Fractal and Multifractal Analysis of Pore Size Distribution in Low Permeability Reservoirs Based on Mercury Intrusion Porosimetry

To quantitatively evaluate the complexities and heterogeneities of pore structures in sandstone reservoirs, we apply single fractal theory and multifractal theory to explore the fractal characteristics of pore size distributions based on mercury intrusion porosimetry. The fractal parameters were calculated and the relationships between the petrophysical parameters (permeability and entry pressure) and the fractal parameters were investigated. The results show that the single fractal curves exhibit two-stage characteristics and the corresponding fractal dimensions D1 and D2 can characterize the complexity of pore structure in different sizes. Favorable linear relationships between log(ε) and log(μ,(ε)) indicate that the samples satisfy multifractal characteristics and ε is the sub-intervals with size ε = J× 2−k. The multifractal singularity curves used in this study exhibit a right shape, indicating that the heterogeneity of the reservoir is mainly affected by pore size distributions in sparse regions. Multifractal parameters, D(0), D(1), and ∆f, are positively correlated with permeability and entry pressure, while D(0), D(1), and ∆f are negatively correlated with permeability and entry pressure. The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range, while the range of the pore size distribution has a definite impact on the multifractal parameters. Results indicate that fractal analysis and multifractal analysis are feasible methods for characterizing the heterogeneity of pore structures in a reservoir. However, the single fractal models ignore the influence of microfractures, which could result in abnormal values for calculated fractal dimension. Compared to single fractal analysis, multifractal theory can better quantitatively characterize the heterogeneity of pore structure and establish favorable relationships with reservoir physical property parameters.


Introduction
A reservoir consists of a complex porous medium composed of pores with different origins, irregular shapes, and self-similarities.The pore structure in a reservoir is mainly affected by three factors: sedimentation, diagenesis, and tectogenesis.Each of these factors result in the formation of pores with attributes which fall into different ranges.Thus, different types of pores are formed and exist in the reservoir in a certain distribution.This complex pore size distribution is dynamically nonlinear and is the result of numerous processes occurring at various scales [1].
Fractal theory, a promising tool for investigating complex structures, has been widely used to quantitatively characterize the complexities and heterogeneities of pore size distributions [2].Fractal theory is considered to be an effective means of quantitatively depicting irregular shapes, and can accurately express the complexity and heterogeneity of geological bodies.Extensive research has proven that reservoirs exhibit fractal characteristics.Pfeifer found that the pore surface area of a reservoir exhibits fractal characteristics by the using molecular adsorption [3].Katz investigated different types of sandstone using scanning electron microscopy; the results indicated that the pore spaces of sandstone possess fractal characteristics [4].Friesen obtained the fractal dimensions of coal particles based on capillary injection data [5].
Mercury intrusion porosimetry is commonly used to determine the pore structure distributions of rocks [6].Fractal studies of capillary pressure data mainly focus on obtaining fractal dimensions to establish the relationship between fractal dimension and reservoir physical properties [7,8].There are several widely used fractal models to obtain the fractal dimension of mercury injection data based on single fractal theory [5,[9][10][11].Single fractal theory describes the integrity characteristics of pore structures [7]; it is suitable for application to a homogeneous reservoir, but cannot define the local pore structure or reflect other comprehensive and detailed information [12].The segmented fractal phenomenon occurs when the fractal theory is applied to mercury injection data [13][14][15].This fact demonstrates that describing the complexity of the whole pore system is difficult with a single fractal dimension.
Multifractal analysis was conducted to describe the complexity of pore size distribution because single fractal theory has limitations in describing the local characteristics of pore size distribution.Multifractal theory partitions objects at different scales to obtain distribution characteristics [16,17].As a result, it likely characterizes the complex and heterogeneous behavior of a reservoir more effectively than the single fractal theory [7].Multifractal theory has been widely applied to study effect of pore size changes on reservoir physical properties [18].Multifractal analysis of soil pore structure has been widely studied to characterize soil structure stability and soil surface evolution stages [19].Multifractal theory has also been confirmed as a useful tool for characterizing the internal complexity and amplifying the differences in pore size distributions between different coals [1].Research on the quantitative characterization of the irregular microscopic pore structures of different rock types has been performed on the basis of 2-D images [20][21][22][23].From the above literature review, we can conclude that multifractal analysis of pore size distribution measured by mercury intrusion porosimetry has not been extensively applied to sandstone reservoirs.Comparative studies of single fractal theory and multifractal theory based on mercury intrusion porosimetry have not been performed.
In the present study, single fractal theory and multifractal theory are applied to investigate variability and heterogeneity in pore structures based on mercury intrusion curves.Single fractal and multifractal parameters were analyzed for correlation with reservoir physical parameters.We compared single fractal theory and multifractal theory to test which method is more suitable for characterizing reservoir physical parameters.Multifractal analysis of pore structures can expand our understanding of the pore structures of reservoirs.

