Pore Size Distribution Characterization by Joint Interpretation of MICP and NMR: A Case Study of Chang 7 Tight Sandstone in the Ordos Basin

: Pore size distribution characterization of unconventional tight reservoirs is extremely signiﬁcant for an optimized extraction of petroleum from such reservoirs. In the present study, mercury injection capillary pressure (MICP) and low-ﬁeld nuclear magnetic resonance (NMR) are integrated to evaluate the pore size distribution of the Chang 7 tight sandstone reservoir. The results show that the Chang 7 tight sandstones are characterized by high clay mineral content and ﬁne grain size. They feature complex micro-nano-pore network system, mainly composed of regular primary intergranular pores, dissolved pores, inter-crystalline pores, and micro-fractures. Compared to the porosity obtained from MICP, the NMR porosity is closer to the gas-measured porosity (core analysis), and thus can more accurately describe the total pore space of the tight sandstone reservoirs. The pore throat distribution (PTD) from MICP presents a centralized distribution with high amplitude, while the pore size distribution (PSD) derived from NMR shows a unimodal distribution or bimodal distribution with low amplitude. It is notable that the difference between the PSD and the PTD is always related to the pore network composed of large pores connecting with narrow throats. The PSD always coincides very well with the PTD in the very tight non-reservoirs with a much lower porosity and permeability, probably due to the pore geometry that is dominated by the cylindrical pores. However, a signiﬁcant inconsistency between the PSD and PTD in tight reservoirs of relatively high porosity and low permeability is usually associated with the pore network that is dominated by the sphere-cylindrical pores. Additionally, Euclidean distance between PSD and PTD shows a good positive correlation with pore throat ratio (PTR), further indicating that the greater difference of pore bodies and pore throats, the more obvious differentiation of two distributions. In summary, the MICP and NMR techniques imply the different proﬁles of pore structure, which has an important implication for deep insight into pore structure and accurate evaluation of petrophysical properties in the tight sandstone reservoir.


Introduction
With the growth in energy needs and advances in horizontal drilling and hydraulic fracturing techniques, tight oil reservoirs have become a focus for hydrocarbon exploration and exploitation in recent years [1,2]. In China, the Chang 7 of the Ordos Basin was regarded as one of the most typical lacustrine tight reservoirs, which possessed enormous potential for exploration and exploitation [3]. A large amount of hydrocarbon accumulates in the pore systems of these tight reservoirs, and these pore structures are one of the dominant  Sixteen core samples were collected from six wells in the southwest area of the Ordos Basin ( Figure 1). These samples are from the Chang 7 tight sandstone succession of the Upper Triassic Yanchang Formation, with a well depth interval of about 1720-2070 m. Cylindrical plugs drilled from those samples were prepared with about 2.5 cm in diameter and length, and were cleaned by hot extraction solvent. The porosity and permeability analysis, low-field NMR, and MICP analysis were then performed successively on these samples. In addition, thin section analysis, SEM, mineral composition, and grain size analysis were conducted to determine petrology and pore characteristics.

MICP
The MICP method is based on the fact that mercury behaves as a nonwetting liquid when in contact with most substances [11]. Consequently, mercury does not spontaneously penetrate into the openings and pores of these substances unless pressure is applied. With increasing pressure, smaller pore throats are invaded by mercury [35].
The capillary pressure (P c ) required for mercury to penetrate pores is a function of the contact angle (θ Hg ) between mercury and the porous material to be intruded, its gas/liquid surface tension (σ Hg ), and pore radius (r p ) [12]. This relationship was provided by the Young-Laplace equation with the assumption of cylindrical pores as the Washburn equation [36,37]: 2σ Hg cos θ Hg r p (1) Processes 2022, 10, 1941 4 of 23 Equation (1) indicates that, with increasing pressure, mercury progressively intrudes into narrower pores for constant values of σ Hg and θ Hg . The volume of mercury penetrating the pores is measured directly as the increasing applied pressure.
The mercury porosimetry experiments were performed with a Micromeritics AutoPore IV 9520 porosimeter. The results of mercury porosimetry are generally displayed as graphs of capillary pressure versus cumulative mercury intrusion. The equivalent pore radius was computed according to the capillary pressure using the Washburn equation with P c ranging from approximately 0.003 to 413 MPa, using a contact angle of 130 • [38] and surface tension of 485 dyne/cm [39]. Therefore, the corresponding pore radius ranges from 1.8 nm to 218 µm.

