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

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## Abstract

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## 1. Introduction

_{2}spectrum can be used to describe the entire pore distribution range, but the conversion of transverse relaxation time to pore size is constrained by many factors, such as lithology, paramagnetic minerals, and fluid types [17,20]. Consequently, it is necessary to combine these two methods to interpret pore size distribution of tight sandstone reservoir.

## 2. Materials and Methods

#### 2.1. Geological Setting and Samples

#### 2.2. MICP

_{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]:

_{Hg}and θ

_{Hg}. The volume of mercury penetrating the pores is measured directly as the increasing applied pressure.

_{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.

#### 2.3. NMR

_{2}of NMR measurement is composed of surface relaxation, bulk relaxation, and diffusion relaxation, which can be expressed as Equation (2) [17,40]:

_{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.

_{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):

_{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):

_{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.

_{2}) distributions were computed by multi-exponential inversion of the echo data with 64 preset 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.

## 3. Results

#### 3.1. Petrology and Pore Characteristics

#### 3.1.1. Petrology Characteristics

#### 3.1.2. Pore Characteristics

#### 3.2. MICP Curves and Parameters

_{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.

_{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).

#### 3.3. NMR T_{2} Spectrum

_{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.

_{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.

_{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.

## 4. Discussion

#### 4.1. Comparison of Pore Volume and Size from MICP and NMR

#### 4.1.1. Porosity

_{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.

#### 4.1.2. Pore and Pore Throat Size

_{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}.

#### 4.2. Pore Size Distribution

#### 4.2.1. Calibration of PSD

_{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:

_{2i}) in a T

_{2}relaxation distribution, the corresponding pore size (R

_{i}) can be determined by Equation (4).

_{s}) and surface relaxivity (ρ

_{2}).

_{2i}, the T

_{2cutoff}can be used to calculate the conversion coefficient C. Thus, the following formula is converted from Equation (5):

_{c}is the cut-off pore radius, i.e., the minimal pore radius (μm) for water to discharge at the centrifuge pressure.

_{c}can be obtained from the following formula based on the Washburn equation:

_{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.

_{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.

_{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).

#### 4.2.2. The PSD and PTD

_{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.

#### 4.3. The Difference between MICP-PTD and NMR-Derived PSD

#### 4.3.1. The Pore Network Model

#### 4.3.2. Difference of Pores and Throats

_{edu}is the Euclidean distance, x

_{i}is the pore radius or pore throat radius, and ${\mathrm{y}}_{\mathrm{i}}^{\mathrm{NMR}}$ and ${\mathrm{y}}_{\mathrm{i}}^{\mathrm{MICP}}$ are the incremental porosity of NMR-derived PSD and MICP-PTD in fraction, respectively.

_{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.

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**Mineral composition (

**a**), median grain size (

**b**), porosity (

**c**) and permeability (

**d**) of the Chang 7 tight sandstones.

**Figure 3.**Petrology and pore characteristics of the Chang 7 tight sandstone reservoirs in the Ordos Basin. (

**a**,

**b**) Z168, 1754.56 m, fine-grained, subrounded to subangular, poorly sorted lithic arkose. Qtz—quartz; F—feldspar; RF—rock fragment. (

**c**) H144, 2554.10 m, ankerite cement (blue) is the most common pore-filling constituents, thin section colored by the mixture of sodium alizarinsulfonate and potassium ferricyanide. (

**d**) X257 1927.10 m, micro-scale regular residual primary intergranular pores. (

**e**) Z388, 2080.77 m, large amounts of dissolved pores, including feldspar dissolved pores and lithic fragment dissolved pores. (

**f**) W100, 1968.00 m, micro-fractures. (

**g**) X257, 1927.10 m, microscale regular residual primary intergranular pores, SEM. (

**h**) Z70, 1617.10 m, feldspar dissolved pores, SEM. (

**i**) G8, 1795.60 m, authigenic albite inter-crystalline pores. (

**j**) Z168, 1725.40 m, clay inter-crystallite pores developed in rolled sheet illite, SEM. (

**k**) X257, 1927.10 m, nanoscale triangular or planar clay inter-crystallite pores developed in foliated chlorite aggregates, argon ion polishing, SEM. (

**l**) X257, 1927.10 m, nanoscale clay inter-crystalline pores developed in foliated chlorite aggregates, SEM.

**Figure 4.**Intrusion and extrusion curves (

**a**) and pore throat distribution curves (

**b**) of the Chang 7 tight sandstones in the Ordos Basin derived from MICP.

**Figure 5.**The 100% water-saturated and irreducible T

_{2}spectrum distribution of the Chang 7 tight sandstones in the Ordos Basin.

