3.1 Calculation model and benchmark

An X-ray logging tool model (Fig. 1) was built using the MCNP5 code based on the ENDF70 nuclear data library. This tool is similar to the X-ray litho-density logging tool published by Simon et al. and Qu et al. [12, 16], and a high voltage of 350 kV is applied across the cathode and gold target of the anode in X-ray tubes. The theoretical tool responses were simulated in sandstone and limestone formations of varying porosities [17, 18]. A 20.16 cm diameter borehole was filled with fresh water with a density of 1.0 g/cm³. The formation was cylindrical with a radius of 100 cm and a height of 180 cm. The tool has two gamma-ray detectors located at different distances from the source. The source-to-near detector spacing was 11 cm, and the far detector spacing was 22 cm. In some cases, there was a thin mudcake between the tool and the formation. To ensure the reliability of the calculated results, 4×10⁹ photons were sampled and the statistical errors were reduced to a level below 1.0% using IMP cards during each simulation. The MCNP simulated the X-ray energy spectrum, as shown in Fig. 2. The response of the far detector, which was approximately equal to that of the D4 detector in the study by Simon et al. [12], is shown in Fig. 3.

Fig. 1Full-screen View

(Color online) Tool-borehole-formation Monte Carlo calculation model

Fig. 2Full-screen View

X-ray energy spectrum released by 350 kV X-ray tube

Fig. 3Full-screen View

(Color online) Logarithm of counts in the high-energy (0.15-0.35 MeV) window vs. density for simulation of this paper’s far detector and simulation of Simon’s D4 detector

As shown in Fig. 3, the simulation data of the far detector in this study and the D4 detector were benchmarked with the data simulated by Simon et al. [12]. The relationship between the density window (0.15-0.35 MeV) was simulated in various sandstone and limestone formations; the formation bulk density is basically the same as that obtained by Simon et al., which demonstrates that the results of the proposed numerical simulation in this study are reliable and can be used for subsequent studies on density and P_e measurement methods [12].

3.2 X-ray energy spectrum and cross-section analysis

To detailly study the principle of formation density and P_e measurement using X-ray, the X-ray energy spectrum received by the detectors was first analyzed. Then, the detector responses in dolomite, limestone, and sandstone formations with an electron density of 2.341 g/cm³ and respective P_e levels of 4.53 b/e, 2.72 b/e, and 1.64 b/e were obtained using the tool model (Fig. 1). The energy spectra of the scattering X-rays were also obtained, as shown in Fig. 4.

Fig. 4Full-screen View

(Color online) Scattering X-ray energy spectra of far detector in sandstone, limestone, and dolomite formations with the same density

As shown in Fig. 4, when the photon energy is lower than 0.1 MeV, the count rate varies significantly in different formations and is sensitive to lithology. As photon energy increases, the difference in counts in the three formations decreases gradually and the curves of count rates overlap. For the two density windows that are commonly used in chemical source density logging, when the photon energy is within the range of 0.15–0.24 MeV, the count rates are high, but there is a certain difference in the count rates of the three formations. This is because the photons whose energies are within this range not only undergo Compton scattering (density), but are also affected by the photoelectric effect [19, 20], while photons with energies higher than 0.24 MeV basically undergo Compton scattering only and are not significantly affected by the photoelectric effect. Therefore, the counts in the three formations basically overlap. However, the count rates are quite low, indicating a large statistical error. If only photons with energies higher than 0.24 MeV are used to calculate density, the calculation accuracy cannot meet the project requirements.

In order to determine the extent to which the counts in different energy windows are affected by the photoelectric effect of X-ray litho-density logging, the ratios $R$ (%) of the photoelectric absorption coefficient ${\mu }_{\text{ph}}$ to the total attenuation coefficient $\mu$ (sum of Compton attenuation coefficient ${\mu }_{\text{c}}$ and photoelectric absorption coefficient μ_ph) with respect to the cross-sections of the three formations were calculated using photons with energies within a range of 0–0.35 MeV. The calculation results are listed in Table 1.

From Table 1, it can be seen that when the limestone formation is selected as the reference formation, there is a difference of 1.58% in the values of R in the energy range of 0.15–0.35 MeV for the sandstone formation and 0.53% in the energy range of 0.24–0.35 MeV for the limestone formation; the former is nearly three times the latter, indicating that the impact of photoelectric effect (lithology) on the counts in the energy range of 0.15-0.35 MeV is three times its impact on the counts in the energy range of 0.24-035 MeV. However, the counts in the former range are much more (approximately 20 times) than the counts in the latter.

4.1 Impact of photoelectric effect

The 350 kV X-ray generator released X-rays in the 0–0.35 MeV continuous energy spectrum into the formation; the X-rays interacted with the formation in the form of photoelectric effect and Compton scattering and were received by the detectors. Formation density and P_e were calculated using formulas (1)–(3); the selected energy range in the lithology window was 0.04–0.08 MeV, and the threshold energy in the density window was generally 0.15 MeV. In order to verify the impact of photoelectric effect on GLD method during X-ray litho-density logging, the responses of the logging tool (Fig. 1) in sandstone, limestone, and dolomite formations with 0–30% porosity saturated with water were simulated. Then, based on the scattering X-ray energy spectra received by the detectors, the counts in the lithology window (0.04–0.08 MeV) and the counts in the density window (0.15–0.35 MeV) were respectively calculated, which were then respectively substituted into formulas (1) and (3) to calculate formation density and P_e. The calculation results are shown in Fig. 5.

