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The Magic of LaméBill Goodway
Location: University of Saskatchewan Student Geophysical Society Saskatoon.
Date: September 11th 2009
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Gabriel Lamé (1795-1870): French engineer, mathematician and elastician.Introduced λ and μ in 1828, named after himself, in a series of lectures titled:
“Mémoire sur l’éqiuilibre intérieur des corpssolides homogènes”
Lamé formulated the modern version of Hooke’s law relating stress to strain in general tensor form, creating the basis for the science of materials, including rocks.Interestingly and most notably, only Lamé’smoduli λ and μ appear in Hooke’s law and not Young’s modulus, the bulk modulus, or any other modulus or modulus ratio.
Who was Lamé and what is the physical significance of his parameters Lambda (λ) and Mu (μ) ?Who was Lamé and what is the physical significance of his parameters Lambda (λ) and Mu (μ) ?
Any apparent likeness to myself is purely coincidental!
Assertions
Lamé moduli of rigidity μ and incompressibility λ allow the fundamental parameterization of seismic waves used to extract information about rocks in the Earth.
These parameters link many fields of Earth Science at different scales, from Petroleum Exploration to Earthquake Seismology.
Other common formulations result in contradictions which are removed by restating equations using Lamé parameters.
Disclaimer for use of numerous equations that follow: “If geophysics requires mathematics for its treatment itis the Earth that is responsible not the geophysicist.”
from Sir Harold Jeffreys, University of Cambridge
OutlineP-wave, S-wave propagation, Hooke’s Law and Lamé moduli
Motivation from logs
AVO equations and methods to invert for Lamé moduli
Examples of AVO inversion for elastic parameters from 3D data
Rock Mechanical Properties and Closure Stress
Relating stress (fracture) intensity and orientation to Passive Microseismology and Earthquake Seismology
Conclusions
CompressionalCompressional VelocityVelocity(Sonic Log)(Sonic Log)
UndeformedUndeformed crosscross--sectionsectionof porous sandstoneof porous sandstone
UniUni--axial axial CompressionalCompressionalP Wave P Wave
Modes of Seismic Wave PropagationModes of Seismic Wave Propagation
Shear VelocityShear Velocity(Dipole Log)(Dipole Log)
Shear S Wave Shear S Wave shear
resistanceshea
r re
sist
ance com
pressive resistance
Note λ+2μ, the unnamed P-wave modulus is a direct consequence of bound rocks in the Earth
μμρρ
Vs = λ+2μλ+2μρρ
Vp =
Basic relations: Hooke’s law, moduli and wave equationsHooke’s law as formulated by Lamé:
ijvijij e2e μ+λδ=σ
for shear stress σxz in x direction on the z normal face (i ≠ j) so δij = 0:
xzxz e2μ=σ
Wave equations for P-wave and S-wave propagation “sense” attributes of the medium by equating Newton’s 2nd law (F = Ma) to Hooke’s law.
S-wave propagation of the shear strain exz term in Hooke’s law :
XZ2
2XZ
2
edted
∇μ=ρ
( ) V2
2V
2
e2dt
ed∇μ+λ=ρ
P-wave propagation of the volume strain eV term in Hooke’s law
for normal axial stress in z direction (i = j) so δij = 1:
( ) yyxxzzzz eee2 λ+λ+μ+λ=σ
Lamé parameters: Rigidity Mu (μ) and “Pure Incompressibility” Lambda (λ)
Common moduli resulting from medium’s measurement condition: “Compressional P-wave Modulus” M = λ + 2μ(Bound uni-axial compression)
Young’s Modulus E = μ(3λ+2μ)/(λ+μ) E = M – 2λν(Unbound uni-axial compression)
Bulk Modulus K = λ + (2/3)μ K = M – (4/3)μ
Poisson’s ratio ν = λ /(2 λ + 2μ)
Vp/Vs ratio √(2 + λ/μ)
A given material has various moduli that are purely a function of measurement conditionsLamé parameters λ and μ are invariant and form the basic elements within moduli, giving a simpler physical meaning
Static Moduli and Moduli Ratio Definitions in Lamé terms
Confusing implication of –ve Lambda from Bulk Modulus
μ+λ=Κ
⎟⎠⎞
⎜⎝⎛ μ+λ=−⇒μ+λ=−⇒
++=μ+λδ=σ=−
32 :modulusbulk herew
e32Pe)23(P3
directions stress z y, x,3 themsu
eeee :strain volume wheree2eP
VV
yyxxzzVijVijij σzz
σxx
σyy
Hydrostatic pressure P or tri-axial stress σ in Cartesian coordinates
Lower limits: as μ≥0 and Κ>0 gives a negative Lambda limit as λ> −2/3 μ
Kappa, Mu and Lambda Moduli vs Porosity for 100% BRINE saturation
0
5000
10000
15000
20000
25000
30000
35000
40000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4Porosity
Mod
uli [
MPA
]
Log Kappa: BrineLayer 2 Kappa: BrineLog MuKappa Sw 1.0MuLambda Sw 1.0
Kappa, Mu and Lambda Moduli vs Porosity for 100% GAS saturation
0
5000
10000
15000
20000
25000
30000
35000
40000
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Porosity
Mod
uli [
MPA
]
Log Kappa: GasLayer 2: GasLog MuKappa gasMuLambda Gas
Saturated moduli vs porosity: Gassmann fluid substitution, Lower Hashin-Shtrikman bound porosity
From theoretical rock physics based on log data Lambda is never negative (dry or wet)
P, S Impedance vs. LambdaRho, MuRho Worldwide Log Data Gas Sand discrimination Castagna mudrock Line and LambdaRho,MuRho threshold cutoff
Gas Sand Brine Sand or Shale
0
2
4
6
8
10
0 2 4 6 8 10 12 14P Impedance km/s.gm/cc
S Im
peda
nce
km/s
.gm
/cc
Vp/Vs = 1.5
Vp/Vs = 2
0
60
-10 0 10 20 30 40 50
MuR
hoG
.Pa.
gm/c
c
20
40
-ve Poisson’s+ve Bulk Mod
Vp/V
s =
1.5
Positive Poisson’s RatioPositive Bulk Modulus
+ve Poisson’s-ve Bulk Mod
Vp/Vs = 2
LambdaRho G.Pa.gm/cc
(Castagna & Smith 1994)
MuR
hoG
.Pa.
