Utilizing Differences in Mercury Injection Capillary Pressure and Nuclear Magnetic Resonance Pore Size Distributions for Enhanced Rock Quality Evaluation: A Winland-Style Approach with Physical Meaning

: Pore structure is a fundamental parameter in determining the hydrocarbon storage capacity and flow characteristics of a reservoir. Mercury injection capillary pressure (MICP) and nuclear magnetic resonance (NMR) are two commonly utilized techniques for characterizing rock pore structures. However, current studies indicate that disparities in testing methodologies due to distinct physical characteristics lead to a partial misalignment in pore size distributions. We conducted MICP (dynamic) and NMR (static) experiments on eight tight sandstone and eight shale samples and proposed a method to utilize information from the differences in MICP and NMR pore size distributions, aiming to enhance the accuracy of rock quality analysis. We observed that in rock cores where large pores are interconnected with smaller pore throats, MICP tends to overestimate the proportion of these smaller pores and underestimate the larger ones. Furthermore, we integrated information from both dynamic and static experimental processes based on physical significance and found that the fitting accuracy of the newly proposed method is superior to the Winland r 35 equation. Compared to the Winland r 35 equation, our new method significantly improves fitting accuracy, increasing the R-squared value from 0.46 to 0.93 in sandstones and from 0.80 to 0.87 in shales. This represents a potential high-precision, comprehensive tool for rock quality analysis, offering a new perspective for an in-depth understanding of rock properties.


Introduction
Tight oil and gas, as crucial components for increasing petroleum resources [1,2], are important resources in oil and gas exploration and development.However, the development of tight reservoirs is challenging due to their poor petrophysical properties, strong heterogeneity, and ultra-low permeability [3,4].The porosity, permeability, pore types and micro-pore structures in tight reservoirs play a crucial role in determining the fluid flow and storage capacity of a rock, which are critical factors in rock quality analysis [5][6][7].These fundamental petrophysical studies are commonly conducted using the Winland method and its derivatives [8,9].Hence, an accurate understanding of the micro-pore and throat structures in tight reservoirs will benefit the study of their storage capacity and flow patterns.
At present, micro-CT imaging [10][11][12], mercury injection capillary pressure (MICP) [13][14][15] and nuclear magnetic resonance (NMR) [16][17][18][19] experiments, as methods capable of accurately reflecting micro-to-nano scale pore features, are extensively employed for microscopic porosity studies in tight reservoirs.All techniques have certain limitations.The micro-CT imaging method utilizes 2D CT scan data to construct 3D visualizations of pore structures within core samples.However, high-resolution CT equipment is expensive, and completing 3D imaging often requires multiple scans, which will increase the cost and duration of research.The minimum throat radius measured using MICP is contingent upon the injection pressure; a heightened injection pressure allows for the measurement of a smaller throat radius.Additionally, the permeability of the core can also be predicted based on the capillary pressure curve obtained from MICP [20].Yet, excessive injection pressure can easily damage the original pore structure of the sample.In addition, 100% mercury saturation is hard to reach since the mercury is not in the wetting phase [6,21].
NMR experiments can effectively detect non-connected pores and nano-scale pores in core samples [22].Analyzing the distribution and peak positions of the transverse relaxation time, T 2 , enables the determination of pore types and their size distribution.Additionally, the continuity of NMR signals in the T 2 relaxation time distribution spectrum provides insights into the connectivity between pores and fractures [23][24][25].However, NMR experiments cannot intuitively reflect pore diameter size.Fuyong et al. amalgamated findings from NMR and MICP methods and uncovered that the relationships between NMR relaxation time T 2 and pore size are more in accordance with the power function relations than the linear relation [26].Hugh and Andrew utilized accessibility functions for cubic lattices, combined with coordination numbers obtained from the MICP method, to determine surface relaxivity and percolating cluster size [27].Chaohui et al. identified a novel method that utilizes the transverse relaxation time value (T 2c ) of the right endpoint of negative intervals and the corresponding centrifugal radius (r c ) to determine the coefficient C, enabling the conversion of NMR T 2 relaxation times into pore sizes [28].Overall, integration with other experiments or the use of empirical parameters is necessary to convert the T 2 relaxation time distribution spectrum into a pore size distribution diagram.
In comparative analyses conducted by some researchers on the curves obtained from MICP and NMR techniques, a degree of discrepancy between the two methods was observed, with the pore size distributions revealed by each not aligning perfectly.
As shown in Table 1, researchers have observed in various porous media that there exists a crosspoint in the longer T 2 relaxation times when comparing NMR and MICP experimental methods.The MICP method tends to underestimate the proportion of macropores where T 2 is greater than the crosspoint, while it overestimates the proportion of mesopores.We have prepared a schematic diagram to depict this phenomenon, as illustrated in Figure 1.
Table 1.The discrepancy between MICP and NMR methods.
Some scholars believe that NMR measures the pore size distribution (PSD), while MICP assesses the pore throat distribution (PTD) [34][35][36].Mercury injection is a dynamic process with recorded pressure, while NMR PSD is a static process where brine pre-saturates the core to almost 100%.NMR PSD could reflect all the pores saturated with brine, while MICP is more sensitive to the throat due to the ink bottle effect [37,38].Multiple researchers have reported that MICP overestimates the percentage of the meso-pores, corresponding to the relatively low capillary pressure range [32,39].This phenomenon reflects the detailed pore throat matching pattern and is closely linked to the permeability of the core.We believe that this information could be used to gain a better understanding of pore structure.Figure 1 illustrates a schematic where the MICP method overestimates meso-pores and underestimates macro-pores.The overestimated part is almost equal to the underestimated part.The occurrence of this phenomenon is commonly ascribed to "ink-bottle" phenomenon [32].
Some scholars believe that NMR measures the pore size distribution (PSD), while MICP assesses the pore throat distribution (PTD) [34][35][36].Mercury injection is a dynamic process with recorded pressure, while NMR PSD is a static process where brine pre-saturates the core to almost 100%.NMR PSD could reflect all the pores saturated with brine, while MICP is more sensitive to the throat due to the ink bottle effect [37,38].Multiple researchers have reported that MICP overestimates the percentage of the meso-pores, corresponding to the relatively low capillary pressure range [32,39].This phenomenon reflects the detailed pore throat matching pattern and is closely linked to the permeability of the core.We believe that this information could be used to gain a better understanding of pore structure.
In this study, we focused on Tuha tight sandstones and Hechuan shales, performing both NMR and MICP assays on a collection of 16 tight reservoir core specimens.Firstly, the observed difference between mercury injection and NMR PSD is consistent with the reported cases [29,30,34].Furthermore, we integrated information from both methods to explain the reasons behind these discrepancies.Secondly, we proposed a rock quality assessment method to utilize this information from such differences.
In the Winland r35 method, the percolation threshold [40] is determined based on statistical principles.Although our method is also based on the principle of the Winland percolation threshold, our percolation threshold is determined based on the phenomenon of differences in pore size distribution between MICP and NMR, as demonstrated in Table 1.Our defined percolation threshold originates from the difference between MICP and NMR; specifically, it is determined by smaller throats that control the connectivity of larger pores.By applying Winland fitting results that have actual physical significance, we found it to be superior to fitting results based on statistical principles.Our innovation lies in utilizing the differential information between dynamic (NMR) and static (MICP) pore size distributions to find a percolation threshold that can be applied to the Winland method, providing a more effective tool for the analysis of rock pore structures.In this study, we focused on Tuha tight sandstones and Hechuan shales, performing both NMR and MICP assays on a collection of 16 tight reservoir core specimens.Firstly, the observed difference between mercury injection and NMR PSD is consistent with the reported cases [29,30,34].Furthermore, we integrated information from both methods to explain the reasons behind these discrepancies.Secondly, we proposed a rock quality assessment method to utilize this information from such differences.