Fig. 5Full-screen View

Comparison between formation properties calculated using GLD method (density window: 0.15–035 MeV) and assumed formation properties

In practical cases of litho-density logging, limestone formation is usually selected as the reference formation. For sandstone, limestone, and dolomite formations, the density values calculated using formulas (1) and (2) differed significantly from the values obtained through simulation; particularly for the sandstone formation, the mean error was 0.075 g/cm³, and the maximum error was 0.093 g/cm³. The formation P_e calculated using formula (3) for the lithology window (0.04–0.08 MeV) and density window (0.15–0.35 MeV) differed significantly from the actual formation P_e, and the mean error of the measured P_e of the three formations was 0.21 b/e. For X-ray litho-density logging, the counts in the density window (0.15–0.35 MeV) are affected by the photoelectric effect. For this reason, the accuracy of density measurement by the GLD method in X-ray litho-density logging was low.

4.2 Impact of statistical error

Knoll observed a statistical error with respect to the receipt of photons by the detectors under the same measurement environment, that is, $\text{Δ}N=\sqrt{N}$, where $N$ denotes the counts received by the detectors [21]. Based on formulas (1) and (3), density measurement uncertainty $\Delta \rho$ and P_e measurement uncertainty $\Delta P_{\text{e}}$ can be expressed as

and

where $\Delta N_{\text{L}}$ and $\Delta N_{\text{H}}$ denote the statistical errors of the counts in the lithology window and the density window, respectively; and $f\left(N_{\text{H}}\right)$ and $f\left(N_{\text{L}},N_{\text{H}}\right)$ denote the functional relationship between formation density and N_H, and the functional relationship between P_e and $N_{\text{L}}$, $N_{\text{H}}$, respectively.

From the aforementioned energy spectrum analysis, it can be seen that the density window of 0.24–0.35 MeV is less affected by the photoelectric effect (lithology) than the density window of 0.15–0.35 MeV, but the counts are too low. In order to verify the impact of the density window statistical error on the GLD method during X-ray litho-density logging, the simulation data in Fig. 5 and the scattering X-ray energy spectra received by the detectors were used to calculate the counts in the lithology window (0.04–0.08 MeV) and the counts in the density window (0.24–0.35 MeV). In addition, the density measurement uncertainty $\Delta \rho$ and P_e measurement uncertainty $\Delta P_{\text{e}}$ were calculated using formulas (1) and (4) along with formulas (3) and (5). The calculation results are shown in Fig. 6. Based on "JJG. (Military) 42-2014 Specification for Calibration of Density Logging Tool," Li et al. believed that the standard density measurement uncertainty $\Delta \text{ρ}$ should be ±1% of the measured density value and the standard P_e measurement uncertainty should be ±0.1 b/e [22].

Fig. 6Full-screen View

(Color online) Comparison between measurement uncertainties of GLD method (density window: 0.24–0.35 MeV) and specified measurement uncertainties

From Fig. 6, it can be seen that when the lithology window is 0.04–0.08 MeV and the density window is 0.24–0.35 MeV, the accuracy of density and P_e calculation using formulas (1), (2), and (3) increase, but the measurement uncertainties do not meet the requirements. The analysis reveals that the impact of photoelectric effect on the GLD method can be reduced by choosing a density window of 0.24–0.35 MeV. However owing to the low counts in this density window, the detector statistical error has a significant impact on the formation density and P_e calculation using the GLD method, and the density measurement uncertainty and P_e measurement uncertainty are not within the range of the specified measurement uncertainty levels.

The above analysis indicates that the number of scattering photons received by the detectors is directly proportional to the measuring time; the longer the measuring time, the smaller the detector statistical error will be. Table 1 shows that the count received by the detector in the 0.15–0.35 MeV energy window is 20 times the count in the 0.24–0.35 MeV energy window at a fixed measuring time. In order to reduce the uncertainty of density and P_e calculations based on the counts in the density window (0.24–0.35 MeV), the detector measuring time needs to be significantly increased, which means the logging speed will be reduced significantly.