gm/c
c
wet sand cloud
In-situ wet sand log
lowlow
porosityporosityhighhigh
LambdaRhoLambdaRho G.Pa.gmG.Pa.gm/cc/cc
background shales
0
LambdaRho vs.MuRho cross-plots for in-situ wet and gas substituted logs
+ve κ, -ve or zero λ
Gas substituted log
gas sand cloud
MuR
hoG
.Pa.
gm/c
c
lowlow
porosityporosityhighhigh
LambdaRhoLambdaRho G.Pa.gmG.Pa.gm/cc/cc
background shales
0
Negative LambdaRhovalues reveal log errors
Histograms of Histograms of VpVp/Vs ratios/Vs ratios(wet sand zones) (wet sand zones)
1.51.5 2.02.01.751.75
LambdaRho vs.MuRho cross-plots for in-situ wet and gas substituted logs
comparable log
log for fluid substitution
Motivation from well logs
Ostracod shale has negative Lambda and Poisson ratio but positive Bulk modulusThis is an error and indicates a log problem
Alberta Cretaceous Clastics/Carbonate facies(Glauconite) Vp m/s Vs m/s
Rho gm/cc
P Impedance m/s. gm/cc
(Rayls)
S Impedance m/s. gm/cc
(Rayls)
LambdaRhoG.Pa. gm/cc
(Rayls2)
MuRhoG.Pa. gm/cc
(Rayls2)
Bulk modulus
G.Pa. Vp/Vs Poisson Lambda/Mu Ostracod Limestone 5464 2186 2.70 14753 5902 147.97 34.84 63.41 2.50 0.40 4.25
Ostracod Shale 2851 2115 2.30 6557 4865 -4.33 23.66 4.98 1.35 -0.11 -0.18Channel Shale Plug 4372 2342 2.60 11367 6089 55.06 37.08 30.68 1.87 0.30 1.48Regional Shale/Silt 4098 2186 2.55 10450 5574 47.05 31.07 26.58 1.87 0.30 1.51
Porous Channel Sand 4098 2342 2.35 9630 5504 32.16 30.29 22.28 1.75 0.26 1.06Tight Channel Sand 4684 2623 2.55 11944 6689 53.19 44.74 32.55 1.79 0.27 1.19
AVG. % CHANGE POROUS/TIGHT SAND 21% 19% 49% 39% 37% 2% 5% 11%
Motivation from well logs
VVpp/V/Vss (V(Vpp/V/Vss))22νν μμ
Gas SandGas SandShaleShale
% change% change
2.252.251.711.712727
5.15.12.92.95555
0.380.380.240.244545
λ+2μλ+2μ20.3720.3718.5318.539.29.2
4.0354.0356.3146.314
4444
λλ12.312.35.95.97070
λ/μλ/μ3.13.10.90.9110110
Vs Vp ρ
1290 m/s 2898 m/s 2.425 gm/ccShale
2.275 gm/cc2857 m/s1666 m/sGas Sand
Averaged log values for Shales, and Gas Sands from Alberta
λ = Vp2*ρ - 2 Vs
2*ρ
μ = Vs2*ρ
Relations to transform velocities, density to Lamé parameters λ and μ
Towed streamer P-wave data
Converted wave Converted wave PP--S OBC dataS OBC data
From MacLeod et al 1999 Chevron/WEGCOFrom MacLeod et al 1999 Chevron/WEGCO
OWC
Shale
Shale
Sand
Shear waveVs
OWCSand
Shale
Shale
P-wave Vp
North Sea P-wave and Shear wave Logs Alba Field Seismic P-P and P-S comparison
P-, S-impedance
S ImpedanceP Impedance
metres1240
1260
1280
1300
1320
2000
6000
1000
0
1400
0
1800
0Shale
Carbonates
(m/s.gm/cc)
LambdaRho, MuRho
MuRhoLambdaRho
0 8040 120
160
Silt
ShaleGas Sand BTight Streak
Gas Sand ACarbonates
(GPa.gm/cc)
metres1240
1260
1280
1300
1320
Petroleum industry log tracks for Sands, Shales and Carbonates showing improved LambdaRho, MuRho crossover discrimination of gas zones and lithologies compared to P-, S-impedance
Relations to transform impedances to Lamé parameters λρ, μρ
Impedance = Velocity*DensityP-impedance = Vp*ρS-impedance = Vs*ρ
λρ = (Vp*ρ)2 - 2 (Vs*ρ)2
μρ = (Vs*ρ)2
Fluid, Porosity & Lithology directions in LambdaRho (λρ), MuRho (μρ) spaceFluid, Porosity & Lithology directions in LambdaRho (λρ), MuRho (μρ) space
5%
10%
15%
20%
5%
10%
15%
20%
5%
10%
15%
20% Shale
Poro
sity
Lithology
Carbonates
Sands
FluidWaterOilGas
Dolomite
Shale
QuartzLimestone
λρ (GPa.gm/cm3 or Rayls2)
μρ(G
Pa.g
m/c
m3
or R
ayls
2 )
cons
tant
Poi
sson
ratio
constant P-impedance
Porosity % bubble
20%
15 300
150
125
100
75
25
50
45 60 75 90 105
(Adapted from Hoffe, Perez and GoodwayCSEG convention 2008)
Gas Sand Zone
Wet Sand Zone
Marine Shale above Gas Zone
Tight Sandy Shale between Gas & Wet Sand Zones
Calcareous Shale/Silt
Sensitivity of Vp/Vs, Poisson’s ratio vs. Lambda/Mu ratio
0.000.250.500.751.001.251.501.752.002.252.502.753.003.253.503.754.004.254.50
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6
4.0
4.4
4.8
5.2
5.6
6.0
6.4
6.8
7.2
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Vp/Vs Poisson ratio
Lambda/Mu ratio
Vp/
Vs
ratio
Poisson’s ratio
( )1
VsVp5.0
/d)Vs/Vp(d −
⎟⎠⎞
⎜⎝⎛=
μλ
2)21(5.0)/(d
dν−=
μλν
Comparison to Vp/Vs
Comparison to Poisson’s ratio
0%10%20%30%40%50%60%70%80%90%
100%110%120%130%140%150%160%170%
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Water Saturation (Sw)
Nor
mal
ized
avg
. % c
hang
e“Fizz Water” (Low Gas Saturation) Discrimination
(data points from Han et al 2001)
LambdaRhoLambda/Mu ratioPoisson ratioIp/Is ratioFluid Factor Smith&Gidlow
Relative sensitivity to water saturation:Ip/Is, Poisson’s, Lambda/Mu ratios and LambdaRho, “Fluid Factor”
0.0
2.0
4.0
6.0
8.0
10.0
-10 -6 -2 2 6 10 14 18 22 26 30 34
LambdaRho-MuRho Difference
Lam
bda/
Mu
Rat
io
Gas Sands
Shales
Brine Sands
Lam
bda/
Mu
ratio
LambdaRho-MuRho difference
slope ∝1/μρ
decreasing Ip
λ/μ ratio vs. λρ−μρ difference with lines of constant P- and Shear Impedance
Lambda/Mu Ratio vs. LambdaRho-MuRho difference crossplotsWorldwide Log Data (Castagna and Smith 1994)
Lam
bda/
Mu
ratio
LambdaRho-MuRho difference
(Dave MacKidd 1996)
linear trend
Fluid discriminationLambda/Mu ratio vs Lambda-Mu difference crossplot templates:
Lithology and Porosity discrimination
-10 0 10-15 -5 50.40
0.60
0.80
1.00
1.20
Glauconitic Gas SandGlauconitic Wet Sand
-10 0 10-15 -5 50.40
0.60
0.80
1.00
1.20
Glauconitic Gas SandGlauconitic Wet Sand
λ−μ (GPa.)
