Sw= [H_s,r^1/2/H^1/2_S,r,max]^{[3-Df]                                 (1)}

Where Sw the water saturation, H the seismo magnetic field in ampere/meter generated from the shear wave velocity, H_max the maximum seismo magnetic field in ampere/meter generated from the shear wave velocity and Df the fractal dimension.

“Eq. (1)” can be proofed from:

                  H_{S, ϴ}= [Ф * kf * ζ * ρf √G/ρ * dU_S,r/dt / α_∞ * η]                     (2)

Where H_S,ϴ the seismo magnetic field generated from shear wave velocity in ampere/meter, Φ the porosity, ε0 permittivity of free scape in Faraday/meter, kf dielectric constant of the fluid, ζ the zeta potential in volt, ρf the fluid density in kilogram/meter, G the shear modulus in pascal, ρ bulk density in kilogram/m³, dU_S,r/dt the radial grain velocity in meter/second, α_∞ the tortuosity and η the fluid viscosity in pascal * second.

The zeta potential can be scaled as:

         ζ = [C_s * σf * η/ εf]                                (3)

Where ζ the zeta potential in volt, CS streaming potential coefficient in volt/pascal, σf the fluid electric conductivity in Siemens/meter, η the fluid viscosity in pascal * second, εf the fluid permittivity in Faraday/meter.

Insert “Eq. (3)” in “Eq. (2)”:

                H_{S, ϴ}= [Ф * ε0 * kf * C_s * σf * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η]   (4)

The streaming potential coefficient can be scaled as:

             C_S = [reff^{2 *} C_E/ 8 * σf * η]                        (5)

Where CS the streaming potential coefficient in volt/pascal, r_eff the effective pore radius in meter, CE the electro osmosis coefficient in pascal/volt, σf the fluid conductivity in Siemens/meter and η the fluid viscosity in pascal * second.

Insert “Eq. (5)” into “Eq. (4)”: 

           H_{S, ϴ}= [Ф * ε0 * kf * reff² * C_E σf  * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η * 8 * σf * η]   (6)

If the pore radius is introduced equation 6 will become,

H_{S, ϴ}= [Ф * ε0 * kf * r² * C_E * σf  * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η * 8 * σf * η]   (7)

The maximum pore radius can be scaled as:

H_S,r,max = [Ф * ε0 * kf * r²_max * C_E * σf  * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η * 8 * σf * η]  (8)

Divide “Eq. (7)” by “Eq. (8)”: 

[H_{S, ϴ} / H_S,r,max ] = [ [Ф * ε0 * kf * r² * C_E * σf  * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η * 8 * σf * η] / [Ф * ε0 * kf * r²_max * C_E * σf  * η * ρf √G/ρ * dU_S,r/dt / α_∞ εf * η * 8 * σf * η] ]

  (9)

“Eq. (9)” after simplification will become,

                      [H_{S, ϴ} / H_S,r,max ] = [r² / r²_max]                                      (10)

Take the square root of “Eq. (10)”:

                   √ [H_{S, ϴ} / H_S,r,max ] = √ [r² / r²_max ]                       (11)

“Eq. (11)” after simplification will become,

          [H_{S, ϴ}^1/2 / H_S,r,max ^1/2] = [r / r_max]                                                (12)

Take the logarithm of “Eq. (12)”:

             log [H_{S, ϴ}^1/2 / H_S,r,max ^1/2] = [r / r_max]                                 (13)

But;                            log [r / r_max] = log Sw / [3-Df]                            (14)

Insert “Eq. (14)” into “Eq. (13)”:

                            log Sw / [3-Df] = log [H_{S, ϴ}^1/2 / H_S,r,max ^1/2]                                        (15)

After log removal “Eq. (15)”" will become,

                 Sw = [H_{S, ϴ}^1/2 / H_S,r,max ^1/2] ^{[3-Df]                                                        (16)}

“Eq. (16)” the proof of equation 1 which relates the water saturation, seismo magnetic field, maximum seismo magnetic field generated from the shear wave and the fractal dimension.

The capillary pressure can be scaled as:

              Sw = [Df -3 ] * Pc * constant                   (17)

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 Figure 1. These units from bottom to top are: Lower Shajara Reservoir, Middle Shajara reservoir and Upper Shajara Reservoir. Their acquired results of the seismo magnetic field fractal dimension and capillary pressure fractal dimension are displayed in Table 1. Based on the attained results it was found that the seismo magnetic field 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 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 displayed in Table 1. The seismo magnetic field fractal dimension and capillary pressure fractal dimension were observed to increase with increasing permeability as proofed in Table 1 owing to the possibility of having interconnected channels.  

The Lower Shajara reservoir was denoted by six sandstone samples (Figure 1), four of which label as SJ1, SJ2, SJ3 and SJ4 were carefully chosen for capillary pressure measurement as established in Table 1. Their positive slopes of the first procedure log of the seismo magnetic field (H) to maximum seismo magnetic field (H_max) versus log wetting phase saturation (Sw) and negative slopes of the second procedure log capillary pressure (Pc) versus log wetting phase saturation (Sw) are explained in Figures 2-5 and Table 1. Their seismo magnetic field 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 seismo magnetic field 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 seismo magnetic field fractal dimension and capillary pressure fractal dimension was found to be 2.6843 as pronounced 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 nominated by four samples (Figure 1), three of which named as SJ7, SJ8 and SJ9 as clarified in Table 1 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 and Table 1. Furthermore, their seismo magnetic field 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 seismo magnetic field 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 seismo magnetic field 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