Molecular Dynamic Simulation of Wettability between ...

Molecular Dynamic Simulation of Wettability between Immiscible Phases Using Fractal Theory

EG5911 (2016-17): Individual Project in Petroleum Engineering

By

ARAZ RANJINEH KHOJASTEH 51664043

A dissertation submitted in partial fulfillment of the requirements for the award of Master of Science in Petroleum Engineering at the University of Aberdeen

August, 2017

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ACKNOLEDGEMENT

I would like to begin in the name of God by thanking him for filling me with confidence and enthusiasm during my research and standing side by side with me in every stage of my life.

I appreciate dear Dr. Yingfang Zhou for his generous support and his tolerance every time that I’ve knocked his door for advice and guidance. My special regards to Dr. Rouzbeh Rafati for his advice.

Thanks to Dr. Amir Golparvar for giving his precious time for discussions on fluid configuration. Thanks to Mr. Rajat Saxena for his company and reminding me of my milestones time to time. Thanks Mr. Thejus Hari for helping me with the edits and getting my way around MS Word. I must thank Mr. Patrick Olutola for his consideration and introducing the structure of the writing of dissertation to me. My deeply regard to Mr. Barra Kona Gerard who had a big impact on the shape of this dissertation. Thanks to Mr. Sepehr Nematollahi for his aspirations during the period of researching in Sir Duncan Rice library. Lisbeth Cano for her kindness and motivation for finishing this paper until the last minute.

Lastly, special thanks to my mother who built a peaceful and amazing atmosphere for me for my growth and for her sacrifices in her every single moment of life for her children. I admire her for her strength and powerful character.

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TABLE OF CONTENTS ACKNOLEDGEMENT ....................................................................................................... ii TABLE OF CONTENTS .................................................................................................... iii LIST OF FIGURES .............................................................................................................. v CHAPTER ONE ................................................................................................................... 1 1.1 Introduction ................................................................................................................ 1 1.2 Problem Statement and motivation .......................................................................... 2 1.3 Objectives .................................................................................................................... 2 1.4 Scope of the work........................................................................................................ 2 CHAPTER TWO .................................................................................................................. 3 Literature Review ................................................................................................................. 3 2.1 Background ................................................................................................................. 3 2.2 Typical characteristics of Petroleum Reservoirs ..................................................... 3 2.3 Rock Parameters......................................................................................................... 4 2.3.1 Relative permeability .......................................................................................... 4 2.3.2 Capillary Pressure ............................................................................................... 5 2.3.3 Wettability ............................................................................................................ 7 2.4 Different kind of modelling...................................................................................... 22 2.4.1 Grain ................................................................................................................... 22 2.4.2 Pore network model........................................................................................... 22 2.4.3 Grain-based modelling ........................................................................................ 24 2.4.4 Invasion percolation theory: ............................................................................. 25 2.4.5 Fractal theory ..................................................................................................... 27 CHAPTER THREE............................................................................................................ 33 Methodology ........................................................................................................................ 33 3.1 Relationship between the Capillary pressure 𝑷𝒄 and the volume saturation 𝑺𝒘 of wetting phase and 𝑺𝒏𝒘 of non-wetting phase ......................................................... 33 3.2 Relative permeability ............................................................................................... 40 CHAPTER FOUR .............................................................................................................. 44 Discussion and results ........................................................................................................ 44 iii

4.1 Discussion ...................................................................................................................... 44 4.1

Fractal dimension ............................................................................................... 49

Nomenclature ...................................................................................................................... 56 REFERENCES ................................................................................................................... 57

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LIST OF FIGURES Figure 1 Fractal distribution of a sample reservoir (Source: Azomining ............................... 1 Figure 2 Petroleum Reservoir Cross-section .......................................................................... 3 Figure 3 Capillary pressure .................................................................................................... 6 Figure 4 Wettability vs Roughness ......................................................................................... 7 Figure 5 Measurement of contact angles for water-oil systems ............................................ 8 Figure 6 Dependency of IFT, wettability, capillary pressure, relative permeability .............. 9 Figure 7 Concept of interfacial tension between two immiscible phases............................. 10 Figure 8 Initial water saturation vs. wettability indices........................................................ 14 Figure 9 Residual oil saturation vs. wettability indices ........................................................ 15 Figure 10. Wetting in pores. ................................................................................................. 17 Figure 11. Three fluid configurations ................................................................................... 18 Figure 12 Numerical examples of the semi analytical model in an extracted pore space in Bentheim sandstone……………….………………………………………………… 19 Figure 13 A sketch of arc meniscus (AM) and main terminal meniscus (MTM) in a triangle capillary tube ............................................................................................................... 19 Figure 14 Fluid configurations of oil and water for the imbibition process at mixed-wet conditions in SEM image of Bentheim sandstone simulated with the semianalatycal modeld(1) ..................................................................................................................... 20 Figure 15 Fluid configurations of oil and water for the imbibition process at mixed-wet conditions in SEM image of Bentheim sandstone simulated with the semi analytical model.(2) ...................................................................................................................... 21 Figure 16 Primary drainage and imbibition capillary pressure curves in the SEM image of Bentheim sandstone mixed-wet conditions .................................................................. 21 Figure 17 Schematic of cubic pore network. ........................................................................ 23 Figure 18. Invasion patterns for various values of the hydrophilic pore-throat fraction ...... 23 Figure 19 Schematic of one pore .......................................................................................... 25 Figure 20 Detail of a bond percolation on the square lattice in two dimensions with percolation probability ................................................................................................. 26 Figure 21 The Sierpinski gasket ........................................................................................... 28 v

