1st Lamé parameter
The 1st Lamé parameter, sometimes called Lamé's first parameter, but is more usually referred to simply as lambda, λ. It is an elastic modulus and used extensively in quantitative seismic interpretation and rock physics. It was first described by the French mathematician, Gabriel Lamé (right). Lamé's second parameter is equivalent to shear modulus, μ.
It is often said that λ has no physical interpretation, and most people find it hard to visualize.
In terms of V_{P} and V_{S}
Other expressions
λ can also be expressed in terms of Young's modulus, E, and Poisson's ratio, ν. This could be thought of as 'the engineer's perspective':
The 'fluid substitution perspective' casts λ in terms of bulk modulus and shear modulus μ:
Typical values
Rock | λ, GPa |
---|---|
Quartz | 8 |
Feldspar | 28 |
Calcite | 56 |
Dolomite | 65 |
Anhydrite | 26 |
Siderite | 90 |
Pyrite | 59 |
Sandstone, 10 pu | 1–3 |
Limestone, 10 pu | 18–53 |
Shale, 5 pu | 3–24 |
Brine | 2.3 |
Oil, 40°APIApplication programming interface | 1.6 |
Analysis and interpretation
Goodway and others^{[1]} have described an alternative to (or augmentation of) classic impedance inversion and interpretation. The parameters are closely related:
This approach—estimating Lamé's parameters indirectly via impedance—is problematic^{[2]} so Gray recommended estimating λ and μ contrasts directly from seismic data^{[3]}. See discussion in Avseth et al^{[2]}.
How can we understand lambda?
Extend a rod or a linear spring. Its extension (strain) is linearly proportional to its tensile stress σ, by a constant factor, the inverse of its Young's modulus E, hence,
- .
We can extend this to three dimensions, but then we need Poisson's ratio ν, which accounts for the change in shape in the cross-sectional plane.
- ,
We get similar equations to the loads in directions 2 and 3,
- ,
and
- .
Summing the three cases together () we get
or by adding and subtracting one
and further we get by solving
- .
Calculating the sum
and substituting it to the equation solved for gives
- ,
which simplifies if we substitute and , the Lamé parameters.
- .
Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.
which expression can be simplified thanks to the Lamé constants :
Can we simplify further by just considering the principal axes?
References
- ↑ Goodway, B, T Chen, and J Downton (1997), Improved AVOAmplitude vs Offset fluid detection and lithology discrimination using Lamé's petrophysical parameters, λρ, μρ, and λ/μ fluid stack from P and S inversions. SEG Annual Meeting, Expanded Abstracts, 183–186.
- ↑ ^{2.0}^{2.1} Avseth, P, T Mukerji and G Mavko (2006), Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk, Cambridge University Press.
- ↑ Gray, D, B Goodway, and T Chen (1999), Bridging the gap: Using AVOAmplitude vs Offset to detect changes in fundamental elastic constants, SEG Annual Meeting, Expanded Abstracts, 852–855.
External links
- First Lamé parameter — Wikipedia entry
Conversion formulas — edit | |||||||||||
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The elastic properties of homogeneous isotropic linear elastic materials are uniquely determined by any two moduli. Given any two, the others can thus be calculated. Key reference: Mavko, G, T Mukerji and J Dvorkin (2003), The Rock Physics Handbook, Cambridge University Press.
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P-wave velocity |
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S-wave velocity |
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Velocity ratio |
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1st Lamé parameter |
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Shear modulus |
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Young's modulus |
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Bulk modulus |
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Poisson's ratio |
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P-wave modulus |