On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia
The New Capillary Number Parameterization for Simulation in Surfactant FloodingPremier Publishers
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-wet rock is steered by capillary number that is typically within specific ranges (〖10〗^(-5) to〖 10〗^(-4)). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (〖10〗^(-6) to〖 10〗^(-4)) in comparison with available models with a critical capillary number (〖10〗^(-5) to〖 10〗^(-4)). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter F_kr as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.
Effect of Height and Surface Roughness of a Broad Crested Weir on the Dischar...RafidAlboresha
Weir is usually incorporated as control or regulation devices in hydraulic systems,
with flow measurement as their secondary. It is normally intended for use in the field and thus
to regulate broad discharges. Broad-Crested weir is among the oldest common weir types. In this
paper, the effect of height and surface roughness for different Board Crested weirs models were
studied on discharge coefficient (Cd) in a horizontal open channel. In the crest of the weir,
certain materials may be combined with concrete (e.g., boulders) or may be used as cladding to
minimize the effect of water overflow (e.g. stone). The weir surface should not be considered
smooth in this case, and the discharge coefficient (Cd) must be re-estimated. For these purposes, laboratory flume was used to study the effect of height and surface roughness on the discharge coefficients with four of the different weir models dimensions of the concrete blocks. In this study, the flow conditions were considered to be free water flow and the viscosity effect was neglected. In all cases, the weir height effect was directly proportional to the discharge coefficient while the surface roughness effect was found to be inversely proportional to the coefficient Cd of the case study.
Presentation given by Sai Gu of Cranfield University on "Computational Modelling and Optimization of Carbon Capture Reactors" at the UKCCSRC Gas CCS Meeting, University of Sussex, 25 June 2014
Mesoscopic simulation of incompressible fluid flow in porous mediaeSAT Journals
Abstract
Lattice Boltzmann method is used to simulate cavity driven fluid flow in porous media. A square cavity is considered with the top
lid moving with uniform velocity and other sides kept stationary. Simulation is carried out for values of Darcy number ranging
from 10-6 to10-2 at Reynolds number 10 and 100. Influence of Darcy number and Reynolds number is investigated on velocity
profiles and the streamline plots. Half-way bounce back boundary conditions are employed in the numerical simulation. The
numerical code is first verified with the results available in the literature and then used to simulate the Newtonian fluid flow in
porous media. The Darcy number and the Reynolds number were observed to have great influence on the flow properties and the
location of the primary vortex. Simulation was carried out for a 100100 mesh grid and a fine agreement is established theories
in incompressible fluid flow.
Keywords: Lattice Boltzmann method, incompressible flow, porous media
The New Capillary Number Parameterization for Simulation in Surfactant FloodingPremier Publishers
The Capillary number hypothesis is very empirical in Surfactant flooding Enhanced Oil Recovery (EOR) method. which is of modest experience on the North Sea and many other offshore platforms. The capillary number drives the force wetting processes, which is controlled by the balance between capillary and viscous forces. The mobilization of oil trapped in pores of water-wet rock is steered by capillary number that is typically within specific ranges (〖10〗^(-5) to〖 10〗^(-4)). There is high uncertainty and confusion in the parameterization of capillary number formula, as every quantity is given on a macroscale level. As demonstrated herein, a new microscopic capillary number parameterization was proposed. This paper is written to improve the numerical formulation of capillary number in surfactant flooding model. The new formula for capillary number was derived based on existing equations as a function of residual oil saturation and tested. Thus, the proposed mobility mechanism easily accounts for a broader critical range of capillary number (〖10〗^(-6) to〖 10〗^(-4)) in comparison with available models with a critical capillary number (〖10〗^(-5) to〖 10〗^(-4)). We used an existing model to quantify the effect of capillary number on a miscible and immiscible relative permeability curves by computing the interpolation parameter F_kr as a tabulated function of the Logarithm (base 10) of the capillary number using a new capillary number formulation.
Effect of Height and Surface Roughness of a Broad Crested Weir on the Dischar...RafidAlboresha
Weir is usually incorporated as control or regulation devices in hydraulic systems,
with flow measurement as their secondary. It is normally intended for use in the field and thus
to regulate broad discharges. Broad-Crested weir is among the oldest common weir types. In this
paper, the effect of height and surface roughness for different Board Crested weirs models were
studied on discharge coefficient (Cd) in a horizontal open channel. In the crest of the weir,
certain materials may be combined with concrete (e.g., boulders) or may be used as cladding to
minimize the effect of water overflow (e.g. stone). The weir surface should not be considered
smooth in this case, and the discharge coefficient (Cd) must be re-estimated. For these purposes, laboratory flume was used to study the effect of height and surface roughness on the discharge coefficients with four of the different weir models dimensions of the concrete blocks. In this study, the flow conditions were considered to be free water flow and the viscosity effect was neglected. In all cases, the weir height effect was directly proportional to the discharge coefficient while the surface roughness effect was found to be inversely proportional to the coefficient Cd of the case study.
