CO2 sequestration crosswell monitoring based upon …cees.stanford.edu/docs/CEES-Morency.pdf · CO2...
Transcript of CO2 sequestration crosswell monitoring based upon …cees.stanford.edu/docs/CEES-Morency.pdf · CO2...
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