NMR signal intensity
In the Winland r 35 method, the percolation threshold [40] is determined based on statistical principles.Although our method is also based on the principle of the Winland percolation threshold, our percolation threshold is determined based on the phenomenon of differences in pore size distribution between MICP and NMR, as demonstrated in Table 1.Our defined percolation threshold originates from the difference between MICP and NMR; specifically, it is determined by smaller throats that control the connectivity of larger pores.By applying Winland fitting results that have actual physical significance, we found it to be superior to fitting results based on statistical principles.Our innovation lies in utilizing the differential information between dynamic (NMR) and static (MICP) pore size distributions to find a percolation threshold that can be applied to the Winland method, providing a more effective tool for the analysis of rock pore structures.

Rock Samples
MICP and NMR tests were conducted on rock core samples sourced from Jurassic wells in the Tuha Santanghu basin [41], a tight oil production region [42], and the Permian wells in the Sichuan Hechuan-Tongnan block [43], a shale gas production region [44].Specifically, 8 sandstone samples from three distinct wells in the Tuha oil field and 8 shale samples from four different wells in the Hechuan-Tongnan block were analyzed; the representative samples are shown in Figure 2.

Rock Samples
MICP and NMR tests were conducted on rock core samples sourced from Jurassic wells in the Tuha Santanghu basin [41], a tight oil production region [42], and the Permian wells in the Sichuan Hechuan-Tongnan block [43], a shale gas production region [44].Specifically, 8 sandstone samples from three distinct wells in the Tuha oil field and 8 shale samples from four different wells in the Hechuan-Tongnan block were analyzed; the representative samples are shown in Figure 2. In the Tuha oil field, wells S1-2, S1-3, and S1-4 contained conglomerate sandstones, while the remaining samples from this field were identified as tight sandstones.The dominant minerals in these samples were quartz and feldspar, accompanied by trace amounts of clay minerals and dolomite.The helium porosity of these sandstone samples ranged In the Tuha oil field, wells S1-2, S1-3, and S1-4 contained conglomerate sandstones, while the remaining samples from this field were identified as tight sandstones.The dominant minerals in these samples were quartz and feldspar, accompanied by trace amounts of clay minerals and dolomite.The helium porosity of these sandstone samples ranged between 13.66% and 15.88%, with permeabilities spanning from 1.06 mD to 8.20 mD.For the samples from the Hechuan block, an average helium porosity of 6.86% was recorded, with an average permeability value of 0.0394 mD.Detailed physical properties of these samples are presented in Table 2.

Experimental Setups
The NMR measurements were conducted using a MesoMR12-060H-I instrument from Suzhou Niumag Analytical Instrument Corporation (Suzhou, Jiangsu, China).The magnetic field strength of the instrument is 0.3 T, and the spectrometer frequency is 12 MHz.

Experimental Procedures 2.3.1. Permeability and Porosity Measurements
The porosity of all 16 core samples was determined using the helium porosity measurement method.The permeability of sandstone was measured using the steady-state method, while the permeability of shale was determined using the pulse decay method.Specific measurement steps refer to the Chinese National Standards "GB/T 34533-2023 Determination of porosity, permeability, and saturation of shale [45]".

NMR and MICP Measurements
The NMR method primarily involves scanning the core for T 2 spectra to detect hydrogen nuclei in the core, thereby determining the volume of fluid within the core.The specific experimental procedure follows the standards of the Chinese petroleum and natural gas industry: Chinese oil and natural gas industry standard "SY/T 6490-2023 Specification for laboratory measurement of NMR parameter of rock samples [46]".
The MICP method involves applying pressure to liquid mercury, allowing it to overcome capillary resistance and enter the pores.Then, based on the fraction of pore volume filled with mercury and the corresponding pressure, the pore size distribution of the core is obtained.The specific experimental procedure follows the Chinese National Standard "GB/T 29171-2023 Rock capillary pressure measurement [47]".
The primary signals detected using NMR in the rock core were mainly water signals (with shales containing kerogen, asphaltene, and structural water).The maximum mercury intrusion pressure for the MICP could reach up to 200 MPa.The pore throat distribution obtained from MICP and the T 2 relaxation time distribution from the NMR test can be further analyzed to examine the fluid distribution in different pore sizes of the rock core.The specific experimental steps are shown in Figure 3.
The NMR method primarily involves scanning the core for T2 spectra to detect hydrogen nuclei in the core, thereby determining the volume of fluid within the core.The specific experimental procedure follows the standards of the Chinese petroleum and natural gas industry: Chinese oil and natural gas industry standard "SY/T 6490-2023 Specification for laboratory measurement of NMR parameter of rock samples [46]".
The MICP method involves applying pressure to liquid mercury, allowing it to overcome capillary resistance and enter the pores.Then, based on the fraction of pore volume filled with mercury and the corresponding pressure, the pore size distribution of the core is obtained.The specific experimental procedure follows the Chinese National Standard "GB/T 29171-2023 Rock capillary pressure measurement [47]".
The primary signals detected using NMR in the rock core were mainly water signals (with shales containing kerogen, asphaltene, and structural water).The maximum mercury intrusion pressure for the MICP could reach up to 200 MPa.The pore throat distribution obtained from MICP and the T2 relaxation time distribution from the NMR test can be further analyzed to examine the fluid distribution in different pore sizes of the rock core.The specific experimental steps are shown in Figure 3. (1) All samples underwent an extensive cleaning process using Dean-Stark apparatus.
After cleaning, the core samples were dried at 80 °C for 72 h.The status of the heated samples was ascertained using the T2 relaxation time distribution spectrum.If the T2  (1) All samples underwent an extensive cleaning process using Dean-Stark apparatus.
After cleaning, the core samples were dried at 80 • C for 72 h.The status of the heated samples was ascertained using the T 2 relaxation time distribution spectrum.If the T 2 signal intensity is high, an additional 72 h of vacuum heating is required until the T 2 signal intensity becomes constant.(2) After confirming that the rock core is thoroughly cleaned, a piece is cut for the MICP experiment with a thickness of 2 cm.The experimental maximum injection pressure is selected based on the lithology.The compressive strength of tight sandstone is usually higher than that of shale.Therefore, we have set the maximum injection pressure for tight sandstone and shale at 200 MPa and 100 MPa, respectively.(3) The clean rock core is placed in a container and subjected to evacuation treatment for 24 h.Subsequently, simulated formation brine is injected into the container, and the pressure is gradually increased to 60 MPa using the step-wise pressure build-up method [48] at a rate of 5 MPa/h.Water porosity is compared with helium porosity to confirm full saturation of the rock core (with an error of less than 2% [49]).The T 2 relaxation time distribution spectrum of the rock sample is then measured.(4) Based on the cumulative frequency distribution from both MICP and the NMR T 2 relaxation time distribution spectrum, translate the T 2 relaxation times into a pore size distribution.

