The impact of the acoustic approximation on time-lapse FWI Bram Willemsen, M.I.T, Jun Cao, ConocoPhillips and Baishali Roy, ConocoPhillips

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1 Bram Willemsen, M.I.T, Jun Cao, ConocoPhillips and Baishali Roy, ConocoPhillips SUMMARY Conventional time-lapse analysis methods are adequate for relatively simple geologies, but may degrade in the presence of a complex subsurface. The performance of Full Waveform Inversion (FWI) in complex geologies has sparked interest in applying it to the time-lapse problem. Due to the challenges of full D elastic FWI, most FWI applications use acoustic inversion. Real seismic waves travel through an elastic or viscoelastic Earth, making an acoustic approximation in the inversion result in inaccuracies. In this paper, we investigate the impact of this approximation on time-lapse FWI, more specifically Double Difference FWI (DDFWI), by using three synthetic datasets derived from a North Sea geologic model with increasingly realistic physics, starting from constant density acoustics to variable density acoustics and ending with isotropic elasticity. Numerical results show that the acoustic time-lapse estimate of the elastic data contains degraded amplitude and also significant artifacts. To mitigate such effects, we propose and evaluate several potential solutions, including using a better baseline model, limiting the maximum offset and regularizing time lapse FWI with D confidence information. Together, these steps reduce the detrimental effects of the acoustic approximation on the time-lapse estimate. INTRODUCTION Production induced changes in pore-fluids, pressure and porosity alter the seismic velocity and impedance. The goal of timelapse seismology is to detect these changes in seismic properties over time to infer the behavior of the reservoir and optimize field development accordingly. Changes in the thickness and seismic velocity of geological layers induce time-shifts in the recorded data. The measured time-shift at a particular location is caused by the cumulative change in the overburden above that point. An useful indicator of time-lapse change over an interval is the derivative of the time-shift, also known as the time-strain (Rickett et al., 6, 7). Landrø and Stammeijer () relate time-strain to compaction and velocity change by looking at time-strain in both near and far-offset traces. They also introduce an alternative using the combination of time-shift and change in amplitude. Examples of field applications where amplitude changes or time shifts are used to infer reservoir changes are given by for instance Ghaderi and Landrø (9), Eggenberger et al. () and Lyngnes et al. (). The methods presented so far deteriorate in complicated geologies where the Earth deviates strongly from D models (Ayeni and Biondi, ). For that reason there is a need for full prestack wave equation based methods, which more accurately represent wavefields in complicated geologies. In this paper we focus on the application of Full Waveform Inversion (Tarantola, 98) to the time-lapse problem. Double Difference FWI (DDFWI) (Watanabe et al., ) is a popular technique which inverts directly for the time-lapse model change from the time-lapse seismic data difference. Many aspects of DDFWI have been investigated, ranging from the effect of repeatability issues (Yang et al., ) to that of regularization (Zhang and Huang, ; Maharramov and Biondi, ). The number of field data applications is limited, with Yang et al. () showing an example applied to a North Sea field. Most field applications of time-lapse FWI make the acoustic approximation. The effect of this approximation on the time-lapse FWI estimate is poorly understood. To contribute to this understanding we perform a synthetic study on a realistic North Sea model where we investigate the effect of the acoustic approximation by incrementally increasing the physics of the recorded data. METHODOLOGY We study the performance of full waveform based time-lapse inversion algorithms on a realistic synthetic model. A reservoir simulator and rock physics model provide a D cube of approximate time-lapse changes of seismic properties. A D slice is extracted from this cube and will serve as the true Earth model in this study. Figure shows the true P-wave velocity V p. The blocky low velocity features are the gas clouds and the high velocity layer at km depth is the chalk reservoir. The reservoir anticline has a slightly lower V p due to the presence of hydrocarbons. Density ρ and shear S-wave velocity V s are not shown here. Figure shows the true V p time-lapse change investigated in this study. This change is limited to the zoomed-in region represented by the black box in Figure. The largest changes in V p are seen in the reservoir. The two elliptical overburden perturbations represent the response of planned waste-water injection wells. In addition to the true time-lapse V p, both ρ and V s change in the reservoir as well. The elliptical overburden anomalies have only V p perturbations. P-velocity true Figure : The true baseline V p model. The black box outlines the expanded region shown in following figures. To investigate the impact of the acoustic approximation we use

