Improving Perturbation Designs for Gradient-based Optimization Methods in History Matching

Assisted history matching is now widely used to constrain reservoir models by integrating well production data and/or 4D seismic data. Among the optimization methods for performing history matching, gradient-based approaches are often applied. However, history matching is a complex inverse problem, and the computational effort (in terms of the number of reservoir simulations, which are very expensive in CPU time) increases with the increasing of the number of matching parameters. For a problem with N parameters, we need generally N perturbations (or N+1 reservoir simulations) to calculate all the gradients in order to find an optimized solution direction. It is always a big challenge to history match large fields with a large number of parameters. In this paper, we present a new technique based on approximate derivative computations, which can considerably reduce the number of simulations for the gradient-based optimization. In this new approach, the objective function is first split into local components, and the dependence of each local component on principal parameters is analyzed to minimize the number of influential parameters. The interaction between parameters and local components is allowed in this approach. Then, we define a perturbation design, based on the minimisation of errors on a test function for the derivative calculation and the technique of graph colouring. The proposed perturbation design can compute the derivatives of the objective function with only a few simulations. This method is particularly interesting for regional and well level history matching, and it is also suitable to match geostatistical models by introducing numerous local parameters. This new technique makes history matching with large numbers of parameters (large field) tractable. Some numerical examples, including a large field with 400 parameters, are presented to illustrate the efficiencies of the new method. The commonly-used gradient-based optimization method is unpractical and even unfeasible to handle the large fields, which require 400 perturbations to compute the derivatives. However, using the new technique proposed in this paper, only 5 perturbations are performed to get all the required gradients. Therefore, the CPU time on this large field can be reduced by a factor of 80 and the history match is feasible.