Figure 22 The various iterations of the Koch curve ............................................................. 28 Figure 23 The various iterations of the Koch curve ............................................................. 29 Figure 24 Julia's set .............................................................................................................. 30 Figure 25 Mandelbrot sets .................................................................................................... 30 Figure 26 Random Fractal surface........................................................................................ 31 Figure 27 capillary bundle model ......................................................................................... 36 Figure 28 mixed-wet fluid configuration ............................................................................. 39 Figure 29transient two-phase flow in a single fractal capillary ............................................ 42 Figure 30. Schematic of different size of capillary tubes of unit cell ................................... 45 Figure 31.Schematic of straight circular tube ....................................................................... 46 Figure 32. Bentheim sand stone ........................................................................................... 48 Figure 33. Bentheim sand stone ........................................................................................... 48 Figure 34. Fractal dimension ................................................................................................ 49 Figure 35. Experimental vs. Numerical ................................................................................ 50 Figure 36, Sensitivity (1) ...................................................................................................... 51 Figure 37. Sensitivity (2) ...................................................................................................... 52 Figure 38. Relative permeability .......................................................................................... 53

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CHAPTER ONE Introduction Disordered structures and random processes that are self-similar on certain lengths and time scales are very common in nature [1]. They can be found from the macro scales of galaxies to the nano scales of atoms. The galaxies and landscapes, earthquakes and fractures, aggregates and colloids, rough surfaces and interfaces, glasses and polymers, proteins and other large molecules; all possess fractals. Due to the wide occurrence of self-similarity in nature, the scientific community including astronomers, geoscientists and petroleum engineers have been interested in this phenomenon since a very long time. Among the major achievements in recent years that have strongly influenced our understanding of structural disorder and the derived random processes include the fractal concepts devised and developed by B. B. Mandelbrot. According to his postulate “Fractal geometry is a mathematical language used to describe complex shapes and is particularly suitable for the computer based models because of its iterative nature.” [1] A major advancement of in the field of fractal theory was its application to simulate heterogeneous petroleum reservoirs. Fractal distribution of a sample reservoir (Source: Azomining)

Figure 1 Fractal distribution of a sample reservoir (Source: Azomining

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1.1 Problem Statement and motivation It has been demonstrated by previous studies that fractal theory is a promising concept to generate flow function of multiphase flow in porous media. It has also found applications in generating correlations for flow functions, capillary pressure, relative permeability curves and to physically simulate a continuous multiphase flow equation in the large-scale domains. Generally, most reservoirs encountered in the real world contain mixed wet rocks in contrast to uniformly wet surfaces assumed in theories. Only a very few theories and concepts have been devised which have accounted for the mixed wet conditions. Via this research, I have made an attempt to formulate mixed wet flow function using fractal theory. Additionally, a comparative study between the analytical and experimental data was also conducted for validation of the function developed.

1.2 Objectives The objectives of this dissertation are: 1. To extract pore information, such as pore size distribution and fractal dimension, from rock images 2. To formulate analytical expression of flow functions using fractal theory 3. To compare the analytical flow function with that from lab measurements and/or numerical modelling.

1.3 Scope of the work This work assumes that the geometry of the pore spaces are cylindrical and the tortuosity of the pore spaces is assumed to be unity.

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CHAPTER TWO Literature Review 2.1 Background The knowledge on reservoir rocks and fluid properties are necessary to understand reservoirs better and these understandings are the back bone to production, reservoir management, field development, well testing and so on. This section intends to provide an insight into the literature published on fractal theory & its applications to multiphase flow dynamics of mixed wet systems. 2.2 Typical characteristics of Petroleum Reservoirs Petroleum reservoirs are made through three main steps: 

Deposition



Conversion/migration



Entrapment

Most Petroleum reservoirs contain from 3 phases (Oil, Gas, Water) coexisting with each other and separated based on their density difference into water at the bottom, oil on top of water and finally gas on top of oil. However, we can find reservoirs which include only two-phase which is the scenario assumed in this dissertation.

Figure 2 Petroleum Reservoir Cross-section

The reservoir rocks that hold petroleum are mostly sandstone and limestone. Less than 1% of petroleum exists in fractured igneous or metamorphic rocks [2]. Hydrocarbons has been transformed from an organic matter known as kerogen into petroleum by undergoing the following processes:

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

Diagenesis (100 to 200 ͦ F) resulting in biochemical methane



Catagenesis (200 to 300 ͦ F) resulting in oil or wet gas



Metagenesis (300 to 400 ͦ F) resulting in dry gas

It is virtually impossible to obtain the visual information of rock structures because of their different shapes and complexities in the movements of the water and rock layer underground. Properties of rock-fluid interactions are extremely important to the study of multiphase flow which is discussed in the following section.

Based on the foregoing, specific amount of knowledge about reservoir rock and fluid properties is necessary for understanding better the reservoir and these understanding are back bone of production, reservoir management strategy, economic issues and also field development plan, well testing, production engineering, production methods and so on. 2.3 Rock Parameters

2.3.1 Relative permeability Darcy’s law is applied when a porous media is 100% saturated with a homogenous singlephase fluid. Usually a reservoir rock is filled with two or more phases and these multiphase fluid systems have a great impact on reservoir flow processes when petroleum reservoirs are produced by primary recovery mechanism or immiscible displacement methods involving the injection of gas or water [3]. It is under these circumstances that more than one fluid phase is flowing or is mobile through a porous medium; thus, the flow of one fluid phase interferes with the other. This interference is the competition for the flow paths and must be described accurately for hydrocarbon recovery form petroleum reservoirs. Therefore, for understanding multiphase flow characteristics in reservoirs we have to introduce and define the concept of Relative permeability. The concept of relative permeability is measuring the amount of each phase in a multiphase condition. To some extent, relative permeability has the biggest impact than any other parameter used in reservoir engineering. Thus, a reservoir engineer has to have a good understanding of the relative permeability behavior of a given porous media.