Single Fractal Theory
Several fractal models were established to obtain the fractal dimensions of mercury injection data.Su's fractal model, which considers both the fractal characteristics of pore space and pore length and was proposed based on the capillary bundle model, was applied in this paper [11]: where S Hg is the mercury saturation, D f is the fractal dimension for pore space, with 2 < D f < 3 in three dimensions [24], D T is the fractal dimension for tortuosity, with 1 < D T < 3 in three dimensions [25], and P c is the capillary pressure.The sum of the fractal dimension of pore space and the fractal dimension of tortuosity can be obtained from the double logarithmic curve of mercury saturation and capillary pressure.Theoretically, the sum of the fractal dimensions of pore space and tortuosity should be between 3 and 6; the larger the sum of fractal dimension, the more complicated the pore structure [11].

Multifractal Theory
Multifractal theory describes the local conditions of a fractal structure through the singularity strength [26], and the overall characteristics are investigated from a local perspective.The prerequisite for applying the multifractal analysis method is that the measurement interval must be divided equally [27].
The mercury injection curves obtained in this study have fewer measurement points than is required for the multifractal method.To obtain enough data for multifractal analysis of mercury intrusion curves, we used cubic spline interpolation to interpolate mercury intrusion curves, allowing us to obtain more data points.
In this study, the whole measured pore size was defined as I, I = [0.015,35.6 µm].The difference of capillary pressure data in the aforementioned pore size range is measured over steps of 0.005 µm.Thus, sub-intervals can be obtained.I i represents the i-th sub-interval, and ν i represents the percentage of pore volume in the sub-interval I i .A new measure can be obtained (J = [log0.015,log35.6 µm]) by plotting the aforementioned pore size range I on a logarithmic scale.J was divided into 2 k equal sub-intervals with size ε = J × 2 −k .The whole interval is divided into reduced sub-intervals with increasing k; thus, the effects of pore space changes within a small interval can be investigated.J i represents the i sub-interval in the interval J. P i (ε) represents the percentage of pore volume in the sub-interval J i ; it is equal to the sum of ν i that falls within the sub-interval J i .
The partition function χ(q,ε) can be defined using P i (ε): where q is a real parameter that describes the moment order of the measure.For q < 1, χ(q,ε) emphasizes the regions determined by a small P i (ε) or minimally concentrated region of a measure.For q > 1, χ(q,ε) emphasizes the regions determined by a large P i (ε) or wide concentrated region of a measure.The q used in this study is between −20 and 20.χ(q,ε) and ε follow a power law relationship as follows: where τ(q) is the mass exponent, which can also be expressed by the following formula: The mass exponent can also be expressed as follows, according to previous research results [19]: where D(q) is the generalized dimension.Correspondingly, D(q) can be expressed as follows: For q = 1, D(q) is defined as follows [26]: If ε is sufficiently small, then P i (ε) is nearly evenly distributed within each subinterval, where P i (ε) and ε show the following relationship: where α is the singularity exponent.Different subintervals may have the same α.N α (α) represents the subinterval numbers of the singularity exponent between α and α + dα ; it satisfies the following fractal power law relationship: where f (α) is a multifractal spectrum with singularity exponent α.Different α values and corresponding f (α) constitute the multifractal spectrum that describes multifractal properties.
The singularity exponent can also be expressed as follows: The multifractal spectrum of pore distribution f (α) relative to α is defined as follows: The first step of the multifractal analysis of capillary pressure data is to interpolate mercury intrusion curves to obtain sufficient points.The equidistant division of a logarithmic pore size range is the basis for obtaining the probability density P i (ε) and the partition function χ(q,ε).τ(q), D(q), α(q), and f (q) can be obtained from Equations (2), (4), ( 6), (10) and (11), respectively.τ(q) and D(q) describe the multifractal characteristics, whereas α(q), and f (q) characterize the local characteristics of the multifractal structure.

