Electric potential energy fractal dimension for characterizing shajra reservoir

1. A SciTechnol JournalResearch Article
Alkhidir, Expert Opin Environ Biol 2018, 7:2
DOI: 10.4172/2325-9655.1000156
Expert Opinion on
Environmental Biology
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Electric Potential Energy Fractal
Dimension for Characterizing
Permo-carboniferous Shajara
Formation
Khalid Elyas Mohamed Elameen Alkhidir*
Abstract
In this research, porosity was measured on real collected
sandstone samples and permeability was calculated theoretically
from capillary pressure profile measured by mercury intrusion
contaminating the pores of sandstone samples in consideration.
Two equations for calculating the fractal dimensions have been
employed. The first one describes the functional relationship
between water saturation, electric potential energy, maximum
electric potential energy and fractal dimension. The second
equation implies to water saturation as a function of capillary
pressure and fractal dimension. Two techniques for obtaining
the fractal dimension have been used. The first was completed
by plotting the logarithm of the ratio between electric potential
energy and maximum electric potential energy versus logarithm
water saturation. The second was prepared by plotting logarithm
capillary pressure versus logarithm water saturation. The
slope of the first technique=3-Df (fractal dimension). Whereas,
the slope of the second procedure=Df-3 The results exhibited
similarity between electrical potential energy fractal dimension
and capillary pressure fractal dimension.
Keywords
Electric potential energy; Fractal dimension; Water saturation
*Corresponding author: Khalid Elyas Mohamed Elameen Alkhidir, Department
of Petroleum and Natural Gas Engineering, College of engineering, King Saud
University, Riyadh, Saudi Arabia, Tel: +996114679118; E-mail: kalkhidir@ksu.edu.sa
Received: March 21, 2018 Accepted: May 01, 2018 Published: May 07, 2018
The quality of the Shajara Formation reservoirs increases with the
increase in grain size and grain sorting was reported by Al-Khidir et
al. [7]. Fractal dimensions indicate the heterogeneity of the Shajara
Reservoirs which account for a wide range of pore sizes was described
by Al-Khidir et al. [8]. An increase of bubble pressure fractal
dimension and pressure head fractal dimension with decreasing
pore size distribution index and fitting parameters m*n owing to
possibility of having interconnected channels was confirmed by Al-
Khidir et al. [9]. An increase of fractal dimension with increasing in
grain size and permeability was testified by Al-khidir [10].
Materials and Methods
Sandstone samples were collected from the surface type section of
the Permo-Carboniferous Shajara Formation (Figure 1). 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 electric potential energy fractal
dimension and to confirm it by capillary pressure fractal dimension.
The fractal dimension of the first procedure is determined from
the slope of log (UE
/UEmax
) versus log log Sw
. However, the fractal
dimension of the second procedure is obtained from graphical plot of
log Pc versus log Sw whose slope =fractal dimension–3.
Introduction
The water saturation can be described as function of capillary
pressure and fractal dimension [1]. A model to calculate relative
permeability by choosing a suitable capillary pressure technique,
especially in drainage processes was developed by Li et al. [2]. A model
whichhasbeenderivedtheoreticallytocorrelatecapillarypressureand
resistivity index based on the fractal scaling theory was demonstrated
by Li et al. [3]. A fractal dimension value was successfully applied in
a model of permeability prediction that is based on formation factor
and specific surface area (Spor) was investigated by Zhang et al. [4]. A
high fractal dimension value which be used to represent the degree of
heterogeneity of pore structure, and the presence of dissolution pores
and large intergranular pores donates to a heterogeneous pore throat
distribution was studied by Wang et al. [5]. A segmentation method
which is more suitable for low permeability samples was adopted by
Guo et al. [6] to obtain fractal dimension.
Figure 1: Surface type section of the Permo-carboniferous Shajara
Formation latitude 26°52′ 17.4″, longitude 43° 36′ 18″.

