An Asymptotic Model of Seismic Reflection from a Permeable Layer

Abstract

Analysis of compression wave propagation in a poroelastic medium predicts a peak of reflection from a high-permeability layer in the low-frequency end of the spectrum. An explicit formula expresses the resonant frequency through the elastic moduli of the solid skeleton, the permeability of the reservoir rock, the fluid viscosity and compressibility, and the reservoir thickness. This result is obtained through a low-frequency asymptotic analysis of Biot's model of poroelasticity. A review of the derivation of the main equations from the Hooke's law, momentum and mass balance equations, and Darcy's law suggests an alternative new physical interpretation of some coefficients of the classical poroelasticity. The velocity of wave propagation, the attenuation factor, and the wave number are expressed in the form of power series with respect to a small dimensionless parameter. The absolute value of this parameter is equal to the product of the kinematic reservoir fluid mobility and the wave frequency. Retaining only the leading terms of the series leads to explicit and relatively simple expressions for the reflection and transmission coefficients for a planar wave crossing an interface between two permeable media, as well as wave reflection from a thin highly permeable layer (a lens). Practical applications of the obtained asymptotic formulae are seismic modeling, inversion, and attribute analysis. Keywords Hooke's law · Darcy's law · Poroelasticity · Low frequency · Permeability · Asymptotic analysis · Seismic imaging This work is dedicated to the memory of V. M. Entov.

Low-frequency seismic analysis and direct hydrocarbon indicators

“…Figure 5: Pattern of P-wave reflections from thin permeable layer (layer 2) based on converted fast-slow-fast waves (Silin and Goloshubin, 2010) ..... 7 (Deegan et al, 1977;Casey et al, 1993;Lee et al, 1993;Blystad et al, 1995;Brasher et al, 2013;Rossland et al, 2013;Zhong, 2014 Madonna et al (2010) and Nakagawa et al (2016). Numerical modeling studies have been published by e.g.…”

“…Numerical modeling studies have been published by e.g. Castagna et al (2003), Carcione et al (2010), Silin and and Goloshubin (2010), Chen et al (2012), Lefeuve-Mesgouez et al (2012) and Huang et al (2015). Several authors have observed low-frequency anomalies associated with hydrocarbon reservoirs in reflection-seismic exploration data, e.g.…”

Converted slow waves in heterogeneous permeable fluid-saturated medium

“…Изменения пористости, скорости P-волны и S-волны внутри пласта. модули минералов, формирующих скелет горной породы [Silin, Goloshubin, 2010]. Проницаемость вмещающей сре-ды была взята равной 0.1 мД, что соответствует значениям проницаемости глинистых сланцев.…”

“…В асимптотическом представлении теории пороупру-гости Био [Silin, Goloshubin, 2010] коэффициенты отраже-ния и преломления на проницаемой границе могут быть представлены в виде ряда по малому безразмерному пара-метру ε = (iρ f kω/η), где ρ f -плотность флюида, k -про-ницаемость, η -вязкость флюида, ω -угловая частота, i -мнимая единица. Если в случае нормального падения на границу быстрой Р-волны коэффициенты отражения и прохождения быстрой Р-волны обозначить как R FF и T FF , а коэффициент отражения и прохождения обменной медлен-ной волны -как R FS и T FS , то формулы для их расчета можно записать в виде …”

Frequency-dependent seismic reflection coefficient for discriminating gas reservoirs

“…The mass and momentum balance imply that the skeleton displacement, the Darcy velocity of the fluid, the total stress, and the fluid pressure must be continuous at the interface. The asymptotic expressions of normal-incidence reflection and transmission coefficients from this interface are (Silin and Goloshubin, 2010) …”