Samples and Experiments
A total of 13 samples were obtained from a well located in Western Sichuan, China.The physical properties of the samples are relatively variable (Figure 1), which is convenient for comparison using multifractal analysis.All samples were tested for porosity, and permeability and subjected mercury injection experiments in accordance with Chinese Petroleum Industry Standards SY/T 6385-1999 and SY/T 5346-2005.Porosity and permeability were obtained using routine rock property measurement techniques.The average permeability is 6.11 mD; the range extends from 0.14 mD to 42.29 mD.The average porosity is 11.59%; it ranges from 6.31% to 16.65%.The experimental results are summarized in Table 1.The entry pressure (the point on the curve at which the mercury first enters the pores of the samples) varies from 0.037 MPa to 1.450 MPa, with an average of 0.726 MPa.r 50 varies from 0.018 µm to 0.247 µm, with an average of 0.089 µm.An analysis of physical properties shows that the reservoir exhibits strong heterogeneity and complexity in its microscopic pore structure.The pore size distributions of four samples are shown in Figure 2. Samples 64 and 72 have a wide range of pore sizes, with about half larger than 1 µm and half smaller than 1 µm, respectively.Samples 52 and 142 have a small range of pore size distributions, and the pores are mainly distributed below 1 µm.

Single Fractal Characteristics
Equation ( 1) was applied to obtain the fractal dimensions of the 13 mercury injection data.The results (Table 2) show that the high R-squared value demonstrates that fractal method is useful for mercury intrusion porosimetry (Figure 3).The fractal curves exhibit a two-stage characteristic and the corresponding fractal dimensions D1 and D2 can characterize the complexity of pore structure in different sizes.Despite the different pore size distributions, all 13 mercury intrusion curves exhibit a two-stage fractal characteristic.D1 of 13 samples varies widely, while D2 is mainly distributed around a value of 3.2.

Single Fractal Characteristics
Equation ( 1) was applied to obtain the fractal dimensions of the 13 mercury injection data.The results (Table 2) show that the high R-squared value demonstrates that fractal method is useful for mercury intrusion porosimetry (Figure 3).The fractal curves exhibit a two-stage characteristic and the corresponding fractal dimensions D 1 and D 2 can characterize the complexity of pore structure in different sizes.Despite the different pore size distributions, all 13 mercury intrusion curves exhibit a two-stage fractal characteristic.D 1 of 13 samples varies widely, while D 2 is mainly distributed around a value of 3.2.To clarify the factors controlling the fractal dimension, pore size distributions of four samples are shown in Figure 4. Samples 64 and 72 have a wide range of pore size distributions, with about half of the pore volume larger than 1 μm and half less than 1 μm.Samples 52 and 142 have a small range of pore size distributions and the pores are mainly distributed below 1 μm.The mercury pressure of macropores is lower, meaning that a smaller fractal dimension can be obtained under the same mercury saturation condition.Therefore, the D1 values of samples 64 and 72-4.1698 and 4.3472, respectively-are significantly larger than the D1 value of samples 52 and 142, which are 5.0715 and 6.3375, respectively.Under the same conditions, the larger the proportion of large pores, the easier it is to obtain smaller fractal dimensions; this can be confirmed by examining the correlation between D1 and r50.D1 has a good negative correlation with r50 when two abnormal points affected by microfractures are neglected.The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range.
The sums of fractal dimensions D1 of samples 13, 46, and 142 are 6.87, 7.10 and 6.34, respectively; these values are all beyond the theoretical value.Despite simultaneously considering the fractal dimension for pore space and tortuosity, the sum of fractal dimension D1 may be greater than the theoretical value.Friesen's model, Angulo's model, Shen's model, and Su's model assume that only porous media are present in the reservoir and ignore the influence of microfractures.However, there may be microfractures in the samples, and the existence of the microfractures causes the abnormalities in the sum of fractal dimension D1 [11].To clarify the factors controlling the fractal dimension, pore size distributions of four samples are shown in Figure 4. Samples 64 and 72 have a wide range of pore size distributions, with about half of the pore volume larger than 1 µm and half less than 1 µm.Samples 52 and 142 have a small range of pore size distributions and the pores are mainly distributed below 1 µm.The mercury pressure of macropores is lower, meaning that a smaller fractal dimension can be obtained under the same mercury saturation condition.Therefore, the D 1 values of samples 64 and 72-4.1698 and 4.3472, respectively-are significantly larger than the D 1 value of samples 52 and 142, which are 5.0715 and 6.3375, respectively.Under the same conditions, the larger the proportion of large pores, the easier it is to obtain smaller fractal dimensions; this can be confirmed by examining the correlation between D 1 and r 50 .D 1 has a good negative correlation with r 50 when two abnormal points affected by microfractures are neglected.The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range.
The sums of fractal dimensions D 1 of samples 13, 46, and 142 are 6.87, 7.10 and 6.34, respectively; these values are all beyond the theoretical value.Despite simultaneously considering the fractal dimension for pore space and tortuosity, the sum of fractal dimension D 1 may be greater than the theoretical value.Friesen's model, Angulo's model, Shen's model, and Su's model assume that only porous media are present in the reservoir and ignore the influence of microfractures.However, there may be microfractures in the samples, and the existence of the microfractures causes the abnormalities in the sum of fractal dimension D 1 [11].

