Qinzhuo Liao

Probabilistic collocation method for strongly nonlinear problems: 2. Transform by displacement

The probabilistic collocation method (PCM) is widely used for uncertainty quantification and sensitivity analysis. In paper 1 of this series, we demonstrated that the PCM may provide inaccurate results when the relation between the random input parameter and the model response is strongly nonlinear, and presented a location-based transformed PCM (xTPCM) to address this issue, relying on the transform between response and location. However, the xTPCM is only applicable for one-dimensional problems, and two or three-dimensional problems in homogeneous media. In this paper, we propose a displacement-based transformed PCM (dTPCM), which is valid in two or three-dimensional problems in heterogeneous media. In the PCM, we first select collocation points and run model/simulator to obtain response, and then approximate the response by polynomial construction. Whereas, in the dTPCM, we apply motion analysis to transform the response to displacement. That is, the response field is now represented by the displacement field. Next, we approximate the displacement instead of the response by polynomial, since the displacement is more linear to the input parameter than the response. Finally, we randomly generate a sufficient number of displacement samples and transform them back to obtain response samples to estimate statistical properties. Through multiphase flow and solute transport examples, we demonstrate that the dTPCM provides much more accurate statistics than does the PCM, and requires considerably less computer time than does the Monte Carlo (MC) method.

Data Assimilation for Strongly Nonlinear Problems by Transformed Ensemble Kalman Filter

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 by use of the EnKF may lead to nonphysical 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. Because the relation between displacement and observation is relatively linear, this new method provides a physically meaningful updated state without resolving the forward model. The TEnKF is tested in the history matching of multiphase flow in a 1D homogeneous medium, a 2D heterogeneous reservoir, and a 3D PUNQ-S3 model. The case studies show that the TEnKF produces physical results without the oscillation problem that occurs in the traditional EnKF, whereas the computational effort is reduced compared with the REnKF.

A two-stage adaptive stochastic collocation method on nested sparse grids for multiphase flow in randomly heterogeneous porous media

A new computational method is proposed for efficient uncertainty quantification of multiphase flow in porous media with stochastic permeability. For pressure estimation, it combines the dimension-adaptive stochastic collocation method on Smolyak sparse grids and the Kronrod-Patterson-Hermite nested quadrature formulas. For saturation estimation, an additional stage is developed, in which the pressure and velocity samples are first generated by the sparse grid interpolation and then substituted into the transport equation to solve for the saturation samples, to address the low regularity problem of the saturation. Numerical examples are presented for multiphase flow with stochastic permeability fields to demonstrate accuracy and efficiency of the proposed two-stage adaptive stochastic collocation method on nested sparse grids.

Constrained probabilistic collocation method for uncertainty quantification of geophysical models

The traditional probabilistic collocation method (PCM) uses either polynomial chaos expansion (PCE) or Lagrange polynomials to represent the model output response. Since the PCM relies on the regularity of the response, it may generate nonphysical realizations or inaccurate estimations of the statistical properties under strongly nonlinear/unsmooth conditions. In this study, we develop a new constrained PCM (CPCM) to quantify the uncertainty of geophysical models accurately and efficiently, where the PCE coefficients are solved via inequality constrained optimization considering the physical constraints of model response, different from that in the traditional PCM where the PCE coefficients are solved using spectral projection or least-square regression. Through solute transport and multiphase flow tests in porous media, we show that the CPCM achieves higher accuracy for statistical moments as well as probability density functions, and produces more reasonable realizations than does the PCM, while the computational effort is greatly reduced compared to the Monte Carlo approach.