A Method of Inverting Rock Grain Size Based on Nuclear Magnetic Resonance Logging Data and Application

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Introduction
Grain size data are widely used in the research of stratigraphic petrophysics and play an important role in the theory of sediment classification and nomenclature, identification of paleogeographic environment, analysis of sedimentation, evaluation of reservoir quality, etc. [1,2]. In addition, the grain size of rock has a great influence on diagenesis and plays an important role in restricting the physical properties of rock and the changing characteristics of its pore structure. It can be seen that the rock grain size parameter is the key evaluation parameter of reservoir petrophysical analysis. Since high-temperature and high-pressure formations [3][4][5][6] are widely developed in the Yinggehai-Qiongdongnan Basin and the mud proportion used in the drilling process is up to 2.0, this high-temperature and high-pressure mud system has a high content of barite and heavy ore components, which causes obvious problems in determining the grain size of logging rock, and the error of logging profile obtained is large, bringing a series of problems to the subsequent petrophysical analysis. At present, the techniques for obtaining rock grain size mainly include experimental analysis and logging data. The experimental analysis techniques mainly include laser grain size analysis and sieve analysis. Zhao et al. [7], Chen et al. [8], and Luo et al. [9] used logging curves such as natural gamma ray and flushing resistivity to model with median grain size, and Ning et al. [10] proposed to use natural gamma ray curves to obtain grain size parameters through wavelet transform. However, the above methods obtain few grain size parameters, which cannot truly reflect the vertical combination characteristics of the formation. Given the existing characteristics and problems of the reservoir in the Yinggehai-Qiongdongnan Basin in the western sea area of the South China Sea, this paper studies the relationship between the nuclear magnetic resonance T 2 spectrum of the core and the rock grain size distribution curve and calculates the rock grain size distribution from the nuclear magnetic T 2 distribution for each core by using the subsection nonlinear scale conversion method, obtains the conversion scale parameters of each core, classifies the core in combination with the porosity and permeability parameters, and determines the scale parameters of each type of core. A model for continuous calculation of rock grain size distribution from nuclear magnetic logging data is established. The reliability of this method is proved by the test and application of actual data.

Methodology
According to the NMR relaxation mechanism, the lateral relaxation time T 2 obtained from nuclear magnetic logging [11] is as follows: where T 2B is the volume (free) relaxation time of fluid, D is the diffusion coefficient, G is the magnetic field gradient, T E is the echo interval, S is pore surface area, V is the pore volume, ρ 2 is the transverse surface relaxation strength of rock, and T 2 is the transverse relaxation time.
The value of T 2B in the actual stratum is usually in the range of 2000-3000 ms, which is much larger than T 2 , i.e., T 2B > >T 2 . Therefore, the first term on the right side in Equation (1) can be neglected. If there is a significant diffusion effect on the relaxation of water in spin echo measurement, due to the diffusion effect, the T 2 distribution shifts towards the short T 2 direction, and in order to correctly evaluate the aperture distribution, it is necessary to use a short echo time for measurement. The current nuclear magnetic logging tool has a minimum echo time of only 0.1 to 0.2 ms, at which time the diffusion effect can be completely ignored. The nuclear magnetic resonance data in the study area described in the paper are all obtained using the CMR-PLUS instrument of Schlumberger Company. The magnetic field of the logging tool is uniform, and the formation wettability in the study area is of hydrophilic type. Therefore, the G value corresponding to Equation (1) is very small, and T E is short enough [12]. Therefore, the third term in Equation (1) can also be ignored. For the simple mineralogy of 100% quartz sand, the transverse relaxation time is directly related to the specific surface S/V of the pores, which can be expressed as follows: From Equation (2), it can be seen that the relaxation time T 2 is related to the size and shape of the pore space [13][14][15][16][17][18][19][20][21][22][23][24]. It is assumed that the rock in the formation consists of an ideal accumulation of spherical grains, as shown in Figures 1 and 2.
The ratio of the specific surface of rock grains to the total volume of rock grains can be derived from the volume model (see Equation (3)). And the parameters of rock pore surface area and pore volume can be obtained by converting the specific surface of rock grains, the total volume of grains, and rock porosity, as shown in Equation (4). Therefore, the relationship between rock pore surface area, pore volume, and rock grain size can be obtained from Equation (3), Equation (4), and Equation (5). Combining Equation (2), the calculation model of the nuclear magnetic transverse relaxation time T 2 and the rock grain size can be obtained, which is shown in Equation (6). Rock grains are influenced by sediment sources as well as late diagenesis, resulting in the regular accumulation of rock grains in the stratum that are not exactly round. Therefore, to correct for these effects,   2 Geofluids an influence factor CF parameter is added to Equation (6), which leads to Equation (7). It is clear that the rock grain size is related to the NMR T 2 time, ρ 2 rock transverse sur-face relaxation strength, rock porosity ϕ, and the correction factor CF. When calculating the rock grain size distribution using Equation (7) for the NMR T 2 data, it is also necessary   to obtain accurate ρ 2 and CF parameters. Chen et al. [25] pointed out that the ρ 2 parameter is affected by the rock mineral fraction, resulting in a wide range of distribution of this value, which is not easy to determine. Therefore, the conversion factor C value in Equation (8) needs to be deter-mined by interscaling the NMR and grain size experiments. Forty-one representative cores in the region were selected for simultaneous NMR and grain size analysis experiments to obtain the aforementioned C values. 1 1 where S matrix is the surface area of rock grains, V matrix is the volume of rock grains, r grain is the grain size of rock grains, π is the circumference, S is the pore surface area, V is the pore volume, ϕ is the porosity of rock, ρ 2 is the transverse surface

