A Newton-Raphson iterative scheme for integrating multiphase production data into reservoir models

This paper presents a new linearized iterative algorithm for building reservoir models conditioned to multiphase production data and geostatistical data. The significant feature of the proposed algorithm is that the computation of the sensitivity coefficients of production data, with respect to model parameters, can be avoided. This leads to dramatic reduction of computational cost. Instead of generating the sensitivity matrix as required for the least-squares algorithms, the proposed approach relies on solving the inversion equations, which are derived from the necessary conditions for a functional extremum. It is proved that the proposed method requires considerably less computational effort than the traditional algorithms such as the Gauss-Newton method, the Levenberg-Marquardt method, and the generalized pulse-spectrum technique (GPST). This is because the computation of the sensitivity coefficients makes the traditional algorithms computationally intensive, However, for the proposed linearized iterative scheme, the computational requirement only depends on the timesteps used in the reservoir simulator rather than the number of parameters or the number of observed data. The linearized iteration scheme converges quickly because the inversion equations can be solved through the Newton-Raphson method. At each iteration, the new approach requires solving the finite difference equations and the linear adjoint equations only once, respectively. Since the solver for the flow equations can be used to solve both the adjoint equations and the inversion equations, the proposed algorithm can be easily applied to commercial reservoir simulators. In this paper, two numerical examples for incorporating water oil rate data into geostatistical models are given to prove the efficiency of the proposed algorithm.