C (I-1B) Reservoir Fundam Fluid Flow

8/11/2019 C (I-1B) Reservoir Fundam Fluid Flow

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Copyright 2008, NExT, All rights reserved

Basics of Reservoir Engineering Module I

I.1.B Reservoir Fundamentals of Fluid Flow


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Porosity

Porosity of a measure of the

void space within a rock

Range or Typical Values:

30%, unconsolidated well-sorted

sandstone

20%, clean, well-sorted consolidated

sandstone8%, low permeability reservoir rock

0.5%, natural fracture porosity


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Reservoir Make-up

Rock matrix

Pore space Fluids: Water,

Oil and gas


4/97Copyright 2008, NExT, All rights reserved

Rock Matrix and Pore Space

Rock matrix Pore space



Rock Matrix and Pore Space

Rock matrix Water Oil and/or gas



Porosity Definition

Porosity: The fractional void space within a rock that isavailable for the storage of fluids

b

mab

b

p

V

VV

V

VPorosity

===



Effect Grain Size and Packing

Cubic Packing of SpheresPorosity = 48%



Porosity Calculations Cubic/Uniform Spheres

Bulk volume = (2r)3 = 8r3

Matrix volume =

Pore volume = bulk volume - matrix volume

34

3

r

Calculations for Cubic Packing



Porosity Calculations Cubic/Uniform Spheres

( ) %6.47

321

8

3/48

3

33

==

=

=

=

r

rr

VolumeBulk

VolumeMatrixVolumeBulk

VolumeBulk

VolumePorePorosity




Rhombic Packing of SpheresPorosity = 27 %




Packing of Two Sizes of Spheres

Porosity = 14%

P S Cl ifi ti



Pore-Space Classification

Total porosity, t =

Effective porosity, e =

VolumeBulk

SpacePoreTotal

VolumeBulk

SpacePorectedInterconne

C i f T t l d Eff ti P iti


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Comparison of Total and Effective Porosities

Very clean sandstones : t = e

Poorly to moderately well -cemented intergranular

materials: t e

Highly cemented materials and most carbonates: e

< t

F t Th t Aff t P it


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Factors That Affect Porosity

Particle shape

Packing

Particle sizes Cementing materials

Overburden stress

Vugs and fractures

Example Porosity/Overburden Pressure Relationship


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Example Porosity/Overburden Pressure Relationship

50

Overburden pressure, psi

Po r o

s ity ,

%30

40

20

10

00 1,000 3,0002,000 4,000 5,000 6,000

Sandstones

Shales

Permeability


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Permeability

Permeability is a measure of the rocks ability to transmit fluids

pA

Lqk

= Aq q

p1p

2

L

Permeability Values


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Permeability Values

The quality of the reservoir, as it relates to permeability, can be

classified as follows:

k < 1 md poor

1 < k < 10 md fair

10 < k < 50 md moderate

50 < k < 250 md good

250 md < k very good

Example


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Example

Permeability-Porosity Relationship

From Tiab and Donaldson

Factors affecting permeability


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Factors affecting permeability

1.0

.8

.6

.4

.2

.0

0 2000 4000 6000 8000 10000

Permeability:fractionofo

riginal

Net overburden pressure: psi

A

B

C

A

Well cemented

Friable

Unconsolidated

Scales of Geological Reservoir Heterogeneity


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Scales of Geological Reservoir Heterogeneity

FieldWide

Interw

ell

Well-Bore

(modified from Weber, 1986)

Hand Lens or

Binocular Microscope

Unaided Eye

Petrographic orScanning Electron

Microscope

DeterminedFrom Well Logs,Seismic Lines,

StatisticalModeling,etc.

10-100'sm

10-100'smm

1-10'sm

100's

m

10'sm

1-10 km

100's m

Well WellInterwellArea

ReservoirSandstone

Scales of Investigation Used in Reservoir


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Scales of Investigation Used in Reservoir

Characterization

Gigascopic

Megascopic

Macroscopic

Microscopic

Well Test

Reservoir ModelGrid Cell

Wireline LogInterval

Core Plug

GeologicalThin Section

Relative Volume

1

1014

2 x 1012

3 x 107

5 x 102

300 m

50 m

300 m

5 m 150 m

2 m

1 m

cm

mm -m

(modified from Hurst, 1993)

Permeability Exercise 1


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What are the units of permeability?

