Multidimensional reservoir description and data integration using time-space wavelet analysis

Developing computationally efficient nonlinear regression algorithms is central to reservoir parameter estimation. The problem of inverse analysis is further complicated when the constraining data exist at different scales of resolution. In this case, integrating the data to reduce the scale effect becomes necessary. A commonly adopted approach is to use different scaling factors to bring the different datasets to the same resolution. However, there is no unique way to select scaling factors and this introduces some bias into the inverse modeling giving more importance to some datasets. In this paper, we propose the integration of multiwell production data using the wavelet transform. The method involves the use of the two-dimensional wavelet transformation of the data space to reduce the linear adjoint system for a multiphase flow sensitivity computation. The approach proved very effective at reducing the cost of computing sensitivity coefficients. Multiple datasets from different wells, representing different production responses (pressure, water cut, etc.), were treated as a single matrix of data rather than separate vectors that assume no correlation amongst datasets. This enabled us to transform the multiwell production data into a two-dimensional wavelet domain and subsequently select the most important wavelets for history match. By minimizing the square of the Frobenius norm of the residual matrix we were able to match the calculated response to the observed response. We derived the relationship that allows us to replace a conventional minimization of the square of the l ₂ norms of multiobjective functions with the minimization of the square of the Frobenius norm of the integrated data. The usefulness of the approach is demonstrated using several examples.