# Fractal analysis of tight shaly sandstones using nuclear magnetic resonance measurements

## Abstract

Assessing quantitatively the microscopic pore structures of porous rocks, including irregularities of pore shapes and pore size distributions, is becoming one of the most challenging efforts. Nuclear magnetic resonance (NMR) measurements were used to provide insights into the pore geometry (pore size and shape) and pore connectivity of the Chang 7 tight shaly sandstones (in situ permeability <0.1 md) in Ordos basin. The incremental transverse relaxation time (*T*_{2}) distributions for the 100% brine-saturated samples display unimodal and bimodal behaviors, presenting a geometrical arrangement composed of small to large pore size domains. The NMR parameters such as bulk volume irreducible (BVI) (capillary and clay-bound water), free fluid index (FFI) (movable water), the value of *T*_{2} separating the BVI from FFI, the amplitude weighted mean on a logarithmic scale (*T*_{2gm}), and the value of *T*_{2} that shows the highest frequency on the *T*_{2} spectrum (*T*_{2peak}) for each sample were determined. Then the fractal theory was adopted to quantitatively express the complexity and heterogeneity of the sandstones. The results show that only minor primary intergranular porosity remains, and variable amounts of micropores and secondary intragranular porosity with poor connectivity occur in the Chang 7 tight shaly sandstones. Assuming spherical pores, a new model to calculate the fractal dimension of pore structure from the NMR *T*_{2} distributions is proposed. The fractal dimensions of all the samples are calculated, and the accuracy of the proposed model is verified by the regression coefficients. The microscopic pore structures are heterogeneous in these tight sandstones according to the high value of fractal dimensions. Micropores are the primary causes of heterogeneity in tight sandstones, and samples with unimodal *T*_{2} distribution behaviors and high content of short components have the highest fractal dimension and heterogeneity. The calculated fractal dimension is strongly correlated with *T*_{2peak} and *T*_{2gm}; therefore, the fractal model proposed in this study can be used to calculate the fractal dimensions and evaluate the heterogeneities of the porous rocks satisfactorily. The fractal model proposed in this study helps to quantitatively assess the pore structures of tight sandstones using NMR measurements.

## Introduction

Knowledge of pore size distribution is critical for understanding both storage and transport properties of porous sedimentary rocks (Pape et al., 2006; Anovitz et al., 2013; Clarkson et al., 2013). As complex geological materials, there are many irregular pores that occur at different scales in the porous sedimentary rocks (Wang et al., 2012). Because of the complexity and irregularity of the pore size distribution (pore size and shape, tortuosity, and pore connectivity), it is difficult to quantitatively characterize pore structure by traditional Euclidean geometry (Wang et al., 2012). Several studies have suggested that the pore systems of sedimentary rocks have a fractal nature (Friesen and Mikula, 1987; Hansen and Skjeltorp, 1988; Krohn, 1988; Broseta et al., 2001; Li and Horne, 2006; Li, 2010a; Giri et al., 2012; Daigle et al., 2014a; Lai and Wang, 2015; Sakhaee-Pour and Li, 2016). Fractals are defined as virtual, self-similar objects that appear identically independent of the scale of magnification (Kulesza and Bramowicz, 2014), and these self-similar fractal objects could be characterized by the fractal dimension (*Df*) (Giri et al., 2012; Lai and Wang, 2015; Weller et al., 2016; Zhou and Kang, 2016). The fractal theory builds up the relationships between microscopic pore structures and macro petrophysical parameters (porosity and permeability) (Hu et al., 2012). By calculating the fractal dimensions of the sedimentary rocks, the complexity and heterogeneity of the highly disordered pore systems could be quantitatively characterized (Cai et al., 2010; Li, 2010a; Wang et al., 2012), and the fractal descriptions of pore systems can also be used for prediction of petrophysical properties such as permeability, wettability, tortuosity, diffusivity, and electrical conductivity (Daigle et al., 2014a).

