A new parameter for characterizing pore-fracture structure heterogeneity: fractal dimension based on the mercury extrusion curve

Abstract

Pressure mercury intrusion test is (MIP) one of the most commonly used methods to characterize pore-fracture structure. Here, we use the fractal dimension of the mercury intrusion curve to analyze the heterogeneity of pore and fracture distribution. Differing from the intrusive mercury curve, the extrusive curve provides a better representation of the seepage capacity of a reservoir. In this paper, the division method of sample types using both mercury invasive parameters (pore volume, pore volume percentage, porosity, permeability) and extrusive parameters (mercury removal efficiency) is discussed. The fractal dimension values of mercury intrusive and extrusive curves are calculated for all samples using the Menger, Thermodynamics, and Multifractal fractal models. Moreover, the fractal significance of the mercury withdrawal curve is examined. The results are as follows. 1) The samples can be divided into three types based on the mercury removal efficiency and total pore volume. Type A is characterized by lower total pore volume (< 0.08 cm³·g⁻¹) and removal efficiency (< 30%), type B has lower total pore volume (< 0.08 cm³·g⁻¹) and higher removal efficiency (> 30%), and type C has larger total pore volume (> 0.08 cm³·g⁻¹) and higher removal efficiency(> 30%). 2) Mercury removal efficiency does not correlate with the mineral composition or total pore volume, but it does show a clear positive correlation with pore volume in the range of 100 to 1000 nm. Unlike the Menger model, the mercury removal curve analyzed using the thermodynamics and multifractal model shows good fractal characteristics. 3) In contrast to the injective curves, the fractal dimension of mercury removal curves exhibits an obvious linear negative correlation with pore structure parameters and mercury removal efficiency. Moreover, the multifractal dimensions D₀–D₁₀ obtained from the mercury removal curves show a negative correlation with porosity and permeability. This indicates that fractal dimension based on the mercury extrusion curve can be used as a new parameter for characterizing pore-fracture structure heterogeneity.

Graphical abstract

Keywords

pore-fracture structure / mercury extrusion curve / pore structure / porosity / permeability

1 Introduction

The pores and fracture structures in tight sandstone play a crucial role in restricting oil-water-gas seepage and production. Various technologies, such as high pressure mercury injection experiments (HPMI), low-temperature nitrogen and carbon dioxide gas adsorption (LPN₂/CO₂ GA), and low field nuclear magnetic resonance (LF-NMR), are currently used to characterize unconventional reservoirs (Lai et al., 2018; Su et al., 2018; Hou et al., 2020, 2023; Zhang et al., 2020; Wang et al., 2023). Given the range in pore sizes in tight sandstone, different methods are selected accordingly.

In contrast to other testing methods, HPMI provides a quantitative characterization of the nano-micro-pore structure by measuring the mercury injection volume under different injection pressures. Parameters such as pore volume and specific surface area are then calculated (Schmitt et al., 2013; Guo et al., 2014; Peng et al., 2017; Zhang et al., 2022c). Because of its advantage of being faster, simpler, and more cost-effective, it has become one of most commonly used methods for characterizing the structure of unconventional reservoirs. At present, extensive literature examines the shape of the mercury injection curve and injective volume, pore types, pore size distribution, and the dynamic variation of porosity-permeability influenced by this curve. However, research on the mercury removal curve remains relatively scarce, with only the mercury removal efficiency being investigated (Zhang et al., 2022c). Compared with the mercury injection curve, it has been established that the mercury withdrawal curve is better suited to characterize the migration of gas and water (Cai et al., 2018). Therefore, further research on mercury removal curves is needed.

Since quantitative analysis of pore-fracture systems using traditional geometric methods is challenging, fractal theory based on HPMI tests has emerged as the preferred approach for quantitative characterization of pore structure heterogeneity (Knackstedt et al., 1996; Cai et al., 2016, 2018; Su et al., 2018; Li et al., 2020). Typical fractal models include Menger, Thermodynamic fractal (Zhang et al., 2006), and Multifractal models (Song et al., 2018). The mentioned fractal models rely on mercury injection curves as data sources, and fractal dimension is calculated to examine the pore-fracture structure heterogeneity of the reservoir itself. However, the fractal nature of the mercury removal curve and its impact on changes in porosity and permeability remains to be investigated.

