Abstract
In this study we explore the use of multilevel derivative-free optimization for history matching, with model properties described using PCA-based parameterization techniques. The parameterizations applied in this work are optimization-based PCA (O-PCA) and convolutional neural network-based PCA (CNN-PCA). The latter, which derives from recent developments in deep learning, is able to represent accurately models characterized by multipoint spatial statistics. Mesh adaptive direct search (MADS), a pattern search method that parallelizes naturally, is applied for the optimizations required to generate posterior (history matched) models. The use of PCA-based parameterization reduces considerably the number of variables that must be determined during history matching (since the dimension of the parameterization is much smaller than the number of grid blocks in the model), but the optimization problem can still be computationally demanding. The multilevel strategy introduced here addresses this issue by reducing the number of simulations that must be performed at each MADS iteration. Specifically, the PCA coefficients (which are the optimization variables after parameterization) are determined in groups, at multiple levels, rather than all at once. Numerical results are presented for 2D cases, involving channelized systems (with binary and bimodal permeability distributions) and a deltaic-fan system, using O-PCA and CNN-PCA parameterizations. O-PCA is effective when sufficient conditioning (hard) data are available, but it can lead to geomodels that are inconsistent with the training image when these data are scarce or nonexistent. CNN-PCA, by contrast, can provide accurate geomodels that contain realistic features even in the absence of hard data. History matching results demonstrate that substantial uncertainty reduction is achieved in all cases considered, and that the multilevel strategy is effective in reducing the number of simulations required. It is important to note that the parameterizations discussed here can be used with a wide range of history matching procedures (including ensemble methods), and that other derivative-free optimization methods can be readily applied within the multilevel framework.
