# Modeling of Characteristics of Complex Microstructure and Heterogeneity at the Core Scale
## 3 Results
### Modified Squrt Model Application  
In the previous section, the porosity distribution characteristics of five carbonate cores were obtained via inversion (Figure 12), and the P-wave and S-wave velocities were predicted via the obtained ultrasonic frequency squirt flow model considering multiple sets of microfractures with different aspect ratios. For comparison, Figure 14 shows measurements, predictions from the new model, calculations from the Biot model using the modulus of dry cores, and numerical simulation data from the new model. The simulation data suggest that the predictions of the new model, which combine the squirt flow model considering multiple microcracks and the mixed random medium model, agree well with the experimental data, and the accuracy is improved compared to the predicted values of the biot model. In Figure 14, Modified-G is the \"wet skeleton\" modulus obtained from the dry core moduli using the modified squirt flow model and the prediction at saturation using the Gassmann model. The Biot model is the saturated prediction obtained by applying the dry core modulus to the Biot model. The simulations are the result of numerical simulation by adding the Modified-G predictions to the random medium model. The measurements are the saturated velocities measured in the ultrasonic experiments. As shown in Figure 14, the saturation velocities predicted by the modified squirt flow model at high frequencies agree well with the ultrasonic experimental measurements, whereas the predictions made by traditional Biot theory are grossly underestimated, especially for sample 26. The fit is slightly worse for dolomite sample 132, but the error is also much less than 5%, as shown in Figure 14. The difference lies first in the fact that all the limestones in this study contain solution pores and solution fractures, while limestone 132 is characterized by intergranular  
Figure 13: Core-scale random medium model: a and b are the autocorrelation lengths in the x and z directions, and n is the rounding coefficient. pores. The differences in pore size between large pores and small pores are also more significant, as are the differences in lithology. However, the distribution of the aspect ratio of microfractures obtained through inversion did not consider these differences. By controlling the size of the stiff pores and achieving a more uniform distribution of the matrix in the simulations, the simulated results became closer to the measured results. In Figure 14c,d, the curves plotted for both the calculated results of the new model and the measured data of the experiment are flatter, and the trends are more compatible. For the other three samples, the experimental data show a greater rate of change in the low-pressure section than that of the new model. Combined with the analyses of Figures 10-12, this suggests that the new model is more applicable to rocks with a uniform distribution of microporosity aspect ratios and a smaller and more uniformly distributed stiff pore radius. The errors of the predictions and saturated measurements for a variety of models are given in Figure 15. The dry and the saturated predictions of the modified squirt flow model are indicated by the blue and the red dots, and the predictions of the Gassmann and Biot models and the simulation results of the model considering the random medium are shown in green, purple, and orange, respectively, with different shapes representing different samples. In Figure 15a, the errors of the P-wave predictions obtained from all models are within 10%, and only the predictions of the modified squirt flow model in the dry case and the Biot model have errors of more than 5%. The predictions of the Modified-G model for the saturated cores (in red) are almost always close to both sides of \(y=x\), with small errors. The new model, which combines the squirt flow model and random medium models, has smaller errors in the simulations (orange), which are closer to \(y=x\).