Geofluids
relaxation strength of rock, T 2 is the transverse relaxation time, CF is the correction factor, and C is the conversion factor, C = CF × ρ 2 × 3. From Equation (8), the rock size distribution can be approximated under the condition that the T 2 distribution is known.

NMR and Grain Size Analysis Experiments
3.1. Sampling Background. The buried depth of the Huangliuzu Formation reservoir in the Yinggehai-Qiongdongnan Basin in the west of the South China Sea is 2700-3200 m, and the compaction diagenesis is strong, resulting in the complex pore structure of the Huangliuzu Formation reservoir. In the early stage, operators obtained abundant physical core data from the Huangliuzu Formation reservoir by drilling or sidewall coring and selected a certain amount of cores to carry out nuclear magnetic resonance, particle size analysis, and other experiments, forming the core analysis database of Liushagang Formation. In this study, 41 core samples that can represent the characteristics of regional reservoir and measure nuclear magnetic resonance and grain size were selected from the database as the research data. The core lithology is argillaceous siltstone, silty fine sandstone, siltstone, fine sandstone, etc. The porosity distribution range is 11.5%~20.7%, as shown in Table 1. The XRD  Figure 9: Diagram of the relationship between the conversion coefficient C and the core physical properties.  Table 2. From the table, it can be seen that the mineral composition of the core is mainly composed of quartz, there is a certain amount of feldspar and clay minerals in the low permeability rock core. Figure 3 shows four representative core photos. The core nuclear magnetic experiment described in this paper is measured by the CoreSpec-1000 NMR core analyzer of NUMAR Company in the United States. The echo interval selected in the experiment is 0.1 ms. The waiting time, receiving grain, and echo string length are selected according to different cores. The gain size experiment was measured by laser method.

Interrelationship between T 2 Distribution and Grain Size
Distribution. From Equation (8), it can be seen that under the condition that the nuclear magnetic resonance T 2 spectrum is known, the rock grain size distribution curve can be approximately obtained, which is called the rock grain size distribution curve. The nuclear magnetic resonance T 2 spectrum has a close correlation with the rock grain size distribution curve. To illustrate the rationality of Equation (8), the nuclear magnetic resonance T 2 spectrum and grain size distribution curve of core are analyzed. Figures 4 and 5 show the nuclear magnetic resonance T 2 spectrum of core 6# and obtain the grain size distribution curve of rock through grain size experiments. In order to more intuitively see the relationship between core nuclear magnetic resonance T 2 spec-trum and grain size distribution curve, it is necessary to preprocess these two experimental data. Firstly, convert the porosity increment parameter in the ordinate of Figure 4 into a porosity percentage distribution parameter, and convert T 2 to 1/T 2 . Then, convert the rock grain size r in Figure 5 to 1/r. Finally, the nuclear magnetic resonance T 2 spectrum and grain size distribution curve after conversion coordinate calibration are made into a cross plot, as shown in Figure 6. In Figure 6, there are two ordinates, with 1/T 2 value on the left, and the unit is ms. On the right is the reciprocal 1/r value of grain size, in mm. The abscissa is the percentage content. Similarly, the nuclear magnetic resonance T 2 and rock grain size distribution curves of 41 cores described in Table 1 are processed. Figure 7 shows the overlapping comparison diagram of T 2 distribution and grain size distribution of 4 rock samples (nos. 1, 7, 20, and 36 in Table 1). From the figure, it can be seen that the change patterns of the two are very similar or similar, indicating that there is a close correlation between the two, so it is feasible to convert the rock grain size distribution curve through nuclear magnetic resonance T 2 .

