NMR based fractal analysis in shale under different confining pressure, and its correlation to Archie's cementation exponents

Abstract

This study delves into the complexity of shale pore structures through fractal dimension analysis of nuclear magnetic resonance (NMR) data under varying confining pressures. Focusing on nine illite-rich shale samples, we investigate how confining pressure influences the pore size distribution, particularly narrowing meso- and macropores. Our analysis utilizes two distinct models to calculate fractal dimensions: Model 1 categorizes pores into micro and meso + macro based on T₂ cutoffs, while Model 2 considers all pore sizes collectively. Results show that fractal dimensions increase with confining pressure, indicating enhanced pore complexity. Notably, Model 2 proves more adept for shales, revealing no significant correlation with quartz or total clay content or NMR porosity. However, a robust correlation (R² = 0.9416) between fractal dimension and Archie's cementation exponent m suggests a novel method for estimating Archie's m through pore distribution analysis without electrical resistivity evaluation. This approach offers new insights for petroleum exploration and pore structure characterization in shales, particularly within the Goldwyer Formation and similar geological settings.

1. Introduction

The success of commercial hydrocarbon production from shale reservoirs has necessitated the development of petrophysical methods better suited for shales [1,2]. Unlike sandstones, shales feature more complex and heterogeneous pore structures due to the presence of nano-scale pores [3,4], which include micropores (with diameter less than 2 nm), mesopores (2–50 nm), and macropores (greater than 50 nm) following the pore size classification proposed by the International Union of Pure and Applied Chemistry (IUPAC) [5,6]. Shales generally contain broad pore size distributions with micro-porous organic pores and macro-scale inorganic matrix [7,8]. The properties of shale pore structure, such as pore size distributions, pore morphology, and pore connectivity, have been demonstrated to play a crucial role in controlling the transport mechanisms of hydrocarbons, and consequently the estimation of reservoir storage capacity and production potential [9,10]. Thus, improved characterization of the pore structure in shales is essential.

The pore structure of shales has been investigated using various techniques, such as scanning electron microscopy [11] and field emission scanning electron microscopy [12]. However, these techniques compromise the pore structure investigation since they can only characterize the surface or a thin section on the surface of the sample [13,14]. Other fluid invasion methods, such as low-pressure nitrogen adsorption [15,16] and mercury intrusion method [17,18], are unable to provide full-range pore structure information and the latter may cause artificial micro-fractures under high confining pressures [19,20]. More recently, nuclear magnetic resonance (NMR) techniques associated with fractal theory have been demonstrated as effective approaches for the investigation of pore structure heterogeneity [14,21].

NMR mainly investigate the relaxation process of hydrogen nuclei of the pore-bearing fluids, providing information such as pore size distribution, wettability, permeability, and porosity [[22], [23], [24]]. The fractal theory associated with NMR transverse relaxation times (T₂) has proven successful in evaluating the complexity and regularity of pore networks and pore throat structures through a fractal dimension index, D [25,26]. Higher fractal dimensions typically represent more irregular pore networks, and hence more complex and heterogeneous pore structures. Fractal analysis based on NMR T₂ data has been previously applied to sandstones [27], coals [28,29], and tight sandstones [9,19], demonstrating that pore structure information acquired using NMR can be effectively characterized by fractal theory. Fractal analysis using NMR has also been conducted on shales, although previous studies are rare. Yuan and Rezaee [20] investigated the relationship between clay-bound water and fractal dimension using both NMR and low-pressure nitrogen adsorption methods. Yan et al. [30] established a comparably positive correlation between fractal dimension and total organic content (TOC), quartz content, and clay mineral content by conducting NMR experiments on shales. However, the influence of shale pore structure on fractal dimension under reservoir pressure, specifically mesopores and macropores, remains unclear and there is also a need for further investigation of the relationship between fractal dimension of NMR T₂ and Archie's cementation exponent, m, which also strongly correlates to pore structure [31]. Archie's equation has been accepted as an industry standard for determining water saturation in well-log interpretation. The original equation is relevant to clay-free rocks and considers formation water as the only conducting medium in the rock. For shales, this parameter has been derived from the lab work [32,33], and found to be pressure sensitive due to pore structure rather than pore water salinity [34]. It is therefore likely that a correlation exists between Archies m and the NMR-derived fractal dimensions for shales.