2 P-velocity true time-lapse Initial model EL baseline inverted 6 Figure : The true time-lapse V p change. The region corresponds to the black box in Figure. an elastic solver to generate three different types of data which will be represented by acronyms. Constant Density Acoustic (CDA). V p (x), constant ρ and V s. Variable Density Acoustic (VDA). V p (x) and ρ(x), constant V s. ELastic (EL). V p (x), ρ(x) and V s (x) We simulate shots with m spacing at a depth of.m. The pressure receivers at depth 87.m are stationary with a spacing of m and represent a permanent monitoring system (Bertrand et al., ). Anisotropy and attenuation are not considered in the modeling in order to limit the number of variables in this study. Another approximation is that we use an absorbing boundary on the top of the model. The source wavelet is a Ricker wavelet with peak frequency of 6.Hz. We use the same constant density acoustic solver to invert for the CDA, VDA and EL datasets to investigate the impact of increasingly more realistic recorded data. Amplitude normalization is used during the inversion to emphasize on phase, since the constant density acoustic inversion cannot accurately model the energy loss due to mode conversions. We first invert for the baseline model in each of the three scenarios starting from a heavily smoothed true V p model; while in the real case we can start with a tomographic model. For this step we use the standard least-squares FWI methodology (Tarantola, 98) in the timedomain. No processing was done on any of the datasets. Starting from their corresponding baseline models we minimize the DDFWI objective function χ(m ) = (d d ) ( u(m ) u(m ) ), () where subscript refers to the monitor state and the subscript refers to the baseline state. The objective function depends on the monitor model m with baseline model m fixed at the previous baseline inversion result. d and u are the true and modeled data respectively. RESULTS The first step of the workflow is the baseline inversion whose results for the EL case are shown in Figure together with the Figure : The starting V p model and the inverted baseline model using the EL data. starting model. The results for CDA and VDA are not shown due to limited space. They are very similar but of slightly higher quality because the physics in the data are more compatible with the forward model used in the inversion. For the EL case we observe a migration artifact around the reservoir and there are small perturbations around the receiver locations. The low wavenumber updates are mostly limited to region above the reservoir due to the lack of deeper diving waves. Despite these deficiencies we resolve the overall structure of the overburden relatively well. As quality control we inspect the modeled traces and see that important events such as the top of reservoir reflection are not cycle skipped. The next step in the workflow is DDFWI for each of the three scenarios starting from the baseline model m corresponding to that scenario. The results of this procedure are shown in Figure. As an additional comparison we also show the so called inverse crime result where the data is generated using the same forward model as is used in the inversion. This approach can lead to a trivial inversion (Colton and Kress, ), but is often used in publications and therefore useful to include for evaluating the more realistic scenario against that ideal case. Figure shows that the CDA inverse crime result is the best, with the reconstruction gradually deteriorating as the physics become more realistic. Even in the ideal (inverse crime) circumstances we see oscillatory artifacts below the true reservoir time-lapse change. This is due to limited wavenumber sampling caused by limited acquisition geometry and bandwidth in this case (Sirgue and Pratt, ). Comparing the EL DDFWI results in Figure with the true V p time-lapse in Figure, we see that both the shape and amplitude of the reconstruction at the reservoir are inadequate. For instance, the maximum amplitude event is now positive (blue) instead of negative (red). Upon closer inspection of the data we see that at large offsets the phase of the top of reservoir reflection in the EL scenario

