| Based on the polynomial approximation, the traditional probabilistic collocation method (PCM) approximates the model output response, which is a function of the random input parameter, from the Eulerian point of view in specific location. In some cases especially when the advection dominates, the model response has a strongly nonlinear profile with discontinuous shock or large gradient. This nonlinearity in space domain is then translated to the nonlinearity in random parametric domain, which causes oscillation and inaccurate estimation using the traditional PCM.;To address this issue, a new transformed PCM (TPCM) is developed in this study, where the model response is now represented by the an alternative variable (e.g., the location of particular response value, or displacement vector, or arrival time of particular response value), which is relatively linear to the random parameter with a smooth profile. This alternative variable is then approximated by the polynomial, from which a sufficient number of location samples are randomly generated and transformed back to obtain the response samples. Finally, the statistical moments and probability density functions of model response are estimated from these samples.;The advantage of the TPCM is demonstrated through applications to multiphase flow and solute transport in porous media. It is shown that the TPCM achieves higher accuracy for the statistical properties than does the PCM, and produces more reasonable realizations without oscillation, while the computational effort is greatly reduced compared to the direct sampling Monte Carlo method.;The ensemble Kalman filter (EnKF) has been widely used for data assimilation. It is challenging, however, when the relation of state and observation is strongly nonlinear. For example, near the flooding front in an immiscible flow, directly updating the saturation using the EnKF may lead to non-physical results. One possible solution, which may be referred to as the restarted EnKF (REnKF), is to update the static state (e.g., permeability and porosity) and rerun the forward model from the initial time to obtain the updated dynamic state (e.g., pressure and saturation). However, it may become time consuming, especially when the number of assimilation steps is large.;In this study, we develop a transformed EnKF (TEnKF), in which the state is represented by displacement as an alternative variable. The displacement is first transformed from the forecasted state, then updated, and finally transformed back to obtain the updated state. Since the relation between displacement and observation is relatively linear, this new method provides a physically meaningful updated state without re-solving the forward model. The TEnKF is tested in history matching of multiphase flow in a one-dimensional homogeneous medium, a two-dimensional heterogeneous reservoir, and a three-dimensional PUNQ-S3 model. The case studies show that the TEnKF produces physical results without the oscillation problem that occurs in the traditional EnKF, while the computational effort is reduced compared to the REnKF. |
