Example

We illustrate this technique with a synthetic model that resembles a typical producing reservoir in the North Sea. The model is depicted in Figure : the reflectivity on the left, and the reference slowness on the right. The model consists of several fractured horizontal reservoirs which are in production. The reference slowness is smooth and not conformant with the stratigraphy. We assume that the reference slowness ${\bf S}$ is derived from the reference survey and that it perfectly focuses the reference data $\d_0$ to create the reference image $\r_0$.

 

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Figure 2 Synthetic model: reflectivity (left) and slowness (right).

Figure  shows the slowness perturbations we introduce in the slowness model. For each of the two scenarios, we generate data using the same reflectivity (Figure ) but different slowness models generated by adding the respective slowness perturbations to the reference slowness. We then image using the reference slowness to create repeat survey images. Finally, we subtract the reference image from each of these two images and obtain the image perturbations (a.k.a. the 4-D seismic data) depicted in Figure .

Figure  enables us to make two observations:

• Although the changes in the reservoirs occur at only two levels, the changes in the images occur at all levels underneath. This situation is common for 4-D seismic surveys. Typically only the top reservoir can be properly analyzed since the 4-D effects created by the deeper reservoirs are either masked or seriously shadowed by the top reservoir.
• Completely different changes in the reservoirs yield fairly similar perturbations of the images. Even for such a simple model, as the one we use in this analysis, it is really hard to visually analyze the image perturbation and distinguish among the two cases (Figure ). In practice, this distinction is virtually impossible, and the only place where we can extract reliable information is at the top-most producing reservoir.

 

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Figure 3 Image perturbation: scenario 1 on the left and scenario 2 on the right.

We address the ambiguity of the 4-D interpretation using WEMVA. Figure  shows the slowness perturbations obtained by the adjoint of the WEMVA operator ${\bf L}$ in Equation () applied to the image perturbation $\Delta \r$in Figure . The two cases can be better distinguished now, although the information is not yet localized at the producing reservoirs.

The least-squares inversion result, shown in Figure , is much better focused at the reservoirs. Despite the inherent vertical smearing mainly caused by the limited data aperture, we can precisely indicate the location of the producing reservoirs, the sign of the slowness change, and even the relative magnitude of the change from one reservoir to the other.