Seismo Electric Field Fractal Dimension for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia

Introduction

Seismo electric effects related to electro kinetic potential, dielectric permitivity, pressure gradient, fluid viscosity, and electric conductivty was first reported by Frenkel J [1]. Capillary pressure follows the scaling law at low wetting phase saturation was reported by Li K, et al. [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 Revil A, et al. [3]. The magnitude of seismo electric current depends porosity, pore size, zeta potential of the pore surfaces, and elastic properties of the matrix was investigated by Dukhin A, et al. [4]. The tangent of the ratio of converted electic field to pressure is approximately in inverse proportion to permeability was studied by Guan W, et al. [5]. Permeability inversion from seismoelectric log at low frequency was studied by Hu H, et al. [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 reportet by Borde C, et al. [7]. An increase of seismo electric transfer function with increasing water saturation was studied by Jardani A, et al. [8]. An increase of dynamic seismo electric transfer function with decreasing fluid conductivity was described by Holzhauer J, et al. [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 Rong P, et al. [10]. Seismo electric coupling is frequency dependent and decreases expontialy when frequency increases was demonstrated by Djuraev U, et al. [11]. An increase of permeability with increasing pressure head and bubble pressure fractal dimension was reported by Alkhidir KEME [12]. An increase of geometric and arithmetic relaxtion tiome of induced polarization fractal dimension with permeability increasing was described by Alkhidir KEME [13].

Material and Methods

Porosity was measured on collected sandstone samples and permeability was calculated from the measured capillary pressure by mercury intrusion techniques. Two procedures for obtaining the fractal dimension have been developed. The first procedure was done by plotting the logarithm of the ratio between seismo electric field and maximum seismo electric field versus logarithm wetting phase saturation. The slope of the first procedure = 3- Df (fractal dimension). The second procedure for obtaining the fractal dimension was completed by plotting the logarithm of capillary pressure versus the logarithm of wetting phase saturation. The slope of the second procedure = Df -3. The seismo electric field can be scaled as [𝟑−𝐃𝐟]

𝐒𝐰= [ 𝐄[𝟏

𝟑]

Where Sw the water saturation, E the seismo electric field in volt / meter, Emax, the maximum seismo electric field in volt / meter, and Df the fractal dimension. Equation 1 can be proofed from 𝐄= [𝛆𝐟∗𝛇∗𝛒𝐟∗ϋ 𝛈∗𝛔𝐟 ] 𝟐 Where E the seismo electric field in volt / meter, εf dielectric permittivity of the fluid, ζ the zeta potential in volt, ρf density of the fluid in kilogram / cubic meter, ϋ the seismo electric acceleration in meter / second square, η the fluid viscosity in pascal second, and σf the fluid conductivity in Siemens /meter.

𝐁𝐮𝐭, 𝛆𝐟∗𝛇

𝛈∗𝛔𝐟= 𝐂𝐒 𝟑

Where CS, the streaming potential coefficient in volt / pascal Insert equation 3 into equation 2 𝐄= 𝐂𝐬∗𝛒𝐟∗ϋ 𝟒 The streaming potential can be scaled as 𝐂𝐒= 𝐕 𝐐 𝟓 Where V the volume in cubic meter, Q the electric charge in coulomb Insert equation 5 into Equation 4 𝐄= [𝐕∗𝛒𝐟∗ϋ 𝐐 ] 𝟔 The volume V can be scaled as 𝐕= 𝟒

𝟑∗𝟑. 𝟏𝟒∗𝐫𝟑 𝟕 Where r the pore radius in meter Insert equation 7 into equation 6

𝐄= [𝟒∗𝟑. 𝟏𝟒∗𝐫𝟑∗𝛒𝐟∗ϋ

𝟑∗𝐐 ] 𝟖 The maximum pore radius rmax can be scaled as

𝟑 ∗𝛒𝐟∗ϋ 𝟑∗𝐐 ] 𝟗 𝐄𝐦𝐚𝐱= [𝟒∗𝟑. 𝟏𝟒∗𝐫𝐦𝐚𝐱

Divide equation 8 by equation 9

𝟒∗𝟑.𝟏𝟒∗𝐫𝟑∗𝛒𝐟∗ϋ [ 𝐄

𝟑∗𝐐 ] 𝟏𝟎 ] = [

𝟒∗𝟑.𝟏𝟒∗𝐫𝐦𝐚𝐱 𝟑 ∗𝛒𝐟∗ϋ 𝐄𝐦𝐚𝐱

𝟑∗𝐐 Equation 10 after simplification will become ] = [ 𝐫𝟑 [ 𝐄

𝐫𝐦𝐚𝐱 𝟑 ] 𝟏𝟏 𝐄𝐦𝐚𝐱

Take the third root of equation 11

= √[ 𝐫𝟑 √[ 𝐄

] 𝟑 𝐫𝐦𝐚𝐱 𝟑 ] 𝟑

𝟏𝟐 𝐄𝐦𝐚𝐱

Equation 12 after simplification will become [ 𝐄[𝟏

𝟑]

