Investigation of the Pore Structure of Tight Sandstone Based on Multifractal Analysis from NMR Measurement: A Case from the Lower Permian Taiyuan Formation in the Southern North China Basin

: Understanding the pore structure can help us acquire a deep insight into the ﬂuid transport properties and storage capacity of tight sandstone reservoirs. In this work, a series of methods, including X-ray di ﬀ raction (XRD) analysis, casting thin sections, scanning electron microscope (SEM), nuclear magnetic resonance (NMR) experiment and multifractal theory were employed to investigate the pore structure and multifractal characteristics of tight sandstones from the Taiyuan Formation in the southern North China Basin. The relationships between petrophysical properties, pore structure, mineral compositions and NMR multifractal parameters were also discussed. Results show that the tight sandstones are characterized by complex and heterogenous pore structure, with apparent multifractal features. The main pore types include clay-dominated micropores and inter- and intragranular dissolution pores. Multifractal parameters of sandstone samples were acquired by NMR and applied to quantitatively describe the pore heterogeneity in higher and lower probability density regions (with respect to small and large pore-scale pore system, respectively). The multifractal parameter ( D − 10 ) of lower probability density areas has better correlation with the petrophysical parameters, which is more suitable for evaluating the reservoir properties of tight sandstone. However, the multifractal parameter ( D 10 ) of higher probability density areas is more conducive to characterize the pore structure of tight sandstone. Additionally, the mineral compositions of sandstone have a complex e ﬀ ect on multifractal characteristics of pores in di ﬀ erent probability density areas. The D 10 increases with the decrease of quartz content and increase in clay mineral content, whereas D − 10 decreases with the increase in clay minerals and decrease of authigenic quartz content and feldspar content. coated with carbon in advance, were examined with a ZEISS SIGMA field emission scanning electron microscope equipped with energy-dispersive X-ray spectra (EDX), to identify clay minerals’ type, the morphologies of clay occurrence within the pore spaces and micropores associated with clay minerals [39]. The mineral compositions of the samples were determined by XRD analysis, following the Oil


Introduction
Due to their huge geological reserves and wide distributions, tight sandstone gas greatly alleviates the contradiction between the world's increasing demand on energy and the depletion of conventional resources [1]. In China, tight sandstone gas reservoirs are characterized as one with the porosity and permeability less than 10% and 0.1 mD, respectively [2], and these tight reservoirs require extensive hydraulic fracture or special gas extraction techniques to achieve commercial production [3]. In comparison with conventional sandstones, the pore structures of tight sandstones are usually more complex and heterogeneous, because of their various pore size (nano-scale to micro-scale), poor connectivity and irregular pore geometry [4][5][6][7]. The pore structure replaces porosity and The continuous uplift of the entire North China platform during the Caledonian movement led to the absence of the Upper Ordovician, Silurian and Devonian, and Lower Carboniferous strata [35]. From the Late Carboniferous to the Late Permian, affected by the regional tectonic movement, the study area underwent the depositional process from the epicontinental sea to the continental basin ( Figure 2) [36]. The Taiyuan Formation (P1t), which is composed of black shale, coal, limestone and sandstone, was deposited during the early Permian, with a thickness of about 30-175 m ( Figure 2) [37]. Sedimentary facies of the Taiyuan Formation are mainly lagoon, tidal flat, barrier island, and carbonate platform (Figure 2). A previous study has shown that shale and coal from Taiyuan Formation possess high organic matter content (TOC), moderate maturity, mainly type III kerogen, and high gas generation potential, which can serve as good source rocks for tight sandstone reservoirs, as well as the potential reservoirs for shale gas and coal bed methane [38]. The continuous uplift of the entire North China platform during the Caledonian movement led to the absence of the Upper Ordovician, Silurian and Devonian, and Lower Carboniferous strata [35]. From the Late Carboniferous to the Late Permian, affected by the regional tectonic movement, the study area underwent the depositional process from the epicontinental sea to the continental basin ( Figure 2) [36]. The Taiyuan Formation (P 1 t), which is composed of black shale, coal, limestone and sandstone, was deposited during the early Permian, with a thickness of about 30-175 m ( Figure 2) [37]. Sedimentary facies of the Taiyuan Formation are mainly lagoon, tidal flat, barrier island, and carbonate platform ( Figure 2). A previous study has shown that shale and coal from Taiyuan Formation possess high organic matter content (TOC), moderate maturity, mainly type III kerogen, and high gas generation potential, which can serve as good source rocks for tight sandstone reservoirs, as well as the potential reservoirs for shale gas and coal bed methane [38].  [38]) and more details of Taiyuan Formation (well wc1), including burial depth, lithology, sedimentary facies.

Samples and Experiments
A total of 5 tight sandstone samples were selected from cores obtained in wells wc1 and wpd1 ( Figure 1). Cylindrical plugs, 2.5 cm in diameter and 3-4 cm in length, were cut from the cores. The core plugs were carefully cleaned and dried, to remove contamination from remnant hydrocarbon and drilling fluids. The permeability of samples was first measured using a calibrated instrument DX-07G, with a steady flow of N2 based on the Chinese Core Measurement and Analysis Method Standard (GB/T29172-2012). Secondly, the samples were placed in the brine with a salinity (~1200 mg/L), similar to formation water under vacuum for at least 24 h, until the brine saturated state. NMR signals were generated using the brine saturated samples. The samples were put in a pulsed magnetic field, and then after a brief pulse, the NMR signal gradually decayed, with a characteristic relaxation time T2. During this process, the T2 spectra under saturated conditions were recorded with a lowfield NMR instrument Rec core 2500. Magnetic field strength and resonance frequency were 1200 G and 2.38 MHz, respectively. Thereafter, the saturated samples were centrifuged via an HR2500-2 apparatus, to obtain an ideal irreducible water state at a rotational speed of 5000 r/min. The experimental parameters were set as follows: the echo number 2048, the echo space 300 μs, the waiting time 6 s, and the test temperature was 25 °C.
Subsequently, the samples were dried again, and divided into three sections, for analysis of pore structure and mineralogical compositions, respectively. Thin sections, impregnated with blue epoxy resin under vacuum, were prepared to observe the type, geometric shapes, and distribution of the pores. In the SEM experiment, the freshly broken samples, which were polished and coated with carbon in advance, were examined with a ZEISS SIGMA field emission scanning electron microscope equipped with energy-dispersive X-ray spectra (EDX), to identify clay minerals' type, the morphologies of clay occurrence within the pore spaces and micropores associated with clay minerals [39]. The mineral compositions of the samples were determined by XRD analysis, following the Oil  [38]) and more details of Taiyuan Formation (well wc1), including burial depth, lithology, sedimentary facies.

