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1
Pore Fluid and Elastic Properties of Rock
GP170/2001 #2
Change in Elastic Properties -- Han's Data
8
10
12
14
8 10 12 14
Sat
ura
ted-
Roc
k P-
Impe
dan
ce
Dry-Rock P-Impedance
HAN 40 MPa
WATER
OIL
OIL:K = 0.5 GPaRHO = 0.8 g/ccWATER:K = 2.5 GPaRHO = 1 g/cc
.05
.10
.15
.20
.25
.30
6 8 10 12 14
Pois
son
's R
atio
P-Impedance
HAN 40 MPa
WATER
OILDRY
Change in Elastic Properties -- Soft Sand
0.2
0.3
0.4
4 6 8
Poi
sson
's R
ati
o
P-Impedance
UpperShale
Reservoirw/Water
Reservoirw/Hydrocarbons
2
Physics of Pore Fluid Effect on Elastic Properties
GP170/2001 #2
In static (low-frequency) approximation, pore fluid interactswith rock through pore pressure
σ ij = λδ ijεαα + 2Gε ij ⇒ ε ij = [(1 + ν )σ ij − νδ ijσαα ] / E ⇒
ε ij =1
2G(σ ij −
1
3δ ijσαα ) +
1
9Kδ ijσαα . K: Bulk Modulus; G: Shear Modulus
Hooke’s Law of Linear Isotropic Elasticity(Compression Corresponds to Positive Stress and Strain)
Adding Pore Pressure: Pore pressure only affects volumetric deformation
ε ij =
1
2G(σ ij −
1
3δ ijσαα ) +
1
9Kδ ijσαα −
1
3HδijPp
Volumetric Deformation (Hydrostatic)
Pp
PcPore
Pressure
ConfiningPressure
θ ≡ εαα ≡ ε11 + ε22 + ε33 = Pc / K − Pp / H =
(Pc − αPp ) / K
Pc = σ11 = σ22 = σ33; α = K / H.
Effective Pressure and Stress
Pe =
Def
Pc − αPp ; σ ije =
Def
σ ij − αδij Pp ⇒ ε ij = 1
2G(σ ij
e − 1
3δ ijσαα
e ) + 1
9 Kδ ijσαα
e ⇒ θ = Pe
K
α = 1 −KDry
KSolid
3
GP170/2001 #2
Physics of Pore Fluid Effect on Elastic Properties
In static (low-frequency) approximation, pore fluid affects only the bulk modulus of rock
Gassmann's Equation -- Basis of Fluid Substitution
KSat
Ks − KSat
=KDry
Ks − KDry
+K f
φ (Ks − K f )
Bulk Modulus ofRock w/Fluid
Bulk Modulus ofMineral Phase
Bulk Modulus ofDry Rock
Bulk Modulus ofPore Fluid
Porosity
GSat = GDryShear Modulus of
Rock w/FluidShear Modulus of
Dry Rock
KSat = K s
φKDry − (1 + φ )K f KDry / Ks + K f
(1 − φ )K f + φKs − K f KDry / Ks
KDry = K s
1 − (1 − φ )KSat / K s − φKSat / K f
1 + φ − φK s / K f − KSat / Ks
Vp = (K Sat + 4
3GDry ) / ρSat
V s = GDry / ρSat
ρSat = ρDry + φ ρFluid > ρDry
4
GP170/2001 #2
Fluid Effect on Velocity, Impedance, and Modulus
3
4
5
3 4 5
Satu
rate
d-R
ock V
p (km
/s)
Dry-Rock Vp (km/s)
HAN 40 MPa
WATER
OIL
OIL:K = 0.5 GPaRHO = 0.8 g/ccWATER:K = 2.5 GPaRHO = 1 g/cc
3.5
4.0
4.5
3.5 4.0 4.5
Satu
rate
d-R
ock V
p (km
/s)
Dry-Rock Vp (km/s)
HAN 40 MPa
WATER
OIL
