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Estimation of Time-lapse Velocity Changes Using Gaussian Reconstruction

Conference Paper · June 2017

DOI: 10.3997/2214-4609.201701204

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Phung K. T. Nguyen

Heriot-Watt University

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Colin Macbeth

Heriot-Watt University

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Maria-Daphne Mangriotis

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79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

We B1 01

Estimation of Time-Lapse Velocity Changes Using

Gaussian Reconstruction

P.K.T. Nguyen* (Heriot-Watt University), C. MacBeth (Heriot-Watt University), M.D. Mangriotis

(Heriot-Watt University)

Summary

A key task in 4D seismic analysis is to resolve changes in subsurface velocity ΔV/V that affect both imaging in the

monitor data and our interpretation of time-lapse amplitudes. This study introduces a new approach to recover

ΔV/V using a Gaussian mixture model. The Gaussians are found to be better representative of the property fields

than other choices such as B-splines. This approach is tested by application to a North Sea field, where

geomechanical effects are active. Recovery of ΔV/V from three different time-shift estimates, using three

approaches is firstly compared with Gaussian reconstruction. A second comparison estimates ΔV/V directly from

the trace data. In these tests, the new approach compares favourably in the presence of noise, and is relatively

simple to implement.

79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Introduction

Time-lapse velocity change ∆V/V is a fundamental subsurface parameter, representing the changes inpore-fluid and rock properties within the reservoir, and reflecting geomechanical effects. Moreover, itprovides the input to imaging correction for enhanced time-lapse interpretation. Many different methodshave been developed to extract this information, either from measured time-shift data by differentiationor from the seismic trace data directly using trace warping and full pre-stack tomography. This studypresents a new approach called Gaussian reconstruction for recovering ∆V/V from seismic data andcompares this with two commonly used methods, which are layer stripping and damped least squaressolution. The advantages and disadvantages of this approach are discussed with application to a fielddataset.

Methodology

Here we recover ∆V/V using the Gaussian mixture model Reynolds (2009) in two ways. The first startswith estimated time-shifts (on the assumption of negligible physical displacement). The second beginswith time-lapse trace data. These two categories of approach are implemented in time domain.

(a) Recovery of ∆V/V from time-shift - for a given baseline velocity field V (t) and time-lapse velocitychanges ∆V (t), in which T is two-way time, time-shift ∆t is calculated as

∆t =T∫

0

VV +∆V

dt−T∫

0

dt (1)

This collapses to the integral of quantity ϑ(t)= [1/(1+∆V/V )−1] which is used in this study, instead of∆V/V (derived from first-order Taylor approximation), to obtain good accuracy in time-lapse velocitychange estimation. After time-shift measurement from the seismic data we employ three methods toobtain ∆V/V :

• Method 1, Gaussian reconstruction - Gaussians are chosen as basis functions to represent time-lapse velocity changes. This approach is similar in strategy to the B-splines method but we believeis more representative of the physical property field. Time-shift and ∆V/V data are generatedusing Gaussian reconstruction, which guarantees a stable and analytic solution. The key to thismethod is to decide a suitable Gaussian grid. A sensitivity analysis of grid and Gaussian dimensionwith respect to computing time and mean square error is essential. The study determines a givenpre-set Gaussian grid and width, which allows easy implementation to the seismic problem. Givena location µ and width σ , the quantity is rewritten as a linear mixture of Gaussians Gi, in whichwi is the weighting factor

ϑ(t) =k

∑i=1

wiGi(t|µi,σi) (2)

The corresponding ∆t is:

∆t =k

∑i=1

wi [Ri(t|µi,σi)−Ri(0|µi,σi)] (3)

where the cumulative distribution function R is

Ri(0|µi,σi) =

T∫−∞

G(0|µ,σ)dt =12

[1+ er f

(T −µ

σ√

2

)](4)

79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Instead of inverting for velocity changes, this method inverts for the weighting factors wi andthe final solution of ϑ can then be easily calculated. Comparison of the Gaussian reconstruc-tion method to the another two existing methods is an important task to understand the nature ofGaussian reconstruction and benefits that it may offer.

• Method 2, Layer stripping - (1) can be easily discretised as the linear equation Lϑ = ∆t in whichoperator L is a lower triangular matrix, ϑ and ∆t are now matrices of discretised componentsaccording to the sampling rate. This is an even-determined matrix equation and the L operator iscompletely invertible. ϑ is easily solved. The inverse of the L operator is exactly the differentialoperator hence this method is also called differentiation.

• Method 3, Damped least squares solution - there is always noise present in time-shift data so thatthe solution above will also perfectly fit the noise. Providing a regularisation term that balancesthe resolution of the solution and noise contribution is necessary. This study uses the second orderTikhonov regularisation and the optimal damping factor is determined by the knee point curveusing Hansen (1994).

