The Math, Science and Computation of Hydraulic Fracturing

Transcription

1 The Math, Science and Computation of Hydraulic Fracturing 03/21/2013 LLNL-PRES-XXXXXX This work was performed under the auspices of the U.S. Department of Energy by under contract DE-AC52-07NA Lawrence Livermore National Security, LLC

2 Detection Empirical Matched Field processing Location Bayesian-based algorithm Characterization Adjoint full waveform inversion Schematic of enhanced geothermal systems using an injection and production well. Source: MIT, Geomechanical model fracture network simulator Creating optimal fracture networks! Model fracture networks! To optimize reservoir performance! Improve fracture monitoring techniques! To increase production! Evaluate induced seismicity risks LDRD-SI Physical and Life Sciences Directorate (PI: Rick Ryerson) 2

3 Forward wave propagation: ρ 2 t s = T + f f = M δ(x x s )S(t) Adjoint wave propagation: ρ 2 t s = T + f f = Isotropic Fréchet derivatives: K ρ = K µ = K κ = T 0 T 0 T 0 ρ(x)[s (x,t t) 2 t s(x,t)]dt N [s(x r,t t) d(x r,t t)] δ(x x r ) r=1 2µ(x)[D (x,t t) :D(x,t)]dt κ(x)[ s (x,t t) s(x,t)]dt f = w r(t t) N r t s(x r,t t)δ(x x r ) Kρ = K ρ + K κ + K µ K β =2 K µ 4 µ 3 κ K κ κ K α =2 µ K κ κ [Tromp et al., 2005] 3 LLNL-PRES

4 Waveform misfit function: χ = 1 w rp (t)s(x r,t; m) d(x r,t) 2 dt 2 rp Forward wave propagation: ρ t 2 s = T M δ(x x s )S(t) Adjoint wave propagation: ρ 2 t s = T + rp w rp (t)(s d)(t 2t 0 t)δ(x x r ) Fréchet derivatives: χ = ij M (x s,t)s(t 2t 0 t)dt i,j χ = x s i [M : (x s,t)]s(t 2t 0 t)dt x s i χ = t s χ = h s M : (x s,t) ts S(T 2t 0 t)dt M : (x s,t) hs S(T 2t 0 t)dt [Kim et al., 2011] 4

5 GASB 40 Synthetic case: Moment tensor coefficients & depth 39 HOPS MNRC SUTB 39 Event location Real data: 1) Moment tensor coefficients based on linear inversion 2) Moment tensor coefficients & depth based on iterative adjoint method CVS MCCM 38 BKS BDM! 1D model GIL7 [Pasyanos et al, 1996]! topography! 225 x 280 x 36 km^3! 1 event Mw 4+! 8 receivers

6 m00 Data (Black) and Synthetics (Red) model M00 m12 Data (Black) and Synthetics (Red) model M12! Starting model m00: error introduced in Mij coefficients & depth Z 4.74e Z 4.74e ! Signals filtered between 2-8s R 4.47e 05 R 4.47e 05! Misfit reduction ~97% T T e e Time (s) Moment (x1e22 dyne/cm) Mw Moment Mw depth Misfit Depth (km) Misfit (x1e 9)

7 ! Waveform 3-components data from NCEDC, January 4 th 2009 event (Mw 4+)! 1D velocity model [model GIL7 after Pasyanos et al, 1996] + topography! Signals filtered between 20-50s elementary MT [Kikuchi & Kanamori, 1991] Linear relationship between Green s functions, moment tensor & data: Gm = d Generalized inverse solution: m =(G T G) 1 G T d m00 7

8 m00 m08! Starting model m00: linear inversion! 1D velocity model! Signals filtered between 15-20s! Misfit reduction 27% & Variance reduction 70% Moment (x1e22 dyne/cm) Moment Mw Mw Depth (km) depth Misfit (x1e 11) Misfit

9 m00 m08! 1D velocity model! Signals filtered between 20-50s! Comparison to Dreger et al., 2011 adjoint inversion depth = 9.3 km M0 = 4.26e22 dyne.cm Mw = 4.38 VR = 82% (20-50s) after Dreger et al., 2011 depth = 4.5 km M0 = 3.86e22 dyne.cm Mw = 4.4 VR = 82% (20-50s) 9

10 Implementation of the USGS 3D velocity model including topography in SPECFEM3D Improves waveform fitting Improves algorithm convergence Improves depth/location resolution => path to improve higher frequency resolution, smaller events Geologic rock units incorporated in the model (detailed model view) 10

11 Event depth location: red < 30km green between 30-80km blue > 80km Data: 182 events and 356 stations are used for the inversion Initial model: 3D model S2.9EA (Kustowski et al., 2008 shear wave velocity model beneath Eurasia) Computational cost: 1 iteration = 66,000 CPU hours gray line = modeled region [BAA funding Savage (URI), Tromp & Peter (Princeton Univ.), Rodgers & Morency (LLNL)] 11

12 Non-smoothed event kernel 1 = 1 source + multiple stations K s c Smoothed misfit kernel = multiple sources + multiple stations K c = s K s c 12

13 Misfit 3 components seismograms filtered between s black = data red = synthetics M00 M13 13

14 ! Synthetic case: high accuracy in the inversion because the velocity model is perfectly known! MT inversion strategy: 1) linear inversion for starting moment tensor coefficients 2) adjoint inversion for moment tensor coefficients and depth/location remaining waveform discrepancies: 3D crustal heterogeneities?! Strategy Needed: 1) implement 3D earth model 2) adjoint tomography of area of interest to reach higher frequencies 3) source parameters inversion for small events 4) information on fracture openings 14

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