Error models for uncertainty quantification

In groundwater modeling, uncertainty on the permeability field leads to a stochastic description of the aquifer system, in which the quantities of interests (e.g., groundwater fluxes or contaminant concentrations) are considered as stochastic variables and described by their probability density functions (PDF) or by a finite number of quantiles. Uncertainty quantification is often evaluated using Monte-Carlo simulations, which employ a large number of realizations. As this leads to prohibitive computational costs, techniques have to be developed to keep the problem computationally tractable. The Distance-based Kernel Method (DKM) [1] limits the computational cost of the uncertainty quantification by reducing the stochastic space: first, the realizations are clustered based on the response of a proxy; then, the full model is solved only for a subset of realizations defined by the clustering and the quantiles are estimated from this limited number of realizations. Here, we present a slightly different strategy that employs an approximate model rather than a proxy: we use the Multiscale Finite Volume method (MsFV) [2,3] to compute an approximate solution for each realization, and to obtain a first assessment of the PDF. In this context, DKM is then used to identify a subset of realizations for which the exact model is solved and compared with the solution of the approximate model. This allows highlighting and correcting possible errors introduced by the approximate model, while keeping full statistical information on the ensemble of realizations. Here, we test several strategies to compute the model error, correct the approximate model and achieve an optimal PDF estimation. We present a case study in which we predict the breakthrough curve of an ideal tracer for an ensemble of realizations generated via Multiple Point Direct Sampling [4] with a training image obtained from a 2D section of the Herten permeability field [5]. [1] C. Scheidt and J. Caers, "Representing spatial uncertainty using distances and kernels", Math Geosci (2009) [2] P. Jenny et al., "Multi-Scale finite-volume method for elliptic problems in subsurface flow simulation", J. Comp. Phys., 187(1) (2003) [3] I. Lunati and S.H. Lee, "An operator formulation of the multiscale finite-volume method with correction function", Multiscale Model. Simul. 8(1) (2009) [4] G. Mariethoz, P. Renard, and J. Straubhaar "The Direct Sampling method to perform multiple-point geostatistical simulations", Water Resour. Res., 46 (2010) [5] P. Bayer et al., "Three-dimensional high resolution fluvio-glacial aquifer analog", J. Hydro 405 (2011) 19