resengch7

77Fundamental Properties of Reservoir Rocks

CONTENTS

INTRODUCTION

1. CHARACTERISTICS OF RESERVOIR ROCKS

2. PHYSICAL CHARACTERISTICS OFRESERVOIR ROCKS

3. POROSITY3.1 Range of Values3.2 Factors Which Affect Porosity3.2.1 Packing and Size of Grains3.2.2 Particle Size Distribution3.2.3 Particle Shape3.2.4 Cement Material3.3 Subsurface Measurement of Porosity3.3.1 Density Log3.3.2 Sonic Log3.3.3 Neutron Log3.4 Average Porosity

4. PERMEABILITY4.1 Darcy's Law4.2 Factors Affecting Permeability4.3 Generalised Form of Darcy's Law4.4 Dimensions of Permeability4.5 Assumptions For Use of Darcy's Law4.6 Applications of Darcy's Law4.7 Field Units4.8 Klinkenberg Effect4.9 Reactive Fluids4.10 Average Reservoir Permeability

5. STRESS EFFECTS ON CORE MEASUREMENTS5.1 Stress Regimes5.2 Compressibility of Poros Rock5.3 Types of Compressiblilty5.4 Measurements of Pore Volume Compressiblity5.5 Effect of Stress on Permeability

6. POROSITY - PERMEABILITY RELATIONSHIPS

7. SURFACE KINETICS7.1 Capillary Pressure Theory7.2 Fluid Distribution in Reservoir Rocks7.3 Impact of Layered Reservoirs

8. EFFECTIVE PERMEABILITY8.1 Definition8.2 Water Displacement of Oil8.2.1 Water - Oil Relative Permeability8.3 Gas Displacement of Oil and Gas - Oil Relative

Permeability

2

LEARNING OBJECTIVES

Having worked through this chapter the Student will be able to:

• Define porosity and express it as an equation in terms of pore, bulk and grainvolume.

• Explain the difference between total and effective porosity.• Define permeability and present an equation, Darcy’s Law, relating flow rate

to permeability in porous media.• List the assumptions for the applicability of Darcy’s Law.• Derive an equation based on Darcy’s Law relating flow of gas in a core plug and

the upstream and downstream pressures.• Derive an equation relating flow rate to permeability for a radial incompressible

system.• Comment on the difference between gas and liquid permeability (Klinkenberg

effect ).• Sketch a figure relating liquid permeability to gas permeabilities plotted as a

function of reciprocal mean pressure.• Briefly describe the impact of reservoir stresses on permeability and porosity• Draw a sketch demonstrating the result of interfacial tension between oil, water

and a solid, and locate the contact angle and define its values for wetting andnon-wetting phases.

• Express the capillary pressure Pc as two equations, one in terms of interfacialtension, contact angle and pore radius, and the other in terms of height anddensity of fluids.

• Define the free water level.• Draw the Pc or height vs. saturation capillary pressure curve and identify

significant features.• Sketch and explain the impact of saturation, history, density difference and

interfacial tension in capillary pressure curves.• |Sketch the impact of capillary pressure effects on the saturation distribution of

stratified formations• Define effective and relative permeability and plot typical shapes.• Define imbibition and drainage in the context of capillary pressure and relative

permeability curves.• Sketch the pore doublet model and use it to explain the retention of trapped oil

in large pores and briefly relate it to the principle behind some enhanced oilrecovery methods.

• Define mobility ratio.• Sketch a shape for gas- oil relative permeability curves.

Department of Petroleum Engineering, Heriot-Watt University 3


INTRODUCTION

The properties of reservoir rocks with respect to the fluids they contain and withrespect to the fluids which will be injected into them are important when characterisinga reservoir in terms of its reserves and the mobility of the fluids. This next sectiongives a brief over view of these properties, and is followed by chapters on theirmeasurement and variation. In relation to the detailed description of rock characteristicsthe reader is referred to the Geology module of this Petroleum Engineering course.

The reservoir engineer is concerned with the quantities of fluids contained within therocks, the transmissivity of fluids through the rock and other related properties.

1. CHARACTERISTICS OF RESERVOIR ROCKS

The specifications of a reservoir rock are such that there must be a large enoughcapacity to store economically viable amounts of hydrocarbon and the hydrocarbonmust flow at economical rates when penetrated by a well. The factors which may affectthe capacity and the flow properties are the porosity, permeability, capillary pressure,compressibility and fluid saturation. In the case of a reservoir rock, these are notstandard characteristics determined before formation of the rock, but are closelylinked to the geological processes that brought the sediments together and depositedthem in the sequences and under the chemical and physical changes inherent in thesystem.

In order to contain enough oil or gas to make production economically viable, areservoir rock must exceed: a minimum porosity, a minimum thickness, a minimumpermeability, and a minimum area.

In order to extract the fluids the rock must be permeable which requires that there besufficiently large, interconnecting pores.

Although a permeable rock must also be porous, a porous rock is not necessarilypermeable. Certain volcanic rocks are porous but not permeable because the voids arenot interconnecting; shale may be quite porous but impermeable because the pores areextremely small, thereby preventing free movement of the fluids contained within.

2. PHYSICAL CHARACTERISTICS OF RESERVOIR ROCKS

Considering a common reservoir rock, sandstone, the grains making up this rock areall irregular in shape. The degree of irregularity, or lack of roundness reflects thesource sediments and the physical and chemical processes to which they weresubsequently exposed. Violent crushing or grinding action between rocks causesgrains to be very irregular and sharp-edged. The tumbling action of grains along thebottom of streams or seabeds smoothes sand grains. Wind-blown sand, as occurs inmoving dunes in deserts, results in sand grains that are even more rounded. Sandgrains that make up sandstone beds and fragments of carbonate materials that makeup limestone beds do not fit together congruently: the void space between the grainsforms the porosity.

4

The pore spaces (or interstices) in reservoir rock provide the container for theaccumulation of oil and gas and these give the rock its characteristic ability to absorband hold fluids. Most commercial reservoirs of oil and gas occur in sandstone,limestone or dolomite rocks, however, some reservoirs occur in fractured shale andeven in basement rocks such as in Vietnam. Knowledge of the physical characteristicsof the pore space and of the rock itself (which controls the characteristics of the porespace) is of vital importance in understanding the nature of a given reservoir.

For the reservoir engineer, porosity is one of the most important rock properties as ameasure of the space available for accumulation of hydrocarbon fluids.

3. POROSITY

The first step in forming a sandstone, for example, is to have a source of material whichis eroded and transported to low lying depressions and basins such as would be foundoff the coasts of a landmass. The material would consist of a mixture of minerals, butfor a sandstone, the majority would be made of quartz in the form of grains. When thesewere deposited, they would be surrounded by sea water or brine, and as the sedimentthickness increased, the weight or the pressure produced by the overlying sedimentswould force the grains together. Where they contacted each other large stresses wouldbe produced and a phenomenon called pressure solution would occur which dissolvedthe quartz at the points of contact between the grains until the stresses reduced to a levelwhich was sustainable by the grains. The dissolved material would be free toprecipitate in other regions of the sediment. In this way the initially loose materialwould be solidified with discrete connections between the grains.

Initially, if subsea, the pore spaces would be filled with brine, and as the lithificationprocess occurred, some pore spaces would be isolated with the brine trapped inside.If the vast majority were interconnected then the initial pore fluid would be free to beswept through the rock by other fluids such as hydrocarbons. In this way the geometryof the grains produces an assembly of solids with voids in between them. The grainsvary in diameter but may be from a few microns to several hundred microns. Thegeometry of the pore spaces is such that they have narrow entrances (pore throats)where the edges of the grains touch each other and larger internal dimensions (betweenthe grains). The complicated nature of these interconnected pores is illustrated infigure 1 which is a metal cast of the pores in a sandstone rock.

Figure 1

Metallic Cast of Pore

Spaces in a Consolidated

Sand



One method of classifying reservoir rocks, therefore, is based on whether pore spaces( in which the oil and gas is found) originated when the formation was laid down orwhether they were formed through subsequent earth stresses or ground water action.

The first type of porosity is termed original porosity and the latter, secondary orinduced porosity. This is illustrated in figure 2.

Cementing material

Sand grain

Effective porosity 25%

Isolated porosity 5%

Total porosity 30%

Secondary porosity in limestone beds occurred as a result of fracturing, jointing,dissolution, recrystallisation or a combination of these processes.

Where water is present in a carbonate formation, there is a continuous process ofsolution and deposition or recrystallization. If solution is greater than deposition inany zone, porosity will be developed between the crystal grains. An important typeof porosity of this kind is found in dolomite zones which occur in conjunction withlarge limestone deposits. Dolomite may be deposited originally as a sedimentary rock,or it may be formed by replacing the calcium carbonate in limestone rock withmagnesium.

The impact of isolated pore space clearly cannot contribute to recoverable reserves offluid nor contribute to permeable pore space as illustrated in figure 3.

Total Pore Space

Dead EndPore

Isolated Pore SpaceEffective Pore Space

Permeable Pore Space

Figure 2

Effective, isolated and total

porosity

Figure 3

Total, effective, isolated

permeable and dead end

pore space

6

Porosity is defined as the ratio of the void space or pore space (Vp) in a rock to the bulk

volume (Vb) of that rock and it is normally expressed as a percentage of total rock

volume. The porosity is usually given the symbol φ. The matrix volume is the volumeof the solid grains, V

m.

