6.CapillaryPressure

Introduction to

Relative Permeability

and

Capillary Pressure

• Absolute permeability: is the permeability of a

porous medium saturated with a single fluid

(e.g. Sw=1)

• Absolute permeability can be calculated from

the steady-state flow equation (1D, Linear Flow;

Darcy Units):

Review: Absolute Permeability

L

pAkq

Multiphase Flow in Reservoirs

Commonly, reservoirs contain 2 or 3 fluids

• Water-oil systems

• Oil-gas systems

• Water-gas systems

• Three phase systems (water, oil, and gas)

To evaluate multiphase systems, must

consider the effective and relative

permeability

Effective permeability: is a measure of

the conductance of a porous medium

for one fluid phase when the medium is

saturated with more than one fluid.

• The porous medium can have a distinct and

measurable conductance to each phase present in

the medium

• Effective permeabilities: (ko, kg, kw)

Effective Permeability

Amyx, Bass, and Whiting, 1960; PETE 311 Notes

• Oil

• Water

• Gas

L

Akq

o

ooo

L

Akq

w

www

L

Akq

g

gg

g

Effective Permeability

Steady state, 1D, linear flow

equation (Darcy units):

qn = volumetric flow rate for a

specific phase, n

A = flow area

n = flow potential drop for

phase, n (including pressure,

gravity and capillary pressure

terms)

n = fluid viscosity for phase n

L = flow length

Modified from NExT, 1999; Amyx, Bass, and Whiting, 1960; PETE 311 NOTES

Relative Permeability is the ratio of the effective

permeability of a fluid at a given saturation to some

base permeability

• Base permeability is typically defined as:

– absolute permeability, k

– air permeability, kair

– effective permeability to non-wetting phase at irreducible wetting

phase saturation [e.g. ko(Sw=Swi)]

– because definition of base permeability varies, the definition

used must always be:

• confirmed before applying relative permeability data

• noted along with tables and figures presenting relative

permeability data

Relative Permeability

Amyx, Bass, and Whiting, 1960

• Oil

• Water

• Gas

k

kk

o

ro

)3.0,5.0(

)3.0,5.0(

k

kk

w

rw

)3.0,5.0(

)3.0,5.0(

k

kk

g

rg

)3.0,5.0(

)3.0,5.0(

Relative Permeability

Modified from Amyx, Bass, and Whiting, 1960

So =0.5

Sw =0.3

Sg = 0.2

Relative Permeability Functions

0.40

0

0.20

0.400 1.000.600.20 0.80

Water Saturation (fraction)

Re

lati

ve

Pe

rme

ab

ilit

y (

fra

cti

on

) 1.00

0.60

0.80

Water

krw @ Sor

Oil

Two-Phase Flow

Region

kro @ Swi • Wettability and direction of

saturation change must be

considered

•drainage

•imbibition

• Base used to normalize this

relative permeability curve is

kro @ Swi

• As Sw increases, kro decreases

and krw increases until

reaching residual oil

saturation

Modified from NExT, 1999

Imbibition Relative Permeability

(Water Wet Case)

Effect of Wettability

for Increasing Sw

0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

Rela

tive P

erm

eab

ilit

y, F

racti

on

1.0

0.6

0.8

Water

Oil

Strongly Water-Wet Rock

0.4

0

0.2

400 1006020 80

Water Saturation (% PV)

Rela

tive P

erm

eab

ilit

y, F

racti

on

1.0

0.6

0.8

WaterOil

Strongly Oil-Wet Rock

• Water flows more freely

• Higher residual oil saturationModified from NExT, 1999

• Fluid saturations

• Geometry of the pore spaces and pore

size distribution

• Wettability

• Fluid saturation history (i.e., imbibition

or drainage)

Factors Affecting Relative Permeabilities

After Standing, 1975

Characteristics of Relative

Permeability Functions

• Relative permeability is unique for

different rocks and fluids

• Relative permeability affects the flow

characteristics of reservoir fluids.

• Relative permeability affects the

recovery efficiency of oil and/or gas.

Modified from NExT, 1999

Applications of

Relative Permeability Functions

• Reservoir simulation

• Flow calculations that involve

multi-phase flow in reservoirs

• Estimation of residual oil (and/or

gas) saturation

Hysteresis Effect on Rel. Perm.

0

20

40

60

80

100

0 20 40 60 80 100

Drainage

Imbibition

krnw

Wetting Phase Saturation, %PV

Rela

tive P

erm

eab

ilit

y,

%

Residual non-wetting

phase saturation

Irreducible wetting phase saturation

Non-wetting

phaseWetting

phase

krnw krw

What is kbase for this case?

