Bayesian Deconvolution of Oil Well Test Data Using Gaussian Processes

We use Bayesian methods to infer an unobserved function that is convolved with a known kernel, using noisy realizations of an observable process. Our method is based on the assumption that the function of interest is a Gaussian process. Thus, the resulting convolution is also a Gaussian process. This fact is used to obtain inferences about the unobserved process, effectively providing a deconvolution method. We apply the methodology to the problem of estimating the parameters of an oil reservoir from well-test pressure data. From a system of linear ordinary differential equations, we write the equation that governs the dynamics of pressure in the well as the convolution of an unknown process with a known kernel. The unknown process describes the structure of the well. This is modeled with a Gaussian process whose mean function is obtained as a linear combination of specific bases. A purposedly designed directional Monte carlo method is used to sample from the posterior distribution of the parameters. Applications to data from Mexican oil wells show very accurate results.