Direct modelling of coda wave interferometry Jérôme Azzola (1), Jean Schmittbuhl (1), Dimitri Zigone (1), Vincent Magnenet (2) and Frédéric Masson (1) (1)Institut de Physique du Globe de Strasbourg, Université de Strasbourg/EOST, CNRS, France (2) ICUBE, UMR CNRS 7357, 72, Route du Rhin, F-67411 ILLKIRCH, France

Introduction Coda Wave Interferometry (CWI) aims at tracking small changes in solid materials II. Mechanical behavior and scattering properties : Effective medium selection (hole’s number – radium) and reproducing of the scattering properties of a typical granite sample like rocks where elastic waves are diffusing. They are intensively sampling the medium, making the technique much more sensitive than those relying on direct wave arrivals. Application of CWI to ambient seismic noise has found a large range of applications over the past Fig.3: Fig. 4 years like for multiscale imaging but also for monitoring complex structures such as regional faults or reservoirs (Lehujeur et al., 2015). Fig. 3 a) Fig. 3 b) a) Photography: granitic sample from a core of the Soultz-sous-Forêts deep This emerging technique could interestingly be applied to geothermal reservoirs to monitor stress changes occurring at depths, for geothermal reservoir (depth of 3790m). A plutonic rock composed of large example. crystallographic grains behaving as numerous scatters ; Physically, observed changes are typically interpreted as small variations of seismic velocities. However, this interpretation remains b) They are modelled as a set of cylindric holes randomly distributed in the block (112mm x 95mm). questionable. Here, a specific focus is put on the influence of the elastic deformation of the medium on CWI measurements. The goal of Fig.4: the present work is to show from a direct numerical and experimental modeling that deformation signal also exists in CWI Characterization of the scattering in effective medium vs in the granite medium: measurements which might provide new outcomes for the technique. calculation of the linearized energy density functions W(r,t) – comparison to On this purpose, we model seismic wave propagation within a diffusive medium using a spectral element approach (SPECFEM2D) simple diffusing model – estimation of mean free path l during an elastic deformation of the medium. The mechanical behavior is obtained from a finite element approach (Code ASTER) keeping the mesh grid of the sample constant during the whole procedure to limit numerical artifacts. The CWI of the late wave arrivals in the synthetic seismograms is performed using both a stretching technique and a cross-correlation method. Both show that the elastic deformation of the scatters is fully correlated with time shifts of the CWI differently from an acoustoelastic effect. As an illustration, the Methods and principle: modeled sample is chosen as an effective medium aiming to mechanically and acoustically reproduce a typical granitic reservoir rock. 1. Energy density W(r, t) is computed from seismograms: The modeled sample is shown to reproduce the behavior of a granite sample from Soultz-sous-Forêt. - Obtained experimentally from an acoustic propagation in the granite sample Our numerical approach is compared to experimental results where multi-scattering of an acoustic wave through a perforated loaded - Obtained numerically for different distributions of holes in effective medium Au4G (Dural) plate is performed at laboratory scale. Experimental and numerical results of the strain influence on CWI are shown to be consistent. This results could support the interpretation of coda wave interferometry and have implications for ambient seismic noise monitoring. 2. Observed energy density function W(r, t) is compared to a wave diffusion model for body waves (Fig.4)

I. Principle and details of the setup – numerical aspects Fig. 5 3. Grid search (Fig.5) for radius ranging from 2.5mm to 5mm, which could be realistic sizes of grains inclusions in a rock mass and for number varying from 25 to 110. Estimation of the fit between modeled radiated energy Fig. 1 Mechanical Loading obtained experimentally and numerically, for each configuration. 4. For the retained configuration (70 holes of radium of 3mm): Fig. 6 δ  Assessment of the scattering properties ; development of the diffuse wavefield (Fig.4) (comparison of W(r, t) and a wave diffusion model to estimate the mean free path ℓ by least square fitting) >Multi-scattering if the wave front sees the heterogeneities several times before the receiver, when wavelength λ is of same length or shorter than the heterogeneity size d >Strong scattering if the mean free path ℓ, the size of the defects d (varying here from 5mm to 10mm), the size of sample D (here, 95mm), and the wavelength λ are satisfying the inequality:

λ ≤ d ≤ ℓ < D (Planès and Larose, 2013). ℓ = 8.8mm, satisfies the inequality  Assessment of the mechanical behavior (Fig.6) Linear variation of stress with strain: the block has a perfectly elastic behavior during deformation Source Receiver An effective Young modulus of 51 GPa, smaller than the bulk modulus (54 GPa)

Source Parameters Receiver Parameters Fig.5: Grid search in function of number and radium of cylindrical holes Ricker temporal shape Sampling rate dt=10-10 Fig.6: Mechanical behavior of the plate: strain is calculated as variation of height of sample Central freq : f0 = 400kHz Signal length: 120µs III. Results of CWI: estimation of time-shifts by stretching and windowed cross-correlation, acousto-elastic effects, variation with strain of relative time shift and estimation of error Wavelength λ = 2- 8mm Fig.7 Fig.8 Fig.1: Principle of the numerical experiment We impose a mechanical loading δ to a rock sample imposing a rigid step by step displacement of the upper face. During the loading, a Fig. 2 Ricker Wavelet (see left insert) is sent in the medium from the source and recorded at the opposite face by the receiver (see right insert);