OIL:K = 0.5 GPaRHO = 0.8 g/ccWATER:K = 2.5 GPaRHO = 1 g/cc
2
3
2 3
Satu
rate
d-R
ock V
s (k
m/s)
Dry-Rock Vs (km/s)
HAN 40 MPa
WATER
OIL
OIL:K = 0.5 GPaRHO = 0.8 g/ccWATER:K = 2.5 GPaRHO = 1 g/cc
1.5
1.6
1.7
1.8
1.5 1.6 1.7 1.8
Satu
rate
d-R
ock V
p/V
s
Dry-Rock Vp/Vs
HAN 40 MPa
WATER
OIL
OIL:K = 0.5 GPaRHO = 0.8 g/ccWATER:K = 2.5 GPaRHO = 1 g/cc
Vp = (K Sat + 4
3GDry )/ ρSat
Vs = GDry / ρSat
ρSat = ρDry + φρ Fluid > ρDry
Han's Laboratory Data
North Sea Log Data
2.4
2.6
2.8
3.0
3.2
3.4
TopPayBottom
Vp (km
/s)
Fluid-Substituted
6
7
P-I
mped
an
ce
15
20
25
0.1 0.2 0.3
M-M
odu
lus
(GPa)
Density-Porosity0.1 0.2 0.3
Density-Porosity
Fluid-Substituted
5
GP170/2001 #2
Approximate Fluid Substitution Equations -- Vp Only
M is the compressional modulus
M = ρbV p2 = K +
4
3G
MSat = M s
φMDry − (1 + φ )K f MDry / Ms + K f
(1 − φ )K f + φMs − K f MDry / Ms
MDry = Ms
1 − (1 − φ )MSat / M s − φMSat / K f
1 + φ − φMs / K f − MSat / Ms
A soft sand sample of 35% porosityThe dry-rock density is 1.722 g/cm3
Fluid bulk modulus 2.5 GPa; density 1 g/cm3
1.2
1.4
1.6
1.8
2.0
2.2
10 20 30
Vel
ocit
y (k
m/s)
Pressure (MPa)
Dry Sandstone35% Porosity
Vp
Vs
2
3
4
5
6
7
8
10 20 30
Ela
stic
Mod
uli (G
Pa)
Pressure (MPa)
Dry Sandstone35% Porosity
M-Modulus
K
G
Dry-rock lab data -- velocity versus pressure
1.8
2.0
2.2
2.4
2.6
10 20 30
Vp (km
/s)
Pressure (MPa)
Vp-Only
Dry
Gassmann
100% Quartz
10 20 30Pressure (MPa)
Vp-Only
Dry
Gassmann
70% Quartz30% Clay
Water-Substituted
Solid:(a) Pure quartz -- K = 36.6 GPa; G = 45 GPa(b) 70% quartz + 30% clay -- K = 30 GPa; G = 25.5 GPa
EXAMPLE
6
GP170/2001 #2
Partial Saturation -- Fluid's Bulk Modulus
SOLID
WATER
GAS
For any number N of fluid phases of saturation Si , the effective bulk modulus is the Reuss low bound of their bulk
moduli:
1
K f
= Si
Kii=1
N
∑ .
0 0.2 0.4 0.6 0.8 10
1
2
Sw
Mix
ture
Bu
lk M
odu
lus
(GPa)
Water 2.25 GPaGas 0.005 GPa
0 0.2 0.4 0.6 0.8 1
1.0
1.5
2.0
Sw
Vel
ocit
y (k
m/s)
OTTAWA SAND
Water 2.25 GPaGas 0.005 GPa
Vp
Vs
If pore pressure increment is ∆P then the volume change of water is − VSw∆P/ Kw , and the volume change of
gas is − V(1 − Sw )∆P/ Kg , where Kw and Kg are the bulk moduli of water and gas, respectively.
The total change of volume is then ∆V = −[VSw∆P / Kw + V(1 − Sw )∆P/ K g ]. The bulk modulus of the
water-gas mixture K f can be now calculated from this total volume change and pressure increment:
∆V
V= −
∆P
K f
⇒1
K f
=Sw
Kw
+1 − Sw
Kg
.
