Oluwaseun Fadugba

Geodynamics 2016 CERI, The University of Memphis, TN

A Review of Seismic

Anisotropy

One thing I would like you to remember after this presentation

• Isotropy assumptions is an approximation of the earth’s system.

• A better knowledge of seismic anisotropy in our models makes our interpretations close to reality.

(Paulssen, 2015)

Motivation

(Paulssen, 2015)

An orthorhombic model used for thin horizontal layers with parallel vertical faults Tsvankin (1997).

Difference between observed and desired results

(Anderson, 1989).

Purpose of the Term Project

• To review seismic anisotropy in order to understand the seismic observations and

• How it is used in different fields of geoscience for better interpretation of observed anomalies

Presentation Outline

• Types of anisotropy

• Causes of anisotropy

• Evidence of anisotropy

• Governing equations • Case studies

• Summary and conclusion

Types of Anisotropy

Vertical Traverse Isotropy (VTI) • Isotropic media are stacked

vertically e.g. shales and other undisturbed sedimentary rocks.

• Fast velocities in horizontal direction

• Complicated for dipping layers.

Horizontal Traverse Isotropy (HTI) • characterizes vertical faulting

such as in fractured reservoir. • Faster in the vertical direction

and parallel to the strike of the fractures than when travelled across the rock boundaries. (Wild, 2011)

Causes of anisotropy

• Anderson (1989 and references therein) noted that when a mineral with dominant slip system undergoes homogeneous deformation

• This preferred orientation caused by deformation lead to anisotropy in the media. P-, SV- and SH- velocities are faster along the preferred orientation the perpendicular to it. e.g. olivine and orthopyroxene in the upper mantle.

• In igneous rocks, the preferred orientation can be caused by grain rotation due to the ambient stress field. recrystallization of minerals in a nonhydrostatic stress field

or in the presence of a thermal gradient. crystal setting in magma chambers, flow orientation and

dislocation-controlled slip. Fractures (Anderson, 1989).

In sedimentary rocks such as shales.

textural alignments due to applied stress to form alignment of platy clay minerals in a ‘clay domain’,

alignment of domains, alignment of non-spherical pores and micro-cracks

alignment of fractures at scales larger than the scale of pore and grains, but smaller than the seismic scale, and

fine-scale lamination of shaly minerals, silty materials and organic materials (Bandyopadhyay, 2009).

• In metamorphic rocks, Partial melting and segregation of melt-enriched bands

Evidence of anisotropy in the lithosphere and mantle

Shear wave splitting (S-wave birefringence)

Love/Rayleigh incompatibility. Pn velocities are azimuth dependent

Base-state flow in the deformation of partially

molten rocks

Shear wave splitting (S-wave birefringence) (1/4) the time delay between the arrivals of the shear waves: the

extent of the zone of anisotropy. the shear wave splitting parameter, φ: gives the azimuth of

the fastest shear wave.

Evidence of anisotropy in the lithosphere and mantle (Cont’d)

(Maureen and Becker, 2010)

Love/Rayleigh incompatibility (2/4) The observed Love wave velocities are significantly higher

than those predicted from Rayleigh waves, indicating the presence of significant radial anisotropy.

Evidence of anisotropy in the lithosphere and mantle (Cont’d)

Average dispersion curves for Rayleigh and Love waves (grey shading and error bars) (Pederson et al., 2006).

Pn velocities are azimuth dependent (3/4)

Evidence of anisotropy in the lithosphere and mantle (Cont’d)

The deviation from mean velocity of 8.159 km/s with azimuth. Zero degrees coincide with the strike of the mid-ocean ridge

Schematic diagram of a divergent plate boundary showing the symmetry axes of olivine.

(Anderson, 1989)

Base-state flow in the deformation of partially molten rocks (4/4)

Evidence of anisotropy in the lithosphere and mantle (Cont’d)

E

(Qui et al., 2015)

Governing Equations

𝐶𝑖𝑗𝑘𝑙𝜕𝑗𝜕𝑘𝑢𝑙 = 𝜌𝜕𝑡𝑡𝑢𝑖

The wave equation in homogeneous anisotropic medium

The plane wave solution of the above partial differential wave equation

𝑢 𝑟 , 𝑡 = 𝑎 𝑓(𝑡 − 𝑛. 𝑟

𝑐)

𝐶𝑖𝑗𝑘𝑙𝑎𝑙𝑛𝑗𝑛𝑘 1

𝑐2= 𝜌𝑎𝑖

This equation can be rewritten as:

𝑚𝑖𝑙𝑎𝑓 = 𝑐2𝑎𝑖

Where 𝑚𝑖𝑙 = 1

𝜌 𝐶𝑖𝑗𝑘𝑙𝑛𝑗𝑛𝑘 . mil is referred to as Christoffel matrix.

