Complete WF Manual Cobb

7/23/2019 Complete WF Manual Cobb

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11/10

WATERFLOODING

By

William M. Cobb

James T. Smith


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05/11

COPYRIGHT

By

William M. Cobb & Associates, Inc.

12770 Coit Road, Suite 907

Dallas, TX 75251

Telephone: (972) 385-0354

Fax: (972) 788-5165E-Mail: [email protected]

ALL RIGHTS RESERVED

This book, or any part thereof, may not be reproduced

in any form without permission of William M. Cobb & Associates, Inc.


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TABLE OF CONTENTS

PAGE

I. INTRODUCTION

The End of Primary Depletion ................................................................. 1-2Factors Controlling Waterflood Recovery .............................................. 1-3

Waterflooding versus Pressure Maintenance ......................................... 1-5

Other References ....................................................................................... 1-6

II. REVIEW OF ROCK PROPERTIES AND FLUID FLOW

Wettability .................................................................................................. 2-1

Definition............................................................................................... 2-1

Importance ............................................................................................ 2-3

Determination ....................................................................................... 2-4

Factors Affecting Reservoir Wettability ............................................ 2-5

Sandstone and Carbonates .................................................................. 2-6

Native-State, Cleaned, and Restored-State Cores............................. 2-6

Capillary Pressure ..................................................................................... 2-7

Definition............................................................................................... 2-7

Importance ............................................................................................ 2-7

Sources of Data ..................................................................................... 2-7

Effect of Reservoir Variables .............................................................. 2-9

Fluid Saturation ............................................................................... 2-9

Saturation History ........................................................................... 2-10

Pore Geometry ................................................................................. 2-11Averaging of Data ................................................................................ 2-11

J-function ......................................................................................... 2-12

Correlate with Permeability ........................................................... 2-14

Relative Permeability ................................................................................ 2-17

Definition............................................................................................... 2-17

Air Permeability .............................................................................. 2-18

Absolute Permeability ..................................................................... 2-18

Effective Permeability ..................................................................... 2-18

Relative Permeability ...................................................................... 2-18

Importance ............................................................................................ 2-19Sources of Data ..................................................................................... 2-19

Effect of Reservoir Variables .............................................................. 2-20

Saturation History ........................................................................... 2-20

Wettability ........................................................................................ 2-21

End-Point Values ................................................................................. 2-23

Averaging of Data ................................................................................ 2-24

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Date Averaging Methods ................................................................ 2-24

Adjust Average Data to Account for Different Irreducible

Water Saturations ......................................................................... 2-25

Default Relative Permeability Relationships ..................................... 2-29

Problem ...................................................................................................... 2-38

III. WATERFLOODABLE OIL IN PLACE

Oil Saturation ............................................................................................. 3-2

Porosity ....................................................................................................... 3-6

Net Pay ........................................................................................................ 3-8

Net Pay Determination Using Air Permeability versus Oil

Permeability ....................................................................................... 3-12

Net Pay Determination after Accounting for Data Scatter .............. 3-18

George and Stiles Fieldwide Net Pay Method .............................. 3-19

George and Stiles Individual Well Net Pay Method

(Net to Gross Method) ................................................................... 3-24

Waterflood Permeability Cutoff Determination Using a

Water Cut Method ............................................................................ 3-29

Comparison of Original Oil-In-Place Material Balance Versus

Volumetric Estimates ........................................................................ 3-40

Primary Production Net Pay versus Secondary Floodable

Net Pay ............................................................................................... 3-41

Summary .................................................................................................... 3-45

Problems ..................................................................................................... 3-47

IV. MECHANISM OF IMMISCIBLE FLUID DISPLACEMENT

Introduction ............................................................................................... 4-1

Reservoir Response Incompressible vs. Slightly Compressible

Liquids ..................................................................................................... 4-5

Fractional Flow Equation ......................................................................... 4-8

Effect of Wettability ............................................................................. 4-17

Effect of Formation Dip and Direction of Displacement.................. 4-19

Effect of Capillary Pressure ................................................................ 4-20

Effect of Oil and Water Mobilities ..................................................... 4-21Effect of Rate ........................................................................................ 4-22

Variations of Fractional Flow Equation ............................................ 4-23

Frontal Advance Equation ....................................................................... 4-24

Welge Analysis of the Buckley-Leverett Theory in Linear Systems .... 4-26

Welge Method Saturation at Flood Front ...................................... 4-27

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Welge Method Average Water Saturation ..................................... 4-30

Performance at Water Breakthrough ........................................... 4-32

Performance after Breakthrough .................................................. 4-40

Application to Radial Flow ................................................................. 4-47

Effect of Free Gas Saturation ............................................................. 4-48

Production Performance................................................................. 4-55Displacement Efficiency .................................................................. 4-55

Conditions for Development of an Oil Bank ................................ 4-56

Properties Fluid PVT ............................................................................. 4-59

Reservoir Pressure Distribution ............................................................... 4-64

Maintain Optimum Reservoir Pressure to Minimize orS ................... 4-74

Gravity Under-Running ............................................................................ 4-75

Summary .................................................................................................... 4-76

Appendix A Development of Frontal Advance Equation ................... 4-79

Appendix B Buckley-Leverett Theory .................................................. 4-83

Buckley-Leverett Theory ..................................................................... 4-83Stabilized Zone Concept ...................................................................... 4-85

Welge Solution to Buckley-Leverett ................................................... 4-89

Water Saturation at the Front ....................................................... 4-89

Average Water Saturation ............................................................. 4-92

Problems ..................................................................................................... 4-98

V. FLOOD PATTERNS AND AREAL SWEEP EFFICIENCY

Introduction ............................................................................................... 5-1

Mobility Ratio ............................................................................................ 5-1Water Displacing Oil ........................................................................... 5-2

Water-Oil Mobility Ratio after Breakthrough ................................. 5-6

Oil Displacing Gas ................................................................................ 5-7

Basic Flood Patterns .................................................................................. 5-9

Direct Line Drive .................................................................................. 5-10

Staggered Line Drive ........................................................................... 5-11

Five-Spot ............................................................................................... 5-12

Nine-Spot............................................................................................... 5-13

Seven-Spot............................................................................................. 5-14

Areal Sweep Efficiency ............................................................................. 5-15

Causes and Effects ............................................................................... 5-16

Areal Sweep Efficiency at Breakthrough ................................................ 5-23

Isolated Pattern ............................................................................... 5-24

PAGE

Developed Pattern ........................................................................... 5-24


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Normal Pattern ................................................................................ 5-24

Inverted Pattern ................................................................................... 5-24

Areal Sweep Efficiency after Breakthrough ..................................... 5-29

Effect of Free Gas Saturation on Areal Sweep .................................. 5-37

Water Zone ...................................................................................... 5-37

Oil Zone (Oil Bank) ......................................................................... 5-38Re-Saturation Effects ...................................................................... 5-42

Other Factors Affecting Areal Sweep Efficiency .............................. 5-43

Fractures .......................................................................................... 5-43

Directional Permeability ................................................................. 5-43

Areal Permeability Variations ....................................................... 5-44

Formation Dip ................................................................................. 5-44

Off-Pattern Wells ............................................................................ 5-44

Sweep Beyond Edge Wells .............................................................. 5-45

Isolated Patterns .............................................................................. 5-45

Irregularly Spaced Wells ................................................................ 5-46

Peripheral and Line Floods ...................................................................... 5-47

Selection of Waterflood Pattern ............................................................... 5-48

Summary .................................................................................................... 5-49

Problems ..................................................................................................... 5-51

VI. INJECTION RATES AND PRESSURES

Factors Affecting Water Injection Rate .................................................. 6-1

Radial System, Unequal Mobilities .......................................................... 6-2

Regular Patterns ........................................................................................ 6-6

Unit Mobility Ratio .............................................................................. 6-6Non-Unit Mobility Ratio ...................................................................... 6-10

Regular Patterns, Unequal Mobilities ................................................ 6-16