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2.3.2 Capillary Pressure

A difference in the pressure across the interface because of interfacial energy between two immiscible phases causes curvature of the interface. The separation between two immiscible fluids will occur when they are in contact with each other because of the discontinuity in pressure among them. This difference in pressure is called CAPILLARY PRESSURE and is normally known by Pc. Capillary forces are a result of the combination of interfacial tension, pore size, geometry and wettability.

The existence of capillary forces in a porous medium causes retention of fluids in the pore space against gravity forces in a porous medium, even though in large containers, such as tanks and pipes of large diameters, immiscible fluids usually completely segregate because of gravity. For example, if on drop of oil released from the seafloor it would move to the sea surface, this movement is only dependent on the difference between density of seawater and drop of oil. While, if the same situation happens in the porous medium there is an extra force existing in the pores and acts as a resistant for the upward movement of Oil and it is Capillary.

Capillary forces also play a key role for controlling the displacement of one fluid by another in the pore space of porous medium and this displacement is either positive or negative by the capillary pressure. As a result, if we want to maintain a porous medium partially saturated with a non-wetting fluid while the medium is also exposed to the wetting fluid, it is necessary to maintain the pressure of the non-wetting fluid at a value greater than that in the wetting fluid [4]. Especially during water flooding, the capillary forces may act together with frictional forces to resist the flow of oil. It is beneficial to understand the nature of these capillary forces both from a reservoir structure (fluid contacts, transition zones, and free water level) and also hydrocarbon recovery points. Capillary pressure = Pressure in the non-wetting phase – Pressure in the wetting phase 𝑃𝑐 = 𝑃𝑛𝑤 − 𝑃𝑤 Pressure in the wetting phase and the non-wetting phase is indicated by 𝑃𝑤 and 𝑃𝑛𝑤 5

In the case of oil and water, either phase could preferentially wet the rock and thus assuming water is a wetting phase and oil is the non-wetting phase: 𝑃𝑐𝑜𝑤 = 𝑃𝑜 − 𝑃𝑤 Capillary pressure is a combined effect surface and interfacial tensions, pore size, geometry and wetting characteristics of a given system. The development of such equations, have already been discussed in numerous books and journals.

𝑃𝑐 =

2σcosθ r

Equation in last section just illustrates that capillary pressure of an immiscible pair of fluids illustrated in terms of interface forces, wettability, and capillary size. Capillary pressure is a function of adhesion tension (σcosθ) and propotionl to the radius of the capillary tube. Now an examination of the effect of pore size and the adhesion tension on capillary pressure must be made. Figure. (3) indicates the effect of different wetting phases versus different capillary tube radius.

We will see further in this paper that we have another equation based on this equation for fractal theory.

Figure 3 Capillary pressure

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2.3.3 Wettability “Wetting” is the ability of a liquid to maintain contact with a solid surface through intermolecular interactions. In other words, wetting is the balance between adhesive and cohesive Forces; cohesive forces within a liquid cause it to avoid contact with the surface by forming a spherical shape when all three phases are in equilibrium, whereas adhesive forces between the liquid and the solid surface cause the liquid to spread across the surface. If a surface of a material is attracted to a certain liquid, it is “philic” for that substance; for example, a surface that is attracted to water is called “hydrophilic”, and attracted to oil is called “oleophilic”. However, if a surface tends to force a certain liquid, it is “-phobic” for that substance; for example, “hydrophobic” and “oleophobic” mean that the surface pushes back water or oil, respectively [3].

Figure 4 Wettability vs Roughness

The contact angle (measured through the denser phase) is:

𝐶𝑜𝑠θ =

σso − σsw σwo

Where σso = interfacial tension between the solid and oil σsw = interfacial tension between the solid and water σwo = interfacial tension between water and oil 7

If the liquid is water, if the apparent θ is smaller than 90°, the surface is considered to have high wettability, or hydrophilic; larger than 90° is considered low wettability, or hydrophobic; and over 150°, the surface is classified as super hydrophobic. Hydrophobic; and over 150°, the surface is classified as super hydrophobic. These terms are defined similarly for other types of liquids

Figure 5 Measurement of contact angles for water-oil systems; (a-c) show measurements using a drop of water surrounded by oil, and (d-f) show drops of oil surrounded by water

When the solid surface is perfectly flat, rigid, smooth and chemically homogenous (in other words, an ideal surface), the contact angle obtained from the droplet is called Young’s contact angle, which is related to the surface tension () of each substance through the following formula [8]:

SV SL LV cos  WhereSV is the surface tension between the solid and gas (vapour), SL is between solid and liquid, and LV is between liquid and gas. In general, θ increases when LV decreases; in other words, the lower LV is, the easier it is for that liquid to wet the surface. [6]

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2.3.3.1 Interfacial tension and wettability

Generally petroleum reservoirs are composed of 2 fluids which may exist as 3 phase systems will or in some cases 4 phases including oil, water, gas and a solid phase such as aspartame. When we have only one fluid in our system, only one set of forces are predominant and that is the attraction between the rock and the fluid. In any reservoir when we have only one single fluid such as an aquifer these forces may not be that important because porosity and absolute permeability are adequate to define the characteristics of the reservoir [3]. However, when we have two-fluid system the forces for consideration are: Fluid 1  Fluid 2 Fluid 1  Rock Fluid 2  Rock

These forces give rise to interfacial tension, wettability, capillary pressure and also relative permeability.