PorosityVoid volumeBulk volume

PorosityB volume

Bulk volume

=

=−

=+

×

x

ulk Grain volumex

Porositypore volume

void volume grain volume

100

100

100

Bulk VolumeRepresentation

Grain VolumeRepresentation

Pore VolumeRepresentation

These components are illustrated in figure 4 for monosize spheres.

Total porosity is defined as the ratio of the volumes of all the pores to the bulk of amaterial, regardless of whether or not all of the pores are interconnected. Effectiveporosity is defined as the ratio of the interconnected pore volume of a material.

If the grains are represented by spheres stacked together as in figure 4, then the porespace can be seen between the solid grains.

Total Porosity = Total Void SpaceVb

Effective Porosity = Interconnected Void SpaceVb

Induced or Secondary Porosity = porosity from fractures or vugs (large chambersformed in certain carbonates and limestones caused by groundwater flow anddissolution).

Figure 4

Representation of bulk,

grain and pore volumes



3.1 Range Of ValuesThe maximum porosity of porous media can be considered in relation to an assemblyof spheres arranged as a cubic packing of spheres. If the sides of a cube are assumedto be formed by the lines drawn from the centre of each sphere to the adjacent spheres,the cube in figure 5 would be produced.

The length of each side would be 2x radius, giving the bulk volume as:

Vb = (2r)3 = 8r3

The grain volume would be the equivalent of the volume of one sphere

Vm = r4

3

3π

and the porosity (given the symbol φ) would be

=−

=−

=V V

V

rr

rb m

b

84

38 6

33

3φ

ππ

1 - = 0.476

If the spheres fit in the cusps generated by the lower layer then a porosity of 26%occurs. For a size distribution of spheres the ultimate minimum porosity would bezero which would be the case if sufficient grains were available to completely fill thepore spaces as shown in figure 6 for part filling of the void.

Several factors may combine to affect the porosity of a rock, but the main distinctionto be made is as follows based on the amount of connected pore volume, and whetherthe pore space has been altered by dissolution or by fracturing after deposition andlithification.

Figure 5

Cube defined by the centres

of each adjacent sphere

Figure 6

Minimum porosity when all

pore spaces are filled

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3.2 Factors Which Affect PorosityThe porosity (and permeability) of sandstone depend upon many factors, amongwhich are the packing, size and shape of the grains, variations in size of grains,arrangement in which grains were laid down and compacted, and amount of clay andother materials which cement the sand grains together.

3.2.1 Packing And Size Of GrainsThe absolute sizes of the sand grains which make up a rock do not influence the amountof porosity occurring in the rock. However variations in the range of sand grains sizesdo influence considerably the porosity.

3.2.2 Particle Size DistributionIf spheres of varying sizes are packed together, porosity may be any amount from 48per cent to a very small amount approaching 0 per cent as shown in figure 7.

3.2.3 Particle ShapeIf the sand grains are elongated or flat and are packed with their flat surfaces together,porosity and permeability may both be low we will discuss further in the context ofpermeability.

Pore Space

3.2.4 Cement MaterialSandstones are compacted and usually cemented together with clays and minerals.The porosity and permeability of a sandstone are both influenced to a marked degreeby the amount of cementing material present in the pore space and the way thismaterial occupies the pore space between the sand grains. The cementing materialmay be uniformly located along the pore channels to reduce both porosity andpermeability or the cementing material may be located at the pore throats whichreduces the ability of fluid to enter the pore, but may not reduce the overall porosityof the rock by a significant amount.

Limestone formations may have intergranular porosity. However, the pore openingsare more often inter-crystalline, that is spaces between microscopic crystals. Theyalso may take the form of pits or vugs caused by solution and weathering, or byshrinkage of the matrix. These forms of porosity are called secondary porosity.Another type of secondary porosity is that caused by fracturing and is very important

Figure 7

Reduction in porosity due to

a range of particle sizes



in that it permits many limestone rocks of otherwise low porosity to become excellentreservoirs.

Porosity may range from 50% to 1.5% and actual average values are listed below:

Recent sands (loosely packed) 35 - 45%Sandstones (more consolidated) 20 - 35%Tight/well cemented sandstones 15 - 20%Limestones (e.g. Middle East) 5 - 20%Dolomites (e.g. Middle East) 10 - 30%Chalk (e.g. North Sea) 5 - 40%

A point that needs to be emphasised is that the concept of ‘porosity’ is complex andtherefore difficult to define and determine. It may refer to spaces between sand grainsor it may refer to limestone caves: it may even exclude a fraction of the free water(water not bound chemically) present in the rock. Sometimes good estimates, (i.e.relevant to reservoir development problems) may be obtained from laboratory studies,or core samples, and sometimes such measurements are irrelevant.

In summary, the amount of porosity is principally determined by shape and arrange-ment of sand grains and the amount of cementing material present, whereas perme-ability depends largely on the size of the pore openings and the degree and type ofcementation between the sand grains. Although many formations show a correlationbetween porosity and permeability, the factors influencing these characteristics maydiffer widely in effect, producing rock having no correlation between porosity andpermeability.

3.3. Subsurface Measurement Of PorosityPorosity is measured directly from recovered rock samples as part of core analysis andalso downhole by special tools which indirectly measure a property which can berelated to the formation porosity. These downhole measurement techniques are verysophisticated in both their engineering and in their practice. For example, the porosityof a formation can be logged while the hole is being drilled, giving almost real timeindications of the nature of the reservoir. Core analysis procedures will be reviewedlater.

In general the downhole porosity may be related to the acoustic and radioactiveproperties of the rock.

3.3.1 Density LogThe density log is derived from the response of the atoms in the minerals in the rockto bombardment with gamma radiation. The atoms accept energy of a specificfrequency and emit energy of a different frequency; this energy is detected. Theenergy density is related to the number of atoms and therefore to the density of the rockbeing bombarded. If the formation under test is known, for instance a sandstone, thenchanges in the density measured within the sandstone result from a change in theporosity of the formation rather than a change in the mineralogical nature of thesandstone. This obviously relies on a good description of the geology of the formation.In a porous formation, the pore fluid will also affect the response of the tool in that the

10

atoms of the fluid will also react to the bombardment and affect the energy detected.With reference to calibration samples of different rock types, the effect of bothmineralogy and pore fluid content can be accounted for. Empirical relationships havebeen developed to relate the porosity to the values of density which have been logged.In the following relationship, the logged density, ρ

L, matrix density, ρ

m , and the fluid

density, ρf , are related to the porosity, φ

−L m

ρ ρ φ ρ φ

φρ ρρ

= (1- ) +

=

L m f

ff m− ρ

The contribution of the matrix and the pore fluid are in relation to the relative amountsof each, and these are related to the porosity. Typically, matrix densities and freshwater density are as follows

ρQuartz = 2.65 gcm-3

ρLimestone = 2.71 gcm-3

ρWater = 1.00 gcm-3

3.3.2 Sonic LogThis log is similar in concept to the density log, however, it is acoustic energy whichis radiated into the formation from sonic transducers in the logging tool. Theseproduce compression waves which travel along the side of the borehole in theformation. The time taken for the wave to travel from the transmitter to the receiver(travel time) is related to the acoustic properties of the formation. As for the case ofthe density log, if the formation is known and its mineralogy is not changing, thenvariations in the travel time must result from the changes in the formation acousticproperties, the most significant of which is the density which is related to the porosity.As with the density tool, the density of the formation fluids in the pore spaces willaffect the travel time and this must be accounted for. Calibration samples of differentrock types have lead to an empirical relationship between the logged travel time, ∆T

L ,

matrix travel time, ∆Tm , the fluid travel time, ∆T

f , and the porosity, φ .

L m

f m

T TT T

−−

φ φ

φ

∆ ∆ ∆

∆ ∆∆ ∆

T = T (1- ) + T

=

L m f

The contribution of the matrix and the pore fluid are in relation to the relative amountsof each, and these are related to the porosity. Typically, matrix travel times and freshwater travel time are as follows

∆TQuartz = 55µs ft-1

∆TLimestone = 47µs ft-1

∆TWater = 190µs ft-1

3.3.3 Neutron LogThis is another radioactive logging technique which measures the response of the



hydrogen atoms in the formation and can give an indication of the porosity. Neutronsof a specific energy are fired into the formation and they disrupt the steady stateactivity of hydrogen atoms. They then radiate energy which is detected by the tool: theenergy returned is related to the number of hydrogen atoms which is related to thehydrocarbon and water in the pore spaces. By calibration, the porosity can bedetermined.

3.4 Average PorosityPorosity is normally distributed and an arithmetic mean can be used for averaging. Forunclassified data,

= =∑

φφ

na

ii 1

n

(1)

where φa is the mean porosity, φ

i is the porosity of the ith core measurement and n is

the number of measurements.

4 PERMEABILITY

4.1 Darcy's LawThe permeability of a rock is the description of the ease with which fluid can passthrough the pore structure.

At one extreme, the permeability of many rocks is so low as to be considered zero eventhough they may be porous. Such rocks may constitute the cap rock above a porousand permeable reservoir and they include in their members clays, shales, chalk,anhydrite and some highly cemented sandstones.

The permeability is a term used to link the flowrate through and pressure differenceacross a section of porous rock. The problem is complicated in that the number of porespaces, their size and the interconnections is not standard. Thus the application of thegeneral energy equation, for example as in the case of flow through pipes, becomesvery difficult for flow through porous media.

In petroleum engineering the unit of permeability is the Darcy, derived from theempirical equation known as Darcy’s Law named after a French scientist whoinvestigated the flow of water through filter beds in 1856. His work provided the basisof the study of fluid flow through porous rock.

