Backward simulation of the probability of a trace particle reaching a given region

This paper describes the solution of the backward-type Kolmogorov equation (backward method) as an alternative to the forward method to determine the probability (P) of a trace particle reaching a given region within a time τ, which is of interest in groundwater contamination investigation, such as pollution monitoring and controlling. The backward-type Kolmogorov equation describing P was theoretically derived under the assumption that the divergence of the velocity field is zero. We here verify that this assumption is unnecessary and therefore the equation can be used in general cases when various recharge/discharge sources in the flow equation are considered. The backward method is validated by comparing its results with that from the forward method. We find that boundary conditions must be properly imposed in the backward method in order to obtain the same results as from the forward method. The backward method is much more efficient than the forward method.