Study of Perturbation Designs for the Partially Separable Objective Function With Gradient-Based Optimizations in History Matching

Summary Assisted history matching is widely used to constrain reservoir models by integrating well-production data and/or 4D seismic data. However, history matching generally requires a large number of reservoir simulations, which are very CPU time-consuming, in the optimization procedure. In this paper, we present a new technique that allows us to reduce the number of reservoir simulations for the gradient-based optimization method in history matching. The new method is based on the partial separability of the objective function with local components referred to the wells and/or the seismic zones. How to choose a good perturbation design for the gradient computation is the main issue. In this paper, the graph-coloring algorithm is applied to determine “independent” components and parameters for a partially separable function, and analytical test functions are proposed for the selection of perturbation designs. This method can significantly reduce the number of reservoir simulations for a partially separable objective function in history matching.

Convex separable minimization problems with a linear constraint and bounded variables

Consider the minimization problem with a convex separable objective function over a feasible region defined by linear equality constraint(s)/linear inequality constraint of the form “greater than or equal to” and bounds on the variables. A necessary and sufficient condition and a sufficient condition are proved for a feasible solution to be an optimal solution to these two problems, respectively. Iterative algorithms of polynomial complexity for solving such problems are suggested and convergence of these algorithms is proved. Some convex functions, important for problems under consideration, as well as computational results are presented.