Qk P A

L=

µ∆ . (2)

where:

Q = flow rate in cm3/secA = cross sectional area of sample in cm2

∆P= pressure different across sample, atm

12

µ = viscosity in centipoiseL = length of sample in cmk = permeability in Darcy

Darcy’s law of fluid flow states that rate of flow through a given rock varies directlywith the pressure applied, the area open to flow and varies inversely with the viscosityof the fluid flowing and the length of the porous rock. In terms of equating theparameters, the constant of proportionality in the equation is termed the permeability.The unit of permeability is the Darcy which is defined as the permeability which willpermit a fluid of one centipoise viscosity (= viscosity of water at 68°F) to flow at alinear velocity of one centimetre per second under a pressure gradient of oneatmosphere per centimetre. Permeability has the units Darcys. Figure 8 illustrates theconcept and the units of permeability

L = 1 cmk = 1 darcy

1cm2Q = 1 cm3

µ = 1 cp

∆p = 1 atmos

sec

Darcy’s Law experiment consisted of a sandpack through which water flowed at aconstant rate (figure 9).

Manometricheads of water

Length, L

Flowrate, Q

Flowrate, Q

h1 h2

San

d

Area of the end of the sandpack

His results showed that the flowrate was directly proportional to the area open to flow,the difference in pressure and inversely proportionate to the length of the sandpack,i.e.

Q A hL

∝ ∆, ,1

Figure 8

Concept of permeable rocks

Figure 9

Schematic of Darcy’s

experiment



or

Q kA h h

L=

−( )1 2

where Q is the flow rate, A is the area of the end of the core, h1 and h

2 are the static

heads of water at the inlet and outlet of the core (the equivalent of the static pressure),L is the length of the core. K is the constant of proportionality. It is constant for aparticular sand pack. When other workers replicated the experiment, the results weredifferent to those of Darcy. This was accounted for by inclusion of the viscosity of theflowing fluid and the equation becomes:

QkA h h

L=

−( )1 2

µ

where the original terms have the same meaning and µ is the viscosity of the fluid incentipoise.

On a more theoretical basis, Poiseuille formulated the relationship between flow rateand pressure drop for fluid flowing in a pipe. The form of the relationship is

=π

µQ

r P8 L

4∆

(3)

where Q is the flowrate, r is the radius of the tube, µ is the viscosity of the fluid andL is the length of the tube. In this case the dependence of the flowrate / pressure droprelationship can be seen to be dependent on the radius of the tube. In a similar manner,the radius of the pores in a rock dictate the nature of the relationship, specifically, theradius of the pore throats is of most significance, since these are the smallest radii andtherefore affect the flowrate/ pressure drop relationship most.

The practical unit is the millidarcy (mD) which is 10-3 Darcy. Formationpermeabilities vary from a fraction to more than 10000 milli-Darcies. At the low endof the range, clays and shales have permeabilities of 10-2 to 10-6 mD. These very lowpermeabilities make them act as seals between more permeable layers.

4.2 Factors Affecting Permeability Permeability along the flat surfaces will be higher, than the permeability in a directionperpendicular to the flat surfaces of the grains. In a reservoir, the permeabilityhorizontally along the bed is usually higher than the permeability vertically across thebed because the process of sedimentation causes the grains to be laid down with theirflattest sides in a horizontal position (minimising the area exposed to the prevailingcurrents during deposition). Figure 10 illustrates the concept.

If sand grains of generally flat proportions are laid down with the flat sides non-uniformly positioned and located in indiscriminate directions, both porosity andpermeability may be very high. To illustrate, if bricks are stacked properly, the spacebetween the bricks is very small; if the same bricks are simply dumped in a pile, thespace between the bricks might be quite large.

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Horizontal permeability 400mDVertical permeability 200mD

Horizontal permeability 900mDVertical permeability 500mD

Porosity 16% Porosity 32%

The shape and size of sand grains are important features that determine the size of theopenings between the sand grains. If the grains are elongated, large and uniformlyarranged with the longest dimension horizontal, permeability to fluid flow through thepore channels will be quite large horizontally and medium-to-large vertically. If thegrains are more uniformly rounded, permeability will be quite large in both directionsand more nearly the same. Permeability is found generally to be lower with smallergrain size if other factors (such as surface tension effects) are not influential. Thisoccurs because the pore channels become smaller as the size of the grains is reduced,and it is more difficult for fluid to flow through the smaller channels.

This directional perspective to any property is termed anisotropy. As shown abovepermeability is a directional property and gives rise to different permeabilitiesdepending on the shape and depositional characteristics. Very dramatic anisotropy isgenerated if a rock is fractured. These anisotropic perspectives are illustrated in figure 11.Porosity is a non directional property and therefore is isoptropic.

Sandstone Fractured Core

4.3 Generalised Form Of Darcy’s LawA three dimensional rock can be defined within the co-ordinate system illustrated infigure 12.

Figure 10

Directional Permeability

Figure 11

Directional permeability.



-Z

+y

+Z

+x

Vss

0

The x and y co-ordinates increase from zero to the left and out from the page; the z co-ordinate increases downwards. The flow velocity in a particular direction can bedefined as the flowrate in that direction divided by the area open to flow. In anydirection, s, the flow velocity is termed Vs and is equated to the static pressure gradientin that direction (i.e. the change in pressure, dP, over a small element of length, ds inthat particular direction) minus a contribution from the difference in head (becauseof the difference in elevation) of the fluid across the section ds. Therefore,

dpds

gx

dzds

−(.

)61 0133 10µρ

V = -k

s (4)

and the change in elevation head is equal to the sine of the angle to the horizontal

= sine θ, where θ is in degrees.

The Darcy units are:

Vs

= velocity along path s - cms-1

k = permeability - Darcysµ = viscosity - centipoiseρ = density of fluid - gcm-3

g = acceleration due to gravity - 980 cms-2

pds

− = pressure gradient along s - atm cm-1

1.0133 x 106 converts from dynes cm−2 to atmospheres

4.4 Dimensions Of Permeability

From Darcy’s equation,dpds

gx

dzds

−(.

)61 0133 10µρ

V = -k

s the dimensions of

each term can be deduced in terms of length, L, mass, M and time, T

LT

MLT

ML

3µ ρV = = =

P =

s

MMLT

LT

dPds

ML T2 2 2 2g = =

Figure 12

Co-ordinate system for

rock permeability

16

Therefore, the equation in terms of the dimensions (and keeping permeability as k) is

LT

kLTM

ML T

MLL T

LT

KLT

2 2 3 2=

=

K = L2

( )−

(5)

It can be seen that the dimensions reflect the nature of the constant of proportionalityand it should not be confused with, for example, the area open to flow, A, of the endof a core or a sand pack. In terms of metric units, since 1 atm = 14.73 psi = 1.013 barand 1 cp = 10-3 Pas it follows that

1D = 9.87 x 10-13m2 ~ 1 x 10-12m2

1mD = 9.87 x 10-16m2 ~ 1 x 10-15m2

Other units of inches2 or cm2 could be used but they are all too large for porous mediaand they would also require conversion to relate to permeabilities quoted in otherunits. Darcys and milliDarcys are most commonly used.

4.5 Assumptions For Use Of Darcys LawThe simple Darcy Law, as used to determine permeability, only applies when thefollowing conditions exist:

(i) Steady state flow(ii) Laminar flow;(iii) One phase present at 100% pore space saturation.(iv) No reaction between fluid and rock;(v) Rock is homogenous

1. Steady state flow, i.e. no transient flow regimes. This becomes unrealistic in termsof flow in a reservoir where the nature of the fluids and the dimensions of the reservoirmay produce transient flow conditions for months or even years. For laboratory basedtests, the cores are small enough that transient conditions usually last only a fewminutes.

2. Laminar flow, i.e. no turbulent flow. For most reservoir applications this is validhowever near to the well bore when velocities are high for example in gas productionturbulent flow occurs. Sometimes it is termed non- darcy flow. Figure 13



Laminar Flow

Turbulent Flow

=

∆PQA

µL

∴ K = . .

QA

kµ

. ∆PL

QA

∆PL

3. Rock 100% saturated with one fluid, i.e. only one fluid flowing.

In the laboratory this can be achieved by cleaning cores, however, there will be acertain connate water saturation in the reservoir, and there may be gas, oil and mobilewater flowing through the same pore space. The concept of relative permeability canbe used to describe this more complex reservoir flow regime. Relative permeabilityis discussed later.

4. Fluid does not react with the rock, i.e. it is inert and there is no change to the porestructure through time.

There are cases when this may not happen, for example when a well is stimulatedduring an hydraulic fracturing workover. The fluids used may react with the mineralsof the rock and reduce the permeability. In such cases, tests on the rock to determinethe compatibility of the treating fluids must be conducted before the workover.

5. Rock is homogeneous and isotropic, i.e. the pore structure and the materialproperties should be the same in all directions and not vary. In reality, the layerednature and large areal extent of a reservoir rock will produce variations in the verticaland horizontal permeability.

4.6 Applications of Darcys LawTo examine the applicability of this simple relationship, approximations to the typeof flow encountered in a reservoir can be made: linear flow along a reservoir sectionand radial flow into a wellbore. More complex geometries cannot be analysed usingthis simple analytical equation and forms of approximating the geometry and flow arerequired.

Figure 13

Effect of Turbulent Flow

on Measured Permeability

18

In the following expressions, the nomenclature is identical to that used above.

(i) Horizontal, linear, incompressible liquid system (figure 14)

A

L

P1

P2Q

From the basic Darcy equation

K dPds

gx

dzds

gx

dzds6 61 0113 10 1 0113 10

V = - , = zeros (.

