IntroductionToledo et al. [1] reported that the wetting phase saturation can be described as function of capillary pressure and fractal dimension. Li and Horne [2] demonstrated that the Purcell model was found to be the best fit to the experimental data of the wetting phase relative permeability for the cases as long as the measured capillary pressure curve had the same residual saturation as the relative permeability curve. They also reported that in the reverse procedure, capillary pressure could also be computed once relative permeability data are available. Li and Willams [3] derived theoretically a model to correlate capillary pressure and resistivity index based on the fractal scaling theory. Their results demonstrated that the model could match the experimental data in a specific range of low water saturation. Zhang and Weller [4] showed the fractal dimension resulting from longer transverse NMR relaxation times and lower capillary pressure reflects the volume dimension of larger pores. They also reported that the fractal dimension derived from the short NMR relaxation times is similar to the fractal dimension of the internal surface. Wang et al [5] reported that the fractal dimensions can be used to represent the complexity degree and heterogeneity of pore structure, and the coexistence of dissolution pores and large intergranular pores of Donghetang sandstones contributes to a heterogeneous pore throat distribution and a high value of fractal dimension. Guo, et al. [6] studied the relationship among capillary pressure (PC), nuclear magnetic transverse relaxation time (T2) and resistivity index (I). An increase of bubble pressure fractal dimension and pressure head fractal dimension and decreasing pore size distribution index and fitting parameters m * n due to possibility of having interconnected channels was confirmed by Al-khidir [7,8] An increase of fractal dimension with increasing arithmetic, geometric relaxation time of induced polarization, permeability and grain size was investigated by Alkhidir [9-11]. An increase of seismo electric field and resistivity fractal dimensions with increasing permeability and grain size was described by Alkhidir [12-14]. An increase of electro kinetic fractal dimension with increasing permeability and grain size was reported by Alkhidir [15].
Material and Method
Samples were collected from the surface type section of the Shajara reservoirs of the Permo-carboniferous Shajara formation at latitude 26 degree 52' 17.4", longitude 43 degree 36' 18". Porosity was measured and permeability was derived from the measured capillary pressure data.
The electric power can be scaled as
Where Sw the water saturation, P the electric power in watt, Pmax the maximum electric power in watt, and Df the fractal dimension.
Equation 1 can be proofed from
Where J the electric current density in ampere / square meter, σ the electric conductivity in Siemens / meter. and E the electric field volt / meter.
But the electric conductivity σ can be scaled as
Where Σ the surface conductance in Siemens, and r the grain radius in meter.
The electric field E can be scaled as
Where V the electric potential in volt, and L the distance in meter.
Insert Equation 5 and Equation 4.
Where U the electric potential energy in Joule, and Q the electric charge in Coulomb.
Insert equation 7 into Equation 6.
The electric potential energy U can be scaled as
Where P the electric power in watt and t the time in second.
Insert equation 9 into Equation 8
Equation 10 after rearrange of grain radius r will become
The maximum grain radius rmax can be scaled as
Equation 13 after simplification will become
Take the logarithm of Equation 14
Insert equation 16 into Equation 15
Equation 17 after log removal will become
Equation 18 the proof of equation 1 which relates the water saturation Sw, the electric power P, the maximum electric power
Pmax , and the fractal dimension Df.
The capillary pressure can be scaled as
Where Sw the water saturation, Pc the capillary pressure and
Df the fractal dimension
Result and Discussion
Based on field observation the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units as designated in Figure 1. These units from bottom to top are: Lower Shajara Reservoir, Middle Shajara reservoir, and Upper Shajara Reservoir. Their developed results of the electric power fractal dimension and capillary pressure fractal dimension are shown in Table 1. Based on the achieved results it was found that the electric power fractal dimension is equal to the capillary pressure fractal dimension. The maximum value of the fractal dimension was found to be 2.7872 assigned to sample SJ13 from the Upper Shajara Reservoir as confirmed in Table 1.
Whereas the minimum value 2.4379 of the fractal dimension was recounted from sample SJ3 from the Lower Shajara reservoir as displayed in Table 1. The electric power fractal dimension and capillary pressure fractal dimension were witnessed to increase with increasing permeability as proofed in (Table 1) owing to the possibility of having interconnected channels.
The Lower Shajara reservoir was symbolized by six sandstone samples (Figure 1), four of which considered as SJ1, SJ2, SJ3 and SJ4 as confirmed in Table 1, were carefully chosen for capillary pressure measurement. Their positive slopes of the first procedure and negative slopes of the second procedure are delineated in Figures 2-5 and Table 1. Their electric power fractal dimension and capillary pressure fractal dimension values are proofed in Table 1. As we proceed from sample SJ2 to SJ3 a pronounced reduction in permeability due to compaction was reported from 1955 md to 56 md which reflects decrease in electric power fractal dimension and capillary pressure fractal dimension from 2.7748 to 2.4379 as specified in Table 1. Again, an increase in grain size and permeability was recorded from sample SJ4 whose electric power fractal dimension and capillary pressure fractal dimension was found to be 2.6843 as described in Table 1.
In contrast, the Middle Shajara reservoir which is separated from the Lower Shajara reservoir by an unconformity surface as shown in Figure 1. It was designated by four samples (Figure 1), three of which named as SJ7, SJ8, and SJ9 as illustrated in Table 1, were picked for capillary pressure measurement. Their positive slopes of the first procedure and negative slopes of the second procedure are displayed in Figures 6-8 and Table 1. Their electric power fractal dimensions and capillary pressure fractal dimensions show similarities as defined in Table 1. Their fractal dimensions are higher than those of samples SJ3 and SJ4 from the Lower Shajara Reservoir due to an increase in their permeability as elucidated in Table 1.
On the other hand, the Upper Shajara reservoir is separated from the Middle Shajara reservoir by yellow green mudstone as revealed in Figure 1. It is defined by three samples so called SJ11, SJ12, SJ13 as explained in Table 1. Their positive slopes of the first procedure and negative slopes of the second procedure are exhibited in Figures 9-11 and Table 1. Moreover, their electric power fractal dimension and capillary pressure fractal dimension are also higher than those of sample SJ3 and SJ4 from the Lower
Shajara Reservoir due to an increase in their permeability as explained in Table 1.
Overall a plot of electric power fractal dimension versus capillary pressure fractal dimension as shown in Figure 12, reveals three permeable zones of varying Petrophysical properties. Such variation in fractal dimension can account for heterogeneity which is a key parameter in reservoir quality assessment. This heterogeneity was also confirmed by plotting positive slopes of the first procedure versus negative slopes of the second procedure as proofed in Figure 13.
Conclusion
The sandstones of the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units based on electric power fractal dimension. The Units from bottom to top are: Lower Shajara Electric Power Fractal dimension Unit, Middle Shajara Electric Power Fractal Dimension Unit, and Upper Shajara Electric power Fractal Dimension Unit. These units were also proved by capillary pressure fractal dimension. The fractal dimension was found to increase with increasing grain size and permeability.
Acknowledgement
The author would like to thank King Saud University, College of Engineering, Department of Petroleum and Natural Gas Engineering, Department of Chemical Engineering, Research Centre at College of Engineering, and King Abdullah Institute for Research and Consulting Studies for their supports.