λ/μ
(from T. Chen et al 1998)
-20 0 20 40 600.00
0.80
1.60
2.40
3.20
Porous Sand
Carbonates
Shale
Sandstone line
-20 0 20 40 600.00
0.80
1.60
2.40
3.20
Porous Sand
Carbonates
Shale
Sandstone line
λ−μ (GPa.)
λ/μ
Alberta Glauconitic Gas Sand Log
2 4 6 8 100
2
4
6
λ (GPa)
Hydrocarbons(in yellow)
TargetSand Lines
0 11 - Water SaturationWet Oil
λ/μ
GOM log data
Geology, Logs and Classes of AVO methods
Real Earthrock layers
(impedances)
Reflectivitymethods
Seismic Trace
Inversion methods
Impedance model of the earth
Earth’s seismic reflection response from log measured
impedance contrasts
INVERSION
SeismicWavelet
⊗CONVOLUTION
Aki-Richards Shuey
Smith-Gidlow-FattiHilterman-VermWang-Goodway
HampsonGoodwayConnolly
AVO Reflectivity MethodsAVO Reflectivity Methods
Common Reflection PointMultiple Shots and Receivers
Offset Gather with Reflection Amplitude Variation with Offset
Seismic Ray Paths
Far Offset Near Offset
Increasing Offset
ShotsGeophones
WalkawayWalkaway VSP Geometry VSP Geometry Direct proof of AVODirect proof of AVO
AVO AVO ““GathersGathers”” from VSPfrom VSP3C
Geo
phon
es
Source Positions
FarOffset
Near Offset
PPii PPjj
PPii~ ~ PPjj
RRpp--pp==PP--PPr r /P/Pii
P-Pr
Reflector
PP--P reflectionP reflection
OffsetOffset0.50.5
0.60.6
0.70.7
0.80.8
0.90.9
1.11.1
Near FarNear Far200m 1700m200m 1700m
Gas Sand Gas Sand AVO AVO
responseresponse1.01.0
Linear P-P AVO three term reflectivity equations and approximations following Aki & Richards 1980
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+
Δ==
Δρρ
VpVp
21)0(Rpp
Ip2Ip
⎟⎟⎠
⎞⎜⎜⎝
⎛ Δ+
Δ==
Δρρ
VsVs
21)0(Rss
Is2Is
where the link is:
( )ρρθθθθ Δ
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
Δ−
Δ+= 2
2
22
2
22 sin
VpVs41
21
VsVssin
VpVs4
VpVptan1
21)(Rpp Aki & Richards
1980
Rpp(θ) = f1(θ)∗(Vp change) - f2(Vp/Vs,θ)∗(Vs change)+ f3(Vp/Vs,θ)∗(density change)
Gidlow,Smith,Fatti 1992 ignore Δρ/ρ 3rd term
ρρθθθθθ Δ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−−
Δ⎟⎟⎠
⎞⎜⎜⎝
⎛−
Δ+= 2
222
22 sin
VpVs2tan
21
Is2Issin
VpVs8
Ip2Ip)tan1()(Rpp
Rpp(θ) = f1(θ)∗(Ip change) - f2(Vp/Vs,θ)∗(Is change)+ [f(Vp/Vs,θ)∼0]∗(density change)
Other P-P reflectivity AVO equations and approximations
( )Δθ−θ+θ⎟⎟
⎠
⎞⎜⎜⎝
⎛ν−νΔ
++=θVp2Vpsintansin
)1()0(ARpp)0(Rpp)(Rpp 222
2
Shuey 1985 ignore ΔVp/Vp term
Note ΔVp/Vp ~ 4Δρ/ρ Gardner 1974
Vp2Vp
)tan1(2
)(Rpp 2 Δ++
Δ= θ
ρρ
θ − μμθ
2sin
VpVs4 2
2 Δ⎟⎟⎠
⎞⎜⎜⎝
⎛ Wang 1999 (Goodway 1998)ignore Δρ/ρ term wide angle gradient approximationintroduced Mu reflectivity into AVO
General form: Rpp(θ) ≈ A + B sin2(θ) A= intercept, B=gradient
NI = Rpp(0) normal incidence reflectivity, PR = Δν/(1−ν)2 Poisson reflectivity
Hilterman & Verm 1995 ignore ΔVp/Vp term( )
θν−νΔ
+θ=θ 22
2 sin1
cos)0(Rpp)(Rpp
Wiggins 1984 ignore Δρ/ρ termθθ 2
2sin)0(Rss
VpVs8)0(Rpp)0(Rpp)(Rpp ⎟
⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−+=
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65
Comparison of 3 term Aki & Richards to 2 term approximations
00.010.020.030.040.050.060.070.080.09
0.10.110.120.130.140.150.160.170.180.19
0.20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Angle
P-P
refle
ctio
n co
effic
ient
Aki & Richards 3 termGidlow, Smith FattiShueyWang
P re
flec t
ion
coef
ficie
n t
Angle
30° 45°
-0.038-0.036-0.034-0.032-0.030-0.028-0.026-0.024-0.022-0.020-0.018-0.016-0.014-0.012-0.010-0.008-0.006-0.004-0.0020.0000.0020.0040.0060.0080.0100.0120.0140.016
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
Angle
Ref
lect
ivity
Rμ(θ)
Rρ(θ)
Rλ(θ)
Rfull(θ)
Aki & Richards AVO reflectivity curve with separate Lambda, Mu and Rho termsR(lambda)= -0.1 R(mu)=0.1 R(rho)=-0.03
Decomposition of AVO reflectivity equations into Lambda,Mu,Rho(Gray et al 1999, Goodway 1997, 2002)