All these properties of petroleum reservoir rocks spread with multiple fluid saturation should be determined to accurately describe the situation of production of a given petroleum reservoir. The first force that needs to be considered is the surface forces or interfacial tension. Because the wettability depends on interfacial tension, capillary pressure relates on interfacial tension and wettability, while relative permeabilities are dependent on interfacial tension, wettability, and capillary pressure along with some other properties [3].

Figure 6 Dependency of IFT, wettability, capillary pressure, relative permeability

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2.3.3.2 Interfacial tension and surface tension

In petroleum reservoir, 3 immiscible phases are in contact with each other and exist gas-oil, gas-water, and oil-water pairs. The thickness of this interface is only few molecular diameters [4]. For studying multiphase systems, it is necessary to consider effect of the forces that exist at the interface when two immiscible fluids are in contact. To understand the concept of interfacial tension or surface tension, consider a system of two immiscible fluids, oil and water, as shown in fig (7).

Figure 7 Concept of interfacial tension between two immiscible phases

An oil or water molecule far from the interface is surrounded by another same molecule resulting net attractive force on the molecule of 0as it is pulled in all directions. However, a molecule at the interface has a force acting upon it from the oil lying immediately above the interface and water molecules lying below the interface. The resulting forces are not balanced due to difference in forces from above and below and gives rise to interfacial tension.

The interfacial tension of two liquids is less than the highest individual surface tension because the mutual attraction is moderated by all molecules involved [4]. 2.3.3.3 Fundamental concepts of wettability

To understand the concept of wettability, we have to know about these definitions: 10

1. The relative ability of a fluid to spread on as solid surface in presence of another fluid 2. The tendency of a surface to be wet with each fluid 3. The tendency of one fluid pair to coat the surface of the solid 4. The tendency of a fluid to adhere or wet a solid surface in the presence of the other immiscible phase 5. Reservoir wettability has complex boundary conditions within the pore space 6. The term used to describe the relative adhesion of two fluids to a solid surface. 7. When two immiscible fluids contact a solid surface, one of them tend to adhere or spread more than the other Considering these conditions, can make us clear that in one surface one of the fluids has a greater degree affinity toward the solid surface. Thus, the tendency of a fluid phase to the solid is an indication of the wetting characteristics of the fluid for the solid. Adhesion tension and how to calculate the contact angle is already discussed in previous pages.

2.3.3.4 A discussion on practical aspects of wettability

The understanding of the wettability of a petroleum reservoir rock and its effect on different areas becomes important with the multiple phase flowing in the reservoir. Wettability influences the productivity and oil recovery during primary recovery. The original wettability and altered wettability during the first migration (from source rock to reservoir rock) or second one (from reservoir to the trap) can affect initial water saturation. In the early days of petroleum engineering it was assumed that almost all of the reservoirs are strongly water wet (𝞱=0) and that water completely coated the rock surface. This belief was from the beginning because of the history of saturation that engineers knew that reservoirs were completely saturated with water prior to the migration of the hydrocarbon from the source rock. This assumption led to a lot of problems in reservoir engineering later, that is reservoirs were expected to behave in a certain fashion. Questions a raised about natural wettability of hydrocarbon reservoirs ad numerous examples of wettability were discovered.

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As a matter of a fact, vast number of reservoirs are identified as oil wet. Cuiec has identified many reservoirs as partly oil-wet. Also based on other research, wettability was recognized, in some but not many reservoirs, as an intermediate between oil-wet and water-wet.

However, a well-defined classification is necessary to relate the wetting characteristics of a reservoir rock to the various SCAL properties and fluid flow processes on a pore level.

2.3.3.5 Factors affecting wettability

The most significant factors are the following aspects: 

Composition of the reservoir oil

Composition of reservoir oil clearly has impact on wettability. While, it is generally agreed that the existence of asphaltene in the reservoir oil attributed to the adsorption of asphaltenes onto the mineral surface of reservoir rocks. In spite of the significance of asphaltenes on wettability alteration in reservoir rocks, it is difficult to evaluate the mechanism from core tests because of the coupled effects of wetting and pore morphology. Many studies are focused on observing the interactions of crude oils and their components with smooth solid surfaces, via contact angle measurements [4]. Rayes et al. studied about oil-water-rock systems for a Libyan and a Hungarian oil field to understand the effect of the asphaltenes. His result illustrated a change in the contact angle from 40 to 120 degrees. 

Composition of the brine

According to researchers (Vijapurapu and Rao), the initial oil-wet nature of the system was changed to intermediate wettability simply by diluting the reservoir brine with deionized water. Tang and Morrow had studied on Berea Sandston cores and different oil brine systems, to study the effect of brine concentration on oil recovery and wettability. Their results indicated that salinity of the connate and brines influences wettability and oil recovery at reservoir temperature. They result illustrates that oil recovery increased with dilution of the connate brine and invading brine. 12



Reservoir pressure and temperature

Reservoir pressure and temperature alteration can have impact on the crude oil composition and in turn influences the precipitation of asphaltenes from crude oil. Effect of pressure and temperature studies indicate that there is a decrease in contact angle with increasing temperature. This means that the systems become slightly more water-wet.