Presentation given by Sai Gu of Cranfield University on "Computational Modelling and Optimization of Carbon Capture Reactors" at the UKCCSRC Gas CCS Meeting, University of Sussex, 25 June 2014
Mesoscopic simulation of incompressible fluid flow in porous mediaeSAT Journals
Abstract
Lattice Boltzmann method is used to simulate cavity driven fluid flow in porous media. A square cavity is considered with the top
lid moving with uniform velocity and other sides kept stationary. Simulation is carried out for values of Darcy number ranging
from 10-6 to10-2 at Reynolds number 10 and 100. Influence of Darcy number and Reynolds number is investigated on velocity
profiles and the streamline plots. Half-way bounce back boundary conditions are employed in the numerical simulation. The
numerical code is first verified with the results available in the literature and then used to simulate the Newtonian fluid flow in
porous media. The Darcy number and the Reynolds number were observed to have great influence on the flow properties and the
location of the primary vortex. Simulation was carried out for a 100100 mesh grid and a fine agreement is established theories
in incompressible fluid flow.
Keywords: Lattice Boltzmann method, incompressible flow, porous media
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Unsteady state series CSTR modeling of removal of ammonia nitrogen from domes...IJECEIAES
This work shows simulation results for subsurface vertical flow constructed wetland (VFCW) using a series CSTR model. The VFCW considered received the outflow from a domestic wastewater treatment plant. In addition, it was planted with Cyperus sp. and filter media was unsaturated. The model was based on an unsteady state mass balance for ammonia, nitrites, and nitrates, using one to three series CSTRs. Nitrogen transformation mechanisms considered were ammonification, nitrification, plant uptake and denitrification. The following effects were evaluated: the number of reacting CSTRs from one to three; the occurrence of the reaction in second and third CSTRs for the case that three CSTRs hold; the use of either equal or different values of reaction rate parameters between CSTRs; and the discretization of the reaction rate parameters. The inflow and outflow measurements of ammonium, nitrites, and nitrates were used for model calibration. The estimated parameters included the reaction rate coefficients and reactor water volume. The coefficient of determination (R ) evidenced a satisfactory capability of simulating outlet pollutant concentrations. Two and three reacting CSTRs achieved similar R value (0.54-0.55), whereas one reacting CSTR achieved an R 2 of 0.39, and three CSTRs with reaction only in the first tank achieved an R of 0.42. Discretization of the nitrification rate for the case of two reacting CSTRs led to an R 2 of 0.94. The parameter sensitivity analysis revealed a significant effect of model parameters on the R 2 value. 2 2 2
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Revisit To Forchheimer Equation Applied In Porous Media FlowIJRES Journal
A brief reference to various non-linear forms of relation between hydraulic gradient and velocity of
flow through porous media is presented, followed by the justification of the use of Forchheimer equation. In
order to study the nature of coefficients of this equation, an experimental programme was carried out under
steady state conditions, using a specially designed permeameter. Eight sizes of coarse material and three sizes
of glass spheres are used as media with water as the fluid medium. Equations for linear and non-linear
parameters of Forchheimer equation are proposed in terms of easily measurable media properties. These
equations are presented in the form of graphs as quick reckoners.