Samples and Experiments
A total of 5 tight sandstone samples were selected from cores obtained in wells wc1 and wpd1 ( Figure 1). Cylindrical plugs, 2.5 cm in diameter and 3-4 cm in length, were cut from the cores. The core plugs were carefully cleaned and dried, to remove contamination from remnant hydrocarbon and drilling fluids. The permeability of samples was first measured using a calibrated instrument DX-07G, with a steady flow of N 2 based on the Chinese Core Measurement and Analysis Method Standard (GB/T29172-2012). Secondly, the samples were placed in the brine with a salinity (~1200 mg/L), similar to formation water under vacuum for at least 24 h, until the brine saturated state. NMR signals were generated using the brine saturated samples. The samples were put in a pulsed magnetic field, and then after a brief pulse, the NMR signal gradually decayed, with a characteristic relaxation time T 2 . During this process, the T 2 spectra under saturated conditions were recorded with a low-field NMR instrument Rec core 2500. Magnetic field strength and resonance frequency were 1200 G and 2.38 MHz, respectively. Thereafter, the saturated samples were centrifuged via an HR2500-2 apparatus, to obtain an ideal irreducible water state at a rotational speed of 5000 r/min. The experimental parameters were set as follows: the echo number 2048, the echo space 300 µs, the waiting time 6 s, and the test temperature was 25 • C.
Subsequently, the samples were dried again, and divided into three sections, for analysis of pore structure and mineralogical compositions, respectively. Thin sections, impregnated with blue epoxy resin under vacuum, were prepared to observe the type, geometric shapes, and distribution of the pores. In the SEM experiment, the freshly broken samples, which were polished and coated with carbon in advance, were examined with a ZEISS SIGMA field emission scanning electron microscope equipped with energy-dispersive X-ray spectra (EDX), to identify clay minerals' type, the morphologies of clay occurrence within the pore spaces and micropores associated with clay minerals [39]. The mineral compositions of the samples were determined by XRD analysis, following the Oil and Gas Industry Standards (SY/T5463-2010). Samples were initially crushed to 100 mesh size, and then they were mixed with ethanol, ground into mortar, and placed on glass slides. The measurement was carried out using the Ultima IV X-ray diffractometer. The mineral content was quantified using Jade software.

NMR Theory
NMR can effectively reveal the important information about pore structure of rock, based on the T 2 transverse relaxation time [15]. The total transverse relaxation T 2 time is associated with three relaxation members: bulk relaxation T 2B , surface relaxation T 2S , and diffusion of pore fluid T 2D , described as [40]. 1 Generally, the bulk relaxation and diffusion relaxation are usually ignored when magnetic field is uniform. In this case, T 2 can be related directly to pore size: [16].
where ρ (µm/s) is the transversal surface relaxation rate; S (µm 2 ) and V (µm 3 ) are the surface area and fluid volume of pore space, respectively; the surface/volume ratio (S/V) is a function of pore radius r (µm), and a is the pore shape factor (a = 3 for spherical pore, while a = 2 for tubular pore). Thus, the T 2 distribution under fully brine-saturated conditions can be converted to the curve of pore size distribution by Equation (2), with the help of ρ. Details of this method have been shown in the previous studies [41].

Multifractal Methods Based on NMR T 2 Distributions
Numerous studies have introduced the algorithm of multifractal theory in detail [17,25,42,43]. In this study, the popular box counting method [31] was employed for the implement of multifractal algorithm on the basis of the 100% water-saturated T 2 distributions of the samples. T 2 distributions are split into N square boxes of size r, here r = 2 m (m = 0, 1, 2 . . . ). The probability mass distribution function P i (r) of the ith box could be represented as where M i is the pore volume in the ith box and is the total porosity. If porosity has a multifractal distribution, and then P i (r) has a power exponent relationship to r, as follows: where α i is the singularity strength for boxes [23]. Furthermore, the number of boxes with a similar α value is defined as N α (r), by the relationship: where f (α) is a multifractal or singularity spectrum, expressing the fractal dimension of boxes with similar values of α [19]. Furthermore, f (α) could reach its maximum value when the following conditions are met: In order to accurately acquire the distribution properties, the partition function is expressed as: where q is a moment expressing the contribution to X(q, r) of boxes with diverse P i (r), which is commonly defined as [−10, 10]; When q < 0, X(q, r) represents the density probability of the area with low concentration of porosity; when q > 0, X(q, r) denotes the density probability of the area with high concentration of porosity [44]. Moreover, τ(q), known as mass exponent, could be depicted by: The generalized multifractal dimension D q , another way to characterize singularity, which can be defined as: On the other hand, the α(q), f (α) can also be determined from τ(q) with the Legendre transformation, respectively [43]: Generally, four sets of parameters, such as D q , α(q), f (α) and ∆α (=α max −α min ), are commonly applied to characterize pore structure heterogeneity. The more complex and nonhomogeneous pore structure corresponds to the larger value of D q and ∆α.