Depth of the reservoir structure

Jerauld and Rathmell studied about the Orudhoe Bay reservoir as a function of the depth of the reservoir structure. Core samples were taken from different levels of structure and their wettability measured using Amott test [5].

2.3.3.6 Relationship between wettability and irreducible water saturation and residual oil saturation The main reason that focus on the effect of wettability on 𝑆𝑤𝑖 and 𝑆𝑜𝑟 is that these two points are the two end points in the recovery of hydrocarbon. 1-𝑆𝑤𝑖 illustrates the number of initial hydrocarbons in place that can be recovered, while 𝑆𝑜𝑟 indicates that the oil left in the pore space after the primary production.

2.3.3.7 Wettability and irreducible water saturation

Based on researchers, water-wet rocks have connate water saturation greater than 20 to 25 percent, while oil-wet rocks connate water is less than 15 percent of pore volume. Based on the plot in figure 8 generated, initial water saturation tends to decrease with increasing oil-wetness, as shown in figure (13). Also, Bennionet al. presented data on wettability in Western Canadian sedimentary Basin. The data indicated the same result, decreasing initial water saturation with increasing oil-wetness [5].

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Figure 8 Initial water saturation vs. wettability indices

2.3.3.8 Wettability and residual oil saturation

This part focuses on the relationship between wettability and water flood residual oil saturation, 𝑆𝑜𝑟 , because the relationship between primary recovery and residual oil has not been developed yet. The plot of wettability (x axis in terms of δ𝑜 and δ𝑤 as afunction of oil recoveryat 2.4 pore volume shows increasing oil recoveries for δ𝑤 from 1 to 0.6, a plateau of high oil recovery for δ𝑤 from 0.6 to 0.05, and further decrease in the oil recovery with increasing value of δ𝑜 . [14] Figure 9 illustrates that low recoveries or high S𝑜𝑟 are obtained at either wettability extremes, while higher recoveries or low δ𝑜𝑟 ’s are obtained in the weakly water-wet to neutral wettability condition. As illustrated in fig () results discussed based on the relationship between wettability and residual oil saturation or oil recovery represent a crest shaped curve when oil recovery is plotted against wettability. And although -shaped curve when residual oil saturation is plotted against wettability. For mixed-wet wettability systems, if oil-wet paths were continuous through the rock, water could displace oil from the large pores and little or no oil could be held by capillary forces in small pores or at grain contacts. [9]

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Figure 9 Residual oil saturation vs. wettability indices

2.3.3.9 Classification/Type of wettability

A variety of wettability types exists for petroleum reservoirs, depending on both reservoir fluid and rock properties. The classifications are presented in the following text: Water-wet In this mode, the rock surface has more tendency for the eater phase rather than the hydrocarbon phase. Therefore, hydrocarbon contexts are contained in the center of the pores and do not cover any of the rock surfaces. 

Oil-wet

In this wettability state, the situation is exactly opposite of the water-wet. It is assured that asphaltenic components cause this wetting stat. 

Intermediate-wet

In this classification rock has tendency for both phases. This nature is not defined; therefore, it includes the subclasses of both fractional and mixed wettability. In neutral wettability, the rock tendency is equal for both water and oil to wet the rock. (Angle of contact is 0)

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

Fractional wettability

In this type, some of the pores are water-wet, whereas others are oil-wet. Fractional wettability happens when our porous medium is composed of many minerals having various chemical properties leading to difference in wettability through the internal surface of the pores. The fractional wettability has not been discovered in real fields yet, although they can be created artificially in laboratory. 

Mixed-wettability

This concept was proposed by Salathiel (oil recovery by film drainage in mixed wettability rock) [9]. In this type, the smaller pores are filled with water and are water-wet, while the oil is occupied the larger pores preferentially. It is believed that the deposition of asphaltenic compounds renders the surface to oil-wet. this term is commonly used to refer to the condition where the smaller pores are occupied by water and are water-wet, whereas the larger pores of the rock are oil-wet, and a continuous filament of oil exists throughout the core in the larger pores.

Mixed wettability can occur as a result of the invasion of oil containing interracially active polar organic compounds into a water-wet rock saturated with brine. After having displaced the brine from the larger pores, the interracially active compounds react with the rock’s surface, displacing the remaining aqueous film and producing an oil-wet lining on the surface within the larger pores. The water film between the rock and the oil in the pore is stabilized by a double layer of electrostatic forces. As the thickness of the film is diminished by the invading oil, the electrostatic force balance is destroyed and the film ruptures, allowing the polar organic compounds in the oil to react directly with the rock surface.

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Figure 10. Wetting in pores. Water-wet (left) oil remains in the center of the pores. Oilwet(right) water remains in center of the pores. Mixed-wet (middle), oil displaced water from some of the surface, while is still in the centers of water-wet

The fluid with smaller surface energy of interaction with the rock will wet a porous media which contains immiscible fluids. Because of this issue, wettability of reservoir rock indicates the distribution of fluids within the pore space. Based on many researchers, reservoirs are initially water-wet before they are filled with oil. However, the chemical minerals which exists in oil can change the wettability of the reservoir to the oil-wet during geological time. Reservoirs can be partly oil-wet and partly water-wet because of wettability alteration which occurs in the reservoir. This matter will introduce heterogenous wettability, fractionally wettability, and mixed wettability.

