A novel framework for neutron-gamma density logging: semi-empirical modeling, directional neutron sources, and experimental benchmarking

Abstract

Neutron-gamma density (NGD) logging is a vital technique in subsurface exploration, enabling accurate determination of formation density and supporting hydrocarbon reservoir characterization. This study proposes a novel NGD logging method that utilizes a directional neutron source to enhance measurement precision while reducing the number of required detectors. A semi-empirical formula was developed to calculate formation density using directional neutron flux, and detailed simulations were conducted to evaluate the method’s performance under varying neutron emission angles, mudcake thicknesses, and environmental conditions. To validate the simulation results, an experimental test was designed using a scaled well model with controlled porosity levels and a calibrated Am-Be neutron source. The experimental measurements were benchmarked against the simulation results, demonstrating strong alignment and confirming the accuracy of the proposed approach. The findings highlight the advantages of steep neutron angles in achieving superior measurement accuracy, with performance comparable to NGD-TC methods and improved over NGD-FC methods. This study underscores the reliability of the proposed method and provides a framework for further optimizing NGD tools for diverse subsurface conditions, advancing hydrocarbon exploration and reservoir evaluation.

Introduction

Neutron Gamma Density (NGD) well-logging is an essential technique in exploring and characterizing subsurface formations, providing detailed insights into lithology and porosity^1,2. Over the decades, this method has undergone significant advancements^3–5. These developments have improved the accuracy and applicability of NGD well-logging in various geological settings, leading to more precise subsurface models, better resource estimation, and more efficient extraction processes tools^6–8.

In gamma-gamma density (GGD) well-logging, collimators are typically used to direct the gamma rays produced by the source^9,10. The number and energy of gamma rays reaching the detector depend on the collimator’s design, which has been widely studied^11,12. On the other hand, neutron-gamma density (NGD) logging increasingly employs neutron generators instead of chemical sources for environmental and safety reasons, with the sources typically being isotopic^13–15. Unlike GGD, NGD logging relies heavily on the neutron transport equation and the directionality of neutrons¹⁵, making the study of collimator effects more significant. However, comprehensive studies on collimator impacts in NGD logging are still lacking.

In NGD logging, the generated neutrons interact with the formation environment, producing gamma rays that are detected after scattering within the medium^16,17. Analyzing the impact of collimators is complex due to the directionality and energy distribution of these secondary gamma rays influenced by the neutron collimator. Differences in interaction processes further heighten this complexity; NGD involves higher energy gamma rays, which increases the likelihood of interactions such as pair production¹⁸, unlike the predominantly Compton scattering observed in GGD with gamma energies around 0.66 MeV for cesium and 1.25 MeV for cobalt sources¹⁹.

Given these intricacies, there is a clear need for a thorough study to fully understand the effects of collimators in NGD logging. The differences in gamma-ray energy levels and interaction types between NGD and GGD highlight the necessity for such research. Addressing this gap would enhance the accuracy and reliability of NGD measurements, ultimately benefiting hydrocarbon exploration efforts. Therefore, this paper aims to investigate these effects, providing critical insights into the role of collimators in neutron-gamma density well-logging.

Additionally, environmental hydrogen, with its high cross-sectional interaction, alters the neutron flux and, consequently, the gamma flux. Moreover, the angle of radiation incidence in a medium greatly influences the depth of penetration (DOI) of that radiation.

In this study, we develop and evaluate a novel Neutron-Gamma Density (NGD) logging method that utilizes a directional neutron source to enhance measurement accuracy while reducing the number of detectors required. Through detailed simulations, we investigate the impact of neutron emission angles, mudcake thickness, and environmental factors on tool performance, focusing on key parameters such as sensitivity, statistical error, and depth of investigation. Additionally, we benchmark the results against conventional NGD-FC and NGD-TC methods to validate the reliability and efficiency of the proposed approach.

Results

Distribution of energy

In this section, a simulation was conducted using a sandstone model with dry conditions. The neutron beam was collimated at four angles (0°, 25°, 50°, and 75°) and the fluxes of both neutrons and gamma rays were recorded at near and far detectors and the results are shown in Fig. 1.

