Difference between revisions of "1st Lamé parameter"

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:<math>\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}</math>
 
:<math>\lambda = \frac{E\nu}{(1+\nu)(1-2\nu)}</math>
  
The '[[fluid substitution perspective]]' casts ''&lambda;'' in terms of [[bulk modulus]] and [[shear modulus]] ''&mu;'':
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The '[[fluid substitution]] perspective' casts ''&lambda;'' in terms of [[bulk modulus]] and [[shear modulus]] ''&mu;'':
  
 
:<math>\lambda = K - \frac23 \mu</math>
 
:<math>\lambda = K - \frac23 \mu</math>
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This approach—estimating Lamé's parameters indirectly via impedance—is problematic<ref name=avseth>Avseth, P, T Mukerji and G Mavko (2006), ''Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk'', Cambridge University Press.</ref> so Gray recommended estimating ''&lambda;'' and ''&mu;'' contrasts directly from seismic data<ref>Gray, D, B Goodway, and T Chen (1999), Bridging the gap: Using AVO to detect changes in fundamental elastic constants, SEG Annual Meeting, Expanded Abstracts, 852–855.</ref>. See discussion in Avseth et al<ref name=avseth />.
 
This approach—estimating Lamé's parameters indirectly via impedance—is problematic<ref name=avseth>Avseth, P, T Mukerji and G Mavko (2006), ''Quantitative Seismic Interpretation: Applying Rock Physics Tools to Reduce Interpretation Risk'', Cambridge University Press.</ref> so Gray recommended estimating ''&lambda;'' and ''&mu;'' contrasts directly from seismic data<ref>Gray, D, B Goodway, and T Chen (1999), Bridging the gap: Using AVO to detect changes in fundamental elastic constants, SEG Annual Meeting, Expanded Abstracts, 852–855.</ref>. See discussion in Avseth et al<ref name=avseth />.
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==How can we understand lambda?==
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{{divhide|Click SHOW to open &gt; &gt; &gt; }}
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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,
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:<math>\varepsilon = \frac{E}{\sigma}</math>.
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We can extend this to three dimensions, but then we need [[Poisson's ratio]] ''&nu;'', which accounts for the change in shape in the cross-sectional plane.
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:<math>\varepsilon_1' = \frac{1}{E}\sigma_1,\ \ \ \ \varepsilon_2' = -\frac{\nu}{E}\sigma_1,\ \ \ \ \varepsilon_3' = -\frac{\nu}{E}\sigma_1</math>,
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We get similar equations to the loads in directions 2 and 3,
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:<math>\varepsilon_1'' = -\frac{\nu}{E}\sigma_2,\ \ \ \ \varepsilon_2'' = \frac{1}{E}\sigma_2,\ \ \ \ \varepsilon_3'' = -\frac{\nu}{E}\sigma_2</math>,
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and
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:<math>\varepsilon_1''' = -\frac{\nu}{E}\sigma_3,\ \ \ \ \varepsilon_2''' = -\frac{\nu}{E}\sigma_3,\ \ \ \ \varepsilon_3''' = \frac{1}{E}\sigma_3</math>.
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Summing the three cases together (<math>\varepsilon_i = \varepsilon_i' + \varepsilon_i'' +\varepsilon_i'''</math>) we get
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:<math>\varepsilon_1 = \frac{1}{E}(\sigma_1-\nu(\sigma_2+\sigma_3))</math>
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:<math>\varepsilon_2 = \frac{1}{E}(\sigma_2-\nu(\sigma_1+\sigma_3))</math>
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:<math>\varepsilon_3 = \frac{1}{E}(\sigma_3-\nu(\sigma_1+\sigma_2))</math>
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or by adding and subtracting one <math>\nu\sigma</math>
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:<math>\varepsilon_n = \frac{1}{E}((1+\nu)\sigma_n-\nu(\sigma_1+\sigma_2+\sigma_3))</math>
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and further we get by solving <math>\sigma_1</math>
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:<math>\sigma_1 = \frac{E}{1+\nu}\varepsilon_1 + \frac{\nu}{1+\nu}(\sigma_1+\sigma_2+\sigma_3)</math>.
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Calculating the sum
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:<math>\sum_{i=1,2,3}\varepsilon_i = \frac{1}{E}((1+\nu)\sum_{i=1,2,3}\sigma_i - 3\nu(\sum_{i=1,2,3}\sigma_i)) = \frac{1-2\nu}{E}\sum_{i=1,2,3}\sigma_i</math>
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:<math> \sigma_1 +\sigma_2+\sigma_3 = \frac{E}{1-2\nu}(\varepsilon_1 + \varepsilon_2 +\varepsilon_3)</math>
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and substituting it to the equation solved for <math>\sigma_1</math> gives
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:<math>\sigma_1 = \frac{E}{1+\nu}\varepsilon_1 + \frac{E\nu}{(1+\nu)(1-2\nu)}(\varepsilon_1 + \varepsilon_2 +\varepsilon_3)</math>,
