Analysis of geometric selection of the data-error covariance inflation for ES-MDA

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.

Data-space inversion with ensemble smoother

“…For nonlinear problems, the choice of the α k 's and N a has a major impact in the performance of the method. There are recent works proposing methods to select α k 's and N a [27,9,39,30,10]. However, here we use simplest choice, which consists of selecting N a in advance and setting α k = N a , for k = 1, .…”

Evaluating the Impact of the Inflation Factors Generation for the Ensemble Smoother With Multiple Data Assimilation

“…Their results suggest that the geometric generation of these factors enhances the ES-MDA final results. The studies of Evensen (2018) [11] and Emerick (2019) [12] supported this observation. Rafiee and Reynolds (2017) [10] also claim that it is possible to yield good ES-MDA final results even with a low number of assimilations, but with proper inflation factors selection.…”

“…From Equation (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13), we obtain a value of m such that ∇O(m) = 0. This vector corresponds to the maximum a posteriori estimate of the vector of model parameters and is denoted as m map .…”

“…Opposite to the left-hand side of Equation (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15), the right-hand side of the same equation displays a matrix problem of size N d × N d , which is more feasible when N d N m , as explained before. Therefore, one may compute m map as follows:…”

“…5. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19)(20)(21)(22) for different ensembles.…”