To better understand injection and post-injection flow processes and the entrapment of supercritical CO2 during geological carbon sequestration in a carbonate reservoir, the pore systems of sixty-six Cambrian-Ordovician carbonate samples from multiple states in the Midwest United States were analyzed.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
Unsteady state series CSTR modeling of removal of ammonia nitrogen from domes...IJECEIAES
This work shows simulation results for subsurface vertical flow constructed wetland (VFCW) using a series CSTR model. The VFCW considered received the outflow from a domestic wastewater treatment plant. In addition, it was planted with Cyperus sp. and filter media was unsaturated. The model was based on an unsteady state mass balance for ammonia, nitrites, and nitrates, using one to three series CSTRs. Nitrogen transformation mechanisms considered were ammonification, nitrification, plant uptake and denitrification. The following effects were evaluated: the number of reacting CSTRs from one to three; the occurrence of the reaction in second and third CSTRs for the case that three CSTRs hold; the use of either equal or different values of reaction rate parameters between CSTRs; and the discretization of the reaction rate parameters. The inflow and outflow measurements of ammonium, nitrites, and nitrates were used for model calibration. The estimated parameters included the reaction rate coefficients and reactor water volume. The coefficient of determination (R ) evidenced a satisfactory capability of simulating outlet pollutant concentrations. Two and three reacting CSTRs achieved similar R value (0.54-0.55), whereas one reacting CSTR achieved an R 2 of 0.39, and three CSTRs with reaction only in the first tank achieved an R of 0.42. Discretization of the nitrification rate for the case of two reacting CSTRs led to an R 2 of 0.94. The parameter sensitivity analysis revealed a significant effect of model parameters on the R 2 value. 2 2 2
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
A Revisit To Forchheimer Equation Applied In Porous Media FlowIJRES Journal
A brief reference to various non-linear forms of relation between hydraulic gradient and velocity of
flow through porous media is presented, followed by the justification of the use of Forchheimer equation. In
order to study the nature of coefficients of this equation, an experimental programme was carried out under
steady state conditions, using a specially designed permeameter. Eight sizes of coarse material and three sizes
of glass spheres are used as media with water as the fluid medium. Equations for linear and non-linear
parameters of Forchheimer equation are proposed in terms of easily measurable media properties. These
equations are presented in the form of graphs as quick reckoners.
To better understand injection and post-injection flow processes and the entrapment of supercritical CO2 during geological carbon sequestration in a carbonate reservoir, the pore systems of sixty-six Cambrian-Ordovician carbonate samples from multiple states in the Midwest United States were analyzed.
On the equality of electric power fractal dimension and capillary pressure fr...Khalid Al-Khidir
On the equality of electric power fractal dimension and capillary pressure fractal dimension for characterizing shajara reservoirs of the permo carboniferous shajara formation
Transverse relaxation time fractal dimension of nuclear magnetic resonance fo...Khalid Al-Khidir
Transverse relaxation time fractal dimension of nuclear magnetic resonance for characterizing shajara reservoirs of the permo carboniferous shajara formation, saudi arabia
Reservoirs Simulations of Gel Treatments to Control Water Production, Improve...Andrea Urdaneta
eservoirs Simulations of Gel Treatments to Control Water Production, Improve the Sweep Efficiency and the Conformance Factor in Eastern Venezuelan HPHT Fractured Reservoirs
Electric potential gradient fractal dimensions for characterizing shajara res...Khalid Al-Khidir
Electric potential gradient fractal dimensions for characterizing shajara reservoirs of the shajara formation of the permo carboniferous unayzah group, saudi arabia
Immunizing Image Classifiers Against Localized Adversary Attacksgerogepatton
This paper addresses the vulnerability of deep learning models, particularly convolutional neural networks
(CNN)s, to adversarial attacks and presents a proactive training technique designed to counter them. We
introduce a novel volumization algorithm, which transforms 2D images into 3D volumetric representations.
When combined with 3D convolution and deep curriculum learning optimization (CLO), itsignificantly improves
the immunity of models against localized universal attacks by up to 40%. We evaluate our proposed approach
using contemporary CNN architectures and the modified Canadian Institute for Advanced Research (CIFAR-10
and CIFAR-100) and ImageNet Large Scale Visual Recognition Challenge (ILSVRC12) datasets, showcasing
accuracy improvements over previous techniques. The results indicate that the combination of the volumetric
input and curriculum learning holds significant promise for mitigating adversarial attacks without necessitating
adversary training.
Cosmetic shop management system project report.pdfKamal Acharya
Buying new cosmetic products is difficult. It can even be scary for those who have sensitive skin and are prone to skin trouble. The information needed to alleviate this problem is on the back of each product, but it's thought to interpret those ingredient lists unless you have a background in chemistry.
Instead of buying and hoping for the best, we can use data science to help us predict which products may be good fits for us. It includes various function programs to do the above mentioned tasks.
Data file handling has been effectively used in the program.
The automated cosmetic shop management system should deal with the automation of general workflow and administration process of the shop. The main processes of the system focus on customer's request where the system is able to search the most appropriate products and deliver it to the customers. It should help the employees to quickly identify the list of cosmetic product that have reached the minimum quantity and also keep a track of expired date for each cosmetic product. It should help the employees to find the rack number in which the product is placed.It is also Faster and more efficient way.