).

−µ

ρ ρ

The flow rate and area open to flow is substituted for the flow velocity. The variablesare separated and integrated over the length (for the flow rate) and the pressures P

1 to

P2 for the change in pressure. The pressure drop P

2 minus P

1 is negative and is corrected

by the negative sign on the left hand side of the equation.

QA

kA dPdx

Q dxkAL

0

V = V =

Q = -

= -

s x

∫

µ

µdPdP

kAP P

kA P PL

P

P

1

2

2 1

1 2

∫

−

−

Q(L - 0) = -

Q =

µ

µ

( )

( )

(6)

The final form is as formulated by Darcy and the permeability will have the units ofDarcys if the other units are:

flow rate, Q - cm3s-1 pressure, P - atmarea open to flow, A - cm2 length, L - cmviscosity, µ - centipoise

(ii) Horizontal, linear, compressible ideal gas system

Figure 14

Linear flow regime



The flow regime is the same as for the linear liquid system and from the basic Darcyequation:

kdPds

gx

dzds

gx

dzds

QAkA dP

dx

6 61 0113 10 1 0113 10−V = V = - , = zero

V =

Q = -

s x

s

ρ ρ

µ

(.

).

In this case, the laboratory measurement of the gas flow would usually be conducteddownstream from the core at almost atmospheric conditions (i.e. there would not bea large pressure drop across the flow meter). It is assumed that the gas used is ideal,however, there needs to be a correction to the volumetric flow rate measured toaccount for the higher pressure in the core. Figure 15.

Valve Qb

P1 P2 Pb

Flowmeasurement

Core

L

A

P

The flowrate measured, Qb at ambient pressure, P

b is related to the flowrate, Q in the

core at the pressure in the core, P via the ideal gas law. If the assumption is made thatthe temperature is constant, then

QP = Q P

Q =Q P

P

b b

b b

and substituting into the equation, separating the variables and integrating produces

Q PP

= -

Q P = -

Q P (L - 0 = -

Q =

b b

b b

b b

b

kA dPdx

dxkA

PdP

kA P P

kA P PLP

L

P

P

b

µ

µ

µ

µ

0 1

2

22

12

12

22

2

2

∫ ∫

−

−

( )

( )

(7)

Figure 15

Configuration for gas

permeability measurements.

20

kQbL

A P P=

−2

12

22

µ( ) (8)

Comparing the two expressions equations 6 and 7, it is seen that the gas flow rate isproportional to the difference in the pressure squared, whereas the liquid flowrate isproportional to the difference in the pressure. In well testing, the flow rates aremeasured at the surface and for gas wells one of the diagnostic plots is the flowrateversus difference in pressure squared plot. Neglecting the fact that the gas is real, itgives an indication of the ability of the reservoir to produce gas.

Gas Q = Liquid Q = b

kA P PLP

kA P P

bµ µ12

22

1 2

2− −( ) ( )

LL

In certain circumstances, the mean flow rate, Q is measured at a mean pressure, Pwhich, in the case of a laboratory experiment on a core, is the mean of the upstreamand downstream pressure, i.e.

PP P

= 1 2

2+

and

Q = Volume flow rate at P

P Q = PbQ

b

substituting this into the above gas equation 7.

PQkA P P

LP Q =b b

12

22

2=

−( )µ

and since

12

P P Q12

kAL

(P P )(P P )1 2 1 2 1 2+( ) = − +µ

QkA P P

L=

−( )1 2

µ (9)

The ideal gas permeability can be calculated from the liquid equation using meanflowrate, Q measured at mean pressure.

(iii) Horizontal, radial, incompressible liquid system (figure 16)



Well

Radial flow

Plan Elevation

Pe

Pw

rw

re

re

rw

re

is the outer boundary radiusr

wis the inner boundary radius (well)

Pe

is the pressure at the external boundaryP

wis the pressure at the inner boundary

Starting from the basic Darcy expression again,

k dPds

gx

dzds

gx

dzds

−(.

).6 61 0113 10 1 0113 10µ

ρ ρV = - , = zeros

Substituting for flow velocity, QA

V = V = s r

In this case the direction of flow is in the opposite sense to the co-ordinate system,therefore

ds = -drFor radial geometry, the area, A, is now radius dependent therefore

A = 2πrhSubstitution into the basic expression gives

Qrh

k dPdr−2π µ

= - (10)

separating the variables and integrating

Qh

drr

kdP

Qh

r rk

P P

rw

re

Pw

Pe

e w e w

=

− −

∫ ∫

(ln ln ) (

2

2

π µ

π µ= ))

which gives the final form

Figure 16

Radial geometry with radial

flow from the outer

boundary to the wellbore

22

( )

lnQ =

2π

µ

kh P Prr

e w

e

w

− (11)

(iv) Horizontal, radial, compressible real gas system

In this case the geometry is identical to that of the radial flow of incompressible fluidwith the modifications for the compressibility of a gas as per the linear gas flowsystem.

(.

).

V = - , = zero

= -

s 1 0113 10 1 0113 10

2

6 6µρ ρ

π µ

k dPds

gx

dzds

gx

dzds

Qrh

k dPdr

−

−

If the assumption is made that the temperature is constant, then

QP = Q P

Q =

b b

Q PPb b

and substituting into the equation, 10

Q PP

2 rhk dP

drb b = π

µ

separating the variables

Q Pdrr

2 khPdPb b

r

r

P

P

w

e

w

e

=∫ ∫πµ

and integrating produces

Q P lnrr

2 kh P P2

Qkh

P lnrr

P P

b be

w

e2

w2

b

be

w

e2

w2

=

−

=

−( )

πµ

π

µ

(10)



4.7 Field UnitsMeasurements made in the field are often quoted in field units and to ensurecompatibility with the Darcy equation, a conversion is required. The field units areusually as follows:

Flow rate, Q - bbl/day or ft3/dayPermeability, k - DarcyThickness or height of reservoir, h - feetPressure, P - psiaViscosity,µ - centipoiseRadius, r - feetLength, L - feet

I oder to convert the Darcy equation for liquid flow, Q = −

µKA P P

L( )1 2 to oil field

units, the following conversion factors are used:

Q bblday

(.5 615615 1728 16 39

24 3600

92914 696

30 48

3 3

3

3

3

22

2

ftbbl

inft

cmin

dayhr

hrs

K Aftcm

ftPpsia

atmpsia

Lftcm

ft

)( )(.

)( )( )

( )( )( )( )(.

)

( )( )(.

) =

∆

µ

and these produce the following version of Darcy’s equation in field units:

1 2bblday

KA P PL

( )Q = 1.1271

µ−

(11)

4.8 Klinkenberg EffectDarcy’s Law would indicate that the permeability should be the same irrespective ofthe fluid transmitted, since viscosity is included in the equation. Measurements madeon gas as against liquid for some conditions give higher permeabilities than the liquid.

This phenomenon is attributed to Klinkenberg, who attributed the behaviour to theeffect of the slippage of gas molecules along the solid grain surfaces. This occurs whenthe diameter of the capillary opening (pore throat diameter) approaches the mean freepath of the gas (i.e. there is in effect only one gas molecule per capillary). Darcys Lawassumes laminar flow and viscous theory specifies zero velocity at the boundary of theflow channel. This is not valid when the mean free path of the gas approaches thediameter of the capillary and the result is that low pressure permeability measurementsare unrealistically high because there is insufficient gas molecules to form a zerovelocity boundary layer at the edges of the pores and to form a mass of flowing gaswithin the pores. In this case, too many gas molecules flow through the pores and thepermeability appears to be higher than it actually is: the effect reported by Klinkenberg.

24

Since the mean free path is a function of the size of the molecule, the permeability isa function of the type of gas used in the permeability measurement. This gaspermeability is corrected for the Klinkenberg effect by plotting the gas permeabilityat each reciprocal mean pressure. This is illustrated for hydrogen, nitrogen and carbondioxide in figure 17:

100

80

0 5

Reciprocal Mean Pressure: (Atm.)

Ga

s P

erm

ea

bili

ty:

Mill

ida

rcie

s

60

40

20

01 2 3 4

Hydrogen

Nitrogen

Carbon Dioxide

Liquid permeability

Pm is the mean pressure of the gas (the mean of the upstream and downstream pressures

either end of the core orp in figure 15). In effect, if the gas pressure is raised infinitelyhigh, the gas will perform as an incompressible liquid would, therefore if severalmeasurements of permeability are made at different mean pressures, the relationshipbetween mean pressure and permeability can be extrapolated to the equivalentpressure conditions of a liquid. In reality, extrapolation to infinity is impossible, so thereciprocal mean pressure is used and the results are extrapolated to zero reciprocalmean pressure (i.e. 1/infinitely high mean pressure). This point corresponds to theliquid permeability. The different gasses have different slopes, but they all extrapolateto the same equivalent liquid permeability.

The form of the equation developed by Klinkenberg is of the form

kk

lb

P

LG

m

=+ (12)

Figure 17

Variation in gas

permeability with reciprocal

mean pressure



wherek

L = equivalent liquid permeability

kG = permeability to gas

Pm = mean flowing pressure

b = Klinkenberg constant for a particular gas and rock (slope of the gas permeability,inverse mean pressure relationship).

The Klinkenberg effect is greatest for low permeability rocks and low mean pressures.

