Determination of the Fractal Dimension of CO<sub>2</sub> Adsorption Isotherms on Shale Samples

Qi, Hongyan; Wang, Zhenlin; Yu, Jianglong; Yu, Peirong; Zheng, Guoqing

doi:https://doi.org/10.1155/2022/2206989

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Characterization of the Unconventional Continental Shale Oil Reservoirs

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Volume 2022 | Article ID 2206989 | https://doi.org/10.1155/2022/2206989

Determination of the Fractal Dimension of CO₂ Adsorption Isotherms on Shale Samples

Hongyan Qi,¹Zhenlin Wang,¹Jianglong Yu,¹Peirong Yu,¹and Guoqing Zheng¹

Academic Editor: Kouqi Liu

Received19 Jan 2022

Accepted31 Jan 2022

Published22 Feb 2022

Abstract

The fractal theory has been widely applied to the analysis of gas adsorption isotherms, which are used for the pore structure characterization in unconventional reservoirs. Fractal dimension is a key parameter that can indicate the complexity of the pore structures. So far, most fractal models for gas adsorption are for N₂ adsorption, while fractal models for CO₂ adsorption are rarely reported. In this paper, we built a fractal model for CO₂ adsorption by combining a thermodynamic model and the Dubinin–Astakhov model. We then applied the new model to three CO₂ adsorption isotherms measured on shale samples. The results show that the fractal dimension from the new model lies between 2 and 3, which agrees with the fractal geometry. The new model presented in this paper can be used for the CO₂ adsorption analysis, which allows characterizing micropore structures in shales.

1. Introduction

Knowing the pore structures of shale rocks is an essential part for the reservoir characterization which could assist in understanding the original oil/gas in place and the flow characteristics of the shale rocks [1–3]. The gas adsorption method is now a standard method for pore strcuture correction. The gas adsorption method involves bringing the gas/vapor into contact with the solid surface [4–6]. For shale rocks, N₂ and CO₂ are the two gases that are typically used for gas adsorption. N₂ adsorption (at 77 k) can be used to derive the specific surface area (using the Brunauer, Emmett, and Teller equation) and pore size distribution (using the Barrett–Joyner–Halenda model or density functional theory) [7, 8]. N₂ adsorption can mainly get the meso-macropore information (pore size >2 nm) and cannot provide the micropore information. Under low temperature, the N₂ molecule is kinetically restricted from accessing the micropores [9]. In order to overcome the limitations of N₂ adsorption, CO₂ adsorption is commonly performed. The critical dimensions of the CO₂ molecule and the N₂ molecule are very similar (0.28 nm for CO₂ and 0.30 nm for N₂), but the higher working temperature for CO₂ adsorption (273 k for CO₂ adsorption) helps the CO₂ molecule to enter into the micropores [9]. CO₂ adsorption (273 k) is usually complemented by the N₂ adsorption (77 K) to get a wider pore size information in shale reservoir characterization.

The pore structure of shale samples is very complicated, which has been shown by many researchers using the scanning electron microscope [10–15]. In order to understand the complexity of the pore structures, the fractal theory can be applied. Avnir et al. [16] found that at the molecular level, the surface of most materials has a fractal behavior with fractal dimension varying from 2 to 3, where 2 means a perfectly smooth surface and 3 denotes significantly rough and a disordered surface. Several fractal models have been developed for N₂ adsorption, such as the Frenkel–Halsey–Hill (FHH) theory [17] or the thermodynamic model by Neimark [18]. These models for N₂ adsorption are mainly focused on the meso-macro pore (>2 μm) capillary condensation by using Kelvin’s equation. However, Kelvin’s equation is not valid for the pores with sizes smaller than 7.5 nm [19], which indicates that these current fractal models cannot be used for CO₂ adsorption. Most researchers only analyzed the fractal dimension from the N₂ adsorption, even when they performed both N₂ adsorption and CO₂ adsorption experiments [3, 20, 21]. To the best of the authors’ knowledge, the fractal model for the CO₂ adsorption in shale studies has not been yet reported. In this paper, we present a fractal model for CO₂ adsorption built by combining the thermodynamic model and the Dubinin–Astakhov analysis model.

2. Model Description

From the thermodynamic viewpoint, the differential of the interface area ds can be calculated from the balance between the work of forming the interface and the work from the adsorption of CO₂ [18]:where σ is the surface tension, µ is the differential chemical potential of CO₂, and N is the adsorption amount of CO₂.

For CO₂, the differential chemical potential under pressure p can be calculated using the following equation [22, 23]:where R is the universal gas constant, 8.314 Jmol⁻¹K⁻¹; T is the temperature, 273 K; p is the working pressure; and p₀ is the CO₂ saturation pressure under 273 K.

By combing equations (1) and (2), we can obtain the following equation:where N_max is the maximum cumulative adsorption quantity and N_(p/p0) is the cumulative adsorption quantity under the relative pressure (p/p0).

The correlation of the area for a fractal surface and the volume circumscribed by the surface obeys the following equation [24]:where D is the fractal dimension.

If the fractal surface is measured on a Euclidean area, equation (4) can be changed to the following equation by the dimensional analysis [25]:where k is a correlation factor between the surface and the volume, r is the radius, and V is the volume.