Mineralogical Compositions of Tight Sandstone
The XRD analysis results of samples are shown in Table 1. Overall, the studied P 1 t tight sandstone samples mainly consist of quartz and clay. Therein, quartz content ranges from 58.9% to 74.1% average as 68.42%, and clay content varies from 16.3% to 35.8%, with an average of 24.32%. The clay minerals are dominated by the mixed illite/smectite (I/S) and illite, attended by a few kaolinites, and chlorites ( Table 1). The high content of clay minerals may result from the great heterogenous composition and strong alteration of detrital feldspars in the tight sandstones [45]. There is only trace amount of feldspar (0.9-8.3%, averaging 5.22%). The lack of feldspar content is possibly due to the dissolution caused by factors such as basin subsidence, thermal events and acid produced by coal-bearing strata [46]. Other minerals, including ankerite, pyrite and calcite, can be identified only in individual samples, and their contents are extremely low (<2.5%).

Pore Type and Characteristics
Casting thin section and SEM image analysis results show that there are three dominant pore types in the P 1 t tight sandstone samples, including micropores associated with clay minerals, secondary intergranular and intragranular dissolution pores ( Figure 3). Primary intergranular pores are rarely observed. Quartz overgrowth and authigenic quartz grains are commonly developed in P 1 t tight sandstones (Figure 3a,b,d), and pyrite crystals are also observed in some pores (Figure 3e). Secondary dissolution pores mostly occur on detrital feldspars, as a result of partial to complete dissolution (Figure 3a,b). These dissolution pores are typically enveloped by authigenic clay minerals derived from the dissolution of detrital feldspars (Figure 3f). Additionally, a few dissolution pores occur on detrital grain boundaries (Figure 3c). Secondary dissolution pores mostly occur on detrital feldspars, as a result of partial to complete dissolution (Figure 3a,b). These dissolution pores are typically enveloped by authigenic clay minerals derived from the dissolution of detrital feldspars ( Figure 3f). Additionally, a few dissolution pores occur on detrital grain boundaries ( Figure 3c). The pore structures of samples are severely impacted from clay cementation, because the pore throat system is mainly filled by a large amount of authigenic clay minerals such as the slablike or booklet kaolinite (Figure 3g), and the flaky illite and hair-like mixed illite/smectite (I/S) (Figure 3h,i). SEM image analysis showed that abundant micropores are developed within these clay minerals (Figure 3g,h,i). These micropores are continuously distributed with a multi-scale pore size (mainly < 1 μm), which provide the necessary percolation path, connecting other relatively larger pores for tight sandstone reservoirs to a certain extent.

Petrophysical Properties and NMR T2 Distributions
The porosity of the five samples ranges from 1.95% to 3.41%, with an average of 2.7%, and permeability varies from 0.037 mD to 0.494 mD ( Table 2). The petrophysical properties of samples are lower than those observed in the Chang 7 reservoir, with an average value of porosity of 7.2% and permeability of 0.18 mD. The Chang 7 reservoir is an important tight oil reservoir from Yanchang Formation in the Ordos basin [47]. This indicates that the samples have relatively poorer pore The pore structures of samples are severely impacted from clay cementation, because the pore throat system is mainly filled by a large amount of authigenic clay minerals such as the slablike or booklet kaolinite (Figure 3g), and the flaky illite and hair-like mixed illite/smectite (I/S) (Figure 3h,i). SEM image analysis showed that abundant micropores are developed within these clay minerals (Figure 3g-i). These micropores are continuously distributed with a multi-scale pore size (mainly < 1 µm), which provide the necessary percolation path, connecting other relatively larger pores for tight sandstone reservoirs to a certain extent.

Petrophysical Properties and NMR T 2 Distributions
The porosity of the five samples ranges from 1.95% to 3.41%, with an average of 2.7%, and permeability varies from 0.037 mD to 0.494 mD ( Table 2). The petrophysical properties of samples are lower than those observed in the Chang 7 reservoir, with an average value of porosity of 7.2% and permeability of 0.18 mD. The Chang 7 reservoir is an important tight oil reservoir from Yanchang Formation in the Ordos basin [47]. This indicates that the samples have relatively poorer pore structures and reservoir quality. Nevertheless, a positive exponential relationship can be observed between porosity and permeability of samples ( Figure 4). Table 2. The petrophysical parameters and pore structure parameters of the tight sandstone samples from NMR measurements. structures and reservoir quality. Nevertheless, a positive exponential relationship can be observed between porosity and permeability of samples ( Figure 4).  Reservoir quality index (RQI) and flow zone indicator (FZI) are two ideal macroscopic petrophysical parameters used to evaluate the micro pore structure and reservoir properties of tight sandstone [48]. RQI and FZI were calculated by the following formulas, respectively [33]: where K is the permeability, mD; φ is the porosity, %. As shown in Table 2, the values of RQI vary from 0.0041 μm to 0.012 μm, while FZI values range from 0.1498 μm to 0.341 μm, averaging as 0.228 μm. These values are close to the tight oil reservoir researched by Zhao et al., 2017, whereas they are lower than the Chang 7 tight reservoir [47].
For the fully water-saturated rock, the T2 distributions provide information about the pore size distributions. The T2 relaxation time is in proportion to the pore size [15,49], and the signal amplitude of T2 distributions reflect the pore fluid content and pore volume. The 100% brine-saturated T2 spectra of five samples are shown in Figure 5. Except for sample 1 (unimodal T2 spectrum), all samples show the bimodal characteristics of T2 spectra, and almost all pore sizes of tight sandstones present in the range from 0.1 to 100 ms. There are no pores with the relaxation time larger than 100 ms, attributed to the absence of residually large intergranular pores. The main peaks are distributed between 0.1 ms and 10 ms, and their signal amplitudes are far larger than the secondary peak. The relative amplitudes of T2 peaks indicate that sample porosities are dominated by smaller pore sizes, and the larger porosity are relatively few. The pores of sample 1 are all smaller pores. Reservoir quality index (RQI) and flow zone indicator (FZI) are two ideal macroscopic petrophysical parameters used to evaluate the micro pore structure and reservoir properties of tight sandstone [48]. RQI and FZI were calculated by the following formulas, respectively [33]: where K is the permeability, mD; ϕ is the porosity, %. As shown in Table 2, the values of RQI vary from 0.0041 µm to 0.012 µm, while FZI values range from 0.1498 µm to 0.341 µm, averaging as 0.228 µm. These values are close to the tight oil reservoir researched by Zhao et al., 2017, whereas they are lower than the Chang 7 tight reservoir [47].
For the fully water-saturated rock, the T 2 distributions provide information about the pore size distributions. The T 2 relaxation time is in proportion to the pore size [15,49], and the signal amplitude of T 2 distributions reflect the pore fluid content and pore volume. The 100% brine-saturated T 2 spectra of five samples are shown in Figure 5. Except for sample 1 (unimodal T 2 spectrum), all samples show the bimodal characteristics of T 2 spectra, and almost all pore sizes of tight sandstones present in the range from 0.1 to 100 ms. There are no pores with the relaxation time larger than 100 ms, attributed to the absence of residually large intergranular pores. The main peaks are distributed between 0.1 ms and 10 ms, and their signal amplitudes are far larger than the secondary peak. The relative amplitudes By analyzing NMR T2 data, some NMR pore structure parameters, including movable-fluid porosity (φm), bound-fluid porosity (φb), T2cutoff, T2gm (amplitude weighted logarithmic mean), T35 and T50, are also summarized in Table 2. Porosity in the rock can be separated into bound-fluid porosity (T2 < T2cutoff) and movable-fluid porosity (T2 > T2cutoff) in the cumulative T2 spectrum by T2cutoff value ( Figure 6). The bound-fluid porosity of tight sandstone usually exists in clay-dominated micropores, which contain capillary and clay-bound water, while movable-fluid porosity tends to reside in large pores which are connected by effective pore throat [18]. Then, the movable-fluid porosity is determined by removing the proportion of the bound fluid from the 100% brine-saturated NMR signal, varying from 0.38% to 1.85%. Compared to movable-fluid porosity, bound-fluid porosity is commonly high, with the range of 1.08%-2.25%, indicating that the sandstone samples have the complex pore structure with poor pore connectivity. T35, T50 are corresponding to the T2 value, where the samples reach 35% and 50% brine saturation in the cumulative T2 distributions, respectively [3].
Overall, compared to Chang 7 and Xujiahe tight sandstone reservoirs [48,50], T2cutoff, T2lm, T35, and T50 of samples are characterized by relatively lower values, indicating a narrower pore size distribution in the samples.

