## 5 Inversion of Fluid Content from Field NMR Measurements and Error Assessments  
In this section, we apply the robust inversion scheme developed in previous sections on the inversion of fluid contents from NMR measurements. We will also discuss the application of \(L_{p}\)-norm inversion on the data without complete quality control, and we demonstrate that the errors of the inverted parameters are localized during \(L_{p}\)-norm inversion. We first measure the response map of the coil using the dual equation of (5). After calibration, the result is shown in Figure 5. The spacing between two sampling points is 1 in, the same resolution as that of the fluid content samples. Note that the response function is not homogeneous within the entire coil. This reflects the impacts of the limited size of the coil on the magnetic field, and therefore, produces negative impacts on the measurements of NMR data that must be removed during fluid content inversion. After obtaining the response map, the processing and inversion are carried out by the procedures in Figure 6. If large deviations and non-physical noises can be removed in the pre-processing step, the relatively large value of \(p\)'s can be used for inversion. Once  
Figure 5: The digitized response function of the coil. the fluid contents have been inverted, it can be used for the estimation of other more production-related parameters such as maximum producible hydrocarbon. Figure 7 shows the inversion of three cores. We plot the inverted results of the fluid content and also insert the core photos. The color of the core represents the laminated distribution of kerogen along the core, a key factor for the estimation of the oil and gas reserves. The darker color indicates higher kerogen content in the core. In the pre-processing step, considering the ending effects, the first two and very last NMR signals are removed for all the core measurements. We will show the inverted results without pre-processing later on. These three cores (a-c) represent three typical scenarios. For core A (Figure 7a), the inverted fluid content varies in a complete cycle of a down-up-down pattern. This variation pattern closely follows the color changes of the core. The resolution of the inverted fluid contents is about 2 inches. The color of core B (Figure 7b) is almost constant. This fact is correctly expressed in the inverted fluid content. For core C (Figure 7c), the kerogen consistently increases along the core, and the color darkens, which is perfectly matched by the inverted fluid content. The \(p\) values used in the inversions in Figure 7 are all 1s, and the unreliable data measurements have been removed. According to Figures 3c and 4, slightly larger values of \(p\) can still be used if all the data samples follow the normal trend, which can be predicted by Equation (8). Next, we will show that \(L_{p}\)-norm inversion with small \(p\) can automatically remove the impact of data points with large measurement errors and correctly recover the effective fluid contents. Figure 8a,c show the NMR measurements for two cores, core A and core D. Both the pre-processed (curves with triangles) and non-pre-processed (red curves) NMR data are plotted. For core A, the first two and the very last data points are negative (but with a small magnitude). For core D, the first two data points show significant deviations from the rest. Figure 6: Workflow of NMR inversion. Figure 7: Comparison of inverted HSR-NMR fluid contents in the whole cores with the photos for three cores (**a–c**). Figure 8: _Cont_. Figure 8b,d show the inverted fluid content using different values of \(p\) using the non-pre-processed data, and they are compared with those using pre-processed data (solid red curves). For core A, since the erroneous data is slightly different from the major trend, there are no large differences between the inverted results with different \(p\)'s. This indicates that for sufficiently well-prepared data, \(L_{2}\) inversion shall be able to provide reasonable results.