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Vwater ρ water − Vair ρ air R= = 0.9986 ≈ 1 Vwater ρ water + Vair ρ air
air water
S-wave transmission and reflection
Vs1,ρ1 Vs2,ρ2
polarization
medium 1 medium 2
The elastic moduli can be extracted from measurements of density and any two wave velocities
µ = ρVs2
4 2 K = ρ (V − Vs ) 3 V p2 − 2Vs2 σ= 2(V p2 − Vs2 )
2 p
σ is Possion’s ratio.
If Pore fluids (saturation,salinity), pressure, temperature change? Clay changes? Sorting changes? Cementation or compaction changes? Fractures appear? Fractures are filled fluids or shale? Sandstone, limestone (dolomite), volcanic rock?
Hilterman, F. J., 2001, Seismic amplitude interpretation, Seismic Amplitude Interpretation: Soc. of Expl. Geophys., 235.
Per Avseth, Tapan Mukerji, Gary Mavko, 2010, Quantitative Seismi Interpretation-Applying Rock Physics Tools to Reduce Interpretation Risk, Cambridge University Press, Cambridge.
- Louis Pasteur, 1822-1895
Different Authors⇒Different Emphasis
Bortfeld (1961,Geophysical Prospecting)
Zoeppritz Linear Approximation
sin θ1 2 2 ln( ρ 2 / ρ1 ) 1 α 2 ρ 2 cos θ1 2 R (θ ) = ln[ ]+ ( ) ( β1 − β 2 )(2 + ) 2 ln( β 2 / β1 ) α 1 ρ1 cos θ 2 α1
From Fred J.Hilterman,2001
AVO Intercept (I) and Gradient (G) crossplot
The intercept and gradient fit to the top and base reflections from a gas sand. (From Brian Russell, AAPG Explorer).
Aki & Richards (1980)
2 1 4β 2 ∆ ρ 1 ∆ α 4 β ∆β 2 2 R (θ ) = (1 − 2 sin φ ) + − 2 sin φ 2 2 α ρ 2 cos φ α α β
Where,
Δα= difference in P-wave velocity = α2-α1 Δβ= difference in S-wave velocity = β2-β1 Δρ= difference in density velocity = ρ2-ρ1
K dry K fl K sat = + , K m − K sat K m − K dry φ ( K m − K fl )
Where, Kdry is the effective bulk modulus of dry rock; Ksat is the effective bulk modulus of the rock with pore fluid; Km is the bulk modulus of mineral material making up rock; l is the effective bulk modulus of pore fluid Φ is the porosity µdry is the effective shear modulus of dry rock; µsat is the effective shear modulus of rock with pore fluid.
2 α 2 + α1 α= 2 β 2 + β1 β= 2
； ；
ρ=
ρ 2 + ρ1
安艺敬一
Shuey’s approximation (1985, Geophysics) which is most famous equation for AVO
R(θ ) = Rp + [ Rp A0 + ∆σ ∆σ 2 2 2 ]sin θ + (tan θ − sin θ) 2 (1 − σ ) 2α
Outline
• AVO theory • Rock physics theory • Homework Assignment #1
AVO-Amplitude Versus Offset (Angle)
Reflection coefficient 1-Transmission coefficient
AVO and Rock Physics Theory
•
Gary Mavko, Tapan Mukerji, Jack Dvorkin, 2009, The Rock Physics Handbook-Tools for Seismic Analysis of Porous Media, Cambridge University Press
R (θ ) = R p + G sin 2 θ
Where Rp and G are intercept and gradient.
Variation of a P-wave reflectivity with angle of incidence (Sheriff and Geldart)
No change in Poisson’s Decreasing Poisson’s ratio Increasing Poisson’s ratio at the interface (solid (solid curves, σ1=0.4, ratio (solid curves, curves, σ1=σ2=0.3; σ1=0.1,σ2=0.4;dashed, σ2=0.1; dashed, dashed, σ1=σ2=0.2); σ1=0.3,σ2=0.1); σ1=0.1,σ2=0.2);
From Fred J.Hilterman,2001
Rock Physics Theory
Seismic velocity
Vp = K + 4 / 3µ
ρ
λ + 2µ = ρ
µ Vs = ρ
Where Vp is the P-wave velocity Vs is the S-wave velocity ρ is density K is the bulk modulus μ is the shear modulus λ Lame’s coefficient.
