# Difference between revisions of "1st Lamé parameter"

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

${\displaystyle \lambda =\rho (V_{\mathrm {P} }^{2}-2V_{\mathrm {S} }^{2})}$

## 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':

${\displaystyle \lambda ={\frac {E\nu }{(1+\nu )(1-2\nu )}}}$

The 'fluid substitution perspective' casts λ in terms of bulk modulus and shear modulus μ:

${\displaystyle \lambda =K-{\frac {2}{3}}\mu }$

## 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:

${\displaystyle \lambda \rho =I_{\mathrm {P} }^{2}-2I_{\mathrm {S} }^{2}}$
${\displaystyle \mu \rho =I_{\mathrm {S} }^{2}}$

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].

## In terms of Hooke's law

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,

${\displaystyle \varepsilon ={\frac {E}{\sigma }}}$.

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.

${\displaystyle \varepsilon _{1}'={\frac {1}{E}}\sigma _{1},\ \ \ \ \varepsilon _{2}'=-{\frac {\nu }{E}}\sigma _{1},\ \ \ \ \varepsilon _{3}'=-{\frac {\nu }{E}}\sigma _{1}}$,

We get similar equations to the loads in directions 2 and 3,

${\displaystyle \varepsilon _{1}''=-{\frac {\nu }{E}}\sigma _{2},\ \ \ \ \varepsilon _{2}''={\frac {1}{E}}\sigma _{2},\ \ \ \ \varepsilon _{3}''=-{\frac {\nu }{E}}\sigma _{2}}$,

and

${\displaystyle \varepsilon _{1}'''=-{\frac {\nu }{E}}\sigma _{3},\ \ \ \ \varepsilon _{2}'''=-{\frac {\nu }{E}}\sigma _{3},\ \ \ \ \varepsilon _{3}'''={\frac {1}{E}}\sigma _{3}}$.

Summing the three cases together (${\displaystyle \varepsilon _{i}=\varepsilon _{i}'+\varepsilon _{i}''+\varepsilon _{i}'''}$) we get

${\displaystyle \varepsilon _{1}={\frac {1}{E}}(\sigma _{1}-\nu (\sigma _{2}+\sigma _{3}))}$
${\displaystyle \varepsilon _{2}={\frac {1}{E}}(\sigma _{2}-\nu (\sigma _{1}+\sigma _{3}))}$
${\displaystyle \varepsilon _{3}={\frac {1}{E}}(\sigma _{3}-\nu (\sigma _{1}+\sigma _{2}))}$

or by adding and subtracting one ${\displaystyle \nu \sigma }$

${\displaystyle \varepsilon _{n}={\frac {1}{E}}((1+\nu )\sigma _{n}-\nu (\sigma _{1}+\sigma _{2}+\sigma _{3}))}$

and further we get by solving ${\displaystyle \sigma _{1}}$

${\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {\nu }{1+\nu }}(\sigma _{1}+\sigma _{2}+\sigma _{3})}$.

Calculating the sum

${\displaystyle \sum _{i=1,2,3}\varepsilon _{i}={\frac {1}{E}}((1+\nu )\sum _{i=1,2,3}\sigma _{i}-3\nu (\sum _{i=1,2,3}\sigma _{i}))={\frac {1-2\nu }{E}}\sum _{i=1,2,3}\sigma _{i}}$
${\displaystyle \sigma _{1}+\sigma _{2}+\sigma _{3}={\frac {E}{1-2\nu }}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})}$

and substituting it to the equation solved for ${\displaystyle \sigma _{1}}$ gives

${\displaystyle \sigma _{1}={\frac {E}{1+\nu }}\varepsilon _{1}+{\frac {E\nu }{(1+\nu )(1-2\nu )}}(\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})}$,

which simplifies if we substitute ${\displaystyle \mu }$ and ${\displaystyle \lambda }$, the Lamé parameters.

${\displaystyle \sigma _{1}=2\mu \varepsilon _{1}+\lambda (\varepsilon _{1}+\varepsilon _{2}+\varepsilon _{3})\ }$.

Similar treatment of directions 2 and 3 gives the Hooke's law in three dimensions.

## 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. 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.