Welcome to WIPAC Monthly the magazine brought to you by the LinkedIn Group Water Industry Process Automation & Control.
In this month's edition, along with this month's industry news to celebrate the 13 years since the group was created we have articles including
A case study of the used of Advanced Process Control at the Wastewater Treatment works at Lleida in Spain
A look back on an article on smart wastewater networks in order to see how the industry has measured up in the interim around the adoption of Digital Transformation in the Water Industry.
Saudi Arabia stands as a titan in the global energy landscape, renowned for its abundant oil and gas resources. It's the largest exporter of petroleum and holds some of the world's most significant reserves. Let's delve into the top 10 oil and gas projects shaping Saudi Arabia's energy future in 2024.
CFD Simulation of By-pass Flow in a HRSG module by R&R Consult.pptxR&R Consult
CFD analysis is incredibly effective at solving mysteries and improving the performance of complex systems!
Here's a great example: At a large natural gas-fired power plant, where they use waste heat to generate steam and energy, they were puzzled that their boiler wasn't producing as much steam as expected.
R&R and Tetra Engineering Group Inc. were asked to solve the issue with reduced steam production.
An inspection had shown that a significant amount of hot flue gas was bypassing the boiler tubes, where the heat was supposed to be transferred.
R&R Consult conducted a CFD analysis, which revealed that 6.3% of the flue gas was bypassing the boiler tubes without transferring heat. The analysis also showed that the flue gas was instead being directed along the sides of the boiler and between the modules that were supposed to capture the heat. This was the cause of the reduced performance.
Based on our results, Tetra Engineering installed covering plates to reduce the bypass flow. This improved the boiler's performance and increased electricity production.
It is always satisfying when we can help solve complex challenges like this. Do your systems also need a check-up or optimization? Give us a call!
Work done in cooperation with James Malloy and David Moelling from Tetra Engineering.
More examples of our work https://www.r-r-consult.dk/en/cases-en/
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
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Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
2. page 2 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
Brooks-Corey model (B-C model). A model that predicts
the hydraulic conductivity for un- saturated soil-water
retention curve and conductivity saturation was derived
by Mualem YA [3]. Later, based on Mualem’s formula,
Van Genuchten MTh [4] described a relatively simple
expression for the hydraulic conductivity of unsaturated
soils. The Van Genuchten model (V-G model) contained
three independent parameters, which can be obtained
by fitting experimental data. A function to estimate the
relationship between water saturation and capillary
pressure in porous media was proposed by Oostrom M
and Lenhard RJ [5]. This function is test data of sandstone
rocks and carbonate rocks with high permeability were
described as a new capillary pressure expression [6].
A fractal approach can be used to model the
pc measured with mercury intrusion in Geysers grey
wacke rock; however, the B-C model could not be used,
according to a study by Li [7]. Sub-sequently, a theoretical
analysis using fractal geometry was conducted by Li K and
Horne RN [8] to deduce the B-C model, which has always
been considered as an empirical model. Subsequently,
fractal modeling of porous media was used to develop a
more generalized capillary pressure model was studied
by Li K [9]. With the new model [9] also evaluated the
heterogeneity of rocks [10]. Physical parameters of soils in
term of fractal dimension were studied by GlobusAM [11].
An increase of volumetric water content with increasing
hydraulic conductivity was reported by Alfaro Soto MA,
et al. [12]. Bimodal Pore Size behavior of the Shajara
Formation reservoirs of the permo-carboniferous Unayzah
group was studied [13]. Subdivision of the Shajara
reservoirs into three units based on thermodynamic fractal
dimension approach and 3-D fractal geometry model of
mercury intrusion technique was reported by Al-Khidir K,
et al. [14]. The work published by Al-Khidir, et al. [14]
was cited as Geoscience; New Finding reported from
King Saud University Describe advances in Geoscience.
Science Letter (Oct 25, 2013): 359. Al-khidir, et al. [15].
An increase of bubble pressure fractal dimension and
pressure head fractal dimension with decreasing pore
size distribution index and fitting parameters m×n due to
possibility of having inter connected channels was proofed
by Al-khidir KEME [16]. An increase of fractal dimension
with increasing permeability, relaxation time of induced
polarization, due to an increase in pore connectivity was
confirmed by Alkhidir KEME [17].
