co2 sequestration crosswell monitoring based upon …cees.stanford.edu/docs/cees-morency.pdf · co2...
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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
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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
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Forward wave propagationbased on SEM
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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
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Three & 1/2 rheologies continued
Elastic with Gassmann’s formulae [Gassmann, 1951]:
In an isotropic case:and
Effective saturated bulk & shear moduli:
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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
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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:
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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)
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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 ]
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Strong form:
Weak form:
weak form valid for any test vectors
Poroelastic governing equations
with
boundary integral naturally unfolds
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Finite-elementsMapping from reference square/cube to quad/hexahedral element:
Jacobian of the mapping:
shape functions
3D mesh
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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:
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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
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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)
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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
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Finite-frequency sensitivity kernelsbased on adjoint method
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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
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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:
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Elastic & Poroelastic Sensitivity Kernels
(Tromp et al., 2005)
(Morency et al., 2009)etc…
Isotropic elastic medium => 3 parameters
Isotropic poroelastic medium => 8 parameters
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Traveltime anomaly kernel construction
(Tape et al., GJI 2007)
Adjoint source construction:
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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
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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)
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CO2 sequestration monitoring
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CO2 sequestration
Importance of monitoring
(http://energy.er.usgs.gov/health_environment/co2_sequestration)
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CO2 sequestrationNagaoka (Japan) site: crosswell seismic data
after Onishi et al, 2009
after 1st injection
P-wave time-lapse anomaly
baseline after 1st injection
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CO2 sequestration
CO2 saturation
Frio (Texas, US) site: crosswell seismic data
after Daley et al, 2008
P-wave time-lapse anomaly
Geometry
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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
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“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
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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
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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
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Final model update
P-wavespeed
S-wavespeed
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Final model update
Bulk density
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Final model update
Fluid bulk modulus
Fluid density
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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