30° 45°
( ) ( ) ( ) tan125.0V
Vsin2
22tan125.0)(R 2
2P
2S22
ρρΔ
θ−+μμΔ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛θ−
μ+λμ+λΔ
θ+=θ
( ) cos4
2cos
V
V
cos2
2cos
V
V21
cos4
1)(R22
P
2S
2
2
2P
2S
2 ρρΔ
θ
θ+
μμΔ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
θ
θ+
λλΔ
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
θ=θ
Note behavior at 45ºΔρ/ρ term disappears
with “polarity” flipΔμ/μ term has an
inflection and is zeroΔλ/λ alone remains
Log track Low frequency background model
Inversion result
AVO gather log model
AVO gather inversion
AVO gather error
Base of zone
45ºpolarity flip
P-impedance S-impedance Density
Log based AVO model inversion for density (Gidlow method with error term retained in constrained AVO inversion)
45º angle requirement and polarity flip
Density term is rejected in the error plot as seen in the polarity flip at 45º
( )
KK
KK
1
KK
1 KK
fluid
1solid
solid
dry
2
solid
dry
drysat
⎟⎟⎠
⎞⎜⎜⎝
⎛ φ+⎟⎟
⎠
⎞⎜⎜⎝
⎛−φ−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
+=−
Biot-Gassmann fluid replacement equation in Lamé terms and relation to “Pore Space Modulus” (Hedlin, Russell, Hilterman and Lines 2003)
Biot-Gassmann Equation:K is bulk modulus, “sat” is saturated rock, φ is porosity
o45at AVO fromdirectly obtained becan Remember K
K
KK K and ngsubstituti and romF
sat2solid
2fluid
drysoliddrysatsatdry
λφ
Δλ≈λΔ⇒
−=Δλ−λ=λΔμ=μBiot-Gassmann Equation in Lamé terms
( ) ( )
K
Kf
above from equating so )(f
2VV
C whereC2ICIf modulus space Pore
sat2solid
2fluid
satdrysat
dry2
)dry(S
)dry(Psatsatsat
2S
2P
ρφ
Δλ≈ρ⇒
λΔρλ−λ=ρ⇒
μ
μ+λ=⎟
⎟⎠
⎞⎜⎜⎝
⎛=μρ−ρμ+λ=−=ρ
Relating “Pore Space Modulus” ρf to Biot-Gassmann Equation in Lamé terms
AVO Classes in Rp, Rs Reflectivity Space(Goodway, Hoffe, Perez 2002, 2008)
III
II
θ
0.20
ΔI p/
I p(θ)
0.10
0.00
-0.10
-0.20
IVV
Rutherford&WilliamsCastagnaYoung&LoPiccoloI
100% φ intercepts: gas sand (Vp≈0) wet sand (Vp≈1500m/s) Vp=A+BVs where Vs=0
gas sands
Mudrock
line
P-wave velocity (km/s)1 2 3 4 5
1
2
3
S-w
ave
velo
city
(km
/s)
shale
swet sa
nd
Linear dry
sand relation
6
Grinsburg and Castagna:Vp(sand)=1065 +1.25 Vs Vp(shale)=1126 +1.3 Vs(velocities in m/s)
background mudrock line relations (Vp=A+BVs)
III
II I
False Type 1
-0.3 0.3
-0.3
+0.3
+ΔIp/Ip = Rp
+ΔIs/Is = Rs
reference pt
V
Increase in Vp/Vs
Zero Gradient Line(ΔIp/Ip=2ΔIs/Is)
Decrease in Vp/Vs
IIIIV
BackgroundRs = 1.43 Rp
-ΔIp/Ip
-Δ Is/Is
Rotation of gas sands away from background
Backg
roun
d wet
sand
/shale
Top gas sand anomaly
Basegas sand anomaly
Backg
roun
d wet
sand
/shale
mud
rock
trend
GAS
BRINE
Rp Rs
TagluT-C11
Rp Rs
Brine (in-situ)
Gas (substituted)
Well: 300F486930134017-Nov-2002 01:44:38
2900.0
2900
3000
3100
3200
3300
3400
3500
DepthMetres
TWTIME_GAS.DT_1US/M500 100
TWTIME_GAS.DTSM_1US/M1200 200
AEC.APPLY_30 10
AEC.VSH_3V/V0 1
TWTIME_GAS.RHOB_1K/M31950 2950
TGLU
T-C8A
T-C11T-C9
T-C11
Rs vs. Rp xplots: angle range 5-40o
Substituted T-C11 gas sand zone identified from blue (top) and orange
(base) x-plot polygons of AVO anomaly
Mac Delta gas discovery: pre-drill AVO model reflectivity prediction Rp,Rs synthetic: Brine (in-situ) vs Gas (substituted)
AVO Inversion Methods
Gidlow, Smith, Fatti methodAVO extraction of Rp, Rs
Goodway methodRp,Rs inversion to Ip,Is
λρ = Ip2 - 2Is2
μρ = Is2
λρ, μρλρ, μρ
P & S ImpedanceP & S Impedance
AngleAngleGathersGathers
P & S P & S ““ReflectivityReflectivity””
Connolly methodRp(θ) inversion to Ip(θ)
AngleAngleGathersGathers
λ, μ, ρ
ElasticElastic ImpedanceImpedanceAI, EIAI, EI
Elastic Impedance extended in Laméparameter terms
Elastic Impedance in Lamé (LMR) terms following Connolly 1999