Scanning electron microscopy (SEM) (Katz and Thompson, 1985), small-angle neutron scattering (Wong et al., 1986; Clarkson et al., 2013), thin sections (Xu and Wang, 2013), nitrogen adsorption desorption (Yang et al., 2014; Bu et al., 2015), pore-scale images (Sakhaee-Pour and Li, 2016), mercury intrusion (Li, 2010a; Nooruddin et al., 2014), and nuclear magnetic resonance (NMR) measurements (Daigle et al., 2014a) could be used to infer fractal dimensions of porous rocks. The NMR measurements record the spin axis relaxation times of hydrogen atom or protons ^{1}H in the presence of external and static magnetic fields (Coates et al., 1991; Daigle et al., 2014b; Dillinger and Esteban, 2014). Besides permeability prediction, fluid type evaluation, assessment of different volumes of pore fluid (free water, clay-bound water, and hydrocarbons), and determination of pore size distribution can be obtained from NMR data (Pape and Clauser, 2009; Daigle et al., 2014b). The NMR measurements of relaxation time made in the laboratory on saturated core plugs are widely used to characterize the pore systems and fluids in porous media (Daigle et al., 2014a; Sigal, 2015). Although direct determination of pore sizes from relaxation times requires calibration with mercury injection capillary pressure (MICP) methods to determine the surface relaxivity, the NMR relaxation time distributions could be used to determine the pore size distribution (Daigle and Johnson, 2016) and therefore offer a rapid, nondestructive method to determine fractal dimensions of the sedimentary rocks (Daigle et al., 2014a).

The major goals of this paper are to derive a fractal model from NMR measurements and to investigate the fractal characteristics of Chang 7 tight shaly sandstones from the Ordos basin of west China. Pore size distributions are determined from the NMR transverse relaxation time (*T*_{2}) spectrum, and the NMR parameters such as capillary and clay-bound water content (bulk volume irreducible [BVI]), mobile water content (free fluid index [FFI]), the value of *T*_{2} separating the BVI from FFI (*T*_{2cutoff}), the amplitude weighted mean on a logarithmic scale (*T*_{2gm}), and the value of *T*_{2} that shows the highest frequency on the *T*_{2} spectrum (*T*_{2peak}) are then determined from the NMR measurements. The fractal dimension, which was used to present the fractal characterization, was then calculated from NMR *T*_{2} distribution data. Then the relationships between NMR parameters and fractal dimensions were analyzed. A fractal model is proposed in this study to calculate the fractal dimensions of tight sandstones using NMR measurements. The work presented here helps extend the application of NMR measurements and our understanding of the microscopic pore structure and heterogeneity of reservoir rocks, particularly for tight sandstones having experienced extensive diagenetic modifications.

## Geological Backgrounds

The Ordos basin, with an area of 320,000 km^{2} (123,000 mi^{2}), is a large sedimentary basin located in the western part of the North China block (Yang et al., 2005), and it can be divided into six geological structural units: the Yimeng uplift, Jinxi fold belt, Yishan slope, Tianhuan depression, Western thrust belt, and Wei River uplift (Figure 1A) (Lai et al., 2016b). Abundant hydrocarbon resources are present in the Upper Triassic Yanchang Formation, which consist of lacustrine deltaic and fluvial siliciclastics with a thickness of 1000–1500 m (3281–4921 ft) (Zeng and Li, 2009; Cao et al., 2016b). The development of the Ordos basin can be divided into three evolutionary stages: Cambrian–Early Ordovician cratonic basin, Middle Ordovician–Middle Triassic cratonic basin, and Late Triassic–Early Cretaceous intraplate remnant cratonic basin (Yang et al., 2005). The Ordos basin became an isolated lake during the Middle and Late Triassic, and a series of lacustrine and deltaic clastic sediments were deposited in the basin, the Yanchang Formation (Ji et al., 2008). The margins of the Ordos basin experienced intense tectonic movements; however, its interior was not strongly deformed because of the rigid basement, and the Ordos basin is characterized by a stable tectonic setting (Yang et al., 2005). The Yanchang Formation can be subdivided into 10 members, namely, Chang 1 to Chang 10 from top to bottom (Figure 1B) (Lyu et al., 2016). Among them, the Chang 7 member (oil layer) consists of oil shale and black mudstone interbedded with fine-grained sandstone and siltstone (Li et al., 2011; Lai et al., 2016b). Traditionally, the sixth and eighth members (Chang 6 and Chang 8) of the Yanchang Formation are the main Mesozoic reservoirs in the Ordos basin. Currently, with the exploration and development of tight oil resources, hydrocarbons could also be produced in tight shaly sandstones and siltstones in the Chang 7 oil layers (Lai et al., 2016b). Tight oil has become the focus of research in global petroleum geology (Cao et al., 2016a). The dark gray lacustrine mudstones and shales within Chang 7 oil layers act as source rocks in this hydrocarbon system, and these tight sandstones interbedded with Chang 7 mudstones and shales have the advantage for trapping oil because of closer proximity to lacustrine source rocks (Figure 1B).