Fig.2: Details of the numerical scheme The elastic deformation of the medium at ASTER: Specfem2D: each step of the loading is obtained from a 2D Finite Element approach elastic wave (Code_ASTER). Seismic wave propagation is modeled within the medium using a deformation propagation spectral element approach (SPECFEM2D) with the same mesh. Synthetic seismograms produced at the receiver are analyzed using Coda Wave Interferometry. Plate characteristics; Propagation Data Elastic parameters Coda Wave Granite Vp = 5070 m/s Interferometry 112mm*95mm Vs = 2888 m/s

ρ=2570kg/m3 Fig.8: Evolution of relative time shift (ε=δt/t) in fct of macroscopic uniaxial strain (εyy) ; for the stretching technique (black) or the time window correlation No intrinsic attenuation E=54GPa ; ν=0.26 technique (red) a) ε is measured from signals acquired at the middle of plate - one receiver. The bars account for the error on the measure of the relative time shift b) ε is obtained from a spatial homogenization. The bars account for the spatial variation References Discussion Aki, K, 1969. Analysis of the seismic coda of local earthquakes as scattered waves, J. Geophys. Res., 74, 6215-6231 Conclusion: Measurement of ε carry a real deformation signal, from highly Lehujeur, M., Vergne, J., Schmittbuhl, J., Maggi, A., 2014. Investigating a deep geothermal reservoir using ambient noise correlation, in: EGU sensible CWI techniques (assessment of the independency towards parameters of the simulation such as time-resolution – estimation technique – General Assembly Conference Abstracts. p. 13798 source parameters). Relative time shifts show to vary linearly in function of vertical strain: Fig.8-a) δt/t= 0.41 εyy ; Fig.8-b) δt/t= 0.37 εyy. Dainty A M. Toksöz M.N., 1977. Elastic wave propagation in a highly scattering medium—a diffusion approach, J. Geophys , 43, 375–388. Discussion and interpretation of the reversible time shifts measured here with a constant wave speed in the light of classical models: Hughes, D.S., Kelly, J.L., 1953. Second-Order Elastic Deformation of Solids. Phys. Rev. 92, 1145–1149. - A model based on (Snieder et al., 2002): most used, relying on the measure of irreversible wave velocity variations typically related to damage Schurr, D.P., Kim, J.-Y., Sabra, K.G., Jacobs, L.J., Thompson, D.O., Chimenti, D.E., 2011. Monitoring damage in concrete using diffuse ultrasonic development in the sample (e.g. micro-cracks); coda-wave interferometry. pp. 1283–1290. Grêt, A., R. Snieder, and J. Scales , 2006. Time-lapse monitoring of rock properties with coda wave interferometry, J. Geophys. Res., 111, B0330 - A model relying on acousto-elasticity (Aoki, 2015): measure of reversible time delays linked to seismic velocity changes that are produced by Snieder, R., A. Grêt, H. Douma, and J. Scales, 2002. Coda wave interferometry for estimating nonlinear behavior in seismic velocity, Science, 295, reversible stress perturbations in a non-linear rheology. Here, we do not consider such a rheology. 2253–2255. Fig.7: Coda Wave Interferometry : Comparison of waveforms recorded for a given imposed displacement The comparison to experimental results is necessary to distinguish the physical origins of time shifts. At the laboratory, we use a perforated Au4G Snieder, R., 2006. The theory of coda wave interferometry, Pure and Applied Geophysics, vol. 163, no 2-3, pp. 455-473. a) δ = 0µm (black), δ = 75µm (blue) and δ = 150µm (red)) plate under a 10T press and CWI from acoustic records at given displacement δ. Results currently exploited. Weaver, R.L., Hadziioannou, C., Larose, E., Campillo, M., 2011. On the precision of noise correlation interferometry: Precision of noise correlation b) Zoom in a 6.5 µs window: negligible time-shift Opening - feasible infield application: Measurements could be interpreted in field by slight changes in the stress state in a reservoir rock mass. It interferometry. Geophys. J. Int. 185, 1384–1392. c) Zoom at the end of the waveform where the time-shift is clearly sensitive to the imposed displacement. Wegler, U., Lühr, B.G., 2001. Scattering behaviour at Merapi volcano (Java) revealed from an active seismic experiment. Geophys. J. Int. 145, d) Time-shifts within the coda are measured either with a time window correlation technique (circles) or a frequency stretching technique could have implications for monitoring of stress changes at depths - monitoring the response of the rock mass to pumping variations - monitoring 579–592 (solid line): waveforms at 75µm vs 0µm (blue) and at 150µm vs 0µm (red). The windows are 6.5µs long activities at depths to make changes to the stimulation design.