NMR
In porous rocks, the transverse relaxation T 2 of NMR measurement is composed of surface relaxation, bulk relaxation, and diffusion relaxation, which can be expressed as Equation (2) [17,40]: where T 2B (ms) is the bulk relaxation time, T 2S (ms) is the surface relaxation time, T 2D (ms)is the diffusion-induced relaxation time, ρ 2 (µm/ms) is the surface relaxivity, S is the surface area of pore and V is the volume of the pore, D (µm 2 /ms) is the molecular diffusion coefficient of the pore fluid, G (G/cm) is the field-strength gradient, γ is the gyromagnetic ratio and T E (ms) is the inter-echo spacing used in the CPMG sequence. The bulk relaxation is produced by interactions of hydrogen nuclei of fluid, which can be negligible due to its long relaxation time compared to the surface relaxation [17]. Diffusion relaxation occurs in the inhomogeneous (gradient) magnetic field and it is almost close to zero in the homogeneous magnetic field [41]. So, the transverse relaxation T 2 can be approximated as the surface relaxation generated by interactions between fluid nuclei and solid interface (pore walls). That is, Equation (2) is converted to Equation (3): This relationship is based on an important assumption that diffusion within the pore is in the fast diffusion, which is valid when ρ2γ/D << 1 [40]. In the condition of the fast diffusion, the limit diffusion across the pore is much more rapid than the relaxation at the pore surface [42] and is met in most reservoir rocks [43].
The ρ 2 is close to a constant coefficient when the measured samples come from the same region, same formation, and similar lithology [20]. Therefore, the surface relaxation is a function of the surface-to-volume ratio (S/V) of the pores, which means that the small pores have short T 2 times and the large ones have long T 2 times. Additionally, the S/V depends on pore geometry. Consequently, Equation (3) is converted to Equation (4): where F s is the shape factor, which is a constant with values of 3 and 2 for spherical and cylindrical pores [44], respectively. R is pore radius (µm). Therefore, there is a linear relationship between the value of T 2 and pore size for the same area and formation and negligible lithological difference in clastic rocks. The NMR measurements were conducted on a SPEC-023 instrument. It has a constant magnetic field strength of 1200 Gauss and a resonance frequency of 2.38 MHz. The measurement parameters were set as follows: echo spacing, 0.2 ms; waiting time, 3 s; echo numbers, 2048; and number of scans, 64 [27]. After the measurements, transverse relaxation time (T 2 ) distributions were computed by multi-exponential inversion of the echo data with 64 preset Processes 2022, 10, 1941 5 of 23 decay times logarithmically spaced from 0.1 ms to 10 s [17]. The samples were fully saturated in water for 48 h under an ambient pressure of 30 MPa. In addition, the centrifugation was performed on the 100% water-saturated core plugs under a centrifugal force of 2.76 MPa (400 psi) for 1.5 h to obtain irreducible water condition. The centrifuge pressure used in the centrifuge experiment, which is an empirical value derived from the repeated experiment. The higher centrifugal force may induce a change in the pore structure [45]. The T 2 spectrum at the water-saturated and irreducible conditions are obtained successively.

Petrology Characteristics
The Chang 7 tight sandstones in the study area are mainly made up of lithic arkose and feldspathic litharenite [12,34]. The content of quartz mainly ranges from about 50% to 60%, except for one sample with a lower quartz content of about 35% (Figure 2a). As a sedimentary debris, feldspar generally have lower content of about 10-15% compared with quartz. The content of carbonate minerals shows large fluctuations, ranging from 5% to 20%, while that of clay minerals is generally high and relatively stable from 15% to 20% (Figure 2a). In addition, skeleton particles are mainly composed of quartz, feldspar, and lithic fragment (Figure 3a,b), while the interstitial material is mainly composed of high content clastic clay matrix, as well as cements dominated by carbonate, silicate, illite, and chlorite minerals (Figure 3c,j-l). Laser grain size analysis shows that the median grain size of most samples mainly ranges from 80 to 100 µm (Figure 2b), which reveals a predominantly fine size fraction in the Chang7 tight sandstones (Figure 3a-f) [46]. Therefore, the Chang 7 tight sandstones are composed of silt fine clastic grains and are characterized by high content of clay minerals or carbonate carbon minerals. Moreover, fine grain structure and high clay content lead to strong compaction and the precipitation of carbonate cements in the pores, which finally leads to complex pore structure and micro-nano pore throat system in the Chang 7 tight sandstones ( Figure 3).  The porosity varies from 4% to 12% and can be divides into two parts, mainly ranging from 4% to 6% and from 8% to 10% (Figure 2c). The permeability shows significant change, mainly ranging from about 0.01 mD to 0.10 mD (Table 1, Figure 2d). Additionally, the lower part of porosity (4-6%) always corresponds to the ultra-low permeability, less than 0.02 mD, indicating that such tight sandstones are generally not effective in the tight oil accumulation [47] and are non-reservoir rocks, such as samples of No.1 to No.4. So: sorting coefficient. P d : displacement pressure. P c50 : median capillary pressure. r max : maximum pore throat radius. S max : maximum mercury saturation. We: mercury withdrawal efficiency. PTR: the pore throat ratio, the average ratio of the pore volume and the throat volume derived from the mercury injection curve and ejection curve, respectively; PTR = S R Smax−S R , S R is the residual mercury saturation in the pores after completely mercury withdrawal. ϕ Hg : the porosity derived from MICP. r 2.5 , r 25 , and r 75 (µm) are the pore-throat radius corresponding to 2.5%, 25% and 75% of mercury injection cumulative saturation, respectively.

