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

Authors: Hongyan Qi; Zhenlin Wang; Jianglong Yu; Peirong Yu; Guoqing Zheng
Journal: Geofluids
 (2022)

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## 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.


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## Body

## 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]:(1)σds=ΔμdN,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]:(2)Δμ=RTlnpp0,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:(3)S=1σ∫Np/p0NmaxRTlnpp0dN,where Nmax 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]:(4)S1/D∼V1/3,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]:(5)S=kDr2−DVD/3,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:(6)V=Nmax−NNp/p0VL,where VL is the molecular volume of CO2.By combining equations (3) and (6), we obtain the following expression:(7)1σ∫Np/p0NmaxRTlnpp0dN=kDrp/p02−DNmax−Np/p0D/3VLD/3,which can be further rewritten as(8)∫Np/p0Nmaxlnp/p0dNrp/p02=σRTkDVLD/3Nmax−Np/p0rp/p0D,where rp/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, rp/p0 can be derived from the Dubinin–Astakhov model [26]:(9)WW0=exp−−RTlnp/p0En,where w0 is the limiting adsorption volume, w 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:(10)dW/W0drp/p0=3nD0Enrp/p0−3n+1exp−D0Enrp/p0−3n,where D0 is the dispersion interaction energy.If we combine equations (9) and (10), we can express the radius:(11)rp/p0=−RTlnp/p0D0−1/3.For equation (8), let Ap/p0=ln∫Np/p0Nmaxlnp/p0dN/rp/p02 and Bp/p0=lnNmax−Np/p0/rp/p0; then, equation (8) can be written in the following form:(12)Ap/p0=C+DBp/p0.Thus, ifA 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).Figure 1
Measured CO2 adsorption isotherms of three shale samples. (a) Sample 1. (b) Sample 2. (c) Sample 3.
(a)(b)(c)


### 3.2. Impact ofD0 on the Results

We applied the new model to calculate theA 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.Figure 2
Fractal analysis of the CO2 adsorption isotherms of the three shale samples. (a) Sample 1. (b) Sample 2. (c) Sample 3.
(a)(b)(c)In the previous examples, we had to assume a value forD0 in order to calculate rp/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.Figure 3
The impact ofD0 on the fractal dimension (sample 1). (a) D0 = 1000 Jnm3mol−1. (b) D0 = 1400 Jnm3mol−1. (c) D0 = 1500 Jnm3mol−1. (d) D0 = 1600 Jnm3mol−1. (e) D0 = 2000 Jnm3mol−1.
(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.


## 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).Figure 1
Measured CO2 adsorption isotherms of three shale samples. (a) Sample 1. (b) Sample 2. (c) Sample 3.
(a)(b)(c)


## 3.2. Impact ofD0 on the Results

We applied the new model to calculate theA 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.Figure 2
Fractal analysis of the CO2 adsorption isotherms of the three shale samples. (a) Sample 1. (b) Sample 2. (c) Sample 3.
(a)(b)(c)In the previous examples, we had to assume a value forD0 in order to calculate rp/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.Figure 3
The impact ofD0 on the fractal dimension (sample 1). (a) D0 = 1000 Jnm3mol−1. (b) D0 = 1400 Jnm3mol−1. (c) D0 = 1500 Jnm3mol−1. (d) D0 = 1600 Jnm3mol−1. (e) D0 = 2000 Jnm3mol−1.
(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 energyD0 on the fractal dimension calculation and demonstrated that the choice of the D0 value does not affect the model outcomes.



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*Source: 2206989-2022-02-22.xml*