NMR Pore Size Distribution Calibration
The techniques for measuring nuclear magnetic resonance (NMR) spectroscopy enable the rapid acquisition of reservoir properties, such as porosity, permeability, oil saturation, and water saturation, by analyzing the volume, properties, and NMR T 2 relaxation time distribution spectrum characteristics of fluids within rock core pores.The transverse relaxation time, T 2 , of the fluid in the rock core can be determined via surface relaxation, bulk relaxation, and diffusion relaxation [50,51], as illustrated in Equation ( 1): where T 2 is the transverse relaxation time in milliseconds (ms), T 2S denotes the surface relaxation time in ms, T 2B stands for the bulk relaxation time in ms, and T 2D represents the diffusion relaxation time in ms.Kleinberg et al. discovered that in scenarios where the pore fluid is water, the contribution of T 2B is typically insignificant.Conversely, in cases where the pore fluid consists of a viscous oil or a solution with a high concentration of paramagnetic ions, bulk relaxation mechanisms may predominate [52].When the pore fluid is a single-phase wetting fluid, the bulk relaxation time T 2B is much larger than the surface relaxation time T 2S , which can be neglected in Equation (1).In an NMR system utilizing a homogeneous magnetic field, the effect of diffusion relaxation becomes negligible.Furthermore, when the gradient of the magnetic field is minimal and the echo time in the Carr-Purcell-Meiboom-Gill (CPMG) sequence is adequately brief (less than 1 ms), diffusion effects can also be effectively ignored [53].The expression for the transverse relaxation time, T 2 , of porous media can then be expressed to Equation ( 2), as per references [31,54] where ρ 2 is the transverse relaxation rate, which characterizes the strength of fluid relaxation at the pore surface, in µm/ms.S/V denotes the specific surface area of a single pore, in µm 2 /µm 3 .Equation ( 2) suggests that the transverse relaxation time T 2 is related to the specific surface area of the porous medium.If we simplify the pore structure to be either spherical or cylindrical, then Equation ( 2) can be transformed into Equation (3): where F S is the pore shape factor (F S is 3 for spherical pores and 2 for tubular pores, dimensionless), and r represents the pore radius, measured in micrometers (µm).According to Equation (3), the transverse relaxation time T 2 of the rock core is linearly proportional to pore radius r.However, after extensive experimental matching, it has been observed that due to the complexity of the formation pore structure, T 2 is proportional to the pore radius r with a power index (n) [26,33,55].Therefore, Equation ( 3) is modified to obtain Equations ( 4) and ( 5): In experimental settings, obtaining definitive values for ρ 2 and F S is complex.Consequently, converting the transverse relaxation time T 2 distribution into a pore diameter distribution curve by determining the values of ρ 2 and F S presents significant challenges.By referencing the T 2 distribution diagram of the 100% water-saturated rock core and the high-pressure mercury injection pore throat distribution curve, a conversion between T 2 distribution and pore throat radius can be achieved.Let C = (ρ 2 F S ) In porous rock cores, due to the varied sizes of the pores and throats, high-pressure mercury intrudes preferentially into the larger pores.As the pressure escalates, it gradually permeates into the smaller-sized pores.Given that the MICP method has a pressure ceiling, pores below these thresholds cannot be discerned.On the other hand, the T 2 relaxation time distribution spectrum of rock cores saturated with water at 60 MPa can reflect information on almost all the pores present within the rock core.Directly juxtaposing the MICP curve with the NMR pore size distribution curve can lead to significant deviations.
To minimize the discrepancies between the pore size distribution curves obtained from MICP and NMR, one should begin by plotting the cumulative distribution curves for both methods.Based on these cumulative curves, calculate the radii of the pore throats from mercury intrusion and the T 2 relaxation times corresponding to the same cumulative frequencies.Subsequently, by determining the relationship between these two sets of values, one can derive the constants C and n, facilitating the conversion between the T 2 relaxation time and the pore throat radius.