In this study, 20 sandstone samples from the Xujiahe Formation were selected from wells in the Sichuan Basin (Fig.1). The method to classify sample types using both mercury intrusion parameters (pore volume, pore volume percentage, porosity, permeability) and extrusive parameters (mercury removal efficiency) is discussed. The fractal dimension values of the mercury injection and removal curves are calculated for all samples using the Menger, Thermodynamics and Multifractal fractal models. Moreover, the fractal significance of the mercury withdrawal curve is explored. Finally, the relationship between the fractal dimension values obtained from the mercury removal curve and porosity-permeability is examined.

2 Geological setting of sampling site

The Sichuan Basin is a superimposed basin, developed on the upper Yangtze Craton, located in south-west China (He et al., 2011). The study area is located north-east of Sichuan Basin (Fig.1(a)), structurally bounded by the Longmen mountain thrust belt to the west, the Micang mountain orogenic belt to the north, and the Daba mountain thrust nappe belt to the east. It has undergone multiple tectonic movements since the Triassic, including the Indosinian, Yanshanian, and Himalayan structural movements, and can be divided into five substructural units. Previous studies have indicated that the provenance of the Xujiahe Formation is from the three peripheral orogenic belts and that the formation has characteristics of a multi-source mixture (Fig.1(b)). The Xujiahe Formation (T₃x) consists of coal-bearing strata developed in a lake delta environment (Fig.1(b)). Its main facies include underwater distributary channels, braided river delta plains, shore-shallow lacustrine facies, distributary bays, peat swamp facies and others. The total strata thickness of the T₃x strata ranges from 270 to 730 m, and can be divided into five members sections, i.e., from T₃x¹ to T₃x⁵ (Zhang et al., 2009; Wang et al., 2023). A 1D basin model shows that the Xujiahe Formation has experienced a maximum burial depth 5600−6800 m and reached a maximum paleotemperature of 180°C−220°C. It has undergone almost 100 Ma tectonic uplift with a total exhumation of 1200−2800 m (Zhao et al., 2022).

Twenty fresh shale samples (approximately 15 × 15 × 15 cm³) were collected from a single well, and their distribution and parameters are shown in Fig.1 and Tab.1. Gas porosity (%) and permeability (mD) of the core samples were measured using an Ultrapore-200A helium core porosimeter and CMS-300 Automated Permeameter. The mineral composition of each sample was analyzed, and the pore volume, specific surface area and pore size distribution were quantitatively studied using HPMI tests. The analysis scheme is as follows. HPMI analysis was conducted on the samples using an IV9500 mercury intrusion instrument. One hundred 20 pressure points were measured for each sample with an analysis time of 5 s for each end. The working pressure of the mercury intrusion analyses ranged from 0.124 to 270.79 MPa.

The following section discusses three fractal models, with detailed process information can be found in Zhang et al. (2022a, 2022b).

Menger fractal model:

(1)\lg (\mathrm{d}{V}_{\mathrm{o}\mathrm{b}\mathrm{s}}/\mathrm{d}P)\mathrm{\infty }(D-4)\lg P,

where D represents the value of a fractal dimension, dimensionless; P represents the injection pressure, MPa; V represents injection volume of mercury, cm³·g⁻¹;

Thermodynamic fractal model:

(2)\mathrm{d}W=-P\mathrm{d}v=-r_{\mathrm{L}}\cos \theta \mathrm{d}S,

where v is pore volume, cm³·g⁻¹; W is the surface energy; r_L is the surface tension between mercury and surface, J/m; θ is the contact angle between mercury and pore surface (approximately 140°); and S is the specific pore surface, m²·g⁻¹

Multi-fractal model. a–f (a) is a set of primary languages describing the local features of multifractals, called multifractal spectrum. Another set q–D (q) is introduced from the information theory perspective and is known as the generalized fractal dimension. A detailed description is available in previous studies (Hu et al., 2020; Zhang et al., 2022a, 2022b).

