1st Lamé parameter

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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

\lambda = \rho(V_\mathrm{P}^2 - 2 V_\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':

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

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

\lambda = K - \frac23 \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°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:

\lambda\rho = I_\mathrm{P}^2 - 2I_\mathrm{S}^2
\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].

How can we understand lambda?

References

  1. 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. 2.02.1 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 AVOAmplitude vs Offset 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.

\dagger \ X = \sqrt{9\lambda^2 + 2E\lambda + E^2}

(V_\mathrm{P},\,V_\mathrm{S}) (\mu,\,\lambda) (E,\,\lambda)\,\dagger (E,\,\mu) (K,\,\lambda) (K,\,\mu) (K,\,E) (\nu,\,\lambda) (\nu,\,\mu) (\nu,\,E) (\nu,\,K)
P-wave velocity
V_\mathrm{P}=\,
V_\mathrm{P} \sqrt{\tfrac{\lambda+2\mu}{\rho}} \sqrt{\tfrac{E - \lambda + X}{2\rho}} \sqrt{\tfrac{\mu\,(E-4\mu)}{\rho\,(E-3\mu)}} \sqrt{\tfrac{3K - 2\lambda}{\rho}} \sqrt{\tfrac{K+\frac43\mu}{\rho}} \sqrt{\tfrac{3K \left(3K + E\right)}{\rho \left(9K - E\right)}} \sqrt{\tfrac{\lambda (1 - \nu)}{\rho\nu}} \sqrt{\tfrac{2 \mu (1-\nu)}{\rho (1 - 2\nu)}} \sqrt{\tfrac{E(1 - \nu)}{\rho(1 + \nu)(1 - 2\nu)}} \sqrt{\tfrac{3K(1 - \nu)}{\rho(1 + \nu)}}
S-wave velocity
V_\mathrm{S}=\,
V_\mathrm{S} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{E - 3\lambda + X}{4\rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{3 (K - \lambda)}{2 \rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{- \tfrac{3EK}{\rho \left(E - 9K\right)}} \sqrt{\tfrac{\lambda}{2 \nu \rho} - \tfrac{\lambda}{\rho}} \sqrt{\tfrac{\mu}{\rho}} \sqrt{\tfrac{E}{2\rho(1+\nu)}} \sqrt{-\tfrac{3K (2\nu - 1)}{2\rho (\nu + 1)}}
Velocity ratio
\Gamma=\,
\frac{V_\mathrm{P}}{V_\mathrm{S}} \sqrt{\tfrac{\lambda+2\mu}{\mu}} \sqrt{\tfrac{3E + 3\lambda + X}{2E}} \sqrt{\tfrac{E - 4 \mu}{E - 3 \mu}} \sqrt{\tfrac{\tfrac43 \lambda - 2 K}{\lambda - K}} \sqrt{\tfrac{K+\frac43\mu}{\mu}} \sqrt{\tfrac{E + 3K}{E}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}} \sqrt{\tfrac{2\nu - 2}{2\nu - 1}}
1st Lamé parameter
\lambda=\,
\rho (V_\mathrm{P}^2 - 2V_\mathrm{S}^2) \lambda \lambda \tfrac{\mu(E-2\mu)}{3\mu-E} \lambda K-\tfrac{2\mu}{3} \tfrac{3K(3K-E)}{9K-E} \lambda \tfrac{2 \mu \nu}{1-2\nu} \tfrac{E\nu}{(1+\nu)(1-2\nu)} \tfrac{3K\nu}{1+\nu}
Shear modulus
\mu=\,
\rho V_\mathrm{S}^2 \mu \tfrac{E - 3\lambda + X}{4} \mu \tfrac{3(K-\lambda)}{2} \mu \tfrac{3KE}{9K-E} \tfrac{\lambda(1-2\nu)}{2\nu} \mu \tfrac{E}{2(1+\nu)} \tfrac{3K(1-2\nu)}{2(1+\nu)}
Young's modulus
E=\,
\tfrac{\rho V_\mathrm{S}^2 (3V_\mathrm{P}^2 - 4V_\mathrm{S}^2)}{V_\mathrm{P}^2 - V_\mathrm{S}^2} \tfrac{\mu(3\lambda + 2\mu)}{\lambda + \mu} E\ E \tfrac{9K(K-\lambda)}{3K-\lambda} \tfrac{9K\mu}{3K+\mu} E\ \tfrac{\lambda(1+\nu)(1-2\nu)}{\nu} 2\mu(1+\nu)\, E\ 3K(1-2\nu)\,
Bulk modulus
K=\,
\rho (V_\mathrm{P}^2 - \tfrac43 V_\mathrm{S}^2) \lambda+ \tfrac{2\mu}{3} \tfrac{E + 3\lambda + X}{6} \tfrac{E\mu}{3(3\mu-E)} K K K \tfrac{\lambda(1+\nu)}{3\nu} \tfrac{2\mu(1+\nu)}{3(1-2\nu)} \tfrac{E}{3(1-2\nu)} K
Poisson's ratio
\nu=\,
\tfrac{V_\mathrm{P}^2 - 2V_\mathrm{S}^2}{2(V_\mathrm{P}^2 - V_\mathrm{S}^2)} \tfrac{\lambda}{2(\lambda + \mu)} \tfrac{- E - \lambda + X}{4\lambda} \tfrac{E}{2\mu}-1 \tfrac{\lambda}{3K-\lambda} \tfrac{3K-2\mu}{2(3K+\mu)} \tfrac{3K-E}{6K} \nu \nu \nu \nu
P-wave modulus
M=\,
\rho V_\mathrm{P}^2 \lambda+2\mu\, \tfrac{E - \lambda + X}{2} \tfrac{\mu(4\mu-E)}{3\mu-E} 3K-2\lambda\, K+\tfrac{4\mu}{3} \tfrac{3K(3K+E)}{9K-E} \tfrac{\lambda(1-\nu)}{\nu} \tfrac{2\mu(1-\nu)}{1-2\nu} \tfrac{E(1-\nu)}{(1+\nu)(1-2\nu)} \tfrac{3K(1-\nu)}{1+\nu}