Introduction

Seismo electric effects related to electro kinetic potential, dielectric permittivity, pressure gradient, fluid viscosity, and electric conductivity was first reported by [1]. Capillary pressure follows the scaling law at low wetting phase saturation was reported by [2]. Seismo electric phenomenon by considering electro kinetic coupling coefficient as a function of effective charge density, permeability, fluid viscosity and electric conductivity was reported by [3]. The magnitude of seismo electric current depends on porosity, pore size, zeta potential of the pore surfaces, and elastic properties of the matrix was investigated by [4]. The tangent of the ratio of converted electric field to pressure is approximately in inverse proportion to permeability was studied by [5]. Permeability inversion from seismoelectric log at low frequency was studied by [6]. They reported that, the tangent of the ratio among electric excitation intensity and pressure field is a function of porosity, fluid viscosity, frequency, tortuosity, fluid density and Dracy permeability. A decrease of seismo electric frequencies with increasing water content was reported by [7]. An increase of seismo electric transfer function with increasing water saturation was studied by [8]. An increase of dynamic seismo electric transfer function with decreasing fluid conductivity was described by [9]. The amplitude of seismo electric signal increases with increasing permeability which means that the seismo electric effects are directly related to the permeability and can be used to study the permeability of the reservoir was illustrated by [10]. Seismo electric coupling is frequency dependent and decreases expontialy when frequency increases were demonstrated by [11]. An increase of permeability with increasing pressure head and bubble pressure fractal dimension was reported by [12,13]. An increase of geometric relaxation time of induced polarization fractal dimension with permeability increasing and grain size was described by [14,15].

Materials and Methods

Sandstone samples were collected from the surface type section of the Permo-Carboniferous Shajara Formation, latitude 26° 52’ 17.4”, longitude 43° 36’ 18” (Figure1). Porosity was measured on collected samples using mercury intrusion Porosimetry and permeability was derived from capillary pressure data. The purpose of this paper is to obtain luminous efficacy fractal dimension and to confirm it by capillary pressure fractal dimension. The fractal dimension of the first procedure is determined from the positive slope of the plot of logarithm of the ratio of luminous efficacy to maximum luminous efficacy log (Le^1/2/Le^1/2_max) versus log wetting phase saturation (logSw). Whereas the fractal dimension of the second procedure is determined from the negative slope of the plot of logarithm of log capillary pressure (log Pc) versus logarithm of wetting phase saturation (log Sw).

The Luminous efficacy can be scaled as

Where Sw the water saturation, Le luminous efficacy in lumen / watt, Le_max the maximum luminous in lumen / watt, Df the fractal dimension

Equation 1 can be proofed from

Where LP the luminous power in lumen, L the luminance in lumen / (square radian × square meter), SA the solid angle in square radian, A the projected area in square meter.

The luminous power can also be scaled as

Where LP the luminous power in lumen, RP the radiant power in watt, Le the luminous efficacy in lumen / watt.

Insert equation 3 into equation 2

The area A can be scaled as

Where A the area in square meter, r the pore radius in meter

Insert equation 5 into equation 4

The maximum pore radius can be scaled as

Divide equation 6 by equation 7

Equation 8 after simplification will be

Take the square root of equation 9

Equation 10 after simplification will be

Take the logarithm of equation 11

Insert equation 13 into equation 12

Equation 14 after log removal will become

Equation 15 the proof of equation 1 which relates the water saturation, luminous efficacy, maximum luminous efficacy, and the fractal dimension.

The capillary pressure can be scaled as

Where Sw the water saturation, Pc the capillary pressure and Df the fractal dimension.

Results and Discussion

Based on field observation the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation were divided here into three units as described in (Figure1). These units from bottom to top are: Lower Shajara Reservoir, Middle Shajara reservoir, and Upper Shajara Reservoir. Their attained results of the luminous efficacy fractal dimension and capillary pressure fractal dimension are shown in (Table 1). Based on the achieved results it was found that the luminous efficacy fractal dimension is equal to the capillary pressure fractal dimension. The maximum value of the fractal dimension was found to be 2.7872 allocated to sample SJ13 from the Upper Shajara Reservoir as verified in (Table 1). Whereas the minimum value of the fractal dimension 2.4379 was reported from sample SJ3 from the Lower Shajara reservoir as shown in (Table 1). The Luminous efficacy fractal dimension and capillary pressure fractal dimension were detected 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 label as SJ1, SJ2, SJ3 and SJ4 were carefully chosen for capillary pressure measurement as proven in (Table 1). Their positive slopes of the first procedure log of the Luminous efficacy to maximum Luminous efficacy versus log wetting phase saturation (Sw) and negative slopes of the second procedure log capillary pressure (Pc) versus log wetting phase saturation (Sw) are clarified in (Figures 2-5) and (Table 1). Their Luminous efficacy fractal dimension and capillary pressure fractal dimension values are revealed in (Table 1). As we proceed from sample SJ2 to SJ3 a pronounced reduction in permeability due to compaction was described from 1955 md to 56 md which reflects decrease in Luminous efficacy fractal dimension from 2.7748 to 2.4379 as quantified in (Table 1). Again, an increase in grain size and permeability was proved from sample SJ4 whose luminous efficacy 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 revealed in (Figure 1). It was nominated by four samples (Figure 1), three of which named as SJ7, SJ8, and SJ9 as illuminated in (Table1) were chosen for capillary measurements as described in (Table 1). Their positive slopes of the first procedure and negative slopes of the second procedure are shown in (Figures 6-8) (Table 1). Furthermore, their Luminous efficacy 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 explained in (Table 1).

On the other hand, the Upper Shajara reservoir was separated from the Middle Shajara reservoir by yellow green mudstone as shown 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 displayed in (Figures 9-11) and (Table 1). Moreover, their luminous efficacy 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 simplified in (Table 1).

Overall a plot of positive slope of the first procedure versus negative slope of the second procedure as described in (Figure 12) reveals three permeable zones of varying Petrophysical properties. These reservoir zones were also confirmed by plotting luminous efficacy fractal dimension versus capillary pressure fractal dimension as described in (Figure 13). Such variation in fractal dimension can account for heterogeneity which is a key parameter in reservoir quality assessment.

Conclusion

The sandstones of the Shajara Reservoirs of the permo-Carboniferous Shajara Formation were divided here into three units based on luminous efficacy fractal dimension. The Units from base to top are: Lower Shajara Luminous Efficacy Fractal Dimension Unit, Middle Shajara Luminous Efficacy Fractal Dimension Unit, and Upper Shajara Luminous Efficacy 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 owing to possibility of having interconnected channels.

Acknowledgement

The author would 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, College of science, Department of Geology, and King Abdullah Institute for research and Consulting Studies for their supports.

Figure 1: Surface type section of the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation at latitude 26° 52' 17.4" longitude 43° 36' 18".

Figure 2: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ1.

Figure 3: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ2.

Figure 4: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ3.

Figure 5: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ4.

Figure 6: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ7.

Figure 7: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ8.

Figure 8: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ9.

Figure 9: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ11.

Figure 10: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ12.

Figure 11: Log (Le1/2/Le1/2max) & log pc versus log Sw for sample SJ13.

Figure 12: Slope of the first procedure versus slope of the second procedure.

Figure 13: Luminous efficacy fractal dimension versus capillary pressure fractal dimension.

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