Injectivity in Five-Spot Patterns .............................................................. 6-16

Prats, et al Method ............................................................................... 6-16

Craig Method........................................................................................ 6-17

Problem ...................................................................................................... 6-20

VII. RESERVOIR HETEROGENEITY

Areal Permeability Variations ................................................................. 7-1Detection of Areal Permeability Variations ...................................... 7-2

Effect of Areal Permeability Variations ............................................ 7-3

Vertical Permeability Variations ............................................................. 7-3

PAGE

Detection of Stratification ................................................................... 7-4

Quantitative Evaluation of Permeability Stratification ................... 7-4


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Single-Value Representation .......................................................... 7-5

Permeability Variation ................................................................... 7-6

Stiles Permeability Distribution ..................................................... 7-14

Lorentz Coefficient .......................................................................... 7-20

Miller-Lents Permeability Distribution ........................................ 7-21

Selection of Layers ............................................................................... 7-24Geological Zonation ........................................................................ 7-25

Natural Barriers .............................................................................. 7-25

Equal Thickness............................................................................... 7-25

Equal Flow Capacity ....................................................................... 7-25

Statistical Zonation ......................................................................... 7-25

Effect of Crossflow between Layers ................................................... 7-26

Vertical Sweep Efficiency ......................................................................... 7-26

Mobility Ratio ....................................................................................... 7-27

Crossflow............................................................................................... 7-27

Gravity Forces ...................................................................................... 7-27

Capillary Forces ................................................................................... 7-27

Problems ..................................................................................................... 7-29

VIII. PREDICTION OF WATERFLOOD PERFORMANCE

Simple Methods ......................................................................................... 8-1

Analogy ................................................................................................. 8-2

Rules of Thumb .................................................................................... 8-3

Empirical Relationships ...................................................................... 8-3

Reservoir Stratification ............................................................................. 8-3

Dykstra-Parsons Method ..................................................................... 8-5Stiles Method ........................................................................................ 8-8

Flow Capacity (C) and Permeability Distribution (k) ................. 8-9

Vertical Sweep Efficiency (Ev) ........................................................ 8-12

Water Cut and Water-Oil Ratio .................................................... 8-14

Oil and Water Producing Rates ..................................................... 8-16

Cumulative Oil Recovery ............................................................... 8-17

Procedure for Predicting Performance ......................................... 8-18

Confined Patterns with Stratification, Areal Sweep, and

Displacement Methods ............................................................................ 8-18

Numerical Simulation ............................................................................... 8-20

PAGE

CGM CRAIG-GEFFEN-MORSE METHOD

Introduction .......................................................................................... CGM-1


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Initial Calculations - Single Layer ...................................................... CGM-3

Stage 1: Performance Prior To Interference .................................... CGM-7

Stage 2: Performance from Interference To Fillup ......................... CGM-12

Stage 3: Performance from Fillup To Breakthrough ...................... CGM-15

Stage 4: Performance after Water Breakthrough ........................... CGM-18

Multi-Layer Performance ................................................................... CGM-35Problems ............................................................................................... CGM-40

IX. WATERFLOOD SURVEILLANCE

Introduction ............................................................................................... 9-1

Production and Injection Test Analyses .................................................. 9-2

Maps ...................................................................................................... 9-2

Production Well Test Procedures ....................................................... 9-3

Production and Injection Trend Analysis ......................................... 9-3

Production Wells ............................................................................. 9-4

Coordinate Graph ..................................................................... 9-5

Exponential Decline Curves (and Hyperbolic and

Harmonic) ............................................................................... 9-6

Injection Wells ................................................................................. 9-9

Patterns ............................................................................................ 9-10

Voidage Replacement Ratio (VRR) (Monthly and

Cumulative)................................................................................... 9-12

Spaghetti Graph .............................................................................. 9-14

Water/Oil Ratio Plot ....................................................................... 9-18

Oil Cut .............................................................................................. 9-20

X Plot ................................................................................................ 9-21Oil Cut versus Cumulative Production (Coordinate Graph)...... 9-24

Recovery Factor versus Hydrocarbon Pore Volumes Injected .. 9-25

Multiple Trend Forecasting With Field Production

Constraints .................................................................................... 9-27

Summary of Production Graphs .................................................... 9-27

Pressure Transient Testing ....................................................................... 9-28

Pressure Buildup and Pressure Falloff Testing ................................ 9-29

Step Rate Test ....................................................................................... 9-30

Hall Method of Analyzing Injection Well Behavior ......................... 9-37

PAGE

Pattern Balancing ...................................................................................... 9-45

Volumetric Sweep Efficiency .................................................................... 9-58

Injection Profile Testing ........................................................................... 9-75

Interval Selection for Waterflood Monitoring ........................................ 9-78


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Injection Profiles ........................................................................................ 9-80

Alteration of Injection Profiles ................................................................. 9-84

Flood Front (Bubble) Maps ...................................................................... 9-85

Injection Analysis ...................................................................................... 9-91

Analysis Without Free Gas ( 0Sg ) ................................................... 9-93

Analysis With Free Gas ( 0Sg ) ........................................................ 9-101Numerical Simulation .......................................................................... 9-114

Water Testing Program ............................................................................ 9-115

Dissolved Gases .................................................................................... 9-115

Microbiological Growth ...................................................................... 9-116

Minerals ................................................................................................ 9-116

Total Solids ........................................................................................... 9-116

Produced Water ................................................................................... 9-117

Pie Charts ................................................................................................... 9-117

Integrated Waterflood Monitoring .......................................................... 9-119

Project Review ........................................................................................... 9-124Problem ...................................................................................................... 9-129


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INTRODUCTION

Waterflooding is the most widely used fluid injection process in the world today. It has

been recognized1 since 1880 that injecting water into an oil-bearing formation has the

potential to improve oil recovery. However, waterflooding did not experience fieldwide

application until the 1930s when several injection projects were initiated,2,3

and it was not

until the early 1950s that the current boom in waterflooding began. Waterflooding is

responsible for a significant fraction of the oil currently produced in the world. In fact, in

the 21st century, most operators begin to investigate the feasibility of water injection

within a short time following the initial field discovery.

Many complex and sophisticated enhanced recovery processes have been developed

through the years in an effort to recover the enormous oil reserves left behind by

inefficient primary recovery mechanisms. Many of these processes have the potential to

recover more oil than waterflooding in a particular reservoir. However, no process has

been discovered which enjoys the widespread applicability of waterflooding. The

primary reasons why waterflooding is the most successful and most widely used oil

recovery process are4,5,6

:

general availability of water

low cost relative to other injection fluids

ease of injecting water into a formation

high efficiency with which water displaces oil

The purpose of these notes is to discuss the reservoir engineering aspects of

waterflooding. It is intended that the reader will gain a better understanding of the

processes by which water displaces oil from a reservoir and, in particular, will gain the

ability to calculate the expected recovery performance and to manage the project to

maximize oil recovery with a minimum number of wellbores and injection volumes.


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While written materials will be limited to the displacement of oil by water, the

displacement processes and computational techniques presented have application to other

oil recovery processes.

I.

The End of Primary Depletion.If the cumulative water injection exceeds the cumulative production since the start

of injection, the reservoir pressure is no longer declining and in most instances

reservoir pressure begins to increase. For each time period (usually on a monthly

basis) in which the injection equals or exceeds production, measured at reservoir

conditions, the average reservoir pressure is maintained or increased. In those

instances where average pressure is maintained or increased, the primary depletion

stops. This is due to the fact that the predominate primary drive mechanisms

including liquid or rock expansion, gas evolving from solution, gas cap expansion,

or natural water influx are the result of declining reservoir pressure. When pressure

is being maintained or increased, these primary drive mechanisms no longer

function.