Fractal dimension models are well-known and widely used in the study of conventional rocks, and some applications have been extended to shale using N₂ adsorption data or MICP data. However, the application of fractal theory using NMR data requires further investigation. Additionally, the petrophysical significance of the fractal dimension (D) and Archie's parameter (m) shows some similarities, yet this aspect is rarely highlighted in current studies. Therefore, this study focuses on estimating changes in shale pore structure under various pressures and their impact on fractal dimensions. This is achieved using NMR Carr-Purcell-Meiboom-Gill (CPMG) measurements and regularized transverse relaxation time distributions. Two fractal models with different kernel functions were applied to calculate fractal dimensions, and comparisons were made to determine the suitability of each model for fractal analysis of shales. The derived fractal dimensions from the two models were compared to identify a suitable fractal model for micropore-dominant shale samples. The most reliable D was correlated with Archie's parameter (m) from our previous results. This study suggests a potential possibility of replacing Archie's parameter (m) with fractal dimension (D) in well-log interpretation to save on electrical tests, as D requires only pore size distribution analysis, whereas m requires additional electrical testing.

2. Methodology

2.1. Samples and experiments

A total of nine illite-rich shale samples were collected at various depths from the Ordovician Goldwyer Formation of Theia#1 in the Canning Basin, Australia (Table 1). The detailed geological information was documented in our previous studies [32,33]. All samples underwent full saturation with 250,000 ppm NaCl at 2000 psi. The results of NMR at ambient pressure were first recorded before transferring the samples into a Hassler cell for the resistivity tests at both ambient and reservoir conditions (2800 psi confining pressure). After reaching reservoir pressures in the Hassler cell, the samples were transferred back to the NMR instrument and measured again for comparison using an experimental procedure of Zhong et al. [33]. Similarly, the resistivity was measured for two samples (Th24, Th25) under elevated confining pressures incrementally from ambient up to 8500 psi as per the experimental procedure presented in Zhong et al. [32]. Due to the lack of high-pressure core flooding NMR facilities, the resistivity measurement was paused at each pressure step and NMR analysis was performed. The process was repeated at each confining pressure increment. NMR T₂ data was measured using Carr, Purcell, Meilboom, and Gill (CPMG) pulse sequence [[35], [36], [37]], with a 100 μs inter-echo spacing (TE), a 10,000 ms inter-experiment delay, 5000 number of echoes and a minimum of 200 signal-to-noise ratio (SNR). The resistivity measurements were conducted in potentiostatic mode with a constant 1 V excitation at 1 kHz, where the phase ($\theta$) reaches its lowest value ($\theta$) < 5°.

Table 1.

Composition of the studied shale samples from XRD.

Sample ID	Depth (m)	Quartz (%)	Kaolinite (%)	Illite/Mica (%)	Smectite (%)	Chlorite (%)	Total Clay (%)	Calbonate (%)	Pyrite (%)	Other minerals (%)
Th17	1472.13	17.42	3.43	55.8	N/A	10.99	70.2	N/A	N/A	12.37
Th23	1508.89	14.94	0.14	35.5	0.69	N/A	36.3	32.02	2.28	14.46
Th24	1510.54	18.34	0.3	24.3	0.98	5.64	31.3	35.57	2.32	12.51
Th25	1512.7
Th28	1520.42	11.89	0.1	44.3	0.39	10.69	55.5	10.69	0.1	21.87
Th37	1554.73	21.61	0.15	45.5	0.71	N/A	46.3	14.21	2.42	15.43
Th39	1563.3	18	1	56	N/A	3	60	7	3	12
Th40	1570.83	17.09	0.65	49.5	N/A	N/A	50.1	12.29	2.25	18.25
Th45	1591.9	17.7	0.16	40.7	0.88	N/A	41.7	9.64	3.33	27.6