T 2 Relaxation Time Distribution Spectrum
In the state where the sandstone core is fully saturated with formation brine (TDS: 4500 ppm), the characteristics of the NMR T 2 relaxation time distribution spectrum distribution can effectively elucidate the distribution features of its pore throats.The greater the number of peaks in the spectrum, the more complex the distribution of the pore throats [56].Concerning the Tuha oilfield, the fully water-saturated T 2 spectra of the eight tight sandstones are shown in Figure 4. Based on the T 2 spectra, the sandstone cores can be categorized into two types: monomodal (comprising S1-1, S1-2, S2-5, S2-6, S3-7, and S3-8) and bimodal (S1-3 and S1-4).The six monomodal cores demonstrate a wide distribution range for T 2 relaxation time, with nuclear magnetic signals mainly detected from 0.03 ms to 1000 ms.However, their peak values are relatively concentrated, primarily distributed between 1.5 and 5 ms, indicating that they predominantly feature micropores.The T 2 spectra of the two bimodal cores are strikingly similar, both showcasing a left peak with a signal intensity slightly higher than the right one, and the two peaks are continuous, suggesting that the reservoir's overall pore throat distribution is continuous.Both bimodal cores exhibit a left peak corresponding to a T 2 relaxation time of 0.8 ms, which is lower than the peak T 2 time observed in the monomodal state.The right peak's T 2 relaxation time is 20 ms, aligning with the peak T 2 time of the monomodal state.It is noteworthy that the porosity distribution range for these eight cores is between 13.66% and 15.88%; however, there is a considerable variation in the total amount of nuclear magnetic signals.This variation is not due to errors in core saturation but is attributable to the disparities in the core volume, which leads to significant differences in signal magnitude.
Figure 5 illustrates the 100% formation brine (TDS: 174,000 ppm) saturated T 2 relaxation time distribution spectra of shales from eight different core samples of four wells in the Hechuan area.From the T 2 spectra, it is evident that the T 2 distribution of the Hechuan shale primarily spans from 0.02 ms to 1000 ms.The T 2 relaxation time distribution spectra of cores H1-2, H2-4, and H3-5 exhibit a distinct bimodal distribution.Notably, core H2-3 is considerably distinct from the other seven cores, with its signal peak T 2 relaxation time being 0.6 ms, which is lower than the rest.Moreover, the peak position of the H2-3 core has shifted significantly to the left compared to the other cores, suggesting that this specific core mainly comprises smaller pore sizes than the others, which implies a relatively lower permeability.Consulting the physical properties of the eight Hechuan shale samples (Table 2), it is evident that the H2-3 core exhibits the lowest permeability amongst the eight, measuring 0.0065 mD, which aligns with the T 2 spectra measurements.
than the peak T2 time observed in the monomodal state.The right peak's T2 relaxation time is 20 ms, aligning with the peak T2 time of the monomodal state.It is noteworthy that the porosity distribution range for these eight cores is between 13.66% and 15.88%; however, there is a considerable variation in the total amount of nuclear magnetic signals.This variation is not due to errors in core saturation but is attributable to the disparities in the core volume, which leads to significant differences in signal magnitude.Figure 5 illustrates the 100% formation brine (TDS: 174,000 ppm) saturated T2 relaxation time distribution spectra of shales from eight different core samples of four wells in the Hechuan area.From the T2 spectra, it is evident that the T2 distribution of the Hechuan shale primarily spans from 0.02 ms to 1000 ms.The T2 relaxation time distribution spectra of cores H1-2, H2-4, and H3-5 exhibit a distinct bimodal distribution.Notably, core H2-3 is considerably distinct from the other seven cores, with its signal peak T2 relaxation time being 0.6 ms, which is lower than the rest.Moreover, the peak position of the H2-3 core has shifted significantly to the left compared to the other cores, suggesting that this specific core mainly comprises smaller pore sizes than the others, which implies a relatively lower permeability.Consulting the physical properties of the eight Hechuan shale samples (Table 2), it is evident that the H2-3 core exhibits the lowest permeability amongst the eight, measuring 0.0065 mD, which aligns with the T2 spectra measurements.Dhananjay pointed out that the horizontal anisotropy of porous media and the chemical reaction of fluid and core in the reservoir will affect core permeability.Yadav pointed out that the horizontal anisotropy of porous media and the chemical reaction of fluid and core in the reservoir will affect core permeability [57,58].
Further observation reveals a trend in the NMR signal peak corresponding to T2 relaxation times for the following core samples: H2-3 < H4-8 < H3-6 ≈ H1-2 ≈ H4-7 < H1-1.This pattern aligns with the distribution of core permeabilities, indicating that for the Hechuan shale samples, as permeability increases, the peak value of T2 also rises.However, a notable exception is observed in the case of the H3-5 sample.Even though H3-5 has the highest permeability of all the samples, clocking in at 0.0753 mD, the NMR experiment did not display the longest T2 relaxation time.On the contrary, for core H2-4, it can be observed that the T2 relaxation time distribution spectrum has a bimodal shape, and the left peak is about 0.18 ms, indicating that this core has lower permeability.
From Figure 5, it can be observed that although the curve of the H3-5 core exhibits a lower peak T2 value, its signal distribution in the short T2 range from 0.02 to 0.3 ms is Dhananjay pointed out that the horizontal anisotropy of porous media and the chemical reaction of fluid and core in the reservoir will affect core permeability.Yadav pointed out that the horizontal anisotropy of porous media and the chemical reaction of fluid and core in the reservoir will affect core permeability [57,58].
Further observation reveals a trend in the NMR signal peak corresponding to T 2 relaxation times for the following core samples: H2-3 < H4-8 < H3-6 ≈ H1-2 ≈ H4-7 < H1-1.This pattern aligns with the distribution of core permeabilities, indicating that for the Hechuan shale samples, as permeability increases, the peak value of T 2 also rises.However, a notable exception is observed in the case of the H3-5 sample.Even though H3-5 has the highest permeability of all the samples, clocking in at 0.0753 mD, the NMR experiment did not display the longest T 2 relaxation time.On the contrary, for core H2-4, it can be observed that the T 2 relaxation time distribution spectrum has a bimodal shape, and the left peak is about 0.18 ms, indicating that this core has lower permeability.
From Figure 5, it can be observed that although the curve of the H3-5 core exhibits a lower peak T 2 value, its signal distribution in the short T 2 range from 0.02 to 0.3 ms is minimal, suggesting the least presence of micro-pores.Furthermore, its signal quantity in the medium to long T 2 range from 20 to 600 ms is substantially greater than other cores, indicating a greater number of macro-pores.These two factors might contribute to the elevated permeability observed in core H3-5.