Multifractal Characteristics
In this study, multifractal characteristics of pore structures were obtained from 100% brinesaturated T2 spectra. The range of moments q is defined in the interval from −10 to 10. The generalize dimension spectra (Dq~q) and the relationship of mass exponent τ(q) versus q are presented in Figures  7 and 8, respectively. The Dq with respect to variable q shows an inverse S-shaped curve. Dq has a larger variation for q < 0, while a minor variation for q > 0. Moreover, τ(q) follows a monotone increase as q increase, and the increasing trend gradually becomes smoother with increasing q. Figure 9 represents the multifractal spectra or singularity spectra of samples, where α(q) are also strongly correlated to the variable q. Overall, these spectra of different samples all show two different variation trends, that reveal that the pore size distributions of tight sandstone samples are multifractal. Therefore, the heterogeneity of pore volume distribution can be represented via the generalized dimension and singularity spectra shape and its characteristic parameters, which further reveal the local differences in the whole.
Generally, the heterogeneity of the whole pore size distribution is assessed by the total width of singularity spectra α-10-α10 and generalized dimension spectra D−10-D10 [19]. Higher values of α-10-α10 By analyzing NMR T 2 data, some NMR pore structure parameters, including movable-fluid porosity (ϕ m ), bound-fluid porosity (ϕ b ), T 2cutoff , T 2gm (amplitude weighted logarithmic mean), T 35 and T 50 , are also summarized in Table 2. Porosity in the rock can be separated into bound-fluid porosity (T 2 < T 2cutoff ) and movable-fluid porosity (T 2 > T 2cutoff ) in the cumulative T 2 spectrum by T 2cutoff value ( Figure 6). The bound-fluid porosity of tight sandstone usually exists in clay-dominated micropores, which contain capillary and clay-bound water, while movable-fluid porosity tends to reside in large pores which are connected by effective pore throat [18]. Then, the movable-fluid porosity is determined by removing the proportion of the bound fluid from the 100% brine-saturated NMR signal, varying from 0.38% to 1.85%. Compared to movable-fluid porosity, bound-fluid porosity is commonly high, with the range of 1.08%-2.25%, indicating that the sandstone samples have the complex pore structure with poor pore connectivity. T 35 , T 50 are corresponding to the T 2 value, where the samples reach 35% and 50% brine saturation in the cumulative T 2 distributions, respectively [3]. Overall, compared to Chang 7 and Xujiahe tight sandstone reservoirs [48,50], T 2cutoff , T 2lm , T 35 , and T 50 of samples are characterized by relatively lower values, indicating a narrower pore size distribution in the samples. By analyzing NMR T2 data, some NMR pore structure parameters, including movable-fluid porosity (φm), bound-fluid porosity (φb), T2cutoff, T2gm (amplitude weighted logarithmic mean), T35 and T50, are also summarized in Table 2. Porosity in the rock can be separated into bound-fluid porosity (T2 < T2cutoff) and movable-fluid porosity (T2 > T2cutoff) in the cumulative T2 spectrum by T2cutoff value (Figure 6). The bound-fluid porosity of tight sandstone usually exists in clay-dominated micropores, which contain capillary and clay-bound water, while movable-fluid porosity tends to reside in large pores which are connected by effective pore throat [18]. Then, the movable-fluid porosity is determined by removing the proportion of the bound fluid from the 100% brine-saturated NMR signal, varying from 0.38% to 1.85%. Compared to movable-fluid porosity, bound-fluid porosity is commonly high, with the range of 1.08%-2.25%, indicating that the sandstone samples have the complex pore structure with poor pore connectivity. T35, T50 are corresponding to the T2 value, where the samples reach 35% and 50% brine saturation in the cumulative T2 distributions, respectively [3].
Overall, compared to Chang 7 and Xujiahe tight sandstone reservoirs [48,50], T2cutoff, T2lm, T35, and T50 of samples are characterized by relatively lower values, indicating a narrower pore size distribution in the samples.