**Figure 6.**Cross-plots of φ and φ

_{Hg}(

**a**), φ and φ

_{NMR}(

**b**), φ

_{Hg}and φ

_{NMR}(

**c**). φ: Gas-measured porosity. φ

_{Hg}: MICP porosity. φ

_{NMR}: NMR porosity. Data of φ

_{NMR}is from Li et al. [27].

**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.

**Figure 8.**Plots of pore radius and pore throat radius. (

**a**) The plot of r

_{75}and R

_{75}show a good match except for sample 14. (

**b**) The plot of r

_{50}from Li et al. [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.

**Figure 9.**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.

**Figure 10.**The pore network model of the Chang 7 tight sandstones in the Ordos Basin. (

**A**) (

**a**) No. 4, L231, 2027.90 m, large amounts of tubular/cylindrical throats or long strip throats are developed in the very tight sandstones, including: (

**a1**) the curved lamellar throat, (

**a2**) the laminated throat; (

**a3**) the tubular throat. (

**b**) Necking throats are fine throats connecting large-size pore bodies and small-size pore throats, origin from the close contact of grains caused by fierce compaction and grain coating chlorite. (

**b1**) No. 15, L231, 2022.25 m; (

**b2**) No. 5, L231, 2108.13 m. (

**B**) Pore network model. (

**C**) The cumulative PSD (or PTD) from MICP and NMR. No. 4, the cylindrical pore model (CPM), the PSD and PTD match well with each other. No. 5, the sphere-cylindrical pore model (SPM), a significant difference between the PSD and PTD is presented due to large amounts of necking throats.

Sample No. | So | P_{c50}(MPa) | P_{d}(MPa) | r_{max}(μm) | S_{max}(%) | We (%) | PTR | φ_{Hg}(%) | r_{2.5}(μm) | r_{25}(μm) | r_{75}(μm) |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 1.69 | 46.11 | 6.82 | 0.108 | 71.83 | 32.42 | 2.08 | 3.24 | 0.104 | 0.034 | 0.002 |

2 | 1.87 | 59.51 | 4.39 | 0.167 | 77.35 | 44.04 | 1.27 | 2.85 | 0.173 | 0.043 | 0.003 |

3 | 1.59 | 19.37 | 3.56 | 0.206 | 86.70 | 36.99 | 1.70 | 5.74 | 0.203 | 0.084 | 0.010 |

4 | 1.43 | 18.97 | 3.56 | 0.206 | 85.21 | 24.22 | 3.13 | 6.06 | 0.195 | 0.075 | 0.012 |

5 | 1.51 | 26.88 | 6.02 | 0.122 | 89.33 | 11.85 | 7.44 | 7.07 | 0.213 | 0.048 | 0.009 |

6 | 1.49 | 11.02 | 2.87 | 0.256 | 86.99 | 35.04 | 1.85 | 8.45 | 0.212 | 0.118 | 0.016 |

7 | 1.69 | 11.42 | 1.84 | 0.400 | 90.31 | 32.78 | 2.05 | 8.18 | 0.415 | 0.151 | 0.019 |

8 | 1.53 | 11.30 | 2.60 | 0.283 | 91.92 | 41.67 | 1.40 | 9.39 | 0.258 | 0.129 | 0.021 |

9 | 1.65 | 12.21 | 1.83 | 0.401 | 91.20 | 20.41 | 3.90 | 5.97 | 0.471 | 0.117 | 0.020 |

10 | 1.23 | 10.06 | 4.40 | 0.167 | 87.87 | 27.73 | 2.61 | 8.96 | 0.214 | 0.110 | 0.028 |

11 | 1.54 | 5.90 | 2.25 | 0.326 | 91.59 | 19.83 | 4.04 | 9.14 | 0.293 | 0.180 | 0.033 |

12 | 0.98 | 7.11 | 2.26 | 0.326 | 89.88 | 26.09 | 2.83 | 9.20 | 0.287 | 0.155 | 0.051 |