Generalized fractal dimension are selected as an example, parameter D_q includes D₋₁₀, D₁₀, D₋₁₀–D₁₀, D₀–D₁₀, and D₋₁₀–D₀. D_q is a monotonically decreasing function with a sigmoidal shape. D₋₁₀ is influenced by the lowest probability measure areas, whereas D₁₀ is effected by highest probability measure areas. D₀–D₁₀ and D₋₁₀–D₀ are the amplitudes of the right and left branches of D_q, which represents the high and low probability measure areas heterogeneity, respectively (Zhang et al., 2020).

The data sources of the mentioned fractal models are all mercury injection curves, and the fractal dimension is calculated to analyze the heterogeneity of the pore-fracture structure in the reservoir. It is important to investigate whether the mercury removal curve exhibits fractal characteristics, and its impact on changes in porosity and permeability. The fractal dimensions of the three fractal models were calculated based on the mercury removal curve of the same set of samples. The differences in fractal characteristics between the mercury injection curve and withdrawal curve of these samples are discussed.

3 Results

3.1 Sample type classification and pore-fracture system of all samples

The mineralogical composition of all sandstone samples is presented in Tab.1. The clay content ranges from 1% to 11%, whereas the quartz content varies from 10% to 94%, and the feldspar content ranges from 0 to 18%. These results indicate that the samples are primarily composed of quartz, which is favorable for reservoir reconstruction. The porosity-permeability results show that the porosity of the samples ranges from 0.50% to 4.22%, whereas the permeability varies from 0.002 to 5.42 mD. Except for sample 9, most of the samples have low porosity and low permeability. Based on differences in research objectives, literature classifies the sample types according to factors such as pore structure parameters and mineral components. Fig.2 shows four types of sample classification schemes based on pore structure parameters (total pore volume, pore volume percentage of 10−100 nm and < 100 nm), porosity and mercury removal efficiency. Since this study focuses on the mercury removal curve, the mercury removal efficiency and total pore volume are selected to classify the samples into three categories.

Fig.3(a) shows that the mercury injection curve of type A samples remains nearly horizontal when the mercury pressure is less than 100 MPa, indicating a lack of developed pore-fracture structure. The mercury removal curve for type A samples is almost flat, suggesting a complex pore structure with minimal mercury removal. Type B samples exhibit a nearly horizontal mercury injection curve at pressures below 1 MPa, indicating the absence of larger pores. However, the mercury injection curve becomes nearly vertical for the pressure range of 1 to 100 MPa, indicating the presence of smaller pores. The mercury removal curve for type B samples is almost vertical, suggesting a more straightforward pore structure with a larger amount of mercury removal (Fig.3(b)). Type C samples display intermediate characteristics between type A and type B, representing a transitional type.

The pore volume of type B samples is greater than that of types A and C, which is consistent with the results in Fig.3. Moreover, the mercury removal efficiency of type B samples is larger than that of types A and C. This indicates that type B samples have a larger pore volume, higher mercury removal efficiency, and a more favorable pore structure, making them the focus of further research (Fig.4(a) and 4(b)). Compared with Fig.4(c), the pore volume percentage of micro-pores (pore diameter less than 100 nm) of all samples ranges from 0.2 to 0.8, with minimal variation among different types. The meso-pore (100−1000 nm) volume percentage of type B and C samples is higher than that of type A samples, and the macro-pore (more than 1000 nm) volume percentage of type A and C samples is higher than that of type B samples (Fig.4(c)).