Multifractal Characteristics
In this study, multifractal characteristics of pore structures were obtained from 100% brinesaturated T2 spectra. The range of moments q is defined in the interval from −10 to 10. The generalize dimension spectra (Dq~q) and the relationship of mass exponent τ(q) versus q are presented in Figures  7 and 8, respectively. The Dq with respect to variable q shows an inverse S-shaped curve. Dq has a larger variation for q < 0, while a minor variation for q > 0. Moreover, τ(q) follows a monotone increase as q increase, and the increasing trend gradually becomes smoother with increasing q. Figure 9 represents the multifractal spectra or singularity spectra of samples, where α(q) are also strongly correlated to the variable q. Overall, these spectra of different samples all show two different variation trends, that reveal that the pore size distributions of tight sandstone samples are multifractal. Therefore, the heterogeneity of pore volume distribution can be represented via the generalized dimension and singularity spectra shape and its characteristic parameters, which further reveal the local differences in the whole.
Generally, the heterogeneity of the whole pore size distribution is assessed by the total width of singularity spectra α-10-α10 and generalized dimension spectra D−10-D10 [19]. Higher values of α-10-α10

Multifractal Characteristics
In this study, multifractal characteristics of pore structures were obtained from 100% brine-saturated T 2 spectra. The range of moments q is defined in the interval from −10 to 10. The generalize dimension spectra (D q~q ) and the relationship of mass exponent τ(q) versus q are presented in Figures 7 and 8, respectively. The D q with respect to variable q shows an inverse S-shaped curve. D q has a larger variation for q < 0, while a minor variation for q > 0. Moreover, τ(q) follows a monotone increase as q increase, and the increasing trend gradually becomes smoother with increasing q. Figure 9 represents the multifractal spectra or singularity spectra of samples, where α(q) are also strongly correlated to the variable q. Overall, these spectra of different samples all show two different variation trends, that reveal that the pore size distributions of tight sandstone samples are multifractal. Therefore, the heterogeneity of pore volume distribution can be represented via the generalized dimension and singularity spectra shape and its characteristic parameters, which further reveal the local differences in the whole.
Generally, the heterogeneity of the whole pore size distribution is assessed by the total width of singularity spectra α −10 -α 10 and generalized dimension spectra D −10 -D 10 [19]. Higher values of α −10 -α 10 and D −10 -D 10 usually suggest a more heterogeneous pore size distribution within samples, and vice versa. The right part of the generalized dimension spectra and the singular spectra (q > 0) corresponds to the areas with higher probability density of porosity distribution (concentrated areas). However, the left part of the generalized dimension spectra and the singular spectra (q < 0) represent the areas with lower probability density (sparse areas) [32,51]. Therefore, multifractality parameters D 10 , D 0 -D 10 , α 10 , and α 0-α 10 can describe the pore characteristics in higher probability areas, and the parameters D −10 , D −10 -D 0 , α −10 , and α −10 -α 0 play the same role in lower probability areas. For the studied sandstone samples, the porosities in higher probability density areas mainly consist of the smaller pores, such as clay-dominated micropores, whereas the porosities in lower probability density areas mainly refer to the larger pores, such as intergranular dissolution pores, with a relatively larger pore size.
Energies 2020, 13, x FOR PEER REVIEW 10 of 18 and D−10-D10 usually suggest a more heterogeneous pore size distribution within samples, and vice versa. The right part of the generalized dimension spectra and the singular spectra (q > 0) corresponds to the areas with higher probability density of porosity distribution (concentrated areas). However, the left part of the generalized dimension spectra and the singular spectra (q < 0) represent the areas with lower probability density (sparse areas) [32,51]. Therefore, multifractality parameters D10, D0-D10, α10, and α0-α10 can describe the pore characteristics in higher probability areas, and the parameters D−10, D−10-D0, α-10, and α-10-α0 play the same role in lower probability areas. For the studied sandstone samples, the porosities in higher probability density areas mainly consist of the smaller pores, such as clay-dominated micropores, whereas the porosities in lower probability density areas mainly refer to the larger pores, such as intergranular dissolution pores, with a relatively larger pore size.   D0 is defined as a capacity dimension or box-counting dimension. D1, as an information dimension, is a measure of concentration degree of pore size distribution [28]. D2 is the correlation dimension, which explains the scaling behavior of the second sampling moments [25]. For a monofractal structure, D0 = D1 = D2 [44]. However, as shown in Table 3, all samples show a same order of D0 > D1 > D2, suggesting that the pore size distribution of every studied tight sandstone sample has a tendency toward a multifractal type of scaling. Additionally, the calculated D−10, D10, D−10-D0, D0-D10 and D−10-D10 are also listed in Table 3. For all samples, D−10 > D10 and D−10-D0 > D0-D10, indicating and D−10-D10 usually suggest a more heterogeneous pore size distribution within samples, and vice versa. The right part of the generalized dimension spectra and the singular spectra (q > 0) corresponds to the areas with higher probability density of porosity distribution (concentrated areas). However, the left part of the generalized dimension spectra and the singular spectra (q < 0) represent the areas with lower probability density (sparse areas) [32,51]. Therefore, multifractality parameters D10, D0-D10, α10, and α0-α10 can describe the pore characteristics in higher probability areas, and the parameters D−10, D−10-D0, α-10, and α-10-α0 play the same role in lower probability areas. For the studied sandstone samples, the porosities in higher probability density areas mainly consist of the smaller pores, such as clay-dominated micropores, whereas the porosities in lower probability density areas mainly refer to the larger pores, such as intergranular dissolution pores, with a relatively larger pore size.   D0 is defined as a capacity dimension or box-counting dimension. D1, as an information dimension, is a measure of concentration degree of pore size distribution [28]. D2 is the correlation dimension, which explains the scaling behavior of the second sampling moments [25]. For a monofractal structure, D0 = D1 = D2 [44]. However, as shown in Table 3, all samples show a same order of D0 > D1 > D2, suggesting that the pore size distribution of every studied tight sandstone sample has a tendency toward a multifractal type of scaling. Additionally, the calculated D−10, D10, D−10-D0, D0-D10 and D−10-D10 are also listed in Table 3. For all samples, D−10 > D10 and D−10-D0 > D0-D10, indicating and D−10-D10 usually suggest a more heterogeneous pore size distribution within samples, and vice versa. The right part of the generalized dimension spectra and the singular spectra (q > 0) corresponds to the areas with higher probability density of porosity distribution (concentrated areas). However, the left part of the generalized dimension spectra and the singular spectra (q < 0) represent the areas with lower probability density (sparse areas) [32,51]. Therefore, multifractality parameters D10, D0-D10, α10, and α0-α10 can describe the pore characteristics in higher probability areas, and the parameters D−10, D−10-D0, α-10, and α-10-α0 play the same role in lower probability areas. For the studied sandstone samples, the porosities in higher probability density areas mainly consist of the smaller pores, such as clay-dominated micropores, whereas the porosities in lower probability density areas mainly refer to the larger pores, such as intergranular dissolution pores, with a relatively larger pore size.   D0 is defined as a capacity dimension or box-counting dimension. D1, as an information dimension, is a measure of concentration degree of pore size distribution [28]. D2 is the correlation dimension, which explains the scaling behavior of the second sampling moments [25]. For a monofractal structure, D0 = D1 = D2 [44]. However, as shown in Table 3, all samples show a same order of D0 > D1 > D2, suggesting that the pore size distribution of every studied tight sandstone sample has a tendency toward a multifractal type of scaling. Additionally, the calculated D−10, D10, D−10-D0, D0-D10 and D−10-D10 are also listed in Table 3. For all samples, D−10 > D10 and D−10-D0 > D0-D10, indicating D 0 is defined as a capacity dimension or box-counting dimension. D 1 , as an information dimension, is a measure of concentration degree of pore size distribution [28]. D 2 is the correlation dimension, Energies 2020, 13, 4067 11 of 19 which explains the scaling behavior of the second sampling moments [25]. For a monofractal structure, D 0 = D 1 = D 2 [44]. However, as shown in Table 3, all samples show a same order of D 0 > D 1 > D 2 , suggesting that the pore size distribution of every studied tight sandstone sample has a tendency toward a multifractal type of scaling. Additionally, the calculated D −10 , D 10 , D −10 -D 0 , D 0 -D 10 and D −10 -D 10 are also listed in Table 3. For all samples, D −10 > D 10 and D −10 -D 0 > D 0 -D 10 , indicating that the pore distributions of higher probability areas may be more homogeneous than that of lower probability areas.  For singularity parameters, the similar trend, α −10 > α 10 and α −10 -α 0 > α 0 -α 10 , are also found in Table 4, indicating that the pore system in lower probability density areas owns more obvious multifractal characteristics than that in a higher probability density area. As presented in Table 4, the values of α 10 -α −10 are in the range of 1.15-3.69, indicating that the pore structures of samples are highly heterogeneous. The parameter A = (α 0 -α 10 )/(α −10 -α 0 ) is referred to express the asymmetry of singularity spectrum, and A > 1 demonstrates a strong fluctuation in pore size distribution. The values of A for sample 1 and sample 2 are lower than 1, which exhibit more stable pore size distributions compared to those of other samples. The multifractal analysis of tight sandstone pores shows that the pore distribution is complicated, multifractal and heterogeneous.

