CO2 sequestration crosswell monitoring based upon spectral-element and adjoint methods

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1 CO2 sequestration crosswell monitoring based upon spectral-element and adjoint methods Christina Morency Department of Geosciences, Princeton University Collaborators: Jeroen Tromp & Yang Luo Computational Geosciences Seminar Series- Sept. 27, Stanford

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

3 Forward wave propagation based on SEM

Three rheologies Elastic: In an isotropic case: and Poroelastic [Biot, 1962]: B, C and M are the Biot coefficients defined in terms of solid, fluid, and frame

4 Three rheologies Elastic: In an isotropic case: and Poroelastic [Biot, 1962]: B, C and M are the Biot coefficients defined in terms of solid, fluid, and frame properties where { Acoustic, inviscid fluid and neglecting gravity effects [Chaljub and Valette, 2004; Komatitsch et al., 2005]: where the displacement and the acoustic pressure

5 Three & 1/2 rheologies continued Elastic with Gassmann s formulae [Gassmann, 1951]: In an isotropic case: and Effective saturated bulk & shear moduli:

6 Poroelastic governing equations Microscopic equations for the solid and fluid phase: Conservation of mass Constitutive relationships (Hooke s law, Navier-Stokes) Conservation of momentum Averaging method (Pride & Berryman, 1998; Whitaker, 1999) Macroscopic equations of the biphasic porous medium: (i) Microscopic material properties are constant on the scale of the averaging volume, but they can vary at the macroscale (ii) Wavelengths of waves of interest are large compared to the averaging volume

7 Characteristic parameters Acoustic: 1 compressional wave: P 2 characteristic parameters: Elastic: 1 compressional wave: P 1 shear wave: S 3 characteristic parameters: Poroelastic: 2 compressional waves: fast P & slow P 1 shear wave: S 8 characteristic parameters:

8 Frequency dependence of fluid flow regime 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 Poiseuille flow Characteristic frequency (Biot, 1956; Auriault et al., 1985; Carcione, 2007)

9 Elastic governing equations Strong form: e.g., moment tensor earthquake source : Weak form: [ for finite-fault kinematic rupture ] weak form valid for any test vector w boundary integral naturally unfolds

10 Poroelastic governing equations Strong form: with Weak form: weak form valid for any test vectors boundary integral naturally unfolds

11 Finite-elements Mapping from reference square/cube to quad/hexahedral element: shape functions Jacobian of the mapping: 3D mesh

12 Lagrange polynomials and Gauss-Lobatto-Legendre (GLL) points Lagrange polynomial definition: The 5 degree 4 Lagrange polynomials Lagrange polynomial property: GLL points are the n+1 roots of degree 4 GLL points where is a Legendre polynomial of degree.

13 Interpolation & Integration rule Representation of a functions on an element using the Lagrange polynomials: Integration of a function using the GLL quadrature rule: Poroelastic: diagonal mass matrices: Elastic: Newmark time marching

14 Parallel implementation velocity field Globe partitioning 6 chunks of n*n mesh slices Regional (SEG/EAGE) irregular partitioning of mesh slices Regional (S. California) regular partitioning of n*m mesh slices Resolution & Stability Criteria - 5 points per shortest wavelength - Courant number < 0.3 (Komatitsh & Vilotte, Bull. Seism. Soc. Am.1998)

15 e.g., 9 Sept. 2001, Hollywood eartquake Mw 4.2 Vertical component: black = data & red = SEM SEM snapshots (Komatitsch et al., Bull. Seism. Soc. Am. 2004)

16 Finite-frequency sensitivity kernels based on adjoint method

17 Misfit functions Least-squares waveform misfit: Traveltime misfit: Amplitude misfit: => Lagrange multiplier method to minimize the misfit function constrained by wave equations => Lagrange multiplier = adjoint field (Dahlen et al., 2000; Liu & Tromp, 2006; Tromp et al., 2005)

18 Forward & Adjoint wavefields forward wavefield adjoint wavefield Waveform adjoint source: Traveltime adjoint source: Amplitude adjoint source: (Elastic: Tromp et al., 2005; Poroelastic: Morency et al., 2009)

19 Elastic & Poroelastic Sensitivity Kernels Isotropic elastic medium => 3 parameters Isotropic poroelastic medium => 8 parameters (Tromp et al., 2005) etc (Morency et al., 2009)

20 Traveltime anomaly kernel construction Adjoint source construction: (Tape et al., GJI 2007)

21 e.g., S. California new crustal model East of LA basin & within Ventura basin Mw = 5.4 MC = Malibu Coast fault SY = Santa Ynez fault Standard 1D model Initial 3D model Final model (Tape et al., Science 2009 & GJI 2010)

22 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) In continual development: Princeton University (US) & Pau University (FR) Freely available for non-commercial purposes via the Computational Infrastructure for Geodynamics (

23 CO2 sequestration monitoring

CO2 sequestration (http://energy.er.usgs.

24 CO2 sequestration ( Importance of monitoring

25 CO2 sequestration Nagaoka (Japan) site: crosswell seismic data baseline after 1st injection after 1st injection P-wave time-lapse anomaly after Onishi et al, 2009

26 CO2 sequestration Frio (Texas, US) site: crosswell seismic data Geometry CO2 saturation P-wave time-lapse anomaly after Daley et al, 2008

27 2-D SEM model geometry, synthetic data Material type: red & yellow = elastic blue & green = poroelastic 4 sources (Ricker, 50Hz) and 20 receivers Baseline (BSL) = before injection Data = after injection Model characteristics: 150 x 165 elements total time steps 1d-5 s iteration time step 0.2 s seismogram time length => Importance of the physical theory used to model the aquifer on how accurate the imaging & inversion can be

Data parameters -12% +5% -51% 2035 1845-9% Injection of CO2 changes material

28 Data parameters -12% +5% -51% % Injection of CO2 changes material properties and how waves propagate We use these differences to track the CO2

29 Results Input model m00 Event 2, Receiver 10 Receivers Event 1 Event 2 Event 3 Event 4 Measurements investigated (1) P-wave traveltime (2) P- & S-wave traveltimes (3) P- & S-wave traveltimes and amplitudes

30 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

31 Final model update P-wavespeed S-wavespeed

32 Bulk density Final model update

33 Final model update Fluid bulk modulus Fluid density

34 Conclusions 1) Forward & adjoint wave propagation: - SEM highly suitable for parallel computation - Sensitivity kernels defined based upon an adjoint method SPECFEM packages for forward & inverse problems 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

Spectral-Element and Adjoint Methods in Seismology

Spectral-Element and Adjoint Methods in Seismology Jeroen Tromp Min Chen, Vala Hjorleifsdottir, Dimitri Komatitsch, Swami Krishnan, Qinya Liu, Alessia Maggi, Anne Sieminski, Carl Tape & Ying Zhou Introduction