Numerical simulation of first-order backscattered P and S waves for time-lapse seismic imaging in heterogeneous reservoirs

Abstract

First-order elastic wave scattering theory, validated in general features by deep-borehole seismic coda wave observations, is used to numerically simulate time-lapse borehole seismic data aimed at monitoring oil-water substitution in heterogeneous hydrocarbon reservoirs. A first-order perturbation solution of the vector wave equation leads to P-P and S-S backscattered displacement vector motion expressed as the angular summation of the second radial derivative of P- and S-wave velocity and density fluctuations α'/α, β'/β, ρ'/ρ over expanding spherical wave fronts of radii ξ=αt/2 and ξ=βt/2. P- and S-wave velocity and density spatial heterogeneity is modelled as long-range correlated random fluctuations consistent with the power-law-scaling character of crustal rock physical properties measured by borehole logs. The spectra of model scattering displacement seismograms for a power-law-scaling volumetric noise distribution duplicate the frequency enrichment observed in the spectra of broadband earthquake coda waves recorded in a deep well. Modelling of time-lapse scattering in power-law-scaling permeability structures suggests that a stable borehole seismic source can locate oil-water substitution in reservoir volumes tens of metres on a side at distances of up to a few hundred metres from an observation well using currently available borehole seismic technology. Time-lapse tracking of oil-water substitution and the monitoring of reservoir stress conditions can lead to spatially well-constrained reservoir models, despite large-scale, large-amplitude correlated random heterogeneity.