History Matching with Dual-Loop Techniques
Authors: Ates Harun , Kelkar Mohan 
Published: December 2007

Abstract
In most oil fields, production data are available in abundance. However, most reservoir characterization studies rely on static reservoir descriptions because efficient methods are not available to incorporate realistic field production data. In this study, two-phase production history data was incorporated into reservoir characterization by inverse modeling. We have developed a procedure for optimizing both relative permeability and heterogeneity characteristics for history matching. The most probable model of reservoir heterogeneity is obtained by dual-loop inversion technique that combines Gauss-Newton and conjugate gradient algorithms. This technique allows one to avoid the construction of the sensitivity coefficient matrix. Thus, a substantial reduction in the computational cost and storage requirements is achieved. This technique has been shown to be very efficient for reservoirs with large number of grid blocks, which in turn makes it feasible for field case studies.
The approach we consider in optimizing the relative permeability parameters is based on the premise that the relationship between fractional flow versus saturation is unique at each well and is represented by relative permeability curves. This is particularly suitable for true field studies since fractional flow as a function of time may appear different at each producer, and exact breakthrough time is often missing in field data. The approach has been tested with synthetic runs and a significant improvement in future predictions after optimizing relative permeability parameters has been observed.
Finally, this approach was also validated using a field case. The field case study was conducted on a North Sea field, which has been subjected to water flood. The field is a sandstone reservoir, and is partially influenced by water influx.

Introduction
Since production data are usually available for every production well, the successful implementation of the inverse procedure under multiphase flow conditions will substantially increase our ability to characterize a mature reservoir for history matching purposes. Two-phase production data and prior geostatistical information can be integrated using a Bayesian inversion technique together with an efficient optimization algorithm based on gradient methods. This technique has been applied in a number of cases1,2for single-phase flow.
Chu et al.3 proposed a dual loop procedure, which utilizes conjugate gradient relaxation in order to solve the inverse problem, efficiently. The use of conjugate gradient relaxation in the inversion allows bypassing the actual computation of the sensitivity coefficient matrix and the inversion of the product of sensitivity coefficient matrix and its transpose. Ates and Kelkar4 developed and implemented the dual loop procedure for two-phase flow. A substantial reduction in computational cost and storage requirements for solving the inverse problem was achieved. Proposed procedure uses sensitivity matrix equation whose terms can be an alytically computed outside forward simulation run. Thus, the method is independent of flow simulator that is used (i.e., it can be incorporated to any finite difference or streamline simulators). Only one simulation run is required to supply all the necessary information for sensitivity coefficient matrix, which is not explicitly constructed.