NMR Pore Size Distribution
Porous media exhibit a wide range of pore throat sizes.In MICP, external pressure is gradually increased on a rock sample to force mercury into its pores, thereby obtaining a distribution of pore sizes.In contrast, NMR detects the magnetic resonance signals from hydrogen nuclei within the rock, providing insight into its pore size distribution.Importantly, NMR is capable of effectively representing the entire volume of pores in the rock, including disconnected pores.However, the PSD from NMR requires assistance from MICP for calibration.
To ensure precision in the calculations, it is necessary to remove the influence of the pockmark effects from the MICP results.Hence, only portions of the T 2 relaxation time distribution spectrum that match the MICP pore throat radius distribution are chosen for comparison.At points in the NMR T 2 cumulative distribution curve that coincide with the cumulative distribution frequency of the MICP pore throat curve, multi-point sampling is executed [59].Using sandstone S1-1 and shale H3-5 as examples, the cumulative distribution curves for NMR T 2 relaxation time and MICP pore throat radius are illustrated in Figure 6.Subsequently, based on the cumulative distribution frequencies from both testing methodologies, the relationship between the rock's T 2 relaxation time and pore throat radius is determined, as presented in Figure 7. comparison.At points in the NMR T2 cumulative distribution curve that coincide with the cumulative distribution frequency of the MICP pore throat curve, multi-point sampling is executed [59].Using sandstone S1-1 and shale H3-5 as examples, the cumulative distribution curves for NMR T2 relaxation time and MICP pore throat radius are illustrated in Figure 6.Subsequently, based on the cumulative distribution frequencies from both testing methodologies, the relationship between the rock's T2 relaxation time and pore throat radius is determined, as presented in Figure 7.  Based on the fitting results from Figure 7, the conversion coefficients C and n values correlating the NMR T 2 relaxation time and pore throat radius for the sandstone S1-1 and shale H3-5 samples were derived.This pore size matching method has received consensus among multiple researchers [34,60,61].By method from Section 2.3.3, the C and n values for the 8 sandstone samples from Tuha and the 8 shale samples from Hechuan were determined.The specific conversion coefficients are listed in Table 3.Based on the fitting results from Figure 7, the conversion coefficients C and n correlating the NMR T2 relaxation time and pore throat radius for the sandstone S1 shale H3-5 samples were derived.This pore size matching method has received con among multiple researchers [34,60,61].By method from Section 2.3.3, the C and n for the 8 sandstone samples from Tuha and the 8 shale samples from Hechuan we termined.The specific conversion coefficients are listed in Table 3.

Comparative Analysis of Sandstone Pore Throat Distributions
Using the conversion coefficients C and n and Equation ( 6), the transverse relaxation time T 2 was converted into a pore size distribution.As illustrated, for core S1-1, a comparison chart of the pore throat distribution derived from the Tuha sandstone's NMR T 2 relaxation time distribution spectrum and the MICP pore throat distribution was constructed, as depicted in Figure 8.
From Figure 8, it can be observed that the signal frequency curves of pore size distribution converted from both MICP and NMR do not align perfectly.Comparing the two methods, it is noticeable that the NMR-detected signal frequency is higher than that of the MICP method for pores smaller than 0.0037 µm (micropores).This discrepancy is attributed to the sandstone mercury injection pressure set at 200 MPa, which is incapable of detecting pores smaller than 0.0037 µm.However, NMR can detect all hydrogen-containing fluid volumes in the rock sample; hence, for pores smaller than 0.0037 µm, the distribution frequency detected using NMR is higher than MICP.Using the conversion coefficients C and n and Equation ( 6), the transverse relaxation time T2 was converted into a pore size distribution.As illustrated, for core S1-1, a comparison chart of the pore throat distribution derived from the Tuha sandstone's NMR T2 relaxation time distribution spectrum and the MICP pore throat distribution was constructed, as depicted in Figure 8. From Figure 8, it can be observed that the signal frequency curves of pore size distribution converted from both MICP and NMR do not align perfectly.Comparing the two methods, it is noticeable that the NMR-detected signal frequency is higher than that of the MICP method for pores smaller than 0.0037 μm (micropores).This discrepancy is attributed to the sandstone mercury injection pressure set at 200 MPa, which is incapable of detecting pores smaller than 0.0037 μm.However, NMR can detect all hydrogen- In the pore throat radius range of 0.0037~0.2µm, while the distribution frequency of the NMR signal mostly agrees with that of MICP, the discrepancy between the two methods increases with decreasing pore throat radius.This is likely due to the inability of mercury intrusion to access some of the smaller and non-connected pores.In region 1 (0.2~1.8 µm), the distribution frequency from MICP is notably higher than NMR, whereas in region 2 (1.8~7 µm), it is lower than NMR.This divergence is due to the inherent differences in the two detection methods.When the core is fully saturated, the NMR T 2 relaxation time distribution spectrum distribution only reflects the distribution of pore size and pore throats in the rock sample.The intensity of the NMR signal represents the proportions of differently sized pores, uninfluenced by the connectivity of pores and throats.In contrast, the MICP method calculates the frequency distribution based on the volume of mercury that enters the rock sample at a specific injection pressure.For homogeneous rock samples, the MICP method yields accurate pore throat results.During the formation of reservoirs, depositional environments, diagenesis, tectonic actions, fluid activities, formation temperature, and pressure all have significant impacts on rock pore structures [3].Due to the complex nature of stratification, especially in heterogeneous reservoirs where large pores connect with small throats, discrepancies arise, as shown in Figure 9.

