Based on numerical simulation, research on the characteristics and patterns of dual laterolog resistivity responses in fractured coal seams

Abstract

Laterolog (DLL) logging, utilizing deep and shallow detection depths, is instrumental in identifying and evaluating complex reservoirs such as coal measures. It effectively captures the resistivity contrast between the virgin zone and the invaded zone, providing crucial data for fluid identification, reservoir quality assessment, and fracture system characterization. However, the DLL response in practice is a complex function of coupled factors including borehole environment and bed thickness. Existing studies have predominantly focused on conventional sandstone or carbonate reservoirs, leaving the response characteristics in low-resistivity reservoirs like coal seams poorly understood, which consequently limits the accuracy of quantitative evaluations in fractured reservoirs. To address this gap, this study established a two-dimensional numerical model for deep and shallow laterolog responses using the COMSOL finite-element platform. We systematically investigated the influence of key parameters—including sonde coefficients, borehole diameter, coal seam thickness, and fracture properties (count, width, and spacing)—on DLL resistivity responses. The results indicate that sonde coefficients are significantly influenced by model dimensions; the shallow laterolog sonde coefficient stabilizes rapidly within the detection range, whereas the deep laterolog sonde coefficient requires a substantially larger model domain to achieve convergence. An increase in borehole diameter causes a simultaneous decrease in both deep and shallow resistivities, characterized by a “positive separation” where the shallow resistivity exhibits a more pronounced drop. A positive correlation exists between coal seam thickness and apparent resistivity; the vertical resolution of the DLL for coal seams is approximately 2.5 m, below which the seam’s response characteristics are readily masked by adjacent formations. Fracture development markedly alters resistivity curve profiles, inducing characteristic depressions. Increasing the number of fractures or their width exacerbates these depressions and lowers the overall resistivity. The shallow laterolog curve often displays a distinct “spike-depression” morphology at fracture zones. Notably, increased fracture spacing facilitates the formation of macroscopic conductive pathways, leading to a further reduction in resistivity. The simulation results demonstrate excellent agreement with field-measured data, validating the high reliability of the model. The findings provide a robust theoretical foundation and methodological reference for refined DLL interpretation, environmental correction, and reservoir quality assessment in coal seams and fractured reservoirs.

1 Introduction

Within the framework of well logging technologies, acoustic logging, nuclear logging, and electrical logging constitute the three principal methodological pillars (Chen et al., 2025). To mitigate the influence of drilling fluid shunt effects and low-resistivity surrounding rock interference on measurement accuracy in conventional electrical logging, the Laterolog technique, based on electrical logging principles, was developed in the 1950s. This technology, by precisely measuring formation resistivity parameters, demonstrates unique advantages in lithology identification, reservoir delineation, and oil-bearing potential evaluation. Technological advancements have led to the evolution of Laterolog into three primary configurations: the Laterolog-3, Laterolog-7, and Dual Laterolog, which collectively form the technical lineage of focused current resistivity logging (Wu, 2023).

The three-lateral laterolog (laterolog-3) effectively mitigates interference from mud and surrounding rock through a current focusing mechanism involving a main electrode and guard electrodes. However, its single depth of investigation fails to distinguish resistivity differences between the virgin formation and the invaded zone (Wang, 2018; Shiguan et al., 2017). Although the seven-lateral laterolog (laterolog-7) further improves focusing efficiency and vertical resolution, it remains constrained by the single-detection mode and cannot meet the need for radial resistivity comparison (Qian, 2024; QC-71A Seven-Laterolog Tool, 1975). The dual laterolog employs a dual-guard electrode array structure. By alternately supplying current to the main electrode and the guard electrodes, it generates focused current fields with two distinct depths of investigation: a deep laterolog (RLLD) whose detection range extends to the virgin formation, thereby acquiring true resistivity unaffected by mud-filtrate invasion; and a shallow laterolog (RLLS) with a smaller detection radius, primarily responding to the electrical properties of the invaded or flushed zone (Vasvari, 2011). Based on the amplitude difference between deep and shallow resistivities (a positive difference indicates an oil zone, a negative difference indicates a water zone), the fluid property of the reservoir can be intuitively identified. Furthermore, by integrating Archie’s formula, mud filtrate resistivity (Rmf), effective porosity, and other parameters, oil saturation (Sw) can be quantitatively calculated, providing a quantitative basis for accurate discrimination of oil and water zones (Jin Nam et al., 2009). By overcoming the inherent limitations of single-detection modes, the dual laterolog has become one of the core technologies in resistivity logging suites for complex reservoir evaluation.

