On importance sampling Monte Carlo approach to uncertainty analysis for flow and transport in porous media

In this study we introduce a new approach named importance sampling or quick simulations. The method has been extensively used in communication theory in estimating probability of rare events. The basic idea behind importance sampling techniques is that certain values of the input random variables (or vectors) have more important impact on the parameters being estimated than others, and if these “important” values are sampled more frequently than others, i.e., sampled from a biased density function, the variance of the estimator can be reduced. The outputs from simulations are then weighted to correct such biased sampling. Two illustrative examples are given to show the general procedure of the importance sampling approach as well as its applicability to subsurface flow and transport problems. In one example we estimated the mean and variance of hydraulic head for one-dimensional flow, and in the other we estimated the probability of a particle’s travel time t less than a given critical value T. In both examples, we compared results from analytical solutions, the conventional Monte Carlo (CMC) simulations, and the importance sampling approach. It is shown that when an importance density function is chosen appropriately, importance sampling techniques may be many orders of magnitude more efficient than the CMC simulations and have a great potential in simulating subsurface flow and transport.