H4
In Figure 9a, the pore throat radii are in the order r 1 > r 2 > r 3 .When the injection pressure of MICP is lower than the capillary force required to break through the smaller pore throat r 2 , mercury fluid cannot enter the large pore B. However, when the injection pressure surpasses the capillary force needed for pore throat r 2 , mercury fluid infiltrates into the large pore B, as depicted in Figure 9b.Although pore B gets filled up at the moment the capillary force needed to penetrate throat r 2 is reached, the volume of mercury fluid infiltrated is measured in the MICP method as the mercury volume corresponding to pore throat r 2 ; this could be attributed to ink bottle phenomenon [62].Diamond also indicated in research on cement-based materials that MICP-generated PSD is not reliable [32].Essentially, as nearly all pores are located internally within the sample, the mercury fluid can only make contact with the majority of these pores through narrow, elongated throats of varying sizes and shapes [32].Hence, in the pore size and frequency distribution chart (Figure 8) derived from both MICP and NMR, the fluid volume in region 1 detected by NMR is lower than that by MICP, and in region 2, the volume of hydrogen-containing fluid detected by NMR surpasses the mercury volume measured by MICP.
that enters the rock sample at a specific injection pressure.For homogeneous rock samples, the MICP method yields accurate pore throat results.During the formation of reservoirs, depositional environments, diagenesis, tectonic actions, fluid activities, formation temperature, and pressure all have significant impacts on rock pore structures [3].Due to the complex nature of stratification, especially in heterogeneous reservoirs where large pores connect with small throats, discrepancies arise, as shown in Figure 9.In Figure 9a, the pore throat radii are in the order r1 > r2 > r3.When the injection pressure of MICP is lower than the capillary force required to break through the smaller pore throat r2, mercury fluid cannot enter the large pore B. However, when the injection pressure surpasses the capillary force needed for pore throat r2, mercury fluid infiltrates into the large pore B, as depicted in Figure 9b.Although pore B gets filled up at the moment the capillary force needed to penetrate throat r2 is reached, the volume of mercury fluid infiltrated is measured in the MICP method as the mercury volume corresponding to pore throat r2; this could be attributed to ink bottle phenomenon [62].Diamond also indicated in research on cement-based materials that MICP-generated PSD is not reliable [32].Essentially, as nearly all pores are located internally within the sample, the mercury fluid can only make contact with the majority of these pores through narrow, elongated throats of varying sizes and shapes [32].Hence, in the pore size and frequency distribution chart (Figure 8) derived from both MICP and NMR, the fluid volume in region 1 detected by NMR is lower than that by MICP, and in region 2, the volume of hydrogen-containing fluid detected by NMR surpasses the mercury volume measured by MICP.
Consequently, potential inaccuracies might exist in terms of mercury injection volume estimation when using the MICP method, especially in non-uniform reservoirs characterized by numerous large pores linked to smaller pore throats.This could result in the MICP method overestimating the proportion of smaller throats while underestimating the Consequently, potential inaccuracies might exist in terms of mercury injection volume estimation when using the MICP method, especially in non-uniform reservoirs characterized by numerous large pores linked to smaller pore throats.This could result in the MICP method overestimating the proportion of smaller throats while underestimating the larger ones.Conversely, the NMR technique stands out when delineating the pore size distribution in core samples, with its merits of high precision, speediness in measurement, and an expansive detection scope.
Figure 10 compares the pore throat distribution of the Tuha sandstone core samples measured by both NMR and MICP methods, with the distribution trend being largely similar to that of S1-1.The phenomena observed in these eight core samples are consistent with the mismatch between MICP and NMR reported in the literature [29,30,34].The three core samples are noteworthy: S1-2, S1-3, and S1-4.In the pore size range of 0.001 to 0.0037 µm, the proportion of the NMR signal distributions reached 5.39%, 9.07%, and 6.67%, respectively, significantly higher than the proportions measured via mercury intrusion in this pore size range.
Interestingly, all three of these core samples are conglomerate sandstones.With the increase in permeability, the NMR signal frequency for the pores below 0.0037 µm also increased.This might suggest the presence of many non-flowing micropores in the gravel, which cannot be detected during high-pressure mercury injection.However, once saturated with formation brine, the micropores in the conglomerate might be filled and can be detected via the NMR method.Additionally, gravel particles in rocks are usually larger, resulting in larger inter-granular spaces.Hence, in most scenarios, the higher the gravel content in conglomerate sandstone, the higher the core permeability.This offers an explanation as to why the NMR signal in the microporous segment of the conglomerate sandstone intensifies with increasing permeability.

Comparative Analysis of Shale Pore Throat Distributions
Figure 11 displays the pore throat distribution for the Hechuan block H3-5 shale obtained through both NMR and MICP methods.Observations indicate that within the pore throat radius range of 0.001 to 0.075 µm, the distribution frequency measured by NMR is significantly higher than that measured via the mercury intrusion method.There are primarily two reasons for this phenomenon: Limitations of the MICP Technique: Since shale has a lower compressive strength compared to sandstone [63], a lower pressure is used to ensure the authenticity of MICP results for shale.Consequently, with the application of a 100 MPa pressure, pore diameters smaller than 0.0075 µm remain undetectable when using the MICP method.Characteristically, shales are highly compacted and contain an abundance of micro-pores; these intricate pores, due to their diminutive size, are challenging for the MICP technique to detect accurately.larger ones.Conversely, the NMR technique stands out when delineating the pore size distribution in core samples, with its merits of high precision, speediness in measurement, and an expansive detection scope.These two reasons are key to the notable differences in the micropore distributions obtained from the NMR and MICP methods.In research on core pore size distribution, it is essential to understand and consider these factors, as they directly influence the accuracy and reliability of the measurement results.
Within the pore size range from 0.0075 to 0.19 μm, the NMR method shows a slightly higher porosity distribution frequency than the MICP method.This phenomenon can be attributed to the high heterogeneity within the shale.Given the complex microstructure of the shale, which encompasses numerous micropores and nano-scale throats, it is challenging for the MICP method to achieve full saturation in this pore size range.Specifically, the capillary forces from the throats and the surface tension faced by the liquid mercury prevent it from fully entering or filling certain micro-pores.This disparity is especially pronounced in cores with low permeability.Taking the H2-3 core as an example, the extremely low permeability (0.0065 mD) and T2 relaxation time distribution spectrum distribution suggest a significant presence of micro-pores and nano-scale throat structures.The maximum saturation measured via the MICP method for this core is only 59.246%, further confirming that in cores with low permeability, mercury struggles to saturate the micro pores completely.In contrast, the NMR method primarily relies on nuclear magnetic resonance signals to determine the presence and distribution of fluids within the core.It can more accurately detect the distribution of water or other hydrogen-containing fluids within the micro-pores, offering more detailed porosity distribution information.Thus, when dealing with highly heterogeneous rock samples with intricate microstructures, the NMR method may offer more advantageous insights into pore size distribution analysis.When the pore size exceeds 0.19 μm, a phenomenon similar to that observed in sandstones occurs.Specifically, the cumulative frequency of MICP in region 1 is higher than that of NMR, while in region 2, it is lower than NMR.
As shown in Figure 12, in the multi-sample pore throat characterization of the Hechuan shale core, a porosity distribution was observed that was similar to Figure 11 and the published data.Specifically, for the pore size range in region 1 (refer to Figure 11), the frequency distribution measured by the MICP method was generally higher than the results obtained via the NMR method.In contrast, within the pore size range of region 2, the porosity distribution frequency, as reflected by NMR, exceeded that characterized by the MICP method.This observation further confirmed the potential inaccuracies of the MICP method in porosity distribution characterization.NMR Signatures from Specific Components in Shale: Shales contain a significant amount of non-flowing hydrogen-containing components, such as kerogen, asphaltene, and structural water.These components produce signals in NMR measurements, and these signals mainly represent the smaller pores.This means that these micro-pores can only be detectable when using the NMR method.
These two reasons are key to the notable differences in the micropore distributions obtained from the NMR and MICP methods.In research on core pore size distribution, it is essential to understand and consider these factors, as they directly influence the accuracy and reliability of the measurement results.
Within the pore size range from 0.0075 to 0.19 µm, the NMR method shows a slightly higher porosity distribution frequency than the MICP method.This phenomenon can be attributed to the high heterogeneity within the shale.Given the complex microstructure of the shale, which encompasses numerous micropores and nano-scale throats, it is challenging for the MICP method to achieve full saturation in this pore size range.Specifically, the capillary forces from the throats and the surface tension faced by the liquid mercury prevent it from fully entering or filling certain micro-pores.This disparity is especially pronounced in cores with low permeability.Taking the H2-3 core as an example, the extremely low permeability (0.0065 mD) and T 2 relaxation time distribution spectrum distribution suggest a significant presence of micro-pores and nano-scale throat structures.The maximum saturation measured via the MICP method for this core is only 59.246%, further confirming that in cores with low permeability, mercury struggles to saturate the micro pores completely.In contrast, the NMR method primarily relies on nuclear magnetic resonance signals to determine the presence and distribution of fluids within the core.It can more accurately detect the distribution of water or other hydrogen-containing fluids within the micro-pores, offering more detailed porosity distribution information.Thus, when dealing with highly heterogeneous rock samples with intricate microstructures, the NMR method may offer more advantageous insights into pore size distribution analysis.When the pore size exceeds 0.19 µm, a phenomenon similar to that observed in sandstones occurs.Specifically, the cumulative frequency of MICP in region 1 is higher than that of NMR, while in region 2, it is lower than NMR.
As shown in Figure 12, in the multi-sample pore throat characterization of the Hechuan shale core, a porosity distribution was observed that was similar to Figure 11 and the published data.Specifically, for the pore size range in region 1 (refer to Figure 11), the frequency distribution measured by the MICP method was generally higher than the results obtained via the NMR method.In contrast, within the pore size range of region 2, the porosity distribution frequency, as reflected by NMR, exceeded that characterized by the MICP method.This observation further confirmed the potential inaccuracies of the MICP method in porosity distribution characterization.