Regarding the investigation of underlying physical mechanisms, exploring the response characteristics and patterns of Laterolog tools remains a persistent Frontier research focus. Forward modeling methods offer a systematic approach to elucidating their physical essence and controlling mechanisms. Through integrated studies employing numerical simulation and physical experimentation, domestic and international scholars have achieved a series of breakthrough results in the Dual Laterolog domain (Saboorian-Jooybari et al., 2016), establishing a comprehensive theoretical framework and technical methodology for evaluating complex reservoirs. In the realm of fracture response mechanisms, Saboorian-Jooybari et al. systematically characterized the Dual Laterolog response of fractured formations, laying a theoretical foundation for fractured reservoir identification and evaluation (Pan et al., 2022). Pan Weiguo et al. further advanced this by revealing the unique response mechanisms associated with fracture development characteristics in unconventional reservoirs, thereby enhancing the technique’s practical utility in unconventional hydrocarbon exploration (Qin et al., 2022). Regarding modeling and inversion methodologies, Qin et al. refined response equations and iterative algorithms to achieve more accurate simulation and inversion of true resistivity, fracture porosity, and fracture extent in fractured reservoirs (Shazly and Tarabees, 2013). Tarek F. et al. focused on establishing effective digital models of fractured-vuggy reservoirs, providing crucial data support for quantitative investigations of such complex formations (Mao-Jin et al., 2011). Furthermore, multi-parameter fusion and integrated identification have emerged as significant trends. Researchers such as Tan Mao-Jin and Jie Tian, CA1, et al., by integrating logging data including natural gamma ray, acoustic transit time, and density, have effectively enhanced the accuracy of lithology identification and validation in sand-shale sequences, as well as the characterization of carbonate vugs and pore structures (Jie et al., 2023; Wang et al., 2024). Addressing anisotropy and complex borehole conditions, investigators like Wang Lei and Liu Diren, et al., have elucidated through simulation studies the regulatory patterns and influence mechanisms of factors such as formation boundary dip angle, borehole size, and mud resistivity on logging responses, offering a basis for refined interpretation under complex scenarios including horizontal wells and unconformities (Jiaming et al., 2018; Zhang et al., 2024). Zhang Nianying, Su Jun, et al., through the development of numerical simulation programs and analysis of anisotropic response characteristics, have provided valuable supplementary information and evaluation methods for practical log interpretation (Su, 2015; Wang, 2025). Collectively, these investigations constitute a systematic methodological framework for the application of Dual Laterolog technology in reservoir evaluation, particularly within heterogeneous and fractured reservoirs, holding significant theoretical importance and practical reference value. While these research achievements have deepened the scientific understanding of Dual Laterolog technology and propelled the innovative development of interpretation methodologies, a notable gap remains in the systematic analysis concerning the influence of sonde coefficients, borehole diameter, formation thickness, and fracture parameters—specifically count, width, and spacing—on DLL responses. Furthermore, detailed descriptions of numerical modeling procedures, including model construction, mesh generation, and parameter configuration, are often insufficient.

To address these issues, this paper presents a numerical simulation of the Dual Laterolog response using the COMSOL finite element software. By investigating the sonde coefficients, the study explores the effects of borehole diameter, formation thickness, fracture count, fracture width, and fracture spacing on Dual Laterolog resistivity curves. Furthermore, the simulation results are integrated with and compared against actual Dual Laterolog data from Well XX, aiming to provide a beneficial reference for the application of Dual Laterolog technology.

2 Research approach

Based on the fundamental principles of dual laterolog logging and the finite element method, this study established numerical models of deep and shallow dual laterolog responses in coal seams under fracture invasion conditions using the COMSOL Multiphysics software, aiming to investigate the electrical response characteristics of dual laterolog logging in fractured coal seams. On the basis of unified model parameters, the apparent resistivity curves of deep and shallow dual laterolog measurements were compared and analyzed to ensure consistency of the forward modeling results. Meanwhile, combined with actual dual laterolog data and electrode array coefficients, the critical formation width was determined. Correction of the models was performed by comparing the deep and shallow theoretical models to ensure model accuracy. Subsequently, by systematically varying the coal seam thickness, surrounding rock-to-coal resistivity ratio, borehole enlargement degree, and fracture parameters (e.g., number, width, and spacing) in the two-dimensional model, the variation characteristics of the apparent resistivity curves were analyzed, and the corresponding response patterns were summarized. Finally, the simulation results were compared with measured dual laterolog data to verify the reliability of the established models and the validity of the obtained patterns Figure 1.

FIGURE 1

3 Fundamental theory of dual laterolog modeling

3.1 Fundamental theory of electric field modeling

The three laterolog (LL3) is a focused resistivity logging technique, whose electrode system consists of a main electrode (A₀), guard electrodes (A₁, A₂), and two return electrodes (B₁, B₂) arranged symmetrically (Wang, 2017). Depending on the electrode spacing, it can be configured for either shallow or deep detection modes. As a technical predecessor, LL3 laid the foundation for the development of laterolog methods; however, its single depth of investigation limits its ability to radially distinguish resistivity variations. This limitation drove the evolution toward the dual laterolog (DLL), which integrates both deep and shallow detection modes. In dual laterolog logging, the characterization of formation resistivity is formulated as solving Maxwell’s equations under specific boundary conditions. Based on Gauss’s law for electricity, this method accurately describes the electromagnetic field quantities (D, E, B, H) within a given spatial region, as expressed in Equation 1:

In practical measurement engineering, the dual laterolog tool emits low-frequency alternating current through its central and guard electrodes. The resulting electric field can be treated as a divergent and irrotational steady-state current field problem. According to Ohm’s law and Kirchhoff’s law, the sum of currents entering a node is always equal to the sum of currents leaving that node (Shen Liang et al., 2000; Liang, 2008), and their distribution within the formation remains constant. Based on electrostatic field theory and the characteristics of electric field distribution, the influence of induced magnetic fields can be further neglected, and Equation 1 can be further refined as Equation 2.

Where: D is the electric displacement vector, in C/m²; U is the voltage, in V; ρ is the charge density, in C/m²; E is the electric field intensity, in V/m; Jis the conduction current density vector, in A/m²; ϕ is the electric potential, in V; I i is the current, in A; ρ is the resistivity of the medium, in Ω·m; and is the distance from the primary electrode to the shield electrode, in m.

