On estimating functional average breakthrough curve using time‐warping technique and perturbation approach

Simulated contaminant breakthrough curves (BTC) are often used to predict mass arrival at compliance boundaries at waste storage sites. In numerical simulations that involve uncertainties on input parameters such as randomly heterogeneous rock properties, Monte Carlo simulations are commonly utilized and the mean breakthrough curve is often calculated from the arithmetic average of all realizations. The arithmetic mean breakthrough curve in general overestimates the mass flow rate at early and late time but underestimates the peak mass flow rate. The averaged breakthrough curve usually does not resemble any of individual breakthrough curves. The reason is that BTCs vary not only on amplitude but also on dynamics (time) and therefore it is not appropriate to take the arithmetic average directly. In this study, we consider each BTC as a random curve, and use time‐warping techniques to align all curves in a time‐warped space, compute the sample mean of the curves in the time‐warped space, and transform the means back to the original time space. We show that all BTCs are aligned based on the percentile of mass reaching the compliance boundary, and the functional average is the percentile average of all BTCs. The confidence interval of the sample mean curve is estimated using the perturbation approach. The functional average provides an additional metric that can be used to characterize the breakthrough behavior in addition to more traditional median and arithmetic average curves. The method is illustrated using transport simulations at the Material Disposal Area G, Los Alamos National Laboratory (LANL) in New Mexico.