By modelling fractionally wet media by choosing certain fractions to water-wet or oil-wet, in dependent of pore sizes. They used invasion percolation theory to model the capillary displacement of water and oil phases. Other researchers have studied heterogeneously wet porous medium using pore network modeling. Their network models are typically based on a lattice of pores and throats with different shapes (triangle, square, circular, star-shape, etc.) and randomly chosen sizes. The shapes are inspired by naturally occurring throats, which keep possession of wetting phase at grain contacts after the throats has drained. The angular 17

cross sections throats represent a useful advance over traditional; cylindrical throats by retaining wetting phase in the corners of their throats. It puts in place of heterogenous wettability, e.g. the surface of a passageway containing oil after drainage becomes oil-wet except in the corners where water phase remained at the drainage endpoint.

Figure 11. Three fluid configurations denoted (a) – (c) which could exist in an angular pore during imbibition. Red—means oil phase, blue—water phase, brown lines—surfaces of altered wettability on which a contact angle, how, may be specified.

A semi analytical pore-scale model is used to simulate capillary pressure curves in 2D SEM images of Bentheim sandstone during drainage and imbibition at mixed-wet condition. The pore spaces identified in the rock images are represented as cross section of straight capillary tubes. The fluid configuration occurring during drainage and imbibition in the highly irregular pore spaces are modeled at any capillary pressure and wetting condition by combining free-energy minimization with an AMs (arc meniscus, when oil and water touch)determining procedure that identifies the intersections of two circles moving in opposite directions along the pore boundary. Circle rotation at pinned contact lines accounts for mixed-wet conditions. The valid fluid-configuration change is associated with the most favorable entry pressure among all the allowed displacement scenarios that are generated by combining the identified AMs in different ways. [30]

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Figure 12 Numerical examples of the semi analytical model in an extracted pore space in Bentheim sandstone for (a) uniformly wet and (b through) three mixed-wet conditions.

Figure 13 A sketch of arc meniscus (AM) and main terminal meniscus (MTM) in a triangle capillary tube [29]

Fig (14) shows that, for the small initial water-saturation instance, water, in general, invades the larger pore spaces first, and oil layers of varying size and shape form and exist in the irregular pore spaces in most of the saturation range. Fig (15), which represents the larger initial water saturation, shows that water seems to invade the largest and smallest pore spaces 19

first, leaving the pore spaces of the intermediate size occupied by oil. Helland and Skjæveland (2006), who considered mixed-wet triangular tube cross sections, reported a similar trend [26]. This nonmonotonic invasion order of the pore sizes occurs because the fractional area of oil-wet surfaces is likely to be higher in large pore spaces that therefore exhibit more oil-wet behavior than the smaller pore spaces under these conditions. In addition, the water content that remains in the corners of these geometrics at the end of primary drainage is larger, which indicates that the invading water most likely contacts the water phase present in the corners during invasion. This results in increased capillary entry pressure that depend on the initial water increased capillary entry pressure that depend on the initial water saturation, and oil-layer formation occurs to a smaller extent fig (16).

Figure 14 Fluid configurations of oil (in red) and water(in blue) for the imbibition process at mixed-wet conditions in SEM image of Bentheim sandstone simulated with the semianalatycal model. The images are taken at decreasing capillary pressures ordered from (a

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Figure 15 Fluid configurations of oil (in red) and water (in blue) for the imbibition process at mixed-wet conditions in SEM image of Bentheim sandstone simulated with the semi analytical model. The images are taken at decreasing capillary pressures ordered from

Figure 16 Primary drainage and imbibition capillary pressure curves in the SEM image of Bentheim sandstone mixed-wet conditions simulated with the semi analytical model by use of a contact angle of θ_o=180 and two initial water saturations, S_wi=0.150 and S_wi=0. 21

2.4 Different kind of modelling

2.4.1 Grain Here, we focus on fractional-wettability described as a state in which our porous medium is oil-wet or water-wet to understand more about the grains. According to past papers, they assumed that oil-wet and water-wet grains are distributed randomly within the porous medium. The contact angle on each grain is based on random choice or personal whim, rather than any reason, while for the simplicity they assumed that all the water-wet grains have the same contact angle, and all the oil-wet grains have the same contact angle. Fractionally wet porous media can be found in certain areas, and they are the simplest heterogeneously wet media. In fields, the fractional wettability may come about as the acids may coat grains of varied sizes and mineralogy differently. In the lab, we can easily prepare fractional wet porous media by mixing different fractions of oil-wet grains and water-wet grains. Due to this, fractionally wet medium has been focus of most experimental measurements on properties of heterogeneously wet media and introduces basic knowledge about intermediate wet condition. 2.4.2 Pore network model The PNMs are based on the representation of the pore space in terms of a network of pores (or sites) connected by throats (or bond). The “pores “roughly correspond to the larger voids whereas the throats connecting the pores correspond to the constrictions of the pore space. In the most advanced developments, the pore network is constructed from direct imaging (which we are going to do in this paper), usually by micro-X-ray computerized numerically. This leads to “morphological” pore networks since the pore network constructed directly from the “real” microstructure. The method is for example illustrated in with a two-dimensional network constructed from a model fibrous medium. Although using morphological pore networks have rarely been used in relation with fuel cell related problems see however. As sketched in figure below pores of cubic shape are regularly placed on a 3D cartesian grid (with denoting the lattice spacing, that is the distance between two adjacent pores). Two first neighbor pores are linked by a channel of square cross-section. Such a channel is referred to as a bond or throat.

