Bakkalaurea Thesis
 
Supervised by:
Approval date:
 
 
Table of Contents
Abstract……………………………………………………………………………………………………….. 4
1  Introduction: Reservoir Rocks………………………………………………………………………. 5
1.1     Important properties of reservoir rocks……………………………………………………………………………… 5
1.2     The types of Reservoir Rocks……………………………………………………………………………………………. 6
1.2.1      SANDSTONE RESERVOIR ROCKS……………………………………………………………………………….. 6
1.2.2      CARBONATE RESERVOIR ROCKS……………………………………………………………………………….. 8
1.2.3      Shale…………………………………………………………………………………………………………………………… 9
2      Fluids Saturation in Reservoir Rock………………………………………………………….. 10
2.1     Methods of Determining Fluid Saturation……………………………………………………………………….. 10
2.1.1      Determination of Fluid Saturations from Rock Samples……………………………………………….. 10
2.1.2      Determination of Fluid Saturations by Extraction with a Solvent…………………………………… 13
2.1.3      Determination of Fluid Saturations with Electric Well Logs (Indirect)……………………………. 14
2.2 The Use of Core-Determined Fluid Saturations…………………………………………………………………… 16
3      Capillary Pressure in Reservoir Rock………………………………………………………… 18
3.1     Definition…………………………………………………………………………………………………………………………. 18
3.1.1      Capillary Forces – Wettability……………………………………………………………………………………… 19
3.1.2      Drainage and Imbibition……………………………………………………………………………………………… 19
3.2     Methods of measuring capillary pressure………………………………………………………………………. 20
3.2.1      Desorption Method……………………………………………………………………………………………………. 20
3.2.2      Restored state method……………………………………………………………………………………………….. 21
3.2.3      Mercury injection method (Purcell)……………………………………………………………………………… 22
3.2.4      Conversion of laboratory results………………………………………………………………………………… 25
4      Relationship between capillary pressure and fluids saturation………………………. 26
Summary……………………………………………………………………………………………………. 28
References………………………………………………………………………………………………….. 29
Nomenclature………………………………………………………………………………………………. 30
 
 
List of Figures
Figure 1: Pores in reservoir rocks. 5
Figure 2: Permeability in Reservoir Rocks. 6
Figure 3: Sandstone digenesis (zonation from Scott Oilfield, North Sea, UK) 7
Figure 4: A thin-section photomicrograph of a limestone. 8
Figure 5: The construction of shale in reservoir 9
Figure 6. Retort distillation apparatus. 11
Figure 7. Typical retort calibration curve for water 11
Figure 8. Typical retort calibration curve for oil 12
Figure 9. Laboratory determination of fluid saturation. 13
Figure 10. Modified ASTM extraction apparatus. 13
Figure 11. Soxhlet extractor 14
Figure 12. Limiting values of total core water for oil and gas production. 17
Figure 13. Wettability of fluids. 19
Figure 14. Drainage and imbibition. 20
Figure 15. Semi-permeable diaphragm.. 21
Figure 16. Capillary pressure curve.(restored states method). 22
Figure 17. Mercury injection method. 23
Figure 18. Capillary pressure by mercury injection: homogeneous matrix medium.. 23
Figure 19. Capillary pressure mercury injection: medium v macropores and matrix. 24
Figure 20. Capillary pressure by mercury injection in the case of 2 matrix. 24
Figure 21. Example for the position of the interface. 26
Figure 22. Drainage and imbibition curve in PC and Snm diagram.. 27
 
 

Abstract

The aim of this baccalaureate thesis was to give an overview of reservoir rocks and searching for determination of fluids saturation and capillary pressure in reservoir rock, in order to find out the relationship between fluid saturations and capillary pressure.
The focal point in this baccalaureate thesis was how to determine the value of fluids saturation and capillary pressure in reservoir rocks, what can be concluded to two methods, ”direct”  and “indirect” method. By the equipment, we can read the value of fluids saturation and capillary pressure directly. But we also can calculate them through the other properties in reservoir rocks, e.g. porosity, resistivity from logs, when the condition can not supply the equipment, which we need. This is indirect method. No matter which method is used in the determination, we need a result to know about the relation between capillary pressure and fluids saturation in the end. So that we will know, how will the capillary pressure effect on the fluids saturation. Further, we will get more information.

1  Introduction: Reservoir Rocks

A reservoir rock is capable of storing a fluid and producing it into boreholes. Although the term “reservoir rock” suggests the function of storage only, the ability to produce fluids into wells is equally important. For example, a water-saturated shale or clay may contain as much water per unit volume as an aquifer capable of producing large volumes of water per day. The fluids produced form reservoir rocks are oil, gas, and water, and in general a reservoir rock capable of producing one of these fluids is capable of producing the others. Some writers limit the term “reservoir rock” to rocks which produce oil or gas, but there seems to be no valid reason for restricting the term to rocks which contain a particular fluid. In the interest of clearness and consistency, it seems advisable to define a reservoir rock by its lithologic characteristics, and not by the type of fluid it contains.