Correlation of Rock Core Physical Properties
The NMR method accurately reflects pore size distribution, whereas MICP measurements may underestimate the proportion of macro-pores and overestimate that of mesopores.The discrepancies between the two methods are consistent with numerous studies

Correlation of Rock Core Physical Properties
The NMR method accurately reflects pore size distribution, whereas MICP measurements may underestimate the proportion of macro-pores and overestimate that of meso-pores.The discrepancies between the two methods are consistent with numerous studies conducted by other researchers.We suggest that the differences in information obtained from dynamic and static methods reflect meaningful physical characteristics that can serve as characteristic values for porous media.
Taking the sandstone core S1-1 as an example, the study investigates the volume overestimated in the MICP method and the pore throat radius controlling mercury intrusion into the large pores.
Figure 13 illustrates the relationship between cumulative frequency distribution and pore size in NMR and MICP experiments.It is observed that at larger pore sizes, particularly when mercury intrusion just begins, the cumulative frequency of NMR is higher than that of MICP.As the mercury intrusion pressure increases and mercury enters pores of 1.2 µm, the cumulative frequencies of both methods become equal.The pore radius corresponding to this equal cumulative frequency is defined as controlled throat radius r c .In MICP, r c represents the pore throat radius restricting mercury entry into larger pores.When the injection pressure increases to a point where mercury can enter large pores controlled by r c , the cumulative frequencies of MICP and NMR align.The cumulative frequency distribution corresponding to r c is then twice the volume of pores controlled by r c .Defining the volume controlled by pore throat r c as F rc , it is evident from Figure 13 that in core S1-1, 4.91% of the pore volume is influenced by r c .It is noteworthy that the primary differences between MICP and NMR are observed in the low capillary pressure range.There is a significant size disparity between the throats and pores in this range.However, in the high capillary pressure range, the diameters of the pore throats are similar, and no significant discrepancies between the two methods are observed.Therefore, it is considered that the controlled volume by throat size at r c contributes dominantly to the permeability.In the past, Dr. H.D Winland [64] found that r 35 (the radius corresponding to the point at which mercury intrusion volume reaches 35% of the total pore volume of the core) controlled fluid volume is dominant in the flow though feature and used r 35 to quantify the rock quality from a statistical point of view.Shohreh et al. [65] classified reservoir rock types based on micro-CT results and then evaluated rock quality for different rock types using the Winland method [65].
Appl.Sci.2024, 14, x FOR PEER REVIEW 17 of 25 conducted by other researchers.We suggest that the differences in information obtained from dynamic and static methods reflect meaningful physical characteristics that can serve as characteristic values for porous media.
Taking the sandstone core S1-1 as an example, the study investigates the volume overestimated in the MICP method and the pore throat radius controlling mercury intrusion into the large pores.
Figure 13 illustrates the relationship between cumulative frequency distribution and pore size in NMR and MICP experiments.It is observed that at larger pore sizes, particularly when mercury intrusion just begins, the cumulative frequency of NMR is higher than that of MICP.As the mercury intrusion pressure increases and mercury enters pores of 1.2 μm, the cumulative frequencies of both methods become equal.The pore radius corresponding to this equal cumulative frequency is defined as controlled throat radius rc.In MICP, rc represents the pore throat radius restricting mercury entry into larger pores.When the injection pressure increases to a point where mercury can enter large pores controlled by rc, the cumulative frequencies of MICP and NMR align.The cumulative frequency distribution corresponding to rc is then twice the volume of pores controlled by rc.Defining the volume controlled by pore throat rc as Frc, it is evident from Figure 13 that in core S1-1, 4.91% of the pore volume is influenced by rc.It is noteworthy that the primary differences between MICP and NMR are observed in the low capillary pressure range.There is a significant size disparity between the throats and pores in this range.However, in the high capillary pressure range, the diameters of the pore throats are similar, and no significant discrepancies between the two methods are observed.Therefore, it is considered that the controlled volume by throat size at rc contributes dominantly to the permeability.In the past, Dr. H.D Winland [64] found that r35 (the radius corresponding to the point at which mercury intrusion volume reaches 35% of the total pore volume of the core) controlled fluid volume is dominant in the flow though feature and used r35 to quantify the rock quality from a statistical point of view.Shohreh et al. [65] classified reservoir rock types based on micro-CT results and then evaluated rock quality for different rock types using the Winland method [65].The empirical expression is as follows: log r = 0.732 + 0.588 log K − 0.864 log Φ (7) where r35 represents the radius corresponding to the point at which mercury intrusion volume reaches 35% of the total pore volume of the core, measured in μm.Ka is the The empirical expression is as follows: logr 35 = 0.732 + 0.588logK a − 0.864logΦ (7) where r 35 represents the radius corresponding to the point at which mercury intrusion volume reaches 35% of the total pore volume of the core, measured in µm.K a is the absolute permeability of the core, expressed in mD, and Φ is the porosity of the core, denoted in %.
According to Equation ( 7), the porosity-permeability graph merged with the Winland r 35 method (Figure 14) can be derived.It is observable from the figure that sandstone and shale predominantly occupy two distinct regions.Furthermore, the traditional Winland r 35 method underperforms in characterizing the attributes of our samples.Numerous researchers have also conducted studies using Winland's method, finding that the constants in the Winland equation vary for rock cores with different physical properties [8,9,40].We performed a comparison between our findings and the Winland equation.The results are shown in Table 4.
absolute permeability of the core, expressed in mD, and Φ is the porosity of the core, denoted in %.
According to Equation ( 7), the porosity-permeability graph merged with the Winland r35 method (Figure 14) can be derived.It is observable from the figure that sandstone and shale predominantly occupy two distinct regions.Furthermore, the traditional Winland r35 method underperforms in characterizing the attributes of our samples.Numerous researchers have also conducted studies using Winland's method, finding that the constants in the Winland equation vary for rock cores with different physical properties [8,9,40].We performed a comparison between our findings and the Winland equation.The results are shown in Table 4.     Due to the improper removal of the pockmarks effect, the r c and F rc of S1-4 was not detected.
Based on Equation (8), fitting was conducted for r 35 , K a , and Φ using both sandstone and shale samples in the style of the Winland equation.The results of this fitting are illustrated in Figure 15.Due to the improper removal of the pockmarks effect, the rc and Frc of S1-4 was not detected.
Based on Equation ( 8), fitting was conducted for r35, Ka, and Φ using both sandstone and shale samples in the style of the Winland equation.The results of this fitting are illustrated in Figure 15.From Figure 15, it is observable that the Winland formula fits well for shale samples; however, the matching degree for tight sandstone samples is less satisfactory.Lafage found that the Winland r 35 method is not reliable when evaluating the quality of some carbonate reservoirs [66].Winland r 35 is a trial-and-error method.If r 35 is not working, other r could be tried in the fitting, such as r 40 , r 45, and r 50 .This is a common choice for researchers who use the Winland method [9,67].In addition, different core samples with identical permeability and porosity may share the same r 35 values according to the Winland r 35 equation.However, there can be significant differences in pore size distribution and connectivity within actual reservoirs.Therefore, relying solely on r 35 to assess the relationship between permeability and porosity in cores might not be suitable for all rock samples.Figure 16 reveals the saturation of large pores controlled by the smaller pore throat r c , known as F rc , which reflects the volume of large pores in the core influenced by pore throat r c .Based on Equation ( 9), fitting was conducted for F rc , K a , and Φ in the style of the Winland equation.
where A 2 , B 2 , and C 2 are constants.F rc is the volume controlled by pore throat r c , in %. Figure 16 presents the nonlinear surface fitting analysis of F rc , K a , and Φ for both sandstone and shale samples.It is evident from the figure that the R-Squared is quite high for both types of rock cores, with sandstone exhibiting an R 2 value of 0.9267 and shale at 0.8711.Table 5 summarizes the nonlinear surface fitting formulas for various rock cores, revealing a significant correlation in both sandstone and shale between the permeability, porosity, and the volume of large pores controlled by the small pore radius r c .F rc shows better fitting since it used all the information from MICP and NMR.Whereas the r 35 method is a trial-and-error procedure, other metrics such as r 45 , r 55 , and r 65 may also be used for a better fitting, depending on the requirements.This approach finds a special pore throat that controls fluid flow in a porous media.It provides a comprehensive method to cohesively use information from both MICP and NMR, thereby making rock quality analysis more physically meaningful and accurate.By thoroughly analyzing these relationships, a more accurate assessment and optimization of the development strategies for oil and gas reservoirs can be achieved.Figure 16 presents the nonlinear surface fitting analysis of Frc, Ka, and Φ for both sandstone and shale samples.It is evident from the figure that the R-Squared is quite high for both types of rock cores, with sandstone exhibiting an R 2 value of 0.9267 and shale at 0.8711.Table 5 summarizes the nonlinear surface fitting formulas for various rock cores, revealing a significant correlation in both sandstone and shale between the permeability, porosity, and the volume of large pores controlled by the small pore radius rc.