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Figure 17 Schematic of cubic pore network. The size of network is 40x40x10. As shown in the figure, the GDL thickness is 252 𝞵m and there are 10 pores 25.2 𝞵m apart across the GDL. [7]

Figure 18. Invasion patterns for various values of the hydrophilic pore-throat fraction f in a 2D 50 x 50 square network.

Pore-network modeling is a three-step procedure. In the first step, thin sections, CT images and depositional models are used to extract a pore space model from a sample of sedimentary 23

rock. In the second step, this pore space model is simplified by replacing the actual (e.g., imaged) pore shapes with the well-defined geometrical shapes such as ducts of circular, triangular or square cross section. The output of this step is a pore-network representation of the sediment. Finally, in step three, the simplified pore network is used to simulate a multiphase displacement process, and to obtain the respective macroscopic relative permeability and capillary pressure functions. Therefore, the pore-scale models differ from the large-scale models in that the macroscopic constitutive relationships are not assumed, but emerge from averaging of the relevant pore-scale physics. Using this three-step procedure, Øren et al. [1998] simulated two-phase flow in a sandstone and were able to predict the experimental drainage and imbibition capillary pressure and relative permeability curves. [3]

2.4.3 Grain-based modelling Pores are in the empty place in the rock that grains are not there, this concept was not considered in traditional pore network models. In other words, pore shapes are not elemental; rather, in nature the grain shapes are the geometric primitives which determine pore shapes. 24

Wettability is a property of the grain surface, not a pore. The difference between the pore network and grain based network is this article was derived based on grain locations and contact angles, rather than in terms of the geometry of the pores and throats. The correct geometry and location of a grain is calculated by a straightforward algorithm in a grain based model. Because of that we it is possible to learn more the movement, combination and living together of interfaces in a pore network model.

Figure 19 Schematic of one pore (defined as one tetrahedral cell in the Delaunay tessellation of sphere centers) in a dense disordered packing of equal-sized spheres

2.4.4 Invasion percolation theory: The simultaneous flow of multiple immiscible fluids through porous or fractured rocks, sediments, and soils is of significant interest in many fields, such as hydrology, subsurface carbon sequestration, petroleum engineering, and geothermal energy generation. Natural heterogeneities and limited, expensive sampling of the porous geological material typically cause large uncertainties in such engineering applications. In order to assess these uncertainties, it is crucial to understand how the material’s pore system and wettability affect the effective transport properties which are used as input for large-scale simulations. In this context, approaches inspired by percolation theory have proven extremely useful [14]. 25

Capillary-dominated flow has indeed been related to percolation theory, both on a theoretical basis as by numerical simulations in which porous media are represented as a network of idealized pores and throats. Since the first network model investigation of porous media by Fatt [1956], sophisticated pore network models (PNM) have been developed. It has been shown that such models need to capture the pore space’s real topology, geometry, and the spatial correlations of these properties adequately in order to produce realistic drainage and imbibition behavior [32].

Comparison of percolating theory with pore network theory for relative permeability and capillary pressure can be described as Percolating theory looks to introduce by statistical means the morphology of, and transport through randomly disordered media. The theory pertains to network models that consists of branches (bonds or links) and nodes (sites or junctions). Percolation theory determines the distribution functions for the accessible, allowed or occupied ducts, based on an appropriate distribution for the whole network. For the water permeability and capillary pressure, network modeling and percolation theory give very similar results.

Figure 20 Detail of a bond percolation on the square lattice in two dimensions with percolation probability p = 0.51 26

Percolation is a common model for disordered systems, and its relation to the fractal concept is emphasized.

2.4.5 Fractal theory

Disordered and structures and random processes that are self-similar on certain length and time scales are very common in nature. They can be found on the largest and the smallest scales: In galaxies and landscapes, in earthquakes and fractures, in rough surfaces and interfaces, in glasses and polymers, in proteins and other large molecules, in our lungs and blood vessels. Owing to the wide occurrence of self-similarity in nature, the scientific community interested in this phenomenon is very broad, ranging from astronomers and geoscientists to material scientists and life scientists. [14]

Among the major achievements in recent years that have strongly influenced our understanding of structural disorder and its formation by random processes are the fractal concepts pioneered by B. B. Mandelbrot. Fractal geometry is a mathematical language used to describe complex shapes and its particularly suitable for computers because of its iterative nature. [14] Mandelbrot (1977) was the first to suggest that many natural phenomena are statistically fractal in shape. He coined the term fractal from the Latin participle: fracturs which means broken/irregular – fractal structures being highly irregular in shape. A fractal is structure that is self-similar at all scales. This indicates that if were look at a small subsection of a fractal it would be visually and statistically identical to the whole. There are strictly two forms of fractals, non-random and random fractals. As the name suggests nonrandom fractals are generated by a recursive process that does not involve the use of statistical rules. Two popular examples are the Koch curve and Sierpinski gasket shown below. On the other hand, random fractals are produced by populating a structure with a given probability, p. The percolating cluster is an example of a random fractal. [15].

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Figure 21 The Sierpinski gasket

Figure 22 The various iterations of the Koch curve

Mandelbrot played with pictures instead of formulas. Fractals were always around us but invisible. Making invisible visible and finding order in disorder. He said do not pay attention to these disordered patterns on surface you see, pay attention to that what it took to produce what you see. According to his definition about fractals self-similarity means endless repetition, in other words if you zoom in or zoom out the object will look the same. The similarity of pattern just keeps on going. One of the studies about this issue was about tress in jungles of Costa Rica which indicates that patterns of branching in trees are similar. They measured the branches of one tree and compared them in whole jungle and results were amazing. The pattern of branching from the mother tree and daughters were similar.