Fig. 1.

Energy distribution of neutrons and gamma rays at various collimation angles in dry sandstone model.

The study of neutron and gamma fluxes reveals overall stability, with some distinctions in their sensitivity to beam angle. Neutron flux remains stable regardless of angle, showing minimal sensitivity to directional changes, while gamma flux, particularly in the low-energy range, exhibits a stronger dependence on beam angle. This is most notable at higher angles like 50° and 75°, where neutron-induced gamma production is more sensitive to the incidence direction due to the nature of interactions that generate prompt gamma rays. Despite these differences, the general trends suggest that neutron interactions are the primary drivers of gamma emissions, although gamma flux shows slightly greater angular sensitivity.

Both near and far detectors illustrate differing responses to collimation angles. The near detectors, especially for gamma flux, show higher sensitivity, with flux decreasing at larger angles, indicating a stronger angular effect near the neutron source. In contrast, far detectors demonstrate stable flux across all angles, with the far neutron detector remaining particularly resilient to directional changes. This stability in the far detectors suggests that scattering and diffusion processes over distance mitigate angular effects, particularly for neutrons.

Additionally, the energy spectra of both neutrons and gamma rays shift toward lower energies as the beam angle increases, particularly in the near gamma detector. This energy shift, most noticeable in the higher angles, likely results from increased scattering and energy loss with off-axis neutron incidence. Neutron energy spectra remain relatively stable, although the far neutron detector shows a slight shift to lower energies due to longer travel distances and scattering effects. These trends suggest that higher collimation angles favor the dominance of scattered, lower-energy particles in the overall spectrum.

Spatial distribution of the scattered neutron

Figure 2 illustrates the significant change in scattered neutron flux with different focused neutron beam angles. The directional emission angles of 0°, 25°, and 50° demonstrate distinct spatial variations in the flux distribution, while the isotropic case provides a uniform reference. The focused neutron beam angles cause the flux to concentrate in specific regions, creating asymmetric patterns as the angle increases. This highlights how the beam direction influences the behavior of scattered neutrons within the medium.

Fig. 2.

Effect of focused neutron beam angles on scattered flux distribution: (a) Fast neutron flux at 0° Angle. (b) Fast neutron flux at 25° angle. (c) Fast neutron flux at 50° angle. (d) Fast neutron flux (isotropic).

At 0°, the neutron flux is highly concentrated near the source and spreads symmetrically outward, maintaining a smooth and uniform gradient. At 25°, the flux distribution shifts downward along the beam direction, elongating and introducing asymmetry in the flux profile. By 50°, the neutron flux becomes significantly more dispersed and skewed, with visible regions of high flux concentrated at the upper part of the medium and lower flux areas elsewhere. These changes indicate that higher beam angles cause increased scattering and broader dispersion of neutron flux.

The isotropic case, in contrast, produces a uniform flux distribution without any angular bias. While isotropic scattering ensures even coverage, it sacrifices resolution compared to focused beam angles. Directional neutron beams (e.g., 0°, 25°, and 50°) create sharper flux gradients and localized regions of high flux, which are more effective for generating focused gamma flux. However, as the beam angle increases, neutron dispersion reduces the localized flux intensity, weakening the gamma flux generation. This comparison emphasizes the trade-off between resolution in directional beams and uniformity in isotropic flux.

Depth of investigation

Figure 3 illustrates the Depth of Investigation (DOI) for gamma flux at different directions (0°, 25°, 50°, and 75°). The DOI is determined as the depth where the flux drops to a predefined threshold (e.g., 10% of its maximum value).

Fig. 3.

Comparison of percentage flux vs thickness.

In comparison, the results in Fig. 3 show that the flux at 0° direction penetrates deeper with higher flux retention, indicating a more focused neutron beam. As the direction angle increases (e.g., 25°, 50, and 75°), the flux attenuates more rapidly, reducing the DOI. This trend occurs due to increased scattering and angular dispersion of neutrons at higher angles, leading to energy loss and lower flux at greater depths. Thus, simulations validate the physical behavior of neutron transport, showing that DOI is direction-dependent, with sharper attenuation observed at higher angles compared to 0°.

