Intelligent Fast Calculation of Petrophysical Parameters of Clay-Bearing Shales Based on a Novel Dielectric Dispersion Model and Machine Learning

Abstract

Dielectric dispersion and its interpretation process through clay-bearing shales is very complicated, which makes the saturation evaluation of clay-bearing shales difficult. This paper focuses on developing a model that considers the clay effect on the dielectric dispersion of clay-bearing shales. The effects of water saturation, clay content, and other factors on the dielectric dispersion characteristics of clay-bearing shale rocks are analyzed. By combining a dielectric dispersion response database with backpropagation neural network (BPNN) models, this paper develops a calculation model that can simultaneously calculate five petrophysical parameters, i.e., the water salinity, rock cementation exponent, clay content, clay moisture content, and water saturation. The results indicate that the newly developed dielectric dispersion model can characterize the effects of clay content and clay moisture content. The correlation coefficients of the five parameters can all exceed 99% for each sub-sample database and reach an average of 95.06% in an application case, and the calculation efficiency is also very satisfactory, which significantly outperforms the traditional optimization algorithms. The proposed method provides a practical alternative to traditional inversion approaches for shale evaluation.

1. Introduction

In recent years, global exploration has increasingly focused on unconventional resources, with shale oil being of particular significance [1,2,3]. Despite advances in characterization technologies, accurately determining key parameters like hydrocarbon saturation is crucial for reserves estimation [4]. However, it remains challenging in shale reservoirs due to complex mineralogy including organic matter, pyrite, high clay content, low porosity, and high formation wettability [5].

Dielectric logging addresses this challenge by leveraging the significant permittivity contrast between formation water, rock minerals, and hydrocarbons. Interfacial polarization, orientational polarization, and electronic polarization can occur in shale rocks, with interfacial and orientational polarization being particularly significant. Interfacial polarization represents the macroscopic dipole moment phenomenon caused by the accumulation of free charges at interfaces, while orientational polarization refers to the alignment of polar molecules with an external electric field, transitioning from random orientations. Both mechanisms are closely related to shale rock parameters such as water saturation, pore structure, and salinity, which influence the dielectric characteristics of shale rocks [6]. In other words, a phenomenon known as dielectric dispersion occurs primarily due to interfacial polarization and orientation polarization, which demonstrates that the macroscopic permittivity and conductivity of shale rock are frequency-dependent. The array dielectric logging tool (ADT) can measure permittivity and conductivity at four frequency points within the range of 20 MHz to 1.0 GHz, which provides rich geological information. Particularly suited for compact shale reservoirs where mud invasion is minimal, ADT enables accurate quantification of residual hydrocarbons and formation water salinity [7,8].

Researchers have developed a number of models to interpret the measured data of ADT. The Complex Refractive Index (CRI) model is a representative model and can effectively interpret data measured around a frequency of 1.0 GHz [9]. However, the permittivity-saturation relationship derived from the CRI model ignores the effects of interfacial polarization. In shale reservoirs with high clay content (e.g., montmorillonite and illite), strong interfacial polarization in the frequency range less than 1.0 GHz may render the CRI model unsuitable. Moreover, minerals such as pyrite and cementing materials can strongly influence the dielectric responses of shale reservoirs [10,11]. Researchers introduce the Lichtenecker–Rothter (LR) model [12], the Stroud-Milton-De (SMD) model [13], the Maxwell-Garnett (MG) model [14] and other dielectric dispersion models, and these models perform well in pure sandstone and carbonate reservoirs. Clay minerals significantly influence dielectric measurement, which necessitates specialized models to characterize their dielectric dispersion.

The mathematical complexity of models such as SMD and MG requires solving implicit functions for parameters including water saturation and salinity. Traditionally, inversion or optimization algorithms such as the Levenberg–Marquardt nonlinear inversion, Bayesian inversion, Markov-chain Monte Carlo algorithm, particle swarm optimization (PSO), and simulated annealing optimization, are often employed to reduce the difference between the measured multi-frequency values and the calculated values based on dielectric dispersion models [7,13,15]. The accuracy of these algorithms is easily affected by initial values of input parameters and prone to fall into local optimization. Additionally, their calculation can be time-consuming. Universality, real-time performance, and accuracy have been the core objectives of current artificial intelligence-based reservoir evaluation [16]. Recently, machine learning has played an increasingly important role in reservoir evaluation. Specifically, the backpropagation neural network (BPNN)—a multilayer feedforward neural network trained via error backpropagation—has been applied in reservoir evaluation such as formation resistivity calculation and fracture evaluation [17,18].

Therefore, this study presents a three-stage methodological framework: (1) development of a novel dielectric dispersion model specifically for clay-bearing shales, (2) establishment of an interpretation model enabling rapid and accurate derivation of petrophysical parameters from dielectric measurements, and (3) systematic analysis of dispersion characteristics and model validation through field case studies. The proposed approach significantly advances dielectric evaluation in clay-rich shale reservoirs.

