Estimation of Depths of Fluid Contacts by History Matching Using Iterative Ensemble Kalman Smoothers

Abstract

With the ensemble Kalman filter (EnKF) or smoother (EnKS), it is easy to adjust a wide variety of model parameters by assimilation of dynamic data. We focus on the case where reallizations and estimates of the depths of the initial fluid contacts as well as gridblock rock property fields are generated by matching production data with the EnKS. The objective is to account for uncertainty in the depths of the contacts and provide improved estimates of the depths by conditioning reservoir models to production data. The efficiency of EnKF and EnKS arises because data are assimilated sequentially in time and so "history matching data" requires only one forward run of the reservoir simulator for each ensemble member. For EnKS and EnKF to yield reliable characterizations of the uncertainty in model parameters and future performance predictions, the updated reservoir simulation variables, e.g., saturations and pressures, must be statistically consistent with the realizations of these variables that would be obtained by rerunning the simulator from time zero using the updated model parameters. This statistical consistency can only be established under assumptions of Gaussianity and linearity that do not normally hold. Here, we explore the use of iterative EnKS methods that are statistically consistent to improve the performance of EnKS. Introduction Since the ensemble Kalman filter (EnKF) was introduced by Evensen (1994) in the context of ocean dynamics literature as a Monte Carlo approximation of the extended Kalman filter, it has been used extensively in meteorological and oceanographical systems. Since its introduction into the petroleum engineering literature (Naevdal et al., 2003, 2003), it has been the subject of extensive research investigation in petroleum engineering as summarized recently by Aanonsen et al. (2008). Moreover, it is now been applied successfully to several field cases (Haugen et al., 2006; Evensen et al., 2007; Bianco et al., 2007) in the sense that application of the EnKF improved the history matches obtained by more traditional methods. A comprehensive overview of the methodology and theory can be found in Evensen (2007). One of the strengths of EnKF is the fact that one can easily incorporate virtually any model parameters into the state vector and subsequently adjust these parameters by assimilating production and seismic data. In the field example presented by Evensen et al. (2007), and in the problem considered here, the set of model parameters include the depths of the initial fluid contacts as well as gridblock porosities and permeabilities. As the initial depths of contacts are adjusted at every data assimilation step, the method is an emsemble Kalman smoother (EnKS) even though we do not explicitly update the initial saturation and pressure distributions during data assimilation. The basic problem of updating depths of initial fluid contacts was considered previously by Thulin et al. (2007). The data matches and uncertainty estimates they obtained using EnKF (EnKS) were not nearly as good as expected. As we discuss in more detail later, one reason for that is that they used in inappropriate truncation scheme in singular value decomposition (SVD). Results presented here indicate that a second reason for the relatively poor performance is due to nonlinearity.

Ensemble-based conditioning of reservoir models to seismic data

“…The numerical cost can possibly be reduced by switching from localization of the Kalman gain based on all measurements simultaneously, to assimilating uncorrelated batches of observations sequentially. Wang et al [40] claimed that the EnRML scheme is not guaranteed to produce a downhill gradient direction due to all approximations involved, and that results may be rather sensitive to the choice of the step-size β. Further experimentation is needed to investigate this issue.…”

Inversion of the Fluvial Channel Reservoir Permeability Field Using Ensemble Kalman Filter Based on Discrete Cosine Transform

“…However, due to the application of the average sensitivity matrix, EnRML can not guarantee that the search direction is always downhill. From the experiment results, the better result could not be obtained by doing more iteration [21] . Compared to EnRML method, the computation of the HIEnKF is more efficient.…”

“…To overcome the inconsistency between the updated models and the updated dynamical variables in nonlinear problems, we improve the standard EnKF and get the half iterative EnKF (HIEnKF) [21] . State vector only includes model parameters in the HIEnKF, thus the HIEnKF analysis equation is…”

“…The result is not improved when more iterations are done. Wang [21] proposed a half iterative EnKF (HIEnKF) to estimate the depths of the fluids contacts and relative permeability curves. The result obtained from HIEnKF is similar to that obtained from EnRML.…”

Ensemble Kalman filtering for hydraulic conductivity characterization: Parallelization and non-Gaussianity

“…This iteration is needed because of the strong non-linearities of the forecast model. Example applications can be found in the petroleum engineering literatures (e.g., Liu and Oliver, 2005;Oliver, 2007, 2006;Chen, 2005, 2006;Li and Reynolds, 2007;Wang et al, 2010;Zhao et al, 2008), and also in hydrogeology (Franssen and Kinzelbach, 2008).…”