During the time of constant or increasing reservoir pressure resulting from water

injection, oil recovery is the result of a displacementprocess. It should be clear that

it is possible within localized areas of the field to have situations where injection

may be greater than production (pressure increasing) and in other areas injection

may be less than production (pressure declining). In those areas where injection is

less than production and where average reservoir pressure is only being partially

maintained, the reservoir is experiencing a combination of pressure depletion and

fluid displacement. When both recovery processes occur simultaneously, reservoir

analysis is very complicated and usually must be analyzed using a finely gridded

numerical simulation model which has been properly history matched with field

conditions.

The injection to production ratio on a pattern or fieldwide basis is frequently

referred to as the voidage replacement ratio (VRR). Reservoir voidage is measured


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at reservoir conditions and includes oil, water, and free gas production. As a

reminder, the free gas production should not be assumed negligible. Failure to

account for free gas in the voidage computation can be a major flaw in computing

total reservoir voidage.

II. Factors Controlling Waterflood Recovery

Oil recovery due to waterflooding can be determined at any time in the life of a

waterflood project if the following four factors are known.

A.Oil-in-Place at the Start of Waterflooding-- The oil-in-place at the time of initial

water injection is a function of the floodable pore volume and the oil saturation.

Floodable pore volume is highly dependent on the selection and application of net

pay discriminators such as permeability (and porosity) cutoffs. A successful

flood requires that sufficient oil be present to form an oil bank as water moves

through the formation. An accurate prediction of waterflood performance or the

interpretation of historical waterflood behavior can only be made if a reliable

estimate of oil-in-place at the start of waterflooding is available. Oil-in-place

considerations are discussed in Chapter 3.

B.

Areal Sweep Efficiency-- This is the fraction of reservoir area that the water willcontact. It depends primarily upon the relative flow properties of oil and water,

the injection-production well pattern used to flood the reservoir, pressure

distribution between the injection and production wells, and directional

permeability. The prediction of areal sweep efficiency will be discussed in

Chapter 5.

C.Vertical Sweep Efficiency-- Vertical sweep refers to the fraction of a formation

in the vertical plane which water will contact. This will depend primarily upon

the degree of vertical stratification existing in the reservoir and will be discussed

in Chapter 6.


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D.Displacement Sweep Efficiency-- This represents the fraction of oil which water

will displace in that portion of the reservoir invaded by water. Chapter 4 will

discuss methods of determining the displacement sweep efficiency.

Methods for predicting oil recovery by waterflooding will be presented in Chapter 8.

The cumulative displaced waterflood oil can be computed at any time in the life of a

waterflood project from the following general equation:

D A V D

N N E E E (Eq. 1.1)

where

N = the oil in place in the floodable pore volume at the start of waterinjection, STBE = the fraction of the floodable pore volume area swept by the injected

water

VE = the fraction of the floodable pore volume in the vertical planeswept by the injected water

DE = the fraction of the oil saturation at the start of water injectionwhich is displaced by water in that portion of the reservoir

invaded by water

If at the start of water injection, a free gas saturation has not formed within the oil

column, it can be assumed that the displaced waterflood oil is approximately equal

to waterflood oil production. However, if at the start of injection, reservoir

pressure has declined below the initial bubble point pressure and a free gas

saturation has been developed, then the displaced oil described in Eq. 1.1 is less

than the produced waterflood oil. This subject is described in more detail in

Chapter 4.

Waterflood recovery is dependent on a number of variables. The variables which

usuallyhave the greatest impact on waterflood behavior are listed below:

| Oil saturation at the start of waterflooding, oS


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Residual oil saturation to waterflooding, ( )or orwS S

Connate water saturation, wcS

Free gas saturation at the start of water injection, gS

Water floodable pore volume, pV , BBLS (This takes into account the

permeability or porosity net pay discriminator)

Oil and water viscosity, o and w

Effective permeability to oil measured at the immobile connate water saturation,

( ) Swirok

Relative permeability to water and oil, rwk and rok

Reservoir stratification, (Dykstra-Parsons coefficient, V )

Waterflood pattern (symmetrical or irregular)

Pressure distribution between injector and producer

Injection rate, BWPD

Oil formation volume factor, o

Economics

III. Waterflooding versus Pressure Maintenance

Maximum combined primary and secondary oil recovery occurs when waterflooding

is initiated at or near the initial bubble point pressure. When water injection

commences at a time in the life of a reservoir when the reservoir pressure is at a high

level, the injection is frequently referred to as a pressure maintenance project. On


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the other hand, if water injection commences at a time when reservoir pressure has

declined to a low level due to primary depletion, the injection process is usually

referred to as a waterflood. In both instances, the injected water displaces oil and is

a dynamic displacement process. Nevertheless, there are important differences in

the displacement process when water displaces oil at high reservoir pressures

compared to the displacement process which occurs in depleted low pressure

reservoirs. The differences in the displacement mechanisms will be discussed in

Chapters 4 and 5.

IV. Other References

In June 2002, a search of the SPE e-library was conducted to obtain a listing of the

technical papers on the subject of waterflooding which have been presented at SPE

technical conferences or published in SPE journals. The listing is found at the end

of this chapter.


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CHAPTER 1 REFERENCES

1.Carll, J.F.: The Geology of the Oil Regions of Warren, Venango, Clarion, and Butler

Counties, Pennsylvania, Second Geological Survey of Pennsylvania (1880) III, pp.

1875-1879.

2.History of Petroleum Engineering, API, Dallas, Texas (1961).

3.Fettke, C.R.: "Bradford Oil Field, Pennsylvania and New York," Pennsylvania

Geological Survey, 4th Series (1938) M-21.

4.Craig, F.F., Jr.: The Reservoir Engineering Aspects of Waterflooding, Monograph

Series, SPE, Dallas, Texas (1971) 3.

5.

Willhite, G.P.: Waterflooding, Textbook Series, SPE, Dallas (1986) 3.

6.Waterflooding, Reprint Series, SPE, Richardson, TX (2003) 56.


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REVIEW OF ROCK PROPERTIES AND FLUID FLOW

An understanding of the basic rock and fluid properties which control flow in a porous

medium is a prerequisite to understanding how a waterflood performs and how a

waterflood should be designed, implemented, and managed. The purpose of this sectionis not to teach the fundamentals of rock and fluid properties -- a basic knowledge of these

is assumed. However, certain multiphase flow properties will be discussed as they apply

to waterflood systems.

I. Wettability

A. Definition

In a rock/oil/brine system, wettabilitycan be defined as the tendency of a fluid topreferentially adhere to, or wet, the surface of a rock in the presence of other

immiscible fluids. In the case of a waterflood, the wetting phases can be oil or

water; gas will often be present, but will not wet the rock. When the rock is

water-wet, water occupies the small pores and contacts the rock surface in the

large pores. The oil is located in the middle of the large pores. In an oil-wet

system, the location of the two fluids is partlyreversed from the water-wet case.

Water usually continues to fill the very small pores but oil contacts the majority of

the rock surface in the large pores. The water present in the large pores in the oil

wet rock is located in the middle of the pore, does not contact the large pore throat

surface, and is usually present in small amounts. Water fills the smallest pores

even in the oil-wet system because oil never enters the small pore system due to

capillary forces and consequently, the wettability of the small pores is not

expected to change.

Wettability concepts and the location of oil and connate water in the larger pores

can be illustrated with a simple diagram. Consider the "large" pore in Figure 2-1

which contains both oil and water.


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FIGURE 2-1

PLANE VIEW, CROSS-SECTION VIEW, AND FLUID DISTRIBUTION IN A

HYPOTHETICAL WATER-WET, OIL-WET, AND FRACTIONAL-WET PORE

TORTUOUS PORE

A

A

PORE CROSS-SECTION AT POSITION A-A

WATER-WET OIL-WET FRACTIONAL-WET

CONNATE WATER

OIL

It is important to note, however, that the term wettability is used for the wetting

preference of the rock and does not necessarily refer to the fluid that is in contact

with the rock at any given time. For example, consider a cleansandstone core

that is saturated with a refined oil. Even though the rock surface is coated with

oil, the sandstone core is still preferentially water-wet. Wettability is not a

parameter that is used directly in the computation of waterflood performance.

