Variance-Preserving Ensemble Smoother with Multiple Data Assimilation for History Matching with Diagonal Observation Error Covariance

Summary The ensemble smoother with multiple data assimilation (ESMDA) is widely used for reservoir history matching because of its parallel computational efficiency and ease of implementation. Production data in high-dimensional reservoir applications are often correlated across time, wells, and response types. When a diagonal observation error covariance is used, these correlations are ignored, and redundant observations are treated as independent. Standard ESMDA can therefore overcondition the ensemble and underestimate posterior uncertainty. We address this problem with a variance-preserving ESMDA (VP-ESMDA) based on the effective dimension of production data. At each assimilation step, VP-ESMDA constructs a standardized observation-anomaly Gram matrix from the predicted-response ensemble, computes the effective observation dimension via a participation ratio, and converts this into a redundancy factor that inflates the observation error covariance. When the total number of observations greatly exceeds the diagnosed effective dimension, the update is automatically weakened, preserving ensemble variance. We validate the proposed approach on scalar nonlinear benchmarks, a synthetic 2D reservoir, and the Brugge field case. The results show that under correlated observations, VP-ESMDA recovers a posterior distribution closer to the reference solution, yields larger posterior variance than standard ESMDA, and mitigates the underestimation of posterior uncertainty.