3.2 Operational principle of the dual laterolog instrument

The dual laterolog tool operates based on the principle of focused resistivity logging. It typically comprises a main electrode (A0), guard electrodes (A1, A2), and monitoring electrodes (M1, M2). During operation, the tool can be configured for two distinct modes based on the investigation objective: deep laterolog and shallow laterolog. In deep laterolog mode, the guard electrodes (A1, A2) are electrically connected to function as upper and lower guard electrodes. They emit a focusing current with the same polarity as that emitted by the main electrode (A0). This causes the measurement current emitted from the main electrode to be collimated into a horizontally oriented, disc-shaped current beam that penetrates deep into the formation, thereby achieving a deep investigation (Cao, 2019; Lai et al., 2022). In shallow laterolog mode, a portion of the guard electrodes (A1, A2) serves as the focusing electrodes (with the same polarity as A0), while the remaining portion acts as return electrodes (with opposite polarity to A0). This configuration forces the measurement current to return to the tool shortly after entering the formation, thus enabling a shallow investigation. Throughout the measurement process, the magnitude of the guard current is dynamically adjusted so that the potential difference between the monitoring electrodes (M1 and M2) approaches zero. This ensures optimal focusing of the main current, allowing for a more accurate measurement of the formation’s true resistivity (Wu et al., 2022; Fan et al., 2016). The apparent resistivity is then calculated using the following formula:

Where: represents the apparent resistivity (in), is the dimensionless sonde coefficient, is the potential difference measured at the monitoring electrodes (in V), and is the current emitted by the main electrode (in A).

The operational modes of laterolog tools primarily encompass four types: constant-current, constant-voltage, ratio-measurement, and constant-power modes (Gao, 2017). The constant-voltage mode is particularly suitable for precise measurements in high-resistivity formations. In this mode, the potential difference between the main electrodes is maintained constant, while the tool’s circuitry automatically adjusts the output current in response to formation changes. When the formation resistivity is high, the main electrode current decreases accordingly to maintain the set voltage. This current signal is stable and exhibits a linear relationship with formation conductivity. This operational approach ensures that the signal amplitude remains within the tool’s optimal detection range across the entire high-resistivity measurement spectrum, thereby effectively ensuring measurement sensitivity and signal-to-noise ratio. Furthermore, due to the constant voltage condition, the boundary conditions in numerical simulations are clear and stable. This characteristic makes the forward modeling calculation process more convergent and efficient, providing a reliable and easily implementable mathematical and physical foundation for investigating the response mechanisms of Dual Laterolog and for conducting inversion interpretations. Consequently, the constant-voltage principle was adopted in this study (Qiang et al., 2025).

4 Model construction and processing

4.1 Tool and formation construction

Based on the fundamental principles and measurement methods of the dual laterolog tool, this study establishes a two-dimensional numerical model of its response using the COMSOL Multiphysics finite element simulation platform. The model consists of four main components: a standard dual laterolog tool, the borehole, the formation, and fractures (Figure 2). During simulation, the properties of the formation matrix and fractures can be adjusted according to the target reservoir characteristics. Specifically, fracture widths are set to 0.1–0.35 mm, and spacings range from 0.15 to 0.4 m, based on published data on coal fractures (Laubach et al., 1998; Li et al., 2017). It is worth noting that the actual fracture widths are smaller than the grid resolution limit (see Table 2; the minimum grid cell size is 0.0014 m). Therefore, instead of directly modeling the geometric thickness of fractures, this study adopts an effective medium approach, assigning an extremely low resistivity (0.01 Ω m) to resolvable thin layers to represent mud-filled fractures—a common practice in well-logging simulations (e.g., Deng et al., 2012). The selected range of fracture spacings spans from dense fracture zones (<0.2 m) to moderately fractured zones (∼0.4 m), allowing the model to capture the transition from isolated fractures to an interconnected conductive network. The model is constructed using COMSOL Multiphysics, with key steps including material assignment, boundary condition specification, mesh generation, and finite element solution (Chen et al., 2024).

FIGURE 2

In the two-dimensional dual laterolog model constructed in this study, the following conditions are established: ① The formation matrix is isotropic and consists of multiple layers of equal thickness, with the lithology comprising an interbedded sequence of siltstone, sandstone, coal seam, limestone, and siltstone. ② The formation matrix is homogeneous, and the scenario of theoretical mud invasion into the borehole is simulated. A representative example of the general model parameters is provided in Table 1.

4.2 Mesh generation and instrument movement

In COMSOL Multiphysics, mesh generation discretizes the continuous physical model by partitioning the solution domain into a finite number of small elements. This process is essential for obtaining numerical solutions, as it directly affects the feasibility and accuracy of subsequent calculations. During the construction of the dual laterolog model, this study adopted differentiated mesh generation strategies for different components, following the recommendations of (Gao et al., 2016).

