Abstract
We consider the problem of estimating the properties of an oil reservoir, like porosity and sand thickness, in an exploration scenario where only a few wells have been drilled. We use gamma ray records measured directly from the wells as well as seismic traces recorded around the wells. To model the association between the soil properties and the signals, we fit a linear regression model. Additionally we account for the spatial correla tion structure of the observations using a correlation function that depends on the distance between two points. We transform the predictor variable using discrete wavelets and then perform a Bayesian variable selection us ing a Metropolis search. We obtain predictions of the properties over the whole reservoir providing a probabilistic quantification of their uncertainties, thanks to the Bayesian nature of our method. The cross-validated results show that a very high accuracy can be achieved even with a very small number of wavelet coefficients.
Highlights
Predicting the properties of a reservoir using the information provided by data collected from wells is a fundamental issue in petroleum management and exploration
Log and core data are usually scarce, as they are only available at the few locations where wells have been drilled, whether seismic data are usually available for the whole reservoir
We take a Bayesian approach to estimate the parameters in the model and, following Brown Fearn and Vannucci (2001), we specify a prior for the original regression coefficients b as a p-variate normal with mean 0 and covariance matrix H, denoted as Np(0, H), for H corresponding to the covariance matrix of an autoregressive process of order one. Such a distribution is used to guarantee that the components of b vary smoothly and that the variances of the transformed coefficient β show the typical decay of wavelet coefficients
Summary
Predicting the properties of a reservoir using the information provided by data collected from wells is a fundamental issue in petroleum management and exploration. Data collected from wells usually consists of core analyses, performed, on actual samples of the soil, and recordings of different electromagnetic, physical, chemical or radioactive properties of the soil obtained by inserting various tools into the well These are usually refer to in the industry as well logs or well profiles, (see Hearst and Nelson, 2000; Tiab and Donaldson, 2004). The method we present in this paper consists on regressing the observations obtained from the wells on the wavelet decomposition of a signal, either well logs or seismic traces. Wavelet transformations provide a parsimonious representation of the information in the signal Their multiresolution properties have been successfully applied to quantify the decay of energy from large to small scales in well logs and seismic traces, see Alvarez et al (2003).
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