Sensitivity

The relationship between the ratio of the far-to-near gamma detector count rate and the beam angle at varying formation densities and the Mean Squared Error for each direction is illustrated in Fig. 4 This ratio provides a critical indicator of the response of neutron-gamma logging tools under different geological conditions. The far-to-near count rate ratio depends on the attenuation of gamma rays, which is influenced by both the beam angle and the density of the formation. Higher densities generally result in greater attenuation, leading to a smaller count rate ratio, particularly for larger beam angles. As the angle increases from 0° to 75°, the A coefficient shows a slight upward trend, reflecting a decrease in the sensitivity of the near detector relative to the far detector at higher angles. Similarly, the B coefficient also increases with the angle, suggesting an enhanced proportionality between the gamma count ratio and formation density at larger angles. The Mean Squared Error (MSE) values, which represent the model’s accuracy, vary slightly but remain small across all directions, indicating a reliable fit to the data. However, the fitting error consistently decreases as the angle increases, suggesting the model performs better at higher angles.

Fig. 4.

Fit lines for all direction with swapped axes.

A key parameter for evaluating the accuracy of a density logging tool is its measurement sensitivity, which corresponds to the slope of the lines shown in Fig. 5a Measurement sensitivity determines how effectively the tool can differentiate between formations of varying densities based on the far-to-near count rate ratio. Higher sensitivity indicates greater accuracy in capturing small changes in density, making it a crucial factor in tool design and optimization. The sensitivity can be mathematically expressed as:

1

where S is the measurement sensitivity, R is the far-to-near count rate ratio, and ρ is the formation density.

Fig. 5.

Error propagation and sensitivity analysis across different neutron source angles: (a) Error propagation as a function of density for different neutron source angles. (b) Sensitivity as a function of density for different angles.

As the slope increases, the tool becomes more adept at distinguishing small density variations, which is critical for precise geological analysis. The sensitivity values, which vary with the beam angle, highlight the directional dependency of the tool’s performance. The steeper slopes at higher beam angles (e.g., 50° and 75°) suggest that these configuration provide enhanced resolution, enabling the detection of finer density differences.

Density precision

The measurement uncertainty was estimated based on the simulation results of the detector count rates and the measurement sensitivity. These simulation-based estimations are essential for evaluating the accuracy and reliability of density logging tools under various operational conditions. The results of this analysis, particularly for different source collimator angles, are summarized in Fig. 5b.

The analysis of density precision, calculated for a 1.85*10⁸ neutron source, reveals that the uncertainty decreases as the neutron source angle increases from 0° to 75°. This trend demonstrates that higher angles enhance the interaction between neutrons and the formation, allowing the far-to-near detector count ratio to differentiate density variations better. At higher angles, the larger B coefficients improve sensitivity to density changes, while smaller A coefficients reduce the overall uncertainty. These results highlight that steep neutron source angles provide superior density precision.

In conventional NGD logging, there is often a trade-off between sensitivity (the ability to detect small variations in formation properties) and detection efficiency (the ability to capture enough neutrons for precise measurements). The use of higher emission angles enhances the vertical resolution and sensitivity, making it easier to detect small variations in formation density and porosity. However, this improvement comes at the cost of reduced neutron flux reaching the detectors, leading to increased statistical uncertainty.

When the emission angle increases, the path length of neutrons in the formation increases, and due to the geometry of the measurement system, fewer neutrons are directed towards the detectors. This reduction in neutron flux can lead to less precise measurements because the number of neutron interactions (which is key for determining density) decreases, thereby increasing the measurement error. This has been observed in the literature, where an increase in the emission angle typically leads to a decrease in the raw count rate detected by the gamma detectors.

