturation.

CO₂ flooding in heavy oil reservoir often takes multiple cycle injection or alternate water and CO₂ injection. Therefore, the fluid in wellbore would change from drilling fluid to heavy oil, CO₂ or water. To study the influence of wellbore fluid on the evaluation of CO₂ saturation with σ_f in CO₂ flooding reservoir, based on the instrument and formation model shown in Fig. 3, with wellbore fluid set as water, gas, and heavy oil respectively, the relationships between σ_f and formation porosity under the different wellbore fluids were simulated, and the results are shown in Fig. 8. It can be seen that the curves of σ_f with formation porosity under different wellbore fluid conditions are parallel. The difference in the deceleration ability of water and oil to fast neutrons is small. The value of σ_f when the wellbore is filled with oil is slightly smaller than that when wellbore is filled with water. Compared with water and oil, gas has much weaker deceleration capacity to fast neutrons, so the value of σ_f when the wellbore is filled with gas significantly decreases. Therefore, when evaluating effect of CO₂ flooding in heavy oil reservoir with σ_f, the gas-bearing property needs to be corrected according to the fluid type in the wellbore.

In multiple cycles of CO₂ injection in heavy oil reservoir, if the reservoir contains CH₄, the injected CO₂ would mix with a certain proportion of CH₄, and CH₄ would accumulate during the CO₂ flooding, affecting the evaluation of CO₂ saturation. Based on the instrument and formation model shown in Fig. 3, with the total gas saturations (CO₂ and CH₄) of the formation set at 30% and 80%, and the CH₄ saturations set at 5%, 10% and 15% respectively, the relationships between σ_f and formation porosity under different mixing ratios of the two kinds of gases were simulated, and the results are shown in Fig. 9. It can be seen when the total gas saturation is high, the influence of CH₄ content is smaller. When the total gas saturation is low, the influence of CH₄ content although increases, is still not obvious. Therefore, it is difficult to distinguish CH₄ and CO₂ using evaluating CO₂ saturation. Compared with the formation matrix, heavy oil and water, CH₄ and CO₂ both have lower σ_f values. Although the σ_f of CH₄ is slightly larger than that of CO₂, the two kinds of gases take a relatively small proportion of the total volume of the formation, and small amount of CH₄ in the pores does not affect the deceleration ability of the formation to fast neutrons. In fact, the calculation result of CO₂ saturation is closer to the total saturation of CO₂ and CH₄. Therefore, it is difficult to eliminate the calculation error in CO₂ saturation caused by the presence of CH₄ when evaluating the gas saturation of reservoir with σ_f. To ensure the accuracy of CO₂ saturation monitoring during the CO₂-EOR project, the CH₄ content after multiple cycles of CO₂ injection must be strictly controlled.

An example was simulated to verify the effectiveness of the method presented in this paper. The formation is 234 m thick in total and separated by clay layers. The pore fluids of the formation are different combinations of heavy oil, fresh water and CO₂. Table 4 shows the lithology, porosity, density, shale content, fluid properties and saturation parameters of each reservoir. During the simulation logging process, the tool slided along the wellbore wall from bottom to top to detect and record the inelastic and capture gamma information of the multi-detectors. The information was converted into σ_f, and combined with gas saturation calculation method and correction method, CO₂ saturations of the reservoirs were obtained.

From the interpretation results shown in Fig. 10, it can be seen that the calculation errors of CO₂ saturations of layer A to layer F are 0.3%, 1.7%, 13.5%, 2.9%, 2.4%, and 0.6%, respectively. The formation lithology and shale content do not affect the error of the method in evaluating CO₂ saturation. No matter whether the formation contains shale, the calculation errors of CO₂ saturation can be controlled within 3%. Therefore, monitoring CO₂ saturation of heavy oil sandstone reservoir with σ_f is an effective method. When the formation contains CH₄ (layer C), the calculation error of CO₂ saturation is larger (13.5% for layer C). This phenomenon also confirms this method has the shortcoming of unable to distinguish CO₂ and CH₄.

CO₂ differs widely from oil and water in fast neutron deceleration ability. In this work, the fast neutron scattering cross-section was applied to the quantitative monitoring of gas saturation of CO₂ flooding heavy oil reservoir, the gas saturation evaluation model was established to realize quantitative calculation of CO₂ saturation. This method provides technical support for gas saturation monitoring in heavy oil reservoir developed by CO₂ flooding.

Based on the three-detector pulsed neutron logging technology, the inelastic and capture gamma of the near and far detectors can be combined to realize the characterization of fast neutron scattering cross-section of the reservoir. The shale-bearing reservoir is different from pure lithology reservoir in fast neutron scattering cross- section. Hence, it is more reasonable to evaluate the CO₂ saturation by using shale content and physical model of bulk-volume rock of the reservoir. The changes of miscible oil density caused by the variations of formation temperature and pressure are relatively small and don’t affect the deceleration ability of the heavy oil and CO₂ miscible fluid to fast neutrons, so the influence of the formation temperature, pressure and heavy oil density to CO₂ saturation evaluation can be ignored. Compared with the formation matrix, CH₄ and CO₂ are both low in values of fast neutron scattering cross-section, the presence of CH₄ would cause errors in the evaluation of CO₂ saturation. The CH₄ content must be controlled strictly when monitoring CO₂ saturation. In addition, the type of wellbore fluid greatly affects the measurement of fast neutron scattering cross-section. When using the fast neutron scattering cross-section to judge the effect of CO₂-injected heavy oil reservoir, the gas-bearing property must be corrected according to the type of fluid in the wellbore.

The simulation results of an example show that calculated CO₂ saturations by this method have errors of less than 3% when there is no CH₄ in the reservoir, verifying the effectiveness of the method.

A, B, C—constants, g/cm³;

i—the average number of gamma photons produced by inelastic scattering of a fast neutron colliding with a formation element;

i_k—the number of gamma photons produced by element k in a collision;

p—formation pressure, MPa;

I—the number of gamma rays;

r—the radius of the sphere centered on the neutron source, m;

dr—the thickness of the spherical shell with radius r, m;

R—source distance of detector, m;

S_g—CO₂ saturation, %;

T—formation temperature, K;

V_sh—shale content, %;

α—scale factor related to source distance;

ε—the error between the calculated value and the theoretical value of σ_f, %;

λ_s—fast neutron scattering free path, m;

μ_m—mass attenuation coefficient, m²/kg;

ρ—formation density, kg/m³;

Δρ_p, Δρ_t—the change in the density of the miscible oil caused by changes of formation pressure and temperature, g/cm³;

ρ₀—density of miscible oil under formation pressure of 0.1 MPa and formation temperature of 288.7 K, g/cm³;

σ_f—fast neutron scattering cross-section, m^-1;

σ_f,cal—calculated value of σ_f, m^-1;

σ_f,gas, σ_f,mat, σ_f,oil, σ_f,sh—fast neutron scattering cross-section of CO₂, formation matrix, oil, and shale, m^-1;

σ_f,real—the theoretical value of σ_f set in the formation model, m^-1;

σ_k—the micro-inelastic scattering cross-section of element k in a collision, m^-1;

σ_in—the inelastic scattering cross-section of the formation, m^-1;

ϕ—formation porosity, %;

ϕ₀—neutron source intensity;

ϕ_f(r)—fast neutron flux, m^-2;

ϕ_in(R)—inelastic gamma flux, m^-2;

ϕ_cap1, ϕ_cap3—the gamma counts captured by near and far detectors;

ϕ_in1, ϕ_in3—the inelastic gamma counts of near and far detectors.