A Comprehensive Analysis of the Factors Affecting the Accuracy of U, Th, and K Elemental Content in Natural Gamma Spectroscopy Logs

Abstract

This article discusses how various factors can affect the accuracy of U, Th, and K elemental content measurements in natural gamma spectroscopy logs. These factors include errors in the measured energy spectrum, degradation of energy resolution, and spectrum drift. Currently, there is limited research on quantifying the individual impact of each factor on measurement accuracy. To address this gap, the study proposes a methodology that combines energy spectrum data sampling and single-factor quantitative analysis. This approach allows for a more precise understanding of how each factor influences the accuracy of the measurements. The results of the study have important implications for improving the accuracy of U, Th, and K content measurements in applications such as the oil and gas industry.

1. Introduction

The nuclear logging technique for natural gamma energy spectra finds numerous applications in the geophysical field. Gamma ray spectrometry measures the natural radioactivity of uranium (U), thorium (Th), and potassium (K) elements present in rocks [1]. Subsequent research has expanded the applications of natural gamma energy spectrum logging to various areas, including stratigraphic studies [1,2,3,4], clay content estimation [5,6], the calculation of the gas-bearing volume fraction [7], the identification of shale gas reservoirs [8], and the generation of pseudo-synthetic seismograms [9].

The significance of this article lies in its comprehensive examination of the factors affecting the accuracy of natural gamma energy spectrometry logging techniques. Logging measurements often encounter challenges such as low count rate, resolution spreading, and channel drift, which can be influenced by various factors, including logging speeds, well temperatures [10], borehole conditions, and instrumentation electronics [11]. Despite previous research efforts, there is still a lack of systematic investigations into the specific influence of each factor on the calculated U, Th, and K elemental content. This investigation will help developers find out what the main factor is for measurement inaccuracies in field surveying.

To address this research gap, this study developed a novel approach that combines energy spectrum data sampling and single-factor quantitative analysis. By applying this approach and validating our results using measured natural gamma energy spectrum data, we generate valuable insights into the accuracy of U, Th, and K content measurements. Our findings provide essential reference information for future enhancements in this field, enabling more precise measurements and an improved understanding of natural gamma energy spectrometry logging techniques.

This study contributes significantly to the advancement of the field by addressing the limitations and challenges associated with logging measurements. Industries and scientific communities relying on accurate U, Th, and K elemental content measurements will benefit from our research, as it offers a foundation for developing enhanced methods and technologies. Ultimately, our work facilitates more informed decision-making and more accurate evaluation of resources, leading to improved practices and outcomes.

2. Methodology

2.1. Natural Gamma Ray Sources

Natural radionuclides emit gamma rays of varying energies during radioactive decay. Uranium and thorium are two examples of natural radioactive series that produce a complex natural gamma energy spectrum. In nature, natural radionuclides are often in long-term equilibrium, resulting in equal activity of the intermediates within the radionuclide series. As a result, the gamma rays emitted by an intermediate in the radionuclide series can be considered characteristic of the entire radionuclide, for example, with an emission energy of 1.76 MeV from ²¹⁴Bi in the uranium series, an emission energy of 2.62 MeV from ²⁰⁸Tl in the thorium series, and am emission energy of 1.46 MeV from ⁴⁰K. By measuring these characteristic gamma rays, gamma logging can be used to estimate the content of U, Th, and K elements in the formation.

2.2. Spectrum Solving Method

Energy channel refers to a discrete interval or bin into which a gamma spectrometer divides the energy range of detected gamma rays. Each channel corresponds to a specific energy range, allowing the instrument to record how many gamma photons fall within that energy interval. Based on the analysis of nuclear interaction probability, the gamma counts produced by individual radionuclides are statistically independent and do not interfere with each other. This allows us to consider the measured energy spectrum as a linear superposition of gamma rays originating from different elements. In the logging environment, if there are m radionuclides and n energy channels in the measured energy spectrum, the gamma count ci of the i-th channel can be expressed as follows:

$$c_i={\displaystyle {\sum }_{j=1}^m}{a}_{ij}·y_j+{\epsilon }_i,i=1,2,\dots ,n,$$