Despite the reduction in neutron flux, it is important to note that the increased emission angle leads to a significant improvement in the accuracy of the density approximation model. As the angle increases, the term 2rdcos(θ) in the density calculation model becomes less significant, which leads to a better fit for the data and reduced overall error in the density approximation. The simplification of the model due to the reduced influence of the angular term is one of the key reasons why the proposed method performs better at higher emission angles, as demonstrated in Table 1 and Fig. 5b, where the overall error decreases as the emission angle increases.

Table 1.

Comparison of the impact of mudcake errors in the directional neutron method for different angles with NGD-FC and NGD-TC methods at various mudcake thicknesses.

Mudcake thickness	Angel of neutron direction				MSE in NGD-FC	MSE in NGD-TC
	0	25	50	75
3	0.022436	0.0255815	0.02291	0.019095	0.0737	0.0104
6	0.01199	0.0266748	0.023334	0.020145	0.0857	0.0246
9	0.024956	0.0278083	0.023951	0.021634	0.0888	0.0296
12	0.026641	0.0290598	0.025726	0.023261	0.0846	0.0374

Neutron flux is a key factor in determining the precision of NGD measurements. While increasing the emission angle reduces the flux, the application of high-flux neutron sources, such as accelerator-based D-T generators, can offset this reduction. High-flux sources produce more neutrons, even at higher angles, which reduces the statistical uncertainty by providing a more consistent and higher count rate, making the method more robust and reliable in practice.

In formations that are thinly bedded, gas-bearing, or have heterogeneous composition, conventional NGD tools often fail to detect subtle variations in formation density. Higher emission angles enhance the ability of the tool to detect these fine variations, providing better vertical resolution and allowing for more accurate evaluation of formations where traditional methods might struggle.

Impact of mudcake

Mudcake plays a significant role in well-logging by affecting the accuracy and reliability of density measurements, as it introduces additional attenuation and scattering of neutrons and gamma rays. Understanding the impact of mudcake thickness is essential for optimizing Neutron-Gamma Density (NGD) logging methods and ensuring precise formation density determination.

In this study, we compare the influence of mudcake thickness on two NGD logging methods with neutron directions. The NGD-FC method, based on the coupled field theory as described in²⁰, uses an algorithm involving the inelastic gamma count ratio between near and far detectors and the fast-neutron scattering free path with calibration coefficients. The measurement system for this method requires two gamma detectors and two fast-neutron detectors placed at varying distances. Similarly, the NGD-TC method derives its density algorithm from the thermal neutron count ratio, which correlates to the neutron slowing-down length, using calibration coefficients. This method also involves two gamma detectors and two thermal-neutron detectors positioned differently to enhance sensitivity¹⁶.

To evaluate the effect of mudcake thickness on the error in NGD methods, we conducted simulations in 96 baseline states without mudcake to establish calibration coefficients for both methods using equations (9) and (11). Subsequently, 192 additional simulations were performed for each direction, varying mudcake thickness between 3 mm and 12 mm and incorporating a 10% barite weight fraction. Table 1. Comparing the effect of mudcake thickness of any neutron direction with NGD-FC and NGD-TC methods.

The mudcake thickness impact of neutron source direction (0°, 25°, 50°, 75°) on the Mean Squared Error (MSE) in NGD logging methods demonstrates significant differences in performance, as shown in Table 1. Among these directions, the 75° neutron source consistently achieves the lowest MSE across all mudcake thicknesses. This highlights the advantage of steeper neutron angles, which enhance neutron penetration and reduce scattering, leading to improved measurement accuracy. Conversely, the 25° direction shows the highest MSE, indicating that shallower angles are more susceptible to mudcake effects due to increased attenuation and lower sensitivity.

Comparing the directional neutron source method with NGD-TC and NGD-FC methods, the 75° directional approach performs better than NGD-FC but slightly worse than NGD-TC. This trend persists as mudcake thickness increases, with NGD-TC maintaining the lowest MSE values across all scenarios due to its reliance on thermal neutron count ratios, which are less affected by mudcake interference. In contrast, NGD-FC consistently shows the highest MSE, reflecting its vulnerability to fast neutron scattering and attenuation effects caused by mudcake.