Typical Gassmann Effect -- Partial Gas Saturation
7
GP170/2001 #2
Partial Saturation -- Various Results of Fluid Substitution
2.5
3.0
3.5
4.0
4.5
5.0
0 0.5 1
Vp (km
/s)
Sw
Water/Gas25% Porosity
2.0
2.5
3.0
3.5
4.0
4.5
5.0
0 0.5 1
Vp (km
/s)
Sw
Water/Oil25% Porosity
0.20
0.25
0.30
0 0.5 1Poi
sson
's R
ati
op
Sw
Water/Gas25% Porosity
Sof
ten
ing
Roc
k
Velocity Poisson's Ratio
8
GP170/2001 #2
Recovery Monitoring Principles
0 10 20 301
2
3
Diff. Pressure (MPa)
Vp
Vs
Vel
ocit
y (k
m/s)
0 10 20 30Pore Pressure (MPa)
Vp
Vs
2.0
2.5
5 10 15 20 25 30
Vp (km
/s)
Effective Pressure (MPa)
GAS
OIL
BRINE
Phi = 0.35
5 10 15 20 25 30Pore Pressure (MPa)
GAS
OIL
BRINE
Phi = 0.35
5 10 15 20 25 30
2.0
2.5
Pore Pressure (MPa)
GAS
OIL
BRINE
Vp (km
/s)
WaterFlood
GasInjection
Gas out ofSolution
Such diagrams can be used to produce syntheticwell logs, based on production scenarios, and thenproduce synthetic seismic.
0
.1
.2
.3
.4
3 4 5
Poi
sson
's R
ati
o
P-Impedance (km/s g/cc)
BRINE
PorePressure
PorePressure
GAS
OIL
PorePressure
NORTH SEASAND
9
0.25 0.3 0.35Poisson's Ratio
Nu 3.1
6 7Ip
S_26
Ip 3.1
0.1 0.2 0.3Porosity
S_26
Core
0 0.5 1SwT
1 10 100Resistivity
GP170/2001 #2
Forward Modeling of Saturation -- Well Logs
50 100 150 200
2.2
2.3
2.4
2.5
GR
TV
D (km
)
.25
.30
.35
5 6 7 8 9
Poi
sson
's R
ati
o
P-Impedance
PayOil
PayBrine
Bottom
Top
10
GP170/2001 #2
Effect of Saturation and Tool -- Heavy Oil
40 60 80 100 120GR
Dep
th (ft
)
200 f
t
20 40 60 80 100Resistivity
2.0 2.1 2.2 2.3RHOB
0.2 0.3 0.4 0.5Porosity
NPHI
PhiRHO
1.8 2.0 2.2 2.4 2.6Vp (km/s)
Mono
Dipole
0.6 0.8 1Vs (km/s)
0.40 0.45Poisson's Ratio
1.8
1.9
2.0
2.1
2.2
2.3
0.30 0.35 0.40
Vp (km
/s)
Porosity
Monopole
Dipole
11
GP170/2001 #2
Pore Fluid and Frequency -- Velocity/Frequency Dispersion
2.5
3.0
3.5
4.0
4.5
0 20 40 60 80 100
Vp (km
/s)
Differential Pressure (MPa)
DRY
SATURATED
GASSMANN
1.8
2.0
2.2
2.4
2.6
2.8
3.0
0 20 40 60 80 100
Vs
(km
/s)
Differential Pressure (MPa)
SQUIRTFLOW
2000
3000
4000
5000
4 5 6
Vp
Vs
Vp a
nd V
s (m
/s)
Log Frequency (Hz)
Limestone5MPa Pressure
0
0.1
0.2
4 5 6
P
S
1/Q
(In
vers
e Q
uality
Fact
or)
Log Frequency (Hz)
Limestone5MPa Pressure
LAB MEASUREMENTS HAVE TO BE CONDUCTED ON ROOM-DRY SAMPLES
4400
4500
4600
4700
0 1 2 3 4 5 6
Vp (m
/s)
Log Frequency (Hz)
1 cPs
100 cPs
10000 cPs
Effect of Viscosity
Dispersion may be seenin heavy oil rock