The dual laterolog model mainly consists of five components: borehole, surrounding rock, fractures, coal seam, and logging tool. Among them, the borehole has a regular cylindrical geometry and a rectangular cross-section, and is discretized using a structured mesh to ensure regular arrangement of grid elements and computational efficiency. The surrounding rock is discretized using a “normal refinement” mesh preset to balance computational cost and accuracy, while still capturing rapid variations in key physical quantities such as electric field intensity and resistivity. As the fractures are the primary detection targets and represent fine, small-scale structures, they require high mesh resolution and are therefore locally refined using triangular elements. The coal seam, characterized by well-developed fractures and strong heterogeneity, adopts an unstructured mesh to better adapt to complex boundaries and avoid computational deviations caused by mismatches between the mesh and model boundaries. The logging tool, with its complex electrode configuration, also employs an unstructured mesh to prevent accuracy loss due to such mismatches. Considering the large spatial scale of the formation and the need for finer treatment of the logging tool, a global “finer” mesh preset is adopted.

Detailed mesh parameters are summarized in Table 2. Multiple lines of evidence support that these settings are sufficient to obtain reliable simulation results. First, the mesh resolution follows literature-based guidelines: Deng et al. (2012) obtained converged resistivity responses using a minimum element size of 0.001 m and a growth rate of 1.2–1.3 near electrodes and fractures; Tan et al. (2011) used an unstructured mesh with a maximum element size of approximately 0.5 m near the borehole; Gao et al. (2016) demonstrated that a mesh with a maximum element size of 0.5 m and local refinement around the tool probe is sufficient for dual laterolog forward modeling. Our settings (minimum element size: 0.0014 m; growth rate: 1.1; maximum element size: 0.7 m) are comparable or superior to these benchmarks. Second, key geometric features are adequately resolved: although the actual fracture width (0.1 mm) is below the mesh resolution, we use an equivalent thin-layer method with a minimum resolvable thickness of approximately 1 mm. The minimum element size of 1.4 mm remains smaller than both the fracture spacing (0.15 m) and the main electrode length (0.12 m), ensuring that all critical electrical interfaces are discretized by multiple elements. Third, the stability analysis of the deep and shallow laterolog responses in Section 5.2 provides indirect validation of mesh adequacy: when the model size exceeds the detection range, the tool coefficients converge smoothly—a behavior that would not be observed with an overly coarse mesh. Finally, the curvature factor (0.2) and growth rate (1.1) adopted in this study are more conservative than typical values in logging simulations (e.g., growth rate of 1.3–1.5), thereby avoiding undersampling in regions with high electric field gradients (e.g., electrode edges, fracture tips, borehole wall).

In the simulation of the actual dual laterolog measurement process, where the logging tool moves along the borehole axis, the tool’s movement along the formation is modeled using the following approach. First, given the complexity of meshing the borehole and formation, these components are held stationary to simplify the model. Second, the dual laterolog tool is treated as an independent entity, to which a longitudinal displacement distance is applied. Third, a boundary probe is established to detect the electric potentials at the two pairs of monitor electrodes (M1-M1’ and M2-M2’), and a global variable probe is configured to monitor parameters such as the current emitted by the shielding electrode A and the measured resistivity. Finally, by simulating the dual laterolog response under various displacement distances , a continuous dual laterolog resistivity curve is obtained.

5 Analysis of simulation results

5.1 Analysis of raw results

Based on the existing model and measured data, two-dimensional deep and shallow laterolog models were established in this study. Under the ideal condition that the physical parameters of the models were kept consistent, the models were run to obtain the resistivity logs.

5.2 Electrode array coefficient

In dual laterolog logging, the electrode array coefficient is a critical parameter that normalizes the measured potential difference into a resistivity value independent of the injected current. It is used to convert the measured potential difference and injected current into apparent resistivity, reflecting the influence of the electrode arrangement on the electric field distribution (Zero et al., 1988; Ni and Xu, 2017). In the study of simulation experiments, the electrode array coefficient was investigated. Based on Equation 3, the obtained electrode array coefficient corresponds to the theoretical value, which may lead to certain discrepancies between the derived resistivity and the true resistivity. However, since neither the voltage nor the current in the model is altered, these discrepancies are entirely attributable to differences in the electrode array coefficient. Therefore, by adjusting the physical parameters of the model (i.e., mud, formation, and fractures) to align with actual logging data and modifying the model dimensions, a relatively appropriate model boundary and corresponding electrode array coefficient can be explored. This approach aims to reduce computational costs and improve efficiency in simulation studies. The simulation results are presented in Figure 3.

FIGURE 3

Simulation results indicate that the electrode array coefficient K exhibits a decreasing trend with increasing model size. Moreover, the shallow and deep laterolog responses display distinctly different variation patterns and stabilization characteristics, primarily owing to the fundamental difference in their depths of investigation. As a parameter that converts measured voltage and current into apparent resistivity, the electrode array coefficient is strictly governed by the current field distribution surrounding the electrode array. An increase in model size corresponds to a transition of boundary conditions from the near-field to an infinite domain, thereby diminishing the distorting effect of boundaries on the current field. The shallow laterolog, characterized by weaker focal capability and a relatively close return electrode, has a shallower depth of investigation; its current field is mainly confined to the region near the tool, rendering it less sensitive to far-field boundary conditions. Consequently, once the model size exceeds its investigation range (typically several times the tool length), the value rapidly converges to a stable value. In contrast, the deep laterolog employs strong focusing and a remote return electrode design, allowing the main current beam to penetrate significantly deeper, thereby exhibiting greater sensitivity to model boundaries. In smaller models, the boundary substantially compresses the deep current pathways, and the K value stabilizes slowly until the model size substantially exceeds the size required for shallow laterolog stabilization. Therefore, the shallow laterolog K stabilizes earlier, whereas the deep laterolog necessitates a larger model size to achieve stability. This finding holds significant implications for the numerical simulation and experimental calibration of logging tools: in finite element simulations, the model dimensions must be determined based on the stabilization criteria of the deep laterolog; otherwise, the accuracy of its response calculation will be severely compromised.

