Coupled field theory of NGD measurement

NGD logging technology relies on inelastic gamma rays to measure the formation density [24-29]. The distribution of inelastic gamma-rays involves two interconnected links between neutrons and gamma transport [30-32], as described below.

The pulsed neutron source emitted 14 MeV fast neutrons. According to the neutron diffusion theory, the distribution of fast neutrons in a spherical model can be described as follows:(1)where Q, D_n,L_n, and r denote the number of neutrons emitted by the neutron source per second, neutron diffusion coefficient, fast neutron deceleration length, and distance between the pulsed neutron source and neutron detector, respectively. The inelastic gamma rays recorded by a detector with a radius R can be described by the following Eq. [21]:(2)where i, ∑_in, ρ, and μ_m denote the average number of inelastic gamma rays after the neutron enters the formation, inelastic scattering cross section, formation density, and total mass attenuation coefficient, respectively. The inelastic gamma rays within a distance from the source R are recorded [33], and Eq. (2) can be written as(3)According to the Lagrange interpolation method, Equation 3 can be simplified as follows:(4)Here, ξ belongs to and can be expressed as: , .

Assuming that the source distances of the near- and far-gamma detectors are L1 and L2 (L1<L2), respectively, the following equations are obtained:(5)where(6)The logarithm of the ratio of the near and far inelastic gamma counts is given by(7)If , , Equation 7 can be reorganized:(8)Equation (8) shows that the response of the gamma-ray detector is related to the L_n, ρ, μ_m, and the values of a and b. In typical formations, the fast neutron deceleration length (L_n) is characterized by the formation density ρ, hydrogen index I_H, and initial energy of the fast neutron E₀ [34]. Thus,(9)The key parameter in Eq. (8) is the mass attenuation coefficient μ_m, which directly reflects the attenuation of gamma rays in the formation and is analyzed in the following section.

Development of the NGD method

The attenuation of gamma rays in formations is closely linked to the formation density and mass attenuation coefficient. The attenuation is primarily influenced by various physical processes, including the photoelectric effect, Compton effect, and pair production. The Compton effect and pair production are the primary factors that affect the attenuation of high-energy gamma rays. Thus, the total attenuation coefficient μ_m can be expressed as(10)where μ_co and μ_pa denote the mass attenuation coefficients for the Compton effect and pair production, respectively.

According to the principles of the Compton effect and pair production [35, 36], their mass attenuation coefficients can be expressed as(11)(12)where N_A is the Avogadro constant; is the classical electron radius (); ; E_γ is the gamma-ray energy; m_e is the electron rest mass(); c is the speed of light in vacuum (); Z is the atomic number; and A is the atomic weight. The total mass attenuation coefficient can be rewritten as(13)Equation 13 shows the impact of the gamma ray energy E_γ and formation atomic number Z on the mass attenuation coefficient μ_m, highlighting the response sensitivity among these variables. Unlike previous studies, where the mass attenuation coefficient was typically assumed to be constant, the present study emphasizes its variability. However, changes in the mass attenuation coefficient within the average energy range of inelastic gamma rays in formations are sufficiently small to ignore the effect of gamma energy [37]. Instead, its close correlation with the composition of the formation is emphasized, particularly the effects of lithology and pore content on the macroscopic atomic number (Z). This aspect is further confirmed by Fig. 2, which shows the differences in the macroscopic atomic numbers of the formation when the lithology and porosity content change. These changes in the lithology and pore content indicate variations in the constituent elements of the formation, which directly affect the formation macroscopic atomic number (Z) and alters the total mass attenuation coefficient. This affects gamma ray attenuation, thereby complicating gamma ray transport. Therefore, the mass attenuation coefficient is treated as a function of the formation lithology (l) and pore content (p). Accurate density measurements can be achieved by describing the influence of the different formation components on the gamma transport process.

