CO2 sequestration crosswell monitoring basedupon spectral-element and adjoint methods

Christina Morency

Department of Geosciences, Princeton UniversityCollaborators: Jeroen Tromp & Yang Luo

Computational Geosciences Seminar Series- Sept. 27, 2010 - Stanford

Overview

1) Numerical simulation of wave propagation:spectral-element method (SEM)

2) Imaging and inversion:finite-frequency sensitivity kernels based on adjoint method

3) Application: CO2 sequestration monitoring

=> using seismic data for imaging & inversion of subsurface properties

Forward wave propagationbased on SEM

Three rheologies

Elastic:

Poroelastic [Biot, 1962]:

Acoustic, inviscid fluid and neglecting gravity effects [Chaljub and Valette, 2004;Komatitsch et al., 2005]:

where the displacement and the acoustic pressure

{B, C and M are the Biot coefficientsdefined in terms of solid, fluid, andframe properties

In an isotropic case:and

where

Three & 1/2 rheologies continued

Elastic with Gassmann’s formulae [Gassmann, 1951]:

In an isotropic case:and

Effective saturated bulk & shear moduli:

Poroelastic governing equations

Microscopic equations for the solid and fluid phase:• Conservation of mass• Constitutive relationships (Hooke’s law, Navier-Stokes)• Conservation of momentum

Macroscopic equations of the biphasic porous medium:

Averaging method (Pride & Berryman, 1998; Whitaker, 1999)

(i) Microscopic material properties are constant onthe scale of the averaging volume, but they can vary at themacroscale

(ii) Wavelengths of waves of interest are large compared to theaveraging volume

Characteristic parametersAcoustic:

Elastic:

Poroelastic:

1 compressional wave: P1 shear wave: S

3 characteristic parameters:

2 compressional waves: fast P & slow P1 shear wave: S

8 characteristic parameters:

1 compressional wave: P

2 characteristic parameters:

Frequency dependence of fluid flow regime

Poiseuille flow

At low frequencies:laminar fluid flow (Poiseuille flow) = inertials forces are negligible

compared to viscous forces, which control the flow regime=> Diffusive slow P wave

At high frequencies:more complex fluid flow with viscosity effects only in a thin

boundary layer = inertials forces dominate the flow regime=> Slow P wave propagates

Characteristic frequency (Biot, 1956; Auriault et al., 1985; Carcione, 2007)

Strong form:

Weak form:

weak form valid for any test vector w

Elastic governing equations

boundary integral naturally unfolds

e.g., moment tensor earthquake source :

[ for finite-fault kinematic rupture ]

Strong form:

Weak form:

weak form valid for any test vectors

Poroelastic governing equations

with

boundary integral naturally unfolds

Finite-elementsMapping from reference square/cube to quad/hexahedral element:

Jacobian of the mapping:

shape functions

3D mesh

Lagrange polynomials andGauss-Lobatto-Legendre (GLL) points

The 5 degree 4 Lagrange polynomials

degree 4 GLL points

Lagrange polynomial property:

GLL points are the n+1 roots of

where is a Legendre polynomial of degree .

Lagrange polynomial definition:

Interpolation & Integration rule

Representation of a functions on an element using the Lagrangepolynomials:

Integration of a function using the GLL quadrature rule:

diagonal mass matrices:

Poroelastic:

Elastic: Newmark time marching

Parallel implementation

Globe partitioning6 chunks of n*n mesh slices

Regional (S. California) regularpartitioning of n*m mesh slices

velocity field

Regional (SEG/EAGE) irregularpartitioning of mesh slices

Resolution & Stability Criteria- 5 points per shortest wavelength- Courant number < 0.3(Komatitsh & Vilotte, Bull. Seism. Soc. Am.1998)

e.g., 9 Sept. 2001, Hollywood eartquake Mw 4.2

Vertical component: black = data & red = SEM

(Komatitsch et al., Bull. Seism. Soc. Am. 2004)SEM snapshots

Finite-frequency sensitivity kernelsbased on adjoint method

Misfit functions

(Dahlen et al., 2000; Liu & Tromp, 2006; Tromp et al., 2005)

