## Fluid flow modelling and its seismic differencing in time-lapse

### Vanja Vracar (née Milicevic)

In this thesis I explore alternatives to conventional seismic differencing and their application to seismic time-lapse analysis.

The motivation is my observation that conventional seismic differencing relies on a number of assumptions, and that these may not always represent reality. Systematic error, error associated with the use of conventional (imperfect) imaging algorithms, and error due to source/receiver coupling variations are assumed to be small relative to the seismic response of fluid transport in a reservoir for which source/receiver positioning and coupling must be the same between surveys in time-lapse. The result is that conventional differencing involves simple match filtering followed by subtraction where the interpretable product is an image of the change in fluid location superimposed upon some background noise level. In reality, however, errors are often very large.

I observe that though errors might be large, and with the exception of source/receiver location repeatability, coupling variation and systematic errors result in differences in seismic amplitude and not necessarily seismic phase so that any methodology beyond simple match-filtering and differencing might incorporate this observation.

I base my work on numerical experiments where I compare conventional differencing for seismic time-lapse analysis to a number of new algorithms that I have developed. The numerical experiments incorporate a fluid-flow simulator, a Gassmann-equation based conversion from time-lapse saturation and pressure values to density and elasticity, and then 3D, multi-component seismic simulation.

For comparison to my new approaches to seismic differencing, I perform conventional differencing through matrix subtraction. For simplicity, I assume that source and receiver error is negligible (or have been corrected for), and I compute differences by subtraction alone.

Five non-conventional seismic differencing algorithms are implemented: 1) inverse data space differencing (IDSD), 2) cross-correlation differencing (CCD), 3) pseudo crosscorrelation differencing (PCCD), 4) conventional imaging condition differencing (CICD) and 5) imaging condition differencing (ICD). The IDSD is a filtering algorithm that employs inverse data matrix theory on migrated seismic models. It clears amplitude patterns by focusing differences and dimming similarities of two time-lapse steps. The CCD and PCCD algorithms are performed in the time and frequency domains, respectively. These algorithms consist of a cross-correlation operation, gaussian filtering and inversion. The algorithms results are then multiplied by the conventional seismic differencing and passed to a pre-stack depth migration (PSDM). A disadvantage of both algorithms is dependence on the user to manually move data from differencing to migration. The CICD is a combination of PSDM and conventional differencing. It is a pilot algorithm to combine the PSDM with the PCCD. It proves to be efficient and robust when compared tmigration in one algorithm, hence minimizes the user's dependence and improves commputational time and imaging. The CCD, PCCD and ICD capture almost only fluid flow changes and eliminate mostly all similarities on the final differenced models.

The computational cost of non-conventional differencing methods varies. Assume the data is stored in an M by N matrix. The most expensive one is the CCD that takes O(M5N5+3MN +MNlog(MN)) operations to complete, where the ICD is the cheapest taking O(M2N2 + 2MN + 6MNlogMN) operations to complete.

When compared to conventional seismic differencing, all non-conventional seismic differencing algorithms, except CICD, capture significant imaging improvements, hence aid in geophysical interpretation, reservoir monitoring, characterization and time-lapse studies.