Analysis of Geometric Selection of the Data-Error Covariance Inflation for ES-MDA

TL;DR

This study analyzes geometric inflation strategies in ES-MDA for assisted history matching, showing that constant inflations can cause over- or under-shooting and that a geometrically decreasing schedule improves the balance between data fit and model change. It introduces two procedures: GEO1 (based on a regularization-inspired initial inflation) and GEO2 (based on last-step inflation with MDP checks), with GEO2 exhibiting more robust performance for small and large problems. Across a 2D synthetic model, UNISIM-I-H benchmark, and a field case, GEO2 achieves competitive data-mismatch while preserving posterior variance and geological realism, providing better uncertainty quantification for forecasts. The methods leverage singular-value–driven sensitivity insights, MDP-based stability checks, and localization to enable scalable, data-efficient history matching in complex reservoirs.

Abstract

The ensemble smoother with multiple data assimilation (ES-MDA) is becoming a popular assisted history matching method. In its standard form, the method requires the specification of the number of iterations in advance. If the selected number of iterations is not enough, the entire data assimilation must be restarted. Moreover, ES-MDA also requires the selection of data-error covariance inflations. The typical choice is to select constant values. However, previous works indicate that starting with large inflation and gradually decreasing during the data assimilation steps may improve the quality of the final models. This paper presents an analysis of the use of geometrically decreasing sequences of the data-error covariance inflations. In particular, the paper investigates a recently introduced procedure based on the singular values of a sensitivity matrix computed from the prior ensemble. The paper also introduces a novel procedure to select the inflation factors. The performance of the data assimilation schemes is evaluated in three reservoir history-matching problems with increasing level of complexity. The first problem is a small synthetic case which illustrates that the standard ES-MDA scheme with constant inflation may result in overcorrection of the permeability field and that a geometric sequence can alleviate this problem. The second problem is a recently published benchmark and the third one is a field case with real production data. The data assimilation schemes are compared in terms of a data-mismatch and a model-change norm. The first norm evaluates the ability of the models to reproduce the observed data. The second norm evaluates the amount of changes in the prior model. The results indicate that geometric inflations can generate solutions with good balance between the two norms.

Figures (14)

• Figure 1: True log-permeability (2D model). Circles correspond to oil producing and triangles to water injection wells.
• Figure 2: Discrepancy function (2D model).
• Figure 3: Mean log-permeability (2D model).
• Figure 4: Water rate at well P2. Red dots are the observations, red curve is the prediction from the true model and blue curves are the predictions from the posterior ensemble (2D model).
• Figure 5: Position of the wells (UNISIM-I-H)