Discussion
The results shown in the preceding section manifest that several multifractal parameters can be utilized to characterize the pore heterogeneity. In this study, only two multifractal parameters (D 10 and D −10 ) are selected to describe multifractal characteristics and evaluate pore heterogeneity in different probability measure areas within tight sandstones. The multifractal parameter D 10 are used to account for multifractal behaviors of pore network in higher probability density areas, while the parameter D −10 represents the multifractal characteristics of pore network in lower probability density areas.

Relationship between Petrophysical Parameters of Tight Sandstone and Multifractal Parameters
Petrophysical property is the most direct performance of the pore structure of tight sandstone which can significantly affect fractal characteristics of pores. Figure 10 illustrates that multifractal parameters (D 10 and D −10 ) show a negative correlation with permeability, RQI and FZI, but they have no obvious relationship with porosity. Theoretically, porosity is mainly affected by the content of pore in the rock, especially large pore, which is independent of the complexity of pore distribution. Nevertheless, multifractal parameters mainly reflect the irregularity and complexity of pore geometry and pore network, with a significant influence on permeability, RQI and FZI. Meanwhile, not all pores are suitable for fractal analysis, and fractal analysis from a NMR T 2 spectrum only focus on pores with the pore size larger than dozens of nanometers or hundreds of nanometers [7], because the relaxation mechanism of fluid is quite complex in the smaller pore. In this case, the influence of bulk relaxation and diffusion relaxation should be considered, and T 2 cannot be directly simplified to surface relaxation T 2s . Therefore, the porosity of samples as a quantity cannot effectively constrain the heterogeneity of the pore structure [29].
Besides, D −10 shows a more sensitive response to permeability, RQI and FZI than D 10 , indicating that pore structure in lower probability density areas has a significant impact on petrophysical properties of tight sandstone reservoirs. From the T 2 distributions and petrographic observations of sandstone samples, small-scale clay-dominated micropores associated with short T 2 components (T 2 < 10 ms) constitute the majority of the pore system, and only a few proportions are composed of the large-scale dissolution pores associated with long T 2 components (T 2 > 10 ms). Hence, pore system in lower probability density areas mainly consists of dissolution pores with low content; micropores dominate pore system in higher probability density areas. The formation of dissolution pores with larger pore scale, however, greatly improve reservoir properties. Therefore, the complexity of pore system composed of dissolution pores play a more important role in the petrophysical properties of sandstones.
Energies 2020, 13, x FOR PEER REVIEW 12 of 18 properties of tight sandstone reservoirs. From the T2 distributions and petrographic observations of sandstone samples, small-scale clay-dominated micropores associated with short T2 components (T2 < 10 ms) constitute the majority of the pore system, and only a few proportions are composed of the large-scale dissolution pores associated with long T2 components (T2 > 10 ms). Hence, pore system in lower probability density areas mainly consists of dissolution pores with low content; micropores dominate pore system in higher probability density areas. The formation of dissolution pores with larger pore scale, however, greatly improve reservoir properties. Therefore, the complexity of pore system composed of dissolution pores play a more important role in the petrophysical properties of sandstones. Furthermore, the correlation coefficients of multifractal parameters with permeability, RQI and FZI show an increasing trend ( Figure 10). This can be explained by the fact that RQI and FZI integrate porosity and permeability, which is the better petrophysical parameters for characterizing the pore structure of tight sandstone. Meanwhile, FZI is the combination of several microscopic pore structure properties, e.g., pore specific surface area, morphology and tortuosity, and therefore the highest correlation coefficient [33]. Hence, FZI is a superior indicator of the pore structure heterogeneity of tight sandstone, especially in the lower probability density area.