(1)

where a_ij represents the count rate of the i-th channel in the standard spectrum of the j-th normalized element (with an instrumental background spectrum), y_j denotes the yield of the j-th element, and ε_i represents the error term. Therefore, obtaining different elements’ standard gamma energy spectra is necessary to calculate the elemental yields. With the weighted least-squares method [12], the measured elemental yields can be expressed as

$$Y={\left(A^T·W·A\right)}^{-1}·A^T·W·C$$

(2)

where $C$ represents the counting rate of each energy window in the gamma spectrum (such as the potassium window, uranium window, or thorium window), $A$ represents the detector response matrix, which describes the contribution coefficients of geological elements (K, U, Th) to the energy window count, $W$ is determined by the counting statistical error (the higher the counting rate, the smaller the variance and the greater the weight), and $Y$ represents the actual content of K, U, and Th in the formation (to be solved).

2.3. Acquisition of Data

This study used EMT logging instruments to survey the clear gamma energy distribution spectrum of characteristic peaks obtained using gamma spectrometry standard calibration wells from CNPC Logging Co., Ltd. (CPL) (Xi’an, China), namely the standard gamma energy spectrum. The EMT logging instrument uses BGO gamma detectors. A standard calibration well refers to a purpose-built well constructed using natural ores or other materials to achieve specific concentrations of uranium (U), thorium (Th), and potassium (K). It features known geological characteristics, complete data, and structural stability and is widely used for the calibration, comparison, and standardization of logging instruments. It was awarded certificate number [2007], National Petroleum Metrical Standard Series 176. Two underground pits, A and B, were designed, each featuring three holes with varying diameters.

Table 1. The gamma spectrometry parameter list for standard calibration wells.

Pits	Layer	K [%]	U [ppm]	Th [ppm]	Thickness [m]	Length and Width [m]
A (46, 47, 48)	Background	0.072 ± 0.001	0.035 ± 0.02	0.043 ± 0.01	1.5	3.4 and 1.5
	K	6.07 ± 0.069	0.075 ± 0.003	0.075 ± 0.003	1.5	3.4 and 1.5
	U	0.71 ± 0.003	22.4 ± 0.740	0.800 ± 0.092	1.5	3.4 and 1.5
	Th	0.780 ± 0.001	0.770 ± 0.089	60.40 ± 1.700	1.5	3.4 and 1.5
B (43, 44, 45)	Background	0.10 ± 0.008	0.61 ± 0.030	0.80 ± 0.028	1.5	3.4 and 1.5
	Mix 1	5.06 ± 0.160	5.93 ± 0.110	29.30 ± 0.680	1.5	3.4 and 1.5
	Mix 2	3.88 ± 0.100	17.10 ± 0.660	15.20 ± 0.300	1.5	3.4 and 1.5
	Mix 3	1.87 ± 0.044	11.70 ± 0.070	43.00 ± 0.990	1.5	3.4 and 1.5

presents the parameters, and Figure 1 displays the gamma spectrometry profiles of the standard calibration wells.

This study referred to the measured data of well-9131 and analyzed the influence of different factors on the logging data to verify the accuracy of the experimental results.

3. Experimental Results

3.1. Effect of Statistical Errors

Figure 2 depicts the standard energy spectrum recorded for well-9193 using CPL’s natural gamma logging tools and one of its measured energy spectra. The horizontal coordinates in Figure 2 are instrument readings and represent energy. All our subsequent treatments of the energy spectrum are based on this value. False peaks are observed near channels 130 and 150, which are expected to correspond to the characteristic peaks of U-series radionuclides. Unfortunately, two false peaks are observed due to low counts (<10 per channel) in Figure 2. This inaccurate representation negatively impacts the precision of the element content calculation.

3.1.1. Obtaining Spectra Based on the Probability Distribution Function

The researchers conducted a study to assess the influence of statistical errors on the accuracy of solving the energy spectrum. In this article, an energy spectrum with statistical errors is obtained through statistical sampling of the discrete probability distribution function (PDF) thus obtaining the sampling spectrum with statistical errors. From this sampled spectrum, the researchers solve the spectrum to determine the error distribution of each radionuclide, only considering the case of statistical errors. The following procedure was used:

(1)

The ideal case involved calculating spectrum C, which is the mixture spectrum, using spectrum A, which is the measured standard spectrum, and the elemental yield Y(C = Y·A).