Use Dimensional Analysis:pALqk

=



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Relate the permeability concept to other common fluid flow

situations: laminar fluid flow through a pipe and through parallelplates (fractures).

What is the permeability of a circular opening (vug) of 0.005

inches?

What is the permeability of a fracture of 0.01 in thickness?

Saturations


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Saturations

H2O

Fluid Saturations


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Fluid Saturations

Grain Water Gas Oil

Definition of Fluid Saturation


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Definition of Fluid Saturation

p

w

w V

VS =

Water saturation:

Oil saturation:

Gas saturation:

p

ooV

VS =

wog SSS = 0.1

Net Pay Thickness


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Net Pay Thickness

Shale

Sand

h3

h2

h1

h = h1 + h2 + h3

Rock Wettability


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Rock Wettability

Wettability: Tendency of one fluid to spread on or adhere

to a solid surface in the presence of other immiscible fluidsWettability refers to interaction between fluid and solidphases

Solid

WaterOil

os

ws

ow

Solid

WaterOil

os ws

ow

Solid surface is reservoir rock (i.e., sandstone, limestone,

dolomite or mixtures of each) -- Fluids are oil, water, and/or gas

Interfacial Tension and Adhesion Tension


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Interfacial Tension and Adhesion Tension

Interfacial tension is the force per unit length required to create a new

surface

Interfacial tension is commonly expressed in Newtons/meter or

dynes/cm

Adhesion tension can be expressed as the difference between two

solid-fluid interfacial tensions

cosowwsosTA

Contact Angle


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g

Solid

Water

Oil

Oil Oil

os ws

ow

Wetting Phase Fluid


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Wetting Phase Fluid

A wetting phase preferentially wets the solid rock surface

Because of attractive forces between rock and fluid, the wetting phase

is drawn into smaller pore spaces of porous media

Wetting phase fluid often is not very mobile

Attractive forces prohibit reduction in wetting phase saturation belowsome irreducible value (called irreducible wetting phase saturation)

Many hydrocarbon reservoirs tend to be either totally or partially

water wet

Nonwetting Phase Fluid


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Nonwetting Phase Fluid

A nonwetting phase does not preferentially wet the solid rock

surface

Repulsive forces between rock and fluid cause nonwetting phase tooccupy largest pore spaces of porous media

Nonwetting phase fluid is often the most mobile fluid, especially at

large nonwetting phase saturationsNatural gas is never the wetting phase in hydrocarbon reservoirs

Water-Wet Reservoir Rock


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Reservoir rock is considered to be water-wet if water

preferentially wets the rock surfaces

The rock is water-wet under the following conditions:

ws >os

AT < 0 (i.e., the adhesion tension is negative)0


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Solid

Water

Oil

os ws

ow

Note: 0


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Reservoir rock is considered to be oil-wet if oil preferentially

wets the rock surfaces

The rock is oil-wet under the following conditions:

os >ws

AT > 0 (i.e., the adhesion tension is positive)

90


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Solid

Water

Oil

os ws

ow

Note: 90


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Wettability affects the shape of the relative permeability curves.

Oil moves easier in water-wet rocks than oil-wet rocks.

Primary oil recovery is affected by the wettability.

A water-wet system will exhibit greater primary oil recovery.

Oil recovery under waterflooding is affected by wettability

A water-wet system will exhibit greater oil recovery under

waterflooding.