Pore Characteristics
Four types of pores are recognized in the Chang 7 tight sandstone reservoirs, including residual intergranular pores, dissolved pores, inter-crystalline pores, and micro-fractures ( Figure 3). Residual primary intergranular pores are relatively larger in size but fewer in number, with pore diameter mainly ranging from 10 to 100 µm. The shape of these pores is usually triangular or polygonal with smooth and straight edges (Figure 3d,e,g) [12]. Fine throats (nanoscale) always occur in the position of grain contacts due to a fierce mechanical compaction and presence of grain coating chlorite (Figure 3d,e,g) [2]. These large residual intergranular pores connected with fine throats can form a special pore network with large pore bodies and small pore throats. Dissolved pores are produced by the dissolution of unstable minerals, such as feldspar and carbonate cement (Figure 3e,h) [48], which are an important component to the pore space of the Chang 7 tight sandstones. These pores are characterized by rather irregular pore surfaces and large pore sizes, generally ranging from several hundred nanometers to 100 µm. These pores are usually connected by fine throats due to a heterogeneity of the grain dissolution. Inter-crystalline pores are widely distributed and generated from the precipitation of authigenic quartz, albite, ankerite, and clay minerals (Figure 3i-l), including illite, chlorite, and mixed-layer illite/smectite, in intergranular pores and dissolved pores. The dimensions of these pores are generally smaller than several microns, and the inter-crystalline pores within authigenic quartz or albite are larger than those within clay minerals ranging within the nanoscale (Figure 3i-l). These pores often work as throats connecting other pores, such as residual intergranular pores, dissolution pores, and relatively large inter-crystalline pores. Micro-fractures are also observed in some samples, with widths ranging from 2 µm to 10 µm, while their lengths were up to 10 mm (Figure 3f). These are significant for the percolation capacity of the tight reservoir due to the connection to large of micro pores.

MICP Curves and Parameters
MICP is a useful tool to obtain the information of pore structure and pore throat distribution (PTD) [9,10,49,50]. MICP curves can directly and visually reflect pore geometry characteristics and are easily used to calculate pore structure parameters and PTD. The main mercury injection capillary pressure of the Chang 7 tight sandstone distributes from 1 MPa to 100 MPa (Figure 4a), indicating that the dominant portion of pore throats is in the nanoscale (0.00735-0.735 µm). The displacement pressure (P d ) is between 0.94 MPa and 6.82 MPa, with an average value of 2.92 MPa, while the median capillary pressure (P c50 ) ranges from 2.9 MPa to 59.51 MPa, with an average value of 16.33 MPa, suggesting that hydrocarbon cannot easily enter into these tight reservoirs without overpressure ( Table 1). In addition, the PTD has a wide distribution, mainly ranging from 0.01 µm to 1 µm (Figure 4b). The maximum pore throat radius (r max ) is between 0.94 µm and 6.82 µm, with an average value of 2.92 µm. The median pore throat radius (r 50 ) ranges from 0.012 µm to 0.284 µm, with an average value of 0.085 µm [47]. The line section slope of the MICP curves is a visual indication of pore throat sorting characteristics that the smaller is the line slope, and the better is the sorting quality, which can be quantified by sorting coefficients (So) of approximately 0.81 to 2.03, with an average So of 1.51 (Table 1). In short, Chang 7 tight sandstone reservoir is characterized by high capillary pressure, a dominant nanoscale pore throat, and poorer sorting.

