Indicator-based data assimilation with multiple-point statistics for updating an ensemble of models with non-Gaussian parameter distributions

Abstract

Ground-water flow and the transport of contaminants in groundwater systems are strongly controlled by geologic heterogeneities. Because, the data available to model these subsurface heterogeneities are generally sparse, the observed groundwater flow, piezometric head at well locations, or the concentration profile of solutes provide valuable additional information about these complex subsurface systems. Regardless, there is likely to be considerable uncertainty associated with the predictions of the subsurface heterogeneities. This has motivated the development of several ensemble-based schemes for assimilating flow response data into predictions of subsurface heterogeneities. Most ensemble-based data assimilation methods including ensemble Kalman filter (EnKF) or indicator-based data assimilation (InDA), used for assimilation of dynamic data into geologic models, utilize statistics in the form of covariances calculated using the ensemble of models to perform model updates. These covariance based updates do not preserve the complex spatial characteristics of geologic structures that are better represented using multiple-point statistics. Additionally, in EnKF, the mismatch between the observed and the simulated values are assumed to be linearly related to the parameter updates, and the distribution of the state variable is assumed to be multi-Gaussian. The spatial distribution of the primary variables like facies can be non-Gaussian because of the spatial continuity exhibited by channel facies (sands) and non-channel facies (clay), that have distinctly different properties. Also, the relationship between the mismatch and the parameter updates can be strongly non-linear. In indicator-based data assimilation (InDA) method the indicator transforms of parameter and mismatch variables are used, which affords the treatment of these variables as bi-variate non-Gaussian. Furthermore, in the indicator transformed space, restrictive linear assumption can also be lifted as the indicator transform is invariant under non-linear transformations. However, the updates are still governed by the bivariate interactions between the state variables. A multiple-point extension of InDA is proposed in this paper which utilizes existing multiple-point simulation algorithms like single normal equation simulation (SNESIM) in combination with InDA, to preserve the spatial characteristics of the geologic model while at the same time honor the observed flow or transport response. The proposed method is applied to a synthetic groundwater system with complex channel distributions. The channel features of the final updated ensemble of models is shown to converge towards the reference model and the ensemble flow responses are also shown to match the reference flow response.