*3.3. Development of T2 Spectrum Shape Correction Model for Oil-Based Mud Wells*

Both NMR *T*<sup>2</sup> spectrum data and core mercury injection data can be used to analyze the pore throat size of reservoir rocks, and the two have good consistency [23]. To quantitatively analyze pores of different sizes in rocks, some researchers have proposed a method to characterize the pore size by using the porosity of the *T*<sup>2</sup> spectrum interval of nuclear magnetic resonance logging [24]. Predecessors usually give seven fixed transverse relaxation time values to characterize the pore structure and pore size distribution information of reservoir rock, namely 1.0 ms, 3.0 ms, 10.0 ms, 33.0 ms, 100.0 ms, 300.0 ms, and 1000.0 ms, and the NMR logging *T*<sup>2</sup> spectrum is divided into eight porosity intervals (named 8 bins) [25]. Each interval reflects different pore sizes, the short transverse relaxation time represents small pores, and the long transverse relaxation time represents large pores.

Under the actual drilling differential pressure, macropores contribute the most to the permeability of the reservoir [26]. Mud filtrate primarily invades the macropores to displace the movable fluid (including movable water and oil and gas) within the detection range, while mud filtrate invades the small pores less, and the bound fluid will not change basically. Therefore, the invasion of mud filtrate only has a great impact on the *T*<sup>2</sup> spectrum of NMR logging corresponding to macropores but has little impact on the *T*<sup>2</sup> spectrum of NMR logging of bound fluid in small pores [27]. Therefore, this study corrected the movable fluid part of the *T*<sup>2</sup> spectrum of formation NMR logging after the invasion of oil-based mud filtrate and combined it with the *T*<sup>2</sup> spectrum of NMR logging of the original bound fluid part to obtain the complete NMR *T*<sup>2</sup> spectrum after morphological correction [28].

Considering that the invasion of oil-based mud had an impact on each pore component in the macropore space, and the average *T*2*cut-off* value is 17.48 ms, to characterize the impact of oil-based mud filtrate on the *T*<sup>2</sup> spectrum of NMR logging, this study defines only four transverse relaxation times: 33.0 ms, 100.0 ms, 300.0 ms and 1000.0 ms to divide the NMR *T*<sup>2</sup> spectrum. Combining the *T*2*cut-off* value and the maximum transverse relaxation time value, the *T*<sup>2</sup> spectrum can be divided into five intervals ([17.48, 33.0] ms, [33.0, 100.0] ms, [100.0, 300.0] ms, [300.0, 1000.0] ms, [1000.0, 3000.0] ms), and the pore composition percentage of each interval can be calculated, as shown in Equations (3)–(5) [29]:

$$X1 = \frac{\int\_{T\_{2\text{cutoff}}}^{T\_{2(1)}} S(T)dt}{\int\_{T\_{2\text{min}}}^{T\_{2\text{max}}} S(T)dt} \tag{3}$$

$$Xi = \frac{\int\_{T\_{2(i-1)}}^{T\_{2(i)}} S(T)dt}{\int\_{T\_{2\text{min}}}^{T\_{2\text{max}}} S(T)dt} i = 2,3,4 \tag{4}$$

$$X5 = \frac{\int\_{T\_{2(4)}}^{T\_{2max}} S(T)dt}{\int\_{T\_{2min}}^{T\_{2max}} S(T)dt} \tag{5}$$

where, *Xi* is the percentage of pore components in the nuclear magnetic *T*<sup>2</sup> spectrum. The values of *T*2*min* and *T*2*max* are 0.3 ms and 3000 ms, respectively; *T*2*cutoff* is the optimal *T2cut-off* value obtained from the core NMR experiment, which is 17.48 ms in this study; *T*2*(i)* are the four *T*<sup>2</sup> relaxation time values defined above (33.0 ms, 100.0 ms, 300.0 ms and 1000.0 ms respectively); and *S(T)* are the pore distribution functions of the nuclear magnetic *T*<sup>2</sup> spectrum.

Equations (3)–(5) can be used to calculate the percentages of five pore components according to the *T*<sup>2</sup> spectrum of NMR logging after the invasion of oil-based mud filtrate. The amplitude corresponding to each relaxation time of the measured NMR *T*<sup>2</sup> spectrum under the condition of water-based mud with *T*<sup>2</sup> relaxation time greater than 17.48 ms was defined as the dependent variable, and the percentage of five pore components was used as the independent variable. Therefore, the multivariate linear function relationship between the amplitude of each point of *T*<sup>2</sup> spectrum composition of NMR logging under the condition of water-based mud and the five pore components of *T*<sup>2</sup> spectrum of NMR logging under the condition of oil-based mud was established. Using this functional relationship, the amplitude of the nuclear magnetic resonance *T*<sup>2</sup> spectrum under waterbased mud conditions with different relaxation times can be calculated from the nuclear