Fig. 2Full-screen View

(Color online) Z-results across different formations with various pore contents

According to the preceding analysis, treating the mass attenuation coefficient as a function of the environmental parameters is expected to better depict the gamma formation reaction sensitivity and improve the accuracy of NGD calculations. Thus,(14)(15)(16)where c and d are constants.

After substituting the mass attenuation coefficient μ_m from Eq. 14 into Eq. 8, the equation can be rewritten as(17)This expression contains two key parameters: the fast neutron deceleration length L_n and macroscopic atomic number of the formation Z, where L_n is related to the formation density and hydrogen index. These physical parameters cannot be directly measured; however, they can be derived by analyzing the detector responses. In the next section, we present a real pulsed neutron logging tool and construct a high-fidelity Monte Carlo model for analyzing these physical parameters.

Analysis of physical parameters using NGD tool

Geant4, an open-source Monte Carlo platform, was used for simulations. The tool model (Fig. 3) had a total length of 2328 mm and diameter of 188 mm and featured four neutron detectors and two gamma detectors. A boron-containing shield was positioned at the base of the neutron detectors to minimize the impact of water in the mud pipe on neutron detection. Additionally, the near-gamma detector was used for density measurement and for formation sigma and elemental measurements. A two-layer shielding structure was implemented to reduce interference from the captured gamma rays generated by the interaction of the tool with thermal neutrons. A cubic space measuring 6 m × 6 m × 6 m, with a borehole diameter of 215.9 mm, was designed to simulate the formation environment. The tool was positioned at the center of the borehole. This tool is currently under construction and will be deployed in the field upon completion. Therefore, it was selected for verifying the feasibility of the proposed method.

Fig. 3Full-screen View

(Color online) Tool model: (a) tool overview; (b) key tool parameters; (c) shielding structure

Extensive simulations were conducted using an NGD tool model that incorporated various formation lithologies (limestone, sandstone, and dolomite) and porosity ranges (0 p.u. to 40 p.u.). These simulations aimed to establish the relationship between the detector responses and relevant physical parameters in Eq. (17) and therefore, will be used for concept validation. The specific relationships are:

(a) Hydrogen index (I_H)

The correlation between the hydrogen index and detector responses was analyzed using the simulation data. Figure 4 presents the correlation coefficients between the various detector responses and hydrogen index. These coefficients measure the strength of the linear relationship between variables, with values closer to 1 indicating a strong correlation. This analysis helps identify the optimal response for representing the hydrogen index. The features represent various detector counts: FN1 and FN2 correspond to near- and far-fast neutron counts; ENT1 and ENT2 correspond to near- and far epithermal neutron counts; and CAP1 and CAP1 correspond to near- and far-capture gamma counts. In addition, RFN, RETN, and RCAP represent the respective ratios of fast neutrons, epithermal neutrons, and capture gamma counts. As shown in the figure, the ratio of the near-to-far epithermal neutron counts exhibits the strongest correlation with the hydrogen index, making RETN the most effective indicator of hydrogen content among all detector responses.

Fig. 4Full-screen View

(Color online) Analysis of physical parameters: (a) Hydrogen Index; (b) Density; (c) Macroscopic atomic number of limestone; (d) Polynomial fit

(b) Formation density (ρ)

To accurately represent the formation density, correlation analysis was applied to evaluate the relationships between the various detector responses and density. In the figure, IN1 and IN2 represent the near and far inelastic gamma counts, respectively, whereas CAP1 and CAP2 represent the near and far capture gamma counts, respectively. RIN is the ratio of the near-to-far inelastic gamma counts and is given by RIN=IN1/IN2. In addition, RCAP is the ratio of the near-to-far capture gamma counts. RIN exhibits the strongest correlation with density, which makes it optimal for describing the density in Eq. (9) and is consistent with the principles of NGD physics [38, 39].

(c) Formation macroscopic atomic number (Z)

The macroscopic atomic number Z, which is an inherent characteristic of the formation, is strongly influenced by the lithology and pore content. By analyzing the detector counts within a specific energy window (0.07–0.35 MeV), denoted as N_lith, a relationship can be established to represent the macroscopic atomic number. As shown in Fig. 4, this method allows the derivation of Z from the detector responses, using counts within the designated energy range to effectively characterize the macroscopic atomic number of the formation.

