Application of ensemble transform data assimilation methods for parameter estimation in reservoir modeling

Abstract

Over the years data assimilation methods have been developed to obtain estimations of uncertain model parameters by taking into account a few observations of a model state. The most reliable methods of MCMC are computationally expensive. Sequential ensemble methods such as ensemble Kalman filers and particle filters provide a favourable alternative. However, Ensemble Kalman Filter has an assumption of Gaussianity. Ensemble Transform Particle Filter does not have this assumption and has proven to be highly beneficial for an initial condition estimation and a small number of parameter estimation in chaotic dynamical systems with non-Gaussian distributions. In this paper we employ Ensemble Transform Particle Filter (ETPF) and Ensemble Transform Kalman Filter (ETKF) for parameter estimation in nonlinear problems with 1, 5, and 2500 uncertain parameters and compare them to importance sampling (IS). The large number of uncertain parameters is of a particular interest for subsurface reservoir modelling as it allows to parametrise permeability on the grid. We prove that the updated parameters obtained by ETPF lie within the range of an initial ensemble, which is not the case for ETKF. We examine the performance of ETPF and ETKF in a twin experiment setup, where observations of pressure are synthetically created based on the known values of parameters. For a small number of uncertain parameters (1 and 5) ETPF performs comparably to ETKF in terms of the mean estimation. For a large number of uncertain parameters (2500) ETKF is robust with respect to the initial ensemble while ETPF is sensitive due to sampling error. Moreover, for the high-dimensional test problem ETPF gives an increase in the root mean square error after data assimilation is performed. This is resolved by applying distance-based localization, which however deteriorates a posterior estimation of the leading mode by largely increasing the variance due to a combination of less varying localized weights, not keeping the imposed bounds on the modes via the Karhunen-Loeve expansion, and the main variability explained by the leading mode. A possible remedy is instead of applying localization to use only leading modes that are well estimated by ETPF, which demands a knowledge at which mode to truncate.

Comparison of regularized ensemble Kalman filter and tempered ensemble transform particle filter for an elliptic inverse problem with uncertain boundary conditions

“…1 a and b and as it has been reported in the literature, e.g. [28,29]. When uncertainty is in both permeability and boundary conditions, we investigate methods performance for two numerical setups.…”

“…Hence, one has to decrease the number of degrees of freedom, i.e. by distancebased localization of [27,28] abbreviated here LETPF. Assume we have a numerical grid of N × N size with grid cells denoted by X l for l = 1, .…”

“…It also has a localized version. In [28], we have employed the method to an inverse problem of uncertain permeability. We have shown that though localization makes the ensemble transform particle filter deteriorates a posterior estimation of the leading modes, it makes the method applicable to highdimensional problems.…”

“…This groundwater model was first used as benchmark for inverse modelling in [11]. It has been used as a test model for the identification of parameters, for example with iterative regularization methods [20], an ensemble Kalman approach [4], and in our previous work with particle filtering [28,29]. In this paper in addition to uncertain log permeability defined as a Gaussian process, we assume an error in boundary conditions that is non-Gaussian distributed.…”

Accounting for model error in Tempered Ensemble Transform Particle Filter and its application to non-additive model error

“…It also has a localized version. In [26], we have employed the method to an inverse problem of uncertain permeability. We have shown that though localization makes the ensemble transform particle filter deteriorate a posterior estimation of the leading modes, it makes the method applicable to high-dimensional problems.…”