However, wettability can have a significant impact on such parameters as relative

permeability, connate water saturation, residual oil saturation, and capillary

pressure which directly affect waterflood performance. Anderson1-6

published a

series of excellent papers which discuss wettability and its impact on rock,

saturation, and fluid flow behavior.


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B. Importance

The performance of a waterflood is controlled to a large extent by wettability.

Reasons for this are:

1.The wettability of the rock/fluid system is important because it is a major factor

controlling the location, flow, and distribution of fluids in a reservoir. In

general, one of the fluids in a porous medium of uniform wettability that

contains at least two immiscible fluids will be the wetting fluid. When the

system is in equilibrium, the wetting fluid will completely occupy the smallest

pores and be in contact with a majority of the rock surface (assuming, of

course, that the saturation of the wetting fluid is sufficiently high). The

nonwetting fluid will occupy the centers of the larger pores and form globulesthat extend over several pores. Since wettability controls the relative position

of fluids within the rock matrix, it controls their relative ability to flow. The

wetting fluid, because of its attraction to the rock surface, is in an unfavorable

position to flow. Furthermore, the saturation of the wetting fluid cannot be

reduced below some irreducible value when flooded with another immiscible

fluid. With all other things equal, a waterflood in a water-wet reservoir will

yield a higher oil recovery at a lower water-oil ratio (WOR) than an oil-wet

reservoir. Chapter 4 presents information that allows an engineer to quantify

the effects of wettability on flood performance.

2.Wettability affects the capillary pressure and relative permeability data used to

describe a particular waterflood system. It is found, in measuring multiphase

flow properties, that the direction of saturation change (saturation history)

affects the measured properties. If measurements are made on a core while

increasing the saturation of the wetting phase, this is referred to as the

imbibition direction. Conversely, when the wetting phase saturation is

decreased during a test, it is referred to as the drainage direction. Different


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capillary pressure and relative permeability curves are obtained depending upon

the direction of saturation change used in the laboratory to make measurements.

The direction of saturation change used to determine multiphase flow properties

should correspond to the saturation history of the waterflood. Thus, it is

necessary to know the wettability of the reservoir. For example, a waterflood

in a water-wet reservoir is an imbibition process; whereas in an oil-wet

reservoir, it would be a drainage process. Different data would apply to these

two situations.

C. Determination

Historically, all petroleum reservoirs were believed to be strongly water-wet. This

was based on two major facts. First, most clean sedimentary rocks are strongly

water-wet. Second, most reservoirs were deposited in aqueous environments into

which oil later migrated. It was assumed that the connate water would prevent the

oil from touching the rock surfaces.

Reservoir rock can change from its original, strongly water-wet condition by

adsorption of polar compounds and/or the deposition of organic matter originally

in the crude oil. Some crude oils make a rock oil-wet by depositing a thick

organic film on the mineral surfaces. Other crude oils contain polar compounds

that can be adsorbed to make the rock more oil-wet. Some of these compounds

are sufficiently water soluble to pass through the aqueous phase to the rock.

The realization that rock wettability can be altered by absorbable crude oil

components led to the idea that heterogeneous forms of wettability exist in

reservoir rock. Generally, the internal surface of reservoir rock is composed of

many minerals with different surface chemistry and adsorption properties, which

may lead to variations in wettability. Fractional wettability is also called

heterogeneous, spotted, or Dalmation wettability. In fractional wettability, crude

oil components are strongly adsorbed in certain areas of the rock, so a portion of


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the rock is strongly oil-wet, while the rest is strongly water-wet. Note that this is

conceptually different from intermediate wettability which assumes all portions of

the rock surface have a slight but equal preference to being wetted by water or oil.

Several methods are available to determine the wettability of a reservoir rock.

These methods have been detailed in the literature2,7,8

and will not be discussed

here. They are:

Contact Angle

Imbibition -- Displacement Core Tests

Capillary Pressure Tests

Relative Permeability Tests

Others

D. Factors Affecting Reservoir Wettability

The original strong water-wetness of most reservoir minerals can be altered by the

adsorption of polar compounds and/or the deposition of organic matter that was

originally in the crude oil. The surface-active agents in the oil are generally

believed to be polar compounds that contain oxygen, nitrogen, and/or sulfur.

These compounds contain both a polar and a hydrocarbon end. The polar end

adsorbs on the rock surface, exposing the hydrocarbon end and making the surface

more oil-wet. Experiments have shown that some of these natural surfactants are

sufficiently soluble in water to adsorb onto the rock surface after passing through

a thin layer of water. In addition to the oil composition, the degree to which the

wettability is altered by these surfactants is also determined by the pressure,

temperature, mineral surface and brine chemistry, including ionic composition and

pH.


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E. Sandstone and Carbonates

The types of mineral surfaces in a reservoir are also important in determining

wettability. Studies1 show that carbonate reservoirs are typically more oil-wet

than sandstone reservoirs. Laboratory experiments show that the mineral surface

interacts with the crude oil composition to determine wettability.

F. Native-State, Cleaned, and Restored-State Cores

Cores in three different states of preservation are used in core analysis: native

state, cleaned, and restored state. Anderson1 indicates the best results for

multiphase-type flow analyses are obtained with native-state cores, where

alterations to the wettability of the undisturbed reservoir rock are minimized.

Anderson's1-6

work defines the term native-state as being any core that was

obtained and stored by methods that preserve the wettability of the reservoir. No

distinction is made between cores taken with oil- or water-based fluids, as long as

the native wettability is maintained. Be aware, however, that some papers

distinguish on the basis of drilling fluid. Anderson further defined native-state to

be cores taken with a suitable oil-filtrate-type drilling mud, which maintains the

original connate water saturation. Fresh-state refers to a core with unaltered

wettability that was taken with a water-base drilling mud that contains no

compounds that can alter core wettability.

The second type of core is the cleanedcore, where an attempt is made to remove

all the fluids and adsorbed organic material by flowing solvents through the cores.

Cleaned cores are usually strongly water-wet and should be used only for such

measurements as porosity and air permeability where the wettability will not

affect the results.

The third type of core is the restored-statecore in which the native wettability is

restored by a three-step process. The core is cleaned and then saturated with brine

followed by reservoir crude oil. Finally, the core is aged in reservoir crude at


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reservoir temperature for about 1,000 hours. The methods used to obtain the three

different types of cores are discussed in more detail in References 1 through 6.

II. Capillary Pressure

A. Definition

Capillary pressure can be qualitatively expressed as the difference in pressure

existing across the interface separating two immiscible fluids. Conceptually, it is

perhaps easier to think of it as the suction capacity of a rock for a fluid that wets

the rock, or the capacity of a rock to repel a non-wetting fluid. Quantitatively,

capillary pressure will be defined in this text as the difference between pressure in

the oil phase and pressure in the water phase. For example:

c o wP P (Eq. 2.1)

B. Importance

1. Capillary forces, along with gravity forces, control the vertical distribution of

fluids in a reservoir. Capillary pressure data can be used to predict the vertical

connate water distribution in a water-wet system.

2. Capillary pressure data are needed to describe waterflood behavior in more

complex prediction models and in naturally fractured reservoirs.

3. Capillary forces influence the movement of a waterflood front and,

consequently, the ultimate displacement efficiency.

4. Capillary pressure data are used to determine irreducible (immobile) water

saturation.

5. Capillary pressure data provide an indication of the pore size distribution in a

reservoir.

C.

Sources of Data

Unfortunately, capillary pressure data are not available for most reservoirs,

especially older reservoirs developed with no thought of subsequent enhanced

recovery projects. The only reliable sources of data are laboratory measurements

made on reservoir core samples. These measurements are seldom made due to the


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2 - 8

time and expense of obtaining unaltered core samples and conducting necessary

tests. The laboratory tests4most commonly used are:

Restored State (porous diaphragm) Method

Centrifuge Method

Mercury Injection Methods

Most laboratory measurements are made using either air-brine or air-mercury

systems. Consequently, the resulting data must be converted to actual reservoir

conditions, taking into account the difference between interfacial tensions of

laboratory and reservoir fluids and the difference in wettability effects of the fluids.