Multifractal Characteristics
Multifractal analysis was conducted to describe the complexity of pore size distribution because single fractal theory has limitations in describing the local characteristics of pore size distribution.Figure 5 depicts the partition function for different q values in the double logarithmic coordinates of the four samples.

Multifractal Characteristics
Multifractal analysis was conducted to describe the complexity of pore size distribution because single fractal theory has limitations in describing the local characteristics of pore size distribution.Figure 5 depicts the partition function for different q values in the double logarithmic coordinates of the four samples.Linear relationships exist between log(ε) and log(χ(ε)) for the samples when −20 ≤ q ≤ 20 and the correlation coefficient is higher than 0.94 (Table 3).Favorable linear relationships between log(ε) and log(χ(ε)) indicate that the samples satisfy the multifractal characteristics.In accordance with Equation (4), the slopes of log(ε) and log(χ(ε)) are the mass exponent τ(q) The corresponding τ(q) increases when q increases from −20 to 20, indicating that the samples exhibit multifractal characteristics in a spatial distribution and that the multifractal method can be used to investigate the complexity and scale effects of pore size distribution.In accordance with Equation ( 6), the generalized dimension D(q) was obtained in the range of −20 ≤ q ≤ 20.The relationship between the multifractal generalized dimension D(q) and the moment order q is presented in Figure 6.The corresponding multifractal parameters are listed in Table 4.In the range of −20 ≤ q ≤ 20, the value of D(q) when q is positive is less than the value of D(q) when q is negative, thus indicating that regions with dense pore size distribution provides a better scale than the sparse regions.For a homogeneous fractal, the curves of D(q) and q form a straight line, whereas those of non-uniform fractals have a certain width, and a large curvature indicates a poor homogeneity of samples.All four samples demonstrate a certain degree of curvature and exhibit a certain non-uniformity, but the curvatures were significantly greater in Samples 64 and 72 than in Samples 52 and 142, thereby demonstrating that Samples 64 and 72 are more heterogeneous in their pore size distribution.The multifractal spectrum of the 13 samples was calculated in accordance with Equations ( 10) and (11).Figure 7 illustrates the multifractal spectrum curves of the four samples.The multifractal spectrum functions a-f(a) denote a continuous distribution, indicating that multifractal theory is a common phenomenon of the pore size distribution.Curves a-f(a) are asymmetrical upward convex curves, which demonstrate that the local superposition of the different degrees during the formation of pores leads to the occurrence of reservoir heterogeneity.
In calculating the multifractal spectrum, the calculation domain is divided into different scales, with considerable scale information in the reservoir pore size distribution.Δa describes the characteristics of different regions, levels, and local conditions in a fractal structure.A large Δa value indicates a highly uneven distribution.The parameters of the multifractal spectrum are listed in Table 5.The value of Δa ranges from 0.7167 to 2.2413, with an average value of 1.2361.The maximum Δa of Sample 72 suggests that its heterogeneity is robust.By contrast, the smallest Δa of Sample 9 suggest that its heterogeneity is relatively weak.Capacity dimension D(0), information dimension D(1), and correlation dimension D(2) are listed in Table 4 [21].A large dimension D(0) indicates a wide range of pore size distributions.The capacity dimensions of Samples 64 and 72 are 1.88 and 1.76, respectively, and are relatively larger than those for the 11 other samples.This indicates that the pore size distribution is large.A large pore size distribution suggests that large pores may be observed in the samples and may significantly improve the porosity and permeability of the reservoir; this condition can be confirmed by the large permeability and porosity of Samples 64 and 72.
The information dimension D(1) reflects the degree of concentration of the pore size distribution, which represents the heterogeneity of pore structure.A large information dimension D(1) indicates a highly heterogeneous pore size distribution.The information dimensions D(1) of Samples 64 and 72 are relatively high, demonstrating that the unevenness of the pore size distribution is significant, and that pores are distributed over a wide range of pore sizes.
D(1)/D(0) shows the dispersion of the pore size distribution.An added pore size is concentrated in the dense area when D(1)/D(0) is close to 1, and the particle concentration in the sparse area is close to 0. The D(1)/ D(0) values of Samples 64 and 72 are relatively minimal.This result shows that the pore size distribution of Samples 64 and 72 is discrete and is biased toward the sparse areas of the pore size distribution.The sparsely-grained area mainly refers to the area with large pore sizes in this study.This area can improve the physical properties and increase the seepage and storage capacities of the fluid in a reservoir despite the relatively minimal volume.
The multifractal spectrum of the 13 samples was calculated in accordance with Equations ( 10) and (11).Figure 7 illustrates the multifractal spectrum curves of the four samples.The multifractal spectrum functions a-f (a) denote a continuous distribution, indicating that multifractal theory is a common phenomenon of the pore size distribution.Curves a-f (a) are asymmetrical upward convex curves, which demonstrate that the local superposition of the different degrees during the formation of pores leads to the occurrence of reservoir heterogeneity.
In calculating the multifractal spectrum, the calculation domain is divided into different scales, with considerable scale information in the reservoir pore size distribution.∆a describes the characteristics of different regions, levels, and local conditions in a fractal structure.A large ∆a value indicates a highly uneven distribution.The parameters of the multifractal spectrum are listed in Table 5.The value of ∆a ranges from 0.7167 to 2.2413, with an average value of 1.2361.The maximum ∆a of Sample 72 suggests that its heterogeneity is robust.By contrast, the smallest ∆a of Sample 9 suggest that its heterogeneity is relatively weak.The equation Δf (Δf = f(amin) -f(amax)) reflects the shape features of the multifractal spectrum.The shape of f(a) depicts a right hook when the small probability subset dominates (Δf < 0).The shape of f(a) illustrates a left hook when the large probability subset dominates (Δf > 0).The multifractal singularity curves in this study exhibit a right shape, indicating that the heterogeneity of the reservoir is mainly affected by the pore size distribution in the sparse region.This study emphasizes that large-scale pores contribute considerably to the spatial heterogeneity of a reservoir.