Discussion

In this method, relatively good accuracy can be achieved compared to other approaches by using only two gamma detectors. The first factor contributing to error is the limited investigation radius, which is small relative to the source-to-detector distance. Although increasing the source-to-detector distance enlarges the investigation radius, this improvement enhances the approximation accuracy of the method. However, excessively increasing the detector distance significantly reduces the flux, leading to increased statistical error. Thus, optimizing the detector spacing requires a tradeoff between these two factors.

On the other hand, increasing the neutron source angle reduces the contribution of the second term (2rdcos(θ)) compared to the first term (r²) in equation 6, resulting in better tool responses at higher angles. However, as discussed in Section "Discussion", "Methods" and "Conclusion", larger angles increase statistical error due to reduced flux, necessitating the use of higher neutron flux sources such as accelerator-based systems. Consequently, determining the optimal angle is one of the critical challenges of this method. It is also noteworthy that, compared to conventional NGD methods, the concentrated flux used in this approach results in higher flux reaching the detectors, thereby significantly reducing statistical error.

In the theoretical framework, it is assumed that Compton scattering is the dominant interaction and the attenuation coefficient is considered proportional to the electron density and, consequently, to the material density, independent of the material type. This assumption encounters challenges in formations with high hydrogen weight percentages, as the atomic-to-mass ratio of hydrogen is 0.5. Additionally, pair production interactions at the energy levels discussed present another issue, especially in formations with heavy materials or high barite content in the drilling mudcake.

However, the dependence on optimal angle and detector spacing underscores the importance of thorough calibration and sensitivity analysis during tool development. Incorporating advanced simulation tools and machine learning models can enhance adaptability to various formation conditions and address limitations posed by heavy materials or hydrogen-rich environments. Future work could explore the integration of these techniques to refine the accuracy and applicability of this method.

An equally critical challenge arises in the physical realization of directional neutron emission, particularly the need for effective collimation of high-energy (14 MeV) neutrons. Due to their high penetration capability, collimation requires thick shielding layers of dense materials which inevitably reduce the neutron flux and may compromise logging efficiency. This trade-off has been explored in related fields such as neutron imaging and therapy. For example, Chen et al. demonstrated that a multilayer collimator structure with hydrogenous and heavy-metal materials can achieve a collimation ratio of 29 for compact imaging systems²². Similarly, Li et al. designed a moderator–collimator system to enable mobile neutron radiography using compact D–T sources²³. Although these studies^22–26 have been conducted in different fields, such as neutron imaging and therapy, they provide valuable insights not only into the design and construction of neutron collimators but also into their potential application in well-logging. These developments suggest that, while further studies are required to explore the practical application of such designs in well-logging, there is a clear opportunity to adapt and utilize these techniques to improve the precision and performance of NGD logging tools in challenging geological formations.

In our present work, to isolate the theoretical benefits of angular neutron control, we performed idealized simulations assuming directional emission without explicitly modeling physical collimators. This allowed us to evaluate the upper performance bounds in terms of sensitivity, statistical error, and depth of investigation. However, for practical implementation, future studies must explore: (1) the impact of small variations in neutron emission angle on measurement precision; (2) the optimization of collimator length, shape, and materials for well-logging constraints; and (3) the performance of low-divergence-angle collimators suitable for borehole deployment. By integrating these real-world constraints with the current theoretical framework, this method lays the foundation for a new generation of compact, high-precision NGD logging tools.

Methods

Physical concept

Neutron-gamma density (NGD) logging is a critical method in subsurface exploration, particularly in determining the bulk density of geological formations. For health, safety, and environmental (HSE) reasons, NGD logging leverages a pulsed neutron generator (PNG) to produce high-energy fast neutrons²¹. These neutrons interact with the formation materials to produce inelastic gamma rays, known as a secondary extended source. Consequently, the gamma rays interact with formation material and are detected by gamma detectors, resulting in a distributed energy spectrum. This phenomenon, along with the high energy level of gamma rays, adds complexity to the analysis of this method.

