## 1. Introduction

[2] Numerical models that describe subsurface multiphase flows greatly enhance the process of reservoir characterization and management, as they improve the accuracy of production forecast. The validity of these reservoir models hinges on access to information regarding the geological and petrophysical features of the actual field. Automatic history matching is a commonly used methods for reservoir characterization whereby reservoir parameters such as porosities and permeabilities are estimated so as to minimize a measure of discrepancy between observations and predictions. The heterogeneities of the geological formations and the uncertainties associated with the medium properties contribute to errors incurred in any history matching effort.

[3] Recently, the ensemble Kalman filter (EnKF) [*Evensen*, 2003] has been shown to be an efficient tool for history matching of reservoir models [*Gao et al.*, 2006; *Gu and Oliver*, 2005; *Liu and Oliver*, 2005; *Naevdal et al.*, 2003] enabling useful estimation of various reservoir parameters. The EnKF uses a Monte Carlo simulation scheme for characterizing the noise in the system, and therefore allows the representation of non-Gaussian perturbations. The performance of the EnKF has been assessed in a number of efforts. For example, in the assimilation of data for land surface models [*Zhou et al.*, 2006] conditional marginal distributions and moments estimated from the EnKF have been compared with estimates obtained from a sequential importance resampling (SIR) particle filter. It has been found, expectedly, that the accurate estimation of higher-order statistics requires a large ensemble size that is often computationally prohibitive. Moreover, current data assimilation techniques provide a map between the probability measure of input parameters and that of predicted observables. This distributional mapping does not lend itself to sensitivity analysis and uncertainty apportionment, as the functional dependencies are smeared out when probability measures are estimated. Furthermore, current procedures do not permit the straightforward estimation of modeling errors associated with the structure of the forward model.

[4] This paper presents a history matching methodology whereby the functional relationship between parameters (described as stochastic processes) and predictions (also described as stochastic processes) is itself approximated as a stochastic mapping. A description of this functional relationship permits the rapid assessment of stochastic sensitivities. Furthermore, the structured construction of an approximation to this functional relationship provides a methodology for model validation that will also be explored in the paper. Moreover, relying on concepts of Hilbertian projections and *L*_{2} convergence from approximation theory, it can be shown that a polynomial chaos description [*Ghanem and Spanos*, 2003] of this functional relationship ensures that the complete probabilistic content of the predicted solution is well approximated, including higher-order statistics and joint distributions. A polynomial chaos expansion of a random variable or process is a series expansion of the random quantity in terms of orthogonal (i.e., uncorrelated) random variables, a stochastic basis, each multiplied by a deterministic coefficient (the polynomial chaos coefficients). Thus, by integrating polynomial chaos representations of stochastic processes with the EnKF methodology, it becomes possible to mitigate the computational cost of the standard EnKF while yielding a robust methodology for exploring the effect of errors in model structure on model-based predictions. Efforts along similar lines have already been reported whereby polynomial chaos representations have been used to accelerate the convergence of Markov chain Monte Carlo methods used in Bayesian estimation [*Marzouk et al.*, 2007].

[5] The computational model for the stochastic response is first constructed by representing the unknown model parameters, as well as unknowns describing errors in model structure, via their polynomial chaos decompositions. A solution is then sought in the form of a polynomial chaos expansion. This yields a system of coupled nonlinear stochastic partial differential equations to solve for the polynomial chaos coefficients of the solution. The spectral stochastic finite element method (SSFEM) [*Ghanem and Spanos*, 2003], which relies on a stochastic Galerkin projection scheme, is employed to solve this system using a discrete time-marching scheme. The SSFEM is integrated within Sundance 2.0 [*Long*, 2004], a computational toolkit for the finite element formulation and solution of partial differential equations, for enhanced computational efficiency. Uncertainty propagation is then followed by a filtering step to update the polynomial chaos representations with measurements as they become available. During the filtering step, and since the predicted solution has been described by its polynomial chaos decomposition, its full joint probability distribution is readily computable, thus allowing data assimilation in the presence of non-Gaussian measurement and model errors, as well as non-Gaussian uncertainties in model and parameters. The computational load for a specified accuracy is comparable to that of the EnKF and the storage size is limited to the number of terms in the polynomial chaos expansion of the model states.

[6] The next section presents the details of the particular model of multiphase flow in porous media that is used to demonstrate the foregoing methodology. The equations governing multiphase flow in a random porous medium are derived using the polynomial chaos formulation. Section 3 gives a background of Kalman filtering techniques and their evolution from the standard Kalman filter [*Kalman*, 1960] to the EnKF. Then a detailed derivation of the polynomial chaos-based Kalman filter is presented. In section 4, the uncertainty representation is discussed leading to the derivation of the stochastic reservoir numerical model. Finally, the application of the polynomial chaos Kalman filter for reservoir characterization problems is presented, results are described, and a discussion of the validity of the proposed method is provided.