Relationship between Petrophysical and Single Fractal Parameters
The fractal dimensions D1 and D2 only characterize the complexity of pore structure in different sizes.The fractal dimension Dsw was introduced based on the weighted of the pore volume [28].
( ) where Sinf is the inflection point saturation and Smax is the maximum saturation.Dsw can characterize the complexity of the whole pore size, and has a better correlation with petrophysical parameters (Figures 8 and 9).Larger Dsw values indicate that macropores and microfractures have greater influence on reservoir physical properties.The Dsw values is between 3.42 and 4.83, with average value of 4.03.
To explore the meaning of saturation-weighted fractal dimension Dsw, the correlations between Dsw and petrophysical parameters were investigated.Dsw has a good negative correlation with permeability (Figure 8), while Dsw has a good positive correlation with entry pressure (Figure 9).Permeability is an important indicator of reservoir quality; larger permeabilities are typically  The equation ∆f (∆f = f (a min ) − f (a max )) reflects the shape features of the multifractal spectrum.The shape of f (a) depicts a right hook when the small probability subset dominates (∆f < 0).The shape of f (a) illustrates a left hook when the large probability subset dominates (∆f > 0).The multifractal singularity curves in this study exhibit a right shape, indicating that the heterogeneity of the reservoir is mainly affected by the pore size distribution in the sparse region.This study emphasizes that large-scale pores contribute considerably to the spatial heterogeneity of a reservoir.

Relationship between Petrophysical and Single Fractal Parameters
The fractal dimensions D 1 and D 2 only characterize the complexity of pore structure in different sizes.The fractal dimension D sw was introduced based on the weighted of the pore volume [28].
where S inf is the inflection point saturation and S max is the maximum saturation.D sw can characterize the complexity of the whole pore size, and has a better correlation with petrophysical parameters (Figures 8 and 9).Larger D sw values indicate that macropores and microfractures have greater influence on reservoir physical properties.The D sw values is between 3.42 and 4.83, with average value of 4.03.
To explore the meaning of saturation-weighted fractal dimension D sw , the correlations between D sw and petrophysical parameters were investigated.D sw has a good negative correlation with permeability (Figure 8), while D sw has a good positive correlation with entry pressure (Figure 9).Permeability is an important indicator of reservoir quality; larger permeabilities are typically associated with high-quality reservoirs.The entry pressure is mainly influenced by the pore size; the smaller the pore size, the greater the entry pressure.Larger D sw values indicate that the macropores are more heterogeneous, but this does not guarantee a larger volume of macropores or better reservoir properties.The correlations between D sw and petrophysical parameters show that the increase of D sw is accompanied by the decrease of pore size and permeability, resulting in poorer reservoir properties.Therefore, D sw is a good indicator of reservoir quality.associated with high-quality reservoirs.The entry pressure is mainly influenced by the pore size; the smaller the pore size, the greater the entry pressure.Larger Dsw values indicate that the macropores are more heterogeneous, but this does not guarantee a larger volume of macropores or better reservoir properties.The correlations between Dsw and petrophysical parameters show that the increase of Dsw is accompanied by the decrease of pore size and permeability, resulting in poorer reservoir properties.Therefore, Dsw is a good indicator of reservoir quality.