This conversion can be made using the relationship:

L

RcLcR PP

cos

cos (Eq. 2.2)

where:

cRP = capillary pressure at reservoir conditions, psi

cL = capillary pressure measured in the laboratory, psi

= interfacial tension

= contact angle

Capillary pressure data from another reservoir having similar rock-fluid

characteristics can also be used but is not generally recommended. When this is

necessary, a correlating function such as the "J-function" (to be discussed later) is

generally used.


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D. Effect of Reservoir Variables

1. Fluid Saturation

Capillary pressure varies with the fluid saturation of a rock, increasing as the

wetting phase saturation decreases. Accordingly, capillary pressure data are

generally presented as a function of wetting phase saturation.4 A typical

capillary pressure curve for a water-wet system is illustrated in Figure 2-2.

FIGURE 2-2

EFFECT OF SATURATION HISTORY ON OIL-WATER

CAPILLARY PRESSURE CURVES FOR A WATER-WET ROCK

0

5

10

15

20

0 20 40 60 80 100

CapillaryPressure,psia

Water Saturation, percent

Imbibition

Drainage


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2. Saturation History

As noted previously, the direction in which the fluid saturation of a rock is

changed during measurement of multiphase flow properties has a significant

affect on measured properties. This hysteresis effect is obvious in Figure 2-2.

The direction of saturation change used in the laboratory, or in other models,

must match the direction of saturation change in the reservoir to which the data

will be applied.

3. Pore Geometry

Other factors being equal, capillary pressure is inversely proportional to the

radius of the pores containing the fluids.9

If all pores were the same size in a

rock, the capillary pressure curve would ideally be described by Curve 1 in

Figure 2-3. However, all rocks exhibit a range of pore sizes which causes a

variation in capillary pressure with fluid saturation. In general, the slope of the

capillary pressure curve will increase with increasing pore size heterogeneity.

This is illustrated by Curves 2, 3, and 4 on Figure 2-3 which represent a

homogeneous, moderately heterogeneous, and very heterogeneous reservoir,

respectively.


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FIGURE 2-3

EFFECT OF RESERVOIR HETEROGENEITY ON

CAPILLARY PRESSURE CURVES

0

5

10

15

20

0 20 40 60 80 100

CapillaryPressure,psia


Curve 1

Curve 2

Curve 3

Curve 4

E. Averaging of Data

Even when good capillary pressure data are available, it is generally found that

each core sample tested from a reservoir gives a different capillary pressure curve

than every other core sample. Thus, an obvious question arises. How do we

determine which curve represents the average behavior of the reservoir to be

waterflooded? Two methods are commonly used to resolve this problem: (1) the

J-function and (2) correlation with permeability.


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1. J-function

This function was developed by M. C. Leverett10

in an attempt to develop a

universal capillary pressure curve. The dimensionless J-function relates

capillary pressure to reservoir rock and fluid properties according to the

relationship.

2

1

cw

k

f

PSJ

(Eq. 2.3)

where:

wSJ = J-function at a particular water saturation, dimensionless

cP = capillary pressure, dynes/cm2

= interfacial tension, dynes/cm

k = permeability, cm2 (1.0 cm2= 1.013 x 108D)

= porosity, fraction

f = wettability function, dimensionless

This equation was developed with the idea that, at a given saturation, the value

of wSJ would be the same for all rocks regardless of their individual

characteristics. For example, suppose the capillary pressure is measured for a

rock with permeability 1k , porosity 1 , using fluids with interfacial

tension 1 , and the wettability function is 1.0cosf . The

capillary pressure for the rock will be some value c1P at *wS . Now suppose

we measure the capillary pressure in a second rock with properties 2k , 2 ,

2 and 0.1f . At saturation*wS (same as for Core 1), a value of


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capillary pressure c2P will be obtained. If the J-function correlation works,

the J-function for Cores 1 and 2, at saturation*wS , will be equal even though

the values of capillary pressure are different. For example:

2

1

2

2

2

22

1

1

1

1

1*2

*1

0.10.1

kPkP

SJSJ ccww (Eq. 2.4)

Further, this relationship would be true at all saturations so a plot of Jversus

wS should be the same for all rocks, as depicted by Figure 2-4.

FIGURE 2-4

J-FUNCTION VS WATER SATURATION

0

1

2

3

4

0 20 40 60 80 100



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Ideally then, it would only be necessary to know the interfacial tension, average

porosity, and average permeability of the reservoir to be flooded to obtain the

proper capillary pressure curve for any reservoir.

Unfortunately, the method does not work universally, i.e., capillary pressure for

all cores, or reservoirs, will not plot on a common curve. This is due primarily

to the difference in pore size distributions and rock wettability between cores.

Rock samples of different permeability and porosity characteristics generally

would not be expected to have equivalent pore size distributions. Further,

because of handling, cleaning, and in situ variation in wettability, it is simply

not adequate to assume in Eq. 2.4 that 0.1f . However, for a given

reservoir, or for a group of reservoirs with similar lithology, this plotting

technique is often satisfactory for smoothing capillary pressure data and

determining the capillary pressure curve that applies at average reservoir

conditions. Consequently, this method is probably used more commonly than

other techniques for averaging data.

2. Correlate with Permeability

This method is based on the following empirical observation. If capillary

pressure is determined for several cores from the same reservoir (so that and

f remain relatively constant) and the logarithm of permeability is plottedas a function of water saturation for fixed values of capillary pressure, then

straight lines or smooth curves result. This is illustrated by Figure 2-5. If the

average effective permeability of the reservoir is known, the correct average

capillary pressure curve can be obtained by simply entering the subject graph

with the average permeability to read values of capillary pressure as a function

of saturation.


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FIGURE 2-5

CORRELATION OF CAPILLARY PRESSURE WITH

PERMEABILITY

1

10

100

1,000

0 20 40 60 80 100

Permeabilit

y,md


k

c1Pc2Pc3Pc4Pc5P

_________________________________________________________

EXAMPLE 2:1

Capillary pressure data measured on five cores from a sandstone reservoir are

presented below.


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Water Saturations for Constant Capillary Pressure, percent

k, md 75 psi 50 psi 25 psi 10 psi 5 psi

470.0 18.5 22.0 29.0 39.0 49.5

300.0 22.5 25.5 34.0 45.5 56.0

115.0 30.0 34.0 41.0 53.5 65.0

50.0 36.0 40.5 51.0 64.0 77.0

27.0 41.0 44.0 55.0 69.0 81.5

The geometric mean permeability of the reservoir, based on 43 core samples, is

155 md. The interfacial tension, L of the air-brine system used to measure

capillary pressure, is 71 dynes/cm. The reservoir oil-water system has an

interfacial tension, , equal to 33 dynes/cm. Find a capillary pressure curve

that will apply to average reservoir conditions, i.e., the geometric mean

permeability.

SOLUTION

Figure 2-6 shows that capillary pressure data can be correlated with

permeability. The laboratory values of capillary pressure versus saturation,

corresponding to k = 155 md, are shown in the following table. The values of

capillary pressure, converted to reservoir conditions, are also tabulated.

percentSw , psicL, ,RcR cLL

P P psi

27.2 75 34.931.5 50 23.2

39.2 25 11.6

51.0 10 4.6

62.8 5 2.3


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10

100

1,000

0 20 40 60 80 100


Permeability,md

75 psi 50 psi 25 psi 10 psi 5 psi

FIGURE 2-6

CORRELATION OF CAPILLARY PRESSURE, SATURATION,

AND PERMEABILITY FOR EXAMPLE 2.1

k = 155 md

III. Relative Permeability

A. Definition

Before engaging in a discussion of relative permeability, a brief review of the

different permeability terms which frequently appear in technical reports or as part

of technical conversations is in order. The different permeability terms are:


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2 - 18

air permeability, md

absolute permeability, md

effective permeability, md

relative permeability, dimensionless

1. Air Permeability- the routinepermeability measured on a core sample. This

measurement is conducted using a gas, such as nitrogen or natural gas, and does

not usually take into account the Klinkenberg effect.9 Air permeabilities are

frequently used as estimates of absolute permeability, However, unless the

Klinkenberg correction is performed, air permeability can overstate the absolute

permeability by a factor of 1.5 or more.