2. Methods

2.1. Novel Dielectric Dispersion Model

Shale reservoir rocks are classified into dispersive and non-dispersive phases, as illustrated in Figure 1. The dispersive phase comprises formation water and the clay containing adsorbed water (or bound water), while the non-dispersive phase consists of the rock matrix, the organic matter of oil, and anhydrous clay. The overall dielectric dispersion of the rock represents a composite response resulting from the combined dielectric dispersion characteristics of its various components. Note that this study does not account for the influence of low-content conductive minerals (e.g., pyrite) on the rock’s dielectric dispersion. Appendix A details the dielectric dispersion characteristics of each component and their combination, and a novel dielectric dispersion model for clay-bearing shales is established and expressed in Equation (A11).

2.2. Intelligent Fast Calculation Method

Mathematically, Equation (A11) exhibits considerable complexity, which consequently complicates petrophysical interpretation based on dielectric dispersion measurements. Traditional interpretation methods primarily depend on classical inversion algorithms (e.g., Gauss-Newton) or optimization techniques (e.g., particle swarm optimization), both of which have inherent limitations in terms of efficiency and accuracy. To address these challenges, an intelligent interpretation approach is developed that adopts the strong capability of BPNN for large database fitting, enabling fast and accurate petrophysical interpretation.

2.2.1. Dielectric Dispersion Response Databases

First, all petrophysical parameters related to the dielectric dispersion of shale rocks must be determined. These parameters include the axis ratio of each component in shale rocks, temperature, salinity, cementation exponent, porosity, water saturation, clay content, clay moisture content, etc. According to the common clay-bearing shale rocks, the axis ratio a is assumed to be 10, the temperature T ranges from 50 to 150 °C, the porosity φ ranges from 0.01 to 0.09, the salinity K ranges from 10 to 150 ppk, the cementation exponent m ranges from 1.5 to 3.0, the water saturation S_w ranges from 10% to 100%, the clay content V_c ranges from 10% to 60%, and the clay water S_wc content ranges from 50% to 100%. The discretization intervals for each aforementioned parameter are 1, 11, 9, 8, 7, 10, 6, and 6, respectively, thus yielding a total of 1,995,840 samples. Since T and φ can be obtained through other methods during actual well logging, the samples are divided into 11 × 9 sub-sample databases, and each sub-sample database includes 20,160 samples. At the four frequencies adopted by ADT, the developed dielectric dispersion model for clay-bearing shale rocks is employed to calculate the rock permittivity and the rock conductivity in batches for each sub-sample database, resulting in complete dielectric dispersion response databases. It should be noted that the response databases can be expanded and stored according to practical needs of shale reservoir evaluation.

2.2.2. Backpropagation Neural Network Model

The architecture of the implemented BPNN is illustrated in Figure 2. Each sub-sample database is divided in an 80:20 ratio, with 80% allocated for training and the remaining 20% reserved for testing. The input parameters comprise the rock permittivity levels (${\epsilon }_{f1}$ to ${\epsilon }_{f4}$) and conductivities ($C_{f1}$ to $C_{f4}$) at four distinct frequency points. The output parameters comprise the water saturation S_w, salinity K, cementation exponent m, clay content V_c, and clay moisture content S_wc. The number of hidden layers and the number of neurons in each hidden layer are determined through continuous optimization, and they are assigned to be 3 and 15, respectively. The activation function tansig(x) in the hidden layers, and the activation function purelin(x) in the output layer can be expressed as

$$tansig\left(x\right)={\displaystyle \frac{2}{1+e^{-2x}}}-1$$

(1)

$$purlin\left(x\right)=x$$

(2)

The hyperparameters of the BPNN model are as follows: the training epochs is 300, the train goal (MSE, i.e., Mean squared error) is 1 × 10⁻⁷, the training algorithm is Bayesian regularization, the max verification failure count is 10, the min gradient is 1 × 10⁻⁷, and the initial regularization coefficient is 1 × 10⁻⁴. The optimized BPNN model for each database is saved for subsequent applications.

3. Results and Discussion

3.1. Dielectric Dispersion Analysis of Clay-Bearing Shales

The default condition adopted here is that ${\epsilon }_m$ = 5, ${\epsilon }_{0c}$ = 1000, ${\epsilon }_{\infty c}$ = 5, ${\epsilon }_{\infty w}$ = 4.9, ${\epsilon }_h$ = 2, $\phi$ = 0.0372, $V_c$ = 0.3, $S_{wc}$ = 33.33%, $T$ = 25 °C, $S_w$ = 10%, $m$ = 2, $K$ = 50 ppk, $f_c$ = 200 MHz, $f$ ranges from 1 MHz to 10 GHz.

