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