Density is present in all EI(θ°) Density and Vs always functions of a power of γ = Vs2/Vp2
At 30° and 45° the EI (θ°) values are:
EIVsVpIpIplnpR )psin41(psin8)ptan1( 222
=ρ=⇒= θγ−θγ−θ+θθθ∫
)21(42
)1(23/4
VsVp)45(EI
VsVp)30(EIγ−γ−
γ−γ−
ρ=
ρ=
Formulated in Lamé and density terms:
At 0°, 30°, 45° and 60° the LMRI (θ °) values are:
( )
( )
( ) 1212
21
3/13/3/212
2/12/)21(
)60(LMRI
)45(LMRI
LMRI(30)
LMRI(0)
−γγ−
γ−
γγ−
γγ−
ρμλ=
λ=
ρμλ=
ρμλ= Density independent of γ in all LMRI(θ°)and changes sign at 45°Density and Mu absent in LMRI (45°)At 45° only Lambda “Pore Space Fluid Modulus”(scaled to power of γ)
( ) ( ) (LMRI) Impedance LaméIp2222 cos22coscos2coscos221 =ρμλ= θθθθγθγ−
θ
( ) μρλγ
γ =⎥⎥⎦
⎤
⎢⎢⎣
⎡==−
3/1
3
2321/1
)45(LMRI
LMRI(60))0(LMRI ,
)60(LMRI LMRI(30)
,)45(LMRI
Assuming meaningful LMRI(45°) LMRI(60°) following elastic parameters are extractable:
AVO Classes in Reflectivity and LMR Space(Goodway, Hoffe, Perez 2002, 2008)
Zero Gradient Line(ΔIp/Ip=2ΔIs/Is)
μμ ··ρρ
((Pa·
kg/m
3 )) I
IIIII
λλ··ρρ ((Pa·kg/m3))
5%
10%
20%
5%
10%
15%
20%
5%
10%
15%
reference ptreference pt
Dolomite
Limestone
Shale
Quartz
15%20%
VIV
III
II
θ
0.20
ΔI p/
I p(θ)
0.10
0.00
-0.10
-0.20
IVV
Rutherford&WilliamsCastagnaYoung&LoPiccoloI
Constant P-impedance Ip
Cons
tant
Po
isson
ratio
LMR space adds insight to AVO classes
Fluid shift
LihologyPo
rosi
ty
IIIII I
False Type 1
-0.3 0.3
-0.3
+0.3
+ΔIs/Is=Rs
reference pt
V
Increase in Vp/Vs
Decrease in Vp/Vs
IIIIV
BackgroundRs = 1.43 Rp
-ΔIp/Ip
-Δ Is/Is
+ΔIp/Ip=Rp
MacKenzieMacKenzie Delta gas discoveryDelta gas discoveryPre-drill Fluid Replacement Log Model of Rp,Rs and LambdaRho vs. MuRhocrossplots of original wet zone in legacy well.Note improved separation and fluid/porosity discrimination in λρ vs. μρ that is not possible with Rp vs. Rs
Rs
Gas sand anomalyAll AVO classes 1,2,3Fluid, porosity confusion?
Back
grou
nd w
et-s
and/
shal
e
mud
rock
trend
Rp
Rp vs. Rs
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
55.0
60.0
0 0
10 10
20 20
30 30
40 40
50 50
60 60
70 70
80 80
90 90
100 100
0.0
5.0
10.0
15.0
20.0
25.0
30.0
35.0
40.0
45.0
50.0
55.0
60.0
0 0
10 10
20 20
30 30
40 40
50 50
60 60
70 70
80 80
90 90
100 100
Incr
easin
g φ
Tight streaks
• Shale• Brine Sand• Gas Sand
-1101
Colo
r: V
olu
me S
hale
Large consistent fluid separationdiscriminates oil (green oval) from gas
λρ
μρ
LambdaRho vs MuRho
Walkaway VSP AVO gather
Zone ofZone ofInterestInterest
R1R1R2R2 4040ºº
3535ºº
3030ºº
2525ºº
200m 1700mFar Offset Near OffsetCMP Gather with Offset Balancing
P vs. S Impedance
LambdaRho (GPa. gm/cc)P Impedance (m/s. gm/cc)
CementedTight SandGas Sand
S Im
peda
nce
(m/s
. gm
/cc)
Shale
Threshold cutoff forIp/Is “Fluid Factor” Stack
SiltShalyGasSand
ShalyGasSand
MuR
ho(G
pa. g
m/c
c) CementedTight Sand
ShaleCluster
Threshold cutoff forLambda/Mu Stack
GasSand
ShalyGasSand
Silt
LambdaRho vs. MuRhoSeismic LambdaRho,MuRho vs. P-,S-impedance crossplots and “fluid factor” stacks
P-, S-impedance “fluid factor” LambdaRho, MuRho “fluid factor”
1.11.1
Ambiguous Gas Sand Zone Isolated Gas Sand channel zone
secsec0.80.8
0.90.9
1.01.0
Alberta Gas Sand Channel
Low High
A B C
Migrated stackamplitude ambiguity
(white boxes)
LambdaRho stackclear gas sand
anomalies (white boxes) at wells
well A wet, wells B,C gas
MuRho stackclear sand channel layer
above unconformity (green zones) at gas wells unconformity
unconformity
unconformity
Alberta Upper/Lower Basal Quartz Sand Channel Discrimination
LMR lithology from crossplot polygons600
700
800
900
Upper BQ
Lower BQ
Tim
em
sec
Upper BQPorosity = 25.6%Permeability = 566 MD
Lower BQPorosity = 13.8%Permeability = 14.6 MD
Core Analysis
1220
LambdaRho G.Pa. g/cc (Incompressibility)
MuR
hoG
.Pa.