Conclusions
In the field of rock pore structure characterization, MICP and NMR are two widely adopted key techniques.This study delves into a comprehensive comparison and analysis of the applicability of these two methods in depicting rock pore structures.By leveraging the discrepancies between NMR and MICP measurements, we have established the Winland F rc method.Sandstone samples from Tuha and shale samples from Hechuan were selected for the experimental analysis.After a series of experiments and data evaluations, the following primary conclusions were drawn:

Figure 1 .
Figure 1.The discrepancy between MICP and NMR methods schematic diagram.

Figure 1 .
Figure 1.The discrepancy between MICP and NMR methods schematic diagram.

Figure 3 .
Figure 3.The process of rock quality evaluation.

Figure 3 .
Figure 3.The process of rock quality evaluation.

Figure 4 .
Figure 4.The T2 relaxation time distribution spectrum of Tuha sandstone.

Figure 5 .
Figure 5.The T 2 relaxation time distribution spectrum of Hechuan shale.

Figure 8 .
Figure 8.A comparison of pore throat size distributions in sandstone S1-1 as determined via NMR and MICP measurements.

1 Figure 8 .
Figure 8.A comparison of pore throat size distributions in sandstone S1-1 as determined via NMR and MICP measurements.

Figure 11 .
Figure 11.A comparison of pore throat size distributions in shale H3-5 as determined via NMR and MICP measurements.

Figure 11 .
Figure 11.A comparison of pore throat size distributions in shale H3-5 as determined via NMR and MICP measurements.
logr 35 = A 1 + B 1 logK a − C 1 logΦ (8)where A 1 , B 1 , and C 1 are the constants in the style of the Winland equation.

log r 35 =
A 1 + B 1 log K a − C 1 log Φ (8)where A 1 , B 1 , and C 1 are the constants in the style of the Winland equation.
of the Winland equation.log F rc = A 2 + B 2 log K a − C 2 log Φ (9) where A 2 , B 2 , and C 2 are constants.Frc is the volume controlled by pore throat rc, in %.(a) sandstone Appl.Sci.2024, 14, x FOR PEER REVIEW 21 of 25 (b) shale

Figure 16 .
Figure 16.The relationship between F rc , K a , and Φ: (a) sandstone and (b) shale.

Table 2 .
Basic physical properties of samples.

Table 3 .
Conversion factors between NMR T 2 relaxation time distribution spectrum and pore throat radius and correlation index.

Table 4 .
r c , r 35 , and F rc for 16 samples.

Table 5 .
The nonlinear surface fitting analysis.

Table 5 .
The nonlinear surface fitting analysis.