**Figure 1.**(A) The tectonic divisions of Ordos basin and (B) lithology column of Upper Triassic Yanchang Formation in Ordos basin (Zeng and Li, 2009; Lai et al., 2013a, 2016b).

## Theory And Methodology

### Theory

Namely, NMR is the interaction between the nucleus and the magnetic field (Meng et al., 2016). Two types of relaxation times are associated with the NMR measurements, of which longitudinal relaxation time (*T*_{1}) refers to the characteristic time for the system to return to the equilibrium; in contrast, *T*_{2} refers to the characteristic time for the processing secondary magnetic field to lose coherence in the direction perpendicular to the primary magnetic field (Sigal, 2015). The NMR *T*_{2} spectrum (relaxation time distribution), which describes the decay of magnetization transverse to an applied, time-varying magnetic field, is preferred in the laboratory because it is a more rapid measurement (Daigle et al., 2014a). The *T*_{2} in the NMR measurements can be expressed as (Coates et al., 1991; Kleinberg et al., 1994)

where is the transverse NMR relaxation time of the full signal, is the bulk relaxation, is the diffusion-induced relaxation time, and is the surface relaxation time (Dillinger and Esteban, 2014; Chi et al., 2016). In water-saturated samples, the relaxation time caused by bulk fluid processes is much longer (order of seconds) than that caused by surface processes (order of milliseconds). For 3.5 wt. % NaCl brine at the temperature of 20°C, equals 2.8 sec, which makes 1/*T*_{2B} commonly negligible. In addition, the measurement may be designed such that 1/*T*_{2D} may be minimized to a value less than 0.001 sec^{−1}, and this could also be negligible in the *T*_{2} distributions (Daigle et al., 2014b; Daigle and Johnson, 2016). The third member (*T*_{2S}) has the most important effect on porous rocks (Dillinger and Esteban, 2014). Therefore, the NMR *T*_{2} could be approximated expressed as (Pape and Clauser, 2009; Daigle et al., 2014b; Müller-Huber et al., 2016)

where *S* (μm^{2}) and *V* (μm^{3}) are pore surface and volume, respectively, indicating that *T*_{2} is directly dependent on the pore surface to pore volume ratio; *ρ* (μm/sec) is the surface relaxivity of grains and is the coefficient corresponding to each *T*_{2} value (Daigle et al., 2014b). By cross-correlating the NMR *T*_{2} distributions and MICP pore throat size distributions, Daigle and Johnson (2016) determined that *ρ* = 16.0 μm/sec for the Devonian Berea sandstone (Ohio), whereas *ρ* = 6.5 μm/sec for the Silurian Racine dolomite (Illinois). The ratio *S*/*V* is the surface/volume ratio of the pore space and is inversely proportional to the pore radius *r* where the factor depends on the pore shape (Pape and Clauser, 2009; Daigle et al., 2014b; Chi et al., 2016; Müller-Huber et al., 2016):

where *a* is a constant; *a* = 2 for cylindrical pores, and *a* = 3 for spherical pores (Pape and Clauser, 2009; Daigle et al., 2014b; Sigal, 2015; Müller-Huber et al., 2016).

An assumption of pore shape is required to convert the NMR relaxation spectrum to a pore size distribution (Sigal, 2015). Previous studies confirm that the number of pores in sandstones does not stay the same with progressive diagenesis, and pores become more spherical with increasing diagenetic modifications (Cook et al., 2011). The Chang 7 tight shaly sandstones are characterized with low porosity, ultralow permeability, and strong microscopic heterogeneity, which is the result of the extensive diagenetic modifications (Lai et al., 2016b). The NMR measurements are more sensitive to pore body size instead of pore throat sizes because the majority of the pore volume is assumed to be composed of pore bodies (Daigle et al., 2014b); therefore, this study assumed the pore shapes to be spheres. Assuming spherical pores and that the magnetic field gradient was constant and equal in all pores (Daigle et al., 2014b), then the *T*_{2} distributions are therefore functions of pore sizes where the constants of proportionality are the surface relaxivities (Sigal, 2015):

The *T*_{2} distributions determined from NMR measurements could be converted to pore size distributions; each *T*_{2} relaxation time constant corresponds to a particular pore radius (Mitchell and Fordham, 2014; Sigal, 2015). From equation 4, it could be concluded that larger pores are associated with longer relaxation times (Mitchell and Fordham, 2014), whereas NMR *T*_{2} relaxation time of small pores is short (Meng et al., 2016). The signal amplitude is proportional to the number of protons, i.e., the pore volume, and the higher the signal amplitude, the more the porosity (Dillinger and Esteban, 2014; Meng et al., 2016).