5.3 Logging environment

5.3.1 Borehole diameter variation

Based on actual dual laterolog data, a two-dimensional numerical model for dual laterolog logging was constructed. In this model, the borehole was simplified as a rectangular region. While keeping other initial conditions of the two-dimensional model unchanged, the borehole size in the coal seam section was progressively increased, with borehole radii set to 0.08 m, 0.12 m, 0.14 m, and 0.16 m, respectively, to simulate the influence of different borehole conditions on the logging response and thereby obtain the corresponding apparent resistivity curves. The numerical simulation results are shown in Figure 4.

FIGURE 4

Simulation results indicate that when borehole enlargement occurs in the coal seam section, the apparent resistivity obtained from both deep and shallow dual laterolog measurements shows a decreasing trend, albeit with significant differences in the extent of the decrease. Specifically, the shallow laterolog response is highly sensitive to borehole conditions, exhibiting a sharp drop in apparent resistivity. In contrast, the deep laterolog, owing to its greater depth of investigation, displays relatively gradual variations in value. A negative correlation is observed between coal seam thickness and its apparent resistivity, with the fitted equations as follows: Y_LLS = −67998.2334X+11,513.07786 and Y_LLD = −22631.2642X+8,302.5684. The correlation coefficient R² associated with this fitted relationship further confirms the aforementioned pattern, with R_LLS² = 0.9657 and R_LLD² = 0.9205, as shown in Figure 5.

FIGURE 5

Increasing borehole diameter enlarges the distance between the instrument electrodes and the formation, thereby extending the current flow path through the borehole mud. Since the resistivity of the mud is typically much lower than that of the coal seam, the current tends to concentrate in the low-resistivity mud within the borehole, leading to a significant increase in shunting losses. This ultimately weakens the measurement current and reduces the apparent resistivity readings. This effect is particularly pronounced in shallow laterolog measurements, which have a shallower depth of investigation, and manifests on the logs as a “positive separation”—i.e., deep laterolog resistivity exceeds shallow laterolog resistivity—a typical borehole environmental artifact. The fitted linear equation quantifies this relationship, where the negative slope reflects the sensitivity of apparent resistivity to borehole enlargement, the absolute value of the slope represents the influence coefficient, and the intercept denotes the background resistivity in the absence of borehole effects. In field applications, these empirical equations should primarily serve qualitative trend analysis; for quantitative predictions, local calibration using actual logging data from the same well or nearby wells is required, and the equations are valid only within the simulated borehole radius range (0.08–0.16 m). To improve model generalizability, future work may adopt dimensionless methods, such as plotting normalized resistivity relationships. In summary, borehole enlargement is a key factor contributing to the distortion of coal seam resistivity measurements. To ensure the accuracy of log interpretation, specialized borehole environmental corrections for apparent resistivity must be performed in intervals where borehole diameter variations occur during reservoir evaluation and parameter calculation.

5.4 Formation conditions

5.4.1 Coal seam thickness

Due to the complexity of actual formation lithology, simulations were conducted under idealized conditions. The formation was divided into five parts (siltstone-sandstone-coal seam-limestone-siltstone). The simulation parameters for each formation were primarily based on theoretical data, and all formations, except for the siltstone layer, had equal length and width. The study primarily focused on the coal seam. Under the initial conditions of the two-dimensional model, the thickness of the coal seam was varied. Considering that in actual geological settings, coal seams develop as thin, medium, and thick seams, this study specifically aimed to analyze the resistivity response characteristics of thin coal seams in dual laterolog logging. Therefore, coal seam thicknesses were set to 6 m, 4 m, 3 m, 2.5 m, and 2 m to obtain apparent resistivity values under different formation thicknesses. The numerical simulation results are shown in Figure 6.

FIGURE 6

The simulation results indicate that as the coal seam thickness decreases, the resistivity of the coal seam exhibits a declining trend, with the corresponding apparent resistivity also decreasing. This suggests a positive correlation between coal seam thickness and its apparent resistivity. The fitted equations are as follows: Y_LLS = 8.9205X²+516.2560X-928.9668 and Y_LLD = 3.9111X²+734.2559X-1377.68, The correlation coefficients R² associated with these fitted relationships further validate the aforementioned pattern, with R_LLS² = 1 and R_LLD² = 0.9998, as shown in Figure 7.

FIGURE 7

As the coal seam thickness decreases, the influence of the surrounding rock on the apparent resistivity becomes more significant. A thinner coal seam provides a shorter leakage path for current dissipation into the surrounding rock. The fitted relationship indicates that the greater the coal seam thickness, the higher the apparent resistivity. The slope reflects the sensitivity of resistivity to thickness variations, while the intercept represents the extrapolated resistivity as the thickness approaches zero. It is worth noting that when the coal seam thickness decreases to a certain value, the deep laterolog apparent resistivity curve exhibits only an anomalous bulge. As the thickness further decreases, this bulge persists, but the response characteristics of the coal seam become increasingly indistinct, indicating that its ability to resolve coal seam thickness is approaching its limit. Consequently, it can be inferred that the vertical resolution of the deep laterolog is approximately 2.5 m, and that of the shallow laterolog is also approximately 2.5 m. When the coal seam thickness falls below 2.5 m, the deep laterolog measurement requires coal seam correction for accurate identification; similarly, the shallow laterolog measurement also requires coal seam correction to effectively identify the coal seam. In field applications, these fitting equations are most suitable for predicting relative changes rather than absolute resistivity values, and local calibration is recommended. Future research may adopt dimensionless forms to enhance the reliability of cross-regional comparisons.