To understand the physical concepts of a custom NGD tool, both the fast-neutron field distribution and the inelastic gamma-ray attenuation must be considered. This study employs a simplified spherical, homogeneous point-source model to investigate the mechanisms governing neutron and gamma-ray transport in NGD measurements. The model is designed to provide an analytical framework for understanding the interactions and distributions of particles involved in NGD logging.

Figure 6a illustrates the model, which includes a pulsed neutron source (Point A) that emits mono-energetic fast neutrons at 14.2 MeV within a defined angle θ. A secondary gamma-ray source (Point B), generated by neutron interactions with the surrounding medium, is located at a distance R from the neutron source. A gamma-ray detector (Point C) is positioned at distances d and x from Points A and B, respectively. Figure 1 illustrates the spatial configuration of this model.

Fig. 6.

Neutron-gamma density logging model and simulation setup: (a) spatial configuration of the neutron-gamma density logging model. (b) MCNP simulation setup of the NGD tool and formation.

According to coupled-field theory, NGD measurements are governed by the interplay of two primary transport phenomena: fast neutron transport and gamma-ray transport. These processes are intricately linked and dictate the overall response of the NGD tool. To simplify the calculations and focus on the essential aspects of these phenomena, the following assumptions have been made:

• The medium is homogeneous, ensuring uniform properties throughout the simulation domain.

• Neutrons are emitted exclusively in the direction defined by the angle θ reducing the complexity of angular distribution.

• Neutrons do not transition from lower to higher energy states during transport, reflecting a unidirectional energy degradation.

• The assumptions of the one-dimensional neutron diffusion equation are applicable, allowing for a simplified yet effective representation of neutron behavior.

By solving the one-dimensional neutron diffusion equation and combining it with the gamma-ray diffusion equation, the gamma-ray flux ​ at Point C is derived as equation (2):

2

The intensity of the pulsed neutron source is represented by Q while the diffusion coefficient of fast neutrons is denoted as Dn. The neutron slowing-down length, Ln is calculated as Ln Here, ∑_in is the macroscopic inelastic scattering cross-section, representing the probability of neutron scattering events within the material. while i represents the average number of gamma photons produced per inelastic collision.

In the context of gamma-ray transport, the gamma-ray flux Φn quantifies the number of gamma rays passing through a unit area per unit time. The diffusion coefficient for gamma rays, Dγ, characterizes the spread of gamma rays through the medium. Additionally, the inelastic gamma-ray attenuation length Lγ defines the distance over which gamma rays are significantly attenuated in the formation material. Variable x can be calculated by:

3

To calculate the total gamma-ray flux detected by the detector, the flux expression derived in equation (2) must be integrated over all possible source-to-detector distances r from 0 to ∞.

4

Solving the integral directly is highly complex and typically requires numerical methods due to the nonlinear dependency of x on r and θ. However, considering that the radius of investigation in NGD logging is limited, we can assume that d is much larger than r (at least 3 to 4 times).

Under this assumption, we approximate x in the exponential term as

5

and in the denominator, we approximate x by d. This simplification significantly reduces the complexity of the integral while preserving sufficient accuracy for practical applications.

The remaining integral over r can be evaluated as:

6

Substituting this result, the total gamma-ray flux becomes:

7

This approximation significantly reduces the computational complexity while still providing a reasonable estimate of the gamma-ray flux detected by the NGD tool. By dividing the total gamma flux at two detectors located at distances d₁​ and d₂​ from the neutron source, and applying some simplifications, we arrive at the equation (8) for bulk density:

8

where A and B are calibration coefficients and R_Φ is the ratio of the near-to-far gamma flux.

MCNP simulation

In This study we use MCNPX (2.6) to simulate a well-logging tool and formation, leveraging its advanced capabilities for multi-energy particle transport to simulate neutron and gamma interactions in heterogeneous geological formations precisely.

The sonde considered in this study is a conventional NGD tool that includes two gamma detectors and two neutron detectors for measuring thermal and fast neutron flux. The specifications of the sonde are detailed below, and Fig. 6b also shows a view of the sonde structure.