2.2. NMR theory

NMR Carr-Purcell-Meiboom-Gill (CPMG) measurements have been widely used in both the laboratory and the field for the determination of transverse relaxation (T₂) times [38,39]. The measured T₂ signal can be expressed as follows [39]:

$$\frac{1}{T_2}=\frac{1}{T_{2,bulk}}+\frac{1}{T_{2,surface}}+\frac{1}{T_{2,diffusion}}$$ (1)

where T_{2, bulk} is the T₂ relaxation time of the bulk fluid, T_{2, surface} is the surface relaxation time and T_{2, diffusion} refers to the relaxation sourced from molecular diffusion. In water-saturated porous media, T_{2, bulk} is generally negligible, since it is much longer than T_{2, surface}. T_{2, diffusion} arising from internal magnetic field gradient caused by magnetic susceptibility difference between various phases, is also minimized under laboratory experimental environment [22]. For fluids inside porous media, T_{2, surface} can be expressed as follows [40]:

$$\frac{1}{T_{2,surface}}\cong {\rho }_2\frac{S}{V}$$ (2)

where ρ₂ is the transverse surface relaxivity of the pore surface, S/V is the ratio of the pore surface to pore volume. For simplicity, the pore geometry is assumed spherical with a pore radius of r, though it is known that pore shapes in shales are most likely flat and elongated[41]. T₂ can be then be simplified as per Eq. (3), allowing for the correlation of T₂ to absolute pore size radius as:

$$\frac{1}{T_2}\cong {\rho }_2\left(\frac{3}{r}\right)$$ (3)

As shown in Eq. (3), NMR T₂ times become dependent on pore size. In the collective T₂ spectrum, longer T₂ components represent fluids in larger pores with smaller S/V, while shorter T₂ indicates fluids in smaller pores.

2.3. NMR fractal theory

A power law correlation between pore radius, r, and the total quantity of pores with pore size larger than r, N_r, was established based on the fractal theory as per below [42,43]:

$$N_r=\underset{r}{\overset{r_{\mathit{max}}}{\int }}f\left(r\right)dr=\frac{C}{r^D}$$ (4)

where D is the fractal dimension and C is a constant fractal factor.

The cumulative pore volume with pore sizes smaller than r (V_r) and total pore volume with full pore sizes (V_t) are calculated as below [20,44]:

$$V_r=\frac{-D\times C^2}{3-D}\left(r^{3-D}-r_{\mathit{min}}^{3-D}\right)$$ (5)

$$V_t=\frac{-D\times C^2}{3-D}\left(r_{\mathit{max}}^{3-D}-r_{\mathit{min}}^{3-D}\right)$$ (6)

where r_min and r_max are the smallest and largest pore sizes, respectively.

The cumulative pore volume fraction V is then obtained according to Eq. (7) [45,46]:

$$V=\frac{V_r}{V_t}=\frac{r^{3-D}-r_{\mathit{min}}^{3-D}}{r_{\mathit{max}}^{3-D}-r_{\mathit{min}}^{3-D}}$$ (7)

Considering a much greater r_max than r_min in shales, Eq. (7) is simplified as:

$$V={\left(\frac{r}{r_{\mathit{max}}}\right)}^{3-D}$$ (8)

According to the aforementioned NMR expression as described in Eq. (3) where NMR T₂ times are proportional to pore radius. Eq. (7) is subsequently transferred into:

$$V={\left(\frac{T_2}{T_{2,\mathit{max}}}\right)}^{3-D}$$ (9)

where T_{2, max} is the maximum T₂ time in T₂ distributions.

By taking the logarithm of both sides of Eq. (9), the gradient of the resulting linear correlation can be used to determine the NMR fractal dimension:

$$\log \left(V\right)=\left(3-D\right)\log \left(T_2\right)+\left(D-3\right)\log \left(T_{2,\mathit{max}}\right)$$ (10)

The correlation as shown in Eq. (10) (named as model 1 in this study) has been widely implemented in the characterization of fractal dimension in sandstones [9,47]. Two straight lines are normally observed from the double logarithm plot, indicating the existence of multiple segments from T₂ distributions. This is achieved by employing a T₂ cutoff obtained from T₂ spectrum, indicating fluids saturated in different pore sizes in shales.