Relationship between Pore Structure of Tight Sandstone and Multifractal Parameters
The relationships between multifractal parameters and NMR pore structure parameters of tight sandstone, including movable-fluid porosity (φm), bound-fluid porosity (φb), T2cutoff, T2gm, T35 and T50, are also analyzed, and the correlation coefficients are summarized in Table 5. D10 is highly associated with movable-fluid porosity, bound-fluid porosity, T2cutoff and T2gm, but it has no apparent correlation with T35 and T50 ( Figure 11). Additionally, D−10 shows few or no relationships with pore structure parameters (Table 5). This may be because small-scale micropores in higher probability density areas Furthermore, the correlation coefficients of multifractal parameters with permeability, RQI and FZI show an increasing trend ( Figure 10). This can be explained by the fact that RQI and FZI integrate porosity and permeability, which is the better petrophysical parameters for characterizing the pore structure of tight sandstone. Meanwhile, FZI is the combination of several microscopic pore structure properties, e.g., pore specific surface area, morphology and tortuosity, and therefore the highest correlation coefficient [33]. Hence, FZI is a superior indicator of the pore structure heterogeneity of tight sandstone, especially in the lower probability density area.

Relationship between Pore Structure of Tight Sandstone and Multifractal Parameters
The relationships between multifractal parameters and NMR pore structure parameters of tight sandstone, including movable-fluid porosity (ϕ m ), bound-fluid porosity (ϕ b ), T 2cutoff , T 2gm , T 35 and T 50 , are also analyzed, and the correlation coefficients are summarized in Table 5. D 10 is highly associated with movable-fluid porosity, bound-fluid porosity, T 2cutoff and T 2gm , but it has no apparent correlation with T 35 and T 50 ( Figure 11). Additionally, D −10 shows few or no relationships with pore structure parameters (Table 5). This may be because small-scale micropores in higher probability density areas dominate the entire pore system of tight sandstone reservoirs, and thus, multifractal parameters of higher probability density regions are more useful for pore structure characterization of tight sandstone. The T2cutoff is the efficient boundary which divides the total pore volume into movable-fluid pores and bound-fluid pores, according to whether the fluid within them can flow or not. For the studied sandstone samples, immovable bound-fluid mainly exists in clay-dominated micropores with poor connectivity, while movable fluid tends to exist in those dissolution pores which are connected by effective pore throats. T2cutoff value severely affects the proportion of different type of pores in tight sandstones. The larger the T2cutoff value, the more bound-fluid pores, and the fewer movable-fluid pores (Figure 12), which increase the heterogeneity of pore network in higher probability density areas.  The T 2cutoff is the efficient boundary which divides the total pore volume into movable-fluid pores and bound-fluid pores, according to whether the fluid within them can flow or not. For the studied sandstone samples, immovable bound-fluid mainly exists in clay-dominated micropores with poor connectivity, while movable fluid tends to exist in those dissolution pores which are connected by effective pore throats. T 2cutoff value severely affects the proportion of different type of pores in tight sandstones. The larger the T 2cutoff value, the more bound-fluid pores, and the fewer movable-fluid pores (Figure 12), which increase the heterogeneity of pore network in higher probability density areas.
sandstone samples, immovable bound-fluid mainly exists in clay-dominated micropores with poor connectivity, while movable fluid tends to exist in those dissolution pores which are connected by effective pore throats. T2cutoff value severely affects the proportion of different type of pores in tight sandstones. The larger the T2cutoff value, the more bound-fluid pores, and the fewer movable-fluid pores (Figure 12), which increase the heterogeneity of pore network in higher probability density areas.  Figure 11d illustrates that T2gm has the negative correlation with D10. The T2gm is a comprehensive performance for the whole pore size distribution of rock, and the large T2gm value means a good pore structure in the tight sandstone reservoirs [52]. In this study, a large amount of short T2 components related to micropores not only dominate the whole T2 distributions of sandstone samples, but also constitute the main peak. Therefore, T2gm mainly reflects the characteristics of main peak  Figure 11d illustrates that T 2gm has the negative correlation with D 10 . The T 2gm is a comprehensive performance for the whole pore size distribution of rock, and the large T 2gm value means a good pore structure in the tight sandstone reservoirs [52]. In this study, a large amount of short T 2 components related to micropores not only dominate the whole T 2 distributions of sandstone samples, but also constitute the main peak. Therefore, T 2gm mainly reflects the characteristics of main peak corresponding to pore system in higher probability density areas. Therefore, the heterogeneity of pore network of the high probability density areas tends to decrease as T 2gm increases.

