1. On the equality of resistivity fractal dimension and geometric relaxation time fractal dimension of induced polarization for characterizing Shajara Reservoirs of the Shajara Formation of the Permo-Carboniferous Unayzah Group, Saudi Arabia.” Professor: Khalid Elyas Mohamed Elameen Alkhidir College of Engineering, King Saud university, Saudi Arabia

2. Abstract
Sandstone samples were collected from the surface type section of the Shajara Formation of the Permo-Carboniferous Unayzah Group for detailed reservoir characterization. Capillary pressure experiment was performed to contact porosity and permeability was derived from the Data. Resistivity was calculated from the distribution of pores and the fractal dimension was proven from the relationship between wetting phase saturation and resistivity. In addition to field observation and obtained results of fractal dimension, the Shajara reservoirs of the Shajara Formation of the permo-Carboniferous Unayzah Group were divided here into three fractal dimension units. The Units from base to top are: Lower Shajara Resistivity Fractal dimension Unit, Middle Shajara Resistivity Fractal Dimension Unit, and Upper Shajara Resistivity Fractal Dimension Unit. These units were also proved by geometric relaxation time of induced polarization fractal dimension. It was found that the resistivity fractal dimension is similar to the geometric relaxation time of induced polarization. It was also reported that the obtained fractal dimension speeds with increasing resistivity and relaxation time due to an increase in pore connectivity.    

3. Al-Khidir et al 2010 Reservoirs Heterogeneity Characterization of the Shajara Member: Permo-Carboniferous Unayzah Formation. The 2nd Saudi meeting on Oil and Natural Gas Exploration and production Technologies King Fahad university of Petroleum and minerals Campus, Dhahran, saudi Arabia December , Al-khidir et al 2011 Bimodal pore size behavior of the Shajara Formation Reservoirs of the Permo-Carboniferous Unayzah Group, Saudi Arabia

4. Journal of petroleum exploration and Production technology http://www
Journal of petroleum exploration and Production technology Al-khidir et al 2012 Characterization of heterogeneity of the Shajara reservoirs of the Shajara formation of the Permo-Carboniferous Unayzah group. Arabian Journal of Geoscienes Al-Khidir et al Geoscience; New Findings Reported from King Saud University Describe Advances in Geoscience Science Letter (Oct 25, 2013):

5. Al-Khidir et al 2013 Differential Capacity Fractal Dimension and Water Saturation Fractal Dimension as Parameters for Reservoir Characterization: Shajara Formation of the Permo-Carboniferous Unayzah Group as a Case Study. 10th Meeting of the Saudi Society for Geoscience “Geosciences for Sustainable Development” April, 2013 King Fahad university of petroleum and minerals Campus, Dhahran, Saudi Arabia.

6. Al-Khidir 2015 Induced Polarization Relaxation Time Fractal Dimension Derived from Capillary pressure data for characterizing Shajara Reservoirs of the shajara Formation of the Permo-carboniferous Unayzah Group. The Eleventh International Geological Conference 23 – 25 Rajab – 14 May

7. Al-Khidir 2017 Nuclear magnetic resonance relaxation time as a diagnostic parameter for reservoir .International Conference on Petrochemical Engineering July 10-12, 2017 Dubai, UAE. Al-Khidir 2017 Pressure Head Fractal Dimension for Characterizing Shajara Reservoirs of the Shajara Formation of the Permo-Carboniferous Unayzah Group, Saudi Arabia. Archives of Petroleum Environmental Biotechnology.

8. Theoretical background
The resistivity can be scaled as
𝐒 𝐰 = 𝛀 𝛀 𝐦𝐚𝐱 𝟑−𝐃𝐟 𝟏
Where Sw is the water saturation; Ω is the resistivity in ohm meter; Ω max is the maximum resistivity and Df is the fractal dimension.
Equation 1 can be proofed from
𝐤= 𝟏 𝟏𝟐𝟎 ∗ 𝟏 𝐅 𝟑 ∗ 𝟏 𝛔 𝟐 𝟐
Where k is the permeability in millidarcy (mad) ; 1/120 is a constant ; F is the formation electrical resistivity factor in zero dimension ; and σ is the quadrature conductivity.
𝐁𝐮𝐭; 𝟏 𝛔 𝟐 = 𝛀 𝟐 𝟑
Insert equation 3 into equation 2
𝒌=[ 𝟏 𝟏𝟐𝟎 ∗ 𝟏 𝑭 𝟑 ∗ 𝜴 𝟐 ] 𝟒