g/c
c (R
igid
ity)
22
42
62
50 80 110 140
2.00
2.85
2.42
Upper BQ
Lower BQ
coals
Density OverlayLog Crossplot
15
35
45
40 80 120 1600
Upper BQ
Lower BQ
MuR
ho
LambdaRho
25
Seismic Crossplot
LambdaRho MuRho anomaly (3D Line)LambdaRho MuRho
Low
High
Gas/condensate anomaly
Alberta Deep Basin: Carbonate gas/condensate discovery75 MMCF/day (AAPG Best Paper 2005)
LambdaRho Anomaly MapStrong anomaly delineates gas/condensate discovery
Low λρ
High λρ
Vp/Vs Anomaly MapLimited Vp/Vs attribute response
Low Vp/Vs
High Vp/Vs
Alberta Deep Basin: Carbonate gas/condensate discovery75 MMCF/day (AAPG Best Paper 2005)
100100
8080
4040
2020
00
6060
--2525 00 2525 5050 7575 100100 125125 150150 175175
Porous gas sandPorous gas sand
Silt shaleSilt shale
Mu error = Mu (true) x2Mu error = Mu (true) x2
Mu error = Mu (true) x1/2 Mu error = Mu (true) x1/2 MuR
hoM
uRho
LambdaRhoLambdaRho
LambdaRho/MuRho Crossplot Errors from Mu (S-impedance or velocity) Correlated Error Alone
Gas sand channel seismic LMR crossplot shows clear relative isolation of gas sand points despite inversion error skew (note -ve LambdaRho values)
--2020
6060
5050
4040
3030
2020
00 2020 4040 6060LambdaRho
MuR
ho
λ /μ
rati o
ov e
r lay
00
0.40.4
0.80.8
1.21.2
1.61.6
2.02.0
1313--1616
0909--1717 1212--1616
0505--1616
0101--1717 0404--1616
1414--0909
1111--0808
0808--0808
0101--0808
0909--08081616--0808
0909--0808
1 km1 km
LMR cross-plot gas sandmask on λρ 3D map
Yellow oval in grey cloud mask of known gas sand points, clearly isolates best gas sands from background. Correlated Rp, Rs extraction errors mean gas sand detection in LMR cross-plot space is unaffected
Poison’s ratio decreases with increased Young’s modulus (Grigg SPE 2004)
Geomechanics and Moduliinvolved in fracture prone rocks
Poissons_ratio = - 0.045700 *E + 0.452206
-
0.050
0.100
0.150
0.200
0.250
0.300
0.350
3.00 3.50 4.00 4.50 5.00 5.50
YOUNG'S MODULUS ( MMpsia)
POIS
SON
'S R
ATIO
.
4.753.9 MMpsi=27GPa
0.235
0.27
Poissons_ratio = - 0.045700 *E + 0.452206
-
0.050
0.100
0.150
0.200
0.250
0.300
0.350
3.00 3.50 4.00 4.50 5.00 5.50
YOUNG'S MODULUS ( MMpsia)
POIS
SON
'S R
ATIO
.
4.753.9 MMpsi=27GPa
0.235
0.27
Barnett shale properties (Grigg 2004)
μ=ν+
21
EThis implies a simple increase in a single modulus of rigidity μ.
Lambda vs. Young’s modulus cross-plotoverlaid with: curves of constant Mu lines of constant Poisson’s ratioBarnett Shale trend
02468101214161820222426283032343638
Mu (G
.Pa)
Barnett Trend0.10.140.180.220.260.30.340.380.420.46
Poisson’s ratio 05
1015202530354045505560657075
0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50
Lambda (GPa)
Youn
gs m
odul
us (G
Pa)
Brittle
Ductile
Closure Stress Equation (Engineering to Geophysics)Fracture generation and moduli: E,ν,μ and λ
[ ] ( ) ee1
E PBPB 1 yyxx2pHPVzzxx ν+
ν−++−σ
ν−ν
=σ
( ) e and e2strain uniaxialfor law sHooke'
zzxxzzzz λ=σμ+λ=σ
So in λ and μ terms
PBe
e-ePB pHyy
xx2
yyPVzzxx +
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛μ+−σ
μ+λλ
=σ⇒2
22Effective stress
= Overburden – Pore PressureTectonic strain energy% anisotropy = e2
yy - e2xx
σzz - BvPp
σxx, exx
eyyConfining stress
σyy = σmax
σxx=σmin
But horizontal strains are zero exx= eyy = 0Uniaxial strain in ez only, as rock is bound in the earthSo Poisson’s ratio ν = - exx/ezz= 0 !
Whereσxx = min. horizontal closure stress to open an existing fractureσzz = overburden stress Pp = pore pressure acting vertically (Bv) and horizontally (Bh)
Fluid, Porosity, Lithology, Geomechanics and Closure Stress reduction directions in LMR SpaceFluid, Porosity, Lithology, Geomechanics and Closure Stress reduction directions in LMR Space
MuR
ho(G
.Pa.
gm/c
c)
5%
10%
15%
20%
5%
10%5%
10%
15%
20%
Poro
sity
Lithology
FluidWaterOilGas
LambdaRho (G.Pa.gm/cc)15 30 45 60 75 90 1050
Dolomite
25
50
75
125
150
100
QuartzLimestone
Shale constant P-impedance
cons
tant
Poi
sson
’s ra
tio
= 0.
17
20%
15%
Sand
Porosity % bubble
20%
adapted from Hoffe et al CSEG convention 2008
PBe
e-ePB pHyy
xx2
yyPVzzxx +
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛μ+−σ
μ+λλ
=σ2
22
Effective stress= Overburden – Pore Pressure
Tectonic strain energy% anisotropy = e2
yy - e2xx
reduced Effective Stress
amplification
reduced Tectonic
amplification
Barnett shale relative EUR’s30
53
Gas shales W.CanadaCalcareous shales
Ductile shales
Carbonates
3053
Barnett analogue
Good production high EURWeak non-aligned anisotropy (weak 3D attribute-green/blue tick marks) Broad fracture network
Poor production low EURStrong aligned anisotropy (strong 3D attribute-red/orange tick marks)Narrow fracture network
Grey/green overlay of microseismic event monitoring of fracture stimulated perm
From Y. Simon M.Sc. U of H 2005
N. Texas Barnett Shale Fracture Generation and Anisotropy Maps
Lithology and geomechanical attribute mapping using LMR
Sands
Gas shales
3D line with colour overlay from cross-plot polygons (discovery well rate 16 mmcfpd)
Ductile
shale
Brittle
shale
SandsCarbonates
Log guided seismic cross plot polygons and resulting seismic section(from Eric Keyser)
Higher
poro
sity
gas s
and
Gas shales
LambdaRho
MuR
ho
Gamma log colour overlay
Seismic AVO Gathers, 3D Azimuths and FracturesCommon Reflection Point Multiple Shots (S) and Receivers (R)
Far Offset Near Offset Far Offset
Seismic Record of offset gather with amplitude variation with offset
Seismic Reflection Ray Paths
S S SS R R RR
Increasing Offset
3D seismic acquisition and fractures
Core fracture orientationS
600m
Si, Ri
R
azimuth sectors
Isotropic AVO
AVAZ:Anisotropic AVO
θ = P-wave propagation angle (vertical plane)α, β = P- and S-wave velocityΔα/α = fractional change in P-wave velocityΔμ/μ = fractional change in rigidity Mu
Anisotropy parameters:φ=0: slow (weak) in symmetry axis (across fractures) φ=90: isotropic (parallel to fractures)Δγ = Sh-wave anisotropy ‘velocity splitting’Δε(v) = P-wave anisotropy Δδ(v) = P& Sv-wave anisotropy