            Important properties of reservoir rocks

A fundamental property of a reservoir rock is its porosity. However, for it to be an effective reservoir rock, the fundamental property is permeability. Both porosity and permeability are geometric properties of a rock and both are the result of its lithologic (composition) character. They determine the rate of production of fluids, the amount that can be stored in the reservoir, the ultimate production, and the type of secondary methods which should be applied. Variations in pore size, that are closely related to permeability, determine to a large degree the relative amounts of hydrocarbons and water in each stratum of the reservoir rocks
Figure 1: Pores in reservoir rocks
A rock with pores is referred to as porous. This means it has tiny holes through which oil may flow. Reservoir rocks must be porous, because hydrocarbons can occur only in pores. The definition of porosity is:
Here    is porosity,   is volume of porous in the rock,   is total volume of rock. The porosity depends on the location of the rock (heterogeneity), the compressibility of rock and the pressure.
 
Figure 2: Permeability in Reservoir Rocks
A reservoir rock is also permeable. That means its pores are connected. If hydrocarbons are in the pores of a rock, they must be able to move out of them. Unless hydrocarbons can move from pore to pore, they remain locked in place, unable to flow into a well. A suitable reservoir rock must therefore be porous, permeable, and contain enough hydrocarbons to make it economically feasible for the operating company to drill for and produce them.
 

            The types of Reservoir Rocks

All reservoir rocks are of sedimentary origin; they have been formed by either mechanical deposition of erosion fragments of older rock (fragmental rocks) or by chemical or organic precipitation. Sedimentary rocks may be broadly classified as sandstones, carbonates and shales. Shale is the most abundant of the sedimentary rocks; it makes up roughly 50 per cent of the world’s sedimentary rock. Sandstone and carbonate rocks constitute about 25 and 20 percent, respectively.
 

            SANDSTONE RESERVOIR ROCKS

Sandstones are fragmental rocks consisting of sand grains. The term sand refers to a particular grain size (62 µm – 200 µm), not to a particular composition. The performance of the sandstone as a reservoir rock, its combination of porosity and permeability, depends upon the degree to which it is a truly sand. Texture should reflect similar sized grains, not a combination of coarse and fine grained material. The best sandstone reservoirs are those that are composed primarily of quartz grains of sand size, silica cement, with minimal fragmented particles. The main mineral constituent of sandstones is quartz. In addition, sandstones may contain varied amounts of clay minerals, feldspar, calcite and other rock fragments.
 
 
Figure 3: Sandstone digenesis (zonation from Scott Oilfield, North Sea, UK)
 
The quality of the initial sandstone reservoir is a function of the source area for the materials, the depositional process, and the environment in which the deposition took place. Sandstone reservoirs are generally 25 meters thick, are lenticular and linear spatially, and less than 250 km2 in area. They range in age from the oldest being Cambrian (in Algeria) to the youngest being Pliocene (Caspian region in Ukraine). In the USA, two-thirds of the sandstone reservoirs are Cenozoic in age. [1]
Reservoir sandstone has individual sand grains that are slightly cemented together. Several sand grains could fit on the head of a pin, but there are still many pores or spaces between the grains that can hold oil. These sand grains were originally deposited in river channels and deltas or as sandbars and beaches in a shallow sea. Limestone reservoir rock may consist of sand-sized or larger fragments of corals, sponges, snails, clams, and other marine animals.
 
 

            CARBONATE RESERVOIR ROCKS

Carbonates are predominantly composed of calcite and dolomite, with clay and quartz as common secondary minerals. Carbonates can be both fragmental and precipitated rock. If the main mineral is calcite, carbonate rock is referred to as limestone. Dolomite rock is the term for carbonates with dolomite as their main constituent. Dolomite rock is almost always a secondary rock formed from limestone by replacement of part of the calcium in calcite by magnesium, a process called dolomitization. Carbonate rocks generally form in warm sea water at shallow depths, ankle deep to about 20 ft. The hard, usually calcareous parts of the organisms pile up on the seafloor over time, forming beds of lime particles. Algae, simple plants, are one of the greatest contributors of lime particles, but any shelled animal may contribute whole or fragmented shells to the pile. Reefs, banks of lime mud, and lime sand bars are commonly found preserved in rocks.
 