9. 𝐈𝐟 𝟏 𝟏𝟐𝟎 ∗ 𝟏 𝐅 𝟑 ∗ 𝛀 𝟐 = 𝐫 𝟐 𝟖∗𝐅 =𝐤 𝟓 r in equation 5 is the pore throat radius. Equation 5 after rearrange will become 𝟖∗𝐅∗ 𝛀 𝟐 = 𝟏𝟐𝟎∗ 𝐅 𝟑 ∗ 𝐫 𝟐 𝟔 Equation 6 after simplification will result in 𝛀 𝟐 = 𝟏𝟓∗ 𝐅 𝟐 ∗ 𝐫 𝟐 𝟕 Take the square root of both sides of Equation 7 𝛀 𝟐 = 𝟏𝟓∗ 𝐅 𝟐 ∗ 𝐫 𝟐 8

10. Equation 8 after simplification will become 𝛀= 𝟏𝟓 ∗𝐅∗𝐫 𝟗 The pore throat radius r can be scaled as 𝐯∝ 𝐫 𝟑−𝐃𝐟 𝟏𝟎 Where v is the cumulative pore volume. Differentiate equation 10 with respect to r 𝐝𝐯 𝐝𝐫 ∝ 𝐫 𝟐−𝐃𝐟 𝟏𝟏 Integrate equation 11 𝐝𝐯=𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭∗ 𝐫𝐦𝐢𝐧 𝐫 𝐫 𝟐−𝐃𝐟 ∗𝐝𝐫 𝟏𝟐

11. 𝐯= 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟑−𝐃𝐟 ∗ 𝐫 𝟑−𝐃𝐟 − 𝐫 𝐦𝐢𝐧 𝟑−𝐃𝐟 𝟏𝟑 The total pre volume can be integrated as follows: 𝐯𝐭𝐨𝐭𝐚𝐥= 𝐫𝐦𝐢𝐧 𝐫𝐦𝐚𝐱 𝐫 𝟐−𝐃𝐟 ∗𝐝𝐫 𝟏𝟒 The result of total pore volume integral 𝐯 𝐭𝐨𝐭𝐚𝐥 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟑−𝐃𝐟 ∗ 𝐫 𝐦𝐚𝐱 𝟑−𝐃𝐟 − 𝐫 𝐦𝐢𝐧 𝟑−𝐃𝐟 𝟏𝟓 Divide equation 13 by equation 15 𝒗 𝐯 𝐭𝐨𝐭𝐚𝐥 = 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟑−𝐃𝐟 ∗ 𝐫 𝟑−𝐃𝐟 − 𝐫 𝐦𝐢𝐧 𝟑−𝐃𝐟 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭 𝟑−𝐃𝐟 ∗ 𝐫 𝐦𝐚𝐱 𝟑−𝐃𝐟 − 𝐫 𝐦𝐢𝐧 𝟑−𝐃𝐟 𝟏𝟔 Equation 16 after simplification will become

12. 𝐒𝐰= 𝐫 𝐫 𝐦𝐚𝐱 𝟑−𝐃𝐟 𝟏𝟕 Insert equation 9 into equation 17 𝐒𝐰= 𝛀 𝟏𝟓 ∗𝐅 𝛀 𝐦𝐚𝐱 𝟏𝟓 ∗𝐅 𝟑−𝐃𝐟 𝟏𝟖 Equation 18 after simplification will become 𝐒𝐰= 𝛀 𝛀 𝐦𝐚𝐱 𝟑−𝐃𝐟 𝟏𝟗 Equation 19 is the proof of equation 1

13. 𝐒𝐰= 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓𝐠𝐦𝐚 𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟑−𝐃𝐟 𝟏 𝐤=𝟗.𝟔∗ 𝐈𝐏𝐓𝐠∗ 𝚽 𝟒 𝟏.𝟓𝟕 𝟐
The geometric relaxation time of induced polarization can be scaled as
𝐒𝐰= 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓𝐠𝐦𝐚 𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟑−𝐃𝐟 𝟏
Where Sw the water saturation; IPTg the geometric relaxation time of induced polarization;IPTgmax the maximum geometric relaxation time of induced polarization; and Df the fractal dimension.
Equation 1 can be proofed from
𝐤=𝟗.𝟔∗ 𝐈𝐏𝐓𝐠∗ 𝚽 𝟒 𝟏.𝟓𝟕 𝟐
Where k the permeability in millidarcy; 9.6 constant ; IPTg geometric relaxation time of induced polarization in milliseconds; Φ the porosity; and 1.57 constant.
The maximum permeability can be scaled as
𝐤𝐦𝐚𝐱=𝟗.𝟔∗ 𝐈𝐏𝐓𝐠𝐦𝐚𝐱∗ 𝚽 𝟒 𝟏.𝟓𝟕 𝟑
Divide equation 2 by equation 3
𝐤 𝐤𝐦𝐚𝐱 = 𝟗.𝟔∗ 𝐈𝐏𝐓𝐠∗ 𝚽 𝟒 𝟏.𝟓𝟕 𝟗.𝟔∗ 𝐈𝐏𝐓𝐠𝐦𝐚𝐱∗ 𝚽 𝟒 𝟏.𝟓𝟕 𝟒