NMR T2 Spectrum
The NMR T2 spectra of the Chang 7 tight sandstones mainly exhibit the unimodal or bimodal distributions ( Figure 5), which generally occur in sandstone and shale rocks [27,53]. The T2 spectrum mainly distributes from 0.   The mercury intrusion and extrusion curves exhibit significant hysteresis (Figure 4a), indicating that a large amount of injected mercury remains in the pore network after complete mercury withdrawal. This phenomenon is intimately related to the crucial role of the connectivity of the pore network system in filling the pore space with mercury [51]. During the extrusion process, mercury remains trapped in the pore bodies of the inkbottle pores due to the difference in capillary pressure between the pore bodies and pore throats [26,52]. This difference can be roughly defined by the parameter of the pore throat ratio (PTR). The PTR is the average ratio of the pore volume and the throat volume derived from the mercury injection curve and ejection curve, respectively. The PTR ranges from 1.27 to 7.44, with an average of 3.35 (Table 1). Additionally, a large amount of mercury trapped in the pores can also be quantified by the pore structure parameters, e.g., the maximum mercury saturation (S max ) and mercury withdrawal efficiency (We). The maximum mercury saturation (S max ) ranges from 71.83% to 92.46% and its mean is 87.33%. In addition, mercury withdrawal efficiency (We) is between 11.85% and 44.04%, with an average value of 26.75%. It is apparent that about 40% to 80% of mercury remains trapped in the pore network (Figure 4a).

NMR T 2 Spectrum
The NMR T 2 spectra of the Chang 7 tight sandstones mainly exhibit the unimodal or bimodal distributions ( Figure 5), which generally occur in sandstone and shale rocks [27,53]. The T 2 spectrum mainly distributes from 0.1 to 100 ms, showing a unimodal distribution, unimodal distribution with a positive skewness, or bimodal distribution ( Figure 5). According to Equation (4), the T 2 spectra of saturated water implies the PSD of rocks, and the long T 2 relaxation time indicates the large pores and the short T 2 relaxation time indicates the small pores. Therefore, the samples with unimodal distribution would have larger percentage composition of small pores than those with bimodal distribution.
As is shown in Figure 5, the left peak (or the peak of the unimodal distribution) and right peak of the T 2 distribution present at 1 to 10 ms and 10 to 100 ms, respectively. The peak number and location of the T 2 distribution can reflect the pore type of rocks. The left peak is representative of the micro pores mainly composed of clay inter-crystallite pores and micro dissolved pores (Figure 3), while the right peak is indicative of residual primary intergranular pores and some macro pores generated from particle dissolution, which are the dominant contribution to permeability. In addition, the T 2 geometric mean (T 2gm ) ranges from 0.78 to 5.94 ms, with an average value of 3.46 ms [27], which further indicates a tight pore system has been developed in the Chang 7 tight sandstones.
Fluid flow in a porous media is mainly controlled by PSD, interfacial tension, and wettability [6,28,36]. For porous reservoirs, the interfacial tension and wettability changes would significantly affect oil migration and recovery [54][55][56][57]. However, it is generally believed that the interfacial tension and wettability of the core plugs are consistent, considering that they are from the same stratum and contain similar lithology. Therefore, the fluid flow is mainly controlled by pore throat size during centrifugation.
After centrifugation, the right peak is almost completely absent except in sample No. 5 and No. 11, while the left peak remains, indicating that the fluid in the small pores is not easy to flow ( Figure 5). The fluid trapped in the large pores (>10 ms) for No. 5 and No. 11 is probably due to occurrence of a large amount of the complex pore geometry that is composed of large pores connected with fine throats (or ink bottle pores) [27,58]. The irreducible fluid is mainly around 0.1 to 10 ms ( Figure 5), consisting of clay-bound water and capillary-bound water that are limited in clay micro pores and small capillary pores, respectively. This phenomenon is mainly related with the sufficient molecular adsorption force and capillary pressure generated from the pore fluid interaction with the clay mineral surface or pore walls of the rocks. The T 2cutoff is a relaxation time threshold that divides the T 2 distribution into the movable fluid and irreducible fluid, ranging from 0.87 to 7.73 ms, with an average value of 3.00 ms ( Table 2) [27]. The movable water saturation (S m ) of Chang 7 tight sandstone ranges from 32.01% to 84.84%, with an average value of 50.53%, while the irreducible water saturation (S ir ) ranges from 15.06% to 67.99%, with an average value of 49.47% (Table 2). Consequently, the movable water volume is about a half of the total pore volume. This result suggests that these micro pores contribute a significant portion of the storage space but may not be important for oil or gas percolation in tight sandstone reservoirs.   [27]. S ir : irreducible water saturation. S w : movable water saturation. C is a constant conversion coefficient representative of the shape factor (Fs) and surface relaxivity (ρ 2 ). R 2.5 , R 25 , R 50 , and R 75 (µm) are the pore radius corresponding to 2.5%, 25%, 50%, and 75% of cumulative pore volume percentage from NMR, respectively.

Comparison of Pore Volume and Size from MICP and NMR
Petrophysical parameters obtained from MICP and NMR always present disagreement due to the differences in test principles and corresponding petrophysical properties [10,11,17,19,50]. Therefore, the porosity and pore or pore throat size obtained from both MICP and NMR are compared with each other to clarify the difference between two methods in pore space and size characterization.