3.1.1. Fully Hydrated Clay

The dielectric dispersion characteristics of the fully hydrated clay are studied under the default condition. To describe the influence of polarization relaxation, the relaxation frequency $f_c$ is varied from 50 MHz to 350 MHz. The dielectric dispersion curves of the fully hydrated clay are shown in Figure 3. The relaxation frequency $f_c$ determines the characteristics of the dielectric dispersion. In the low-frequency range, since the polarization can be fully established, the clay permittivity is relatively high and stable. As the frequency gradually increases and approaches the relaxation frequency $f_c$, the clay permittivity begins to decrease. This is because the polarization units are gradually unable to keep up with the change in the electric field. When the frequency is much higher than the relaxation frequency $f_c$, the permittivity approaches a lower and stable value. At this time, only fast polarization mechanisms such as electronic displacement polarization can respond to the changes in the electric field. It can be observed that with the larger relaxation frequency $f_c$, the sensitive region of dielectric dispersion shifts to the high-frequency zone, which is consistent with expectations. Based on the previous study [19], the relaxation frequency $f_c$ is typically 200 MHz based on rock core dielectric spectra observation. It is worth noting that for the relaxation frequency $f_c$ = 200 MHz, the curve continues to decline slowly when the frequency is above 1.0 GHz, which indicates that the dielectric dispersion region of the fully hydrated clay is quite broad and spans the range of 20 MHz to1.0 GHz.

3.1.2. Clay-Bearing Shales

Using the default condition, the dielectric dispersion of shale rocks with water-bearing clay is compared with that of shale rocks without water-bearing clay. The influences of various parameters are also analyzed, including the water saturation, temperature, salinity, cementation exponent, clay content, and clay moisture content. The results are presented sequentially in Figure 4a–f. As shown in Figure 4a, when the shale rock does not contain both the formation water and the water-bearing clay, the baseline curve remains a straight line, which corresponds to the pure shale rock without any dispersive components. The rock permittivity increases with the increase in water saturation and dielectric dispersion occurs. When the shale rock contains water-bearing clay, the dielectric dispersion can be observed even if the water saturation is 0, as shown in Figure 4b. On this basis, when the shale rock also contains formation water, the dielectric dispersion is further enhanced, and the variation amplitudes of rock permittivity increase.

Under the default condition, the dielectric dispersion of shale rocks with water-bearing clay is compared with that of shale rocks without water-bearing clay. The influences of various parameters are also analyzed, including water saturation, temperature, salinity, cementation exponent, clay content, and clay moisture content. The results are presented sequentially in Figure 4a–f. As shown in Figure 4a, when the shale rock contains neither formation water nor water-bearing clay, the baseline curve remains a straight line, corresponding to the pure shale rock without any dispersive components. The rock permittivity increases with the increase in water saturation, and dielectric dispersion occurs. When the shale rock contains water-bearing clay, dielectric dispersion can be observed even if the water saturation is 0, as shown in Figure 4b. On this basis, when the shale rock also contains formation water, the dielectric dispersion is further enhanced and the variation amplitudes of rock permittivity increase.

As the temperature rises, the migration speed of ions in the formation water increases, enhancing dielectric dispersion, as shown in Figure 4c. Similarly, the increase in ion concentration in the formation water also strengthens dielectric dispersion, as shown in Figure 4d. The cementation exponent slightly affects the dielectric dispersion of shale rock, and the larger cementation exponent results in the lower rock permittivity, as shown in Figure 4e. Finally, when the clay moisture content is fixed and the volumetric fraction of clay increases, or when the volumetric fraction of clay is fixed and the clay moisture content increases, the dielectric dispersion of shale rock will increase for both cases, as shown in Figure 4f and Figure 4g, respectively.

In addition to characterizing the dielectric dispersion through the changes in clay-bearing rock permittivity, the dielectric dispersion can also be described by the variations in clay-bearing rock conductivity, as shown in Figure 5. Each subgraph in Figure 5 corresponds one-to-one with each subgraph in Figure 4. Rock permittivity varies mainly at low-to-medium frequencies, whereas conductivity changes prevail at medium-to-high frequencies. This indicates that by combining both the changes in rock permittivity and rock conductivity, rock parameters such as the water saturation can be effectively characterized across the frequency range of 20 MHz to 1.0 GHz.

3.2. Petrophysical Parameter Calculation

To demonstrate the advantages of the petrophysical parameter calculation model, the calculation results for two sub-sample response databases are presented in Figure 6. The first column corresponds to a sub-sample database with the condition that the temperature is 150 °C and the porosity is 0.09. The second column corresponds to the other sub-sample database with the condition that the temperature is 90 °C and the porosity is 0.03. From top to bottom, each row corresponds to the cementation exponent, the salinity, the clay content, the clay moisture content, and the water saturation, respectively. In each subgraph, the horizontal axis represents the sample counts in a sub-sample database, and the vertical axis represents the corresponding parameter values. The red step-like curve represents the true values, and the blue point represents the predicted values based on the corresponding BPNN models. It can be observed that the predicted values show excellent agreement with the true values across all subgraphs, with correlation coefficients consistently exceeding 99%.