Figure 4: A thin-section photomicrograph of a limestone
This particular sample comes from an interval that is not a good reservoir rock. Circular grains composed of calcite (finely crystalline, reddish-stained areas in a grain) and dolomite (clear, coarse crystals) are completely cemented by medium crystalline calcite. No porosity is visible
 
The most interesting and perhaps impressive aspects of carbonate reservoir rocks are their fossil content. Fossils range from the very small single cell to the larger shelled animals. Prior to the 1920’s, carbonate reservoir rocks were relatively rare and prior to 1950 they were all regarded as essentially organic rocks. But this changed when textural studies of carbonates in Iraq and the Bahamas showed that carbonates are also the result of inorganic processes. Most carbonate rocks are deposited at or in very close proximity to the site of creation. Transportation of material is less common and sorting is essentially non-existent. The “best-sorted” carbonate rocks are Oolites in which the “grains” are the same size and shape. But Oolites are not “sorted” at all, but were formed with the sizes and shapes that they have in the carbonate rock and were cemented in place.[1]
 

            Shale

Shale is a common source rock. The source rock is the place where, millions of years ago, tiny sea plants and animals—called phytoplankton and zooplankton—lived, died, and were preserved. Source rock contains the source of the hydrocarbon.
 
Figure 5: The construction of shale in reservoir
 
Shale consists of compacted beds of clay and other fine-grained minerals. Shales are generally tight and impervious rocks that do not classify as reservoir rock. Yet shales are very important in connection with hydrocarbon reservoirs. For one thing, they often provide the sealing streaks and intercalations are very common in hydrocarbon reservoirs and may have a profound effect on the flow characteristics of reservoirs.
 

2      Fluids Saturation in Reservoir Rock

As a result of the origins of the oil and its formation and migration conditions, the reservoir rocks contain the following fluids:

  • Liquid hydrocarbons: oil from the light fraction to asphalts,
  • Gaseous hydrocarbons.
  • Water (salt water).

These fluids which are distributed in a certain manner in the porous medium under reservoir temperature and pressure conditions are, in general, found to have quite different distributions in the cores brought to the surface.
These modifications are due to the following factors:

  • Firstly, to causes which are difficult to avoid:

(1). Invasion of drilling mud or filtrate.
(2). Gas expansion due to the fall in pressure during the raising of the core.

  • Secondly, there are often handling errors such as the washing of the cores in water, or drying at high temperatures or the lack of preservation.

The quantity of fluid contained in the pores, expressed as a percentage of Vp is called fluid saturation.

            Methods of Determining Fluid Saturation

There are two approaches to the problem of determining the original fluid saturations within a reservoir rock. The direct approach is the selecting of rock samples and measuring the saturations of these samples as they are recovered from the parent formations. The indirect approach is to determine the fluid saturation by measuring some other physical property of direct approach, such as using electric logs or capillary-pressure measurements.

            Determination of Fluid Saturations from Rock Samples

In determining fluid saturations directly from a sample removed from a reservoir, it is necessary to understand first how these values are measure; second, what these measured values represent; and third, knowing what they represent, how they can be applied.
In order to measure values of original rock saturations there have been essentially three methods devised. These methods involve either the evaporation of the fluids in the rock or the leaching out of the fluids in the rock by extraction with a solvent.
Figure 6. Retort distillation apparatus.
One of the most popular means of measuring the initial saturations is the retort method. This method takes a small rock sample. By heating the sample and measuring the volumes of water and oil driven off, it measures the fluid saturations in the sample. The sample is crushed and weighed before being placed in the apparatus. It is then heated in stages or directly to 1200°F during which the fluids are vaporized, collected, condensed and separated. Plateaus in the rise of the cumulative water volume with temperature are sometimes analysed to indicate when free water, surface clay-bound water and interlayer clay-bound water have been driven off. An electric retort is shown in Figure 6.
The retort method has several disadvantages. In order to remove all the oil, it is necessary to approach temperatures on the order of 1000 to 1200°F. At temperatures of this magnitude the water of crystallization within the rock is driven off, causing the water-recovery values to be greater than just the interstitial water.
Figure 7. Typical retort calibration curve for water
An example of such a system is illustrated in Figure 7. Here the water removed in the first 30 min was approximately the interstitial water. As the application of heat was continued, the water of crystallization was removed, amounting to approximately 2 cc of water out of a total recovery of 8 cc. Thus, it is seen that an error of 33 per cent is possible if the water of crystallization is not accounted for.
Figure 8. Typical retort calibration curve for oil
The second error which occurs from retorting samples is that the oil itself when heated to high temperatures has a tendency to crack and coke. This change of a hydrocarbon molecule tends to decrease the liquid volume and also in some cases coats the internal walls of the rock sample itself. The effect of cracking and coking in a retort is shown in Figure 6, wherein 0.4 cc of oil actually in the sample yields about 0.25 cc in the receiving vessel. Thus a fluid correction must be made on all sample data obtained with a retort. Before retorts can be used calibration curves must be prepared on various gravity fluids to correct for the losses from cracking and coking with the various applied temperatures. Another correction curve can also be obtained which correlates recovered.
The retort is a rapid method for the determination of fluid saturations, and utilizing the corrections yields satisfactory results. It gives both water and oil volumes, so that the oil and water saturations can be calculated from the following formulas:
 
In order to obtain realistic values of fluid saturation it is necessary to choose the proper drilling fluid or resort to correlations similar to that reported by Kennedy et al. Figure 9 show the correlations that correlate hydrocarbon saturations before and after coring. It is noted that for cores of 5- and 10-millidarcy permeability, the initial and final hydrocarbon saturation yields and final hydrocarbon saturation yields an approximate straight line for initial saturations greater than 15 per cent. Data for cores of from 127- to 3040-millidarcy permeability were correlated in the same manner as the data for the low-permeability samples. These also resulted in a straight-line correlation for initial hydrocarbon saturations greater than 15 per cent.
 