Z=Vpρ= Acoustic Impedance
From Fred
Question
• Calculate the reflection coefficient for a wave that travels up and reflects at the free surface of the sea • ρair= 0.0013 kg/m3, ρwater= 1.025kg/m3 • Vair = 0.3 km/s, Vwater = 1.4 km/s
Gassmann’s Equation: isotropic form
(Gary Mavko,Tapan Mukerji and Jack Dvorkin,2009)
One of the most important problems in the rock physics analysis of logs, cores, and seismic data is using seismic velocities in rocks saturated with one fluid to predict those of rocks saturated with a second fluid, or equivalently, predicting saturated-rock velocities from dry rock velocities, and vice versa. This is fluid substitution problem. Generally, when a rock is loaded under an increment of compression, such as from a passing seismic wave, an increment of pore-pressure change is induced, which resists the compression and therefore stiffens the rock.
The low frequency Gassmann-Biot (Gassmann,1951; Biot, 1956) theory predicts the resulting increase in effective bulk modulus, Ksat, of saturated rock using the following equation:
Transmission and reflection of a wave normally incident to an impedance contrast. The wave splits into a transmitted and a reflected wave. The reflection coefficient depends on both the acoustic impedance contrast and average about the interface, according to the formulas above.
σ = (σ 1 + σ 2 ) / 2 Intercept ∆σ = σ − σ 2 1
1 − 2σ A0 = B − 2(1 + B ) 1−σ ∆α / α B= ∆ α / α + ∆ρ / ρ
Gradient
1 ∆V p ∆ρ ) Rp ≈ ( + 2 Vp ρ
Shuey’s approximation can be further expressed by following linear equation (θ<300) :
Vs2 ρ 2 − Vs1ρ1 reflection = Vs2 ρ 2 + Vs1ρ1
2Vs1ρ1 transmissi on = Vs2 ρ 2 + Vs1ρ1
In the fields of observation, chance favors only the prepared mind.
or curves 1, Vp2/Vp1= ρ2/ρ1=1.25; For curves 2, Vp2/Vp1= ρ2/ρ1=1.11; or curves 3, Vp2/Vp1= ρ2/ρ1=1.00; For curves 4, Vp2/Vp1= ρ2/ρ1=0.90; or curves 5, Vp2/Vp1= ρ2/ρ1=0.80. (from Ostrander,1984)
air water
S-wave transmission and reflection
Vs1,ρ1 Vs2,ρ2
polarization
medium 1 medium 2
The elastic moduli can be extracted from measurements of density and any two wave velocities
µ = ρVs2
4 2 K = ρ (V − Vs ) 3 V p2 − 2Vs2 σ= 2(V p2 − Vs2 )
2 p
σ is Possion’s ratio.
If Pore fluids (saturation,salinity), pressure, temperature change? Clay changes? Sorting changes? Cementation or compaction changes? Fractures appear? Fractures are filled fluids or shale? Sandstone, limestone (dolomite), volcanic rock?
Hilterman, F. J., 2001, Seismic amplitude interpretation, Seismic Amplitude Interpretation: Soc. of Expl. Geophys., 235.
Per Avseth, Tapan Mukerji, Gary Mavko, 2010, Quantitative Seismi Interpretation-Applying Rock Physics Tools to Reduce Interpretation Risk, Cambridge University Press, Cambridge.
- Louis Pasteur, 1822-1895
Different Authors⇒Different Emphasis
Bortfeld (1961,Geophysical Prospecting)
Zoeppritz Linear Approximation
sin θ1 2 2 ln( ρ 2 / ρ1 ) 1 α 2 ρ 2 cos θ1 2 R (θ ) = ln[ ]+ ( ) ( β1 − β 2 )(2 + ) 2 ln( β 2 / β1 ) α 1 ρ1 cos θ 2 α1
From Fred J.Hilterman,2001
AVO Intercept (I) and Gradient (G) crossplot
The intercept and gradient fit to the top and base reflections from a gas sand. (From Brian Russell, AAPG Explorer).
Aki & Richards (1980)
2 1 4β 2 ∆ ρ 1 ∆ α 4 β ∆β 2 2 R (θ ) = (1 − 2 sin φ ) + − 2 sin φ 2 2 α ρ 2 cos φ α α β
Where,
Δα= difference in P-wave velocity = α2-α1 Δβ= difference in S-wave velocity = β2-β1 Δρ= difference in density velocity = ρ2-ρ1
K dry K fl K sat = + , K m − K sat K m − K dry φ ( K m − K fl )
Where, Kdry is the effective bulk modulus of dry rock; Ksat is the effective bulk modulus of the rock with pore fluid; Km is the bulk modulus of mineral material making up rock; l is the effective bulk modulus of pore fluid Φ is the porosity µdry is the effective shear modulus of dry rock; µsat is the effective shear modulus of rock with pore fluid.