Table 1. The MSE of calculated results for two sub-sample databases.

		Parameter	m	K	Vc	Swc	Sw
	MSE
Number
1			9.090 × 10−5	2.513 × 10−1	2.150 × 10−3	1.432 × 10−4	8.350 × 10−6
2			4.172 × 10−5	9.250 × 10−2	8.283 × 10−7	5.697 × 10−6	4.543 × 10−6

presents the mean squared error (MSE) values for both cases, providing quantitative evidence of the model accuracy. This strong correlation is further supported by testing across all 110 sub-sample databases, which demonstrates consistently high predictive performance.

To better verify the effectiveness of the petrophysical parameter calculation model, Figure 7 shows a simulated application example in a shale reservoir section. Traditional optimization, i.e., the PSO algorithm is also adopted here for comparison. In Figure 7, the first channel is the depth, the second channel is the measured permittivity values EPS_F0, EPS_F1, EPS_F2, EPS_F3, the third channel is the measured conductivity Cond_F0, Cond_F1, Cond_F2, Cond_F3, respectively. The fourth to eighth channels correspond to the calculation results of the formation water salinity Sal, water saturation S_w (%), cementation exponent m, clay content V_c, and clay moisture content S_wc. The suffixes ‘_T’, ‘_BP’, and ‘_PSO’ represent the true values, the calculated values based on the BPNN model, and the calculated values based on the PSO algorithm. In the PSO algorithm, the population size is 100 and the evolutionary generation is 200. The main employed hardware environment includes a 12th Gen Intel(R) Core (TM) i9-12900 CPU and a 128GB RAM. It can be observed that the variability of the measured data is very high, reflecting the diversity of reservoir properties. The calculated values of the BPNN model agree very well with the true values, and the five correlation coefficients are, respectively, 95.2%, 91.7%,92.8%, 97.4%, and 98.2% with an average value of 95.06%. However, the correlation between the calculated values of the PSO algorithm and the true values decreases, and the five correlation coefficients are, respectively, 85.7%, 86.9%, 83.3%, 86.8%, and 84.4% with an average value of 85.42%. On the other hand, the processing time of the petrophysical parameter calculation model is only 0.0228 s, which is far less than the processing time of 268.173 s of the PSO algorithm. In comparison, the developed petrophysical parameter calculation model proposed here is far superior to the PSO algorithm in terms of calculation accuracy and efficiency. It should be noted that this is beneficial from the prior establishment of the response databases and corresponding BPNN models.

It should be noted that the proposed dielectric dispersion model for clay-bearing shales is based on the benchmark model of the MG model. Therefore, its application scope is applicable to the shale category that requires consideration of ellipsoidal particle shapes, such as the coarse-grained shale, the siliceous shale, etc. In these types of shale rock cores, when the clay content is non-negligible, the proposed new model can be adopted. In the future, this model can be applied more to handle the actual data of clay-bearing shales, and can be compared and verified with other conventional interpretation results. In practical applications, It is needed to necessary the main types of clay to be evaluated based on representative rock experiments, so as to determine the static permittivity and relaxation frequency of water-bearing clay more accurately. Although the new method based on BPNN has significantly improved the calculation accuracy, the response databases must be established in advance based on parameters such as temperature and porosity and the corresponding trained model should be selected purposefully to achieve the best results. If the actual temperature or porosity range exceed the preset range of the database, the dispersion model needs to be reused to expand the database; otherwise, the accuracy of the predicted results will decrease.

4. Conclusions

This paper develops a dielectric dispersion model that takes into account the effect of clay minerals on the dielectric dispersion characteristics of clay-bearing shale rocks, analyses the impact of different factors including the water salinity, cementation exponent, clay content, clay moisture content, and water saturation, etc., and develops a petrophysical parameter calculation model based on response databases and the BPNN model, resulting in the following conclusions.

• The clay content and the clay moisture content have obvious significance on the dielectric dispersion of shale rocks. The higher clay content and the higher clay moisture content, the more apparent the dispersion phenomenon.
• The correlation coefficients of calculated values of water salinity, cementation exponent, clay content, clay moisture content, and water saturation can all reach above 99% for each sub-sample database based on BPNN models when the temperature and the porosity are obtained by other methods.
• The calculation accuracy and the processing efficiency of the developed petrophysical parameter calculation model are far superior to the traditional optimization algorithms, which opens up a new approach to reservoir evaluation.