(2)

A PDF-based method was used to obtain the sampled mixed spectrum C at different depths. In the actual measurement, each depth point was sampled using the number of events recorded by the detector (n).

(3)

Spectrum C′, which is the obtained mixed spectrum from step 2, was solved to determine the elemental yield Y′.

(4)

The error distribution resulting from statistical errors in the elemental yield calculation was assessed by calculating (Y′ − Y)/Y.

Measurements in well-9193 were carried out to record the number of events for each depth point, ranging from approximately 500 to 1500. The specific details of the count rate distribution per sampling point are illustrated in Figure 3. The X-axis represents the count for each measurement depth point in the well, and the Y-axis represents the frequency of occurrence of this count. A sampling value of n was chosen randomly from the distribution shown in Figure 3.

The sampled spectrum displayed in Figure 4 closely replicates the shape and counts observed in the measured spectrum. However, one notable distinction between the sampled and measured spectra is the absence of interference factors, such as peak drift and resolution changes, except for statistical fluctuations.

3.1.2. Solving Errors Due to Low Count Rates

By following the procedure outlined in Section 2.1, we can acquire a spectrum that solely reflects the impact of statistical errors. The determination of elemental yield (referred to as Y in step 1) was based on the elemental content of the borehole in the experimental area. The number of events sampled (referred to as n in step 2) in each spectrum corresponded to the number recorded by the detector during the actual measurements. By analyzing the sampled spectra, the researchers can assess the extent to which the count rate at this level (around 1000 events per energy spectrum) introduces error into the solved spectrum.

To address spectra with low count rates, we utilized a range of smoothing and filtering methods. For this purpose, the center of gravity (COG) method [13], low-pass filtering [14], and principal component analysis (PCA) [15] were implemented as filtering and smoothing techniques for the energy spectrum. Additionally, in order to improve the efficiency of the aforementioned methods, a sliding smoothing technique was implemented to smooth the energy spectrum along the depth axis. The results of this smoothing process can be seen in Figure 5. It is important to mention that the smoothing effects of the center of gravity (COG) and low-pass filtering techniques were relatively comparable, with the exception of a slight phase shift observed in the low-pass-filtered spectrum. Conversely, the spectrum filtered using principal component analysis (PCA) showed a closer resemblance to the probability density function (PDF).

Figure 6 compares errors from applying different smoothing methods to solve the energy spectrum. This comparison reveals that utilizing PCA yields the most minor errors. Nevertheless, smoothing the energy spectrum through COG and low-pass filtering induced notable deviations. In particular, COG smoothing resulted in a considerable decrease in the yields of Th-series nuclides and a significant increase in the yields of U-series nuclides. On the other hand, low-pass filtering reduced the yields of both Th-series and U-series nuclides. A comparison of error distributions among the three smoothing methods reveals that K’s yield error was the smallest, while U’s yield error was the largest. This discrepancy can be attributed to the relatively low concentration of U-series nuclides in well-9193. Consequently, these nuclides displayed lower count rates for their characteristic peaks, resulting in compromised error fluctuations.

The steps for the PCA we performed are

(1)

Data preprocessing stage:

(a)

Data input: Original energy spectrum data matrix (n_samples × n_features).

(b)

Data format: Each row represents a spectral sample, and each column represents an energy channel.

(c)

Preprocessing: Smooth pre-processing by the center of gravity method (weight method, 11-point center of gravity method).

(2)

PCA normalization stage:

(a)

Centering: Data is subtracted from the mean.

(b)

Feature scaling: Internal normalization.

(c)

Covariance matrix calculation: The covariance matrix between features is calculated.

(3)

Principal component selection rules:

(a)

Number of components: Fixed selection of 45 principal components (n_components = 45).

(b)

Selection principle: The principal components corresponding to the first 45 largest eigenvalues are retained. This number is determined by solving the spectrum of a mixture with known content. The number of principal components that leads to the minimum solving error is selected.

(c)

Dimension reduction strategy: From 256 feature dimensions to 45 principal component dimensions.