2. Citation: Alkhidir KEME (2018) Electric Potential Energy Fractal Dimension for Characterizing Permo-carboniferous Shajara Formation. Expert Opin
Environ Biol 7:2.
• Page 2 of 5 •Volume 7 • Issue 2 • 1000156
doi: 10.4172/2325-9655.1000156
The electric potential energy can be scaled as:
[ ]3 Df
E
w
Emax
U
S
U
−
 
=  
 
(1)
Where Sw
=Water saturation
UE
=Electric potential energy
UEmax
=Maximum electric potential energy
Df=Fractal dimension.
Equation (1) can proofed from
J=σ * E (2)
Where J=Electric current density
σ=Electric conductivity
E=Electric field
But,
V
E
L
= (3)
Where V=Electric potential
L=Distance
Insert Equation (3) into Equation (2)
v
J
l
σ= ∗ (4)
The electric potential can be scaled as
EU
v
Q
= (5)
Where UE=Electric potential energy
Q=Electric charge
Insert Equation (5) into Equation (4)
EU
J
Q l
σ= ∗
∗
(6)
The electric conductivity σ can be scaled as
2
r
σ
∗Σ
= (7)
Where Σ=Surface conductance
r=Grain radius
Insert Equation (7) into Equation (6)
2 UE
J r
Q l
∗Σ∗
∗ =
∗
(8)
The maximum grain radius can be scaled as
2 Emax
max
U
J r
Q l
∗Σ∗
∗ =
∗
(9)
Divide Equation (8) by Equation (9)
2
2
E
Emaxmax
U
J r Q l
UJ r
Q l
∗Σ∗
∗ ∗
=
∗Σ∗∗
∗
(10)
Equation (10) after simplification will become
E
max Emax
Ur
r U
= (11)
Take the logarithm of Equation (11)
max
log log E
max E
Ur
r U
   
=   
   
(12)
But,
[ ]
log
log
3
w
max
Sr
r Df
 
= 
− 
(13)
Insert Equation (13) into Equation (12)
[ ]
log
log
3
w E
Emax
S U
Df U
 
=  
−  
(14)
Equation (14) after logarithm removal will become
3 Df
E
w
Emax
U
S
U
−
 
=  
 
(15)
Equation (15) the proof of equation 1 which relates the water
saturation (Sw
), the electric potential energy (UE
), the maximum
electric energy (UEmax
), and the fractal dimension (Df).
Results
Based on field observation the Permo-Carboniferous Shajara
Formation were divided into three members as described in Figure
1. These members from bottom to top are: Lower Shajara Member,
MiddleShajaraMember,andUpperShajaraMember.Theirdeveloped
results of fractal dimensions are demonstrated in Table 1. Based on the
accomplished results it was found that the slope of the first technique
is positive whereas the slope of the second technique is negative. The
acquired results demonstrated also the equality of electric potential
energy fractal dimensions and capillary pressure fractal dimensions.
The maximum value of the fractal dimension was found to be 2.7872
assigned to sample SJ13 as confirmed in Table 1. Whereas the
minimum value of the fractal dimension 2.4379 was informed from
sample SJ3 as showed in Table 1. The electric potential energy fractal
dimension and capillary pressure fractal dimension were noticed to
increase with increasing permeability as proofed in Table 1 owing to
the possibility of having interconnected channels. The Lower Shajara
reservoir was designated by six sandstone samples (Figure 1) four of
which have been selected for capillary measurements label as SJ1, SJ2,
SJ3 and SJ4 as confirmed in Table 1. Their positive slopes of the first
technique and negative slopes of the second technique are explained
in Figures 2-5. Their electric potential energy 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 potential energy 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 potential energy 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 defined by four sandstone samples three of which
have been chosen for fractal dimension determination namely SJ7,
SJ8, and SJ9 as illustrated in Table 1. Their positive slopes of the first
technique and negative slopes of the second technique were shown