Implications of Wettability


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1 2 3 4 5 6 7 8 9 10 11 120

20

40

60

80

1

2345

Coreno

Percentsilicone Wettability

0.00

0.02000.2002.001.00

0.649

0.176- 0.222- 0.250- 0.333

Curves cut off at Fwd 100

1 23

4

5

Water injected, pore volumes

Recoveryefficiency,percent,Soi

Implications of Wettability


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0

20

40

60

80

1 2 3 4 5 6 7 8 9 10

Squirrel oil - 0.10 N NaCl - Torpedo core ( 33 O W 663,K 0945, Swi 21.20%)

Squirrel oil - 0.10 N NaCl Torpedo Sandstone core,after remaining in oil for 84 days ( 33.0 W 663, K

0.925, Swi 23.28%)

Recoveryefficiency,percentS

pi

Water injection, pore volumes

Capillary Pressure Definition


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Capillary Pressure Definition

Capillary Pressure is the pressure difference existing

across the interface separating two immiscible fluids.It is usually calculated as:

Pc=pnwt-pwt

ow

Pw, Water

Po, Oil

Pw, Water

Po, Oil

ow

os

wsos

ws

Capillary Example 1


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Define capillary pressure in the following systems: Water-gas system

Water-wet water-oil system Oil-gas system

Capillary Tube Model. Air-Water System


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Water

Air

h



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Capillary Tube Model. Air Water System

The height of water is a function of

The adhesion tension between the air and water

The radius of the tube

The density difference between fluids

aw

aw

grh

=

cos2



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Capillary Tube Model. Air Water System

Air

Water

pa2

h

pa1pw1

pw2

Capillary Pressure. Air-Water System


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p y y

Combining the two relations results in the following

expression :

r

P awc cos2

=

Capillary Pressure. Oil-Water System


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From a similar derivation, the equation for capillarypressure for an oil/water system is

rP

ow

c

cos2

=

Imbibition and Drainage & Capillary Pressure


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Imbibition: Fluid flow process in which the saturation of

the wetting phase increases and the nonwetting phasesaturation decreases

Mobility of wetting phase increases as wetting phasesaturation increases

Drainage: Fluid flow process in which the saturation of the

nonwetting phase increases

Mobility of nonwetting fluid phase increases as nonwetting

phase saturation increases

Typical Drainage and Imbibition Capillary Pressure

Curves


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Curves

Drainage (1)

Imbibition (2)

Si Sm

Sw

Pd

Pc

0 0.5 1.0

The pc-drainage curve is

always higher than the pc-imbibition curve.

Capillarity Exercise 2


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50

0

45

40

35

30

25

20

15

10

5

(1)

(2)

Sw

Pc, psi

0 0.5 1.00.25 0.75

Converting Laboratory Capillary Pressure Data to

Reservoir Conditions


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Reservoir Conditions

Based on our previous derivation, we use the following basic

equations:

L

LL

cL rP

cos2

=

R

RRcR

rP

cos2=


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Setting rL = rR and combining equations yields

Capillary pressure at reservoir conditions

cR

RR

cL

LLRL

PPrr

cos2cos2==

cL

LL

RRcR PP

cos

cos=

Capillarity Example 3


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Converting Laboratory Capillary Pressure Data to Reservoir Conditions

0

400

800

1200

1600

2000

020406080

Mercury Saturation, percent pore space

InjectionPress

ure,psia

Example 3

Solution


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Solution

MercurySaturation

(SHg)%

PcLpsia

PcRpsia

70 1,320 80.560 820 50.0

50 560 34.2

40 410 25.030 310 18.920 240 14.610 200 12.2

Effects of Reservoir Properties on Capillary Pressure


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Capillary pressure characteristics in reservoir are affected by

Variations in permeability Grain size distribution

Saturation history

Contact angle

Interfacial tension

Density difference between fluids

Effect of Permeability


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Decreasing

Permeability

A B

C

20

16

12

8

4

00 0.2 0.4 0.6 0.8 1.0

Water Saturation

Capillary

Pressure

Effect of Grain Size Distribution


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Well-sorted Poorly sorted

Cap

illarypres

sure,psia

Water saturation, %

Effect of Saturation History


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Water saturation, %

Cap

illarypres

sure,psia

Imbibition

Drainage

Effect of Contact Angle


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Decreasing R

R = 30

R = 60

R = 0

Capillary

Pressure

R = 80

20

16

12

8

4

00 0.2 0.4 0.6 0.8 1.0

Water Saturation

Effect of Interfacial Tension


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Water Saturation

Heig

htAboveF

reeWaterL

evel

High Tension

Low Tension

0 1.0

Effect of Density Difference


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Water Saturation

Small Density Difference

LargeDensity

DifferenceHeightAbove

FreeWater

Level

0 1.0

Uses of Capillary Pressure Data


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Determine initial water saturation in the reservoir

Determine fluid distribution in reservoir

Determine residual oil saturation for water flooding applications

Determine pore size distribution index

May help in identifying zones or rock types

Input for reservoir simulation calculations.