Figure 5 illustrates that the characteristic peak of ⁴⁰K is observed in the channel range of 100–105, while that of Th is observed in the 170–200 channel range. Uranium displays characteristic peaks within the channel range of 145–165, with a comparatively lower count rate than the characteristic peaks of Th. The interference of gamma rays from Th series radionuclides occurs within this range, resulting in the highest errors in the calculated yields of U series radionuclides. Figure 5 illustrates that the PCA method provides the most effective smoothing, producing the energy spectrum closest to the PDF after processing. On the other hand, the energy spectrum shows fluctuations in the count range of 110–200 channels when using the GOC and low-pass filtering methods, which is caused by the low count rate.

3.1.3. Radionuclide Content

The experiment resulted in an average content of 6.99% for ⁴⁰K, 10.26 parts per million (ppm) for the Th series, and 2.98 ppm for the U series. Natural gamma energy spectra can be obtained using the sampling technique with only statistical errors. By repeating the sampling process and solving these spectra thousands of times, the error distribution attributable to statistics for a specific nuclide content and total spectrum count can be derived. Figure 7 illustrates the error distribution for the average content. Gaussian-fitting can be used to determine the mean value and range of the solving error, enabling the assessment of the magnitude and deviation of the errors.

In practice, the count of the natural gamma energy spectrum is related to the velocity at which logging instruments move during formation and the radionuclide content in the stratum. The energy spectrum comprises three gamma energy spectra, U-series, Th-series, and ⁴⁰K, with each nuclide’s content determining its contribution to the total energy spectrum. We sampled natural gamma energy spectra under varying content conditions to examine the correlation between nuclide content variation and statistical error. By solving the energy spectra, we determined their error distributions. The sampled energy spectrum exhibits varying K, Th, and U content within ranges of 1–10%, 5–50 ppm, and 2–20 ppm, respectively, based on the distribution range of nuclide content in conventional logs. In each energy spectrum, one nuclide has content varying from small to large, while the content of the other two is fixed to the average content of the experimental wells. For each element, σ and μ can be calculated as shown in Figure 7, and the resultant statistical error variation curves and values are illustrated in Figure 8 and

Table 2. σ and μ of each radionuclide with different contents.

K			Th			U
Content (%)	μ	σ	Content (ppm)	μ	σ	Content (ppm)	μ	σ
1.0	0.421 ± 0.047	5.778 ± 0.047	5.0	28.771 ± 0.868	26.540 ± 0.871	2.0	10.290 ± 0.783	35.470 ± 0.789
2.0	0.425 ± 0.015	4.120 ± 0.015	10.0	16.735 ± 0.293	14.259 ± 0.293	4.0	9.595 ± 0.606	32.660 ± 0.608
3.0	0.529 ± 0.024	3.775 ± 0.024	15.0	11.672 ± 0.184	9.766 ± 0.184	6.0	6.909 ± 0.385	22.064 ± 0.385
4.0	0.204 ± 0.015	3.163 ± 0.015	20.0	11.173 ± 0.172	10.008 ± 0.172	8.0	3.919 ± 0.324	18.483 ± 0.324
5.0	0.500 ± 0.024	2.881 ± 0.024	25.0	−9.978 ± 0.102	7.671 ± 0.102	10.0	3.500 ± 0.161	14.051 ± 0.161
6.0	0.281 ± 0.014	2.392 ± 0.014	30.0	−6.811 ± 0.060	6.318 ± 0.060	12.0	4.811 ± 0.299	15.782 ± 0.299
7.0	0.595 ± 0.009	2.175 ± 0.009	35.0	−4.820 ± 0.041	5.094 ± 0.041	14.0	3.750 ± 0.120	11.519 ± 0.120
8.0	0.467 ± 0.015	2.318 ± 0.015	40.0	−6.229 ± 0.047	5.698 ± 0.047	16.0	4.705 ± 0.131	9.603 ± 0.131
9.0	0.310 ± 0.013	2.333 ± 0.013	45.0	−7.026 ± 0.063	5.952 ± 0.063	18.0	1.930 ± 0.119	10.062 ± 0.119
10.0	0.455 ± 0.019	2.369 ± 0.019	50.0	−4.783 ± 0.054	5.036 ± 0.054	20.0	2.515 ± 0.093	9.028 ± 0.093

, respectively, and

Table 3. Statistical confidence intervals for each nuclide’s σ and μ.