3. Citation: Alkhidir KEME (2018) Electric Potential Energy Fractal Dimension for Characterizing Permo-carboniferous Shajara Formation. Expert Opin
Environ Biol 7:2.
• Page 3 of 5 •Volume 7 • Issue 2 • 1000156
doi: 10.4172/2325-9655.1000156
and negative slopes of the second technique are delineated in Figures
9-11. Moreover, their electric potential energy fractal dimension and
capillary pressure fractal dimension are also higher than those of
sample SJ3 and SJ4 due to an increase in their permeability as clarified
in Table 1.
Overall a plot of electric potential energy fractal dimension versus
capillary pressure fractal dimension as shown in Figure 12 reveals
three zones of varying Petro physical characteristics.
in Figures 6-8. Their electric potential energy and capillary pressure
fractal dimensions show similarities as explained in Table 1. Their
fractal dimensions are higher than those of samples SJ3 and SJ4 due
to an increase in their permeability as explained 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, and SJ13
as explained in Table 1. Their positive slopes of the first technique
Figure 2: Log (energy/energymax
) & log pc versus log Sw for sample SJ1.
Figure 3: Log (energy/energymax
) & log pc versus log Sw for sample SJ2.
Figure 4: Log (energy/energymax
) & log pc versus log Sw for sample SJ3.
Figure 5: Log (energy/energymax
) & log pc versus log Sw for sample SJ4.
Figure 6: Log (energy/energymax
) & log pc versus log Sw for sample SJ7.
Figure 7: Log (energy/energymax
) & log pc versus log Sw for sample SJ8.

4. Citation: Alkhidir KEME (2018) Electric Potential Energy Fractal Dimension for Characterizing Permo-carboniferous Shajara Formation. Expert Opin
Environ Biol 7:2.
• Page 4 of 5 •Volume 7 • Issue 2 • 1000156
doi: 10.4172/2325-9655.1000156
Figure 8: Log (energy/energymax
) & log pc versus log Sw for sample SJ9.
Figure 9: Log (energy/energymax
) & log pc versus log Sw for sample SJ11.
Figure 10: Log (energy/energymax
) & log pc versus log Sw for sample SJ12.
Figure 11: Log (energy/energymax
) & log Pc versus log Sw for sample SJ13.
Reservoir
Sample Porosity % Permeability
(millidarcy)
Electric potential energy fractal
dimension
Capillary pressure fractal
dimension
Upper Shajara
Reservoir
SJ13 25 973 2.7872 2.7872
SJ12 28 1440 2.7859 2.7859
SJ11 36 1197 2.7586 2.7586
Middle Shajara
Reservoir
SJ9 31 1394 2.7786 2.7786
SJ8 32 1344 2.7752 2.7752
SJ7 35 1472 2.7683 2.7683
Lower Shajara
Reservoir
SJ4 30 176 2.6843 2.6843
SJ3 34 56 2.4379 2.4379
SJ2 35 1955 2.7748 2.7748
SJ1 29 1680 2.7859 2.7859
Table 1: Petrophysical model showing the three Shajara Reservoir Units with their corresponding values of electric potential energy fractal dimension and capillary
pressure fractal dimension.
Conclusion
• The Permo-Carboniferous Shajara Formation was divided into
three reservoirs, from bottom to top are : Lower Shajara reservoir,
Middle Shajara reservoir, and Upper Shajara reservoir.
• These reservoir bodies were confirmed by electric potential
energy fractal dimension and capillary pressure fractal dimension.
• The heterogeneity increases with increasing in grain size,
permeability, and fractal dimension reflecting an increase in pore
connectivity.

5. Citation: Alkhidir KEME (2018) Electric Potential Energy Fractal Dimension for Characterizing Permo-carboniferous Shajara Formation. Expert Opin
Environ Biol 7:2.
• Page 5 of 5 •Volume 7 • Issue 2 • 1000156
doi: 10.4172/2325-9655.1000156
Acknowledgements
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.
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Author Affiliations Top
Department of Petroleum and Natural Gas Engineering, College of engineering,
King Saud University, Riyadh, Saudi Arabia
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