Relationship between Petrophysical and Multifractal Parameters
The relationships linking D(0), D(1), ΔD and Δf with permeability are determined to further explore the relationships between multifractal parameters and reservoir pore structure.Figure 10 shows that D(0), D (1), and D Δ are positively correlated with permeability, whereas Δf negatively correlates with permeability.D(0) and D(1) correlate well with permeability, with correlation coefficients of 0.8845 and 0.8665, respectively.D(0) represents the range of the pore size distribution, and D(1) characterizes the heterogeneity of the pore size distribution.The permeability improves with increasing D(0) and D (1).The more widely the reservoir particle size is distributed in the sparse region, the larger the reservoir size distribution range and the stronger the degree of heterogeneity.Therefore, the physical properties of the reservoir improve with increases in the range and heterogeneity of the pore size distribution.associated with high-quality reservoirs.The entry pressure is mainly influenced by the pore size; the smaller the pore size, the greater the entry pressure.Larger Dsw values indicate that the macropores are more heterogeneous, but this does not guarantee a larger volume of macropores or better reservoir properties.The correlations between Dsw and petrophysical parameters show that the increase of Dsw is accompanied by the decrease of pore size and permeability, resulting in poorer reservoir properties.Therefore, Dsw is a good indicator of reservoir quality.

Relationship between Petrophysical and Multifractal Parameters
The relationships linking D(0), D( 1), ΔD and Δf with permeability are determined to further explore the relationships between multifractal parameters and reservoir pore structure.Figure 10 shows that D(0), D (1), and D Δ are positively correlated with permeability, whereas Δf negatively correlates with permeability.D(0) and D(1) correlate well with permeability, with correlation coefficients of 0.8845 and 0.8665, respectively.D(0) represents the range of the pore size distribution, and D(1) characterizes the heterogeneity of the pore size distribution.The permeability improves with increasing D(0) and D (1).The more widely the reservoir particle size is distributed in the sparse region, the larger the reservoir size distribution range and the stronger the degree of heterogeneity.Therefore, the physical properties of the reservoir improve with increases in the range and heterogeneity of the pore size distribution.

Relationship between Petrophysical and Multifractal Parameters
The relationships linking D(0), D(1), ∆D and ∆f with permeability are determined to further explore the relationships between multifractal parameters and reservoir pore structure.Figure 10 shows that D(0), D(1), and ∆D are positively correlated with permeability, whereas ∆f negatively correlates with permeability.D(0) and D(1) correlate well with permeability, with correlation coefficients of 0.8845 and 0.8665, respectively.D(0) represents the range of the pore size distribution, and D(1) characterizes the heterogeneity of the pore size distribution.The permeability improves with increasing D(0) and D(1).The more widely the reservoir particle size is distributed in the sparse region, the larger the reservoir size distribution range and the stronger the degree of heterogeneity.Therefore, the physical properties of the reservoir improve with increases in the range and heterogeneity of the pore size distribution.