5.5 Fracture formation conditions

In the two-dimensional model, five fractures were constructed within the coal seam, positioned near its bottom boundary. These fractures share identical parameters, with a spacing of 0.15 m, an aperture of 0.1 mm, and a length of 10 m. A schematic diagram of the model and the mesh generation are shown in Figures 2, 3, respectively.

5.5.1 Coal seam thickness

Under the condition of incorporating fractures into the two-dimensional model, the number of fractures was set to 0, 1, 2, 3, 4, and 5, respectively. For the case with one fracture, it was positioned at 35 m. For two fractures, they were located at 35.15 m and 34.85 m. For three fractures, the positions were 35 m, 35.15 m, and 34.85 m. For four fractures, they were situated at 35.15 m, 35.3 m, 34.85 m, and 34.75 m. Taking the schematic diagram of five fracture locations as an example shown in Figure 8.

FIGURE 8

For five fractures, the locations were 35 m, 35.15 m, 35.3 m, 34.85 m, and 34.7 m. The apparent resistivity curves corresponding to different numbers of fractures were obtained, and the numerical simulation results are shown in Figure 9. Since the actual conductive network of fractures cannot be effectively characterized, the fitting relationship of apparent resistivity under different numbers of fractures is no longer discussed in the theoretical simulation.

FIGURE 9

The simulation results indicate that when fractures are present within the coal seam, the resistivity response exhibits a distinct “depression” feature at the corresponding depth position. The amplitude of this depression is positively correlated with the number of fractures, meaning that as fracture density increases, the overall resistivity curve shows a declining trend. This phenomenon manifests distinct morphological characteristics in deep and shallow laterolog measurements: the deep laterolog curve typically displays only a broad and gentle depression, whereas the shallow laterolog curve exhibits a unique composite response characterized by an initial “peak” followed by a “depression”. The underlying reason lies in the alteration of the intrinsic electrical structure of the coal seam, which is originally a dense, brittle, high-resistivity medium. In an intact coal seam, the current can be considered to propagate within a homogeneous, high-resistivity medium with relatively little obstruction. Once fractures develop, two key effects come into play: First, the fractures disrupt the structural integrity of the coal body, forming electrically discontinuous interfaces. Second, as preferential flow paths, fractures facilitate the invasion of low-resistivity drilling mud filtrate, leading to current redistribution and the formation of a conductive invasion zone of a certain depth. The “peak-depression” composite response observed in the shallow laterolog is determined by the interaction between its detection characteristics and the fracture system. The shallow laterolog has a shallower depth of investigation and is extremely sensitive to changes in the medium near the borehole wall. As the tool passes a fracture interface, it first captures the high-resistivity fracture face that has not yet been invaded by drilling mud. Due to current concentration at the high-resistivity boundary, a localized anomalous high-resistivity “peak” is formed. As the tool continues to move into the main body of the fracture, which has been fully invaded by mud filtrate, the low-resistivity filtrate becomes dominant. The current preferentially flows through this conductive pathway, causing a sharp decline in resistivity and forming a “depression.” This dynamic response process reveals the transition of the fracture from a tight interface to an effective flow pathway and serves as a key logging signature for identifying and evaluating fracture effectiveness.

5.5.2 Effect of fracture aperture on logging resistivity

To systematically investigate the influence of fracture aperture on the electrical response characteristics of coal seams, the fracture width was progressively increased, with values set to 0.15 mm, 0.2 mm, 0.25 mm, 0.3 mm, and 0.35 mm, respectively. This was conducted to simulate the effect of formation conditions with varying fracture aperture developments on coal seam resistivity. The numerical simulation results are shown in Figure 10.

FIGURE 10

The simulation results indicate that as fracture width increases, the resistivity of the coal seam exhibits a declining trend, with the corresponding apparent resistivity also decreasing. This suggests a negative correlation between fracture width and coal seam apparent resistivity. The fitted equations are as follows: Y_LLS = −533.0547X+4,468.5494 and Y_LLD = −1,542.8925X+57,880.2137, The correlation coefficients associated with these fitted relationships further validate the aforementioned pattern, with R_LLS² = 0.9416 and R_LLD² = 0.9624, as shown in Figure 11.

FIGURE 11

The increase in fracture width further compromises the structural integrity of the coal seam, introducing numerous sharp electrical discontinuity interfaces. At the same time, the widening of fractures creates more efficient seepage pathways, which are preferentially occupied by low-resistivity drilling mud filtrate, thereby forcing a global redistribution of the current. Both deep and shallow laterolog resistivities exhibit a characteristic “step-like drop” pattern, reflecting the coupling effect between the detector and the complex fracture network. The negative slope of the fitted linear equation indicates that the wider the fracture, the more significant the resistivity reduction; the larger absolute value of the deep laterolog slope suggests that, under the simulated conditions, the deep measurement is more sensitive to fracture aperture—a point worthy of further investigation. The intercept represents the resistivity when the fracture width approaches zero (i.e., intact coal). This dynamic response process, transitioning from “boundary shielding” to “internal short-circuiting,” accurately captures the evolution of the fracture system from localized fissures to a globally conductive network, serving as a key electrical signature for evaluating the overall seepage capacity of the reservoir. In practice, these empirical equations should be used with caution: they are valid only within the simulated fracture width range (0.15–0.35 mm) and under similar conditions of mud resistivity and coal properties. For field prediction, it is recommended to calibrate the coefficients using local core or image log data. Future dimensionless analysis (e.g., normalized resistivity versus normalized fracture aperture) will help improve its generalizability.