13 | 1.49 | 7.06 | 2.88 | 0.256 | 83.71 | 22.73 | 3.40 | 8.99 | 0.318 | 0.167 | 0.017 |

14 | 0.81 | 3.94 | 0.94 | 0.782 | 91.23 | 13.58 | 6.36 | 8.90 | 0.645 | 0.254 | 0.135 |

15 | 1.59 | 7.86 | 4.39 | 0.167 | 92.46 | 13.13 | 6.61 | 9.37 | 0.834 | 0.139 | 0.032 |

16 | 2.03 | 2.59 | 1.19 | 0.619 | 89.70 | 25.49 | 2.92 | 8.03 | 0.917 | 0.460 | 0.032 |

_{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; $\mathrm{PTR}=\frac{{\mathrm{S}}_{\mathrm{R}}}{{\mathrm{S}}_{\mathrm{max}}-{\mathrm{S}}_{\mathrm{R}}}$, 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.

Sample No. | S_{ir}*(%) | S_{w}*(%) | C (μm/ms) | R_{2.5}(μm) | R_{25}(μm) | R_{50}(μm) | R_{75}(μm) |
---|---|---|---|---|---|---|---|

1 | 67.99 | 32.01 | 0.027 | 0.151 | 0.037 | 0.022 | 0.013 |

2 | 58.57 | 41.43 | 0.047 | 0.068 | 0.025 | 0.016 | 0.010 |

3 | 56.80 | 43.20 | 0.034 | 0.153 | 0.053 | 0.029 | 0.017 |

4 | 56.47 | 43.53 | 0.024 | 0.491 | 0.096 | 0.040 | 0.021 |

5 | 63.85 | 36.15 | 0.006 | 0.826 | 0.261 | 0.090 | 0.034 |

6 | 53.02 | 46.98 | 0.027 | 0.525 | 0.077 | 0.038 | 0.019 |

7 | 58.87 | 41.13 | 0.019 | 0.954 | 0.123 | 0.046 | 0.023 |

8 | 58.37 | 41.63 | 0.011 | 0.630 | 0.221 | 0.084 | 0.030 |

9 | 40.80 | 59.20 | 0.027 | 1.019 | 0.175 | 0.066 | 0.026 |

10 | 42.26 | 57.74 | 0.015 | 1.318 | 0.177 | 0.064 | 0.028 |

11 | 52.47 | 47.53 | 0.009 | 1.184 | 0.308 | 0.092 | 0.032 |

12 | 55.57 | 44.43 | 0.014 | 1.224 | 0.229 | 0.085 | 0.038 |

13 | 39.60 | 60.40 | 0.017 | 1.347 | 0.310 | 0.083 | 0.034 |

14 | 44.94 | 55.06 | 0.012 | 1.680 | 0.381 | 0.122 | 0.038 |

15 | 26.83 | 73.17 | 0.024 | 0.954 | 0.306 | 0.069 | 0.024 |

16 | 15.06 | 84.94 | 0.057 | 3.414 | 0.419 | 0.128 | 0.032 |

_{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.

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## Share and Cite

**MDPI and ACS Style**

Li, C.; Liu, X.; You, F.; Wang, P.; Feng, X.; Hu, Z.
Pore Size Distribution Characterization by Joint Interpretation of MICP and NMR: A Case Study of Chang 7 Tight Sandstone in the Ordos Basin. *Processes* **2022**, *10*, 1941.
https://doi.org/10.3390/pr10101941

**AMA Style**

Li C, Liu X, You F, Wang P, Feng X, Hu Z.
Pore Size Distribution Characterization by Joint Interpretation of MICP and NMR: A Case Study of Chang 7 Tight Sandstone in the Ordos Basin. *Processes*. 2022; 10(10):1941.
https://doi.org/10.3390/pr10101941

**Chicago/Turabian Style**

Li, Chaozheng, Xiangbai Liu, Fuliang You, Peng Wang, Xinluo Feng, and Zhazha Hu.
2022. "Pore Size Distribution Characterization by Joint Interpretation of MICP and NMR: A Case Study of Chang 7 Tight Sandstone in the Ordos Basin" *Processes* 10, no. 10: 1941.
https://doi.org/10.3390/pr10101941