Figure 9. Laboratory determination of fluid saturation.
Correlations such as presented in Figure 9 can be used to correct saturations measured from cores to original conditions. Additional data are required before universal correlations can be established.
Attempts have been made to use tracers in the drilling fluid to determine the amount of water in the core which is due to mud filtrate invasion. The theory was that mud filtrate displaced only oil. Thus, when the core is recovered to the surface, the salt concentration of the core water can be determined. Knowing the salt concentration in the reservoir water and the tracer concentration in the drilling fluid, it was thought possible to calculate the volume of filtrate and reservoir water in the core. A large fraction of the initial reservoir water may have been displaced by the invading filtrate, so the tracer method would give low values of reservoir water saturation.
 
 

            Determination of Fluid Saturations by Extraction with a Solvent

Figure 10. Modified ASTM extraction apparatus
Extraction can be accomplished by a modified ASTM method or a centrifuge method. In the standard distillation test the core is placed so that a vapor of toluene, gasoline, or naphtha rises through the core and is condensed to reflux back over the core. This process leaches out the oil and water in the core. The water and extracting fluid are condensed and are collected in a graduated receiving tube. The water settles to the bottom of the receiving tube because of its greater density, and the extracting fluid refluxes back into the main heating vessel. The process is continued until no more water is collected in the receiving tube. The distillation apparatus is shown in Figure 10. The water saturation can be determined directly.
 
The oil saturation is an indirect determination. It is necessary to note the weight of the core sample prior to extraction. Then, after the core has been cleaned and dried, the sample is again weighed. The oil saturation as a fraction of pore volume is given by
The core can be completely cleaned in the ASTM extraction apparatus, or once all water is removed, the remainder of the cleaning can be done in a soxhlet extractor (Figure 11). The mechanics of the soxhlet extractor are essentially the same as the ASTM extraction apparatus except that no receiving vessel is supplied for trapping water. The cleaning solution is continually vaporized and condensed on the core. This action leaches out the oil and water from the core. The ASTM extraction method does less damage to a core sample and results in perhaps the cleanest core of any of the saturation determinations. The core sample is ready for porosity or permeability determinations after this extraction process.
Before permeability and porosity can be measured, it is necessary to clean the core sample in a device similar to the soxhlet extractor or one which uses centrifugal force. Thus, using the core sample in a device is similar to the soxhlet extractor or one which uses centrifugal force. Thus, using the ASTM distillation only one additional step is required to obtain information from which to calculate fluid saturations in the core.
Figure 11. Soxhlet extractor

            Determination of Fluid Saturations with Electric Well Logs (Indirect)

Well logs are technique used in the oil and gas industry for recording rock and fluid properties to find hydrocarbon zones in the geological formations. At first, the symbols which appear in this section will be shortly described:

  1. Sw = water saturation: the percentage of the pore space filled with water (as opposed to hydrocarbons or air).
  2. R = resistivity: the resistance to electrical current flow presented by a unit volume of rock.
  3. Rw = water resistivity: the electrical resistance of the water filling the pore space in the rock. This value varies with water salinity and temperature.
  4. = porosity: the void space between grains that is generally filled with liquids or gases.
  5. FF: Formation Factor. The ratio between R0 of 100% saturated rock and Rw, and depends upon the lithological characteristics of the rock and the effective porosity.

 
The matrix of a rock which does not contain clay is an insulator. The electrical conductivity of this rock is due solely to the conducting network formed by the interstitial water contained in the pores. For a given rock sample, there is a constant ratio between the resistivity R0 of rock 100% saturated with conducting brine and the resistivity Rw of this brine. This constant which was first introduced is called Formation Factor. We have the equation of  FF:
 
(Ro resistivity of sample 100% saturated with brine whose own resistivity is Rw)
We have the Formation Factor is linked to porosity  by an equation of the form:
Where a and m are constants characterizing the rock (m varying from 1.3 to 2.2 and more, depending upon the state of cementation of the reservoir).
Since oil is an electrical insulator, it can be seen the fact, that a certain quantity of water is replaced by oil in the rock means an increase in resistivity.
Archie has shown experimentally that between the true resistivity (Rt) of the rock partially saturated with oil, the value S of the water saturation corresponding to this resistivity and the resistivity Ro of the rock 100% saturated with oil there is the following equation:
(Resistivity Ratio)
This can be written:
n2, if the rock is water wet,
2<n<4, if the rock is oil wet.
This equation makes it possible to obtain the in situ rock interstitial water saturation on the basis of resistivity measurements.
If the formation is homogeneous and is visibly oil-bearing at the top and water-bearing at the base electrical logs make it possible to determine Ro and Rt immediately
 