P-wavezero offset
Isotropic (in-fracture) AVO terms Anisotropic (cross-fracture) AVO terms
Rp (θ,φ) = Rp(0) + 0.5([ Δα/α − (2β/α)2 Δμ/μ] + [Δδ(v)+ 2(2β/α)2 Δγ]cos2φ) sin2θ
+ 0.5(Δα/α + [Δδ(v) sin2φ cos2φ+ Δε(v)cos4φ]) tan2θ sin2θ
Isotropic layer
Anisotropic layer
3D surface seismic geometry: multi-azimuth offsets
Principal Anisotropy Planes
P
Sv (S2) slow perpendicular
P(slow)
PSv (S1) fast parallel
P(fast)
fract
ure-
para
llel
isotro
pic p
lane
fracture-perpendicular symmetry plane
Anisotropic layerwith vertical fractures
Isotropic layer
Principal Anisotropy Planes
P
Sv (S2) slow perpendicular
P(slow)
PSv (S1) fast parallel
P(fast)
fract
ure-
para
llel
isotro
pic p
lane
fracture-perpendicular symmetry plane
Anisotropic layerwith vertical fractures
Isotropic layer
Principal Anisotropy Planes
P
Sv (S2) slow perpendicular
P(slow)
PSv (S1) fast parallel
P(fast)
fract
ure-
para
llel
isotro
pic p
lane
fracture-perpendicular symmetry plane
Anisotropic layerwith vertical fractures
Isotropic layer
φ=0º
φ=90
º
opposite sign
AVAZ (Amplitude Variation with Azimuth): Ruger et al equation 1997
Simplified VTI Anisotropy: Thomsen’s δ in Lamé Terms
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
⋅
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−−−−−
=
⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
xy
zx
yz
zz
yy
xx
44
44
44
3344334433
4433334433
4433443333
xy
zx
yz
zz
yy
xx
eeeeee
CC
CCC2CC2C
C2CCC2CC2CC2CC
σσσσσσ
⋅
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
++
+
22
2
μμ
μμλλλ
λμλλλλμλ
Isotropic Modulus Tensor Matrix(Voigt Notation)
Anisotropic VTI Tensor Matrix(adapted from Sheriff’s Dictionary of Geophysics)
⋅
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡−
−
=
66
44
44
333131
13116611
13661111
CC
CCCCCCC2CCC2CC
⋅
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
≡
+≡+≡≡
≡≡+≡
⊥
⊥
⊥333131
13
13
22
2
μμ
μμλλλ
λμλλλλμλ
233
233
33132
332
13
443333
24433
24413
462)(2)(
)CC(C2)CC()CC(
⊥⊥
⊥
μ+μλ+λ
λ−λμ+λ−λ=δ⇒
−−−+
=δ
λ13
λ33
Cross-section of gas filled layers or fractures
So δ is generallypositive asλ13 > λ33
stiffer parallel to layers than across
In nearly all measurements δis either zero (within measurement error) or positive and correlated to positive ε
measurement error
positive
correlatio
n
A better 3 term AVAZ equation: each term has equal significance for robust inversion of Δα/α, Δμ/μ, Δγ, Δδ
Where η∗ = (ε−δ)/(1+2ε) obtained from NMO
P-velocityzero offset
Isotropic (in-fracture) AVO terms
Anisotropic (cross-fracture) AVO terms
Rp (θ,φ) = Δα/α + 0.5(Δα/α + [Δδ(v)+ Δη∗cos2φ]cos2φ) tan2θ
− 0.5((2β/α)2 Δμ/μ − [2(2β/α)2 Δγ − Δη∗cos2φ] cos2φ) sin2θ
Offset vs. Azimuth angle cube
Fracture strike @ N10
1800m offset slice105o 285o
AVO azimuth gradients
strong gradients
perpendicular
Fracture strike @ N10
1800m offset slice105o 285o
AVO azimuth gradients
strong gradients
perpendicular
azimuth angle
t
Azimuthal Anisotropy from 3D seismic AVO over known fractures
III
II I
AVO Classes for Fracture ReflectivityAVO Classes for Fracture ReflectivityAVO Classes for Fracture Reflectivity
-ΔVp/Vp-ΔVp/Vp
-Δμ/μ-Δμ/μ
-0.3-0.3 0.30.3
-0.3-0.3
+0.3+0.3
Zero Gradient Line(assumes ΔIp/Ip=2ΔIs/Is)
Zero Gradient Line(assumes ΔIp/Ip=2ΔIs/Is)
+ΔVp/Vp+ΔVp/Vp
+Δμ/μ+Δμ/μ
V IIIIII
IIII
II
(Hoffe, Perez, Goodway, 2002, 2008)(Hoffe, Perez, Goodway, 2002, 2008)θθ
0.200.20
ΔI p/
I p(θ)
ΔI p/
I p(θ)
0.100.10
0.000.00
-0.10-0.10
-0.20-0.20
IVIVVV
IIIIII
IVIVIIIIII
ΔIp/IpΔIp/Ip ΔIs/IsΔIs/Is Vp/VsVp/VsTypeType
VV
parallel to fractures
perpendicular to fractures
parallel to fractures
perpendicular to fractures
Increase in Vp/Vs
Increase in Vp/Vs
Increase in Vp/Vs
Zero Gradient Line(assumes ΔIp/Ip=2ΔIs/Is)
Zero Gradient Line(assumes ΔIp/Ip=2ΔIs/Is)
Decrease in Vp/Vs
Decrease in Vp/Vs
Decrease in Vp/Vs
IIIIV
False Type 1False Type 1
Colorado B AVAZ fracture intensity and orientation map with well A deviation path
Well A deviation path
Well A deviation path
LOW HIGHINTENSITY
Post-drill well A log tracks confirm 3D seismic attribute predictions
MAXXENE_OVERALL0 150( NONE )
MINXENE_OVERALL0 150( NONE )
XENEDIF0 50( NONE )
0
AVAZ 3D section with Well A gamma logand horizons (Clrd A, Clrd B and SSPK)
FMI images Fracture identification: intensity and orientationfrom FMI from crossed-dipole shear
weak early fast P-wave
strong late slow S-wave
P- and S-wave events recorded across an 8-level 3C geophone array Geometry of treatment and monitor wells
P- and S-waves processed to locate events based on arrival time, azimuth polarization
Potential to characterize fracture plane attributes from magnitude and focal mechanism
Microseismic event frequency ~ 500hz
Microseismic downhole monitoring
Events displayed in cross-section and map view to give quantifiable estimates of Stimulated Reservoir Volume
Surface Microseismic Buried Array Event Mapping
Example of P-wave event 1st arrival polarity reversal
(from Pete Smith and Kevin Smith EnCana & Chris Neale MSI 2009)
100 stations
3000’ grid spacing
Single 3C phone cemented at 250’
~16 sq. mi. grid
Event Map: all 8 stages top 10% of eventsMoment tensor solution suggests normal dip-slip faultingEvents colored by stage and sized relative to all stages
In June a massive Earthquake (magnitude 7.8) twisted South Island NZ. and moved its southern tip 30 cm, toward Australia, a process that usuallytakes ‘hundreds of years‘.Its focus occurred in “soft rocks” between two tectonic plates, muffling its power, as the rocks lurched (ductile) rather than snapped (brittle), causing a low-frequency rolling rather than more damaging high-frequency waves
Moment tensor solution fracture planes
2000’
Frac event trends
Circle size proportional to amplitude(red positive, blue negative) Squares are model amplitude fit to data.