### Experimental Measurements

A collection of cores was retrieved from seven wells (G273, Z143, L89, H212, A92, Y63, and B272) drilled in the central Ordos basin. The porosity and air permeability measurements were performed on 458 core plug samples under a confining pressure of 200.0 psi (1.4 MPa). A subset of 100 of these samples was sent to Core Laboratories at PetroChina Research Institute of Petroleum Exploration Development for NMR measurements to determine the *T*_{2} distributions. The NMR experiment processes are according to the Chinese standard “Specification for normalization measurement of core NMR parameter in laboratory” (SYT6490-2007) (National Development and Reform Commission, 2007), with the same experimental procedure as shown in previous research (Yao et al., 2010). First, the weight of the dry core plug is measured, then the weights of 100% saturated and centrifuged core plug are also measured. Thus, the saturation of the irreducible water content can be derived.

The experiment temperature was 25°C. The NMR *T*_{2} distributions were performed on these core plug samples of 1.5 in. (3.8 cm) in diameter and 1 in. (2.5 cm) in length. For all the 100 samples used in this study, the nitrogen porosity, water porosity, and NMR porosity were measured, whereas the air permeability was measured under atmospheric stress and at net confining stress of 800.0 psi (5.5 MPa). Because of the very low–permeability nature of the rock matrix, some samples were unable to continue testing at net confining stress of 800 psi.

The NMR apparatus (MARAN-DRX/2) provides 2 MHz frequency of the main magnetic field, waiting time (TW) for 6000 msec, and echo spacing (TE) of 0.3 msec performed to obtain each relaxation curve. The NMR parameters were installed by maximum echo numbers of 4096 and scanning numbers of 64. This apparatus records the net magnetization of proton ^{1}H in the presence of an external magnetic field (Dillinger and Esteban, 2014). The samples were fully saturated with NaCl brine with a salinity of 41,000 mg/L, and the NMR *T*_{2} distributions (incremental and cumulative) at 100% brine saturation were measured. The signal/noise ratio for the NMR measurements was a minimum of 100:1. Then the core plug samples were removed with the free water by a centrifugal machine. The rotation speed of the centrifugal machine is 9000 rpm, and the 100% brine-saturated core plugs were kept in this machine for 1 hr to remove the free water. The NMR *T*_{2} relaxation time distribution was used to identify total porosity, effective porosity, permeability, irreducible water, mobile water, and geometric mean of the *T*_{2} distribution (*T*_{2gm}). The *T*_{2cutoff} values could be calculated by comparisons of the *T*_{2} distributions at 100% brine saturation and those at bound water.

The SEM analysis was conducted on the fresh surface of the core plug samples, which are coated with gold to help detect the micropores associated with authigenic clay minerals. Thin sections, which were impregnated with red epiflourescent epoxy, were analyzed to determine intergranular and intragranular pores.

### Methodology

Self-affinity with a dimension is an important feature of fractal objects in nature, and this feature can be mathematically illustrated by a power-law function (Mandelbrot, 1977; Li, 2010a; Daigle et al., 2014a; Zhang and Weller, 2014):

where *r* is characteristic length of a unit (for sedimentary rock, it is the pore radius), means “proportional to,” *N*(*r*) is the number of objects whose sizes are greater than the size *r*, and *Df* is the fractal dimension (Li, 2010b). Generally, the *Df* is in range 2.0 < *Df* < 3.0 in porous media (three-dimensional spaces) (Broseta et al., 2001; Cai et al., 2010; Jin et al., 2013; Lai et al., 2013b; Gao et al., 2014). The value of fractal dimension is a representation of rock heterogeneity, and the greater the fractal dimension, the greater the heterogeneity (Li and Horne, 2006). Generally, *Df* approaches 2.0 for smooth, clay-free rocks, whereas values close to 3.0 are typical of strongly argillaceous sandstones with high heterogeneity (Broseta et al., 2001).

