## Modeling Proxy Time-Lapse 4D Seismic Observations

Time-lapse (4D) seismic observations can be used as an additional data source during reservoir production. These results are usually considered in the form of a solution of a separate inverse problem, e.g., full waveform inversion (FWI). In this problem, geophysical model parameters, which are coefficients in the seismic wave equation, are estimated by matching, for example, calculated seismograms and observed data. Performing FWI requires the solution of the seismic wave equation as a forward problem. For current CLRM purposes, however, the process of obtaining the field property data can be emulated using a synthetic model. A degree of geological realism can be achieved by adding noise to the synthetic data, e.g. to phase saturations. As the level of noise increases, however, the information content of these data will decrease. In addition, the seismic wavelength, which typically differs from the dimensions of reservoir grid blocks, can be incorporated by applying spatial filtering.

In our work, time-lapse seismic data enters in the form of measurements for the phase saturation. We use the simulated saturation data, in some cases supplemented with noise, as a proxy. Figure on the right illustrates schematically the general process of constructing proxy seismic data. Three steps are involved:

- Generate "true" data from the solution of the reservoir equation performed with the true field
- Add noise to the simulated data accounting qualitatively for the fact that the seismic measurement is more precise when porosity is higher.
- Apply spatial filtering to the simulated data with added noise; i.e., average observed data over the half wave-length.

Last step can be accomplished by employing a simple spatial filtering operations, e.g., a discrete convolution with a rectangular kernel, which also has a smoothing effect. As an example for the 2D case, figure shows a three-by-three "box blur" discrete kernel (convolution matrix). Filtering with this template replaces values for a given pixel in the original image by the average of the target pixel and its eight direct neighbors. Kernels of this class are also called low pass filters, as they remove high image frequencies and also contribute to noise reduction. Details on this algorithm for the practical implementation are available in References here.