4.9 Reactive FluidsDarcys Law assumes that the fluid does not react with the formation. Many formationwaters react with clays in the rock to produce a lower permeability to liquid thanwould be obtained with gas. Therefore the permeability to water in the formation maybe much lower than would be determined to gas in the laboratory. Any water injectedinto the formation may severely reduce the permeability due to clay swelling. Thechange in permeability may be substantial, for example from several hundredmillidarcys to less than one millidarcy.

4.10 Average Reservoir PermeabilityPermeability is not normally distributed but has an exponential distribution, thereforea geometric mean is used to obtain an average reservoir permeability.

The Geometric Mean of n numbers is the nth root of their product:

5 STRESS EFFECTS ON CORE MEASUREMENTS

5.1 Stress RegimesIn reservoir engineering the impact of reservoir stresses on reservoir flow and capacityparameters has been considered for a number of years but, increasingly, the interestin stress related measurement has grown. The effect of removing a core from theformation is to remove all the confining forces on the sample, allowing the rock matrixto expand in all directions, partially changing the shapes of the fluid-flow paths insidethe core.

It is worth considering the stresses associated with reservoir rock parameters. Figure18 illustrates the likely configuration of a core extracted from a vertical well, and theorientation of the core plug extracted for permeability and porosity measurements.

26

Whole core

Core plugfor horizontal k measurement

4 Inch

Formation

Core plugfor vertical k measurement

Within a reservoir the stresses in the formation can be expressed in three directions,the major and two minor principal stresses. Figure 19a. The major principal stressacting mainly in the vertical direction. Clearly the depositional environment andformation structure will result in slight changes to these orientations.

Major Principal Stress

Minor Principal Stress

Minor Principal Stress

Equal Stresses

Equal Stresses

Kh

(a)

(b)

(c)

Figure 19

Stress States in Reservoirs

and Cores

Figure 18

Trends in Reservoir Rock

Characterisation



In core analysis, service companies have been asked to measure porosity andpermeability under reservoir stress conditions. They have done this by applyingdifferent stresses for the axial and radial stresses. As can be seen in Figure 19b for aconventional plug the radial stress would be a combination of the major and a minorprincipal stress. To enable the true stress field to be represented, a varying radial stressdistribution would be required. If a vertical plug was used, Figure 19c, then a constantradial stress could be an acceptable value for the average minor stresses. In this case,however, the permeability value would be K

v, the vertical permeability.

The effect of the overburden and the pore pressure on the matrix is to produce a netforce between the grains of the matrix (which, when the area over which the force actsis accounted for produces a net stress). If the matrix is considered to be elastic, thatis, there is a unique relationship between the stress and the strain within the matrix,then the matrix will strain as the stress is altered. If the stress increases, the strainreduces the radius of the pore throats and reduces the volume of the pore space. Thiseffect may be different for different rock types and even within the same rock type ifthe amount of cementing material is altered. The significant aspects of this phenom-enon are when cores are removed from subsurface to the laboratory (since theoverburden and pore pressure will change) and when the pore pressure in the reservoirchanges due to local pressure conditions around the wells (drawdown) and within thereservoir as a whole as it is depressurised, for example. The impact of the netoverburden stress which increases as the reservoir pressure ( pore pressure ) decreasesis illustrated in figure 20.

1.0

.8

0 10000

Net Overburden Pressure: PSI

Pe

rme

ab

ility

: F

ract

ion

of

Ori

gin

al

.6

.4

.2

02000 4000 6000 8000

Unconsolidated

Friable

Well Cemented

In general, the stress regime subsurface is considered to be hydrostatic (as in the caseof the pore fluid) and that the stresses can be resolved into one vertical stress, and two

Figure 20

Permeability Reduction

with Net Overburden

Pressure

28

horizontal stresses. For hydrostatic conditions, all of these are the same. In coreanalysis, therefore, the porosity at equivalent subsurface conditions may be deter-mined by applying an external pressure to the core. This is usually done by insertingthe core into a cell rated for pressures up to 10000 psi (68.9MPa) and applying a stressto the ends of the core and to the sides. The nature of these tests are such that usuallythe stress applied to the sides of the core represents the horizontal stress and the stressapplied to the ends represents the vertical stress. Once trapped inside the cell, the porepressure may be increased to a representative level and measurements of pore volumeand permeability made under these stress conditions.

More recently, the effect of non-hydrostatic stress conditions has been shown to beimportant in certain reservoir conditions, such as in tectonically active areas (Columbia,South America where the formation of the Andes mountains is associated with largehorizontal stresses) or in areas associated with faults or very compressible reservoirrocks such as some chalks. In this case the conventional test cells are not appropriateand special true triaxial cells are required. In these cells the ends of the core aresubjected to the vertical stress as per the conventional cells, but the sides of the coreare wrapped in a cage of individual tubes which can be pressurised in banks aroundthe core to represent the different horizontal stresses.

In summary, when the properties of the cores are measured in the laboratory, they canbe subjected to

Zero stresses No effect of the stress on the property

Hydrostatic stresses The effect of the magnitude of the stresses are measured

Triaxial stresses The effect of stresses resolved in the three principal directionsare measured

Real stress beheviour The effect of the magnitude and direction of the stresses aremeasured

This topic is covered in more detail in the subsequent chapter.

5.2 Compressibility Of Porous RockAs the rock matrix is subjected to a stress, it will deform and alter the pore spacevolume as the rock is compressed. For simplicity, the overburden will be consideredto produce hydrostatic stress (called the compacting stress) on the reservoir, i.e. agrain-to-grain stress in the rock. Within the pores, fluid pressure acts on the surfaceof the grains and reduces the grain-to-grain (or compacting) stress. Therefore in a realreservoir there is a balance between the effect of the overburden stress and the porepressure. This can be described by the relationship

Pcompacting

= Poverburden

- Ppore pressure

where Pcompacting

is the grain-to-grain stress, Poverburden

is the stress produced by the weightof the overburden at a particular depth and P

pore pressure is the pressure of the fluids in the

pores. The expression shows the balance between the overburden and the porepressure in compacting the rock matrix: if the pore pressure declines, the compacting



stress increases and the pore volume declines. This assumes that the overburdenremains constant which is logical over the time period of a producing reservoir. Thebalance can be represented by figure 21:

Enlarged view of the pore space

Grains

Pore space filled with fluid

Reservoir

Cap Rock Depth

Pf and Pc

Po

Surface

Pc

Pc

Pf

Pc

Pc

Pc

Pc

Po = P

f + P

cP

o = overburden pressure

Pf = fluid pressure

Pc = compacting stress

The effect of the change in the balance between the overburden stress and the porepressure is to change the compacting stress. If there is an increase in pore pressure,then the pore volume will increase, however, this is rare and in the main, pore pressuredeclines during production and the pore spaces compact under the increasing compactstress. Two issues are significant: the initial porosity in the reservoir (i.e. to correctlydefine the volume of oil in place) and the reduction in that porosity (or pore volume)as the pressure declines (for material balance and simulation studies).Figure 22 showsthe relationship between porosity and depth (or stress). As the depth (and stress)increases, the porosity declines. Care needs to be taken when assessing porosityvalues: were they measured under overburden or at ambient conditions? The shalesample shows a large change in porosity as the plate-like clay minerals are compactedand fit together in a more congruent manner.

Figure 21

The balance between

overburden & rock stress

and fluid pressure

30

Sandstone

Shale

50

40

30

20

10

00 3000 6000

Depth of burial (ft) or stress (psi)

Por

osity

, φ

The rate of change of pore volume with pressure change can be represented by anisothermal compressibility (assuming temperature is constant):

C = - f

1v

dv

dP (15)

where Cf is the isothermal compressibility, v is the volume, dv is the change in volume

and dP is the change in pressure (the negative sign accounts for the co-ordinate system:as the pressure increases, the volume decreases).

5.3. Types Of CompressibilityAn issue with regard to the compressibility is: which part of the reservoir is beingcompressed and which part is significant in calculating the response of the reservoir.

Three types of compressibility can be considered:

(i) Matrix volume compressibility - the change in volume of the rock grains. This isvery small and usually not of interest in sandstones since it is a purely mechanicalchange in volume of the very stiff grains.

(ii) Bulk volume compressibility - the change in the unit volume of the rock. This isof interest in reservoirs near the surface because of the problem of subsidence;

Changes in volume of the reservoir around faults which may cause the fault to slipand alter the conductivity both through the fault and across it;

Reservoirs composed of unconsolidated or very weakly consolidated material wherethe changes in porosity can be significant. The changes in the volume of the reservoirboth in a vertical sense leading to subsidence and in a horizontal sense leading toshearing of the wellbore and the associated loss in integrity.

(iii) Pore volume compressibility - change in pore volume. This is of greatest interestsince the pore volume affects the porosity which affects reservoir performance.

For completeness, all aspects of the reservoir compressibility should be considered,however, in many problems only specific aspects of the compressibility may berequired such as in a well cemented sandstone reservoir where the bulk volume change

Figure 22

Alteration in porosity with

depth of burial (or stress)



is very small and the subsidence is negligible, but the pore compressibility is animportant feature of the drive mechanism.

5.4 Measurement Of Pore Volume CompressibilityThe measurement of pore compressibility is usually conducted in a coreholder whichapplies an equal compacting pressure around the core. An inner liner ensures thepower fluid (usually hydraulic oil) does not contaminate the pores of the sample. Thepore pressure is usually kept at ambient, i.e. the compacting pressure mimics the neteffect of the overburden and the pore pressure in the reservoir. This makes the testsimpler, however, there may be conditions where the compressibility of the grainsthemselves plays a significant role in the system and the test may require to beconducted at true overburden and pore pressure conditions. For the test at ambientpore pressure conditions, an outlet is connected to the core holder and this is lead toa pipette or a balance to measure the amount of pore fluid expelled. The pressure ofthe hydraulic oil is increased in stages and for each stage the amount of fluid expelledis measured after the rock has come to equilibrium. The data can then be analysed toindicate the change in porosity or pore compressibility. Figure 23 shows the concept.