The simulation setup includes two gamma detectors and two neutron detectors positioned at specific distances from a neutron source. The near gamma detector is located 41 cm from the neutron source, while the far gamma detector is 60 cm away. The near neutron detector is situated 50.7 cm from the neutron source, and the far neutron detector is 69.7 cm away. Each detector, whether for gamma or neutron detection, has a length of 8 cm and a radius of 4 cm. The tool is encased in a durable steel body with a 7.5 cm diameter, designed.

The neutron source is shielded from direct contact with the detectors using layers of paraffin (20 cm), boric acid (10 cm), and lead (3 cm), ensuring precise detection of scattered radiation signals.

D-T neutron source emitting particles at 14.1 MeV with timing parameters of 20 μs working time and a 200 μs period.

The formation under examination is a cylindrical structure with a height of 800 cm and a diameter of 300 cm. At the center of this formation is a wellbore filled with air, having a diameter of 8 cm, with the logging tool positioned at its center. Table 2 provides the assumed mineral composition of the formation. The study investigates mudcake thicknesses ranging from 0 to 12 mm, utilizing barite (BaSO₄) as a weighting agent in water-based mud, with a 10 % weight fraction.

Table 2.

Mineral composition and mudcake specifications of the formation.

Material category	Element weight fraction					Density (g/cm3)
	Si (%)	O (%)	Ca (%)	Mg (%)	C (%)
Silica Sand	46.74	53.26	0	0	0	1.7
Calcium Carbonate	0	47.96	40.04	0	12	2.71
Calcium Magnesium Carbonate	0	47.86	21.73	13.18	17.23	2.87
Magnesium Carbonate	0	47.82	0	28.86	23.32	3.037

Tally F8 for gamma flux monitoring and Tally F4 for neutron flux evaluation are applied and Simulations involved 10⁹ particles to ensure accurate scaling and relative error measurement. Gamma flux was recorded across 1024 energy channels (0–8.5 MeV) and neutron flux was categorized into 4 energy groups. Reactions of neutron and gamma detectors were not considered, and no variance reduction methods were used. Neutron cross-sections were sourced from the ENDF/B-VII nuclear data library.

Benchmarking experiment design

In this study, we aimed to benchmark the simulation results against experimental measurements to validate the accuracy of the simulation model. To achieve this, we replicated the probe used in the experimental tests within the simulation, allowing for a direct comparison between the simulated and experimental data. Notably, while the theoretical part of the study utilized a D-T neutron source, the unavailability of this source in our laboratory necessitated the use of an Am-Be (Americium-Beryllium) source for the experimental measurements. Consequently, all experimental tests were re-simulated using the Am-Be source to align with the experimental setup.

The experimental setup consisted of a probe casing, a neutron detector, and a gamma detector. Additionally, The shielding was designed to block residual neutrons and gamma rays, ensuring that only the scattered radiation reaching the detectors was measured. Specifically, the neutron source was shielded from direct contact with the detectors using multiple layers of protective materials: 20 cm of paraffin, 10 cm of boric acid, and 3 cm of lead.

For the source, we employed an Am-Be neutron source with an intensity of 5 mCi. The source emits neutrons isotopically and was positioned inside the central cylinder of the sond. The neutron and gamma fluxes were measured using a NaI scintillator for detecting gamma radiation and a Boron Lined NaI scintillator for detecting thermal neutrons. The thermal neutron flux was measured by the neutron detector, which was set to count gamma rays at an energy of 0.48 MeV. The gamma energy spectrum, on the other hand, was measured by the gamma detector, which was configured to capture photons within the energy range of 1–10 MeV. These measurements were taken at distances of 40 cm and 60 cm from the source, respectively.