Based on equation 5, many scientists inferred the formulas to calculate the fractal dimensions of pore structure from the MICP according to the capillary tube model and Young–Laplace law (Friesen and Mikula, 1987; Broseta et al., 2001; Li, 2010a, b; Schmitt et al., 2013; Lai and Wang, 2015).

As stated above, the distribution of pore sizes obtained from NMR measurements could be described by a fractal model (equation 5). The NMR *T*_{2} relaxation time distributions could be related directly to pore size distributions (Daigle and Johnson, 2016). The NMR *T*_{2} measurements provide relaxation curves that are functions of time. It is generally assumed that each time constant (*T*_{2} value) represents a particular pore size (Sigal, 2015). The signal amplitude at this relaxation time (*T*_{2i} value) is a function of the number of protons and therefore the porosity (pore volume, *V*_{pi}) corresponding to this pore radius *r*_{i} (Dillinger and Esteban, 2014).

In the fractal model, the total volume of the pores (*V*_{p}, %) is expressed as the total porosity, which could be directly measured from the core plug samples. The *V*_{p} values are the sum of the signal amplitudes from the minimum to maximum *T*_{2} values performed on saturated core plugs. Suppose the pore volume corresponding to pore radius *r*_{i}, i.e., *T*_{2i}, is *V*_{pi}; then the *V*_{p} could be given by

where *V*_{p1} corresponds to the minimum *T*_{2} values, *T*_{2min}, whereas *V*_{pn} corresponds to the maximum *T*_{2} values, *T*_{2max}. The pore volume measured at each characteristic size *V*_{pi} is equal to the signal amplitude at the corresponding NMR *T*_{2} relaxation time or, in other words, the pore volume associated with the *i*th time constant *T*_{2i} (Sigal, 2015).

The pore shape is assumed to be spherical, and then the numbers of pores of a given size *r*_{i} (*T*_{2i} value) are expressed in

Therefore, the pore numbers composed of pore size larger than *r*_{i} are given by

where *j* = *i* + 1.

Equation 9 can be obtained by combining equations 4, 5, and 8:

Equation 9 could be expressed as follows:

where and . Therefore, log(1/*A*) is a constant, and is also a constant.

Equation 9 generally demonstrates that the relationship between the numbers of pores greater than *r*_{i}, i.e., *N*(*r*) derived from NMR measurements and the pore radius (*r*) is linear in a double-logarithm coordination (log(*N*(*r*)−log(*r*))). Equation 10 demonstrates that the relationship between and the pore radius (*r*) is linear in a double-logarithm coordination. The fractal dimension can then be derived from the slope of the best fit line in the log–log plot of pore numbers (*N*(*r*)) against pore radius *r* (*T*_{2}).

## Results

### Porosity, Permeability, and Pore Systems

Core-measured porosity and air permeability of the 444 samples show that porosity ranges from 2.28% to 17.74% and averages 7.48%. Air permeability ranges from 0.0025 to 4.42 md with an average of 0.13 md (Figure 2). Some samples are characterized by high porosity but relative low permeability or low porosity but high permeability, and the coefficient of determination (*R*^{2}) between permeability and porosity is low, possibly reflecting wide differences in dominant pore types (Figure 2) (Lai et al., 2016b). Observations by thin section petrography and SEM indicate that pore systems in the Chang 7 tight shaly sandstones are of both primary and secondary (partial to complete dissolution of framework grains) origins (Figure 3) (Lai et al., 2015c). Minor primary intergranular porosity remains (Figure 3A), and variable amounts of secondary intragranular porosity caused by dissolution of detrital framework grains occur in many samples (Figure 3B). Abundant micropores are associated with the authigenic clay minerals such as illite and chlorite (Figure 3C). Most of the secondary dissolution pores are small pores or micropores with poor connectivity (Figure 3D) (Lai et al., 2016b).

**Figure 2.** Plot of core-measured porosity and air permeability measured at a net confining stress of 200 psi. *R*^{2} = coefficient of determination.

**Figure 3.** Photomicrographs showing pore systems for Chang 7 tight oil reservoirs in Ordos basin. (A) Primary intergranular pore (red arrow) coexisting with the secondary dissolution pores (blue arrow), An 236, 2156.83 m (7076.21 ft), plane polarized light. (B) Secondary dissolution pores, An 237, 2155.22 m (7070.93 ft), plane polarized light. (C) Micropores (red arrow) associated with authigenic clay minerals, scanning electron microscopy (SEM). Information along the bottom includes magnification (mag), voltage (HV), wide diameter (WD), horizontal field width, and the sample depth. (D) Honeycomb-like pores (red arrow) with poor connectivity, SEM.