Hysteresis Effect on Rel. Perm.

• During drainage, the wetting phase ceases to flow at the

irreducible wetting phase saturation

– This determines the maximum possible non-wetting

phase saturation

– Common Examples:

• Petroleum accumulation (secondary migration)

• Formation of secondary gas cap

• During imbibition, the non-wetting phase becomes

discontinuous and ceases to flow when the non-wetting

phase saturation reaches the residual non-wetting phase

saturation

– This determines the minimum possible non-wetting

phase saturation displacement by the wetting phase

– Common Example: waterflooding water wet reservoir

• Oil

• Water

• Gas

L

Akq

o

ooo

L

Akq

w

www

L

Akq

g

gg

g

Review: Effective Permeability

Steady state, 1D, linear flow

equation (Darcy units):

qn = volumetric flow rate for a

specific phase, n

A = flow area

n = flow potential drop for

phase, n (including pressure,

gravity and capillary pressure

terms)

n = fluid viscosity for phase n

L = flow length

Modified from NExT, 1999; Amyx, Bass, and Whiting, 1960; PETE 311 NOTES

• The pressure difference existing across

the interface separating two immiscible

fluids in capillaries (e.g. porous media).

• Calculated as:

Pc = pnwt - pwt

CAPILLARY PRESSURE

- DEFINITION -

Where:

Pc = capillary pressure

Pnwt = pressure in nonwetting phase

pwt = pressure in wetting phase

• One fluid wets the surfaces of the formation

rock (wetting phase) in preference to the other

(non-wetting phase).

• Gas is always the non-wetting phase in both

oil-gas and water-gas systems.

• Oil is often the non-wetting phase in water-oil

systems.

Capillary Tube - Conceptual Model

Air-Water System

Water

Airh

• Considering the porous media as a collection of capillary tubes provides useful

insights into how fluids behave in the reservoir pore spaces.

• Water rises in a capillary tube placed in a beaker of water, similar to water (the

wetting phase) filling small pores leaving larger pores to non-wetting phases of

reservoir rock.

• The height of water in a capillary tube is a function of:

– Adhesion tension between the air and water

– Radius of the tube

– Density difference between fluidsaw

aw

grh

cos2

CAPILLARY TUBE MODEL

AIR / WATER SYSTEM

This relation can be derived from balancing the upward force due to adhesion tension and downward forces due to the weight of the fluid (see ABW pg 135). The wetting phase (water) rise will be larger in small capillaries.

h = Height of water rise in capillary tube, cm

aw = Interfacial tension between air and water,dynes/cm

= Air/water contact angle, degrees

r = Radius of capillary tube, cm

g = Acceleration due to gravity, 980 cm/sec2

aw = Density difference between water and air, gm/cm3

Contact angle, , is measured through the more dense phase (water in this case).

• Combining the two relations results in the following

expression for capillary tubes:

rP aw

c

cos2

CAPILLARY PRESSURE – AIR / WATER

SYSTEM

CAPILLARY PRESSURE – OIL / WATER

SYSTEM

• From a similar derivation, the equation for

capillary pressure for an oil/water system is

rP ow

c

cos2

Pc = Capillary pressure between oil and water

ow = Interfacial tension between oil and water, dyne/cm

= Oil/water contact angle, degrees

r = Radius of capillary tube, cm

Capillary Pressure Curve

Drainage Curve

Effect of Fluids

J-Function Different Rocks

J-Function Carbonates

OWC and FWL

DRAINAGE AND IMBIBITION

CAPILLARY PRESSURE CURVES

Drainage

Imbibition

Swi Sm

Sw

Pd

Pc

0 0.5 1.0

Modified from NExT, 1999, after …

DRAINAGE

• Fluid flow process in which the saturation

of the nonwetting phase increases

IMBIBITION

• Fluid flow process in which the saturation

of the wetting phase increases

Saturation History - Hysteresis

- Capillary pressure depends on both direction

of change, and previous saturation history

- Blue arrow indicates probable path from

drainage curve to imbibition curve at Swt=0.4

- At Sm, nonwetting phase cannot flow,

resulting in residual nonwetting phase

saturation (imbibition)

- At Swi, wetting phase cannot flow, resulting in

irreducible wetting phase saturation (drainage)

Effect of Permeability on Shape

Decreasing

Permeability,

Decreasing

A B

C

20

16

12

8

4

00 0.2 0.4 0.6 0.8 1.0

Water Saturation

Cap

illa

ry P

ressu

re

Modified from NExT 1999, after xx)

Effect of Grain Size Distribution on Shape

Well-sortedPoorly sorted

Ca

pilla

ry p

res

su

re, p

sia

Water saturation, %Modfied from NExT, 1999; after …)

Decreasing

Rise of Wetting Phase Varies with

Capillary Radius

WATER

AIR

1 2 3 4

Ayers, 2001

CAPILLARY TUBE MODEL

AIR/WATER SYSTEM

Air

Water

pa2

h

pa1

pw1

pw2

Water rise in capillary tube depends on the density difference of fluids.