Sealed core

Pump

Pipette

Pressure vessel

The results show the change in pore volume relative to the original pore volume, fora given change in the compacting pressure (this assumes that changes in thecompacting pressure have the same effects as changes in the pore pressure) which canbe substituted in to the isothermal compressibility as

C = - p

1v

dv

dPp

p

c

where:

Cp

= pore volume compressibilityv

p= initial pore volume

dvp

= change in pore volume (amount of fluid expelled)dP

c= change in compacting pressure

Typical values of pore compressibility are in the range 3 x10-6 psi-1 to 10 x10-6 psi-1,however, soft sediments can have compressibilities in the range 10 x10-6 psi-1 to 20x10-6

psi-1 or 30 *10-6psi-1. Figure 24 illustrates the values determined for some limestonesand sandstones.

Figure 23

Measurement of the

reduction inpore volume as

the external stress (or

compacting pressure) is

increased

32

Figure 24

Compressibility of

Sandstones and Limestones

SandstoneLimestone

Porosity %

Por

e co

mpr

essi

bilit

y 10

-6 p

si-1

10

9

8

7

6

5

4

30 10 20

5.5 Effect of stress on permeabilityAs the effect of a stress on the rock matrix affects the pore volume, it also affects thepore throat radii and the permeability of the rock. In general, an increase in stressreduces the pore throat radii and the permeability declines. For most rocks subjectedto an hydrostatic stress, this is the case as the stress is equal in all directions. Figure25 shows typical permeability declines for increase in stress for sandstone.

Permeability stress sensitivity for various sandstones1000

100

10

10 20 40 60 80

Hydrostatic stress (MPa)

Per

mea

bilit

y (m

D)

Unconsolidated material has larger absolute changes in permeability as the total strainis greater.

Figure 25

The reduction in

permeability for a range of

sandstone samples (the

porosity is in the range 15%

to 22%)



In true triaxial stress regimes, the stresses are not identical and the strain (and thereforepore throat radii) may cause the sample to dilate in one direction and increase the porethroat radii therefore enhancing the permeability. This can be illustrated better byconsidering a fractured core (figure 26).

σh maximum

Permeability Fracture

Core

σh maximum σh minimum

σv

σh maximum perpendicular to fracture

Fracture

Core

σh minimum σh maximum

σv

σh maximum parallel to fracture

σh maximum

Permeability

Fracture closing under stress

Fracture opening under stress

If the largest horizontal stress acts across the fracture (i.e. perpendicular to the facesof the fracture) then it will be clamped shut; if the largest horizontal stress acts parallelto the fracture, then it may split open. In this way the anisotropy (or difference in theproperties) may lead to different permeabilities and porosities from the same sampleif the stresses are applied in different ways around the core.

6. POROSITY-PERMEABILITY RELATIONSHIPS

Whereas for porosity there are a number of downhole indirect measurement methods,the same is not the case for permeability. The downhole determination of permeabilityis more illusive. Down hole permeability is mainly obtained by flow and pressuredetermination and requires other characteristics for example the flowing interval.There has been a continued interest in porosity-permeability correlations, on the basisif one has a good correlation of laboratory measured porosity and permeability thendown hole measurements of porosity could unlock permeability values for thoseformations where recovered core has not been practical. Although porosity is anabsolute property and dimensionless, permeability is not and is an expression of flowwhich is influenced by a range of properties of the porous media, including the shapeand dimensions of the grains and the porosity. Since porosity is an importantparameter in permeability it is not surprising for those rocks which have similar

Figure 26

Triaxial stresses applied to

a fractured core

34

Figure 27

Permeability and Porosity

Trends for Various Rock

Types

(Core Laboratories Inc)

particle characteristics that a relationship exists between porosity and permeability.Figure 27 below gives examples of permeability correlations for different rock types.

1000

100

10

1.00 5 10 15 20 25 30 35

Porosity: Percent

Pe

rme

ab

ility

: M

illid

arc

ies

Reef LimestoneSucrosic Dolomite

Oolitic Limestone

ChalkyLimestone

IntercrystallineLimestone andDolomite

Fine GrainedFriable Sand

Well CementedHard Sand

7 SURFACE KINETICS

If core for a particular section cannot be recovered, or for example is formed as a pileof sand on the rig floor, then correlations like these in figure 27 are used. Porositymeasurements obtained indirectly from wireline methods can be used to obtain thelaboratory porosity vs down hole porosity cross plot. Using this laboration porosityvalue the associated permeability value can be determined from an appropriatecorrelation as in figure 27.

The simultaneous existence of two or more phases in a porous medium needs termssuch as the capillary pressure, relative permeability and wettability to be defined. Withone fluid only one set of forces needs to be considered: the attraction between the fluidand the rock. When more than one fluid is present there are three sets of active forcesaffecting capillary pressure and wettability.

Surface free energy exists on all surfaces between states of matter and betweenimmiscible liquids. This energy is the result of electrical forces. These forces causemolecular attraction between molecules of the same substance (cohesion) andbetween molecules of unlike substances (adhesion).



Surface tension (or interfacial tension) results from molecular forces that cause thesurface of a liquid to assume the smallest possible size and to act like a membraneunder tension.

7.1 Capillary pressure theoryThe rise or depression of fluids in fine bore tubes is a result of the surface tension andwetting preference and is called capillarity. Capillary pressure exists whenever twoimmiscible phases are present, for example, in a fine bore tube and is defined as thepressure drop across the curved liquid interface. The equilibrium in force between themolecules of a single phase is disrupted at an interface between two dissimilar fluids.The difference in masses and the difference in the distances between the molecules ofthe different phases produces an initially unbalanced force across the interface. Figure 28shows the interface between oil and water molecules.

Different mass.Different spacebetween molecules.

W

W WW

O

OO O

W: water moleculeO: oil molecule distance between molecules

Interfacial tension deforms the outer surface of immiscible liquids to producedroplets. If the two liquids are present on a surface, the interfacial tension deforms theliquids to produce a characteristic contact angle as shown in Figure 29.

A wetting phase is one which spreads over the solid surface and preferentially wetsthe solid. The contact angle approaches zero (and will always be less than 90˚).

A non-wetting phase has little or no affinity for a solid and the contact angle will begreater than 90˚

Oil

Water

Contact angle, θ

σsoσsw

Solid

σwoθ

Interfacial tension between the water and oil

Interfacial tension between the solid and water

Interfacial tension between the solid and oil

σwo

σso

σ sw

Figure 28

Representation of an oil

water boundary

Figure 29

Interfacial tension between

oil, water and a solid

36

Figure 30

The effect of a change in

the surface on wetting

properties

The contact angle describes the nature of the interaction of the fluids on the surface:for the oil-water system shown above: an angle less than 90˚ indicates that the surfaceis water wet. If the angle were greater than 90˚ then the surface would be oil wet.

The composition of the surface also affects the interfacial tension. Figure 30 shows theeffect of octane and napthenic acid on a water droplet on silica and calcite surfaces.The water is not affected by the change in surface in the water/octane system, however,the napthenic acid causes the water to wet the silica surface, but to be non-wetting onthe calcite surface.

Octane Napthenic acid

30°

30°

106°

35°

Silica

Calcite

Octane Napthenic acid

The Adhesion tension, At is defined as the difference between the solid water and solidoil interfacial tension. This is equal to the interfacial tension between the water and oilmultiplied by the cosine of the contact angle,

At = σ

sw - σ

so = σ

wo Cos θ

wo

If a container of oil and water is considered as in figure 31, the denser water lies belowthe oil.

σσcosθ

h

θ

radius, r

OIL

.c

Water

If a glass capillary tube of radius, r is inserted such that it pierces the interface betweenthe oil and water, the geometry of the tube and the imbalance in forces producedbetween the glass, oil and water cause the interface to be pulled upwards into the tube.If non wetting fluids were used, the interface in the tube may be pushed downwards.

Figure 31

Capillary rise in an oil/

water system



Under equilibrium conditions, i.e. after the tube has pierced the original interface, theadhesion tension around the periphery (2πr) of the tube can be summed to give the totalforce upwards. Since the interface is static, this force must be balanced by the forcesin the column of water drawn up the tube and the equivalent column of oil outside thetube, i.e. at point C, the force (or pressure) must be the same in the tube as outside,therefore the excess force produced by the column of water is balanced by theadhesion tension.

net force upwards = 2πr σwo

Cosθ (16)net force downwards = (ρ

wgh - ρ

ogh)πr2 = gh(ρ

w - ρ

o)πr2 (17)

the interface is at equilibrium, therefore

2πr σwo

Cosθ=gh(ρw - ρ

o)πr2 (18)

The capillary pressure is the difference in pressure across an interface, therefore interms of pressure (the P

c, force acting on area pr2)

gh( ) rr

2 r Cosr

P

gh( ) =2 Cos

r

w o2

2wo

2 c

w owo

ρ ρ ππ

π σ θπ

ρ ρσ θ

−= =

−

It can be seen from the equations, capillary pressure can be defined both in terms ofcurvature and in terms of interfacial tension, as expressed by the hydrostatic head.