(b) Direct inversion ∆V/V from the seismic traces - this section presents the task of recovering ϑ ,directly from the seismic data. The monitor traces are rewritten as a function of the shifted baseline andcompensated with the amplitude changes in a similar fashion to Williamson et al. (2007). Though thiswarping technique generates a velocity change attribute down to zero-frequency, it might not be stableenough to overcome the presence of noise in 4D seismic data. Therefore, here we modify this approachby again applying the Gaussian mixture model to represent ϑ . Given a baseline trace b(t), monitor tracem(t) and estimated wavelet ψ; the difference between the traces is minimized

min∣∣∣∣∣∣∣∣b(t)−m(t +∆t)−ψ ∗ ∂ϑ

∂ t

∣∣∣∣∣∣∣∣2

(5)

where ∆t and ϑ are again defined from pre-set Gaussian grids following from (2) and (3). The problemis also regularised to avoid singular and non-uniqueness of the inversion. Finally, ϑ is found by aniteratively weighted Gauss-Newton scheme for non-linear optimisation.

Application to field data

The above techniques are now applied to a field dataset. This study uses two seismic vintages of baseline(2001) and monitor (2004) from a reservoir in the North Sea. This is a HTHP reservoir and has dominantgeo-mechanical effects in reservoir, overburden and underburden. Time-shift data used in this study(Figure 1) are measured from baseline and monitor using three popular methods, which are: correlatedleakage method (CLM) by Whitcombe et al. (2010), fast cross correlation (DHF) by Hale (2009) andnon-linear inversion (NLI) by Rickett et al. (2007). Selected results for the NLI are presented in Figure2. As expected, the proposed method of Gaussian reconstruction (Figure 2c) produces a smooth, stableimage yet preserves the subsurface variability. The layer stripping method (Figure 2a) inverts for bothnoise and signal; meanwhile the damped least squares solution (Figure 2b) is too smooth and carries anerror. For a better understanding of the behaviour of Gaussian reconstruction, we implement the methodon the three different measured time-shifts. The results are shown in Figure 3, where in spite of thedifferent levels of noise in the input data, the method is observed to remain stable.

Finally, we directly input baseline and monitor traces instead of measured time-shifts for ϑ recovery.Here, the wavelet is extracted statistically using an interval from top overburden to underburden. Thealgorithm quickly converges after 4 to 5 iterations. Figure 4 shows the final results from integratingGaussian reconstruction into a non-linear inversion scheme. This new method has eliminated some levelof noise existing in time-lapse seismic data.

79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Figure 1 Time-shift data measurement from seismic data using methods: (a) NLI; (b) CLM and (c) DHF.

Figure 2 Results of ϑ recovered from: (a) layer stripping method; (b) damped least squares solution;and (c) Gaussian reconstruction. Input is the NLI time-shift measurement.

Figure 3 Results of ϑ recovered from the Gaussian reconstruction method, using: (a) NLI time-shift; (b)CLM time-shift; and (c) DHF time-shift.

79th EAGE Conference & Exhibition 2017 Paris, France, 12-15 June 2017

Figure 4 Results of ϑ recovered from time-lapse seismic traces using the Gaussian reconstructionmethod.

Discussion and Conclusions

In this study we compared two existing methods for recovering velocity changes ∆V/V with a newproposed approach of Gaussian reconstruction. Starting from time-shift data, ϑ is recovered from threedifferent methods revealing: (i) The layer stripping method amplifies noise in the solution. (ii) Balancingbetween honouring the data but also not fitting the noise in the damped least squares inversion methodintroduces error into the solution. (iii) The Gaussian reconstruction method proves stable through differ-ent levels of noise in time-shift estimates. Finally, the integration of the Gaussian reconstruction into anon-linear warping scheme for baseline and monitor traces preserves the ∆V/V subsurface informationand its variability despite coherent noise in the time-lapse seismic data. This new approach appearssimple to implement, and suitable to the time-lapse seismic problem.

Acknowledgements

We thank the sponsors of the Edinburgh Time Lapse Project, Phase VI, for their support (BG, BP,Chevron, CGG, ConocoPhillips, ENI, ExxonMobil, Hess, Ikon Science, Landmark, Maersk, Nexen,Norsar, OMV, Petrobras, Shell, Statoil, Taqa). Special thanks to Shell for dataset and Lu Ji for time-shiftmeasurement.

References

Hale, D. [2009] A method for estimating apparent displacement vectors from time-lapse seismic images.Geophysics, 74(5), V99–V107.

Hansen, P.C. [1994] Regularization Tools: A Matlab package for analysis and solution of discrete ill-posed problems. Numerical Algorithms, 6(1), 1–35.

Reynolds, D. [2009] Gaussian Mixture Models. Encyclopedia of Biometrics, 659–663.Rickett, J., Duranti, L., Hudson, T., Regel, B. and Hodgson, N. [2007] 4D time strain and the seismic

signature of geomechanical compaction at Genesis. The Leading Edge, 26, 644–647.Whitcombe, D.N., Paramo, P., Philip, N., Toomey, A. and Linn, T.R.S. [2010] The correlated leakage

method - It’s application to better quantify timing shifts on 4D data. 72nd EAGE Conference &Exhibition incorporating SPE EUROPEC, Extended Abstract, B037.

Williamson, P.R., Cherrett, A.J. and Sexton, P.A. [2007] A New Approach to Warping for QuantitativeTime-Lapse Characterisation. 69th EAGE Conference & Exhibition, Extended Abstract, P064.

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