6.CapillaryPressure
-
Upload
nur-syaffiqa-mohamad-ruzlan -
Category
Documents
-
view
443 -
download
0
Transcript of 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