Iterative ensemble smoothing scheme for reservoir history matching with unknown observation error covariance

Ensemble -based smoothers are well-known data assimilation tools in reservoir applications, including history matching and geophysical inversion. They entail the knowledge of the covariance of observation errors, which, however, is often poorly known in real applications and commonly estimated subjectively on the basis of a diagonal structure. This neglects the correlations within the observation errors, which may deteriorate the quality of the model estimates. Herein, we relaxed the independent observational error assumption by considering a nondiagonal structure for the covariance. Given that all the elements in the covariance matrix could not be estimated given a limited amount of data, which is typically the case in practice, we reduced the number of degrees of freedom by parameterizing this covariance with two scalar parameters, one for error variance and the other for error correlation length-scale. We then estimated these parameters together with the model variables using a hybrid algorithm that combined a variational Bayesian approach with particle filters and an ensemblebased smoother. The proposed approach was validated with a linear Gaussian model and a nonlinear reservoir flow model. The results clearly demonstrate the potential of the proposed method in effectively addressing the uncertainty of observations in the history -matching process.