2. Absolute Permeability- the permeability of a core sample when filled with a

single liquid such as water or oil. Absolute permeability is independent of the

fluid but is dependent on the pore throat sizes. Absolute permeability is most

applicable in aquifer studies because the aquifer usually contains a single fluid,

water.

3.

Effective Permeability- the permeability to water, oil, or gas ( gow kkk ,, )

when more than one phase is present. Effective permeability of a phase is

dependent on fluid saturation. Application of Darcy's Law for determination of

production ( oq or wq ) or injection ( wi ) rates utilize effective permeability.

Effective permeability to oil and water are most commonly used in waterflood

analysis.

4. Relative Permeability - the ratio of effective permeability to some base

permeability, usually the effective permeability to oil measured at the immobile

(irreducible) connate water saturation, ( ) , /( )wir S wiro S ro o ok k k k


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/( ) wirrw w o S k k k . Since the effective permeability of a rock dependson the fluid saturation, it follows that relative permeability is also a function of

fluid saturation. When the base permeability is ( )wiro S

k , then the relative

permeability to oil at the immobile connate water saturation, ( )wirro S

k , is

1.0. In relative permeability measurements prepared prior to about 1975,

laboratories frequently used the uncorrected air permeability as the base

permeability. The net effect is to cause the ( )wirro S

k value to be less

than 1.0, usually in the range of 0.6 to 0.8.

B. Importance

As the name implies, relative permeability data indicate the relative ability of oil

and water to flow simultaneously in a porous medium. These data express the

effects of wettability, fluid saturation, saturation history, pore geometry, and fluid

distribution on the behavior of a reservoir system.5,6,7

Accordingly, this is

probably the single, most important flow property which affects the behavior of a

waterflood. When using ( )wiro S

k as the base permeability, the relative

permeability to oil and water ranges between 0.0 and 1.0 when plotted versus

water saturation. This scale allows for easy comparison of one set of relative

permeability versus another set from a different core sample. The comparison is

made by a simple overlay.

C.

Sources of Data

1. Laboratory measurement on representative core samples possessing appropriate

reservoir wettability

a. Steady-state method


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2 - 20

b. Unsteady-state method

2. Use data from similar reservoir

3. Mathematical models

4.

History matching

5. Calculate from capillary pressure data

D. Effect of Reservoir Variables

1. Saturation History

Figure 2-7 shows the effect of saturation history on a set of relative

permeability data. It is noted that the direction of flow has no effect on the

flow behavior of the wetting phase. However, a significant difference exists

between the drainage and imbibition curves for the non-wetting phase. This

again points out the need for knowing wettability. For a water-wet system, we

would choose the imbibition data; whereas, drainage data would be needed to

correctly predict the performance of an oil-wet reservoir.


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0

20

40

60

80

100

0 20 40 60 80 100

RelativePermeab

ility,percent

Wetting Phase Saturation, percent

Wetting Phase

FIGURE 2-7EFFECT OF SATURATION HISTORY ON RELATIVE

PERMEABILITY DATA

2. Wettability

Wettability affects the fluid distribution within a rock and, consequently, has a

very important effect on relative permeability data. This is indicated on Figure

2-8 which compares data for water-wet and oil-wet systems.


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0

20

40

60

80

100

0 20 40 60 80 100

RelativePermeabilty,percent


Oil Wet

WaterWet

FIGURE 2-8

EFFECT OF WETTABILITY

ON RELATIVE PERMEABILITY DATA

Several important differences between oil-wet curves and water-wet curves are

generallynoted.

a. The water saturation at which oil and water permeabilities are equal

(intersection point of curves) will generally be greater than 50 percent for

water-wet systems and less than 50 percent for oil-wet systems.

b. The connate water saturation for a water-wet system will generally be

greater than 20 percent; whereas, for oil-wet systems, it will normally be

less than 15 percent


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c. The relative permeability to water at maximum water saturation (residual oil

saturation) will be less than about 0.3 for water-wet systems but will be

greater than 0.5 for oil-wet systems.

These observations may not hold true for intermediate wettability rocks.

Further, for high permeability values 100 mdwir

o Sk these findings

may not be true7. For example, water-wet rocks with large pore throats (high

permeability) sometimes exhibit immobile connate water saturation of less than

10 to 15 percent. Nevertheless, Figure 2-8 indicates the shape and magnitude

of relative permeability curves can give an indication of the wettability

preference of a reservoir for moderate to low levels of permeability; i.e.,( ) 100 md

wiro Sk .

E. End-Point Values

Summary water-oil relative permeability tests are frequently conducted on core

samples. These summary tests are often referred to as "end-point" tests because

they reflect

wirS ,

orS , ( )

Swirok , and( )

Sorwk . Results of these tests are

less expensive than normal relative permeability tests, but they can provide useful

information on reservoir characteristics. Listed below are end-point test data for

three sandstone cores.

Water-Oil End-Point Relative Permeability Tests*

Initial Conditions Terminal Conditions

mdk , %, %wirS , mdok , %orS , mdwk , rok rwk

9.4 14.5 27.5 6.4 35.4 1.8 1.0 0.28

3.7 15.8 37.6 2.4 34.2 0.8 1.0 0.33

18.0 13.8 24.7 13.0 38.3 4.6 1.0 0.35

*Tests conducted at confining overburden pressure


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2 - 24

F. Averaging of Data

1. Data Averaging Methods

Again, we often face the problem of having several permeability curves for a

particular formation, all of which are different. It is desirable to select one set

of curves which will apply at average reservoir conditions, i.e., at the average

formation permeability. Methods to accomplish this are:

a. Determine the saturation at different values of rok or rorw kk / for each

of the different sets of data (use same values of permeability or permeability

ratio in obtaining saturations from the different permeability curves). This

is probably done most often using rorw kk / . The saturations obtained at

equal values of permeability are arithmetically averaged to define the

average set of permeability data.

b. In some cases, a plot of rorw kk / versus water saturation for each core

will yield a correlation with permeability as shown in Figure 2-9. However,

smooth curves rather than straight lines will often result. If the effective

average permeability is known, an average permeability curve can be

determined from the correlation.


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2 - 25

0.1

1

10

100

0 20 40 60 80 100

Water Satuaration, percent

FIGURE 2-9CORRELATION OF RELATIVE PERMEABILITY

DATA WITH ABSOLUTE PERMEABILITY

k1 k2 k3

2. Adjust Average Data to Account for Different Irreducible Water

Saturations

This is not necessary for oil-wet systems, but in the case of water-wet systems,

the situation often occurs where the accepted value of irreducible water

saturation does not agree with the average relative permeability data chosen to

represent the reservoir. The procedure for converting the data to a different

irreducible water saturation is:


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2 - 26

a.From the average relative permeability curves, read values of rok and

rwk at different values of oil saturation.

b.Multiply each of the saturations from Step (a) by 1.0o

wir

S

S

c.Using the normalized curve obtained from Step (b), the permeability data

can be placed back on a total pore volume basis, using any desired value of

initial water saturation, by multiplying the normalized saturations by

wirS0.1 .

It is also possible to normalize the relative permeability data before the data are

averaged.

_______________________________________________________________

EXAMPLE 2:2

Relative permeability curves measured on three cores from the Levelland Field,

San Andres formation, in West Texas are shown in Figure 2-10. The averageinitial water saturation of this reservoir is believed to be 15 percent. Find the

average oil and water relative permeability curves for this reservoir and adjust

the curves to the average connate water saturation.