Least-squares waveform misfit:

Traveltime misfit:

Amplitude misfit:

=> Lagrange multiplier method to minimize the misfit functionconstrained by wave equations

=> Lagrange multiplier = adjoint field

Forward & Adjoint wavefields

(Elastic: Tromp et al., 2005; Poroelastic: Morency et al., 2009)

forward wavefield

adjoint wavefield

Waveform adjoint source:

Traveltime adjoint source:

Amplitude adjoint source:

Elastic & Poroelastic Sensitivity Kernels

(Tromp et al., 2005)

(Morency et al., 2009)etc…

Isotropic elastic medium => 3 parameters

Isotropic poroelastic medium => 8 parameters

Traveltime anomaly kernel construction

(Tape et al., GJI 2007)

Adjoint source construction:

e.g., S. California new crustal model

(Tape et al., Science 2009 & GJI 2010)

East of LA basin &within Ventura basin

Mw = 5.4

MC = Malibu Coast faultSY = Santa Ynez fault

Standard 1D model

Initial 3D model

Final model

SPECFEM 2D & 3D packages

- CUBIT compatible

- 3 modules: (an)elastic, acoustic, poroelastic

- Forward & Adjoint seismic wave propagation

- Topography & Bathymetry

- Parallel computation (SCOTCH for mesh

partitioning & load balancing)

Freely available for non-commercial purposes via the Computational Infrastructure forGeodynamics (www.geodynamics.org)

In continual development: Princeton University (US) & Pau University (FR)

CO2 sequestration monitoring

CO2 sequestration

Importance of monitoring

(http://energy.er.usgs.gov/health_environment/co2_sequestration)

CO2 sequestrationNagaoka (Japan) site: crosswell seismic data

after Onishi et al, 2009

after 1st injection

P-wave time-lapse anomaly

baseline after 1st injection

CO2 sequestration

CO2 saturation

Frio (Texas, US) site: crosswell seismic data

after Daley et al, 2008

P-wave time-lapse anomaly

Geometry

Data = after injection

2-D SEM model geometry, “synthetic” data

Baseline (BSL) = before injection

4 sources (Ricker, 50Hz)and 20 receivers

Material type:red & yellow = elasticblue & green = poroelastic

Model characteristics:150 x 165 elements22400 total time steps1d-5 s iteration time step0.2 s seismogram time length

=> Importance of the physical theory used to model the aquifer on howaccurate the imaging & inversion can be

“Data” parameters

-12%

+5%

-51%

-9%2035 1845

⇒ Injection of CO2 changes material properties and how waves propagate⇒ We use these differences to track the CO2

Event 1

Event 2

Event 3

Event 4

Receivers

Measurements investigated

(1) P-wave traveltime(2) P- & S-wave traveltimes(3) P- & S-wave traveltimes and

amplitudes

Input model m00Event 2, Receiver 10

Results

Elastic kernels model m00(1) P-wave traveltime

(2) P- & S-wave traveltimes

(3) P- & S-wave amplitudes

=> Access to different information depending on the measurements used

Final model update

P-wavespeed

S-wavespeed

Final model update

Bulk density

Final model update

Fluid bulk modulus

Fluid density

Conclusions

1) Forward & adjoint wave propagation:- SEM highly suitable for parallel computation- Sensitivity kernels defined based upon an adjoint method

2) CO2 sequestration monitoring:=> poroelastic signature in data- full iteration procedure- poroelastic inversion: accurate + fluid properties- next: use real data- next: use the full signal (FLEXWIN software, Maggi et al. 2009)- next: strategy to take advantage of all poroelastic kernels

SPECFEM packages for forward & inverse problems