 

2.2 The Use of Core-Determined Fluid Saturations

The saturation values obtained directly from rock samples are usually not reliable for determining the quantity of each fluid in the rock. Other uses exist for fluid-saturation determinations from core samples. Water saturations obtained from core samples cut with oil-base mud are essentially reliable. The saturations of cores cut with water-base mud are used to determine the original oil-gas contact, original oil-water contact, and whether a sand is productive of oil or gas.
The two tables below show these invasions in the cases of two different types of mud:
water base mud and oil base mud.
 

  1. a) Variation in fluid saturation for a core between the reservoir and the surface in the case of water base mud:
Saturation Oil Gas Water
At surface 12% 40% 48%
shrink expand expulse
In core barrel 15% 0% 85%
flush invade
In reservoir 70% 0% 30%

 

  1. b) Variation in fluid saturation of a core between reservoir and surface in the case of oil base mud:
Saturation Oil Gas Water
At surface 40% 30% 30%
shrink and expulse
In core barrel 70% 0% 30%
invade
In reservoir 70% 0% 30%

 
 
The determination of contacts is made by carefully studying the residual oil saturations of the cores as a function of depth. In the oil-saturated regions the samples will have essentially a constant value for residual oil saturations, probably 15 per cent or greater. In the gas region the oil saturation is small or vanishes. Thus the depth of the gas-oil contact is defined by a sharp increase in oil saturation. In the water zone, the oil saturation gradually disappears with depth. By observing these changes in oil saturation, it is possible to choose the depth of the water-oil contact.
It is possible to establish a correlation of the water content of cores and permeability from which it can be determined whether a formation will be productive of hydrocarbons. Such a correlation is shown in Figure 12, wherein it can be noted that low-permeability formations with core water saturations as high as 55 percent may be considered productive. For higher permeability formations the upper limits of water saturations may be slightly less than 50 per cent. Thus, from the investigation of saturation values of cores one can gather that a formation would be productive if the water saturation in the surface samples were less than 50 per cent.
Figure 12. Limiting values of total core water for oil and gas production
 
Another reason for measuring fluid saturations of surface samples is to obtain other correlations such than direct or in direct measurements of other physical properties may also give indications of initial fluid distributions. The measurement of electrical resistivity of the core samples, prior to cleaning, permits correlations of electrical resistivities with other physical properties to aid in electrical log interpretation.
Thus, in summary it is seen that although fluid-saturation determinations made on core samples at the surface may not give a direct indication of the saturations within the reservoir, they are of value and do yield very useful and necessary information.

3      Capillary Pressure in Reservoir Rock

            Definition

In dealing with multiphase systems, it is necessary to consider the effect of the forces acting at the interface when two immiscible fluids are in contact. When these two fluids are liquid and gas, the interface is normally referred to as the liquid surface. All molecules are attracted one to the other in proportion to the product of their masses and inversely as the square of the distance between them. Considering water and oil, fluids commonly found in petroleum reservoirs, it is found that an interfacial tension always exists between the fluids. A water molecule which is remote from the interface is surrounded by other water molecules, thus having a resulting net attractive force on the molecule of zero. However, a molecule at the inter face 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 unbalanced and give rise to interfacial tension. The unbalanced attractive force between the molecules creates a membranelike surface. A certain amount of work is required to move a water molecule from within the body of the liquid through the interface. This work I frequently referred to as the free surface energy of the liquid. Free surface energy, in ergs per square centimetre, may be defined as the work necessary to create a unit area of new surface. The interfacial tension is the force per unit length required to create a new surface. Interfacial tension and surface tension are commonly expressed in dynes per centimetre, which is numerically equal to the surface energy in ergs per square centimetre. Surface tension is measured in the laboratory by standard means such as a tensiometer, the drop method.
Due to the interfacial tensions, a pressure difference exists across the interface. This pressure is called Capillary Pressure. It is related to:

  • The fluid/fluid interfacial tension
  • The relative wettability of the fluids
  • The size of the capillaries

 
We will define local saturation (or, more briefly, saturation) for each fluid at every point in the medium. Note that the volume element necessary to define saturation is undoubtedly greater than that required to define porosity.
Experience shows that, in a rather large range of values for saturation of fluids 1 and 2, each phase is continuous in the porous medium. Let us assume then that the porous block is in equilibrium in the gravity field at a given temperature. Then, according to hydrostatic laws, the pressure in each of the fluids depends only upon the quantity z, and we have:
(4.1)
(4.2)
These relationships can be integrated immediately since, at a fixed temperature,and  are known functions, respectively, of  and (g is the algebraic value of the acceleration of gravity).
 