14. Equation 4 after simplification will become 𝐤 𝐤𝐦𝐚𝐱 = 𝐈𝐏𝐓𝐠 𝟏
Equation 4 after simplification will become 𝐤 𝐤𝐦𝐚𝐱 = 𝐈𝐏𝐓𝐠 𝟏.𝟓𝟕 𝐈𝐏𝐓𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟓 Take the square root of equation 5 𝐤 𝐤𝐦𝐚𝐱 = 𝐈𝐏𝐓𝐠 𝟏.𝟓𝟕 𝐈𝐏𝐓𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟔 Equation 6 can also be written as 𝐤 𝟏 𝟐 𝐤𝐦𝐚 𝐱 𝟏 𝟐 = 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓 𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟕 Take the Logarithm of equation 7 𝐥𝐨𝐠 𝐤 𝟏 𝟐 𝐤𝐦𝐚 𝐱 𝟏 𝟐 = 𝐥𝐨𝐠 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓 𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟖

15. 𝑩𝐮𝐭; 𝐥𝐨𝐠 𝐤 𝟏 𝟐 𝐤𝐦𝐚 𝐱 𝟏 𝟐 = 𝐥𝐨𝐠 𝐒𝐰 𝟑−𝐃𝐟 𝟗 Insert equation 9 into equation 8 𝐥𝐨𝐠 𝐒𝐰 𝟑−𝐃𝐟 = 𝐥𝐨𝐠 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓 𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟏𝟎 If we remove the Log from equation 10 𝐒𝐰= 𝐈𝐏𝐓 𝐠 𝟏.𝟓𝟕 𝟏 𝟐 𝐈𝐏𝐓 𝐠𝐦𝐚𝐱 𝟏.𝟓𝟕 𝟏 𝟐 𝟑−𝐃𝐟 𝟏𝟏 Equation 11 the proof of equation 1 which relates the water saturation; the geometric relaxation time of induced polarization; the maximum geometric relaxation time of induced polarization; and the fractal dimension

16. Sample SJ1;Φ=29%;K=1680mD; Df=2.7859

17. Sample SJ2;Φ=35%;K=1955mD; Df=2.7748

18. Sample SJ3;Φ=34%;K=56mD; Df=2.4379

19. Sample SJ4;Φ=30%;K=176mD; Df=2.6843

20. Sample SJ7;Φ=35%;K=1472mD; Df=2.7683

21. Sample SJ8;Φ=32%;K=1344mD; Df=2.7752

22. Sample SJ9;Φ=31%;K=1394mD; Df=2.7786

23. Sample SJ11;Φ=36%;K=1197mD; Df=2.7586

24. Sample SJ12;Φ=28%;K=1440mD; Df=2.7859

25. Sample SJ13;Φ=25%;K=973mD; Df=2.7872

26. Petrophysical model showing Resistivity Fractal Dimension and Geometric Relaxation Time Fractal Dimension of Induced Polarization
Sample Number
Porosity ( %)
Permeability (mad)
Resistivity Fractal Dimension
Geometric Relaxation Time Fractal Dimension Induced Polarization
SJ1 29
1680
2.7859
SJ2 35
1955
2.7748
SJ3 34 56
2.4379
SJ4 30
176
2.6843
SJ7
1472
2.7683
SJ8 32
1344
2.7752
SJ9 31
1394
2.7786
SJ11 36
1197
2.7586
SJ12 28
1440
SJ13 25
973
2.7872

27. Resistivity fractal dimension versus geometric relaxation time fractal dimension of induced polarization showing the thee Shajara Reservoirs of the shajara Formation of the Permo-Carboniferous Unayzag Group, saudi arabia

28. Conclusions
The sandstones of the Shajara Reservoirs of the Shajara formation of the permo-Carboniferous unayzah group were divided here into three units based on resistivity fractal dimension.
The Units from base to top are: Lower Shajara Resistivity Fractal dimension Unit, Middle Shajara Resistivity Fractal Dimension Unit, and Upper Shajara Resistivity Fractal Dimension Unit.

29. These units were also proved by geometric relaxation time of induced polarization fractal dimension.
The fractal dimension was found to increase with increasing grain size and permeability.

30. Thank You