magnetic resonance logging *T*<sup>2</sup> spectrum under oil-based mud conditions. The function relationship is shown in Equation (6):

$$\begin{aligned} A\_1 &= a\_{11}X\_1 + a\_{12}X\_2 + a\_{13}X\_3 + a\_{14}X\_4 + a\_{15}X\_5 + b\_1 \\ A\_2 &= a\_{21}X\_1 + a\_{22}X\_2 + a\_{23}X\_3 + a\_{24}X\_4 + a\_{25}X\_5 + b\_2 \\ A\_3 &= a\_{31}X\_1 + a\_{32}X\_2 + a\_{33}X\_3 + a\_{34}X\_4 + a\_{35}X\_5 + b\_3 \\ &\tag{6} \end{aligned}$$

$$A\_i = a\_{i1}X\_1 + a\_{i2}X\_2 + a\_{i3}X\_3 + a\_{i4}X\_4 + a\_{i5}X\_5 + b\_i$$

where, *Ai* represents the amplitude value corresponding to the *i* th time distribution point of the *T*<sup>2</sup> spectrum of NMR logging after correction, and the value of *i* is the number of distribution points of the *T*<sup>2</sup> spectrum of NMR logging; *X*1, *X*<sup>2</sup> ...... . *X*<sup>15</sup> is the percentage component of five pore components divided according to the corresponding reservoir type under the condition of oil-based mud; *a*1, *a*<sup>2</sup> ...... . *a*<sup>15</sup> is the coefficient corresponding to the multivariate linear function corresponding to the *i* th distribution point, and its value is calibrated by *T*<sup>2</sup> spectrum data of nuclear magnetic resonance logging of water-based mud and oil-based mud in the sample library; *b*1, *b*<sup>2</sup> ...... . *b*<sup>15</sup> is the constant corresponding to the multivariate linear function corresponding to the *i* th distribution point, and its value (Supplementary Materials, Tables S1–S4)is calibrated by *T*<sup>2</sup> spectrum data of nuclear magnetic resonance logging of water-based mud and oil-based mud in the sample library.

#### **4. Conclusions**

In the process of drilling, the invasion of oil-based mud filtrate has a serious impact on the *T*<sup>2</sup> spectrum of NMR logging, so NMR logging data cannot be directly used to evaluate reservoir parameters. Because the invasion degree of oil-based mud filtrate to reservoirs with different pore structures is different, through the analysis of cast thin sections, physical property experiments, and high-pressure mercury injection experiments, it is proposed to divide the reservoir rocks into four types based on the permeability range.

Based on the difference between the *T*<sup>2</sup> spectrum of nuclear magnetic resonance logging in water-based mud wells and oil-based mud wells, the corresponding morphological correction model of the nuclear magnetic resonance *T*<sup>2</sup> spectrum was established. By comparing the permeability, pore fitting permeability and core analysis permeability calculated by *T*<sup>2</sup> spectrum measured by NMR before and after correction, it was found that the permeability calculated after *T*<sup>2</sup> spectrum morphology correction is the most accurate, which improved the accuracy of calculating permeability by using NMR logging data in oil-based mud wells, it also laid a foundation for further NMR logging of data to evaluate other reservoir parameters.

The proposed correction model has been successfully applied in sandstone reservoirs, which lays a solid foundation for the morphological correction of the *T*<sup>2</sup> spectrum of NMR logging under the condition of oil-based mud in shale oil reservoirs. Different from sandstone reservoirs, when applying the model proposed in this study to NMR *T*<sup>2</sup> spectrum correction of shale oil reservoirs, it is necessary to comprehensively consider the complex pore network composition of shale oil reservoirs and the response mechanism of NMR logging under complex fluid occurrence modes.

**Supplementary Materials:** The following are available online. Tables S1–S4: The coefficients and constant matrix values used in the four types of rocks in the calibration process.

**Author Contributions:** Conceptualization, J.S. and J.C.; methodology, J.C. and J.L. (Jun Li); software, P.F.; validation, F.S., J.L. (Jing Lu); investigation, J.S.; resources, J.C.; data curation, P.F.; writing original draft preparation, P.F. and W.Y.; writing—review and editing, J.S.; visualization, P.F.; supervision, J.L. (Jun Li). All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Natural Science Foundation of China, grant number 41874138 and 42004098, and the National Science and Technology Major Project, grant number 2016ZX05006002-004.

**Data Availability Statement:** Data available on request to the corresponding author.

**Acknowledgments:** The authors would like to thank Shanghai Branch of CNOOC Ltd. for the field logging data.

**Conflicts of Interest:** The authors declare no conflict of interest.

**Sample Availability:** Samples of the compounds are available from the authors.

#### **References**