Porosity
As is shown in Figure 6, there are good correlations between gas-measured porosity (ϕ), MICP porosity (ϕ MICP ), and NMR porosity (ϕ NMR ) despite some differences. ϕ from core analysis is usually considered as the total porosity due to an assumption of gas molecules moving into almost all connected pore spaces. The porosity from MICP is always lower than the total porosity because of incomplete mercury injection, which is mainly caused by limited intruded mercury pressure. The deviation between ϕ and ϕ NMR may be related to several reasons: the presence of paramagnetic minerals [59], gas molecules being smaller than water molecules [19], or external surface water of core plugs. In short, ϕ NMR is much closer to ϕ, and thus can more accurately describe the total pore space of tight sandstone reservoirs when compared to mercury intrusion porosimetry.

Comparison of Pore Volume and Size from MICP and NMR
Petrophysical parameters obtained from MICP and NMR always present disagree ment due to the differences in test principles and corresponding petrophysical propertie [10,11,17,19,50]. Therefore, the porosity and pore or pore throat size obtained from bot MICP and NMR are compared with each other to clarify the difference between two meth ods in pore space and size characterization.

Porosity
As is shown in Figure 6, there are good correlations between gas-measured porosit (φ), MICP porosity (φMICP), and NMR porosity (φNMR) despite some differences. φ from core analysis is usually considered as the total porosity due to an assumption of gas mol ecules moving into almost all connected pore spaces. The porosity from MICP is alway lower than the total porosity because of incomplete mercury injection, which is mainl caused by limited intruded mercury pressure. The deviation between φ and φNMR may b related to several reasons: the presence of paramagnetic minerals [59], gas molecules be ing smaller than water molecules [19], or external surface water of core plugs. In shor φNMR is much closer to φ, and thus can more accurately describe the total pore space o tight sandstone reservoirs when compared to mercury intrusion porosimetry.  Figure 7 shows the method of obtaining the pore radius and pore throat radius cor responding to the different pore volume percentages of the cumulative distribution curve of MICP and NMR. The results are shown in Table 2.
The result shows that the pore radius (R75) and pore throat radius (r75) correspondin  Figure 7 shows the method of obtaining the pore radius and pore throat radius corresponding to the different pore volume percentages of the cumulative distribution curves of MICP and NMR. The results are shown in Table 2.

Pore and Pore Throat Size
Processes 2022, 10, x FOR PEER REVIEW 15 of 25 Figure 7. The method to calculate pore radius and pore throat radius corresponding to 2.5%, 25%, 50%, and 75% of the cumulative pore volume percentage from NMR and MICP.  [47] and R50. r50 are closely equal to R50, indicating R50 can be used in place of r50 to predict permeability [27]. The plot of r25 and R25 (c), r2.5 and R2.5 (d) show that pore radius is much larger than pore throat radius in macropore interval.

Calibration of PSD
In NMR relaxation, the hydrogen atoms in smaller pores experience a greater surface relaxation, and thus relax and decay faster than that in the larger pores. So, the T2 distribution of rocks with water saturation corresponds to the PSD: large pores correspond to a long relaxation time and small pores to a short relaxation time. There was a linear relationship between the T2 value and pore size in single pores of clastic rocks, as Equation (4) verified. Therefore, the T2 distribution can be calibrated to the PSD. Previously, many  Figure 7. The method to calculate pore radius and pore throat radius corresponding to 2.5%, 25%, 50%, and 75% of the cumulative pore volume percentage from NMR and MICP.
The result shows that the pore radius (R 75 ) and pore throat radius (r 75 ) corresponding to the upper quartile provide a good match, except for a slight deviation in some samples (Figure 8a). It is notable that the median pore radius (R 50 ) almost coincides with the median pore throat radius (r 50 ), which is important for accurately predicting permeability by use of R 50 (Figure 8b). The pore radius corresponding to the low quartile (R 25 ) is slightly larger than the pore radius corresponding to the low quartile (r 25 ) (Figure 8c), while the pore radius R 2.5 significantly exceeds the pore throat radius r 2.5 on the location of cumulative percentage of 2.5% (Figure 8d). This indicates that pore size and throat size differ greatly from each other in the large-size pore system. Although there is significant difference between pore and throat in tight sandstones, the medians of the pore and pore throat radius show greatly consistent scalar values, which is highly valuable for the accurate and indirect evaluation of permeability and capillary pressure by use of R 50 .