Methods for Determining the Conversion Factor C
The grain size distribution and NMR T 2 distribution data obtained experimentally from 41 cores were analyzed, and  Figure 10: Comparison of grain size distribution between T 2 NMR distribution and core grain size distribution. 7 Geofluids the flow of the method to determine the conversion factor C value is shown in Figure 8. The main steps are as follows: (i) determine the interval of the conversion factor C value, (ii) change the C value according to the specified step to obtain the pseudo size distribution curve, and (iii) calculate the error between the NMR simulation pseudo size distribution curve and the experimental size distribution. If the error is less than the given cutoff value, the C value is output; otherwise, it returns to (ii) until the calculated error is less than the cutoff value. At this point, the rock size distribution curve is obtained by substituting the C value into Equation (8).
During the actual data processing of the cores, the NMR T 2 distribution needs to be processed in two stages. The T 2 cutoff point is 50 ms, i.e., the conversion factor C values for T 2 ≤ 50 ms and T 2 > 50 ms are not the same. The cores were processed and analyzed using the above method flow to obtain the conversion factor C values for each core, as shown in Table 3.
The flow zone index (FZI) is one of the effective parameters to describe the reservoir pore structure and is defined as follows: where ϕ is the rock porosity, K is the permeability, and FZI is the flow zone index. Combined with the core porosity and permeability parameters, the conversion coefficient C values of 41 cores were analyzed in depth. The cores were classified into three types, as shown in Figure 9: class I: 60 < FZI, 30 mD < K; class II: 10 ≤ FZI ≤ 60, 1 mD ≤ K ≤ 30 mD; and class III: FZI ≤ 10, K < 1 mD. It was found that the conversion factor C values were fixed for each type of core, as shown in Table 4.
The 41 cores in Table 1 were processed using Equation (8) and the conversion factors in Table 3. Figure 10 shows  8 Geofluids the NMR T 2 distribution converted grain size of four of the cores compared with the grain size distribution of the core experiment. The red square dots are the grain size distribution curves obtained from the NMR T 2 distribution calculations, and the blue solid dots are the grain size distribution curves obtained from the core experimental analysis. Analysis of Figure 10 shows that the grain size distribution curves calculated by NMR T 2 distribution and the core experimental grain size distribution curves are in good agreement in terms of morphology and change trend.

An Application into Well Profile
To test the accuracy of the method in this paper, the rock grain size results calculated by this method are compared with the grain size test data. Figure 11 shows the results of the interpretation using the NMR logging data of well XF-11 processed according to the method in this paper; through geological logging, it can be seen that the lithology of the calculated processing layer is fine sandstone, with quartz as the main rock mineral and clay content less than 5%. The fifth panel shows the logging lithology, the sixth panel shows the NMR logging T 2 spectrum, and the seventh panel shows the calculated grain size accumulation probability curve. The solid red dots in the eighth lane are the mean grain size (Φ) of the core analysis, and the solid blue line is the mean grain size (Φ) curve calculated based on the NMR logging data. It can be seen that the logging calculation results are consistent with the core experimental analysis grain size averages. The absolute error between the mean grain size (Φ) of 19 cores and the mean grain size (Φ) calculated from the logs is 0.12. Based on the logs, natural gamma, and triple porosity curves, the lithology of well section 2961.5-2963.2 m is mudstone. The average value of the grain size (Φ) calculated from the logs is different from the reservoir section. There is a core at the depth of 2963 m, and the core analysis results are also in good agreement with the calculation results. It is known that the method of continuous calculation of downhole rock size distribution by NMR T 2 distribution is reliable and can provide accurate continuous rock grain size distribution profiles for petrophysical analysis of geological reservoirs.

Conclusions
(1) The grain size distribution in the actual formation is affected by many factors leading to its complex variation characteristics. For the quartz dominated core samples, core experiments indicate that there is a good correlation between the NMR T 2 distribution and the rock size distribution curve (2) Combined with the core experimental data, the conversion coefficients of NMR T 2 distribution to grain size distribution for three types of reservoirs were obtained by using the segmented nonlinear fitting method. The comparison of the NMR conversion results with the core experimental results has good similarity, which verifies the reliability of the method (3) Applying the method to well logging data, the grain size distribution curve of downhole rocks can be calculated continuously, and then, the grain size parameters of the formation can be evaluated quantitatively. The results are in good agreement with the results of the assay analysis. In the reservoirs dominated by quartz, comparative analysis shows that the method can obtain reliable grain size distribution curves from T 2 distribution with high accuracy. For complex lithologic reservoirs, it is necessary to analyze the applicability of the method based on actual core results to obtain reliable calculation results

Data Availability
The original contributions presented in the study are included in the article/supplementary material. Further inquiries can be directed to the corresponding author.