In this section, MICP measurement data are utilized to build the IFU model, and a satisfying fitting result is obtained (Fig. 7). Additionally, compared with the results in Pia and Sanna (2014b), it can be seen that by introducing variable-ratio factors into the modeling process, the modified model fits the experimental data better, showing a more powerful simulation ability. This provides a solid foundation for subsequent permeability forecasting.
Figure 7. Comparison of cumulative curve of pore size between the IFU model and experimental data for typical samples.
Table 1 shows the parameters of the IFU models of some typical samples. Using these input parameters, the effective flow diameter, tortuosity and cumulative flow of each IFU model can be calculated, and then, the total permeability of the sample can be determined.
Sample ID Number of fractal units Number of pores generated Iteration times Ratio factor Maximum pore diameter (μm) Number of units excluded from iteration FN16-19 Unit A 50 1 2 8 80 1 Unit B 1×104 3 4 4 10 1 Unit C 1.4×108 3 3 4 0.18 0 Unit D 1.6×109 8 1 3 0.03 0 M20-04 Unit A 4.2×105 1 2 5 1.26 1 Unit B 2.5×107 5 3 3 0.26 0 Unit C 2.3×109 3 2 3 0.037 0 Unit D 1.3×1010 1 1 3 0.02 0 B64-33 Unit A 6×103 8 2 3 5 0 Unit B 2.5×106 2 3 3 1.7 0 Unit C 2.2×109 3 3 2 0.05 0 Unit D 0 0 0 0 0 0 B65-8 Unit A 7×104 1 2 3 3.2 0 Unit B 2×106 2 3 4 1.1 0 Unit C 5×108 5 2 3 0.077 0 Unit D 1.6×109 2 2 4 0.04 0 B64-3 Unit A 3.5×104 3 2 3 2.1 0 Unit B 9×106 1 3 2 0.72 0 Unit C 1.2×108 2 3 2 0.19 0 Unit D 1.5×109 3 2 4 0.05 0 B64-38 Unit A 1×103 1 2 3 3.2 0 Unit B 1.4×107 1 6 2 1.1 0 Unit C 3.2×109 1 2 3 0.04 0 Unit D 0 0 0 0 0 0
Table 1. Parameters of the IFU models of some typical samples
Comparison between permeability results by experimental tests and model calculations are presented in Table 2 and Fig. 8, illustrating a good agreement between the predictions of the modified model and the experimental data. Additionally, the calculation accuracy of the improved model is higher than that of the Pia model, demonstrating its validity and reliability in simulating pore space of real porous media, with a relative error of less than 15%. Furthermore, it is found that lower permeability of samples leads to larger deviations in the Pia model. This is because the Pia model does not consider the effect of the boundary layer in tight reservoirs, causing the calculation error to be amplified in permeability prediction of dense rock samples.
Sample ID Experimental permeability(10-15 m2) Model-computed permeability and relative error Modified model Pia model Permeability(10-15 m2) Relative error (%) Permeability(10-15 m2) Relative error (%) FN16-19 18.6 18.698 0.5 21.617 16.2 M20-04 0.041 4 0.040 2 2.9 0.058 2 40.6 B64-33 3.056 3.051 0.2 3.695 20.9 B65-8 0.327 0.347 6.1 0.398 21.7 B64-3 0.183 0.202 10.4 0.244 33.3 B64-38 0.551 0.602 9.3 0.639 16.0 B64-42 0.368 0.387 5.2 0.455 23.6 B64-29 0.033 0.038 15.2 0.044 33.3 M5-1 0.403 0.411 2.0 0.496 23.1 M101-2-2 0.034 0.037 8.8 0.045 32.4 M5-6 0.005 4 0.006 11.1 0.008 48.1
Table 2. Comparison between model-computed permeability and experimental ones
Figure 8. Comparison between permeability results by experimental test and model calculations. Solid line represents the perfect agreement between the experimental data and model predictions.
Also, it is observed that the calculated values of the improved model are generally slightly larger than the experimental ones. This may be due to the development of microcracks in the rock samples caused by the excessive pressure in the MICP measurements, which in turn leads to larger calculations.
Besides, combined with the model parameters in Table 1, it can be seen that the IFU model with more large-sized fractal units will have higher permeability value, basically regardless of the number of small-sized units. This reveals that although a large number of micropores have large reservoir capacity, their contribution to permeability is quite limited, while the developmental degree of macropores with a relatively small volume fraction is an important factor in improving permeability. This is consistent with previous theoretical and experimental studies, demonstrating that the model can provide a better understanding of seepage characteristics and rules of reservoir rocks.
Iteration parameters control the microstructure and permeability of pore systems. An analysis of a specific fractal unit was performed to further illustrate the effects of these parameters on permeability. The primary iteration parameters are iteration times (i), the number of pores produced in the iteration (Np), the number of child squares excluded from iteration (Nsolid), and the ratio factor (b). The permeability of a single fractal unit is determined by Eq. (15).
Let Np=1, Nsolid=0 and b=3, the maximum pore diameter of the model varies between 0.001 to 100 μm. Figure 9 reveals that many small pores are generated in the model with the increase of iteration times, resulting in increases in porosity and permeability, and then, permeability tends to be a constant. This is because the newly produced micropores contribute little to permeability. Namely, it has a diminishing effect on permeability as the number of iteration times increases. Therefore, an excessive iterative process is unnecessary, which can also lead to a significant increase in the calculation amount of the model. In general, iteration times of 2 to 4 may be enough.
Let i=5, Nsolid=0 and b=3. Np has a direct influence on the porosity, and the number of pores in the model. Figure 10 shows that permeability increases significantly with the increase of Np. This is because greater values of Np allow the presence of more large pores, and more pores are generated in each iteration, leading to a higher permeability. Equation (15) indicates that a power function relationship exists between permeability and Np when the other parameters are fixed. Therefore, Np has a significant influence on permeability and should be given priority in the process of parameter adjustments. In other words, when we are going to reproduce the pore size distribution spectrum of some samples with high porosity, a vital issue we should consider is to adjust the value of Np to be large enough, which will facilitate the adjustment of other parameters.
Let i=5, Np=1 and b=3. The permeability tends to decrease slightly as Nsolid increases. This is because an increase in Nsolid means that a portion of the model will be excluded from iteration and retained permanently as matrix, resulting in a decrease in porosity and permeability. Equation (15) exhibits a linear relationship, rather than a power function relationship, between permeability and Nsolid when the other parameters are fixed. As a result, the adjustment of this parameter does not bring significant changes to the permeability (Fig. 11). In some cases, we can even ignore the effect of this parameter. Generally, a value of 0 or 1 is sufficient.
Let i=5, Np=1 and Nsolid=0. The ratio factor has a distinct influence on pore structure in the initial stage and permits a significant decrease in permeability. However, as the ratio factor continues to increase, the decrease in permeability becomes progressively smaller. This is because the larger the ratio factor leads to the smaller generated pores in the model with the same iteration times, and these microcapillaries have limited contribution to the permeability (Fig. 12). Although, the later stage of parameter adjustment for b have little effect on permeability, the effect in the initial stage (i.e., ranges from 2 to 6) is non-negligible and even decisive. The ratio factor allows for fine control of the microscopic pore structure, which theoretically explains the reason why the models with variable-ratio factor has more flexible simulation capabilities than the fixed-ratio factor ones.