It is therefore easy to extend the definition of pressures  and  to the whole space (although only a part of the space is occupied by each of the fluids). In the case of a porous medium in equilibrium, it is permissible, for the pressures, to consider the medium as continuous.
Capillary pressure will then be defined at every point in the porous medium by
(4.3)
From 4.1 and 4.2, the capillary pressure  is a well-defined function of the quantity z:
(4.4)

                     Capillary Forces – Wettability

The fluid distribution in porous media is affected by the forces at fluid/fluid interfaces, and also by forces at fluid/solid interfaces. Wettability is the tendency of one fluid to adhere to a solid surface in the presence of another fluid. When two immiscible fluids are in contact with a solid surface, one fluid is usually attracted more strongly than the other fluid. The more strongly attracted phase is called the wetting phase.
Wettability can be determined when checking for the contact angle:
Figure 13. Wettability of fluids
The solid is considered water-wet, if the contact angle α is smaller than 90°. At contact angles α larger than 90°, the fluid is referred to as oil-wet. Intermediate wettability occurs, when the contact angle α is close to 90° (Figure 13). By convention, contact angles are measured through the water phase. Water-wet is that the entire rock surface of both large and small pores is coated with water. Oil-wet is that the oil completely coats the rock surface. Intermediate wettability tends for both oil and water to wet the rock surface.
In case of wetting fluid, the contact angle is smaller than 90°. At contact angles larger than 90° , the fluid is referred to as non-wetting. In oil/water phase, water is wetting fluid, and oil is non-wetting fluid.

                Drainage and Imbibition

When we talk about capillary pressure, “drainage” and “imbibition” will not avoid to be talked. Depending on the wetting properties of the fluids there are essentially two different types of displacement in two-phase flow in porous media. “Drainage” is the displacement of the wetting fluid by a non-wetting fluid. In the contrary, “Imbibition” is the displacement of non-wetting fluid by a wetting fluid. E.g. in water-oil displacement processes, mostly water will be the wetting fluid.
Figure 14. Drainage and imbibition
Figure 14 shows a typical capillary pressure curve for a water-oil system in a porous rock. The capillary pressure curve consists of two branches: a main drainage, a main imbibition.
At Sw=1, the start of the drainage, an “entrance” pressure needs to be exceeded before oil can enter the sample. Then a plateau is reached. At decreasing water saturations, the capillary pressure rises to very high values. This means that when oil is injected into this system, an ever higher injection pressure is required to force the next bit of water out. The capillary pressure goes to infinity at the connate water saturation Swr.
When the oil pressure is slowly decreased, water will spontaneously imbibe and the saturation will increase. The capillary pressure decreases, and is in general smaller than the drainage capillary pressure for the same saturation, an effect called capillary hysteresis. When the oil pressure is equal to the water pressure (pc=0), the saturation reaches the spontaneous water imbibition saturation Sor. Increasing the saturation from this point can only be accomplished by forcing the water in. An ever higher water pressure is required to force the next bit of oil out, until the residual oil saturation Sor has been reached. Note that pc goes to minus infinity at water saturations near Sw=1-Sor.
 

            Methods of measuring capillary pressure

These measurements are difficult because the progress to ward equilibrium, at which capillary pressure is to be determined, is generally very slow. Therefore, measurements take a very long time, and we can never be quite sure that equilibrium has been effectively established.

             Desorption Method

The sample under study rests on a semi-permeable diaphragm (Figure 10) which allows the wetting phase to flow through, but not the nonwetting phase. The wetting phase of the sample communicates through this diaphragm with the atmosphere. The nonwetting phase bathing the sample is maintained at constant pressure. Equilibrium is reached when the flow of the wetting phase through the semi-permeable diaphragm stops. The pressure is then changed on the nonwetting phase in order to determine the following equilibrium.
Figure 15. Semi-permeable diaphragm

            Restored state method

If we suppose a rock sample from a field located at height h above the zero capillary level, the pair of fluids present in the field is characterized by:
Specific gravity of oil
Specific gravity of water
T = Interfacial tension of pair of fluids.
We have the capillary pressure:
A meniscus radius r corresponds to this capillary pressure and this pair of fluids such that:
 
Group (T﹒) characterizes the pair of fluids (T) and the solid (θ). In order to make the nonwetting fluid penetrate into pores with radius r with system (T, θ), a pressure Pc is necessary, while with system (,) a pressure  is necessary. Pressures Pc and  are

linked by the equation:
This makes it possible to choose one pair of fluids or another in order to study pore morphology or saturation states corresponding to various values for capillary pressure.
The relation between r and water saturation Sw is not rigorously constant if we go from one pair to another, but it will be supposed that there is an invariable relation between Sw and r. Hence the relation which is experimentally obtained in the laboratory between  and Sw can therefore be validly transformed for the real pair by the equation:
If  ,,  are known for one pair, ,  for the other pair and also , , the following curves can be plotted:

  • Capillary pressure in terms of water saturation (Figure 16)
  • Or water saturation in terms of the distance h above the zero capillary level

The height h is given by the equation:
 
Figure 16. Capillary pressure curve.(restored states method).