However, Model 1 becomes invalid in estimating the fractal dimension in porous media with low permeability and a wide range of pore size distributions. This is because micropores, developed from clay minerals with tiny pore throats, are not self-similar to the larger macropore system [47,48]. This discrepancy necessitates the use of fractal dimensions for two distinct pore systems. Furthermore, the fractal dimension is sensitive to the selection of the T₂ cutoff, which is challenging in the presence of coupled NMR T₂ spectral peaks associated with more complex pore structures found in shales.

Lai et al. [48] proposed a model accounting for the contribution of a pore-size distribution to the fractal dimension (named as model 2 in this study). Similar to model 1, model 2 assumes a spherical pore shape and the correlation between T₂ values and pore radius remains valid. The quantity of pores N_i at a given pore size r_i (corresponding to T_2i) is computed as below:

$$N_i=\frac{V_i}{36\pi {\left({\rho }_2{T}_{2i}\right)}^3}$$ (11)

where V_i is the cumulative NMR signal amplitude at corresponding T₂ times (T_2i), ρ₂ is the same transverse surface relaxivity.

Consequently, the number of pores with pore size larger than r_i can be obtained from:

$$N\left(r\right)=\sum _j^n{N}_i=\sum _j^n\frac{V_i}{36\pi {\left({\rho }_2{T}_{2j}\right)}^3}$$ (12)

where j = i+1.

By combining Eq. (4), (12), the correlation can be expressed as follows:

$$\log \left(\sum _j^n\frac{V_j}{{\left(T_{2j}\right)}^3}\right)+\log \frac{1}{36\pi {\left({\rho }_2\right)}^3}=-D\log \left(3{\rho }_2\right)-D\log \left(T_{2i}\right)$$ (13)

Eq. (13) demonstrates the correlation of the total number of pores with pore size greater than r_i to the corresponding NMR T₂ relaxation times. By plotting a double logarithm of those two parameters, a linear relationship can be obtained and the fractal dimension can be determined from the slope.

3. Results and discussion

3.1. NMR T₂ distribution

Fig. 1 shows the variation of the T₂ spectrum of brine saturated shale samples under increasing confining pressure conditions, derived from previous studies [32,33]. Fig. 1a and b show the typical distribution curve of two shale samples (Th23, Th28) under ambient conditions (14.7 psi) and confining pressure, respectively. As expected in shales, the Clay Bound Water (CBW) and microporosity from the T₂ spectrum is the dominant population. This type of pores (T₂ < 1 ms) remain almost the same under two different pressure conditions due to their limited compressibility. While signals (T₂ > 1 ms) representing fluids within mesopores, macropores, and microfractures decrease under the 2800 psi confining pressure, within which the microfracture signal (T₂ > 100 ms) becomes negligible at 2800 psi confining pressure. Similarly, the signal change of samples Th24 and Th25 in Fig. 1c and d are dominated by mesopores, macropores, and microfractures under increasing confining pressure, although the associated change in pore size distribution is somewhat difficult to resolve using NMR T₂ distribution, fractal analysis of the raw data provides a better solution.

Fig. 1.

T₂ spectrum under different confining pressures for 4 tested illite-rich shales from the Ordovician Goldwyer formation. Signals for micropores with T₂ < 1 ms almost overlap each other at different confining pressures. Microfracture signals with T₂ > 100 ms vanish beyond 500 psi. The changes are dominated by meso/macropores signals (T₂ around 10 ms).

3.2. Fractal analysis using model 1

Model 1 is used to analyze the fractal dimension based on the determination of the NMR T₂ cutoff. For sandstone, T₂ cutoff is defined as the T₂ for which the signal transitions from immovable-water-dominant to movable-water-dominant, and it varies from the rock lithology and fluid property [49]. For sandstones, centrifugation is an effective method to achieve water separation [47,50]. While in shales, the centrifuge is less effective due to the inherent complexity of shale [51]. Therefore, Yuan and Rezaee (2019) separated the curves as clay-bound water signals and effective pore signals by drying shale samples at 80 °C and derived two fractal dimensions corresponding to different water types. Wang et al. [46] separated the fractal curves into two regions at a T₂ relaxation time of 1 ms and simply used the high R² region for fractal analysis to evaluate the complexity of the pore structure. Li et al. [52] defined the T₂ cutoff based on T₂ relaxation times corresponding to fluids confined within 10 nm pores and then compared this with N₂ absorption and MICP data.

