Theoretical and efficient practical procedures for the generation of inflation factors for ES-MDA

Abstract

The ensemble smoother with multiple data assimilation (ES-MDA) has proved to be a powerful assisted history-matching method. The main drawback of ES-MDA is that the inflation factors for damping the changes in model parameters have to be determined before starting the historymatch. Although various authors have provided suggestions for determining the inflation factors adaptively as the history-match proceeds, these methods often result in a large number of data assimilation steps which can make ES-MDA too computationally inefficient for practical application to largescale field problems. Here, we provide a theoretical procedure to determine exactly the minimum inflation factor at each data assimilation step that ensures the discrepancy principle is satisfied. Like previous adaptive ES-MDA methods, this method does not allow one to specify a priori the number of data assimilation steps to be done. Thus, using the exact theoretical procedure as a guide, we provide a practical efficient method for determining the inflation factors which allows one to specify a priori the number of data assimilation steps to be done with ES-MDA which still ensures that the initial inflation factor is chosen so that the discrepancy principle is approximately satisfied.

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, .…”

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

“…The next sections present two procedures to compute α 1 . The first procedure was originally presented by Rafiee and Reynolds (2017) and the second is proposed in this paper.…”

“…Moreover, these schemes usually require several data assimilation steps, often more than ten. Recently, Rafiee and Reynolds (2017) proposed to select the inflation factors in a geometrically decreasing order where the first inflation is based on the regularization condition of (Hanke, 1997). For a simple test problem, they showed that the geometric scheme resulted in smoother permeability fields compared to constant inflations.…”

“…The section after that presents the ES-MDA equations emphasizing that the inflation factors plays the same role of the regularization parameter described in the inverse problems literature. This section also reviews the method proposed in (Rafiee and Reynolds, 2017) and presents the new scheme for selecting a geometric sequence of α k . Both methods are tested in three reservoir history-matching problems.…”

“…Both methods are tested in three reservoir history-matching problems. This first problem is similar to the case used in (Rafiee and Reynolds, 2017) and it was designed to show overcorrection in the permeability field using constant inflation factors. The second one is a more realistic history-matching exercise proposed in (Maschio et al, 2013;Avansi and Schiozer, 2015).…”