Squirting flow between saturated pore space and a gas pocket
Squirting flow between soft and stiff pore space
Soft thin fracture
2.8
2.9
3.0
3.1
0 0.2 0.4 0.6 0.8 1
Vp (km
/s)
Saturation
LIMESTONE
1 kHz
50 kHz
Partial Saturation
12
GP170/2001 #2
Saturation and Poisson's Ratio
High-porosity sands -- lab room-dry data
2
3
4
0.2 0.3 0.4
Vp (km
/s)
Porosity
Rock w/GAS
FAST SS
SLOW SS
1.0
1.5
2.0
2.5
0.2 0.3 0.4
Vs
(km
/s)
Porosity
Rock w/GAS
FAST SS
SLOW SS
0
.1
.2
.3
.4
0.2 0.3 0.4
Poi
sson
's R
ati
o
Porosity
Rock w/GAS
FAST
SLOW
0
.1
.2
.3
.4
0.2 0.3 0.4
Poi
sson
's R
ati
o
Porosity
Fast w/Water
Slow w/Water
Poisson's ratio -- fluid substitution
0
.1
.2
.3
.4
3 4 5 6 7 8
Poi
sson
's R
ati
o
P-Impedance
Fast w/GAS
Slow w/Water
Fast w/Water
Slow w/GAS
Seismic detectioncrossplot
13
Forward model
GP170/2001 #2
Using Offset to Differentiate Sand Type
5
6
7
0.2 0.3 0.4
Ip
Porosity
Fast w/GAS
Slow w/Water
.1
.2
.3
.4
0.2 0.3 0.4
Poi
sson
's R
ati
o
Porosity
Fast w/GAS
Slow w/Water
Soft Water Sand:PHI = 0.28;Vp = 2.81 km/s;Poisson's Ratio = 0.278;RHOB = 2.19
Fast Gas Sand:PHI = 0.276;Vp = 3.22 km/s;Poisson's Ratio = 0.127;RHOB = 1.92
SHALE: Vp = 3 km/s; Poisson's Ratio = 0.35; RHOB = 2.3
Θ Forward model
-0.2
-0.1
0
0 10 20 30 40
Rpp
Angle of Incidence
Fast w/GAS
Slow w/Water
14
PATCHY SATURATION CONCEPT
D
GASLIQUID
Slight Shale ContentVariation
2.6
2.7
2.8
2.9
3.0
3.1
3.2
0 0.2 0.4 0.6 0.8 1
Vp (km
/s)
Water Saturation
Drainage
Imbibition
LIMESTONEFREQUENCY 1 kHz
0 0.2 0.4 0.6 0.8 1Water Saturation
50 kHz
0 0.2 0.4 0.6 0.8 1Water Saturation
100 kHz
Rock w/Liquid
Rock w/Gas
Low Frequency: Easy Cross-FlowHomogeneous Saturation
Rock w/Gas
High Frequency: No Cross-FlowPatchy Saturation
Rock w/Liquid
GP170/2001 #2
15
0
0.1
0.2
0.3
0.4
0 0.1 0.2 0.3 0.4
HanJizba
Poi
sson
's R
ati
o
Porosity
Dry Rockat 20 MPa
0 0.1 0.2 0.3 0.4
StrandenesBlangy
Porosity
Ottawa+Clay
Dry Rockat 20 MPa
1
2
3
4
5
6
0 0.1 0.2 0.3 0.4
Vp (km
/s)
Porosity
Dry Rockat 20 MPa
0 0.1 0.2 0.3 0.4Porosity
Dry Rockat 20 MPa
GP170/2001 #2
Patchy Saturation and Elastic Properties in Well Logs
1.5
2.0
0 0.5 1
Vp (km
/s)
Brine Saturation
Patchy
Homogeneous
OTTAWA SAND
0.1
0.2
0.3
0.4
0 0.5 1
Poi
sson
's R
ati
o
Brine Saturation
Patchy
Homogeneous
OTTAWA SAND
0 0.2Poisson's Ratio
Cut-Off
1 2Velocity (km/s)
VpVs
0 0.5 1Sw
0.2 0.4Porosity
0 0.2 0.4
1460
1480
1500
1520
VSHALE and Clay
ClayCore
Dep
th (m
)
Shalefrom GR