3.2 Fractal characteristics using the thermodynamic model

The results of the thermodynamic fractal dimension, based on Eq. (2) and the mercury injection and withdrawal curves are shown in Fig.5. Fig.5(a), Fig.5(b), and Fig.5(c) indicate that mercury removal curve still exhibits fractal characteristics when using a thermodynamic model. However, the fractal dimension obtained from the mercury injection curve is higher than that of mercury withdrawal curve. Fig.5(d) shows no apparent correlation between the fractal dimension obtained from the mercury injection curve and the fractal dimension obtained from the mercury withdrawal curve. Moreover, Fig.5(e) suggests that the difference in fractal dimension obtained from the mercury injection curve among different sample types is relatively minor. However, Fig.5(f) shows that the fractal dimension obtained from the mercury withdrawal curve for type B samples is smaller than that for type A and C samples, indicating that the fractal dimension obtained from the mercury withdrawal curve better reflects the differences among these samples.

3.3 Fractal characteristics using the multi-fractal model

The results of the multi-fractal dimension, using the mercury injection and withdrawal curves, are shown in Fig.6. Fig.6(a), Fig.6(b), and Fig.6(c) indicate that the mercury removal curve still exhibits fractal characteristics when using the multi-fractal model. Moreover, D₋₁₀–D₀ obtained from the mercury injection curve is larger than that obtained from the mercury withdrawal curve, and D₀–D₁₀ obtained from the mercury injection curve is smaller than that obtained from the mercury withdrawal curve. Moreover, within a single sample, the D₋₁₀–D₀ value is larger than the D₀–D₁₀ value. Fig.6(d), Fig.6(e), and Fig.6(f) indicate no apparent correlation between the fractal dimension obtained from the mercury injection curve and the fractal dimension obtained from the mercury withdrawal curve.

Fig.7(a) shows that the difference in fractal dimension obtained from the mercury injection curve among different sample types is relatively smaller. However, Fig.7(b) illustrates that the D₋₁₀–D₀ and D₋₁₀–D₁₀ values obtained from the mercury withdrawal curves for type B samples are larger than those for type A and C samples, indicating that the fractal dimension obtained from the mercury withdrawal curves better reflects the differences among these samples.

3.4 Fractal characteristics using the Menger model

The results of the Menger model, based on Eq. (1) and the mercury injection and withdrawal curves, are shown in Fig.8. The figure indicates that the mercury injection curve still exhibits fractal characteristics when using the multi-fractal model. However, the fractal characteristics of the mercury removal curve are similar to those of the mercury injection curve, indicating that the mercury removal curve, when analyzed using the Menger model, also displays fractal characteristics. It is important to note that the two fractal curves are nearly identical, indicating that the Menger model is not suitable to characterize the fractal characteristics of the mercury removal curves. Fig.9 shows that the fractal dimension obtained from the mercury injection curves is almost above 3, indicating a complex heterogeneity in the pore volume distribution in these samples.

4 Discussion

4.1 Relationship between pore structure parameters and mercury removal efficiency

Fig.10(a) shows that the mercury removal efficiency gradually increases with the increase in total pore volume. This can be attributed to the fact that samples with larger pore volumes exhibit mostly parallel plate-shaped pores, which promote good pore connectivity. Fig.10(b) and 10(c) reveal a certain correlation between the pore volume in the range of 100 to 1000 nm and the total pore volume, indicating that these pores play a significant role in determining the pore structure characteristics of all samples. Moreover, Fig.10(d) shows that pores in the 100 to 1000 nm range restrict the mercury removal efficiency of samples, implying that these pores have control over the overall pore connectivity.

In contrast, Fig.11(a) and Fig.11(b) show no correlation between pore volume and mineral composition. However, there is a weak positive correlation between quartz content and mercury removal efficiency. This can be attributed to the fact that a higher quartz content facilitates the formation of micro-fractures and macro-pores, resulting in stronger connectivity (Fig.11(c) and Fig.11(d)).