There is an order beneath chaos that can describe clouds, flowers. We have to just look at the patterns of nature in the right way. You can play with mathematics and you can write formulas which can describe nature then you will have different kind of geometry. Koch Swedish mathematician introduced a curve fig (23) which was paradox to the eye appeals to be perfectly fine but mathematically it was infinite which means it cannot be measured.

When you start with an equal angles triangle one of the classical geometry figures, he took a piece from each side and substitute to two pieces they are no longer the original pieces, and he did it over and over again. This kind of curve was a big question to mathematician in that time which they called this curves Pythagorean curves. 28

Figure 23 the various iterations of the Koch curve

Mandelbrot’s fresh mind and his enthusiasm about new technology computers made it easy for him to do irritation. The endlessly cycles of calculation which would demanded by mathematical monsters. A problem introduced in World War 1 by a young French mathematician named Gaston Julia. He looked at the simple equation 𝑦 = 𝑥 2 + 1 and he wanted to know what will happen when we take a number you plug into the formula and get number out, and take the number back to the beginning and use it as another input, and do it over and over again, the series of numbers that you get is a set. Julia’s number set. *

𝑦 = 𝑥2 + 1

#

The development of this number had to wait until fast computers got invented. Mandelbrot did something that Julia never could, he used a computer to run an equation millions of times. He turned the numbers from Julia set to the points on the graph:

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Figure 24 Julia's set

Julia set is consisted of hundreds of pictures which lead Mandelbrot to the breakthrough. In 1980, he created his own equation, 𝑓(𝑍) = 𝑍 2 + 𝐶 which combined all of the Julia’s set into a Single image. When Mandelbrot irritated his own equation, he got his own set of numbers graphed on the computers it was like roadmap of the all Julia’s sets and quickly became famous as envelope of FRACTAL geometry.

Figure 25 Mandelbrot sets 30

The set was basically from the classical shape, circle. Now people could see something that was always there but it was invisible. Maybe the secret of the Universe is just in front of us but we are too distracted to notice. Real roughness is often fractal and it can be measured. As we know many natural phenomena follow the fractal theory porous geological rock formations, as well as engineered porous structures, such as chemical reaction columns, have fractal properties, i.e., they are selfsimilar or, more specifically, self-affine over several length scales. Loosely speaking, fractality means that the whole of a porous conglomerate looks like very similar to its parts it is made of, or in other words, there is virtually no visual difference when zooming-in (magnifying) on a porous fractal medium (Mandelbrot, 1982). This “optical confusion” is, for example, clearly recognizable from the scanning-electron-microscopic and thin section images of various sandstone, shales and carbonates taken at different magnifications [30]. Studies show that the fractal, or self-similar range of a porous rock or structure, as measured, for example, by its pore surface or its pore volume, usually starts somewhere beneath the long-length, non-fractal Euclidean regime of the order of the grain size and extends over several orders of magnitude down to the very short-length regime. Several experimental studies in which either wave-scattering- or surface adsorption techniques were used have indicated fractal regimes down even to the molecular level of the crystals that form the porous aggregate.

Figure 26 Random Fractal surface 31

A fractal surface structure has also been found, on a much larger scale, for many naturally occurring geological and geophysical fractures, particularly seismic fault planes and, again, on the small scale, for ruptured metal- or rock surfaces [8]. In addition to the pure geometric fractal property of the porous or fractured medium itself, which is usually quantified by the fractal dimension of its pore surface or its pore volume. Numerous flow processes in a porous medium have also been found to be fractal. The most eminent representatives of this category of porous media flow are the various phenomena of viscous fingering that can occur under specific physical conditions and the understanding of which is of practical importance in many disciplines of groundwater hydrology [8] and, especially, in petroleum reservoir modeling. Based on these definitions we will use fractal theory for simulating wettability in immiscible fluids in mixed-wet condition. [29].

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CHAPTER THREE Methodology 3.1 Relationship between the Capillary pressure 𝑷𝒄 and the volume saturation 𝑺𝒘 of wetting phase and 𝑺𝒏𝒘 of non-wetting phase Many researchers have studied the fractal nature of reservoir rock in the past two decades. It has been found that most natural porous media such as reservoirs rock are fractals and can be characterized using fractal model or a fractal curve which represents the relationship between number of pores and the radius of pores. The magnitude of fractal dimension is a representation of the heterogeneity of the porous medium. The greater the fractal dimension the greater the heterogeneity of porous media. Note that the pore size distribution index in the Brooks-Corey Capillary Pressure model is also a representation of the heterogeneity of porous media. The greater the pore size distribution index, the more homogenous the porous medium. Attention has also been paid to the application of fractal modelling of porous media in reservoir engineering. The applications include the development of relative permeability models, Capillary pressure models and the models to predict oil production rate, etc. The review show that the fractal modeling of porous media is powerful tool to characterize the heterogeneity of porous media to study fluid flow mechanism. [26]

𝑃𝐶 =

4σcosθ ⋋

(3.1)

It is assumed that the porous medium consists of bundle of capillaries, and the cumulative size distribution of capillary sizes, whose sizes are greater than or equal to 𝜆 , has been proven to follow the fractal scaling law:

𝑁(𝐿 ≥ 𝜆) = (

𝜆𝑚𝑎𝑥 𝐷𝑓 ) 𝜆

(3.2)

Where 𝐷𝑓 is the fractal dimension for pore space, 0