K			Th			U
Content (%)	μ	σ	Content (ppm)	μ	σ	Content (ppm)	μ	σ
1.0	[−0.579, −0.492]	[5.809, 5.896]	5.0	[−24.919, −23.311]	[26.803, 28.415]	2.0	[14.802, 17.835]	[53.698, 57.190]
2.0	[−0.598, −0.533]	[4.841, 4.906]	10.0	[−16.307, −15.808]	[11.712, 12.211]	4.0	[4.955, 6.189]	[26.353, 27.588]
3.0	[−0.370, −0.355]	[2.852, 2.866]	15.0	[−13.354, −12.923]	[10.008, 10.439]	6.0	[5.048, 5.687]	[18.708, 19.347]
4.0	[−0.119, −0.083]	[2.563, 2.600]	20.0	[−10.705, −10.409]	[8.285, 8.582]	8.0	[7.727, 8.332]	[17.739, 18.344]
5.0	[0.303, 0.326]	[2.947, 2.970]	25.0	[−8.228, −7.989]	[6.924, 7.163]	10.0	[3.972, 4.307]	[12.360, 12.694]
6.0	[0.631, 0.655]	[2.847, 2.871]	30.0	[−8.590, −8.432]	[7.028, 7.186]	12.0	[3.650, 4.064]	[13.245, 13.658]
7.0	[0.040, 0.073]	[2.199, 2.232]	35.0	[−7.675, −7.561]	[5.561, 5.675]	14.0	[2.445, 2.737]	[11.431, 11.723]
8.0	[0.430, 0.463]	[2.134, 2.168]	40.0	[−9.565, −9.360]	[7.577, 7.782]	16.0	[4.230, 4.508]	[10.035, 10.313]
9.0	[0.563, 0.585]	[2.164, 2.186]	45.0	[−7.432, −7.274]	[6.268, 6.426]	18.0	[2.375, 2.549]	[7.883, 8.057]
10.0	[0.360, 0.374]	[1.618, 1.632]	50.0	[−5.865, −5.805]	[4.529, 4.589]	20.0	[2.402, 2.639]	[9.097, 9.334]

provides statistical confidence intervals for each nuclide, σ and μ.

The distribution of errors (σ) for each element tends to decrease as the content increases. For instance, the error distribution of ⁴⁰K is inherently smaller and decreases from approximately 5.778 ± 0.047% to 2.369 ± 0.019% as the content ranges from 1% to 10%. On the other hand, the error distribution for Th is relatively more extensive, decreasing from 26.540 ± 0.871% to 5.036 ± 0.054% as the content increases from 5 ppm to 50 ppm. Similarly, the error distribution for U also decreases with increased content, going from 35.470 ± 0.789% to 9.028 ± 0.093%. The magnitude of the error variation mainly depends on the content of the nuclides, as shown in Figure 5. Notably, the characteristic peak of ⁴⁰K exhibits more remarkable uniqueness than U and Th. As for the mean value of the error (μ), it progressively diminishes with increasing nuclide content. Already, ⁴⁰K demonstrates a relatively small mean error value. In contrast, Th and U yield results with average errors fluctuating around −35% to −15% and 10% to 4%, respectively, with increased content.

3.2. Degradation of the Energy Resolution

During measurements, the instrument is influenced by temperature, which can lead to changes in spectrum resolution. These changes can occur due to variations in the acquisition circuitry’s performance and the crystal’s properties. As shown in Figure 9, to assess any variations in the resolution of the well-9193 instrument, we employed peak-finding and Gaussian-fitting methods to analyze the measured energy spectrum [16]. Since the characteristic peaks of U-series nuclides may not be readily distinguishable, our focus was solely on the characteristic peaks of ⁴⁰K- and Th-series nuclides. Figure 2 illustrates the peak search process, in which we initially applied a fitting procedure to the background within the characteristic peak range. Subsequently, using the Gaussian-fitting method and background subtraction, we determined these peaks’ positions and the full width at half maximum (FWHM).

Figure 10 displays the FWHM distributions obtained from fitting. The FWHMs of the characteristic peaks of ⁴⁰K- and Th-series nuclides in the standard spectrum (obtained from the gamma spectrometry standard calibration wells of CPL) are 7.83 and 10.26 channels, respectively. The FWHM of the ⁴⁰K characteristic peak varies within the range of 8 ± 1 channels in the measured energy spectrum. In contrast, the FWHM of Th-series nuclides tends to be distributed within the range of 11 ± 2 channels. However, in the majority of instances, it exceeds 10.26 channels.