  • If h is counted from the zero capillary level
  • And if h is counted from the water/oil interface (“water level”) previously defined as

            Mercury injection method (Purcell)

In this method, the non-wetting fluid used is mercury and the wetting fluid, in a manner of speaking, is a vacuum. The sample is placed in a cell which is evacuated (Figure 17). By means of a positive displacement pump, mercury is then injected under a predetermined constant pressure until the mercury no longer enters the sample.
The value corresponding to the saturation in mercury is noted and the experiment is made again with a slightly higher pressure.
Figure 17. Mercury injection method
In the case of the water/air pair where the wetting phase is displaced, there is always a path through which circulation is possible leading to the creation of irreducible saturation in the pores which do not participate in this circulation.
In the case of the mercury/air pair, this condition does not exist and the curves. Obtained do not show the characteristic asymptote for irreducible saturation. The latter can perhaps be defined by the saturation corresponding to the beginning of the rectilinear part of the curve (if this point is marked) or, if not, to the saturation corresponding to a pressure of, for example, 10 to 15 or 20 bars depending upon the samples.
The shape of the curves obtained by plotting the equation for capillary pressure as a function of mercury saturation expressed as a percentage of pore volume is very variable from one sample to another (pore volume is carefully determined by an appropriate method such as immersion in a solvent).
Figure 18 concerns a homogeneous matrix medium. The beginning of the curve corresponds to a surface effect, i.e. the mercury has not yet definitely entered the pores. By means of a “surface correction” it is clearly possible to eliminate this part. The part represented by a broken line is then obtained.
Figure 18. Capillary pressure by mercury injection: homogeneous matrix medium
In certain cases where the pore radii are small, the PS1 threshold can be high: up to 30 bars abs., and even more.
Figure 19 makes it possible to distinguish:
Figure 19. Capillary pressure mercury injection: medium v macropores and matrix

  • The macropores, the part O – a which are invaded under very low pressures.
  • The pores constituting the matrix (as above) which are relatively regularly distributed. The part ab corresponds to channels which can be used for circulation while the part bd corresponds to the windings of the channels.

 
Figure 20 corresponds to the case of two homogeneous matrix media separated by an intermediate medium, for example calcite, coating the pores. The tangential departures of the curve from the abscissa should also be noted. This shape can be interpreted by observing that the macropores have later been filled by another medium
Figure 20. Capillary pressure by mercury injection in the case of 2 matrix.
The speed and accuracy of this method account for the fact that it is very widely used. It is however not suggested for very clayey samples since for a given pressure the saturation is too high because the pores are invaded by clay.
 

            Conversion of laboratory results

Capillary pressure measurements make in the laboratory with mercury/air or water/air pairs must be converted for the water/oil pair which exists in the field(reservoir condition).

If we suppose that the average curvature of an interface in a rock is solely a function of fluid saturation, the ratio of capillary pressures in the case of the mercury/air and water air pairs is:
Experience has shown that we have:

  • For limestones: a ratio of the order of 5.8
  • For sandstones: a ratio of the order of 7.5

There is no common factor for all rocks.
In order to use the laboratory results for capillary pressures, it is necessary to convert them to reservoir conditions. The laboratory results are obtained with a gas/water system which should have the same physical properties as the water, oil or gas of the reservoir. Two techniques differing only by their initial hypothesis are used for converting the laboratory capillary pressure results for reservoir conditions:
or :
 

4      Relationship between capillary pressure and fluids saturation

Let us examine the microscopic significance of capillary pressure. At equilibrium, a difference in pressure proportional to the curvature c of the interface exists between the two sides of the interface separating two immiscible fluids, the stronger pressure being on the concave side:
(4.5)
T is the interfacial tension. It is characteristic of the pair of fluids under consideration. Equation 4.5 is a special case for a fluid velocity everywhere zero.
The capillary pressure therefore depends on the curvature of the interface separating the two fluids and on the interfacial tension. From hydrostatic equilibrium condition 7.4, the curvature of the interface is a function of the quantity z.
In a block of porous medium sufficiently small, the influence of gravity may, on that scale, be neglected, and the interface, in all the pores, has a constant curvature related to the value of the capillary pressure by Equation 4.5.
 
This interface should, according to capillary laws, join the solid surface of the porous medium under a definite angle , the wetting angle. If the capillary pressure is given, the interface between the two fluids is subject to certain conditions. Its curvature is given by Equation 4.5, and the contact angle  at the points where it joins the solid surface is also given.
 