Materials and Methods
Porosity was measured on collected sandstone
samplesandpermeabilitywascalculatedfromthemeasured
capillary pressure by mercury intrusion techniques. Two
procedures for obtaining the fractal dimension have been
developed. The first procedure was concluded by plotting
the logarithm of the ratio between differential capacity and
maximum differential capacity versus logarithm wetting
phase saturation. The slope of the first procedure = 3- Df
(fractal dimension).The second procedure for obtaining the
fractal dimension was resolved by plotting the logarithm
of capillary pressure versus the logarithm of wetting phase
saturation. The slope of the second procedure = Df -3.
The differential capacity can be scaled as
[ ]3
1
2
1
2
max
−
=
Df
C
Sw
C
(1)
Where Sw the water saturation, C the differential capacity
in square meter, Cmax
the maximum differential capacity in
square meter, and Df the fractal dimension.
Equation 1 can be proofed from
*
=
Q
εf
V L
(2)
Where εf the permittivity of the fluid in faraday / meter, Q
the electric charge in coulomb, V the electric potential in
volt, and L the length in meter.
The electric charge can be scaled as
=
S
vol
Q
C
(3)
Where Q the electric charge in coulomb, vol the
volume of the fluid in cubic meter, and CS
the streaming
potential coefficient in volt / pascal.
Insert equation 3 into equation 2
* *
=
S
vol
εf
C V L (4)
3. page 3 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
The volume can be scaled as
Vol = C * H (5)
Where vol the volume of the fluid in cubic meter,
C the differential capacity in square meter, and H the
pressure head in meter
Insert equation 5 into equation 4
*
* *
=
S
C H
εf
C V L
(6)
The streaming potential coefficient can be scaled
as
2
*
8* *
=
E
S
reff C
C
σf ηf
(7)
Where Cs the streaming potential coefficient in
volt / pascal, reff the effective pore radius in meter, CE
the
electro osmosis coefficient in pascal / volt, σf the fluid
conductivity in Siemens / meter, and ηf the fluid viscosity
in pascal * second.
Insert equation 7 into equation 6
2
* *8* *
* * *
=
E
C H σf ηf
εf
reff C V L
(8)
If the pore radius r in meter is introduced equation
8 will become
2 * *8* *
* * *
=
E
C H σf ηf
r
εf C V L
(9)
The maximum pore radius can be scaled as
2
max
* *8* *
* * *
max
=
E
C H σf ηf
r
εf C V L
(10)
Divide equation 9 by equation 10
2
2
max
* *8* *
* * *
* *8* *
* * *
max
=
E
E
C H σf ηf
εf C V Lr
C H σf ηfr
εf C V L
(11)
Equation 11 after simplification will become
2
2
maxmax
=
r C
Cr (12)
Take the square root of equation 12
2
2
maxmax
=
r C
Cr
(13)
Equation 13 after simplification will become
1
2
1
max
2
max
=
r C
r
C
(14)
Take the logarithm of equation 14
1
2
1
max
2
max
log log
=
r C
r
C
(15)
[ ]max
log
;log
3
=
−
r Sw
But
r Df
(16)
Insert equation 16 into equation 15
[ ]
1
2
1
2
max
log
log
3
=
−
Sw C
Df
C
(17)
Equation 17 after log removal will become
[3 Df]
1
2
1
2
max
−
=
C
Sw
C
(18)
Equation 18 the proof of equation 1 which relates
the water saturation, differential capacity, maximum
differential capacity, and the fractal dimension.
The capillary pressure can be scaled as
Sw = [Df - 3] * Pc + constant (19)
Results and Discussion
BasedonfieldobservationtheShajaraReservoirsof
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 developed results of differential capacity
4. page 4 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
and capillary fractal dimensions are presented in (Table 1).
The results display equalities between differential capacity
fractal dimension and capillary pressure fractal dimension.
A maximum fractal dimension value of about 2.7872 was
informed from sample SJ13 as validated in (Table 1). But,
a minimum fractal dimension value 2.4379 assigns to
sample SJ3 from the Lower Shajara Reservoir as verified
in (Table 1). The differential capacity and capillary
pressure fractal dimensions were noticed to increase
with increasing permeability due to the possibility of
having interconnected channels as confirmed in (Table1).
Regarding the Lower Shajara Reservoir, it is represented by
six sandstone samples as shown in (Figure 1), four of which
marked as SJ1, SJ2, SJ3, and SJ4 were taken for capillary
pressure measurements to evaluate the fractal dimension.