Pa2 = pw2 = p2

pa1 = p2 - a g h

pw1 = p2 - w g h

Pc = pa1 - pw1

= w g h - a g h

= g h

• Combining the two relations results in the following

expression for capillary tubes:

rP aw

c

cos2

CAPILLARY PRESSURE – AIR / WATER

SYSTEM

CAPILLARY PRESSURE – OIL / WATER

SYSTEM

• From a similar derivation, the equation for

capillary pressure for an oil/water system is

rP ow

c

cos2

Pc = Capillary pressure between oil and water

ow = Interfacial tension between oil and water, dyne/cm

= Oil/water contact angle, degrees

r = Radius of capillary tube, cm

AVERAGING CAPILLARY PRESSURE

DATA USING THE LEVERETT

J-FUNCTION

• The Leverett J-function was originally an attempt

to convert all capillary pressure data to a

universal curve

• A universal capillary pressure curve does not

exist because the rock properties affecting

capillary pressures in reservoir have extreme

variation with lithology (rock type)

• BUT, Leverett‟s J-function has proven valuable

for correlating capillary pressure data within a

lithology (see ABW Fig 3-23).

EXAMPLE J-FUNCTION FOR

WEST TEXAS CARBONATE

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-fu

nction

Jc

Jmatch

Jn1

Jn2

Jn3

DEFINITION OF LEVERETT J-FUNCTION

( )f

kC PcSJ wcos

• J-Function is DIMENSIONLESS, for a particular rock type:

• Same value of J at same wetting phase saturation for

any unit system, any two fluids, any exact value of k,f

•(k/f)1/2 is proportional to size of typical pore throat

radius (remember k can have units of length2)

•C is unit conversion factor (to make J(Sw) dimensionless)

FlowUnits

Gamma RayLog

PetrophysicalData

PoreTypes

LithofaciesCore

1

2

3

4

5

CorePlugs

CapillaryPressure

f vs k

Pc(Sw) Depends on k,f

High Quality

Low Quality

Function moves up

and right, and

becomes less “L”

shaped as reservoir

quality decreases

LEVERETT J-FUNCTION FOR

CONVERSION OF Pc DATA

Reservoir

c

Lab

cw

k

cosθσ

PCk

cosθσ

PC)J(S

ff

• J-function is useful for averaging capillary pressure data from a given rock type from a given reservoir

• J-function can sometimes be extended to different reservoirs having same lithologies

– Use extreme caution in assuming this can be done

• J-function usually not accurate correlation for different lithologies

• If J-functions are not successful in reducing the scatter in a given set of data, then this suggests that we are dealing variation in rock type

USE OF LEVERETT J-FUNCTION

Capillary Pressure in Reservoirs

A B

Reservoir, o

Aquifer, w

1

2

3

Pc = po-pw = 0

PressureD

ep

th

dpw=wg/D dh

Free

Water

Level

dpo=og/D dh

Fluid Distribution in Reservoirs

Gas & Water

Gas density = g

Oil, Gas & Water

Oil & Water

Oil density = o

Water

Water density = w

„A‟

h1

h2

„B‟

Free Oil Level

Free Water Level

Capillary pressure difference

between

oil and water phases in core „A‟

Pc,ow = h1g (w-o)

Capillary pressure difference

between

gas and oil phases in core „B‟

Pc,go = h2g (o-g)

Modified from NExT, 1999, modified after Welge and Bruce, 1947

Fau

lt

RELATION BETWEEN CAPILLARY

PRESSURE AND FLUID SATURATION

Free Water Level

Pc

Pd

Oil-Water contactHd

Heig

ht

Ab

ove

Fre

e W

ate

r L

eve

l (

Fee

t)

0 50 100Sw (fraction)

0 50 100

Sw (fraction)

0

Modified from NExT, 1999, after …

Pc

0 50 1000

Pd

Sw (fraction)

Lab Data

-Lab Fluids: ,

-Core sample: k,f

J-Function

J-Function

- for k,f

Reservoir Data

Saturation in

Reservoir vs.