In this study, we aim to discuss the change of D under increasing confining pressure. Previous investigations [32] indicate that microfractures are closed beyond 500 psi confining pressure and only the compliant meso/macropores contribute to the pore closure under the increasing confining pressure (Fig. 1). Therefore, we chose the T₂ cutoff around 1 ms to separate the bimodal T₂ spectrum as micropore signals and meso/macropore signals. Fig. 2 shows the fractal analysis of Th23 and Th28 under different confining pressures, and the associated parameters have been determined in Table 2. The fractal dimension associated with micropores are not discussed in this study mainly for three reasons: 1) The T₂ spectrum below around 1 ms almost overlap each other at different confining pressures; 2) The limited resolution of 2 MHz NMR to accurately detect the signals arising from micro-pores; 3) The slope of these T₂ spectra is high before decreasing at higher T₂ relaxation time around T₂ cutoff. The limited data points of T₂ relaxation times and the artificial error may lead to inaccurate estimation of fractal dimension. Besides, T₂ signals beyond 100 ms at ambient conditions are manually neglected to isolate the influence of mesopores and macropores, since these signals vanish above 500 psi (Fig. 1).

Fig. 2.

The fractal dimension derivation of the micropores D(micro) in black, mesopores and macropores under ambient conditions D(meso + macro)-ambient in blue, and mesopores and macropores under 2800 psi confining pressure conditions D(meso + macro)-2800 psi in orange.

Table 2.

Fractal dimensions calculated from model 1 based on NMR T₂ spectrum.

ID	Depth (m)	Micropores		T2cutoff (ms)	Mesopores + Macropores
		D	R2		D at ambient	R2	D at 2800psi	R2
Th17	1472.13	0.97	0.82	1.85	2.92	0.9958	2.94	0.9877
Th23	1508.89	1.01	0.81	1.85	2.96	0.9774	2.97	0.9760
Th28	1520.42	0.96	0.82	1.85	2.96	0.9881	2.97	0.9677
Th37	1554.73	1.00	0.82	1.85	2.90	0.9948	2.90	0.9941
Th39	1563.3	1.00	0.81	1.85	2.97	0.9816	2.97	0.9415
Th40	1570.83	0.89	0.84	1.85	2.98	0.9826	2.98	0.8895
Th45	1591.9	0.81	0.85	1.85	2.98	0.9126	2.99	0.8791

D(meso + macro) is typically 2.90–2.98 (with an average of 2.95) at ambient pressure and 2.90–2.99 (with an average of 2.96) at 2800 psi, presenting a linear increasing trend (Fig. 3) of D(meso + macro) with a high correlation (R² = 0.9842). A strong correlation is likely caused by the compliant clay-rich nature of the samples. Also, this figure shows the pore size distribution dose not changed too much considering we have to move the samples from cell to NMR for the tests, which could let the reopening of the pores. The relationship between D(meso + macro) and pressure is analyzed for the samples (Fig. 4) and it is apparent that most of the increase in D(meso + macro) occurs from ambient pressure to 1000 psi.

Fig. 3.

Relationship of the fractal dimension, derived from model 1, of the meso- and macro-pores, D(meso + macro) with pressure at ambient condition and 2800 psi. Correlation near 1-to-1 (R² = 0.9842).

Fig. 4.

The exponential increase of D(meso + macro) derived from model 1 for increasing confining pressure. Most of the increase in fractal dimension occurs from ambient to 8500 psi confining pressure.

By using model 1, the increase of D(meso + macro) can be found for all tested samples from ambient conditions to reservoir conditions. However, the pressure-induced changes on D are too small for further analysis. Model 1 was developed mainly for sandstone with less pore structure complexity and leads to a derived D value between 2 and 3 by its definition. Shale samples dominated by micropores with high pore structure complexity lead to derived D values close to 3. This phenomenon can also be found in previous studies working on the micropore-dominated porous media [20,27,47,48].