Comparison between Single Fractal and Multifractal Analysis
Single fractal theory has been extensively used to characterize the heterogeneity of the pore structure, while multifractal analysis of mercury intrusion porosimetry in reservoir rocks is less commonly discussed.Comparative studies on single fractal and multifractal analysis are rare.
From the above analysis, we can see that there are some differences between single fractal and multifractal analysis.The Dsw values of samples 64 and 72 are lower than those of other samples, which generally implies that these two samples have low heterogeneity and good reservoir properties.However, samples 64 and 72 have bigger D(0) and D(1) values, as determined through multifractal analysis, demonstrating that these samples have a larger pore size distribution range and strong heterogeneity.Until now, this difference has not been observed or studied, since previous studies focus on the single or multifractal characteristics of mercury intrusion porosimetry.The difference is related to the fact that single fractal analysis and multifractal analysis characterize different aspects of the heterogeneity of a pore size distribution.The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range, while the range of pore size distribution has the greatest effect on the multifractal analysis and multifractal parameters.
Single fractal theory characterizes the heterogeneity of pore structure from the entire range of pore sizes.The fractal curves show segmented fractal characteristics, and the fractal dimensions of each stage represents the complexity of the corresponding pore size range.The segmented fractal phenomenon demonstrates that multifractal theory is appropriate for describing fractal characteristics in a full-scale range.Multifractal theory is applied to explore pore size distribution in different scales.The pore structure exhibits multifractal characteristics in a local distribution, and the multifractal method can be used to investigate the complexity and scale effect of the pore size distribution.
It is worth noting that an intrinsic difference exists with respect to the single fractal models used to compute the fractal dimension.Friesen obtained a fractal model based on the Sierpinski carpet model [5].The capillary bundle model was assumed in Shen's model [10].Su set up a new fractal model considering of the fractal dimensions for pore space and tortuosity [11].Different fractal dimensions may be obtained when these different fractal models are applied [30].Moreover, the fractal dimension of the low pressure stage may exceed the theoretical value.This is because the current fractal models assume that the reservoir is composed only of pores and ignore the influence of microfractures.It has been confirmed that the presence of microfractures can significantly

Comparison between Single Fractal and Multifractal Analysis
Single fractal theory has been extensively used to characterize the heterogeneity of the pore structure, while multifractal analysis of mercury intrusion porosimetry in reservoir rocks is less commonly discussed.Comparative studies on single fractal and multifractal analysis are rare.
From the above analysis, we can see that there are some differences between single fractal and multifractal analysis.The D sw values of samples 64 and 72 are lower than those of other samples, which generally implies that these two samples have low heterogeneity and good reservoir properties.However, samples 64 and 72 have bigger D(0) and D(1) values, as determined through multifractal analysis, demonstrating that these samples have a larger pore size distribution range and strong heterogeneity.Until now, this difference has not been observed or studied, since previous studies focus on the single or multifractal characteristics of mercury intrusion porosimetry.The difference is related to the fact that single fractal analysis and multifractal analysis characterize different aspects of the heterogeneity of a pore size distribution.The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range, while the range of pore size distribution has the greatest effect on the multifractal analysis and multifractal parameters.
Single fractal theory characterizes the heterogeneity of pore structure from the entire range of pore sizes.The fractal curves show segmented fractal characteristics, and the fractal dimensions of each stage represents the complexity of the corresponding pore size range.The segmented fractal phenomenon demonstrates that multifractal theory is appropriate for describing fractal characteristics in a full-scale range.Multifractal theory is applied to explore pore size distribution in different scales.The pore structure exhibits multifractal characteristics in a local distribution, and the multifractal method can be used to investigate the complexity and scale effect of the pore size distribution.
It is worth noting that an intrinsic difference exists with respect to the single fractal models used to compute the fractal dimension.Friesen obtained a fractal model based on the Sierpinski carpet model [5].The capillary bundle model was assumed in Shen's model [10].Su set up a new fractal model considering of the fractal dimensions for pore space and tortuosity [11].Different fractal dimensions may be obtained when these different fractal models are applied [29].Moreover, the fractal dimension of the low pressure stage may exceed the theoretical value.This is because the current fractal models assume that the reservoir is composed only of pores and ignore the influence of microfractures.It has been confirmed that the presence of microfractures can significantly increase the fractal dimension [30].Multifractal analysis explores pore size distribution characteristics without any a priori assumptions, considering the effect of microfractures.
We show that fractal analysis and multifractal analysis are feasible means for characterizing the heterogeneity of pore structures in a reservoir.By comparing the relationships between single fractal and multifractal parameters and reservoir physical parameters, we learn that multifractal parameters have better correlations with reservoir physical properties than single fractal parameters.The aforementioned factors combined mean that multifractal analysis is more suitable for characterizing the heterogeneity of a pore size distribution when the pore size distribution of the samples is addressed without any a priori assumptions.

Conclusions
In this study, single fractal and multifractal theory were applied to investigate pore size distribution characteristics of reservoir rocks as well as the influences of the single fractal and multifractal parameters on pore structure.The following conclusions can be drawn from our work: (1) The single fractal curves exhibit segmented fractal characteristics and the fractal dimensions of the low-pressure section is greater than the fractal dimension of the high-pressure section.The saturation-weighted fractal dimension D sw has a better correlation with permeability and entry pressure than D 1 or D 2 .(2) Linear relationships exist between log(ε) and log(µ,(ε)) for the 13 samples when −20 ≤ q ≤ 20, suggesting that the pore structures of the 13 samples exhibit multifractal characteristics.Multifractal parameters D(0) and D(1) correlate well with permeability and entry pressure.The physical properties of the reservoir improve with increases in the range of pore size distribution.(3) The ratio of larger pores volumes to total pore volumes acts as a control on the fractal dimension over a specific pore size range, while the range of pore size distribution has the greatest effect on the multifractal analysis and multifractal parameters.(4) Fractal analysis and multifractal analysis are feasible methods for characterizing the complexity of pore size distribution in a reservoir.Multifractal analysis with parameters D(0) or D(1) produce better correlations with reservoir physical properties.In conclusion, multifractal analysis is more suitable for characterizing the heterogeneity of pore size distributions when the pore size distribution of samples is addressed without any a priori assumptions.

