Literature Review
The petroleum reservoirs within the earth’s surface, the contents coexist in at least two different phases out of a possible three phases that are immiscible with each other. The possible phases that can be found in the reservoirs are the natural gas components, water deposits, and oil components. The water component due to this immiscibility cannot form a single phase with either the gas or the oil component thus causing a high level of tension within the reservoir. The tension within the reservoir is caused as a result of the forces that separate the different phases due to their immiscibility yielding the presence of new properties in petro-physics, namely relative permeability, wettability and capillary pressure. The gas and oil components also experience this form of tension that causes the two components to be considerably immiscible, forcing the fluids to coexist within the reservoir as droplets. This is because the fluid to fluid interactions are pushed by the force of tension to take up the least surface area as a result of the cohesive and adhesive forces among molecules of the three fluids. This therefore forces the collection of the three fluids in the reservoir to act as if they are subject to surface tension.
While the fluids in the matrix tend to show a greater response to cohesive forces within the fluid matrix to cause the immiscibility, adhesive forces may come into play when the matrix interacts with a solid surface. In this situation, one of the fluids in the matrix tends to demonstrate a higher affinity to the surface of the solid. This implies that the said fluid, will occupy a larger surface thanks to the adhesive forces between the specific fluid and the surface of the solid. This fluid will thus have an impact of displacing the other fluids to bring out the property of wettability. Wettability can therefore be described as the attraction or affinity of one of the fluid phases in the complex immiscible fluid matrix to a solid surface. This property treats the oil and gas components in the matrix as they have a tendency to displace the water component of the matrix together, while the water component displaces the two fluids in the same manner as is illustrated below.
The state of the wettability within the reservoir occurs in three different states depending on the contact angle of the between the solid surface and the fluid molecules. The diagram below depicts the different categories of wettability that could occur when the fluid matrix interacts with the rock surface of the reservoir. The category of wettability is determined by the size of the contact angle between the fluids when a solid surface is introduced.
Relative Permeability and Capillary Pressure
When the contact angle formed between the water component and the surface of the solid is an acute angle, water becomes the fluid component that displaces the other fluid components thus demonstrating a higher affinity to wet the solid surface. This category of wettability is known as water wettability. When the contact angle is obtuse, the situation is described as oil wettability as the oil molecules completely displace the water molecules to wet the surface. Neutral wettability occurs when the contact angle created is a right angle and none of the fluids demonstrate a higher affinity to wetting the surface than the other fluids.
The property of wettability is extremely important in the determination of how the complex immiscible fluid matrix is positioned and distributed within the porous rock of the reservoir. This property also determines how the fluid matrix will flow during the removal of oil water and during the recovery of oil. The interaction between the surface tension forces that promote the immiscibility of the complex fluid matrix and the property of wettability bring about what is referred to as capillary pressure. This pressure can be described as the difference in the pressure between two fluids confined within the same reservoir across an interface. This implies that capillary pressure seeks to establish the difference in the pressure across the interface between two phases or components that exist in the same complex immiscible fluid matrix. The two phases ought to occupy the same pores, so that the forces of tension between the two phases have to be overcome first before flow is initiated. It also greatly determines relative permeability between two phases of the fluid matrix needed to establish the capacity for the fluid matrix to flow during the oil recovery process.
Wettability is an important factor in the determination of multi-phase flow within the porous material of the structure because flow requires that one of the phases remains in contact with the solid surface of the porous material. In this manner, wettability determines the components of flow and the distribution of the phases during oil recovery. Wettability is also the property that determines the saturation levels of the different phases within the porous structure as this is the property that determines the contact of a phase with the surface of the pores of the structure. Saturation is a function of the amount of pore volume occupied by the different phases and thus it is determined by the relationship of the individual phases to the rock surface within the structure of the reservoir, as in its wettability. Wettability can also be related to pore pressure within the porous structure of the reservoir and thus in turn making it a determinant of the capillary pressure and permeability of the structure. This is because the pores of the reservoir can be wetted by the different phases of the fluid matrix responsible for the property of wettability. In the process of wetting the solid surface of the porous material in the reservoir, the responsible fluid phase will fill up the space within the porous material thus reducing the size of the pores and increasing the pore pressure for each phase to create the two pressure components whose difference yield capillary pressure.
During the process of estimating the distribution and amount of natural gas and oil components in a petroleum reservoir, the properties of capillary pressure and relative permeability and their relationships are used to predict the capacity of gas, oil, and water components that can flow from the reservoir. These properties are determined by the chemistry and structure of the fluids and solids that exist in an existing reservoir. These components are determined through theoretical and empirical approaches that have been discovered with continuing research into the science of petroleum exploration and recovery. Permeability is a term that is used to describe the capacity of oil, gas, or water components to flow through the porous rock of the reservoir. The property of permeability in this field is described using Darcy’s law, which relates the flow of the individual phases to the cross-sectional area of the pores where the fluid matrix are to flow through, the difference in pressure between the phases and the and the length of the porous material.
