Physics-based Residual Kriging for dynamically evolving functional random fields

We present a novel approach named Physics-based Residual Kriging for the statistical prediction of spatially dependent functional data. It incorporates a physical model—expressed by a partial differential equation—within a Universal Kriging setting through a geostatistical modelization of the residuals with respect to the physical model. The approach is extended to deal with sequential problems, where samples of functional data become available along consecutive time intervals, in a context where the physical and stochastic processes generating them evolve, as time intervals succeed one another. An incremental modeling is used to account for both these dynamics and the misfit between previous predictions and actual observations. We apply Physics-based Residual Kriging to forecast production rates of wells operating in a mature reservoir according to a given drilling schedule. We evaluate the predictive errors of the method in two different case studies. The first deals with a single-phase reservoir where production is supported by fluid injection, while the second considers again a single-phase reservoir but the production is driven by rock compaction.