Gaussian Broadening

We conducted a study to examine the effects of resolution changes on the solved spectrum. In the first step, we created an energy spectrum consisting entirely of resolution changes by employing Gaussian broadening of the PDF. We analyzed the broadened spectrum in the second step to assess yield errors.

The Gaussian broadening process alters the counts of each channel in the energy spectrum by redistributing them to neighboring channels using a Gaussian distribution centered on each channel. This mathematical operation is defined as follows:

$$B\left(E\right)={\displaystyle \frac{1}{\sigma \sqrt{2\pi }}}{e}^{-{\displaystyle \frac{{\left(E-E_0\right)}^2}{2{\sigma }^2}}}$$

(3)

where E is the spreading energy, E₀ is the energy corresponding to channel broadening, and σ is the standard variance.

The peaks within the PDF exhibit distinct variations. Hence, Equation (3)’s value of σ should not be derived from the variance obtained through Gaussian fitting of the measured energy spectrum. Instead, it should be calculated as follows:

$$\sigma =\sqrt{{{\sigma }_m}^2-{{\sigma }_{PDF}}^2}$$

(4)

where σ_m is the variance of the Gaussian fit for the measured spectrum and σ_PDF is the variance of the Gaussian fit for the PDF.

The PDF after Gaussian broadening is shown in Figure 11.

Based on the Full Width Half Maximum (FWHM) distribution of the measured energy spectra depicted in Figure 10, we conducted multiple levels of resolution degradation on the PDF. Subsequently, we calculated the resulting broadened PDF, and we present the distribution of yield errors in Figure 12. The analysis reveals that variations in resolution affect the solution spectrum. However, considering the resolution change in well-9193, alterations in the FWHM of the characteristic peak by 2 to 3 channels is not the main reason for dealing with yield errors.

3.3. Spectrum Drift

The instrument’s temperature and operating conditions during downhole measurements impact the detector’s energy resolution and lead to channel drift in the spectrum. The PDF contains characteristic peak positions for ⁴⁰K- and Th-series nuclides, which are 105.3 and 185.6, respectively. Figure 13 illustrates the distributions of the characteristic peak positions in the measured spectrum of well-9193. The characteristic peak position of ⁴⁰K varies within 106 ± 0.6 channels. In contrast, the characteristic peak positions of the Th-series nuclides mostly fall within a range of 188.3 ± 1 channels, demonstrating significant channel drift.

We evaluated the impact of channel drift on the solution spectrum. This evaluation involved shifting the characteristic peaks of the PDF using a technique similar to spectral drift correction. As a result, we generated energy spectra that solely exhibited channel drift and assessed the magnitude of the effect caused by channel drift.

3.3.1. Channel Drift Correction Method

Let s(n) be the ideal gamma spectrum. The “energy–channel” correspondence can be expressed as follows:

$${En}_s=a_1·{Ch}_s+b_1$$

(5)

where En_s is the energy value in the ideal gamma spectrum; Ch_s is the channel address; a₁ is the energy interval between each channel; and b₀ is the corresponding energy value when the channel address is 0.

Similarly, the “energy–channel” correspondence of the energy spectrum x(n) to be calibrated can be written as

$${En}_x=a_2·{Ch}_x+b_2$$

(6)

By comparing Equations (5) and (6), the calibrated channel address in x(n) can be found as follows:

$${Ch}_s={\displaystyle \frac{a_2·{Ch}_x+b_2-b_1}{a_1}}$$

(7)

The entire process of spectral drift correction is as follows [17]:

(1)

Assess the energy spectrum to be corrected x(n), with n being the number of channel sites; create a new array p(n) to store the corrected energy spectrum.

(2)

Calculate a₂ and b₂ from the characteristic peak energies and channel addresses in x(n).

(3)

Calculate the new channel address Ch_s(i) corresponding to channel i in the spectrum according to Equation (7).

(4)

Determine whether p(round(Ch_s(i))) is equal to 0. If it is 0, then p(round(Ch_s(i))) = x(i); otherwise, p(round(Ch_s(i))) = x(i) * k + p(round(Ch_s(i))), where k is the width of the intersection between Ch_s(i) and i.

(5)

Repeat (3) and (4) until the entire energy spectrum has been processed.

