# Calibration to well tops

Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 SEG Online Store

The depth structure maps derived from time-to-depth conversion or layer-by-layer inversion invariably will not match the well tops. The sources of discrepancy between the estimated reflector depths and the well tops include limitations in the methods for interval velocity estimation, mispicking of time horizons input to depth conversion, and limitations in the actual depth conversion itself within the context of ray tracing through an earth model that includes complex layer boundaries. For the depth structure maps to be usable in subsequent reservoir modeling and simulation, it is imperative to calibrate them to well tops.

Consider a seismically derived depth structure map zs(x, y) based on time-to-depth conversion, say, for the layer boundary associated with the top-reservoir. Also consider Nw well tops zw (xi, yi) for this horizon at locations (xi, yi), i = 1, 2, …, Nw. Since the velocity-depth model derived from time-to-depth conversion is supposed to be consistent with the input data — the time structure map τ(x, y) created from the interpretation of the time-migrated volume of data, we have

 ${\displaystyle \tau (x,y)=2{\frac {z_{s}(x,y)}{V_{s}(x,y)}},}$ (7a)

where, for the purpose of calibration, Vs(x, y) can be either the average or rms velocity map associated with the horizon zs(x, y). Also, for simplicity in calibration, we consider vertical rays rather than image rays as in equation (7a).

There exists a calibration velocity Vc(x, y) such that, at a well location (xi, yi), it satisfies the relation

 ${\displaystyle \tau (x_{i},y_{i})=2{\frac {z_{w}(x_{i},y_{i})}{V_{c}(x_{i},y_{i})}}.}$ (7b)

Combine equations (7a) and (7b) to get a relation that is satisfied at the well locations

 ${\displaystyle {\frac {V_{c}(x_{i},y_{i})}{V_{s}(x_{i},y_{i})}}={\frac {z_{w}(x_{i},y_{i})}{z_{s}(x_{i},y_{i})}}.}$ (8)

From the knowledge of the well tops zw(xi, yi) and the seismically derived reflector depths zs(xi, yi) at the well locations, equation (8) gives a calibration factor c(xi, yi) = Vc(xi, yi)/Vs(xi, yi) computed at each of the well locations. Next, apply kriging or some other interpolation technique to the sparsely defined calibration factors c(xi, yi), i = 1, 2, …, Nw to derive a calibration factor map c(x, y) specified at all grid locations (x, y). Kriging is a statistical method of determining the best estimate for an unknown quantity such as c(x, y) at some location (x, y) using a sparse set of values such as c(xi, yi) specified at locations (xi, yi) [1] [2].

The final step in calibration is to scale the depth structure map zs(x, y) by the calibration factor map c(x, y)

 ${\displaystyle z_{c}(x,y)=c(x,y)\ z_{s}(x,y),}$ (9)

where zc(x, y) is the calibrated depth structure map. Note that, by way of equations (8) and (9), the calibrated depth zc coincides with the well top zw at well location (xi, yi).

Calibration to well tops is done only after the completion of model building, and just before well planning and reservoir modeling. When estimating an earth model by following a layer-by-layer inversion procedure (next subsection), depth horizon associated with the (n − 1)st layer should not be calibrated before estimating the model for the next layer n. This is because seismically derived layer velocities almost never match with well velocities. The discrepancy between the two is attributable to several factors, including the limited resolution in velocities estimated from seismic data (models with horizontal layers, model with low-relief structure, and model with complex overburden structure) and seismic anisotropy (seismic anisotropy). Additionally, the high-frequency variations in the well velocities are absent from the seismically derived velocities.

The calibrated depth maps can be used to create a solid model of the earth as illustrated in Figure 9.4-13. Each layer is represented by a solid (Figure 9.4-14) with its interior populated by specific layer parameters. These may include compressional- and shear-wave velocities, densities, and rock physics parameters such as porosity, permeability, pore pressure, and fluid saturation. When populated by the petrophysical parameters, the solid associated with the reservoir layer represents a reservoir model. For the purpose of reservoir modeling, the solid for the reservoir layer usually is downscaled in the vertical direction by dividing it into thin slices with a thickness as small as 1 m — much less than the threshold for vertical seismic resolution (seismic resolution). Additionally, the solid for the reservoir layer is upscaled in the lateral direction by dividing each thin slice into finite elements with a varying size of up to 250 m on one side. The reservoir model is eventually fed into a reservoir simulation scheme to predict the geometry of the fluid flow from the given reservoir parameters.

## References

1. Sheriff, 1991, Sheriff, R. E., 1991, Encyclopedic dictionary of exploration geophysics: Soc. Expl. Geophys.
2. David, 1987, David, M., 1987, Geostatistics: in Encyclopedia of Science and Technology, 6, 141–144, Academic Press.