Interpreting azimuthal Fourier coefficients for anisotropic and fracture parameters

Abstract

Amplitude variation with offset and azimuth (AVOAz) analysis can be separated into two separate parts: amplitude variation with offset (AVO) analysis and amplitude variation with azimuth (AVAz) analysis. Useful information about fractures and anisotropy can be obtained just by examining the AVAz. The AVAz can be described as a sum of sinusoids of different periodicities, each characterized by its magnitude and phase. This sum is mathematically equivalent to a Fourier series, and hence the coefficients describing the AVAz response are azimuthal Fourier coefficients (FCs). This FC parameterization is purely descriptive. The aim of this paper is to help the interpreter understand what these coefficients mean in terms of anisotropic and fracture parameters for the case of P-wave reflectivity using a linearized approximation. The FC representation is valid for general anisotropy. However, to gain insight into the significance of FCs, more restrictive assumptions about the anisotropy or facture system must be assumed. In the case of transverse isotropic media with a horizontal axis of symmetry, the P-wave reflectivity linearized approximation may be rewritten in terms of azimuthal FCs with the magnitude and phase of the different FCs corresponding to traditional AVAz attributes. Linear slip theory is used to show that the FCs can be interpreted similarly for the cases of a single set of parallel vertical fractures in isotropic media and in transverse isotropic media with a vertical axis of symmetry (VTI). The magnitude of the FCs depends on the fracture weakness parameters and the background media. For the case of vertical fractures in a VTI background, the AVOAz inverse problem is underdetermined, so extra information must be incorporated to determine how the weights are modified due to this background anisotropy. We evaluated this on a 3D data set from northwest Louisiana for which the main target was the Haynesville shale.

Polynomial amplitude versus azimuth inversion in horizontally transverse isotropic media, as tested on fractured coal seams in the Surat Basin

“…This is also evident as the anelliptic component of Eqn 1 is 1 2 ½D" ðV Þ À Dd ðV Þ sin 2 i tan 2 i cos 4 . Because " (V) does not occur in the elliptic or isotropic components of Rüger's equation, the Fourier method does not easily separate the elliptic component of the anisotropy from the anelliptic component because it is a component of all Fourier coefficients the terms r 0 , r 2 and r 4 which are introduced in (Downton and Roure 2015). In contrast, E and B 1 ani exactly represent elliptic components of the reflectivity and AVO gradient in HTI media respectively, making the methodology presented herein a robust technique of calculating the elliptic and anelliptic anisotropic components.…”

“…The r 2 coefficient (which is introduced by Downton and Roure (2015)) approximates the AVO gradient, B ani , whereby, as reported by Downton and Roure (2015) B ani % 2sin À2 r 2 ðiÞ: ð24Þ…”

“…The azimuth is defined as = az -sym (Rüger 1998) where az is an arbitrary azimuth and sym is the symmetry azimuth of the anisotropy. This can be calculated using the elliptic approximation (Rüger 1998) or by using a Fourier transform (Downton and Roure 2015).…”

“…A high-order Fourier coefficient term, r 4 , can be calculated from Eqn 23 (Downton and Roure 2015) that corresponds to an anelliptic term, Dh (V) /16 where h ðV Þ ¼ " ðV Þ Àd ðV Þ 1þ2d ðV Þ (Alkhalifah and Tsvankin 1995;Downton and Roure 2015). However, if weak anisotropy is assumed, as is the case of the parametrisation of Eqn 23 (Pšencík and Martins 2001), then the anelliptic term in weak anisotropy is h ðV Þ ¼ 1 2 ½" ðV Þ À d ðV Þ (Bakulin et al 2000).…”

“…Rüger (1998) introduced the reflectivity equation in HTI media assuming weak anisotropy. Several AVAZ methods of characterising azimuthal variation in seismic exist, such as assuming ellipticity (Rüger 1998) and using azimuthal Fourier transformations (Downton and Roure 2015).…”