A well with a height of 1.6 meters and a diameter of 1.5 meters was constructed, inside which a cylindrical cavity with a diameter of 30 cm was formed. The experimental tests were conducted using a range of porosity values for the surrounding medium, including 30 p.u., 35 p.u., 40 p.u., 45 p.u., 50 p.u., and 100 p.u., with the pores filled with water. It is important to note that the preparation of samples is crucial for ensuring the accuracy and reliability of experimental results. In our study, significant effort was made to ensure the homogeneity of the samples, guaranteeing that both physical and chemical properties were uniformly distributed throughout the material, which is essential for obtaining consistent measurements. Additionally, proper handling and controlled storage conditions were employed to maintain the integrity of the samples and prevent any environmental factors from altering their properties.

Each experimental test consisted of a 20-minute counting session, with three separate tests conducted for each sample to ensure consistency and reliability. To account for environmental radiation, background counts were recorded in the absence of the source prior to each experiment. These background counts were then subtracted from the total counts measured during the experiments to obtain the net neutron and gamma flux.

The comparison between the simulated and experimental data in Fig. 7b, c reveals key insights. In Fig. 7b, the simulation and experimental neutron flux are compared across different porosity values. The wellbore effect plays a significant role in introducing discrepancies, especially as porosity increases, influencing neutron flow and interactions in ways that are not fully captured by the simulation. The experimental setup, which often includes an uneven distribution of porosity, further contributes to variations in the neutron flux that the simulation, based on idealized conditions, does not account for. This leads to deviations between the experimental and simulated data, although the overall trends remain similar, suggesting the simulation offers a reasonable representation of neutron flux behavior.

Fig. 7.

Experimental Setup and Validation of NGD sonde results: (a) Setup of experimental test for NGD sonde. (b) Simulation versus Experimental Neutron Flux. (c) Comparison of gamma Energy spectrum of Simulation Data and Experimental Data.

In Fig. 7a, c similar strong alignment is seen between the simulated and experimentally measured gamma energy spectra of the Am-Be source. The main peaks align well, confirming the simulation accurately captures the primary features of the gamma photon spectrum. However, the experimental spectrum shows broader peaks compared to the sharper simulated ones, which is attributed to the energy resolution limitations of the NaI detector (with a larger Full Width at Half Maximum). Additionally, the experimental data exhibits statistical fluctuations, especially in low-count regions, which is a common occurrence in real-world measurements. These differences can be linked to the physical characteristics of the NaI detector and the inherent statistical nature of experimental data, reinforcing the reliability and accuracy of the simulation model in replicating key features of both the neutron flux and gamma spectrum.

Conclusion

This study presents a comprehensive analysis of a Neutron-Gamma Density (NGD) logging method leveraging a directional neutron source to improve the accuracy and reliability of formation density measurements. The proposed method achieves precision comparable to conventional NGD approaches while utilizing fewer gamma detectors, simplifying tool design and operation. By simulating various neutron source angles and mudcake conditions, the results underscore the critical influence of neutron directionality, detector spacing, and environmental factors on tool performance.

Key findings highlight the advantages of steep neutron source angles which enhance density precision and reduce statistical error due to improved penetration and directional focus. Additionally, the method demonstrates robustness against mudcake-induced attenuation and scattering, achieving better accuracy than NGD-FC methods and closely aligning with the performance of NGD-TC methods. However, challenges related to optimizing the neutron source angle and detector spacing were noted, requiring a tradeoff between statistical error reduction and flux attenuation.

The theoretical framework, based on simplified coupled-field equations, provides a foundation for understanding neutron and gamma-ray interactions while identifying limitations in environments with high hydrogen content or heavy materials. The study emphasizes the importance of advanced calibration, sensitivity analysis, and the potential integration of machine learning techniques to overcome these challenges and enhance tool adaptability. This research lays the groundwork for developing more accurate and efficient well-logging tools, ultimately contributing to improved resource estimation and extraction processes in hydrocarbon exploration.

Author contributions

A.R. conceived the research, collected the data, performed the simulations, and analyzed the results. S.A.H. provided guidance, supervised the work, and reviewed the manuscript. A.R. and S.A.H. wrote the manuscript.

Data availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author upon reasonable request.

Declarations

Competing interests

The authors declare no competing interests.