Assuming that the gas molecules cannot be compressed, the volume can be calculated using the following equation:where V_L is the molecular volume of CO₂.

By combining equations (3) and (6), we obtain the following expression:which can be further rewritten aswhere r_p/p0 is the pore radius under the relative pressure p/p0.

The form of equation (8) is similar to the equation which was provided by Wang and Li [25] for the N₂ adsorption analysis. However, in their model, they applied the Kelvin equation to obtain the pore radius for the mesopore capillary condensation stage, which is not suitable for the micropores. For the micropores, r_p/p0 can be derived from the Dubinin–Astakhov model [26]:where is the limiting adsorption volume, is the occupied adsorption volume, E is the characteristic energy of the system, and n is an empirical constant.

Then, the pore size of the sample can be calculated using the following equation:where D₀ is the dispersion interaction energy.

If we combine equations (9) and (10), we can express the radius:

For equation (8), let and ; then, equation (8) can be written in the following form:

Thus, if A and B values are calculated under different relative pressure for a CO₂ adsorption isotherm, then the fractal dimension D can be easily determined from the slope of the function in equation (12). C is a constant which can be derived from curve fitting.

3. Results and Discussion

3.1. Model Verification

In order to verify this model, we performed the CO₂ adsorption experiment on three shale samples and calculated their fractal dimension using equation (12). Figure 1 shows the adsorption isotherms of the three samples (the relative pressure is from 0.001 to 0.03).

(a)

(b)

(c)

3.2. Impact of D₀ on the Results

We applied the new model to calculate the A and B values of each sample under different relative pressure values and then plot A as a function of B (Figure 2) (here, we assume that D₀ is 1500 Jnm³mol⁻¹, from Hiden Isochema). Very strong linear relations exist between A and B for all three samples, indicating that the CO₂ adsorption isotherm shows the fractal behavior. The fractal dimensions of these three samples are 2.508, 2.323, and 2.405, respectively. The fractal dimension falls within the expected range of 2 < D < 3, predicted by the fractal geometry [25, 27]. Thus, the model yields robust results and can be used to calculate the fractal dimension of CO₂ adsorption on shale samples.

(a)

(b)

(c)

In the previous examples, we had to assume a value for D₀ in order to calculate r_p/p0. D₀ is the dispersion interaction energy of CO₂, which is not well constrained but is usually set to around 1500 Jnm³mol⁻¹ [28]. In this part, we further studied the effect of D₀ value on the fractal dimension. We set three D₀ values (1000, 1400, 1500, 1600, and 2000 Jnm³mol⁻¹) and then calculated the fractal dimension of sample 1 for all three cases. Figure 3 shows that the absolute values of the A and B values do vary for different D₀, but the slope of the linear regression of A to B remains the same. Therefore, the choice of D₀ value does not affect the fractal dimension calculation. The fractal dimension value of the CO₂ adsorption isotherm from shale samples can be derived even when the exact D₀ value is not well constrained.

(a)

(b)

(c)

(d)

(e)

3.3. Future Research

Clay bound water is an important factor that could affect the fractal dimensions of the gas adsorption which has been studied by many researchers [29–31]. Under different moisture content, the fractal dimension changes. However, in this study, we preheated the samples under 105°C for over 12 hours and we believe that the effect of the clay bound water effect can be neglected. In this study, our focus is to derive a model to describe the fractal model for the analyzing the fractal dimensions of the CO₂ gas adsorption. Thus, the samples we choose are from a single basin and very simple. More samples from the different shale basins will be collected and analyzed to verify the applicability of this model. In addition, based on the studies by other researchers, the fractal dimension from N₂ gas adsorption could be correlated with the pore structures [32]. Whether the fractal dimension from CO₂ gas adsorption is correlated with the microstructures of the samples and how the microstructures affect the fractal dimension will be the target for the next step research.

4. Conclusions

(1)Based on the Dubinin–Astakhov analysis model and the thermodynamic model, we built the fractal analysis model for CO₂ adsorption on shale samples.(2)We applied the new model to calculate the fractal dimensions of the CO₂ adsorption isotherms for three shale samples. We found that the CO₂ adsorption isotherms had the fractal behavior, and the fractal dimension value was between 2 and 3. This agrees with the fractal geometry and indicates the robust performance of the new model.(3)We conducted a sensitivity analysis to investigate the effect of the dispersion interaction energy D₀ on the fractal dimension calculation and demonstrated that the choice of the D₀ value does not affect the model outcomes.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

The authors appreciate PetroChina Xinjiang Oilfield Company and shale oil research team. This study was supported by the Forward-Looking Basic Projects of CNPC (grant no. 2021DJ1803).

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Copyright © 2022 Hongyan Qi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Geofluids

Characterization of the Unconventional Continental Shale Oil Reservoirs

Determination of the Fractal Dimension of CO2 Adsorption Isotherms on Shale Samples

Abstract

1. Introduction

2. Model Description

3. Results and Discussion

3.1. Model Verification

3.2. Impact of D0 on the Results

3.3. Future Research

4. Conclusions

Data Availability

Conflicts of Interest

Acknowledgments

References

Copyright

Determination of the Fractal Dimension of CO₂ Adsorption Isotherms on Shale Samples

3.2. Impact of D₀ on the Results