2 α 2 + α1 α= 2 β 2 + β1 β= 2
； ；
ρ=
ρ 2 + ρ1
安艺敬一
Shuey’s approximation (1985, Geophysics) which is most famous equation for AVO
R(θ ) = Rp + [ Rp A0 + ∆σ ∆σ 2 2 2 ]sin θ + (tan θ − sin θ) 2 (1 − σ ) 2α
Outline
• AVO theory • Rock physics theory • Homework Assignment #1
AVO-Amplitude Versus Offset (Angle)
Reflection coefficient 1-Transmission coefficient
AVO and Rock Physics Theory
•
Gary Mavko, Tapan Mukerji, Jack Dvorkin, 2009, The Rock Physics Handbook-Tools for Seismic Analysis of Porous Media, Cambridge University Press
R (θ ) = R p + G sin 2 θ
Where Rp and G are intercept and gradient.
Variation of a P-wave reflectivity with angle of incidence (Sheriff and Geldart)
No change in Poisson’s Decreasing Poisson’s ratio Increasing Poisson’s ratio at the interface (solid (solid curves, σ1=0.4, ratio (solid curves, curves, σ1=σ2=0.3; σ1=0.1,σ2=0.4;dashed, σ2=0.1; dashed, dashed, σ1=σ2=0.2); σ1=0.3,σ2=0.1); σ1=0.1,σ2=0.2);
From Fred J.Hilterman,2001
Rock Physics Theory
Seismic velocity
Vp = K + 4 / 3µ
ρ
λ + 2µ = ρ
µ Vs = ρ
Where Vp is the P-wave velocity Vs is the S-wave velocity ρ is density K is the bulk modulus μ is the shear modulus λ Lame’s coefficient.
Z=Vpρ= Acoustic Impedance
From Fred
Question
• Calculate the reflection coefficient for a wave that travels up and reflects at the free surface of the sea • ρair= 0.0013 kg/m3, ρwater= 1.025kg/m3 • Vair = 0.3 km/s, Vwater = 1.4 km/s
Gassmann’s Equation: isotropic form
(Gary Mavko,Tapan Mukerji and Jack Dvorkin,2009)
One of the most important problems in the rock physics analysis of logs, cores, and seismic data is using seismic velocities in rocks saturated with one fluid to predict those of rocks saturated with a second fluid, or equivalently, predicting saturated-rock velocities from dry rock velocities, and vice versa. This is fluid substitution problem. Generally, when a rock is loaded under an increment of compression, such as from a passing seismic wave, an increment of pore-pressure change is induced, which resists the compression and therefore stiffens the rock.
The low frequency Gassmann-Biot (Gassmann,1951; Biot, 1956) theory predicts the resulting increase in effective bulk modulus, Ksat, of saturated rock using the following equation:
Transmission and reflection of a wave normally incident to an impedance contrast. The wave splits into a transmitted and a reflected wave. The reflection coefficient depends on both the acoustic impedance contrast and average about the interface, according to the formulas above.
σ = (σ 1 + σ 2 ) / 2 Intercept ∆σ = σ − σ 2 1
1 − 2σ A0 = B − 2(1 + B ) 1−σ ∆α / α B= ∆ α / α + ∆ρ / ρ
Gradient
1 ∆V p ∆ρ ) Rp ≈ ( + 2 Vp ρ
Shuey’s approximation can be further expressed by following linear equation (θ<300) :
Vs2 ρ 2 − Vs1ρ1 reflection = Vs2 ρ 2 + Vs1ρ1
2Vs1ρ1 transmissi on = Vs2 ρ 2 + Vs1ρ1
In the fields of observation, chance favors only the prepared mind.
or curves 1, Vp2/Vp1= ρ2/ρ1=1.25; For curves 2, Vp2/Vp1= ρ2/ρ1=1.11; or curves 3, Vp2/Vp1= ρ2/ρ1=1.00; For curves 4, Vp2/Vp1= ρ2/ρ1=0.90; or curves 5, Vp2/Vp1= ρ2/ρ1=0.80. (from Ostrander,1984)