Calibration of PSD
In NMR relaxation, the hydrogen atoms in smaller pores experience a greater surface relaxation, and thus relax and decay faster than that in the larger pores. So, the T 2 distribution of rocks with water saturation corresponds to the PSD: large pores correspond to a long relaxation time and small pores to a short relaxation time. There was a linear relationship between the T 2 value and pore size in single pores of clastic rocks, as Equation (4) verified. Therefore, the T 2 distribution can be calibrated to the PSD. Previously, many researchers directly overlapped the PTD derived from MICP to the T 2 distribution to obtain the PSD, based on the similarity of PTD and PSD in conventional reservoirs [23,[60][61][62]. However, the PSD and PTD tend to present obvious differences due to the complex pore structure in tight sandstone reservoirs, which makes the reliability of calibrated PSD for this method low or even incorrect. Li et al. [63] found that there was not a noticeable difference within 0.05 µm between pore size and throat size when considering the similarity of pore volumes derived from MICP and nitrogen adsorption. Thus, the T 2 distribution can be converted into the PSD through the "T 2cutoff " method proposed by Yao et al. [19]. Although the method is based on a centrifuge experiment controlled by the Washburn equation, the effect of the significant difference between the pore and pore throat in the large pore system is avoidable to a great extent. The "T 2cutoff " method is as follows: For any relaxation time (T 2i ) in a T 2 relaxation distribution, the corresponding pore size (R i ) can be determined by Equation (4).
where C is a constant conversion coefficient representative of the shape factor (F s ) and surface relaxivity (ρ 2 ). Figure 7. The method to calculate pore radius and pore throat radius corresponding to 2.5%, 25%, 50%, and 75% of the cumulative pore volume percentage from NMR and MICP.  [47] and R50. r50 are closely equal to R50, indicating R50 can be used in place of r50 to predict permeability [27]. The plot of r25 and R25 (c), r2.5 and R2.5 (d) show that pore radius is much larger than pore throat radius in macropore interval.

Calibration of PSD
In NMR relaxation, the hydrogen atoms in smaller pores experience a greater surface relaxation, and thus relax and decay faster than that in the larger pores. So, the T2 distribution of rocks with water saturation corresponds to the PSD: large pores correspond to a long relaxation time and small pores to a short relaxation time. There was a linear relationship between the T2 value and pore size in single pores of clastic rocks, as Equation (4) verified. Therefore, the T2 distribution can be calibrated to the PSD. Previously, many  [47] and R 50 . r 50 are closely equal to R 50 , indicating R 50 can be used in place of r 50 to predict permeability [27]. The plot of r 25 and R 25 (c), r 2.5 and R 2.5 (d) show that pore radius is much larger than pore throat radius in macropore interval.
As a specific T 2i , the T 2cutoff can be used to calculate the conversion coefficient C. Thus, the following formula is converted from Equation (5): where R c is the cut-off pore radius, i.e., the minimal pore radius (µm) for water to discharge at the centrifuge pressure.
In the centrifuge experiment, R c can be obtained from the following formula based on the Washburn equation: 2σ Wr cos θ wr p centr (7) where P centr is the centrifuge pressure in MPa; R c is the cut-off pore radius, the minimal pore radius (µm) for water to discharge at P centr ; θ wr is the contact angle between water molecule and pore surface; and σ wr is the interfacial tension of rocks and water. The σ wr and θ wr of rock to water vary with samples due to the mineral composition difference. They are assigned with values of 0.072 N/m and 0 • according to the fact that the rock is completely water-wet after cleaning [64]. Therefore, the R c that corresponds to T 2cutoff under the centrifuge pressure of 2.76 MPa is 0.05 µm.
The T 2cutoff can be obtained from centrifuge experiment and NMR test. The detailed method to obtain a T 2cutoff is based on Li et al. [27]. Therefore, the PSD from NMR T 2 distribution can be determined according to Equations (6) and (7).