In summary, the density, hydrogen index, and formation macroscopic atomic number Z can be represented using RIN, RETN and , respectively, which can be obtained from the detector counts. Because the root term in the represented equation complicates the acquisition of the calibration coefficients, a polynomial fit approach was used to simplify the equation, as shown in the figure. After substituting the root term with a second-degree expression, Eq. (9) can be rewritten as(18)where τ₀, τ₁ and tm (m=0,1,2) are the constants. By substituting the expression for L_n into Equation 17, the density equation can be obtained as(19)where β₀, β₁, β₂, θ₀, θ₁, and θ₂ are the fitting parameters. Equation (19), indicates that the formation density is determined by the ratio of inelastic gamma counts RIN, ratio of epithermal neutron counts RETN, and count N_lith. The coefficients in the above equation using the Levenberg-Marquardt method are obtained.

Comparison of the two approaches regarding the mass attenuation coefficient

Section 2.2 emphasizes that the proposed method treats the mass attenuation coefficient as a function pertaining to the formation lithology and pore content. To evaluate the effectiveness of this method, a comparison was conducted that primarily focused on two approaches regarding the mass attenuation coefficient: treating it as a constant (denoted as h) versus treating it as a function. Based on Eq. (8) and the analysis of the relevant physical parameters in Sect. 2.3, the following equation is obtained if the mass attenuation coefficient is treated as h:(21)where β₀, β₁, β₂, and h are the fitting parameters. From Eq. (19), the density is determined by the ratios of the inelastic gamma counts RIN and epithermal neutron counts RETN.

Limestone with densities ranging from 2.018 g/cm³ to 2.862 g/cm³ was designed to compare the two approaches. Figure 5 shows the absolute density errors of both approaches. Significant differences were observed. The constant method exhibits a relatively high average absolute error of 0.048 g/cm³, whereas the error calculated by the proposed method is reduced by approximately four times compared with that of the constant method, with an average absolute error of 0.012 g/cm³. This demonstrates the effectiveness and accuracy of the new method for measuring the formation density.

Fig. 5Full-screen View

(Color online) Density results: (a) Comparison of the two approaches; (b) Pore content results; (c) Lithology results

Pore content impact analysis

In NGD measurements, neutron transport is sensitive to the presence of pore content. To evaluate the effect of different pore contents on the accuracy of this method, limestone with porosities ranging from 0.9 p.u. to 40 p.u. was selected, with pores filled with water, gas, or oil. The density results for different pore contents are presented in Table 1 and Fig. 5.

Table 1 and Fig. 5 present the density calculation results for various pore contents. For the analysis, seven porosity types were selected and tested under conditions in which the pores were filled with water, oil, and gas. In limestone with water-filled pores, the Hydrogen Index I_H is equivalent to the porosity of the formation. The results indicate that I_H minimally affects density measurements. Regardless of I_H variations, the errors between the calculated and true densities were less than the threshold of 0.025 g/cm³, demonstrating that the densities calculated using the proposed method were consistent with the true densities. In addition, this method achieved accurate measurements for high I_H formations. A comparison of the density calculations for different pore contents indicated that the absolute density errors were relatively small and remained below 0.015 g/cm³ when the pores were filled with water or gas. However, the errors were relatively large when the pores were filled with oil. Notably, if the pores were filled with water, oil, or gas, the absolute errors were less than 0.02 g/cm³. Thus, the method accurately determined the formation density for various porosities and pore contents.

Analysis of the effect of Lithology

Because different lithologies affect neutron transport and gamma attenuation, 42 models, including limestone, sandstone, dolomite, and one-to-one mixture of any two lithologies, were designed to verify the accuracy of the proposed method. All the model pores were filled with water, with porosities ranging from 0.9 p.u. to 40 p.u. The densities calculated using the proposed method were compared with the true densities used in simulated model construction, which varied between 1.93 g/cm³ and 2.843 g/cm³. The density results are presented in Fig. 5 and Table 2.