3.3.2. Error Analysis of the Effect of Channel Drift

Based on the characteristic peak distribution of the measured spectrum shown in Figure 13, we conducted inverse drift correction on the PDF. This correction aimed to align the peak position in the PDF with the drift observed in the measured spectrum. Figure 14 compares the spectra before and after the correction process.

The solving outcomes after considering channel drift are presented in Figure 15. While a slight deviation in the peak position of ⁴⁰K is observed, it results in a relatively low yield error. In the case of Th, the impact of peak drift is not significant for approximately three channels, likely due to the relatively high counts. However, the computed yield of U exhibits a notable error, which can be attributed to U’s negligible contribution to the PDF. As shown in Figure 14, the distinctive U peak is not prominently visible. On the other hand, the contribution of Th is stacked on the peak of U. The influence of Th should be considered the other cause of U’s notable error in Figure 15a. To address this, we performed another test, as shown in Figure 15b. The content of Th was set to 0 in Figure 15b to exclude the effects of Th on U. As shown in Figure 15b, a significant reduction in the yield error of U appears.

3.4. Effect of Borehole Size and Fluid in the Well

Variations in borehole size, caused by factors such as formation pressure and shot hole, can affect the yields of elements. The influence of well fluid increases as the borehole size increases. We employed Monte Carlo simulation to analyze the impact of borehole size and well fluid on the spectrum. This simulation enabled us to assess the degree of influence that borehole size and well fluid exert on the yields of elements.

In our simulation, we utilized a PDF as the stratigraphic source and manipulated the borehole size and fluid content to obtain an accurate detector energy spectrum. As a result, we solely observed the impact of the borehole and fluid. We conducted tests with borehole sizes of 6.5″, 7.5″, 8.5″, and 9.5″. To examine the influence of borehole size and fluid on the accuracy of yield calculations, the research team conducted experiments using four different fluids: pure water, 10,000 ppm and 20,000 ppm mineralized water (groundwater containing mineral salts), and crude oil. For each test, we utilized 1,000,000,000 particles to minimize statistical errors.

Figure 16 presents the simulation results of the energy spectrum. Figure 16a displays the energy spectrum of the borehole in the absence of fluids, confirming that the sole influencing factor in our simulation is the borehole, consistent with the PDF. Moreover, due to the negligible density differences among the well fluids, the energy spectra of different well fluids with the same well diameter mostly overlap. Figure 16b demonstrates the energy spectrum for various well diameters when pure water is utilized as the well fluid. With increasing wellbore diameter, gamma rays undergo more scattering interactions before reaching the detector, leading to the accumulation of events in the lower energy range.

Figure 17 demonstrates the errors in the calculated yields of K, Th, and U for various well diameters and fluids. Except for an empty borehole, the calculated yields exhibit substantial errors across all conditions. The most significant error occurs in the calculated yield of U, reaching approximately +250%. In contrast, the calculated yields for K and Th exhibit relatively small errors but still approach approximately −40% and −25%, respectively. Figure 4 illustrates the substantial impact of well fluids on the shape of the spectrum and the reduction in the count rate of characteristic peaks, resulting in significant errors in the calculated yields of each nuclide. Due to multiple well fluids having densities within the narrow range of 0.95–1.05 g/cm³, there is minimal variation in density. Consequently, different well fluids result in negligible changes in the spectrum, exerting an insignificant impact on the accuracy of yield calculations. Conversely, as the borehole size increases, the energy spectrum becomes concentrated in the low-energy region, increasing the error of yield calculations.

3.5. Results

Our analysis focused on examining the effects of statistical deficiencies, degradation in resolution, channel drift, borehole size, and well fluid on the accuracy of calculating the natural gamma spectrum. We employed random sampling, Gaussian broadening, channel drift correction, and Monte Carlo simulation to ensure accuracy. Among all the factors considered, insufficient statistics, resolution degradation, and energy channel drift showed agreement with the measured gamma spectrum in well-9193.

Table 4. The range of different influencing factors on the error of K, Th, and U yield calculations.

	Statistics	Resolution	Peak Drifting	Borehole
K	±20%	±1%	±5%	−50%
Th	±30%	−5%	±10%	−30%
U	±75%	±10%	−50%	+250%

presents the extent of error reduction attributed to each factor.