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FIGURE 2-10RELATIVE PERMEABILITY DATA FOR EXAMPLE 2.2

0

20

40

60

80

100

0 20 40 60 80 100


RelativePermeability

1 2 3

1 2 3

SOLUTION

The calculations necessary to average, normalize, and adjust the curves to a

new saturation basis are presented in the following tables for the oil and water

data. The average permeability curves, adjusted to 15 percent irreducible water

saturation, are presented in Figure 2-11.


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2 - 28

Conversion of Oil Permeability Data

(All Values in Percent)

(1) (2) (3) (4) (5) (6) (7) (8)

kro Sw1 Sw2 Sw3 SwAVG

wi

o

S

S

0.1

(6) * (1.0-0.15) (Sw)NEW

1.00 8.0 25.0 37.0 23.3 100.0 85.0 15.0

0.90 11.0 27.5 39.0 25.8 96.7 82.2 17.8

0.80 13.5 30.0 41.0 28.2 93.6 79.6 20.4

0.70 16.5 32.5 44.0 31.0 90.0 76.5 23.5

0.60 20.0 35.0 46.0 33.7 86.4 73.4 26.6

0.50 23.0 37.5 48.5 36.3 83.1 70.6 29.4

0.40 26.5 40.5 51.0 39.3 79.1 67.2 32.8

0.30 30.5 44.0 54.5 43.0 74.3 63.2 36.8

0.20 35.0 47.2 58.0 46.7 69.5 59.0 41.00.10 41.1 51.0 63.2 51.8 62.8 53.4 46.6

0.05 46.0 54.0 67.0 55.7 57.8 49.1 50.9

0.01 52.5 58.0 72.5 61.0 50.8 43.2 56.8

0.00 56.0 60.5 76.0 64.2 46.7 39.7 60.3

Conversion of Water Permeability Data(All Values in Percent)

(1) (2) (3) (4) (5) (6) (7) (8)

krw Sw1 Sw2 Sw3 SwAVGwi

o

S

S

0.1

(6) * (1.0 - 0.15) (Sw)NEW

0.50 62.0 73.0 86.5 73.8 34.2 29.1 70.9

0.40 59.0 70.0 83.5 70.8 38.1 32.4 67.6

0.30 56.0 67.0 80.5 67.8 42.0 35.7 64.3

0.20 52.0 63.5 76.5 64.0 46.9 39.9 60.1

0.10 46.5 58.5 71.0 58.7 53.8 45.7 54.3

0.05 42.5 55.0 67.0 54.8 58.9 50.1 49.90.01 36.0 48.0 62.0 48.7 66.9 56.9 43.1

0.00 8.0 25.0 37.0 23.3 100.0 85.0 15.0


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FIGURE 2-11

NORMALIZED AND ADJUSTED

RELATIVE PERMEABILITY CURVES FOR EXAMPLE 2.2

0

20

40

60

80

100

0 20 40 60 80 100


RelativePe

rmeability

_______________________________________________________________

G. Default Relative Permeability Relationships

The most reliable source of relative permeability data is from laboratory

measurements performed on cores obtained from the reservoir of interest. For the

measurements to be meaningful, considerable care and effort must be expended to

ensure that the in situ reservoir wettability is preserved during coring, surfacing,


46/531

2 - 30

storage, and measurement operations. Failure to preserve native wettability will

cause the measured relative permeability values to be of little use for reservoir

analysis.

Unfortunately, many reservoirs considered for waterflooding are characterized by

the absence of relative permeability or, at best, by unreliable data. In these

situations, it may be necessary to use certain "default" relative permeability

models for data.

Several authors have presented mathematical models which can be used to

describe relative permeability relationships for the simultaneous flow of oil and

water. The relationships are restricted to reservoirs in which flow is through the

matrix. Consequently, those results are not applicable for flow through reservoirs

possessing significant vugs or natural fractures.

Corey11

has suggested that for a drainageprocess (waterflood of an oil-wet rock):

4werw Sk (Eq. 2.5)

where:

wir

wirwwe

S

SSS

0.1

(Eq. 2.6)

with:

wS = water saturation, fraction

wirS = irreducible water saturation, fraction

and:

22 0.1)0.1( wewero SSk (Eq. 2.7)


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2 - 31

Where there is simultaneous flow of oil and water in a water-wet system during an

imbibitionprocess, Smith12

suggests that:

2

1

4

0.1

wirwirw

wrw S

SS

Sk (Eq. 2.8)

and:

2

0.10.1

orwir

wirwro

SS

SSk (Eq. 2.9)

where:

orS = residual oil saturation, fraction

More recently, Hirasaki13

summarized certain relative data compiled by the 1984

National Petroleum Council14

(NPC). As part of a national enhanced oil recovery

study, it was necessary to forecast remaining waterflood recovery in many

reservoirs throughout the United States. In many instances, reservoir data such as

rock wettability and relative permeability were not available. Consequently, an

NPC technical committee recommended default relative permeability relation-

ships similar to those presented by Molina15

. These relationships are listed below.

EXW

wiror

wirwSrwrw SS

SSkk

or

0.1 (Eq. 2.10)

and:

1.0

1.0wir

EXO

w orro ro S

or wir

S Sk k

S S

(Eq. 2.11)

where:


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2 - 32

EXW = water relative permeability exponent

EXO = oil relative permeability exponent

( )Swir

rok = relative permeability to oil at the irreducible water saturation

(usually 1.0)

( )Sor

rwk = relative permeability to water at the waterflood residual oil

saturation (usually about 0.25 to 0.4 depending on

wettability)

orS = residual waterflood oil saturation, fraction

wS = water saturation, fraction

wirS = irreducible water saturation, fraction

In addition to Eq. 2.10 and Eq. 2.11, the NPC also provided certain other default

data which are listed below.

Parameter Sandstone Carbonate

Oil relative permeability end-point 1.0 1.0

Water relative permeability end-point 0.25 0.40Oil relative permeability exponent 2 2

Water relative permeability exponent 2 2

Residual oil saturation, percent 25 37

A comparison of these default end-point values with the statements listed on page

20 of Craig8suggests a possible conclusion that carbonate reservoirs behave as if

they are oil-wet. This observation should not be interpreted as an indication of

rock wettability but the result of attempting to "average" a large amount of data.

Finally, Honapour16

provides a thorough review of the empirical equations used to

compute two phase (oil/water or gas/oil) and three phase (gas/oil/water) relative

permeability.


49/531

2 - 33

__________________________________________________________________

EXAMPLE 2:3

A carbonate oil reservoir is being considered for waterflooding. At the present

time, the immobile (irreducible) water saturation is estimated to be 25 percent.

Compute a pair of oil and water relative permeability curves that could be used in

the evaluation of the waterflood.

SOLUTION

In the absence of specific data, the default relative permeability relationships

described by Eq. 2.10 and Eq. 2.11 will be utilized. The following data are

estimated from analog fields or from the NPC default values.

orwS = 35 percent (analog field)

( )Swir

rok = 1.0 (based on ( ) Swirbase ok k )

( )Sor

rwk = 0.35 (assumes intermediate wettability)

EXO = 2.0 (1984 NPC)

EXW = 2.0 (1984 NPC)

EXW

wiror

wirw

orSrwrw SS

SSkk

0.1

1.01.0wir

XO

w orro ro S

or wir

S Sk kS S


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2 - 34

Substituting:

0.2

25.035.00.1

25.0)35.0(

wrwS

k

and:

0.2

25.035.00.1

35.00.10.1

wroS

k

Finally, rwk and rok can be computed and plotted as a function of water

saturation.