### Pore Size Distribution

The NMR *T*_{2} spectrum provides significant information on the pore structure of rocks (Dillinger and Esteban, 2014). The incremental and cumulative *T*_{2} distributions for the 100% brine-saturated and partially brine-saturated (centrifuged) core plug samples are shown in Figure 4. It is evident from Figure 4 that each *T*_{2} value (*T*_{2min} and *T*_{2max}) corresponds to certain signal amplitudes, which are the pore volumes at this pore radius *r* (*T*_{2}). The signal amplitudes of the saturated samples are much higher than those of the unsaturated samples (Figure 4), indicating that the signal amplitudes are proportional to the water (or hydrocarbon) content of the core sample. The cumulative porosities for both the 100% brine-saturated and unsaturated core plug samples increase with the *T*_{2} values, and the cumulative porosity for the unsaturated samples will reach its maximum values, which equals to the total immobile porosity. For example, the maximum value of the cumulative porosity for the unsaturated sample in Figure 4 is 8.43%. Similarly, in the cumulative porosity curves for the saturated samples, it will also reach this value at some certain *T*_{2} value. Then this *T*_{2} value is called the *T*_{2cutoff}, which separates the capillary and clay-bound water (BVI) residing in micropores and clay minerals from the movable (free) water (FFI) present in macropores (Dillinger and Esteban, 2014). The *T*_{2cutoff} values could be determined in this way; for example, the *T*_{2cutoff} is 4.64 msec in Figure 4. The BVI volume can then be calculated by incrementing the signal amplitude at each relaxation time until *T*_{2} reaches the *T*_{2cutoff} (or in short, the BVI volume equals to the maximum cumulative porosity for the unsaturated sample), whereas the FFI volume can be calculated by subtracting the BVI volume (8.43%) from the total porosity (15.38%) or, in other words, the total cumulative amplitude of the NMR signal (Dillinger and Esteban, 2014). The parameter *T*_{2peak} (Rezaee et al., 2012) can also be determined from the NMR *T*_{2} distribution (Figure 4). However, it should be noted in Figure 4 that there is still a small quantity of water even at larger pores (*T*_{2} > *T*_{2cutoff}) accounted as bound water because the unsaturated curve is not zero (Figure 4).

**Figure 4.** (A) The incremental porosity distribution as a function of the transverse relaxation time (*T*_{2}). The value of *T*_{2} separating the bulk volume irreducible from free fluid index (*T*_{2cutoff}) separates the *T*_{2} spectrum into two parts: movable water and irreducible water. (B) The cumulative porosity distribution as a function of the relaxation time.

Total NMR porosity, capillary and clay-bound water content, mobile water, *T*_{2cutoff}, *T*_{2gm} values, and *T*_{2peak} values from NMR measurements are presented in Table S1 (supplementary material available as AAPG Datashare 89 at www.aapg.org/datashare). The results show that these samples have a relatively high content of capillary and clay-bound water. The BVI is in the range from 46.69% to 92.46% with an average of 68.51% of the total porosity, indicating that more than half the porosity and the number of voids are associated with clay minerals and within capillary pore sizes. The *T*_{2cutoff} reveals a wide range from 1.00 to 56.98 msec and averages as 19.03 msec, whereas the *T*_{2gm} is in the range of 2.17 to 20.90 msec (averaging 9.39 msec) (Table S1, supplementary material available as AAPG Datashare 89 at www.aapg.org/datashare). The *T*_{2peak} shows a wide range from 0.9 to 86.4 msec and has an average of 21.4 msec (Table S1, supplementary material available as AAPG Datashare 89 at www.aapg.org/datashare). The wide range of these NMR parameters implies that these samples have various pore structure and heterogeneity.

The weighting method is introduced here to calculate the content of irreducible water (equation 11). The BVI values (capillary and clay-bound water content), which are derived from the NMR *T*_{2} distribution, are strongly positively correlated with the irreducible water saturation (*S*_{wi}) derived from the weighting method (Figure 5), possibly implying that the method for calculating the *T*_{2cutoff} and BVI value from NMR measurement is reliable.