5.5.3 Effect of fracture spacing on logging resistivity

To systematically investigate the influence of the spatial distribution of fractures on the electrical response characteristics of coal seams, the fracture spacing was progressively increased, with values set to 0.2 m, 0.25 m, 0.3 m, 0.35 m, and 0.4 m, respectively. This was conducted to simulate formation conditions under varying fracture densities. The numerical simulation results are shown in Figure 12.

FIGURE 12

The simulation results indicate that as fracture spacing increases, the resistivity of the coal seam exhibits a significant declining trend, with the corresponding apparent resistivity also decreasing. This suggests a negative correlation between fracture spacing and coal seam apparent resistivity. The fitted equations are as follows: Y_LLS = 3,910.97–583.23Ln(x) and Y_LLD = 4,720.36–844.55Ln(x), The correlation coefficients R2 associated with these fitted relationships further validate the aforementioned pattern, with R_LLS² = 0.9968 and R_LLD² = 0.9995, as shown in Figure 13.

FIGURE 13

When the fracture spacing is small, individual fractures are effectively isolated by the high-resistivity coal matrix, and each fracture merely acts as a localized short-circuit path. However, as the fracture spacing increases, the fractures begin to interconnect, forming a macroscopic “fracture zone” with a significant lateral extent. At this point, the instrument no longer measures the average resistivity of the matrix interspersed with isolated fractures, but instead primarily reflects the bulk conductivity of this continuous low-resistivity fracture zone. The fitted equation shows a negative correlation, meaning that the larger the spacing (wider distribution and eventual interconnection of fractures), the lower the resistivity. The negative slope quantifies the sensitivity of resistivity to fracture spacing, while the intercept approximately represents the coal seam resistivity when the fracture spacing is extremely small (nearly homogenized). The current emitted by the dual laterolog tool is strongly attracted to this conductive zone, resulting in a blocky depression on the resistivity log. In this geological context, the increase in fracture spacing is precisely the prerequisite for the formation of such a preferential conductive pathway. The resultant resistivity decrease serves as a characteristic electrical signature of networked and systematic fracture development, indicating the potential formation of a macroscopic fracture system with excellent permeability within the reservoir. In field applications, the fitting equation is most suitable for trend identification (e.g., detecting the transition from isolated to connected fractures). For quantitative use, local calibration is required, and the equation is strictly valid only within the simulated spacing range (0.15–0.4 m). Adopting dimensionless parameters (e.g., spacing normalized by the tool’s depth of investigation) will significantly enhance the generalizability of these findings in future research.

6 Field data

To validate the accuracy of the simulation results obtained from the dual laterolog model and to assess the reasonableness of the research approach, actual logging data from Well XX were selected as the validation object in this study. By conducting independent data analysis and systematic comparison of the logging curves acquired from this well alongside FMI images, the variation patterns of deep and shallow laterolog responses under different geological parameters—including borehole diameter, formation thickness, number of fractures, fracture width, and fracture spacing—were systematically investigated. The results are shown in Figure 14.

FIGURE 14

As shown in Figure 14, the coal seam in Well XX is located within the depth interval of 1835 m–1845 m, with a thickness of 10 m. The measured data indicate that, within the coal seam intervals identified based on dual laterolog data, the coal seam generally exhibits high resistivity characteristics. When borehole enlargement occurs, the decrease in shallow laterolog resistivity is significantly greater than that of the deep laterolog, and its measured values are generally lower than those of the deep laterolog. This observation is consistent with the pattern revealed in the previous numerical simulations that “the shallow laterolog is more sensitive to borehole variations,” further confirming the significant influence of the borehole environment on shallow investigation resistivity.

Furthermore, in coal seam intervals with fracture development, both the deep and shallow laterolog resistivity curves exhibit pronounced low-resistivity depressions at the corresponding depths. This response characteristic aligns with the conclusion drawn from the simulation results that “fractures induce resistivity depressions,” indicating that the presence of a fracture system creates localized low-resistivity pathways, thereby generating identifiable anomalous signals on the curves.

Integrating the patterns established in the previous analysis, this study attempts a preliminary quantitative analysis of coal seam parameters based on the measured data. By extracting the apparent resistivity values corresponding to typical thick and thin coal seams, it was observed that thick seams (e.g., thickness greater than 2.5 m) exhibit significantly higher resistivity than thin seams (e.g., thickness less than 2.5 m). This finding is consistent with the simulation-derived conclusion of a “positive correlation between seam thickness and resistivity.” Moreover, when the thickness approaches approximately 2.5 m, the resolution capability of the curves for identifying the coal seam diminishes, which closely matches the effective resolution thickness threshold obtained from the simulations. Additionally, by comparing the amplitudes of resistivity depressions in intervals with varying degrees of fracture development, a trend of more pronounced depressions with increased fracture intensity can be observed. This provides a basis for the semi-quantitative assessment of fracture density using resistivity curves.