(Paulssen, 2015)

• The Christoffel matrix, mil, can be used to determine the P-, SV- and SH- wave velocities along different directions, n by solving the equation:

𝑀𝑖𝑙 − 𝑉2𝛿𝑖𝑙 𝑢𝑙 = 0 • The velocities are the square root of the eigenvalues of the

Christoffel matrix.

• The stiffness tensor is a 4th order symmetric tensor with twenty-one independent elements. The notation of Cijkl can also be written in just two indexes using the following rules:

11 12 13. 22 23. . 33

→ 1 6 5. 2 4. . 3

For example: in hexagonal medium with the vertical (x3) symmetry axis, the stiffness tensor is given as:

If the direction n = (1 0 0) i.e. the x-direction, the Christoffel matrix is given as:

𝑚𝑖𝑙 = 1

𝜌 𝐶𝑖𝑗𝑘𝑙𝑛𝑗𝑛𝑘 =

1

𝜌 𝐶𝑖11𝑙

𝑀 = 1

𝜌

𝐶1111 𝐶1112 𝐶1113𝐶2111 𝐶2112 𝐶2113𝐶3111 𝐶3112 𝐶3113

= 1

𝜌

𝐶11 𝐶16 𝐶15𝐶61 𝐶66 𝐶65𝐶51 𝐶56 𝐶55

𝐷𝑖𝑎𝑔𝑜𝑛𝑎𝑙𝑖𝑧𝑒𝑑 𝑀 = 1

𝜌 𝐴 0 00 𝑁 00 0 𝐿

The square root of the eigenvalues of the Christoffel matrix gives the P-, SV- and SH- wave velocities as:

𝛼𝐻 = 𝐴

𝜌 ; 𝛽𝐻 =

𝑁

𝜌 𝑎𝑛𝑑 𝛽𝑉 =

𝐿

𝜌

• Also, if n is (0 0 1) i.e. in the vertical direction, the Christoffel matrix is given as:

𝑚𝑖𝑙 = 1

𝜌 𝐶𝑖𝑗𝑘𝑙𝑛𝑗𝑛𝑘 =

1

𝜌 𝐶𝑖33𝑙

𝑀 = 1

𝜌 𝐿 0 00 𝐿 00 0 𝐶

• The square root of the eigenvalues of the Christoffel matrix

gives the P-, SV- and SH- wave velocities as:

𝛼𝑉 = 𝐶

𝜌 𝑎𝑛𝑑 𝛽𝑉 =

𝐿

𝜌

• A typical stiffness tensor for hexagonal symmetry has A > C

and N > L. Therefore, the P- and S-wave velocities in the horizontal direction are greater than the P- and S-wave velocities in the vertical direction, respectively.

(Paulssen, 2015)

(Paulssen, 2015)

(Anderson, 1989)

Case studies

• Uses of anisotropy in the oil and gas industries

• Deformation of partially melted rocks

Uses of anisotropy in the oil and gas industries

• In reflection seismic survey, the main method is to measure the travel times it takes seismic energy from the seismic source such as vibrator or weight drop in land survey, air gun in marine, to the receivers.

• The migrated data shows a lot of crests and troughs in travel time

• The interpretation of the observed signatures in terms of rock types and containing fluids is challenging, hence well-tie technique.

(Obiadi and Obiadi, 2016)

• Walk-away VSP is often used in practice as it can cover wider area

and is cheap to implement. In walk-away VSP, a number of shots will be fired along a horizontal line into receivers located at regular intervals in a borehole.

(Wild ,2011)

• Due to anisotropy, the synthetic seismograms derived from the borehole data correlate with the migrated seismic section near offset, where the isotropic theory holds, but little or no correlation at far offset.

• Wild et al. (2008) introduced anisotropy in the synthetic model by first verticalizing the borehole derived velocities (deviated propagation direction) to the vertical velocity direction using the previous knowledge of the reservoir geometry.

• The reservoir contains near-vertical fractures which can be modelled as HTI anisotropy.

(Wild ,2011)

Deformation of partially melted rocks

(Takei and Katz, 2013)

• The theory of two-phase flow describes the flow of a low-viscosity liquid (melt) through a permeable and viscously deformable solid matrix (grains).