The PSD and PTD
An average conversion coefficient (C av = 0.023 µm/ms) is used when calibrating the PSD from the T 2 distribution. The PSD of the 16 tight sandstone samples is shown in Figure 9. The result shows that the constructive PSD is not always a good match with the PTD determined by MICP, with some of the samples presenting significant difference between the two distributions.  . The NMR pore size distribution (PSD) and MICP pore throat distribution (PTD) of the Chang 7 tight sandstones in the Ordos Basin. An average conversion coefficient (C av = 0.023 µm/ms) is used when the PSD is derived from the T 2 distribution. When comparing to the PTD and PSD, a discrepancy is found. This inconsistency is mainly because MICP quantifies pore throat size distribution, while NMR reveals the pore body size distribution.
The NMR-derived PSD presents a broad distribution with pore radius spanning three orders of magnitude from nanoscale to several microns. The PSD can be mainly divided into three groups based on the distribution shape and petrophysical properties.
The first group (No. 1,No. 2,No. 3,and No. 4) is a unimodal distribution similar to a logarithmic normal distribution, and some of them present a slightly positive skewness, including a weak tail due to the presence of a few large pores (Figure 9). The peak of the PSD is mainly located in the range of 0.02 to 0.04 µm, while the peak of the PTD ranges within 0.02 to 0.06 µm. It is notable that the PSD of this group always coincides very well with the PTD, probably due to the similar pore geometry. The pores of this group are mainly composed of intergranular triangular or sheet-like micro pores (generally less than 0.2 µm). These pores are related to the fierce compaction and/or tight cementation because petrography composition of these tight sandstones is mainly characterized by very-fine grain and high clay, plastic grain, or carbonate content (Figures 2, 3 and 10). Additionally, samples of this group are the extremely tight non-reservoir rock, which are mainly characterized by a very low porosity and an extremely low permeability, generally with a porosity less than 7% and permeability less than 0.015 mD (Table 1, Figure 2). The second group (No. 6,No. 7,No. 10,and No. 12) is a unimodal distribution with a positive skewness (Figure 9). They include an obvious tail or minor peak which is mainly related to large pores ( Figure 9). The dominant peak is located at 0.04 µm, while its right tail is approximately 0.2 to 3 µm and reveals a second minor population accounting for approximately 20% to 30% of the total pore volume. The pores of this group are composed of a large number of small pores and a small number of large pores. These large pores are probably generated from the limited dissolution. Additionally, the porosity and permeability of this sandstone is generally about 7% to 11% and 0.02 to 0.03 mD, respectively. However, the sample No.8 presents a logarithmic normal distribution (unimodal), with a maximum pore radius exceeding 1 µm and a peak value of approximately 0.1 µm. A large number of clay inter-crystalline pores are observed under SEM (Figure 3j) and the macro petrophysical properties are closer to the second group.
The third group (No. 5,No. 9,No. 11,No. 13,No. 14,No. 15,and No. 16) presents a typical bimodal distribution (Figure 9). The left peak ranges within 0.1 µm with the peak located in 0.02 µm. The right peak distributes from 0.1 to 10 µm with the peak location mainly ranging from 0.2 to 1 µm. Obviously, the pore system of this group possesses two modes, with one corresponding to clay inter-crystalline pores dominated by micro pores and another corresponding to macro pores mainly composed of residual primary intergranular pores and large dissolved pores or inter-crystalline pores (Figures 3 and 10). The two pore types may form the pore geometry of large pores connecting with narrow throats, which further expounds the main reason of the difference between PSD and PTD. The porosity and permeability of the third group is generally more than 9% and 0.04 mD, respectively.
The PTD from MICP always presents a unimodal distribution with a negative skewness including a left tail (Figure 9). In particular, the sample of No.16 presents a second population with small pore throats (<0.1 µm). The PTD is characterized by leptokurtosis and high amplitude, which indicates that the main pore throat presents a concentrated distribution, significantly different from the low-amplitude and platykurtic bimodal distribution of the PSD from NMR. In fact, it is the special pore structure where the finer throats with centralized distribution control much of the larger pores that results in the main difference between the PSD and PTD in tight sandstones. The pore network, which is composed of large pore bodies and fine pore throats, is a system of storage space and percolation pathways for fluids. Macro pores of residual intergranular pores or dissolution pores comprise pore bodies that are connected by the fine passages composed of necking throats and clay inter-crystalline micro pores.
It is noteworthy that two pore network models are identified in the Chang 7 tight reservoir of the study area ( Figure 10). One is the cylindrical pore model (CPM) composed of the flaky throat, curved flaky throat, and tubular throat, while the other is the spherecylindrical pore model (SPM), also known as ink-bottle pores [58], composed of large pore bodies connecting necking throats.
The flaky throat or curved flaky throat originates from the close contact of grains (line contact or even concavo-convex contact) under fierce compaction, and tubular throats are widely distributed in clay inter-crystalline pores and micro dissolved pores ( Figure 10A). The tight sandstones of these throat types tend to have a very fine throat (generally, radius <0.2 µm). Therefore, the samples that are dominated by the CPM are always related to the very tight non-reservoir rock in the study area (e.g., No. 1 to No. 4). However, the pore network of the SPM dominant is composed of large amounts of necking throats, characterized by large pores connecting with narrow throats ( Figure 10A). It is usually connected with the tight reservoirs of relatively high porosity and low permeability (e.g., No. 5,No. 11,and No. 13 to No. 16). Necking throats are generated from the spot contact of grains caused by fierce compaction or grain coating chlorite. Rocks of this type are characterized by interstitial material of low content or dissolution of material and grains, which is favorable to residual intergranular pores, dissolution pores, and relatively large inter-crystalline pores. Additionally, clay inter-crystalline pores are also one of the most important components in the samples of SPM dominant. These pores can act as throats, form ink-bottle pores in combination with macro pores and then control the flow percolation. The abundant micro pores in clay minerals are a main contributor to the left peak of NMR-derived PSD and the fine tail of the PTD from MICP.
These two pore network models make a vast difference between the PSD and PTD. As is discussed above, the PSD and PTD are in a good agreement in the CPM dominant, whereas they show a large discrepancy in the SPM dominant (Figures 9 and 10C).