4. Discussion and Conclusions

4.1. Discussion

The influence of statistical properties on the accuracy of the solution for the natural gamma energy spectrum has been demonstrated. The comparison in Section 3.2 demonstrates the distinct advantage of using principal component analysis (PCA) as a filtering technique for smoothing the natural gamma energy spectrum. The PCA method produces a smoothed energy spectrum that closely resembles the PDF. Furthermore, the PCA method yields fewer errors than other energy spectrum smoothing techniques, such as COG or low-pass filtering.

The resolution of the spectrum varies depending on the conditions of the downhole environment and instrument measurements. However, an analysis of the resolution in well 9193 reveals that the variation in resolution in this well does not significantly impact its accuracy.

While channel drift does impact accuracy, Section 3.3 shows that the magnitude of the error is not strictly proportional to the severity of the drift. The drift in U-series nuclides is found to be less severe compared to Th-series nuclides. However, the lower content of U-series nuclides can result in less prominent characteristic peaks in their spectra. Additionally, the interference of gamma rays from Th-series nuclides affects the calculation of U-series nuclide yields, increasing their vulnerability to the influence of channel drift.

The accuracy of the calculated natural gamma spectrum is significantly influenced by the size of the borehole, which acts as a link between the detector and the formation. The presence of material in the borehole causes scattering of the gamma rays emitted by the stratum, leading to noticeable changes in the spectrum’s shape. As a result, there is a substantial 250% error in calculating U-series nuclide yields. While a slight difference in density has minimal impact on the natural gamma spectrum when the well fluid is pure water, mineralized water, or oil, the presence of mud or sand in the well fluid can greatly affect the density of the fluid, resulting in changes to the shape of the spectrum and errors in the calculation. We utilized 1 billion particles per simulation case to minimize statistical noise, aiming to isolate the pure systematic effects of borehole size and well fluid. However, this approach does artificially suppress Poisson variance. This may lead to an overemphasis on the magnitude of systematic effects. This is a limitation of the current study. In subsequent research, we plan to address this issue by designing simulations that more closely mimic field count rates (e.g., 10⁴–10⁵ counts) to introduce realistic Poisson noise. This will allow us to quantify the combined effects of statistical fluctuations and systematic factors (e.g., borehole size) and better reflect the actual error distribution in field measurements. Additionally, we will conduct comparative analyses between high-count and low-count simulation results to clarify the interplay between statistical noise and systematic effects, thereby enhancing the practical relevance of our findings.

4.2. Conclusions

This study aimed to examine the impact of various factors on the accuracy of natural gamma spectroscopy measurements. In order to achieve this objective, we have put forward a set of techniques to determine the primary factors that impact the precision of the detector during real-time measurements by isolating the influence of each individual factor on the energy spectrum sequentially. Using the methodology outlined earlier, we analyze the data obtained from well-9193 and determine that

(1)

K, Th, and U yields can experience statistical variations of around 20%, 30%, and 75%, respectively, when count rates are low.

(2)

The impact of channel drift on the calculation of K and Th yields is minimal, but it can significantly influence U yields, resulting in deviations of up to 50%. The extent of this effect also depends on the radionuclide content of the formation. The emission of gamma rays from Th-series nuclides can disrupt the characteristic peaks of U-series nuclides.

(3)

Resolution degradation has minimal impact on yield calculations.

(4)

The size of the borehole greatly influences the accuracy of the calculated natural gamma spectrum, as it acts as a crucial intermediary between the detector and the formation.

The methods we introduce enable us to pinpoint the primary factors that impact the precision of the detector in real measurements by isolating the influence of each factor on the energy spectrum. In future research, we will also enhance the instrumentation and refine the measurement process to enhance the instrument’s accuracy in relation to the key influencing factors identified through the analysis.

However, it is crucial to acknowledge that sampling synthesized energy spectrum data may not fully represent the entire population of energy spectra, thus potentially imposing limitations on our methodology. Additionally, focusing on a single factor in a quantitative analysis may oversimplify the complex interactions among different factors. However, this study could be a valuable resource for enhancing the approach to analyzing natural and other gamma energy spectrum logs. It is anticipated that these methods could be expanded to analyze the measured energy spectra of additional instruments, ultimately aiding in the targeted improvement of natural gamma well logging tools to achieve greater accuracy.