4.2 Relationship between fractal dimension of intrusion/removal curves and related parameters

4.2.1 Thermodynamic model

Fig.12(a) shows that there is no obvious relationship between pore volume and fractal dimension when using mercury injection and removal curves. Fig.12(b) indicates a negative linear relationship between pore volume and fractal dimension when using the mercury removal curve, and there is no correlation between pore volume and fractal dimension when using the mercury injection curve. This suggests that the latter parameter better reflects mercury removal efficiency and pore connectivity. Fig.12(e) and Fig.12(f) show that the fractal dimension, when using mercury removal curve, decreases as the pore volume in the range of 100−1000 nm and quartz content increase. However, there is no apparent relationship between pore structure parameters and fractal dimension when using the mercury injection curve, indicating that the fractal dimension derived from the mercury removal curve better reflects the heterogeneity of pore size distribution.

4.2.2 Multi-fractal model

Fig.13 shows a positive linear relationship between quartz content and D₋₁₀–D₀ when using the mercury removal curve, whereas no correlation is observed between pore volume and fractal dimension when using the mercury injection curve. This indicates that latter parameter better reflects mercury removal efficiency and pore connectivity. Fig.13(b) shows a positive linear relationship between mercury removal efficiency and D₋₁₀–D₀ when using the mercury injection curve, and no correlation is found between pore volume and fractal dimension when using the mercury removal curve.

In contrast_,Fig.14 shows that D₀–D₁₀ exhibits a strong linear relationship with pore structure parameters and mercury removal efficiency. As D₀–D₁₀ increase, pore volume, mercury removal efficiency, and quartz content all decrease. Overall, multi-fractal parameters derived from the mercury removal curves better represent the heterogeneity of pore structure distribution.

4.3 Relationship between pore structure and fractal dimension using mercury removal curves

As mentioned above, both the thermodynamic and multifractal models can be used to characterize the fractal characteristics of mercury removal curves. Fig.15(a) shows that there is a good positive linear correlation between mercury removal efficiency and porosity, whereas its correlation with permeability is weak. Fig.15(b) and Fig.15(c) show that the fractal dimension of the removal and injection curves, as determined by the thermodynamic model, has no clear correlation with porosity and permeability. This indicates that the applicability of these two parameters in determining the porosity and permeability characteristics is limited. Furthermore, Fig.15(d) shows that the D₀–D₁₀ parameter of the injection curve, calculated using the multi-fractal model, has no substantial correlation with porosity and permeability, indicating a weak applicability of this parameter in determining the characteristics of porosity and permeability. However, porosity and permeability exhibit a linear decrease with the increase of D₀–D₁₀′ (withdrawal curve using the multi-fractal model), indicating that this parameter can be used to characterize the porosity-permeability structure of the reservoir (Fig.15(e)). The difference in fractal dimension between the mercury injection and withdrawal curves shows a good linear positive correlation with porosity and permeability, indicating that D₀–D₁₀, as determined using the multi-fractal model, is a parameter suitable to characterize pore fracture structure.

5 Conclusions

1) The samples can be divided into three types based on mercury removal efficiency and total pore volume. Type A is characterized by lower total pore volume (< 0.08 cm³·g⁻¹) and removal efficiency (< 30%), type B has lower total pore volume (< 0.08 cm³·g⁻¹) and higher removal efficiency (> 30%), and type C is associated with larger total pore volume (> 0.08 cm³·g⁻¹) and higher removal efficiency (> 30%).

2) Mercury removal efficiency shows no correlation with mineral composition and total pore volume, but it exhibits a clear positive linear correlation with the pore volume in the range of 100−1000 nm. In contrast to the Menger model, the mercury removal curve analyzed using the thermodynamics and multifractal models exhibits robust fractal characteristics. Compared with the mercury injection curve, the fractal dimension calculated based on the mercury withdrawal curve shows a good linear correlation with pore structure parameters and mineral components.

3) The difference in fractal dimension between the mercury injection and withdrawal curves demonstrate a good positive linear correlation with porosity and permeability, indicating that D₀–D_10, as determined using the multi-fractal model, is a parameter suitable for characterizing pore fracture structure. Further research should be conducted on fractal analysis using mercury removal curves.

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