Assessing the Dimensionality of Random Fields with Karhunen-Loève Expansion

In history matching of geostatistical model, we attempt to find multiple realizations that are conditional to dynamic data and representative of the model uncertainty space. The huge dimension of the model makes the history matching problem intractable in practice. In recent years, several parameterization methods are proposed for reducing the dimension of this problem, but it is difficult to quantify the covered uncertainty space. In this paper, we propose a parameterization method based on the Karhunen-Loève expansion of second order random fields. We assess its performance in terms of the reduction of the dimensionality of the random field of interest, i. e. the number of components of its truncated expansion. We show that a small number of components can represent the major part of the variance of the field while preserving the spatial variability. Then from an example of fluid flow simulation in porous media, we show that a permeability field approximated with a small number of components can reproduce the dynamic behaviour of the non-approximated one. This method, allowing both the reduction of the model dimension and the preservation of the model uncertainty space, is then of great interest for solving the history matching inversion problem.