A Multiresolution Analysis of the Relationship Between Spatial Distribution of Reservoir Parameters and Time Distribution of Well-Test Data

Abstract

Summary With recent advances in permanent downhole technology, well-test analysts now have to deal with enormous amounts of data. Using these data requires the development of efficient algorithms that are able to extract the relevant information from the data at minimal cost. We present a multiresolution wavelet approach to estimate the spatial distribution of reservoir parameters, by performing the nonlinear least-squares regression in the wavelet domains of both time and space. Wavelet transforms have the ability to reveal important events in time signals or spatial images. Thus, we transformed both the model space and the time-series pressure data into spatial wavelet and time wavelet domains, respectively, and used a thresholding to select a subset of wavelet coefficients from each of the transformed domains. These subsets were used subsequently in nonlinear regression to estimate the appropriate description of reservoir parameters. The appropriate subset is not only smaller; the problem is also reduced to the consideration of only the important components of the measured data and only the part of the reservoir description that depends on them. As a test of the approach, we first applied the model to well-test problems involving 1D (radially composite) reservoir systems. The inverse problem was solved to estimate the distributed permeability values by performing the nonlinear least-squares regression in the wavelet domains (time and space). Results obtained were compared with those obtained from the conventional nonlinear-regression approach, using all the pressure-time data and the full set of spatial reservoir parameters. The time/space wavelet approach proved to be efficient. By reducing the dimensions of the model and data spaces, the approach eliminates redundancy in the reservoir description and in the data set. Significantly, the approach reveals the true number of reservoir parameters that can be appropriately estimated from a given data set and also reveals which components of the full data set are active in constraining the reservoir model. Thus, the approach provides a good means to integrate different data properly while avoiding the inclusion of irrelevant data during nonlinear regression.