Effect of miNeral Compositions of Tight Sandstone on Multifractal Characteristics
Several studies have testified that mineral compositions, mineral content and contact between minerals have a significant impact on the pore structure of rock, and the influence of each mineral on pores in distinct lithology may also be different [5,8,30,45,53].
As shown in Figure 13a, quartz content shows the negative correlation with D 10 and the weak negative relationship with D −10 . This result demonstrates that quartz plays different roles in the pore structure of different probability density areas. Sandstone petrographic observations show that the studied tight sandstone samples mainly consist of sedimentary quartz grains (Figure 3a,b), which act as the main skeleton minerals of the tight sandstone to resist compaction for the preservation of pores. In general, the higher content of quartz contributes to the higher textural maturity of sandstone and regular pore structure, which decrease the complexity of the pore systems in higher probability density areas. Nevertheless, the tight sandstone samples contain some authigenic quartz grains filling parts of large dissolution pores, which lead to the irregular pore geometry in lower probability density areas and thus larger values of D −10 .
According to Figure 13b, feldspar content is well correlated to D −10 , but has no correlation with D 10 . The above relationships indicate that tight sandstone with high feldspar content tends to have relatively complex and anisotropic pore structure in lower probability density areas. The content of feldspar in tight sandstone is associated with the diagenesis process. After deposition, the Taiyuan Formation experienced a long period of deep burial and diagenetic processes, and the chemical environment of fluid changed since a large amount of organic acids and CO 2 were generated during the early hydrocarbon generation of the organic matter in the source rock, which causes the extensive dissolution of chemically unstable aluminosilicates, such as feldspar. In this study, feldspar contents of all samples are low (average 5.22%), and secondary dissolution porosities as a result of partial/complete dissolution of feldspars are well observed in the sandstone samples (Figure 3a,b), indicating that feldspar experienced the strong alteration. These dissolution pores provide extra pore space for gas storage, while feldspar dissolution is always accompanied by by-products (e.g., authigenic quartz, kaolinite, illite) which precipitate in situ or in adjacent pores, resulting in a rather complex pore network in lower probability density areas.
Different from quartz and feldspar, clay content shows the weak positive correlation with D 10 , whereas it has a better negative relationship with D −10 (Figure 13c). This finding indicates that the effect of clay minerals on the pore structure in different probability density areas is rather complex. A high content of clay can not only reduce complexity of pore in lower probability density areas, but also increases the pore heterogeneity of higher probability density areas. This may be interpreted that tight sandstone samples have the high clay content (average 24.32%), and micropores associated with clay minerals predominate the pore system of the high probability measure areas. The SEM image analysis results show that a large amount of clay minerals fill in between quartz grains with slablike, flaky, and hair-like forms (Figure 3g-i), blocking the pore throat system, causing some large pores to be closed and semi-closed. The micropores gradually replace the large pores, and reduce the amounts of large pores with increasing clay content, which decreases the complexity of the pore network of lower probability density areas. However, clay-dominated micropores have different characteristics within various types of clay minerals, which are also characterized by multi-type pores with multi-scale pore size ranging from nanoscale to microscale, such as intracrystalline micropores (mainly < 2 nm), within aluminosilicate layers, inter-crystalline micropores (2-50 nm) between clay particles, and interparticle micropores (>0 nm) between aggregated clay particles [54]. These factors complicate the pore network in higher probability density areas, resulting in higher D 10 .

Conclusions
In the presented research, the pore structure and its heterogeneity of the Lower Permian Taiyuan Formation tight sandstone in the south China North basin were investigated by a series of experiments and multifractal methods. These conclusions could be obtained: (1) The tight sandstones from Taiyuan Formation are characterized by a high content of quartz and clay minerals, low porosity (1.95-3.41%) and low permeability (0.037-0.494 mD). Pore types mainly include micropores associated with clay minerals, intergranular and intragranular dissolution pores, whereas primary pores are rarely observed. The NMR T2 distributions of samples mainly show bimodal characteristics with less long T2 components, since clay-dominated micropores associated with short T2 components predominate the whole pore system of the tight sandstones.
(2) The multifractal spectrum, generalized dimensions and mass exponent spectrum demonstrate that pore structures of tight sandstone samples have multifractal characteristics. Meanwhile, the multifractal parameters are useful to quantify the pore heterogeneity in the areas with different probability densities.
(3) Multifractal parameter (D−10) of lower probability density areas is more appropriate to determine petrophysical properties. However, the multifractal parameter (D10) of higher probability density areas is the available indicators for evaluating pore structure.
(4) Mineral compositions have various effects on pore structure in different probability density areas. The decrease of quartz content and increase in clay mineral content contribute to high degree of pore anisotropy of higher probability density areas, whereas the decrease in clay minerals and the increase of quartz content and feldspar content can result in more irregular and nonhomogeneous pore network in lower probability density areas.
(5) Overall, the pore structure in lower probability density areas consists of larger dissolution pores, which determine the reservoir properties. Gas storage and seepage usually occur at those pores

Conclusions
In the presented research, the pore structure and its heterogeneity of the Lower Permian Taiyuan Formation tight sandstone in the south China North basin were investigated by a series of experiments and multifractal methods. These conclusions could be obtained: (1) The tight sandstones from Taiyuan Formation are characterized by a high content of quartz and clay minerals, low porosity (1.95-3.41%) and low permeability (0.037-0.494 mD). Pore types mainly include micropores associated with clay minerals, intergranular and intragranular dissolution pores, whereas primary pores are rarely observed. The NMR T 2 distributions of samples mainly show bimodal characteristics with less long T 2 components, since clay-dominated micropores associated with short T 2 components predominate the whole pore system of the tight sandstones.
(2) The multifractal spectrum, generalized dimensions and mass exponent spectrum demonstrate that pore structures of tight sandstone samples have multifractal characteristics. Meanwhile, the multifractal parameters are useful to quantify the pore heterogeneity in the areas with different probability densities.