Capillary Pressure Data Using

the Leverett J-Function


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the Leverett J Function

A universal capillary pressure

curve is impossible to generate

because of the variation ofproperties affecting capillary

pressures in reservoir

The Leverett J-function was

developed in an attempt to

convert all capillary pressuredata to a universal curve

( ) kP

SJ c

wcos

22.0

=

Example J-Function for West Texas Carbonate


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0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00

Water saturation, fraction

J-

fun

ction

Jc

Jmatch

Jn1

Jn2

Jn3


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Capillarity Example 4Calculation of J-function


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A v e r a g e d A ir /B r in e C a p illa r yP re s s u re D a ta

Pcp s i a

Sw%

1 9 8 .32 9 8 .3

4 9 6 .8

8 5 9 .0

1 5 3 6 .33 5 2 5 .4

5 0 0 1 5 .3

Capillarity Example 4Solution


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CalculatedJ-Functions

Sw%

0.22*

J(Sw)

98.3 0.2298.3 0.43

96.8 0.86

59.0 1.7336.3 3.24

25.4 7.57

15.3 108.1



67/97


120

100

80

60

40

0

20

0 20 40 60 80 100

0.22*J

(Sw

)

Sw , %

Capillarity Example 5Estimating Pc from J-function


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c

Estimate capillary pressures from Leverett J-functioncalculated in Example 4 for a different core sample.

Properties of core sample:

k= 100 md

= 10 %



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Estimated Capillary

Pressures for the 100-mdPermeability Core Sample

Sw,

%

Pc,

psia98.3 2.27

98.3 4.45

96.8 8.91

59.0 17.9136.3 36.90

25.4 78.18

15.3 1118.63

Intro to the Relative Permeability Concept


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Permeability is a property of the porous medium and is a

measure of the capacity of the medium to transmit fluids

When the medium is completely saturated with one fluid, then

the permeability measurement is often referred to as

absolute permeability

Calculating Absolute Permeability


71/97


Absolute permeability is often calculated from the flowequation:

L

pAkq

=

Effective Permeability


72/97


When the rock pore spaces contain more than one fluid,

then the permeability to a particular fluid is called the

effective permeability

Effective permeability is a measure of the fluid

conductance capacity of a porous medium to a particularfluid when the medium is saturated with more than one

fluid

Calculating Effective Permeability


73/97


L

pAk

qo

oeo

o

=Oil

Water

Gas

LpAkq

w

weww

=

L

pAkq

g

geg

g

=

Relative Permeability


74/97


Relative permeability is defined as the ratio of the

effective permeability to a fluid at a given saturation to abase permeability

The base permeability is commonly taken as the effective

permeability to the fluid at 100% saturation (absolutepermeability) or the effective non-wetting phasepermeability at irreducible wetting phase saturation

Calculating Relative Permeabilities


75/97


k

kk eo

ro

=Oil

Water

Gas

kkk ewrw =

k

kk

eg

rg =

Fundamental Concepts


76/97


Water phase

Water is located in smaller pore spaces and along sandgrains

Therefore, relative permeability to water is a function of water

saturation only (i.e., it does not matter what the relative

amounts of oil and gas are)

Thus, we can plot relative permeability to water against watersaturation on Cartesian coordinate paper



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Oil phase

Oil is located between water and gas in the pore spaces, and to

a certain extent, in the smaller pores

Thus, relative permeability to oil is a function of oil, water, and

gas saturations If the water saturation can be considered constant (i.e., the

minimum interstitial water saturation), then kro can be plotted

against So on Cartesian coordinate paper



78/97


Gas phase

Gas is located in the center of the larger pores Therefore, the relative permeability to gas is a function of

gas saturation only (i.e., it does not matter what the

relative amounts of oil and water are)

