The primary objective of this lecture is to provide a fundamental understanding of assisted (automatic or semi-automatic) history matching (data assimilation) where history-matching will be viewed primarily in the context of a large-scale inverse problem. We begin with methods for constructing a solution of an ill-posed linear inverse problem in order to gain an understanding how modeling or computational errors can lead to solutions of little use and to motivate the need for regularization. This naturally leads to the use of a prior Gaussian model for regularization of the standard least squares history-matching problem and it is easy to show that minimization of this regularized least squares problem is equivalent to finding a model which maximizes the conditional probability density function (pdf) constructed using Bayesian statistics. With this Bayesian formulation, we present a discussion of procedures for uncertainty quantification. Next, we discuss gradient-based algorithms for minimizing
the regularized least squares objective function where we focus on the adjoint method for calculating the sensitivity rather than the details of optimization algorithms. Ensemble-based methods, e.g., the ensemble Kalman filter and the ensemble smoother, will be motivated directly from gradient-based history-matching techniques. Time permitting, the course will end with a field example of assisted history matching with uncertainty quantification.
A major part of the material presented will be from Oliver, D. S., Reynolds, A. C. and Liu, N.: Inverse Theory for Petroleum Reservoir Characterization and History Matching,
Cambridge University Press, 2008.