The above analysis results, on the one hand, validate the applicability of the response patterns revealed by numerical simulations to actual logging data. On the other hand, they demonstrate that applying the quantitative relationships derived from simulations to measured curves can serve as an auxiliary tool for the identification of coal seam thickness and the preliminary evaluation of fracture development intensity.

7 Discussion

Based on actual dual laterolog data and theoretical models, this study constructed a two-dimensional numerical model for dual laterolog logging. By systematically investigating key geometric and physical parameters—including borehole radius, formation thickness, number of fractures, fracture width, and the spatial distribution characteristics of fractures—the influence mechanisms of these factors on the resistivity responses of deep and shallow laterolog measurements were explored in depth. Through comparative analysis of the response characteristics of deep and shallow laterologs, corresponding variation patterns were summarized. The relevant simulation results and analyses are detailed in Table 3.

The key findings are as follows: An increase in borehole diameter significantly reduces apparent resistivity, and the shallow laterolog is more sensitive to borehole variations than the deep laterolog. The physical origin lies in the extended current path in the low-resistivity mud column and the enhanced shunting effect. Coal seam thickness is positively correlated with apparent resistivity; a decrease in seam thickness intensifies surrounding-rock shunting, weakening the resolving capability of the logging curve. Fracture development manifests as characteristic depressions on the resistivity curves, and an increase in fracture count or width further reduces resistivity. Notably, larger fracture spacing may promote network interconnection, forming a macroscopic low-resistivity zone that more strongly affects current distribution. These patterns are validated by field log data.

To further elucidate the response characteristics from an electromagnetic perspective, at the high-resistivity coal–surrounding-rock interface, current lines are squeezed and focused on the coal side and diverge on the rock side. This focusing and divergence directly control the boundary response shape. Within a mud-filled fracture zone, current lines converge from the coal matrix into low-resistivity channels, creating a short-circuit effect that produces characteristic depressions. This current line behavior is consistent with the resistivity depressions, the shallow laterolog’s peak-depression pattern, and the fracture-parameter trends observed in Section 5.5.

Compared with existing studies, a response model for sand-shale thin interbeds was established (Liu et al., 2012); the electrical characteristics of fractured-vuggy formations and dip-amplified anisotropy were revealed (Deng et al., 2012); dual laterolog responses under extreme conditions were simulated using an hp-adaptive finite element method (Nam et al., 2010). The present study focuses on coal seams, employing a unified numerical model to simulate the effects of borehole diameter, bed thickness, and multiple fracture parameters, and attempts to establish quantitative fitting relationships. It also explores anomalous responses induced by fracture spacing variations, offering a supplementary perspective on spatial fracture configurations.

Although some progress has been made in this study, certain limitations still exist. First, the simplification of a 2D axisymmetric, homogeneous, and isotropic model deviates from actual 3D heterogeneous reservoir conditions. Second, the model does not account for formation dip, dynamic invasion, multiphase fluid distribution, or coal heterogeneity, and assumes that fractures are completely filled with low-resistivity mud. Specifically, electrical anisotropy and dynamic invasion are two key factors that may introduce deviations.

Coal seams with well-developed cleats and bedding often exhibit transverse isotropy, i.e., horizontal resistivity is lower than vertical resistivity. This anisotropy affects dual laterolog responses: the deep laterolog, being more sensitive to vertical resistivity, may underestimate true resistivity, while the shallow laterolog, more sensitive to horizontal resistivity, may alter the observed positive separation. According to published anisotropy ratios (typically 1.5–5), the apparent resistivity can differ from isotropic values by 20%–50%. Previous 3D finite-element simulations of anisotropic and dipping formations (e.g., Shen, Wang and Zhang, 2000) have shown that such deviations are significant and should be accounted for in quantitative interpretations. Qualitative trends remain robust, but the amplitudes of resistivity anomalies may shift.

Furthermore, the model assumes static, fully established mud invasion, whereas actual filtrate invasion is time-dependent, introducing three deviations: (1) a time-varying resistivity profile; (2) mixing of filtrate with formation water; and (3) asymmetric invasion along high-permeability fractures, which cannot be captured by an axisymmetric model. These dynamic effects influence the fitted equations in Table 3. For field applications, anomaly amplitudes should be calibrated using time-lapse logging or invasion modeling, as demonstrated in several dynamic simulation studies.

Despite these limitations, the main qualitative conclusions—positive correlation between coal seam thickness and apparent resistivity; negative correlations with fracture width, count, and spacing; and the peak-depression morphology of the shallow laterolog—remain valid in a relative sense. However, the fitted equations in Table 3 should be used with caution in strongly anisotropic or dynamically invaded formations and must be locally calibrated using site-specific logging data.

Looking forward, to improve generalizability, dimensionless methods (e.g., normalized resistivity versus normalized fracture parameters) should be adopted to eliminate unit dependencies, facilitate cross-regional comparisons, and reduce the need for repeated calibration. At the same time, future work will develop 3D anisotropic geological models and multi-physics coupled models incorporating dynamic multiphase invasion, integrate machine learning for parameter inversion, and combine rock physics experiments to form an integrated “experiment–simulation–logging” methodology. Notably (Shen Liang et al., 2000), systematically validated the transition from 2D to 3D finite-element modeling for laterolog responses in complex 3D environments, providing a solid methodological basis for extending the present study toward more realistic representations.

8 Conclusion

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