Thus ,we can plot krg against Sg (or Sw + So) on Cartesiancoordinate paper

Common Multi-Phase Flow Systems


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Water-oil systems

Oil-gas systemsWater-gas systems

Three phase systems (water, oil, and gas)

Relative Permeability Exercise 1


80/97


What are the relative permeability data sets we need to

use for the following situations? Water flooding an oil reservoir above the bubble point

Production from an oil reservoir with a gas-cap andwater aquifer

Relative Permeability Exercise 1Solution


81/97


For water flooding an oil reservoir above the bubble

point : Water-oil relative permeability

For three phase flow : Water-oil relative permeability

Gas-oil (or gas-liquid) relative permeability 3 phase relative permeability

Oil-Water Relative Permeability


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40

0

20

400 1006020 80

Water Saturation (%)

Re

lativePermeability(%)

100

60

80

Waterkrw@ Sor

Oil

Two-Phase FlowRegion

Irreducible

Water

Saturation

kro @ Swi

Residual Oil

Saturation

Oil-Gas Relative Permeability


83/97


40

0

20

400 1006020 80

Total Liquid Saturation - % of Pore Volume

RelativeP

ermeabilit

y(%)

100

60

80

Gaskro

Oil

krg

SL = So + Swi

Relative Permeability Exercise 2


84/97


0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

RelativePermeability,Frac

tion

1.0

0.6

0.8 (1)

(2)

Importance of Relative Permeability Data


85/97


Relative permeability data affect the flow characteristics of

reservoir fluids.Relative permeability data affect the recovery of oil and/or

gas.

Relative Permeability Example 3Effect of Relative Permeability


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0

20

40

60

80

100

0 20 40 60 80 100

Water Saturation (%)

Rela

tivePerm

eability(

%)

Rock Type 2

Rock Type 1

Relative Permeability Example 3Effect of Relative Permeability


87/97


0

20

40

60

80

100

0 2 4 6 8 10

Pore Volumes Injected

Perce

ntofRec

overable

Oil

Rock Type 1Rock Type 2

Factors Affecting Effective and RelativePermeabilities


88/97


Fluid saturations

Geometry of the rock pore spaces and grain size distribution

Rock wettability

Fluid saturation history (i.e., imbibition or drainage)

Effect of Wettability


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0.4

0

0.2

400 1006020 80


RelativePermeability,Fraction

1.0

0.6

0.8

Water

Oil

Strongly Water-Wet Rock

0.4

0

0.2

400 1006020 80


RelativePermeability,Fraction

1.0

0.6

0.8

WaterOil

Strongly Oil-Wet Rock


90/97

Effect of Saturation History


91/97


0

20

40

60

80

100

0 20 40 60 80 100

Drainage

Imbibition

Wetting Phase Saturation, % PV

R

elativePermeability,%

Residual non-wetting

phase saturation

Interstitial wetting

phase saturation

Choosing the Right Curve


92/97


When simulating the waterflood of a water-wet reservoir

rock, imbibition relative permeability curves should beused.

When modeling gas injection into an oil reservoir, drainagerelative permeability curves should be used.

Three-Phase Relative Permeabilities


93/97


100% Gas

100% Oil100% Water

Relative Permeability to Water in a Three-PhaseSystem


94/97


100% Gas

100% Water 100% Oil

0%

10%

20%

40%

60%

krw= 80%

Relative Permeability to Oil in a Three-Phase System


95/97


100% Gas

100% Water 100% Oil

5%

10%20%

30%

40%

kro = 50%

Uses of Effective and Relative Permeability


96/97


Reservoir simulation

Flow calculations that involve multi-phase flow in

reservoirs

Estimation of residual oil (and/or gas) saturation

References


97/97


1. Amyx, J.W., Bass, D.M., and Whiting, R.L.: Petroleum Reservoir

Engineering, McGrow-Hill Book Company New York, 1960.

2. Tiab, D. and Donaldson, E.C.: Petrophysics, Gulf Publishing

Company, Houston, TX. 1996.

3. Dandekar, A. Petroleum Reservoir Rock and Fluid Properties,

Taylor and Francis, 2006.

Documents

C (I-1B) Reservoir Fundam Fluid Flow