5.1 Density calculation

Simon et al. and Yu et al. believed that X-rays in X-ray litho-density logging can be characterized by lower energy and continuity when compared with 0.662 MeV gamma rays, and the impact of photoelectric effect on the counts in the density window should not be neglected [12, 11]. Then, the relationship between the counts in the density window $N_{\text{H}}$ and the counts in the lithology window $N_{\text{L}}$ can be expressed as

where photoelectric absorption coefficients ${\mu }_{\text{ph}, E_{\text{H}}}=Ka\left(E_{\text{H}}\right)\frac{N_{\text{A}}}{2}U$, ${\mu }_{\text{ph}, E_{\text{L}}}=Ka\left(E_{\text{L}}\right)\frac{N_{\text{A}}}{2}U$; Compton attenuation coefficients ${\mu }_{\text{c}, E_{\text{H}}}=b\left(E_{\text{H}}\right)\frac{N_{\text{A}}}{2}{\rho }_{\text{e}}{\sigma }_{\text{c},\text{e}}$, ${\mu }_{\text{c}, E_{\text{L}}}=b\left(E_{\text{L}}\right)\frac{N_{\text{A}}}{2}{\rho }_{\text{e}}{\sigma }_{\text{c},\text{e}}$; and $d$ is the distance of the source from the detector. $K$ is a constant whose value depends on the photon energy and unit of the cross-section; $a\left(E\right)$ and $b\left(E\right)$ are related to X-ray energy, respectively; $E_{\text{H}}$ and $E_{\text{L}}$ represent the high and low energy in the X-ray energy spectrum, respectively; ${\sigma }_{\text{c},\text{e}}$ denotes the electron scattering cross-section (electron scattering cross-sections of commonly seen formation materials are basically equal); $N_{\text{A}}$ denotes the Avogadro constant; and $U$ is a volumetric photoelectric absorption index defined as ${\rho }_{\text{e}}\cdot P_{\text{e}}$. According to Yu et al. [11], the electron density response formula can be obtained as follows:

When the lithology and density windows are fixed, a, b, and c are constants and can be obtained from a tool experiment or through a simulation. The calculated electron density is substituted into formula (2) to calculate the measured density (apparent density).

Equation (8) in this study for X-ray litho-density logging is similar to the density correction equation, which was introduced by Ellis et al. [1] for the Cs-137 source litho-density logging; however, it is not widely used for this purpose. Both Ellis et al. and Serra et al. [14, 23] believed that when formula (1) and formula (2) are used to calculate the formation density in Cs-137 source density logging, the lithological effect could be eliminated by controlling the boundary of the density window. Its statistical precision and potential well logging speed can meet the engineering requirements.

The previous analysis indicates that as the energy of X-rays released by the X-ray generator is always lower than the energy of gamma rays released by Cs-137, its statistical accuracy will be significantly reduced by increasing the boundary of the density window, and its potential measuring speed will also be reduced. Therefore, formulas (1) and (8) can be used to calculate the formation density in X-ray litho-density logging.

In general, mudcake has a significant effect on density logging, and it is generally corrected by a spine and ribs chart. Similarly, formulas (8) and (1) are used to calculate the near and far spacing densities, and a spine and ribs chart is developed to correct the effect of mudcake in X-ray litho-density logging, as shown in Fig. 7. The "spine" is the locus of near and far spacing densities without mudcake, and the "ribs" trace the two densities for the presence of an intervening mudcake. The physical parameters of different types of mudcake are shown in Table 2.

Fig. 7Full-screen View

(Color online) Spine and ribs chart of the X-ray litho-density tool for mudcakes

In Fig. 7, the red rib points are acquired in the 30% water-saturated sandstone formation, and the blue rib points are acquired in the 5% water-saturated limestone formation. The mudcake density range is 1–2.44 g/cm³, P_e range is 0.36–109.60 b/e, and mudcake thickness is up to 1.6 cm. Therefore, the effect of mudcake on density measurement can be corrected by using the spine and ribs chart in X-ray litho-density logging.

5.2 P_e calculation

The photoelectric absorption coefficient ${\mu }_{\text{ph}}$ is based on the cross-sections of quartz, calcite, and dolomite materials commonly seen in well logging, and the relationship is shown in Fig. 8.

Fig. 8Full-screen View

(Color online) Relationship between Photoelectric absorption coefficient μ_ph and photo energy

The photoelectric absorption coefficient and photo energy (0.02~0.5MeV) in the same formation are directly proportional to each other in a log-log coordinate system [14]; then, we have the following relationship:

So that ${\mu }_{\text{ph},\text{L}}$ can be expressed as:

When density and lithology windows are fixed, $\alpha$ is a constant, and formulas (10), (6), and (7) can be simplified as follows:

where $C_{\text{p}}$ and $D_{\text{p}}$ relate to the attenuation distance d of energy E; $F_{\text{p}}$ relates to energy E and initial photon flux; and $A_{\text{p}}$ and $B_{\text{p}}$ only relate to energy. From formulas (6) and (7), $A_{\text{p}}{U}^{\alpha }$ and $B_{\text{p}}U$ are primarily related to ${\mu }_{\text{ph},\text{L}}$ and ${\mu }_{\text{ph},\text{H}}$, respectively. As shown in Fig. 8, ${\mu }_{\text{ph},\text{L}}$ is two orders of magnitude larger than ${\mu }_{\text{ph},\text{H}}$. Therefore, $A_{\text{p}}{U}^{\alpha }$ is much greater than $B_{\text{p}}U$ and $B_{\text{p}}U$ can be deemed to be zero. Then, formula (11) can be transformed to:

According to the relationship between volumetric absorption index $U$ and P_e, we have:

where A, B, C, and D are constants which can be obtained from tool calibration or through simulation.