%w

S , r w

kro

k

25 0.000 1.000

30 0.001 0.766

35 0.022 0.562

40 0.049 0.391

45 0.088 0.250

50 0.137 0.14155 0.197 0.062

60 0.268 0.016

65 0.350 0.000


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FIGURE 2-12

OIL/WATER RELATIVE PERMEABILITY

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100Water Saturation, percent

RelativePermeability

kro

krw


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CHAPTER 2 REFERENCES

1.Anderson, W.G.: "Wettability Literature Survey - Part l: Rock/Oil/Brime Inter-

actions and the Effects of Core Handling on Wettability," JPT(Oct. 1986) pp. 1125-

44.

2.Anderson, W.G.: "Wettability Literature Survey - Part 2: Wettability Measurement,"

JPT(Nov. 1986) pp. 1246-62.

3.Anderson, W.G.: "Wettability Literature Survey - Part 3: The Effects of Wettability

on the Electrical Properties of Porous Media,"JPT(Dec. 1986) pp. 1371-78.


on Capillary Pressure,"JPT(Oct. 1987) pp. 1283-1300.


on Relative Permeability on Relative Permeability,"JPT(Nov. 1987) pp. 1453-68.


on Waterflooding,"JPT(Dec. 1987) pp. 1605-20.

7.Willhite, G.P.: Waterflooding, Textbook Series, SPE, Dallas (1986) 3.

8.Craig, F.F., Jr.: The Reservoir Engineering Aspects of Waterflooding, Monograph

Series, SPE, Dallas, Texas (1971) 3.

9.Amyx, J.W., Bass, D.M. Jr., and Whiting, R.L.: Petroleum Reservoir Engineering,

McGraw-Hill Book Company (1960).

10.Leverett, M.C.: "Capillary Behavior in Porous Solids," Trans., AIME (1941).

11.Corey, A.T.: "The Interrelation Between Gas and Oil Relative Permeabilities,"

Producers Monthly,(November 1954).

12.Smith, C.R.: Mechanics of Secondary Oil Recovery, Reinhold Publishing

Corporation, New York (1966).

13.Hirasaki, G.J., Morrow, F., Willhite, G.P.: "Estimation of Reservoir Heterogeneity

From Waterflood Performance," SPE Paper 13415, Unsolicited technical papersubmitted for publication during Fall 1984.

14.National Petroleum Council:Enhanced Oil Recovery, (June 21, 1984).


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2 - 37

15.Molina, N.N.: "A Systematic Approach to the Relative Permeability in Reservoir

Simulation," SPE Paper 9234 presented at the 1980 SPE Annual Technical

Conference and Exhibition, Dallas.

16.Honarpour, M., Koederitz, L., and Harvey, A.H.: Relative Permeability of Petroleum

Reservoirs, CRC Press, Boca Raton , FL (1986).


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PROBLEM 2:1

REVIEW OF ROCK AND FLUID PROPERTIES

A series of laboratory studies resulted in the following average relative permeability data

for an oil reservoir. (Note that the base permeability is the air permeability -- it is old

data.)

%w

S , rwk rok

25 0.000 0.565

30 0.002 0.418

35 0.015 0.300

40 0.025 0.218

45 0.040 0.14450 0.060 0.092

55 0.082 0.052

60 0.118 0.027

65 0.153 0.009

70 0.200 0.000

These data indicate the irreducible water saturation in the reservoir is 25 percent. Well

logs and core analysis suggest, however, that the true irreducible saturation isapproximately 15 percent. Adjustthe permeability data so they represent an irreducible

water saturation of 15 percent and present the data in normalizedform on a scale of 0.0

to 1.0.


55/531

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56/5313- 1

WATERFLOODABLE OIL-IN-PLACE

To accurately predict and effectively manage waterflood operations, it is necessary to

estimate the reservoir oil-in- place within the floodable portion of the reservoir at the start

of water injection. As indicated earlier, the basic oil recovery evaluation equation used to

compute displaced oil by waterflooding can be summarized as:

D A V DN N E E E (Eq. 3.1)

where:

D

N = oil displaced by water injection, STB

(It will be shown in later chapters that, in many instances, significant

amounts of displaced oil may not be produced due to gas resaturation

effects.)

N = oil-in-place at start of waterflooding within thefloodablezones, STB

E = areal sweep efficiency, fraction

V

E = vertical sweep efficiency, fraction

DE = unit displacement efficiency, fraction

The oil-in-place at the start of waterflooding is given by:

o

o

7758Ah SN

B

(Eq. 3.2)

where:

= floodable area, acres

h = floodable pay, feet

= porosity, fraction


57/5313- 2

oS = oil saturation at start of the flood, fraction

oB = oil formation volume factor at start of the flood, RB/STB

Three major difficulties encountered in using Eq. 3.2 are the determination of

waterfloodable net pay, porosity, and oil saturation. SinceNrepresents the oil in place at

the start of water injection, it would appear that this value could be obtained by simply

taking the differences in the original oil-in-place and the primary production up to the start

of injection. This simple approach can be misleading due to the fact that the net pay for

primary production is assumed to be the net pay for water injection. As will be shown

later in this chapter, the net pay cutoffs and subsequent net pay for water injection is much

different than for primary depletion. Another deficiency in computing Nby taking thedifference in the original oil-in-place and primary production is that such a simple

calculation does not indicate how the gas saturation has increased and the oil saturation has

decreased if reservoir pressure is below the initial bubble point pressure.

I. Oil Saturation

Most waterfloods are implemented late in the life of the reservoir after significant

primary production has occurred and at a time when the reservoir pressure is below the

bubble-point pressure. As primary production occurs, reservoir pressure declines

below the bubble-point, solution gas evolves from the oil in the reservoir, and a free

gas saturation forms within the oil zone. The development of a free gas saturation is

characterized by the production of a portion of the gas and an increase in the gas-oil

ratio. Despite production of the free gas, a large portion of it remains in the reservoir.

Consequently, the oil saturation at the start of waterflooding can be substantially less

than the oil saturation at the discovery of the field.

The oil saturation in the reservoir is constant at (1 wcS ) during those times when

average reservoir pressure is at or above the initial bubble point pressure. However,

the oil saturation begins to decrease and the free gas saturation begins to increase as

the average reservoir pressure declines below the initial bubble point pressure. To

compute the average oil saturation, let the initial bubble point pressure be the


58/5313- 3

beginning reference point. Consider the following equation at any time when the

reservoir pressure is below the bubble point pressure.

Reservoir Oil Volume

Reservoir Pore VolumeoS (Eq. 3.3)

The reservoir oil volume consists of the number of barrels of oil in the reservoir at the

time of interest and can be estimated as:

Stock Tank Oil Volume = OOIP at bubble-point pressure - Primary Oil

Produced below bubble-point pressure (Eq. 3.4)

or:

Reservoir Oil Volume=ob pp o

( N N )B (Eq. 3.5)

where:

obN = original oil-in-place at the bubble-point pressure, STB

pN = primary oil production between the bubble-point and current

reservoir pressure, STB

oB = oil formation volume factor at prevailing pressure, RB/STB

The reservoir pore volume can be estimated using a volumetric material balance

where:

wc

obob

V (1.0 S )

N B

(Eq. 3.6)

Solving for pore volume gives:

ob obp

wc

N BV

( 1.0 S )

(Eq. 3.7)


59/5313- 4

where:

obB = oil formation volume factor at the bubble-point pressure, RB/STB

wc

S = connate water saturation at the time of discovery, fraction

Substituting Eq. 3.5 and Eq. 3.7 into Eq. 3.3 leads to:

ob pp oo

ob ob

wc

N N BS

N B

1.0 S

(Eq. 3.8)

Rearranging results in the average oil saturation equation.

pp oo wcob ob

N BS 1.0 1.0 S

N B

(Eq. 3.9)

This equation plays a very important role in estimating waterflood potential.

____________________________________________________________________

EXAMPLE 3:1

A reservoir is a candidate for waterflooding. The primary oil recovery factor below

the bubble-point pressure is 12 percent. The connate water saturation is 36 percent,

and the oil formation volume factors oB at the bubble-point and current pressureare estimated fro

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