𝟑] ] = [ 𝐫 ] 𝟏𝟑

[𝟏 𝐫𝐦𝐚𝐱 𝐄𝐦𝐚𝐱

But, Log [r /rmax] = log Sw/[3-Df] 14 Insert equation 14 into equation 13

Log Sw/[3-Df]=log [E1/3 / Emax1/3]15 Equation 15 after log removal will become [𝟑−𝐃𝐟]

𝐒𝐖= [ 𝐄[𝟏

𝟑]

Equation 16 the proof of equation 1 which relates the water saturation, seismo electric field, maximum seismo electric field, and the fractal dimension. The capillary pressure can be scaled as 𝐥𝐨𝐠𝐒𝐰= (𝐃𝐟−𝟑) ∗𝐥𝐨𝐠𝐏𝐜+ 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟏𝟕 Equation 17 can be proofed from the number of pores theorey N (r ) ∝𝐫−𝐃𝐟 𝟏𝟖 Where N( r ) number of pores, r the pore throat radius and Df is the fractal dimension.

𝐁𝐮𝐭,𝐍 (𝐫) = [𝐕𝐇𝐠

] 𝟏𝟗 𝐕𝐜𝐞𝐥𝐥

Where VHg the volume of mercury intruded the pores, Vcell the volume of unit cell. Insert equation 19 into equation 18 [𝐕𝐇𝐠 ] ∝𝐫−𝐃𝐟 𝟐𝟎 𝐕𝐜𝐞𝐥𝐥 If we consider the unit cell as sphere , then the volume of sphere can be scaled as:

𝐕𝐜𝐞𝐥𝐥= 𝟒

𝟑∗𝟑. 𝟏𝟒∗𝐫𝟑 𝟐𝟏 Insert equation 21 into equation 20

[ 𝐕𝐇𝐠 𝟒

𝟑∗𝟑. 𝟏𝟒∗𝐫𝟑] ∝𝐫−𝐃𝐟 𝟐𝟐 Equation 22 can also be written as 𝐕𝐇𝐠∝𝐫𝟑−𝐃𝐟 𝟐𝟑 The pore radius can be scaled as 𝐫= 𝟐∗𝛔∗𝐜𝐨𝐬𝚹

𝟐𝟒 𝐏𝐜

Where σ mercury surface tension, ϴ mercury contact angle and pc the capillary pressure. Insert equation 24 into equation 23 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, Middle, and Upper Shajara Reservoir. Their acquired results of the seismo electric fractal dimension and capillary pressure fractal dimension are displayed in Table 1. Based on the attained results it was found that the seismo electric 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 electric fractal dimension and capillary pressure fractal dimension were observed to increase with increasing permeability as proofed in Table1 owing to the possibility of having interconnected channels.

𝐃𝐟−𝟑 𝟐𝟓 Differentiate equation 25 with respect to Pc

𝐕𝐇𝐠∝𝐏𝐜

𝐝𝐕𝐇𝐠 𝐃𝐟−𝟒 𝟐𝟔

∝(𝐃𝐟−𝟑) ∗𝐏𝐜 𝐝𝐏𝐜 If we remove the proportionality sign from equation 26 we have to multiply by a constant 𝐝𝐕𝐇𝐠 𝐃𝐟−𝟒 𝟐𝟕 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭∗𝐏𝐜 𝐝𝐏𝐜 If wetting phase saturation (water saturation) is introduced, then equation 27 will be written as 𝐝𝐒𝐰 𝐃𝐟−𝟒 𝟐𝟖 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭∗𝐏𝐜 𝐝𝐏𝐜 Integrate equation 28 𝐃𝐟−𝟒∗𝐝𝐏𝐜 𝟐𝟗 ∫𝐝𝐒𝐰= ∫𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭∗𝐏𝐜 Equation 29 after integration will become 𝐃𝐟−𝟑 𝟑𝟎 Take the logarithum of equation 30 𝐒𝐰= 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭∗𝐏𝐜 𝐥𝐨𝐠𝐒𝐰= (𝐃𝐟−𝟑) ∗𝐥𝐨𝐠𝐏𝐜+ 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟑𝟏 Equation 31 the proof of equation 17 which relates the water saturation, capillary pressure and the fractal dimension.

Results and Discussion

Click to enlarge

dimension values are shown 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 seismo electric field fractal dimension from 2.7748 to 2.4379 as specified in Table 1. Again, an increase in grain size and permeability was verified from sample SJ4 whose seismo electric field fractal dimension and capillary pressure fractal dimension was found to be 2.6843 as described in Table 1.

Conclusions

❖ The sandstones of the Shajara Reservoirs of the permo- Carboniferous Shajara formation were divided here into three units based on seismo electric field fractal dimension. ❖ The Units from bottom to top are: Lower Shajara seismo electric Field Fractal dimension Unit, Middle Shajara Seismo Electric Field Fractal Dimension Unit, and Upper Shajara Seismoelectric 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.

Acknowledgment

The author would like to thanks 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.