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which simplifies if we substitute <math>\mu</math> and <math>\lambda</math>, the [[Lamé parameters]].
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:<math>\sigma_1 = 2\mu\varepsilon_1 + \lambda(\varepsilon_1 + \varepsilon_2 +\varepsilon_3)\ </math>.
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Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.
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:<math>
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  \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}
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  = \cfrac{E}{(1+\nu)(1-2\nu)}
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  \begin{bmatrix} 1-\nu & \nu & \nu & 0 & 0 & 0 \\
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                  \nu & 1-\nu & \nu & 0 & 0 & 0 \\
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                  \nu & \nu & 1-\nu & 0 & 0 & 0 \\
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                  0 & 0 & 0 & (1-2\nu)/2 & 0 & 0 \\
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                  0 & 0 & 0 & 0 & (1-2\nu)/2 & 0 \\
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                  0 & 0 & 0 & 0 & 0 & (1-2\nu)/2 \end{bmatrix}
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    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix}
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</math>
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which expression can be simplified thanks to the Lamé constants :
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:<math>
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  \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \\ \sigma_{23} \\ \sigma_{31} \\ \sigma_{12} \end{bmatrix}
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  =
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  \begin{bmatrix} 2\mu+\lambda & \lambda & \lambda & 0 & 0 & 0 \\
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                  \lambda & 2\mu+\lambda & \lambda & 0 & 0 & 0 \\
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                  \lambda & \lambda & 2\mu+\lambda & 0 & 0 & 0 \\
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                  0 & 0 & 0 & \mu & 0 & 0 \\
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                  0 & 0 & 0 & 0 & \mu & 0 \\
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                  0 & 0 & 0 & 0 & 0 & \mu \end{bmatrix}
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    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \\ 2\varepsilon_{23} \\ 2\varepsilon_{31} \\ 2\varepsilon_{12} \end{bmatrix}
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</math>
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Can we simplify further by just considering the principal axes?
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:<math>
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  \begin{bmatrix}\sigma_{11} \\ \sigma_{22} \\ \sigma_{33} \end{bmatrix}
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  =
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  \begin{bmatrix} 2\mu+\lambda & \lambda & \lambda\\
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                  \lambda & 2\mu+\lambda & \lambda\\
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                  \lambda & \lambda & 2\mu+\lambda\\ \end{bmatrix}
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    \begin{bmatrix}\varepsilon_{11} \\ \varepsilon_{22} \\ \varepsilon_{33} \end{bmatrix}
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</math>
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{{divhide|end}}
  
 
==References==
 
==References==
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*[[Wikipedia:First Lamé parameter|First Lamé parameter]] — Wikipedia entry
 
*[[Wikipedia:First Lamé parameter|First Lamé parameter]] — Wikipedia entry
  
{{stub}}
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{{Elastic modulus|{{PAGENAME}}}}
  
 
[[Category:Rock physics]]
 
[[Category:Rock physics]]

Latest revision as of 19:32, 24 September 2012

Gabriel Lamé

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 VP and VS

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°API 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?

References

  1. Goodway, B, T Chen, and J Downton (1997), Improved AVO 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. 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.
  3. Gray, D, B Goodway, and T Chen (1999), Bridging the gap: Using AVO to detect changes in fundamental elastic constants, SEG Annual Meeting, Expanded Abstracts, 852–855.

External links

Conversion formulas — edit
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.

P-wave velocity
S-wave velocity
Velocity ratio
1st Lamé parameter
Shear modulus
Young's modulus
Bulk modulus
Poisson's ratio
P-wave modulus