Stage 4A,B(bridge plug flow)
Stage1Stage 2
Stage 6
Stage 7
Stage 8
Stage 2b
Stage 3
Normal dip slip fault
Barnett
Piceance
Barnett shale is more brittle than Piceance silts: High rigidity (Mu), hence lower Closure stress, results in large fracture planes creating high magnitude, low frequency events
Magnitude vs. distance for Barnett and Pieance
length fracture is L whereLD
xue x
xx ≈∂
∂=Strain:
30
xxr2
M
r2De
π=μ≈μ=σΔ
Stress drop Δσ across circular fracture plane
Earthquake moments and stress drop
Stress drop proportional to moment and inversely proportional to area
73.105.1MlogM
radius fracturer slip, averageD where)r(DM
0w
20
−=
==πμ=
rD
Moment magnitude (Mw) related to seismic moment (M0) penny shaped cracks
30
r
M167
=σΔ
Fracture radius inversely proportional to high corner frequency of displacement spectrum frequencycorner f where
f2V
r 00
S =π
∝
Earthquake amplitude displacementspectrum (1995 Chiapas, Mexico)
f0
Fracture surface area vs. moment and stress drop
1.E+01
1.E+02
1.E+03
1.E+04
1.E+05
1.E+06
1E+2
5
1E+2
6
1E+2
7
1E+2
8
1E+2
9
1E+3
0
1E+3
1
Moment (dyn.cm)
Surfa
ce a
rea
(sq.
km)
1 bar 10 bars 100 bars 1000 bars
y = 0.5655x + 299.36
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
0 1000 2000 3000 4000 5000 6000 7000 8000
1st Confining Stress2nd Confining Stress3rd Confining Stress4th Confining Stress
Y Intercept = Linear CohesionSlope = Coefficient of Internal Friction
UnstableArea Stable
Area
failure slope=μ (friction)
in-situ stress circle
Mu
x sh
ear s
trai
n, p
si.
(she
ar s
tres
s τ=
μ∗e y
y)
Lambda-Mu log cross-plot (scaled by strain)overlaid on Mohr Coulomb Failure Envelopes
Mohr-Coulomb theory related to seismic LMR
σmin σmax
frac stimulation increases pore pressure
Lambda+2Mu x normal strain, psi. (horizontal normal stress σ=[λ+2μ]∗exx)
brittle gas shales withcritically stressed fractures
carbonates
ductile shales
Barnett shale analogue EUR 5330
3D AVAZ with coherence & 3D AVAZ with coherence & MicroseismicMicroseismic event overlays event overlays
%R
efle
ctio
n A
niso
tropy
LOW
HIGH
1600
m
Coherencelineations
Orientation of AVAZmaxinconsistent with SH max?
areas of relatively high reflection AVAZ anisotropy near horizontal borehole
12
3
4
5
6
7
8
9
10
stages
monitor wellhorizontal treatment well path
Microseimic events with AVAZ overlay
Seismic azimuthal anisotropy (AVAZ) intensity and orientation with microseismic overlay
(courtesy of Keith Young)(courtesy of Keith Young)
Fundamental Lamé parameters of rigidity μ and incompressibility λcontrol seismic wave propagation used to extract information about rocks in the Earth.
These parameters link many fields in Geoscience at different scales, from Petroleum Exploration to Earthquake Seismology.
Common contradictory formulations are simplified by restating equations in Lamé parameters e.g. incompressibility Lambda is less confusing than Bulk Modulus in revealing log errors.
Improved petrophysical discrimination and insight linking rock reservoir properties, pore fluid parameters and their seismic responses.
Insight into angle dependent reflectivity and impedance equations reveals the significance of 45º.
More descriptive sensitivity in AVO crossplotting using inverted λρ, μρseismic attributes to isolate effects of lithology, porosity and fluids.
Conclusions
Thomsen’s delta involved in describing VTI and HTI anisotropy is a function of the difference between diagonal and off-diagonal lambda despite Thomsen’s claim that “…lambda has no common name, since it is not useful for much in geophysics…” (DISC 2002)
More intuitive and appropriate for in-situ Geomechanical Closure Stress description of rocks and fractures (brittle vs. ductile deformation) than Young’s modulus and Poisson ratio.
Successful application of seismic attributes in predicting “sweet spots”for unconventional gas shale drilling that intersected predicted fractures and Stimulated Perm as confirmed by Microseismic monitoring.
Potential to extract Microseismic event attributes using Moment tensor solutions from Earthquake Seismology for stress drop on fractureplanes created through well completions.
Conclusions
AcknowledgementsDave Mackidd, Dave Cooper, John Varsek, Christian AbacoEric Keyser, John Parkin, Taiwen Chen, Weimin Zhang, Keith Young, Pete Smith, Kevin Smith, Dan Potocki (EnCana)
Marco Perez (Apache Canada)
Brian Hoffe (Shell Canada)
Dave Gray, Jon Downton (CGGVeritas)
Peter Duncan, Chris Neale (Microseismic Inc.)
Shawn Maxwell (Schlumberger)
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