A finite-difference injection approach to modeling of seismic fluid flow monitoring
Authors: Robertsson J.O.A. , Ryan-Grigor S. , Sayers C. , Chapman C.H. 
Published: October 2008

Abstract
The finite-difference (FD) injection method by Robertsson and Chapman5,6, can be used to efficiently synthesize the seismic response after local model alterations. By integrating the FDinjection method with a reservoir simulator and petrophysical model, a highly efficient and accurate tool for predicting the seismic response of reservoir fluid flow has been developed. In a case study, six complete marine seismic surveys over a reservoir at different production stages were simulated with computational savings greater than a factor of 50.

Introduction
In time-lapse seismics, the difference between seismic data sets acquired at different times during the production process is used to infer changes in the distribution of fluids and pressure due to production. To decide whether a repeat seismic survey is able to detect fluid changes predicted by reservoir simulation, and to check if an existing seismic data set is consistent with the reservoir model, a seismic forward model is required. The predicted distributions of water saturation, pressure and temperature from a reservoir simulator can be combined with rock properties, via a petrophysical model, to provide the elastic moduli for the seismic modeling (e.g., Huang et al. 4 and Biondi et al.2). We combine the realism of such a mechanism for generating seismic models with the advantages of FD modeling. FD methods allow for the full response to be synthesized as the wavefield interacts with a seismic model. This includes wave propagation in heterogeneous anisotropic and anelastic media, scattering, mode conversions, etc. Although accurate, FD calculations can be highly computationally expensive. The FD-injection technique5,6 provides a means to efficiently compute the response from a seismic model subject to changes within subvolumes and is thus ideally suited for time-lapse studies.
We demonstrate our approach through a case study based on the Gullfaks Field in the North Sea, where a successful real time-lapse study was carried out7. The model is not designed to represent the Gullfaks Field, but to demonstrate the application of the finite-difference injection technique in modeling time-lapse seismic data.

FD-Injection
Initially, the FD response from a full reference model is calculated and the wavefield is recorded at receivers and along a closed surface around a sub-volume. This surface is referred to as the injection surface Si. As changes to the model occur inside the sub-volume, the recorded seismograms can be updated by simulating the response on a small FD sub-mesh encompassing the neighborhood of the sub-volume. In the subsequent simulations on the sub-mesh, the wavefield recorded in the first simulation is injected into the sub-mesh along Si. For the case when no model alterations have taken place, the field inside Si will be identical to the field inside Si in the original FD simulation on the full model. The field outside Si will be as close to zero as machine precision allows, since on Si, the injection procedure exactly generates the ingoing waves and cancels the out-going waves.