Given the reservoir models both pre-flood and for each of two time steps of waterflood production, synthetic surface reflection seismic data can be generated for each time snapshot of production.
The synthetic seismograms are calculated based on the generalized Kirchhoff body force scattering theory of Lumley and Beydoun (1993). This theory combines Zoeppritz plane-wave reflection and Rayleigh-Sommerfeld elastic diffraction responses. The modeling theory can generate the correct AVO response for locally planar reflectors, and merge into diffraction where elastic properties vary spatially faster than a seismic wavelength. However, since the Green's tensors are WKBJ ray-valid, only primary reflections and diffractions are computed. Higher order multiple scattering or phenomena such as surface waves are not modeled.
The modeling theory is based on a volume integration of the body force equivalent of the coefficient:
is the P-P scattered vector wavefield measured at the receiver position , and projected onto an arbitrary receiver component direction . As and Ar are the elastic WKBJ Green's function amplitudes from the source and receiver positions respectively, to each subsurface scattering point in the volume . Similarly, is the total traveltime from source to scatterer to receiver. The factor is the non-geometric diffraction angle as shown in Figure , and is unity along the Snell reflection path (specular), and variable otherwise. The factor is the plane-wave Zoeppritz elastic P-P reflection coefficient, and k is the spatial wavenumber .The incident arrival direction vector is drawn in Figure .
Figure shows a noise-free synthetic CMP gather located at an injection well. The left panel is before water injection, and the right panel is after two time steps of waterflood. The reservoir reflections occur at and just below 2 seconds traveltime. In the pre-flood case, there is a strong reflection and AVO response at the reservoir. After two time steps of waterflood, the reservoir reflection is much weaker and there is almost no discernable AVO response. Figure shows the same CMP gathers, but with a significant level of spatially-uncorrelated Gaussian distributed noise, filtered to the same frequency bandwidth as the signal (10-60 Hz). The S/N ratio is about 2:1 at the reservoir reflection in the pre-flood case, and about 1:1 after two time steps of waterflood. It would be very difficult to detect the reservoir reflection after waterflood alone, but the change in reflection character during time-lapse monitoring of the production process seems quite easy to detect for this geometry, petrophysical model, and noise distribution.