Giselle Alvarez and Bruno Sansó
12/31/2007 09:00 AM
Applied Mathematics & Statistics
We consider the problem of joint estimation of some of the soil properties of an oil reservoir, like porosity and sand thickness. In an exploration scenario, only a few wells have been drilled, thus data from the wells are scarce. In contrast, there is an abundance of seismic data. In our example, which corresponds to a Venezuelan oil reservoir, the data available from the wells consist of gamma ray logs measured as a function of depth. The seismic data correspond to traces obtained around the wells.
The average properties of the soil corresponding to a given range of depths for the wells are known from direct measurement. The goal is to predict those properties at points where only seismic data are available.
We fit a multivariate linear regression model that accounts for the spatial correlation using spatial kernels. These are based on the family of Mat\`ern correlations. The kernels provide weighting of the information in the signals that are dependent on spatial locations. We first transform the dependent variable using discrete wavelets. We then perform a Bayesian variable selection procedure using a Metropolis search. This allows for the detection of the most informative wavelet coefficients. Not all the soil properties are available at all wells, so we use a Bayesian approach to handle the missing data. We obtain predictions of all the properties over the whole reservoir. Thanks to the Bayesian nature of our method we are able to provide a probabilistic quantification of the predictive uncertainties. The cross-validated results show that very high accuracy can be achieved even with a very small number of wavelet coefficients.