3.3. Fractal analysis using model 2

The derivation of fractal dimension using model 2 for the T₂ relaxation time below 100 ms is presented in Fig. 5 in order to exclude the influence of microfractures. Model 2 has the advantage of involving all the pores for the evaluation of D, at the expense of estimating a higher complexity in shales when compared with model 1. Using model 2 the fractal dimension has exceeded the theoretical limit of 3 derived from model 1 for some of the samples, according to the previous work [48]. The fractal dimension determined using model 2 (Fig. 6) increases from D_m2 = 2.89–3.12 (averaging 3.01) at ambient pressure to D_m2 = 2.93–3.25 (with an average of 3.11) at 2800 psi confining pressure.

Fig. 5.

The calculation of fractal dimension, D_m2, using model 2 for Th23 and Th28. D_m2 is higher at 2800 psi (orange) than that at ambient pressure (blue). The curves under different pressure conditions almost overlap each other below Log1 ms in x-axial (10 ms in T₂ relaxation time), indicating the main influence on D_m2 is the T₂ signals beyond 10 ms relating to meso/macropores.

Fig. 6.

D_m2 derived using model 2 under ambient and 2800 psi confining pressure. The dashed line is the 1:1 ratio separating the figure into two regions. D_m2 at 2800 psi is higher than D_m2 at ambient pressure for all values tested (R² = 0.6317).

As expected, the positive correlation between D-2800 psi and D-ambient and in model 1 (Fig. 3) and model 2 (Fig. 6) are similar, both indicating increasing pore structure complexity under increasing pressure. The linear trend of D_m2 with increasing confining pressure (Fig. 7) reflects the increasing complexity of pore structure, while only limited changes have been seen in model 1 (Fig. 4). Generally, model 1 underestimates the influence of meso/macropore changes to D_m2 compared to model 2. The main reason is the dominant contribution of micropores in shales. The change of meso/macropores with increasing confining pressure is much smaller than that of micropores. The slope of cumulative curves beyond 1 ms shows less change in Fig. 3. While in model 2, the nearly overlapped signals corresponding to micropores cannot shift the slope of trend lines, and the slope shifts are more related to the meso/macropores signals at the high T₂ relaxation times (Fig. 5), highlighting the contribution of meso/macropores to D_m2. Therefore, we believe model 2 provides a better estimate of D in shales than model 1 regardless of the high D_m2 values up to and above 3.

Fig. 7.

Fractal dimension, D_m2, derived from model 2 increases with increasing confining pressure for both samples Th24 (R² = 0.649) and Th25 (R² = 0.809). The linear regression for each sample (also presented) shows an increase from 2.94 to 3.00 for Th24 and 2.95 to 3.10 for Th25 as the confining pressure is increased from 0 to 8500 psi. Although the correlations aren't strong, the increase in the linear regression is marginally more pronounced here (using model 2) than it was for the same relationship observed in Fig. 4 (using model 1).

3.4. The influences of D derived from model 1 and model 2

By applying model 1, a previous study [48], found positive correlations between D(max) (similar to D(meso + macro)) and both the total clay content and the porosity. Evaluation of the mineralogy of the Goldwyer samples was performed using XRD (Table 1) and indicated quartz content ranging between 11.89 and 21.61 %, total clay content ranging between 46.3 and 70.2 %, and porosity ranging between 10.97 and 15.45 % (data from Zhong et al. [32] and Zhong et al. [33]). Although the ranges of porosity and quartz content were small, the range of total clay content was significant and yet no correlation was found (Fig. 8) between the fractal dimension using neither model 1 nor model 2 with the quartz content, or the total clay content, or NMR porosity. Interestingly, we found positive correlations between D_m2 and Archie's cementation exponent m for multiple samples at 2800 psi (Fig. 9), and for samples Th24 and Th25 under increasing confining pressure derived from our previous studies by using model 2 (Fig. 10). Archie's cementation exponent m was recalculated based on the resistivity data posted in Zhong et al. [33] and the porosity calculation method in Zhong et al. [32] with data from four samples (Th24, Th25, Th28 and Th45). Other samples (Th17, Th23, Th37, Th39 and Th40) presented in Zhong et al. [33], were excluded due to unsatisfactory cleaning and/or lack of necessary data due to sample fracturing.