To verify the effect of lithology on density measurements, the present study focused on a single lithology (clean formations) and mixed lithology comprising sandstone, limestone, and dolomite. Table 2 shows slight differences in the density results across various lithologies, and the average absolute error in the mixed lithology is slightly larger than that in the single lithology, possibly owing to the complexity of the composition of the formation. Regardless of whether it was a single or mixed lithology, the calculated densities were consistent with the true densities, and the absolute density errors were less than 0.02 g/cm³. Overall, the average absolute error in the test database was 0.009 g/cm³, confirming the accuracy of the proposed method for different lithologies.

Multi-parameter impact analysis

The above results quantitatively analyze the impact of lithology and pore content on the accuracy of the density measurement. To further evaluate the proposed method, three cases (Cases 1, 2, and 3) were designed, representing three lithologies (limestone, sandstone, and dolomite) and three pore contents (water, oil, and gas), while considering the mud components (such as chlorite). Specifically, Case 1 simulates water-filled limestone with 20% chlorite; Case 2 simulates oil-filled sandstone with 10% chlorite; and Case 3 simulates dolomite-containing gas with a relatively high chlorite content of 40%. In all these cases, the borehole size was 8.5 in, with a porosity range set from 0 p.u. to 25 p.u., and formation densities between 2.211 g/cm³ and 2.712 g/cm³. The results are shown in Fig. 6.

Fig. 6Full-screen View

(Color online) Results and comparisons. Track 0: Case 1, Case 2, and Case 3. Track 1: Case parameters, including porosity and mud content in the formation. Track 2: Detector counts, including the inelastic gamma count ratio LNRIN, and the epithermal neutron count ratio LNRETN. Track 3: Macroscopic atomic number of the formation. Track 4: Comparison of the true and calculated densities. Track 5: Absolute error

The inelastic gamma and epithermal neutron ratios varied across formations with different lithologies and pore contents, highlighting the effect of formation composition on neutron transport and gamma attenuation. In particular, cases 1 and 2 exhibited smaller measurement errors, demonstrating higher accuracy. In contrast, Case 3 exhibited relatively higher measurement errors, possibly because of its more complex formation composition, characterized by elevated mud content and gas-filled pores. Nevertheless, the absolute density errors in all three cases remained below 0.02 g/cm³, demonstrating the accuracy and reliability of the proposed method for measuring the formation density.

Tool calibration

Owing to the scarcity of commercial tools worldwide, the experimental validation of the NGD tool is briefly discussed in this work using Schlumberger’s work, which introduces an NGD calibration process [40]. The step-by-step process is summarized as follows:

Step 1: Preparation for calibration. A large water-filled calibration tank was used. Access to the aluminum sleeve and the ability to control the fluid in the mud channel (air or water) were ensured.

Step 2: Performing multiple distinct measurements. Measurements across a wide dynamic range were performed under designated configurations: with aluminum sleeve and water in the mud channel, with aluminum sleeve and air in the mud channel, without aluminum sleeve and water in the mud channel, and without aluminum sleeve and air in the mud channel.

Step 3: Fitting a linear model and evaluating the fit The best linear fit was applied to the calibration data points. χ² was calculated to assess the goodness of the fit.

Step 4: Repeatability Check. The calibration process was repeated multiple times without altering the tool or setup. Consistency in the calibration gain was ascertained, particularly for critical parameters.

Step 5: Analyze and Compare. Calibration gains across repeated runs were compared to ensure the reliability of the calibration process.

The multiple configurations employed in this approach effectively established a set of measurement boundary conditions. This design enhanced the dynamic range of the detector count rates and improved the calibration accuracy. The calibration process considers the unique characteristics of NGD tools and can be extended to future NGD tools. Further research, along with calibration and field data acquisition, is planned to validate the practical utility of the proposed method after the development of a new version of the NGD tool.