Inversion of dynamic production data for permeability: fast streamline-based computation of sensitivity coefficients of fractional flow rate

Generation of permeability field in a reservoir model that matchs historical dynamic production data requires an inverse calculation. A gradient method is typically used to solve the inverse minimization problem and requires sensitivity coefficients of reservoir responses, e.g. fractional flow rate or pressure, with respect to the change in the permeability. This paper presents a novel semi-analytical streamline-based method for computing such sensitivity coefficients under the framework of two-phase (oil-water) flow conditions. This method is shown to be significantly faster and generate permeability fields with lower objective function than the traditional perturbation method. The method decomposes the multiple-dimensional full flow problem into multiple 1D problems along streamlines. The sensitivity of fractional flow rate at the production well is directly related to the sensitivity of time-of-flight (TOF) along each individual streamline and the sensitivity of pressure at grid cells along the streamline. The sensitivity of TOF of a streamline can be obtained analytically. The sensitivity of pressure is obtained as part of a fast single phase flow simulation. The proposed method is implemented in a geostatistically based inverse technique, called the sequential self-calibration (SSC) method. Results for fractional flow rate sensitivities are presented and compared with the traditional perturbation method. This new method can be easily extended to compute sensitivity coefficients of saturation (concentration) data.