Fig. 8.

The relationship between D derived from model 1 and other petrophysical parameters for four samples previously published in Refs. [32,33]. The correlations between fractal D and total clay (R² = 0.0064), Quartz (R² = 0.0011), and NMR porosity (R² = 0.0096) are poor.

Fig. 9.

The relationship between D_m2 and Archie m presented for four samples previously published in Refs. [32,33]. The correlation is very strong (R² = 0.9416).

Fig. 10.

The relationship between D_m2 and m under increasing confining pressure for samples Th24 and Th25 from ambient conditions to 8500psi. The projections of D and m show moderate to high correlations between D_m2 and m (R² = 0.6948 for Th25 and R² = 0.7962 for Th24) and demonstrate the possibility of using D_m2 to predict m in shales.

Archies cementation exponent m, has similar petrophysical implications with fractal dimension D. Archies m is related to pore sizes, geometries and connectivity [[53], [54], [55]] and is used to estimate the relationship between rock electrical properties, rock lithology, and rock pore structure. While the fractal dimension is only a pore structure-related parameter, the strong correlation between Archie's m and fractal dimension D_m2 in Fig. 8 supports previous findings that the complexity of pore structure dominates the values of Archie's m [32]. The correlation between D_m2 and m for the 4 available samples was high (R² = 0.9416 in Fig. 9) under 2800 psi confining pressure, indicating the potential to predict Archie's m from the D_m2 analysis within similar rock mineralogy types. The correlations are less strong for the confining pressure dependent experiment performed on samples Th24 and Th25 (R² = 0.7962 and R² = 0.6948 in Fig. 10). This may result from different proportions of meso/macropores, which dominate the pore types affected by confining pressure. The correlations are however still significant and further investigation of the phenomenon using more samples is likely to help improve the pressure-dependent correlation between Archie's m and fractal dimension D derived using model 2. The data ultimately demonstrates that Archie's m may be more easily predicted by simply analyzing the fractal dimension from NMR without measuring the resistivity and further models could be developed to link Archie m and fractal dimension D from NMR T₂ distribution to expand its use to any rock types and under any reservoir conditions.

4. Conclusion

This study delves into the complexity of shale pore structures through fractal dimension analysis of nuclear magnetic resonance (NMR) data under varying confining pressures. By focusing on nine illite-rich shale samples from the Ordovician Goldwyer Formation in the Canning Basin, Australia, we explored how confining pressure influences pore size distribution, particularly narrowing meso- and macropores. Two distinct models were utilized to calculate fractal dimensions: Model 1 categorizes pores into micro and meso + macro based on T₂ cutoffs, while Model 2 considers all pore sizes collectively.

Our results show that fractal dimensions increase with confining pressure, indicating enhanced pore complexity. Notably, Model 2 proves more adept for shales, revealing no significant correlation with quartz or total clay content or NMR porosity. However, a robust correlation (R² = 0.9416) between fractal dimension and Archie's cementation exponent (m) suggests a novel method for estimating Archie's m through pore distribution analysis without electrical resistivity evaluation.

This approach offers new insights for petroleum exploration and pore structure characterization in shales, particularly within the Goldwyer Formation and similar geological settings. The study highlights the potential of using NMR-derived fractal dimensions to replace traditional electrical tests, providing a more efficient and cost-effective method for well-log interpretation and reservoir evaluation. Further research and a larger dataset could refine these correlations and enhance the applicability of fractal analysis in various rock types and reservoir conditions.

CRediT authorship contribution statement

Zhiqi Zhong: Writing – original draft, Resources, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Kaishuo Yang: Writing – original draft, Methodology, Data curation, Conceptualization. Qing Yan: Writing – review & editing, Conceptualization. Minzhen Li: Writing – review & editing, Writing – original draft, Conceptualization. Lingping Zeng: Writing – review & editing, Validation. Lionel Esteban: Writing – review & editing. Yi Wang: Writing – review & editing.

Data availability statement

Data available on request from the authors. The data that support the findings of this study are available from the corresponding author upon reasonable request.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.