3 ... DDFWI CDA inverse crime 7 7 Time (s) VDA baseline EL baseline DDFWI CDA Figure : Comparison of the reservoir top reflection between the VDA and EL case. The source is at x=.km DDFWI VDA DDFWI EL: max offset.km Figure 6: DDFWI EL result with max offset.km. Restricting the offset introduced an artifact in the circled area. The EL inverted baseline model is used. 7 DDFWI EL nounced artifact Figure : DDFWI on corresponding inverted baseline models. is significantly different from the CDA and VDA scenarios, which are similar. Figure shows a comparison of the reservoir top reflection between the VDA and EL scenarios for a source at x =.km. Ray tracing shows this reflection becomes post-critical at offsets approximately km and the phase gradually changes with increasing offset. This phase-shift depends on contrasts in V p, ρ and V s and cannot be reproduced by inverting only for V p. This problem is more pronounced in the time-lapse inversion than in the baseline inversion. We investigate this by restricting the maximum offset to.km in the DDFWI. Figure 6 shows the result of this procedure for the EL scenario. We observe that the overall reconstruction has improved significantly for the reservoir. The reconstruction of the elliptical overburden perturbations has deteriorated because limiting the maximum offset reduces the sampling of this area. A better but more difficult approach would be to mute the postcritical events in all the gathers through processing. The circled area in Figure 6 also shows a more pro- For the EL baseline model we used the full offset range of the data. The inverted baseline model is therefore contaminated by the post-critical reflections, in addition to other elastic effects. To investigate the role of the baseline model we use the CDA baseline model instead of the EL baseline model for the EL DDFWI inversion. This use of the CDA baseline velocity model can be viewed as the result of a scenario where processing perfectly removed the elastic effects from the baseline model inversion. The result of this procedure is shown in Figure 7. We see that the amplitude at the reservoir has improved by using a better baseline model in the inversion. We also investigate if the time-lapse reconstruction can be improved through regularization with the heuristic gradient based confidence map introduced by Willemsen and Malcolm (). The confidence map is obtained from an inversion where both... DDFWI EL: max offset.km on CDA baseline Figure 7: DDFWI EL result with max offset.km. The CDA inverted baseline model is used.

4 Confidence map β g Normalized confidence DDFWI EL: max offset.km and confidence map Illumination compensated gradient Figure 8: Confidence map obtained from preprocessing step. e Gradient evolution Iteration number Figure 9: The squares and circles represent the illumination compensated baseline and monitor gradients for each iteration of the preprocessing inversion. Evolution corresponds to the marked points in Figure 8. Blue confidence β g =.9, magenta β g =.7, red β g =.8, yellow β g =. Figure : DDFWI EL result with max offset.km on CDA baseline using confidence map from Figure 8. P-velocity (m/s).8. DDFWI depth profiles at x=.km True Fig EL Fig 6 Fig 7 Fig...6 Figure : Inverted time-lapse V p depth profiles at x =.km for Figures EL, 6, 7 and..8.. the baseline and monitor data are used simultaneously to invert for a single intermediate model, approximately matching both datas. The mismatch between the gradients from baseline and monitor data during this inversion highlights regions more likely to contain time-lapse changes. The confidence information resulting from using the full offset range of the EL data is plotted in Figure 8. Illumination compensated gradients for four locations marked by colored stars are shown in Figure 9. When the baseline and monitor gradients consistently have opposing sign and large amplitude at a pixel this location is interpreted to be more likely to have time-lapse change. We observe that at the reservoir (blue star) and right overburden perturbation (magenta star) the confidence of time-lapse change is high. At the left overburden perturbation (red star) the confidence is relatively low. This is caused by the low amplitude of the perturbation and that it is much smaller than the dominant wavelength. But overall we observe that the regions of high confidence correspond with regions where the largest time-lapse change takes place. This observation motivates the use of the confidence information to regularize the DDFWI time-lapse estimate, see Willemsen and Malcolm () for details. The results of applying this regularization to the DDFWI inversion of the EL data with the CDA baseline model (i.e. Figure 7) is shown in Figure. We observe that the confidence information partic- ularly suppresses artifacts in low confidence regions, such as the circled artifact introduced in Figure 6. The amplitude of the overburden perturbations is still low, but has improved in the reservoir. Figure shows relevant time-lapse depth profiles. CONCLUSIONS We evaluated the impact of the acoustic approximation on the time-lapse FWI application on data having more complicated physics. By better understanding the impact of this approximation through a synthetic study we can improve the results of applications to real data which include these additional physics and maybe more. We demonstrated the impact of such elastic effects through numerical study with a North Sea geological model. To mitigate their impact we evaluated several potential solutions including using a better baseline model, limiting maximum offset, and regularizing the inversion with timelapse confidence information. The results show significant improvement of the time-lapse velocity estimation. ACKNOWLEDGEMENTS We thank ConocoPhillips for the permission to publish this work. We also thank James Zhang for insightful discussions.

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