Depth• Results from two

analysis methods (after

ABW)

– Laboratory capillary

pressure curve

• Converted to

reservoir

conditions

– Analysis of well

logs

• Water saturation

has strong effect

• Determine fluid distribution in reservoir (initial conditions)

• Accumulation of HC is drainage process for water wet res.

• Sw= function of height above OWC (oil water contact)

• Determine recoverable oil for water flooding applications

• Imbibition process for water wet reservoirs

• Pore Size Distribution Index,

• Absolute permeability (flow capacity of entire pore size

distribution)

• Relative permeability (distribution of fluid phases within the

pore size distribution)

• Reservoir Flow - Capillary Pressure included as a term of flow

potential for multiphase flow

• Input data for reservoir simulation models

Applications of Capillary

Pressure Data

water wet,Z;PD

ZgρpΦ owc,

wow

DRAINAGE AND IMBIBITION

CAPILLARY PRESSURE CURVES

Drainage

ImbibitionS

iSm

Swt

Pd

Pc

0 0.5 1.0

Modified from NExT, 1999, after …

DRAINAGE

• Fluid flow process in which the saturation

of the nonwetting phase increases

• Mobility of nonwetting fluid phase

increases as nonwetting phase saturation

increases

IMBIBITION

• Fluid flow process in which the saturation

of the wetting phase increases

• Mobility of wetting phase increases as

wetting phase saturation increases

Four Primary Parameters

Si = irreducible wetting phase saturation

Sm = 1 - residual non-wetting phase saturation

Pd = displacement pressure, the pressure

required to force non-wetting fluid into largest

pores

= pore size distribution index; determines

shape

DRAINAGE PROCESS

• Fluid flow process in which the saturation of the nonwettingphase increases

• Examples:

• Hydrocarbon (oil or gas) filling the pore space anddisplacing the original water of deposition in water-wet rock

• Waterflooding an oil reservoir in which the reservoir is oilwet

• Gas injection in an oil or water wet oil reservoir

• Pressure maintenance or gas cycling by gas injection in aretrograde condensate reservoir

• Evolution of a secondary gas cap as reservoir pressuredecreases

IMBIBITION PROCESS

IMBIBITION

•Fluid flow process in which the

saturation of the wetting phase increases

•Mobility of wetting phase increases as

wetting phase saturation increases

Examples:

Accumulation of oil in an oil wet reservoir

Waterflooding an oil reservoir in which the reservoir is

water wet

Accumulation of condensate as pressure decreases in

a dew point reservoir

Sw* Power Law Model• Having an equation model for capillary pressure

curves is useful

– Smoothing of laboratory data

– Determination of

– Analytic function for integration (future topic)

• The Sw* Power Law Model is an empirical model that

has proven to work well

– Model parameters: Swi, Pd,

( ) 1/λ*

wdc SPP

wi

wiw*

wS1

SSS

• Sw* rescales x-axis

Sw* Power Law Model

Sw*, fractionSw*=0 Sw*=1

Capillary Pressure vs. Wetting Phase Saturation

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw, fraction

Pc,

psia

Swi=0.20

Pd=3.0

• Power Law Equations plot as Log-Log straight line

Sw* Power Law Model

Capillary Pressure Data Plotted vs. Sw* (Swi=0.20)

1

10

100

0.01 0.1 1

Sw*, fraction

Pc,

psia

slope = -1/ = -1/2.0

Pd=3.0

Sw* Power Law Model• Straight line models are excellent for

– Interpolation and data smoothing

– Extrapolation

– Self Study: review Power Law Equations (y=axb)

and how to determine coefficients, a and b given

two points on the straight log-log line

( ) 1/λ*

wdc SPP

wi

wiw*

wS1

SSS

Sw* Power Law Model

• Pd, can be determined from Log-Log plot

• But, Swi can be difficult to determine from Cartesian plot, if data

does not clearly show vertical asymptote

Capillary Pressure vs. Wetting Phase Saturation

0

5

10

15

20

25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Sw, fraction

Pc,

psia

• Choosing wrong Swi limits accuracy of determining

Pd,

Sw* Power Law Model

Same Capillary Pressure Data Plotted vs. Sw*

1

10

100

0.01 0.1 1

Sw*, fraction

Pc,

psia

Swi value too small

Swi value too large

Swi value correct

Review: Sw* Power Law

Model

• Power Law Model (log-log straight line)

– “Best fit” of any data set with a straight line

model can be used to determine two unknown

parameters. For this case:

• slope gives

• intercept gives Pd

– Swi must be determined independently

• it can be difficult to estimate the value of Swi

from cartesian Pc vs. Sw plot, if the data set

( ) 1/λ*

wdc SPP

wi

wiw*

wS1

SSS