• The low angle of the melt-enriched bands cannot be explained with isotropic viscosity of the melt.

• When viscous anisotropy was considered in experiment in torsional flow, liquid segregates toward lower stress (towards the center). This is referred to as base-state flow and does not occur in the isotropic system.

Takei and Katz (2013) recognized two major differences between isotropic and anisotropic systems. A coupling between shear and volumetric components of stress

and strain rate under viscous anisotropy. They considered a two-dimensional case where 𝑒 𝑧𝑧

∗ = 0. • For an anisotropic system, volumetric stress cannot be directly

related to the volumetric strain rate.

Schematic diagram of the coupling effect caused by viscous anisotropy. (a)Shear stress with zero

volumetric stress causes an expansion while in

(b) an applied shear strain with zero volumetric strain rate, causes a compressive stress.

(Takei and Katz, 2013)

• Qui et al. (2015) tested the viscous anisotropy hypothesis for partially molten rocks using olivine crystals mixed with some alkaline basalt of Hawaii. They performed an experiment with an initially isotropic cylindrical sample and added a uniform melt fraction in a sealed chamber. They deformed the sample in torsion at a constant twist rate.

E

(Qui et al., 2015)

Summary

(Wild, 2011)

(Qui et al., 2011) (Anderson, 1989)

(Paulssen, 2015)

• Isotropy assumptions is an approximation of the earth’s system.

• A better knowledge of seismic anisotropy in our models makes our interpretations close to reality.

(Paulssen, 2015)

References

Anderson, Don L. Theory of the Earth. Boston: Blackwell Scientific Publications, 1989. http://resolver.caltech.edu/CaltechBOOK:1989.001 Bandyopadhyay Kaushik (2009). Seismic anisotropy: geological causes and its implications to reservoir geophysics. PhD Dissertation at Stanford University. Gilligan Amy, Ian D. Bastow, Emma Watson, Fiona A. Darbyshire, Vadim Levin, William Menke, Victoria Lane, David Hawthorn, Alistair Boyce, Mitchell V. Liddell and Laura Petrescu (2016). Lithospheric deformation in the Canadian Appalachians: evidence from shear wave splitting. Geophys. J. Int. vol 206, p. 1273-1280. Lythgoe K.H., A. Deuss, J.F Rudge and J.A. Neufeld (2014). Earth’s inner core: Innermost inner core or hemispherical variations? Earth and Planetary Science Letters vol. 385, pp. 181-189 Obiadi Emina R and CM Obiadi (2016). Evaluation and prospect identification in the Olive Field, Niger Delta Basin, Nigeria. Journal of Petroleum and Environmental Biotechnology vol. 7: 284. doi:10.4172/2157-7463.1000284 Paulssen Hanneke (2015). Theoretical Seismology note: Seismic Anisotropy. Universiteit Utrecht. Pederson A. Helle, Marianne Bruneton, Valerie Maupin and SVEKALAPKO Seismic Tomography Working Group (2006). Lithospheric and sublithospheric anisotropy beneath the Baltic shield from surface-wave array analysis. Earth and Planetary Science Letters vol 244 pp. 590-605.

Qui Chao, David L. Kohlstedt, Richard F. Katz, and Yasuko Takei (2015) Experimental test of the viscous anisotropy hypothesis for partially molten rocks. Proceedings of the National Academy of Sciences of the United States of America. vol 112, no 41, pp. 12616 - 12620. Simons Frederick and Rob D. van der Hilst (2003), Seismic and mechanical anisotropy and the past and present deformation of the Australian lithosphere, Earth and planetary Science Letters 211 (2003) 271-286 Takei Yasuko and Richard F. Katz (2013). Consequences of viscous anisotropy in a deforming, two-phase aggregate. Part 1. Governing equations and linearized analysis. J. Fluid Mech. Vol. 734, pp. 424-455. Tommasi Andrea, Marguerite Godard, Guilhem Coromina, Jean-Marie Duatria and Hans Barsczus (2004). Seismic anisotropy and compositionally induced velocity anomalies in the lithosphere above mantle plumes: a petrological and microstructural study of mantle xenoliths from French Polynesia. Earth and Planetary Sci Letters. vol. 227, p. 539-556. Tsvankin Llay (1997). Anisotropy parameters and P-wave velocity for orthorhombic media. Geophysics, vol. 62, No 4: p. 1292-1309 Wild Phillip (2011). Practical applications of seismic anisotropy. First break volume 29. Pp. 117-124