Where:
.q = rate of flow
.k = permeability
A = cross-sectional area
= viscosity
= the drop in pressure and
L = the length of the material.
This equation demonstrates that a high rate of permeability leads to an increased capacity for the flow of the fluid components within the immiscible matrix. However when analyzing the flow of a matrix with different phases, the expression of relative permeability between two different phases, mainly oil and water are used. The equation used to express relative permeability for multi-phase flow of oil and water phases respectively in the x direction without factoring in gravitational effects is expressed as follows.
For the oil phase and
Where
and are the relative permeability values for the oil and water phases respectively. The values of relative permeability are dimensionless and vary from the range of 0 and 1. These equations used in the computation of relative permeability are responsible for the pressure differences between two phases across an interface, and thus account for the capillary pressure emergent in the reservoir between the oil and water phases. This relationship is expressed in the formula below.
Where:
= capillary pressure between oil and water phases
= pressure in the oil phase
= pressure in the water phase
These two functions are also linked to the property of saturation of the fluid components within the pores of the reservoir. This relationship is based on the definition of saturation as the fraction of the volume of the pores that is occupied by either the oil or water phases. The relationship between capillary pressure saturation and relative permeability are represented in the following equations such that for either the water phase or the oil phase to experience flow, their ought to be the absence of gas.
These relationships presented continue to be difficult parameters to measure due to the cost implications as well as the difficulty of establishing these parameters. The theoretical and empirical correlations are mainly utilized for the calculation and estimation of the values of relative permeability for the two phases. While the process of estimating relative permeability of a phase is a simple and fast method of estimating these parameters, although its accuracy may be questionable. One of the most common methods for estimating the relative permeability of different phases is the Corey correlation approach, which is an entirely theoretical approach to the estimation of this parameter. The Corey correlation approach is one of the most common approaches used in the determination of relative permeability and was derived from the Burdine method of computing permeability and for estimating capillary pressure. The Corey correlation approach makes the assumption that the non-wetting and wetting phases have values of relative permeability that are independent of the saturation of other phases, thus utilizing the saturation data of a single phase, whether oil or water.
The Corey correlation equation is developed as is shown below.
Where:
=relative permeability of the wetting phase component (water)
= relative permeability of the non-wetting phase component (oil)
= relative permeability of the non-wetting phase at the irreducible saturation of the wetting phase
= normalized saturation of the wetting phase
= pore size distribution index (representing the level of uniformity in the sizes of the pores within the reservoir. A small value shows a lack of uniformity while a high demonstrates uniformity. The value used in Corey correlation is 2.
= ( 1-) and is the residual saturation of the non-wetting phase
= saturation of water
= the initial saturation of water
The solutions from this formula differ from the original solutions of the Burdine formula due to the addition of the term which accounts for ensuring that the non-wetting phase solution is obtained the irreducible saturation of the wetting phase. The term was also introduced to represent the first flow of the non-wetting phase, at the point of critical saturation. This is due to the fact that at the beginning of the flow curve of the non-wetting phase, there is a period where there is no connectivity. This means that when the minimal possible number of pores is connected flow starts to take place and the first value of relative permeability can be determined. At this point, the saturation is used to give the most realistic values of relative permeability.
The value of the pore size distribution index () can also be determined empirically from the normalized saturation of the wetting phase as is used in estimating the capillary pressure as is shown below.
Where:
capillary pressure
=pressure at the minimum threshold
= normalized saturation of the wetting phase
The relationships between wettability and capillary pressure can be presented in the figures below.
When considering the distribution of flow as influenced by wettability and the effects is has on the capillary pressure within the porous material, it is important to note that the porous wetting phase will always try to maintain a continuum within the connected series of pores. The effect of this is creation of small pore gullets that connect the main pore spaces to each other. The noon-wetting phase will on the other hand try to maintain a continuum in the liked system of pores that involve large gullets unlike the case of the wetting phase. With increasing saturation of the wetting phase, the pore spaces in the reservoir will continue to get trapped by the wetting phase volume, and in turn filling even the extremely small pore pockets. The effect of this is a reduction in the in the saturation of the non-wetting phase, as the wetting phase will always have an impact of displacing the other phase from having any contact with the solid surface of the reservoir.
This results into the reduction of the relative permeability of the non-wetting phase, () through a process commonly known as imbibition displacement. This continues to occur since the wetting phase is taken into the all the pore spaces of the porous structure, from the smallest pores to the largest systematically. The result of this is the trapping of the non-wetting phase into the biggest pores within the porous structure of the reservoir. The effect of this is an increase in the pore pressure of the reservoir. It also causes significant changes in the relative permeability of the reservoir.
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