Figure 2 .
Figure 2. Pore size distributions of four samples obtained using mercury analysis.

Figure 2 .
Figure 2. Pore size distributions of four samples obtained using mercury analysis.

Figure 3 .
Figure 3. Plots of log(SHg) vs. log(Pc) from the mercury intrusion curve of sample 22.

Figure 3 .
Figure 3. Plots of log(SHg) vs. log(Pc) from the mercury intrusion curve of sample 22.

Figure 4 .Figure 4 .
Figure 4. Plot of the sum of fractal dimension D1 versus r50.Table 2. Parameters of the fractal dimension obtained from the 13 samples.Samples D1 Correlation D2 Correlation Dsw

Figure 6 .
Figure 6.Plot of the multifractal generalized dimension D(q) versus the moment order q.Figure6.Plot of the multifractal generalized dimension D(q) versus the moment order q.

Figure 6 .
Figure 6.Plot of the multifractal generalized dimension D(q) versus the moment order q.Figure6.Plot of the multifractal generalized dimension D(q) versus the moment order q.

Figure 7 .
Figure 7. Plot of multifractal spectrum f(a) versus the singularity strength a.

Figure 7 .
Figure 7. Plot of multifractal spectrum f (a) versus the singularity strength a.

Figure 9 .
Figure 9.A plot of Dsw vs. entry pressure.

Figure 8 .
Figure 8.A plot of D sw vs. permeability.

Figure 9 .
Figure 9.A plot of Dsw vs. entry pressure.

Figure 9 .
Figure 9.A plot of D sw vs. entry pressure.

Figure 10 .
Figure 10.Plots of multifractal parameters and permeability.

Figure 11
Figure 11 presents the relationships linking D(0), D(1), ΔD, and Δf with the entry pressure.Favorable negative correlations exist between D(0), D(1), and ΔD and entry pressure, whereas Δf is positively correlated with entry pressure.Similarly, D(0) and D(1) correlate well with entry pressure, with correlation coefficients 0.9041 and 0.8971, respectively.D(0) and D(1) can be used as important parameters for characterizing and predicting physical properties of a reservoir.

Figure 10 .
Figure 10.Plots of multifractal parameters and permeability.

Figure 11 16 Figure 10 .
Figure 11 presents the relationships linking D(0), D(1), ∆D, and ∆f with the entry pressure.Favorable negative correlations exist between D(0), D(1), and ∆D and entry pressure, whereas ∆f is positively correlated with entry pressure.Similarly, D(0) and D(1) correlate well with entry pressure, with correlation coefficients 0.9041 and 0.8971, respectively.D(0) and D(1) can be used as important parameters for characterizing and predicting physical properties of a reservoir.

Figure 11
Figure 11 presents the relationships linking D(0), D(1), ΔD, and Δf with the entry pressure.Favorable negative correlations exist between D(0), D(1), and ΔD and entry pressure, whereas Δf is positively correlated with entry pressure.Similarly, D(0) and D(1) correlate well with entry pressure, with correlation coefficients 0.9041 and 0.8971, respectively.D(0) and D(1) can be used as important parameters for characterizing and predicting physical properties of a reservoir.

Figure 11 .
Figure 11.Plots of multifractal parameters and entry pressure.

Figure 11 .
Figure 11.Plots of multifractal parameters and entry pressure.

Table 1 .
Parameters of the pore throat structure obtained from the 13 samples.
Note: r 50 corresponds to pore throat diameter at 50% mercury saturation; Sorting coefficient is the dispersion degree of reservoir pores.

Table 2 .
Parameters of the fractal dimension obtained from the 13 samples.

Table 4 .
Multifractal dimension parameters obtained from the 13 samples.

Table 4 .
Multifractal dimension parameters obtained from the 13 samples.

Table 5 .
Multifractal spectrum parameters obtained from the 13 samples.

Table 5 .
Multifractal spectrum parameters obtained from the 13 samples.