Multi-scale Modeling of 1-D Permeability Fields

Permeability plays an important role in subsurface fluid flow studies, being one of the most important quantities for the prediction of fluid flow patterns. The estimation of permeability fields is therefore critical and necessary for the prediction of the behavior of contaminant plumes in aquifers and the production of petroleum from oil fields. In the particular case of production of petroleum from mature fields, part of the available information for the estimation of permeability fields is the production data. To incorporate such information in formal statistical analysis, corresponding likelihood functions for the high-dimensional random field parameters representing the permeability field can be computed with the help of a fluid flow simulator (FFS). Additional information about the permeability field is usually available at different scales of resolution as a result of studies of the geological characteristics of the oil field, well tests, and laboratory measurements. In this paper, in order to incorporate the information available at the different scales of resolution, we use the multi-scale time series model introduced in Ferreira et al. (2001) as a prior for 1-D permeability fields. Estimation of the multi-scale permeability field is then performed using an MCMC algorithm with an embedded FFS running at different scales to incorporate the information given by the production data. We study with simulated data the performance of the proposed approach with respect to the recovery of the original permeability field.