Their positive slopes of the first procedure (log ratio of
differential capacity to maximum differential capacity
versus log water saturation (Sw) and negative slopes of
the second procedure (log capillary pressure (Pc) versus
log water saturation (Sw) were designated in (Figures
2, 3, 4, 5 and Table 1). Their differential capacity fractal
dimension and capillary pressure fractal dimension values
are displayed in Table 1.As we proceed from sample SJ2 to
SJ3 a remarkable reduction in permeability from 1955 md
to 56 md was observed due to compaction which reveals
change in differential capacity fractal dimension from
2.7748 to 2.4379 as explained in (Table 1). Such extreme
change in permeability can account for heterogeneity
which is a key parameter in reservoir quality assessment.
Once more, an increase in grain size and permeability was
informed from sample SJ4 whose differential capacity
fractal dimension and capillary pressure fractal dimension
was found to be 2.6843 as illustrated in (Table 1).
Figure 1: Surface Type Section of the Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Latitude 26 52 17.4, Longitude 43 36 18
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Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
Figure 3: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ2
Figure 4: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ3
Figure 2: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ1
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Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
Table 1: Petrophysical Model showing the Three Shajara Reservoir Units with their Corresponding Values of Differential Capacity
Fractal Dimension and Capillary Pressure Fractal Dimension
Formation Reservoir Sample
Porosity
%
k
(md)
Positive
Slope of
the First
Procedure
Slope=3-Df
Negative
Slope of
the Second
Procedure
Slope=Df-3
Differential
Capacity
Fractal
Dimension
Capillary
Pressure
Fractal
Dimension
Permo-CarboniferousShajara
Formation
Upper
Shajara
Reservoir
SJ13 25 973 0.2128 -0.2128 2.7872 2.7872
SJ12 28 1440 0.2141 -0.2141 2.7859 2.7859
SJ11 36 1197 0.2414 -0.2414 2.7586 2.7586
Middle
Shajara
Reservoir
SJ9 31 1394 0.2214 -0.2214 2.7786 2.7786
SJ8 32 1344 0.2248 -0.2248 2.7752 2.7752
SJ7 35 1472 0.2317 -0.2317 2.7683 2.7683
Lower
Shajara
Reservoir
SJ4 30 176 0.3157 -0.3157 2.6843 2.6843
SJ3 34 56 0.5621 -0.5621 2.4379 2.4379
SJ2 35 1955 0.2252 -0.2252 2.7748 2.7748
SJ1 29 1680 0.2141 -0.2141 2.7859 2.7859
Figure 5: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ4
Concerning the Middle Shajara Reservoir, it is
separatedfromLowerShajaraReservoirbyanunconformity
surface as described in (Figure 1). It was signified by four
samples, three of which named as SJ7, SJ8, and SJ9 were
selected for fractal dimension determination as proved in
(Table 1). Their positive and negative slopes of the first
and second procedures were defined in (Figures 6, 7, 8
and Table 1) Furthermore, their differential capacity and
capillary pressure fractal dimension shows equal values as
presented in (Table 1) Likewise, their fractal dimension
values are higher than those of sample SJ3 and SJ4 from
the Lower Shajara Reservoir due to an increase in their
permeability as identified in (Table 1). On the other hand,
the Upper Shajara reservoir is 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, 10, 11 and Table
1). Also, their differential capacity 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
testified in (Table 1). Global a plot of positive slope of
7. page 7 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
the first procedure versus negative slope of the second
procedure delineates three reservoir zones of varying
petrophysical characteristics as shown in (Figure 12).
These zones were also confirmed by plotting differential
capacity fractal dimension versus capillary pressure fractal
dimension as shown in (Figure 13). Such variation in
fractal dimensions can be used to explain heterogeneity
which is a key parameter in reservoir quality assessment.
Figure 6: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ7
Figure 7: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ8
8. page 8 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
Figure 9: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ11
Figure 10: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ12
Figure 8: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ9
9. page 9 of 10ISSN:XXXX-XXXX SBFBE, an open access journal
Volume 1 · Issue 2 · 1000006SF J Biofuel Bioenerg
Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
Figure 13: Differential Capacity Fractal Dimension Versus Capillary Pressure Fractal Dimension
Figure 11: Log (C1/2
/Cmax
1/2
) & log Pc Versus log Sw for Sample SJ13
Figure 12: Positive Slope of the First Procedure Versus Negative Slope of the Second Procedure
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Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.
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Citation: Khalid Elyas Mohamed Elameen AlKhidir (2018) On Similarity of Differential Capacity and Capillary Pressure Fractal Dimensions for
Characterizing Shajara Reservoirs of the Permo-Carboniferous Shajara Formation, Saudi Arabia. SF J Biofuel Bioenerg 1:2.