Difference of Pores and Throats
The CPM is assumed to study pore geometry of porous material in mercury porosimetry, but in fact, as discussed above, the network of pores and throats in rocks is more similar to the SPM [10,65], also referred to as ink-bottle pores [58] (Figure 10). The ink-bottle pores, which comprise a serial of narrow throats (small pores) connecting to the expanded pore bodies (large pores), are widely distributed in reservoir rocks, especially in tight sandstone reservoirs. Therefore, the real pore geometry is quite different from the cylinder pore assumption, and this assumption can lead to significant differences between the MICP analysis and reality. In the MICP experiment, mercury gradually breaks through the narrow throats with increasing injection pressure, and then enters into the pore bodies connected by these throats. The injected mercury volume under a certain pressure is equal to the total volume of these pores and throats. However, the pore throat radius controlled by this injected pressure equates to the radius of the largest entrance of a pore void and is smaller than the real pore body radius ( Figure 10).
The connectivity of the pore system of ink-bottle pores deteriorates with increasing the PTR in reservoir rocks, and thus the similarity of the PTD and PSD decreases. That is, an increase in PTR reflects a decrease in the volume of cylindrical pores or an increase in the volume of ink-bottle pores (sphere-cylindrical pores), which makes reservoir rocks a transformation from the CPM dominant to the SPM dominant and further presents more significant differences between the two distribution curves.
Euclidean distance is introduced to define the similarity of the two curves of the PTD and PSD. Euclidean distance is a numerical description of the total distance between points or vectors in space [66] and is widely applied for quantification of similarity or for classification in many research fields [67][68][69]. Therefore, it can be used to characterize the similarity of the two curves: the smaller the Euclidean distance is, the higher the similarity between the curves is [70]. Euclidean distance can be expressed as follows [66]: where D edu is the Euclidean distance, x i is the pore radius or pore throat radius, and y NMR i and y MICP i are the incremental porosity of NMR-derived PSD and MICP-PTD in fraction, respectively.
The result shows that the Euclidean distance increases with the increasing PTR of the Chang 7 tight sandstones (Figure 11), which indicates that the difference in pore body and throat size controls the similarity between the PTD and PSD and further reflects the phenomenon of "large pores connecting with narrow throats" in tight sandstones. In the MICP experiment, this phenomenon makes the volume of large pores connected by narrow throats attached to the corresponding throat volume, with the PTD visually shifting the smaller pore size direction compared to the authentic pore size distribution [58]. As discussed above, it is not rigorous or even wrong to directly overlap the curve of the T 2 distribution to the curve of PTD to obtain the PSD of NMR. In addition, Euclidean distance can effectively evaluate the similarity of the PTD obtained from MICP and NMR-derived PSD and indicate pore network characteristics (CPM and SPM) in tight sandstones.
3 Figure 11. The relationship between the Euclidean distance and pore throat ratio (PTR).

Conclusions
In this study, based on the analysis of petrology and pore characteristics, the MICP and NMR techniques were combined to evaluate the PSD and PTD in the Chang 7 tight sandstone reservoirs, and the CPM and SPM (pore network models) were proposed to reveal their differences. The conclusions are as follows: High capillary pressure, dominated nanoscale pore throat, and low mercury withdrawal efficiency indicate that ink-bottle pores and necking throats are developed in the Chang 7 tight sandstones. The micro pores filled with irreducible fluid contribute a significant portion of the storage space but may not be important for oil or gas percolation in tight sandstone reservoirs.
The comparison of pore and throat radius corresponding to different pore volume percentage suggests that pore and throat size dominantly differ in the large-size pore compartment. The constructive PSD does not always match well with the PTD determined by MICP. The main pore throat presents a centralized distribution, while the PSD of NMR is a low-amplitude and platykurtic bimodal distribution. In very tight core samples, the PSD always coincides very well with PTD probably due to the pore network dominated by cylindrical pores, while the samples of high porosity and permeability present a significant difference between the two distributions due to the pore network dominated by the spherecylindrical pores.
The CPM and SPM effectively explain the difference between PSD and PTD of the Chang 7 tight sandstones, providing us a good insight into the heterogeneity of pore geometry (pore and throat). This is of great significance for the application of MICP and NMR techniques to evaluate the pore structure and petrophysical properties of tight sandstone reservoirs. Data Availability Statement: The data presented in this study are available on request from the corresponding author. The data are not publicly available due to patent protection.