Waveform inversion and rock physics methods for quantitative 4D seismics

Morten Jakobsen, University of Bergen and IRIS

Introduction

Fahimuddin (2010):

Outline• Acoustic waveform inversion

• for Vp

• for changes in Vp

• Elastic waveform inversion

• for elastic parameters

• for rock physics parameters

• Uncertainties in the rock physics model

The scalar wave equation

Formal solution

Scattering integral equation

Scattering integral equation

Operator notation

Operator notation

The transition operator

The transition operator

The forward seismic modelling problem

R

V

S

The forward seismic modelling problem

R

V

S

T-matrix perspective

T-matrix perspective

T-matrix vs finite difference

Distorted Born iterative (DBIT) T-matrix inversion method

Distorted Born iterative (DBIT) T-matrix inversion method

Distorted Born iterative (DBIT) T-matrix inversion method

Distorted Born iterative (DBIT) T-matrix inversion method

Marmousi model

21

22

SEG/EAGE salt model

23

24

Inversion of time-lapse seismic waveform data

Muhumuhza (2015).

The Norne field

26

2D Norne baseline model

27

2D Norne monitor model

28

2D Norne time-lapse model

29

Numerical experiment

• The DBIT inversion method in differential time-lapse mode will be used in an attempt to reconstruct the monitor model and the corresponding time-lapse changes, with the baseline model as the initial model.

• The grid sizes for modelling and inversion is (25m x 5m) and (100m x 5m), respectively.

• Local minima are avoided by using a multi-scale regularization technique with the following selected frequencies: (1, 3, 5, 7.5, 12, 15, 18, 20) Hz. 30

Frequency domain seismic waveform data at 20 Hz

31

Inverted monitor model at 1 HZ

32

Inverted monitor model at 3 Hz

33

Inverted monitor model at 5 Hz

34

Inverted monitor model at 7.5 Hz

35

Inverted monitor model at 12 Hz

36

Inverted monitor model at 15 Hz

37

Inverted monitor model at 18 Hz

38

Inverted monitor model at 20 Hz

39

2D Norne monitor model

40

Inverted time-lapse model

41

2D Norne time-lapse model

42

Effects of random noise

43

The true model

Inverted model at 40 Hz40 Hz

Velocity model inverted from noiseless seismic waveform data (DBIT)

Velocity model inverted from noisy seismic waveform data (DBIT, SN = 20 dB)

Elastic wave equation

Actual and reference media

Scattering integral equation

Born approximation

D

x

x’

Inversion friendly form

Isotropic media

Frechet-derivatives

Linear forward model

Linear inversion

DBIT inversion of noiseless seismic waveform data

DBIT inversion of noisy seismic waveform data

Seismic waveform inversion for rock physics parameters

Zooming in on a single grid block:

Pilskog, Lopez and Jakobsen (2014)

References• Pilskog, I., Lopez, M. and Jakobsen, M., 2015.

Linearized waveform inversion for fracture density. Extended abstract, 16th International Workshop on Seismic Anisotropy, Brazil.

• Jakobsen, M. and Pilskog, I., 2016. Gassmann-consistent Born inversion for fracture density. Extended abstract, 78th EAGE annual meeting, Vienna.

Born approximation for the effects of stiffness perturbations

D

x

x’

Seismic wave propagation in cracked porous media

Rock physical parametrization of the reference medium and perturbation

Frechet-derivative of the scattered wavefield wrt. the fracture density

Matrix representation

Regularized linear inversion

True vs inverted fracture density models

Elastic constants of the true model

Elastic constants of the inverted model obained by ignoring the effects of pore

fluid pressure communication

Elastic constants of the inverted model obtained including the effects of pore fluid

pressure communication

Uncertainties in the rock physics model

• Are Gassmann’s formulae always valid?

• How do we model pressure effects?

• How about clay content and other parameters.

• Contact theory vs inclusion models.

Unified inclusion model

Jakobsen et al. (2003)

The t-matrix of a single communicating cavity

Jakobsen and Hudson (2003)

Gassmann-consistent low-frequency limit

Predicted vs observed velocity and attenuation spectra of clayey sandstones

Predicted ultrasonic velocity and attenuation in clay-bearing sandstones

Rock physics modelling of log data from the Norne field

76

77

78

79

80

81

82

83

84

Field scale statistical analysis

85

Concluding remarks• The DBIT method of Jakobsen and Ursin (2015) =

a powerful tool for seismic FWI in time-lapse mode, suitable for uncertainty analysis (Eikrem et al., 2015).

• An elastic version of the DBIT method is currently under development.

• Rock physics can help FWI in anisotropic media.

• Uncertainties in the rock physics model can be dealt with in a systematic manner.