P2 Cos

rcc

gh w o

σ θρ ρ= = −( ) (19)

whereP

c= capillary pressure

σ = surface tensionθ = contact angler

c= radius of the tube

h = height of interfaceρ

w = the density of water

ρo = the density of oil.

For a distribution of capillaries, therefore, the capillary pressure will give rise to adistribution of ingress of wetting fluid into the capillaries. The relative position of thecapillary rise is given with respect to the free water level, FWL, i.e. the point of zerocapillary pressure. Figure 32 illustrates the effect of three different capillary radii onthe rise of water. Figure 33 shows the behaviour for a full assembly of capillaries andalongside the associated capillary pressure curve. In this figure it is important to notefive aspects.

38

• The free water level-the position of zero capillary pressure• The oil -water contact• The 100% water saturation at a distance above the free-water level due to the

capillary action of the largest tube.• The irreducible level representing the limit if mobile water saturation• The different radii segregate the capillary pressure and therefore the height to which

the water is drawn into the oil zone.

The zone of varying water saturation with height above the 100% free water oil contactis called the transition zone.

The formation containing irreducible water will produce only hydrocarbons whereasthe transition zone of varying water saturation will produce water and hydrocarbons.

The shape of the capillary pressure curves in the transition zone will depend on thenature of the rock.

oiloil

oil

oil

θ

θ

θ

h

FREE WATER LEVEL

WATER WATER

Water0%

100%SwSo

100%0%

Oil water contact

Oil

Water

OWC

Pc

Free water level

0FWL

Irre

duci

ble

Wat

er

Tran

sitio

n Z

one

Figure 32

Capillary Rise in

Distribution of Capillaries

Figure 33

Capillary Pressure Curve



It must be remembered that although concepts of capillary pressure were formulatedin terms of fine bore tubes, application in practice deals with a complex network ofinterconnected pores in a matrix carrying surface chemical properties as illustrated infigure 1 of the pore cast of the pore space.

The height at which a wetting liquid will stand above a free level is directlyproportional to capillary pressure which is related to the size and size distribution ofthe pores. It is also proportional to interfacial tension and the cosine of the contactangle and inversely proportional to the tube radius and difference in fluid density. Thesmaller the pores ie. the lower the permeability, the higher the capillary pressure.

7.2 Fluid distribution in reservoir rocksWater is retained by capillary forces as hydrocarbons accumulate in productivereservoirs. The water is referred to as connate or interstitial water and in water wetrocks it coats the rock surfaces and occupies the smallest pores, whereas hydrocarbonsoccupy the centre of the larger pores. The magnitude of the water saturation retainedis proportional to the capillary pressure which is controlled by the rock fluid system.

~Sw Pc = 2σCosθ

re

Rock FluidProperty

Wettability Rock / Fluid Property

Rock Property(Permeability and Porosity)

_

Water wet, coarse grained sand and oolitic and vuggy carbonates with large pores havelow capillary pressure and low interstitial water contents. Silty, fine grained sandshave high capillary pressures and high water contents.

Reservoir saturation reduces with increased height above the hydrocarbon-watercontact. At the base of the reservoir there will usually be a zone of 100% watersaturated rock. The upper limit of this is referred to as the water table or water oilcontact (WOC). However, there is a non identifiable level, the free water levelrepresenting the position of zero capillary pressure.

Figure 34 shows the capillary pressure curve for a reservoir where the water saturationreduces above the aquifer. The 100% water saturation continues some distance abovethe free water level corresponding to the largest pores of the rock, h

D. Above this level

both the oil and water are present and the reservoir water saturation decreases withincreased height above the hydrocarbon water contact, since the larger pores can nolonger support the water by capillary action and the water saturation falls. Betweenthe 100% WOC and the irreducible saturation level is termed the transition zone.

40

Oil

Sand Grain

Pc

Water

WOC

FWL0% Water Saturation 100%

Transition Zone

hp

h

Consider the capillary pressure curves for the two rocks in figure 35. The first sample(case 1) has a small range of connecting pore sizes. The second sample (case 2) hasa much larger range of connecting pore sizes, although the largest pores are of similarsize in both cases. Also, in the first case, the irreducible water saturation is reached atlow capillary pressure, but with the graded system, a much larger capillary pressureis needed.

h

Case 1

Case 2

High Pc needed to reach limitingwater saturation.

Irreducible (or non - communicating)water approach at low Pc

Largest connecting poresabout the same size.Therefore simular hD

hD

Water saturationIrreduciblewater saturation

100%

hI

Pc

= (

Pw

- P

o) g

h

X

In addition to water transition zones, there can also be an oil/gas transition zone, butthis is usually less well defined.

Rock wettability influences the capillary pressure and hence the retentive propertiesof the formation. Oil wet rocks have a reduced or negligible transition zone, and maycontain lower irreducible saturations. Low fluid interfacial tension reduces thetransition zone, while high interfacial tension extends it. Figure 36 illustrates thiseffect.

Figure 34

Capillary Pressure Curve

for Porous media

Figure 35

Capillary Pressure Curves

for Different Rocks



0 100

A

High Interfacial Tension

Low Interfacial TensionHei

ght A

bove

Wat

er L

evel

Water Saturation: Percent Pore Space

Interfacial Tension Effect

Saturation history influences the capillary pressure water saturation relationship andtherefore the size of the transition zone. Drainage saturation results from the drainageof the wetting phase (water) from the rock as the hydrocarbons accumulate. Itrepresents the saturation distribution which exists before fluid production. The levelof saturation is dictated by the capillary pressure associated with the narrow pore andis able to maintain water saturation in the large pore below. Imbibition saturationresults from the increase in the wetting phase (water) and the expulsion of thehydrocarbons. In this case the saturation is determined by the large pore reducing thecapillary pressure effect and preventing water entering the larger pore. This is thesituation which occurs both when natural water drive imbibes into the formationraising the water table level and in water injection processes. Clearly the twosaturation histories generate differnet saturation height profiles. Figure 37 shows thedrainage and imbibition effects on capillary rise.

0 100

ADrainage

ImbibitionHei

ght A

bove

Wat

er L

evel

Water Saturation: Percent Pore SpaceDrainage Imbibition

A large density difference between water and hydrocarbons (water-gas) suppressesthe transition zone. Conversely, a small density difference (water-heavy oil) increasesthe transition zone. Figure 38 shows the differences in density for water/heavy oil andwater/gas on capillary rise. Transition zones between oil and gas are not significantbecause of the large density difference between oil and gas.

0 100

A

Small Density Difference (Water-Heavy Oil)

Large Density Difference (Water Gas)H

eigh

t Abo

ve W

ater

Lev

el

Water Saturation: Percent Pore SpaceFluid Density Difference Effect

Figure 36

Interfacial Tension Effect

Figure 37

Saturation History Effect

Figure 38

Fluid Density Effect

42

7.3. Impact of Layered ReservoirsA characteristic of reservoirs is the various rock types making up the reservoir section.Each rock type has its own capillary pressure characteristics. Wells penetrating suchformations will show a water saturation distribution reflecting the specific capillaryeffects of each formation type. In some cases a 100% water saturation will be abovea lower water saturation associated with a lower elevation material with a higherpermeability, Figure 39.

For example well A would only indicate 100% water. Well B would penetrate thetransition zone of the top layer then a region of 100% water saturation. The saturationprofiles for well B and C are illustrated in figure 39. The transition zone of the nextlayer 2, followed by an interfacial of 100% saturation associated with layers 2, 3 and4 then into 100% for the next two layers. Well D penetrates through the top and nextlayer at the irreducible saturation level, into the transition zone for layer three, theninto irreducible saturation for the 4th layer.

1

23

4

Transitionzone

Water saturationprofile well C onlyWater saturation

profile Well B only

0 = 15K = 40 md

0 = 25K = 190 md

0 = 10K = 5 md

0 = 30K = 200 md

A B C D

SHALE

SHALE

SA

ND

STO

NE

RE

S.

Free Water Level

100% Water Level

FWLFWL

Hei

ght

0% Sw 100%0 100%

Figure 39

Capillary Effects in

Stratified Formations



8 EFFECTIVE PERMEABILITY

8.1DefinitionThe idea of relative permeability provides an extension to Darcy’s Law to the presenceand flow of more than a single fluid within the pore space. When two or moreimmiscible fluids are present in the pore space their flows interfere. Specific orabsolute permeability refers to permeability when one fluid is present at 100%saturation. Effective permeability reflects the ability of a porous medium to permit thepassage of a fluid under a potential gradient when two or three fluids are present in thepore space. The effective permeability for each fluid is less than the absolutepermeability. For a given rock the effective permeability is the conductivity of eachphase at a specific saturation. As well as the individual effective permeabilities beingless than the specific permeability, their sum is also lower.

If measurements are made on two cores having different absolute permeabilities k1

and k2, there is no direct way of comparing the effective permeability k

w and k

o curves

since for the two cores they start at different points k1 and k

2. This difficulty is resolved

by plotting the relative permeability krw

and kro where

Relative Permeability =

permeability to one phase when one or more phases are presentpermeability to one phase alone

kkkr

e=

Relative permeability is dimensionless and is reported as a fraction or percentage. Onrelative permeability plots the curves start from unity in each case, so directcomparisons can be made.

A typical set of effective permeability curves for an oil water system is shown in figure40 and for a gas oil system in figure 41.