According to the NMR laboratory measurements on the 100% brine-saturated core plugs, the cumulative *T*_{2} distributions display unimodal and bimodal behaviors (Figures 6A, B; 7A, B), representing a geometrical arrangement composed of small to large pore size domains. The bimodal behaviors possibly indicate the coexisting of the micropores and macropores, suggesting the presence of a wide range of pore sizes in these heterogeneous sandstones. The short components with small *T*_{2} values (*T*_{2s}) refer to the microporosity associated with clay minerals and capillary pore sizes, whereas the long or large components with large *T*_{2} values (*T*_{2l}) correspond to the macroporosity including large intergranular pores or microfractures (Figure 6A, B) (Dillinger and Esteban, 2014; Müller-Huber et al., 2016). Figure 6A (sample 36) shows the typical bimodal behavior of the NMR *T*_{2} distribution. Strong *T*_{2l} components are presented in this sample, representing the high content of movable water (FFI) existing in the macropores. In contrast, the *T*_{2s} is weak in this sample, indicating a moderate content of capillary or clay-bound immobile water (65.3%). One distinct peak characterized by right-skewed distribution is observed (lower left peak but higher right peak) (Figure 6A). The two distinct peaks in this sample indicate a discontinuous pore size distribution, which could be related to poor sorting of the pore systems. The *T*_{2cutoff} is set at approximately 25.95 msec; therefore, 65.3% of the full NMR signal (i.e., the total porosity) is made of irreducible water (Figure 6A). Figure 6B shows another typical bimodal behavior of the *T*_{2} distribution acquired on sample 35. However, it is different from that of sample 36 in Figure 6A. From the *T*_{2} spectrum, it can be concluded that both *T*_{2s} components and *T*_{2l} components are strong in this sample (Figure 6B). However, the full NMR signal (total porosity) is much lower than that of the sample in Figure 6A. The *T*_{2cutoff} is determined at 27.33 msec, and the BVI value is approximately 85.5%, representing that the fluids in this sample are dominantly the capillary or clay bound immobile water residing in micropores. It is evident that the complexity and heterogeneity of the microscopic pore structure of sample 36 in Figure 6B are greater than those of sample 35 in Figure 6A.

**Figure 5.** Plot of bulk volume irreducible (BVI) from nuclear magnetic resonance measurements versus irreducible water saturation (*S*_{wi}) derived from weighting method.

**Figure 6.** Nuclear magnetic resonance (NMR) transverse relaxation time (*T*_{2}) incremental and cumulative spectra. Note that the *T*_{2} incremental spectra of these samples show bimodal behaviors. (A) Typical bimodal behavior of the *T*_{2} distribution. Both *T*_{2s} components and *T*_{2l} components are strong. (B) Bimodal behavior, the full NMR signal is lower. Black squares indicate saturated, red circles indicate unsaturated.

**Figure 7.** Nuclear magnetic resonance transverse relaxation time (*T*_{2}) incremental and cumulative spectra for samples show unimodal behaviors. (A) Typical unimodal behavior of the *T*_{2} distribution, and short *T*_{2} components are dominant. (B) Typical unimodal behavior of the *T*_{2} distribution, and relatively long *T*_{2} components are dominant. Black squares indicate saturated, red circles indicate unsaturated.

However, there are some samples that display unimodal behaviors (Figure 7A) (sample 27). In the *T*_{2} relaxation time distributions of this sample, there is only one modal representing the *T*_{2s} components, and there are no or very weak *T*_{2l} components (Figure 7A). The values of *T*_{2gm} and *T*_{2cutoff} for this sample are very low (3.63 and 3.78 msec, respectively), indicating that there is minor movable water (31.7%) presented in this sample. Also there are some samples displaying unimodal behaviors that have high *T*_{2gm} and *T*_{2cutoff} values (sample 8), and the *T*_{2} scales are higher (Figure 7B). Generally speaking, both the *T*_{2s} components and *T*_{2l} components are presented (Figure 7B). The NMR parameters such as *T*_{2peak} and *T*_{2gm} of sample 8 in Figure 7B are much higher than those of sample 27 in Figure 7A because of sample 8’s abundance in large pores.

It should be noted that there are minor or even no components greater than 100 msec, indicating the rarity of large pores and absence of microfractures in these samples (Figures 6, 7). Table S1 (supplementary material available as AAPG Datashare 89 at www.aapg.org/datashare) also lists the behavior for the NMR *T*_{2} distribution for all the samples (unimodal or bimodal).

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