Figure 21. Example for the position of the interface
In some simple cases, this is sufficient to establish completely the position of the interface. Consider, for example, the case of a conic capillary tube with an angle of 2 at the apex (Figure 21). The interface in this tube will be spherical. If its curvature c and its angle of contact are prescribed, its position follows immediately. The proportion of, for instance, fluid 1 contained in this pore will thus be directly related to the capillary pressure. This type of reasoning has led to the belief that for a given porous medium there is a relationship between capillary pressure and saturation:
Examination of this simple model also shows that the pressure must be higher in the nonwetting fluid.
 
Unfortunately, the preceding reasoning based upon a conic capillary tube does not immediately apply to a porous medium with complex geometry. We can easily see that complicating the geometry of the system even slightly (for example, by considering capillaries in the form of truncated cones placed end to end) will reveal several different positions of interfaces, giving different saturation values for the same capillary pressure value. We may expect, then, that the observed experimental relationship between capillary pressure and saturation depends on how the experiment is carried out.
This is indeed what is observed. Assume that we begin initially with a sample completely saturated with a wetting fluid which is then very slowly replaced by a nonwetting fluid. This experiment produces a succession of equilibrium states for increasing values of saturation in nonwetting fluids which are very close to each other. Curve 1 (figure 22) represents the capillary pressure/saturation in nonwetting fluid SNM relationship for the process usually called “drainage”.
 
Figure 22. Drainage and imbibition curve in PC and Snm diagram
It will be seen that there is a certain amount of wetting fluid remaining in the sample, even under the highest pressure; this is the irreducible saturation in a wetting fluid.
If now, beginning with the sample at irreducible saturation in a wetting fluid, we slowly displace the nonwetting fluid with a wetting fluid, we obtain curve 2 of Figure 22. This process is called “imbibition.” Note that, for a zero capillary pressure, some saturation remains in a nonwetting fluid. This is the residual saturation in this fluid.
Imbibitions can also be conducted beginning with a state represented by an point on the drainage curve (dotted curves in figure 22), or drainages beginning at points on the imbibition curve (broken curves). Thus any point in the area lying between the drainage and imbibition curves may represent an equilibrium state. Note, however, that for slight saturations in a nonwetting fluid, the drainage curve may represent a state where the fluid is not uniformly distributed in the sample, i.e., only the surface of the sample may be reached by the non-wetting fluid because the saturation in this fluid is insufficient to allow it to create a path to the center.
In practice, we assume that phenomenon studied leads to either a drainage or an imbibition, and that, as a result, the relationship between capillary pressure and saturation is well established.

Summary

On the basis of this bakk. Thesis, The following conclusion is offered:

  1. Laboratory technique has been developed to measure the capillary pressure and fluid saturation from rock sample or rock logs under different equipment. But it is a little difference between the value on the stand condition and the reservoir condition. So the correction factor is searched and given to convert the laboratory condition to field condition.
  2. Drainage and imbibition curve show that the direct relation between wetting fluid saturation and capillary pressure: with the increasing of capillary pressure, the fluid saturation decreases that following drainage curve. In the country, with the decreasing of capillary pressure, the fluid saturation increases that following imbibition curve.
  3. The results of microscopic significance indicate that the interfacial area between fluid phases per volume of porous medium becomes a well-defined macroscopic property at an averaging volume similar to that of saturation. Simulated immiscible displacement experiments were performed to explore how the interfacial area between fluid phases changes during imbibition and drainage in two-fluid system.

References

  • [1]…….. reference from http://leeric.isu.edu/bgbb/4/rocus.html
  • Niesner u. F. Weber:” Geophysikalishe Bohrlochmessungen(Deutsch)”,OEH Leoben,1997
  • Robert P. Monicard: “Properties of reservoir rocks: Core analysis”, 1980 Editions Technip, Paris
  • Charles M. Marle: “Multiphase flow in porous media”, 1981 editions Technip, Paris
  • James W. Amyx: “Petroleum reservoir engineering – Physical Properties”, McGraw-Hill,1960
  • John C. Calhoun,JR: “Fundamentals of reservoir engineering”, Revised edition copyright 1953
  • William D. McCain: “The properties of Petroleum fluids – second edition”, Pennwalt books, Oklahoma,1990

Nomenclature

…………………………..Porosity
…………………………Total volume of rock [m³]
…………………………Porous volume of rock [m³]
…………………………Water saturation
…………………………Oil saturation
………………………..Residual water saturation
…………………………..Resistivity
………………………….Water resistivity
…………………………Formation Factor
……………………………Capillary pressure [psi]
…………………………..Density of oil [kg/m³]
…………………………..Density of water [kg/m³]
…………………………….Interfacial tension of pair of fluids
……………………………..Constant for curvature