44

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01.00.90.80.70.60.50.40.30.20.10

S , Water Saturation, FractionW

Re

lati

ve P

erm

ea

bili

ty

k ro

k rw

1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

01.00.90.80.70.60.50.40.30.20.10

Liquid Saturation = S + S , %WOO

Re

lati

ve P

erm

ea

bili

ty,

Fra

ctio

n o

f A

bso

lute

Co

nn

ate

Wa

ter

plu

s R

esi

du

al

Oil

Sa

tura

tio

n

k rg

k ro

The following points are to be noted:

The introduction of a second phase decreases the relative permeability of the firstphase: for example,

k

or drops as

S

w increases from zero. Secondly, at the point where

the relative permeability of a phase becomes zero there is still a considerablesaturation of the phase remaining in the rock. The value of

So at

k

ro = 0 is called the

residual oil saturation and the value of S

w at

k

rw = 0 is called the irreducible water

saturation.

Figure 40

Relative permeability

curves for water-oil sysrem

Figure 41

Relative permeability

curves for gas-oil sysrem



The shapes of the relative permeability curves are also characteristic of the wettingqualities of the two fluids (figure 42). When a water and oil are considered together,water is almost always the wetting phase. This means that the water, or wetting phase,would occupy the smallest pores while the non-wetting phase, or oil phase, wouldoccupy the largest pores. This causes the shape of the relative permeability curves forthe wetting and non-wetting phase to be different.

Krw

100

90

80

70

60

50

40

30

20

10

01009080706050403020100

Water Saturation, S W

Re

lati

ve P

erm

ea

bili

ty,

%

K ro

Water-Wet Drainage

Water-Wet Imbibition

Oil-Wet Drainage

(Decreasing Sw )

(Increasing Sw )

(Increasing Sw )

This is illustrated by looking at the relative permeability to one phase at the irreduciblesaturation of the other phase. The relative permeability to water at an irreducible oilsaturation of 10% (90% water) is about 0.6, figure 40, whereas the relative permeabil-ity to the non-wetting phase, oil, at the irreducible water saturation of 0.3 approaches1.0. In this case it is 0.95. One practical effect of this observation is that it is normallyassumed that the effective permeability of the non-wetting phase in the presence of anirreducible saturation of the wetting phase is equal to the absolute permeability.Consequently, oil flowing in the presence of connate water or an irreducible watersaturation is assumed to have a permeability equal to the absolute permeability.Similarly, gas flowing in a reservoir in the presence of irreducible water saturationis assumed to have a permeability equal to the absolute permeability.

Relative permeability characteristics are important in the displacement of hydrocar-bons by water, and in the displacement of oil and water by gas. Such displacementsoccur during primary and secondary recovery operations, as well as during coring andcore recovery.

Relative permeability data when presented in graphical form are often referred to asdrainage or imbibition curves. (figure 42)

Figure 42

Oil and Water Relative

Permeability Curves for

Water-Wet and Oil-Wet

Systems

(Core Laboratories Inc)

46

Imbibition relative permeability is displacement where the wetting phase saturationis increasing. For example, in a water flood of a water wet rock, or coring with a waterbase mud.

Drainage relative permeability is where the non-wetting phase saturation is increasing.For example, gas expulsion of oil during primary depletion or gas expansion of fluidsduring core recovery, and the condition existing in the transition zone at discovery.

Water displacement of oil differs from gas displacement of oil since water normallywets the rock and gas does not. The wetting difference results in different relativepermeability curves for the two displacements.

8.2 Water displacement of oilPrior to water displacement from an oil productive sand interstitial water exists as athin film around each sand grain with oil filling the remaining pore space. Thepresence of water as previously stated has little effect on the flow of oil, and oil relativepermeability approaches 100%. Water relative permeability is zero.

Water invasion results in water flow through both large and small pores as the watersaturation increases. Imbibition relative permeability characteristics influence thedisplacement. Oil saturation decreases with a corresponding decrease in oil relativepermeability. Water relative permeability increases as water saturation increases.

Oil remaining after flood-out exists as trapped globules and is referred to as residualoil. This residual oil is immobile and the relative permeability to oil is zero. Relativepermeability to water reaches a maximum value, but is less than the specificpermeability because the residual oil is in the centre of the pores and impedes waterflow.

8.2.1 Water-oil relative permeabilityAccumulation of hydrocarbons is represented by drainage relative permeabilitycurves as the water saturation decreases from 100% to irreducible. Water relativepermeability reduces likewise from 100% to zero while oil relative permeabilityincreases.

Subsequent introduction of water during coring or water flooding results in a differentset of relative permeability curves - these are the imbibition curves. The water curveis essentially the same in strongly water wet rock for both drainage and imbibition. Theoil phase relative permeability is less during imbibition than during drainage.

The oil remaining immobile after a waterflood is influenced significantly by thecapillary pressure and interfacial tension effects of the system. It is of note that a highresidual oil saturation is a result of the oil ganglia being retained in the large pores asa result of capillary forces. Figure 43 illustrates the pore doublet model illustratinghow oil can be trapped in a large pore. The forces to displace this droplet have toovercome capillary forces and are too great to use pressure through pumping. Theforce required can be reduced by reducing the interfacial tension which is the basis formany enhanced oil recovery methods; for example, surfactant and miscible flooding.



Trapped oil

Water penetratingsmaller pores due tocapillary forces

Advancing water

Water In Oil

Water In Oil

Water In Water

An important perspective in a displacement process is the concept of mobility ratio.This relates the mobility of the displacing fluid relative to that of the displaced fluid.It is therefore a ratio of Darcy’s Law for each respective fluid at the residual saturationof the other fluid. In the context of water displacing oil.

M = mobility ratio = k

krw w

ro o

µµ

' /' /

(20)

where krw

is the relative permeability at residual oil saturationk

rois the relative permeability at the irreducible water saturation.

These relative permeabilities are sometimes referred to as end point relativepermeabilities. When M is less than 1 this gives a stable displacement whereas whenM is greater then 1 unstable displacement occurs.

This topic is covered extensively in the chapter on immiscible displacement

8.3 Gas displacement of oil and gas-oil relative permeabilityGas is a non-wetting phase and it initially follows the path of least resistance throughthe largest pores. Gas permeability is zero until a ‘critical’ or ‘equilibrium’ saturationis reached (figure 41).

Figure 43

Pore Doublet Model

48

Gas saturation less than the critical value is not mobile but it impedes the flow of oiland reduces oil relative permeability. Successively smaller pore channels are invadedby gas and joined to form other continuous channels. The preference of gas for largerpores causes a more rapid decrease of oil relative permeability than when waterdisplaces oil from a water wet system. Figure 44 shows the alteration of relativepermeability as gas comes out of solution and flows at increasing saturation throughthe oil reservoir. These gas/oil relative permeability curves are very significant inrelation to the drive mechanism of solution gas drive, which we will discuss in asubsequent chapter.

Cha

ract

eris

tic S

and

Dur

ing

Oil

Dis

plac

emen

tby

Gas

@ 5

% G

as s

atur

atio

nC

hara

cter

istic

San

d D

urin

g O

il D

ispl

acem

ent

by G

as @

20%

Gas

sat

urat

ion

Cha

ract

eris

tic S

and

Dur

ing

Oil

Dis

plac

emen

tby

Gas

@ 4

5% G

as s

atur

atio

n

100

100

80

80

60

60

40

40

20

200

0

Relative Permeability: Percent

Gas

Sat

urat

ion:

Per

cent

Por

e S

pace

Kro

Krg

Gas

Sat

urat

ion:

5%

of P

ore

Spa

ceS

peci

fic

Per

mea

bilit

y (K

s):

250

md.

Effe

ctiv

e P

erm

eabi

lity

to O

il (K

o):

183

md.

Effe

ctiv

e P

erm

eabi

lity

to G

as(K

g):

0.0m

d.R

elat

ive

Per

mea

bilit

y to

Oil

(Kro

) =

183/

250

= 0

.73

Rel

ativ

e P

erm

eabi

lity

to G

as(K

rg)

=0.

0/25

0 =

0.0

Kro

Krg

Gas

Sat

urat

ion:

20%

of P

ore

Spa

ceS

peci

fic

Per

mea

bilit

y (K

s):

250

md.

Effe

ctiv

e P

erm

eabi

lity

to O

il (K

o):

52 m

d.E

ffect

ive

Per

mea

bilit

y to

Gas

(Kg)

: 10

md.

Rel

ativ

e P

erm

eabi

lity

to O

il (K

ro)

=52

/250

=

0.2

1R

elat

ive

Per

mea

bilit

y to

Gas

(Krg

) =

10/2

50

= 0

.04

100

100

80

80

60

60

40

40

20

200

0


Gas

Sat

urat

ion:

Per

cent

Por

e S

pace

Oil

Wat

erG

as

Gas

Sat

urat

ion:

45%

of P

ore

Spa

ceS

peci

fic

Per

mea

bilit

y (K

s):

250

md.

Effe

ctiv

e P

erm

eabi

lity

to O

il (K

o):

6.2

md.

Effe

ctiv

e P

erm

eabi

lity

to G

as(K

g):

70m

d.R

elat

ive

Per

mea

bilit

y to

Oil

(Kro

) =

6.2/

250

= 0

.025

Rel

ativ

e P

erm

eabi

lity

to G

as(K

rg)

=70

/250

=

0.2

8

100

100

80

80

60

60

40

40

20

200

0


Gas

Sat

urat

ion:

Per

cent

Por

e S

pace

Kro

Krg

Figure 44

Gas Oil Relative

Permeabilities ( Core Lab)

resengch7

Documents

capillary pressure curves

capillary pressure pc

capillary pressure theory7

context of capillary

use of darcys law4

flow of gas

applications of darcys

flow rateto permeability