Least-squares wave-equation inversion of time-lapse seismic data sets - A Valhall case study

Case study

We consider a subset of the Life of Field Seismic (LoFS) data sets acquired at Valhall, a giant oil field located in the Norwegian North Sea. There is a wide range of published work on the exploration and development effort in the Valhall field and on different aspects of the LoFS project at Valhall. For example, Munns (1985) discusses Valhall geology in detail; Barkved et al. (2003) discuss the production history and development plans for the field; Barkved (2004) discusses the permanent acquisition array; van Gestel et al. (2008) discuss aspects of the data acquisition, processing, and analysis; and Hatchell et al. (2005) and van Gestel et al. (2011) discuss aspects of the data interpretation and integration with other reservoir data.

In this paper, we consider data from the first (LoFS 1) and the ninth (LoFS 9) surveys acquired in November 2003 and December 2007, respectively. For this study, to avoid imaging challenges caused by a gas cloud located above the crest of the Valhall structure, we choose a subset of the original data covering the Southern flank of the structure. Whereas the original (full) data consists of approximately $50,000$ shots and 2400 receivers, the data subset consists of approximately $33,000$ shots and $470$ receivers. Shots are spaced at $50$ m in both the inline and crossline directions, while the receivers, located along $10$ permanent cables at approximately $70$ m depth, are spaced at $50$ m in the inline and $300$ m in the crossline directions (Figure 1). The maximum absolute source-receiver offset is $5$ km. The data have been preprocessed, preserving only the up-going primary compressional wave data. To simulate an obstruction, we create a $1.44$ sq. km gap in the monitor data at the center of the $9$ sq. km study area (Figure 1(b)). Figure 2 shows the resulting common-midpoint (CMP) fold for the complete (baseline) and incomplete (monitor) geometries. Using reciprocity, shot and receiver locations are swapped, such that receiver gathers are treated as shot records. The data are migrated using 320 frequencies (up to $35$ Hz) with a split-step one-way wave-equation shot-profile migration algorithm. All data are migrated with the baseline velocity model (Figure 3) obtained--to a satisfactory degree of accuracy--by full waveform inversion (Sirgue et al., 2010). The target area is a small ($700$ x $3000$ x $3000$ m) window around the reservoir, located outside the area most affected by the gas cloud. For both the baseline and monitor geometries, we compute the target-oriented Hessian using $64$ frequencies spaced equally within the migration frequency band.

gapped1,gapped2
Figure 1. Acquisition geometry showing locations of all shots and receivers (a) and a zoom showing only the study area (b). Apart from the introduction of a gap, the source-receiver geometry is closely repeated for both data sets. Note that the gap is located at the center of the study area. The coordinate axes in these figures (and in all figures) are distances in meters.

cfold1,cfold2
Figure 2. Surface (CMP) fold for the baseline (a) and monitor (b). Red indicates high fold, whereas blue indicates low fold. Note that whereas the baseline fold is mostly uniform within the study area, the gapped monitor geometry causes significant non-uniformity of fold. The box indicates the same study area shown in Figure 1(b).

vel-1l
Figure 3. Baseline migration velocity obtained by full waveform inversion (Sirgue et al., 2010). Red indicates high velocity, whereas blue indicates low velocity. This velocity model was used to image all data sets in this study. Note that the target area--indicated by the box--is restricted to a small area of interest around the reservoir. The gas cloud, located outside the study area does not cause significant imaging challenge in the target area.

The diagonals of the Hessian matrices (subsurface illumination/fold) for the study area obtained using the complete (baseline) and incomplete (monitor) geometries are shown in Figure 4. Note that in both cases, illumination distribution is highly non-stationary throughout the study area. The ratio between the Hessian diagonals for the two geometries are shown in Figure 5. Note that although the illumination discrepancy is simple at the ocean bottom (Figure 5(a)), this discrepancy becomes highly complex at the reservoir depth (Figure 5(b)).

The migrated baseline and monitor images of the study area are shown in Figure 6. Note that the differences between the images at the reservoir depth are due to a combination of production-related changes and the gap in the monitor data. In addition, the panels in Figure 6 show the target area for inversion. Because of fluid changes caused by production and injection, and compaction caused by pressure depletion, imaging the monitor data with the baseline velocity causes apparent displacements between the baseline and monitor images. Components of the apparent displacements between the baseline and monitor images (Figure 7) are obtained using a cyclic 1D correlation approach (Ayeni, 2011). Before estimating time-lapse images, and prior to inversion, the baseline and monitor are aligned using these apparent displacements. Time-lapse amplitudes extracted within a $60$ m window around the reservoir after migration and inversion are shown in Figure 9.

ilum-1,ilumg-1,ilum-3,ilumg-3,ilum-4,ilumg-4
Figure 4. Hessian diagonal for the complete baseline (left) and incomplete monitor (right). In these (and similar) displays throughout this paper, the top panel is a depth slice and the side panels are the inline and crossline slices. The crosshairs show the position of the slices in the image cube. The depth slices show the illumination at the ocean bottom (a) and (b); above the reservoir (c) and (d); and within the reservoir (e) and (f). Note the locations of the complete receiver lines in (a) and the gap in (b). Red indicates high illumination, whereas cyan indicates low illumination.

ilumr-1,ilumr-4
Figure 5. Illumination ratio between the baseline and monitor at the ocean bottom (a) and at the reservoir depth (b). Note that the simple rectangular illumination disparity at the ocean bottom becomes more complex at the reservoir depth.

mig-1-box,mig-4-box,migg-1-box,migg-4-box
Figure 6. Migrated images showing depth slices at the ocean bottom (left) and at the reservoir depth (right). The box indicates the target area in the baseline image (a) & (b), and in the monitor image (c) & (d). Note the location of the gap in the monitor.

ts-1,ts-2
Figure 7. Vertical (a) and inline (b) components of apparent displacement vectors between the baseline and monitor images within the target area. In both Figures, red indicates positive (downward or rightward) apparent displacements, whereas blue indicates negative (upward/leftward) apparent displacements. Similar results were obtained for the crossline displacement components (not shown). Prior to inversion, the baseline and monitor images are aligned using these apparent displacements.

fine-mod1,gap-mod1,fine-1,gap-1
Figure 8. Migrated (a) & (b), and inverted (c) & (d) monitor images for the target area. Panels (a) & (c) are obtained from the complete monitor data, whereas (b) & (d) are obtained from the incomplete (gapped) monitor data. Note that inverted images (c) & (d) show improved resolution over the migrated images (a) & (b).

hor-4-d3,hor-4-d2,hor-4-d1,hor-4-d
Figure 9. Absolute time-lapse amplitudes in the reservoir obtained from migration (a) & (b), and inversion (c) & (d). Note the discrepancy in the time-lapse amplitude distribution obtained via migration of the complete (a), and incomplete (b) data. Note that this discrepancy has been removed via inversion of the same data sets (c) & (d).

Least-squares wave-equation inversion of time-lapse seismic data sets - A Valhall case study