Data-Partitioning Technique for Gradient-Based-Optimization Methods in History Matching
Authors: Denney Dennis 
Published: February 2015

This article, written by Senior Technology Editor Dennis Denney, contains highlights of paper SPE 130473, ’Development of a Data-Partition Technique for Gradient- Based-Optimization Methods in History Matching,’ by D.Y. Ding, SPE, IFP, prepared for the 2010 SPE EUROPEC/EAGE Annual Conference and Exhibition, Barcelona, Spain, 14-17 June. The paper has not been peer reviewed.
Assisted history matching is used widely to constrain reservoir models. However, history matching is a complex inverse problem, and it is always a challenge to history match large fields that have a large number of parameters. A new technique is presented for gradient-based-optimization methods to improve history matching of large fields. This technique uses data partitioning for the gradient calculations. This perturbation design enables calculating all derivatives of the objective function with only a few perturbations.
Introduction
Assisted history matching constrains reservoir and/or geological models by integrating well-production data and/or 4D-seismic data. The degree of difficulty and the computational effort (in terms of the number of reservoir simulations) increase with the increasing number of matching parameters. Generally, a reasonable history match is acquired at the field level first, then at the regional level, followed by more-rigorous individual-well history matching. The number of parameters usually is increased when the history matching moves to the regional or well levels. Therefore, it is important to study optimization methods for a large number of parameters, especially for large fields.
Among optimization methods for history matching, gradient-based approaches are prevalent. However, gradients of the objective function often are calculated by numerical methods, which need the evaluation of the objective function through reservoir simulations, which, in turn, need considerable CPU time. To optimize M parameters, at least M perturbations (M+1 simulations) are required to calculate all the gradients to obtain an optimized solution. When M is large, serious problems in CPU time can be encountered.
The adjoint method is an efficient technique to reduce the number of simulations for gradient calculation. However, an accurate formulation and solution of the adjoint method requires specific knowledge of the discrete equations in the reservoir simulator. In addition, this method is not always computationally feasible for constructing the Hessian matrix for the Gauss-Newton approach in the optimization.
An alternative approach for reducing the number of simulations in history matching is to split the objective function and/or parameters by subregions. One technique uses principal-component analysis for the Hessian matrix to delineate multiple reservoir regions for history matching. Another method uses independent objective functions with a direct-search algorithm. A polynomial-response surface defines separable subregions in parameter space and applies the neighborhood algorithm for the optimization in 4D-seismic history matching. However, full independence of subregions within a reservoir generally is not possible. Interaction between different regions should be considered.