Temporal-moment matching for truncated breakthrough curves for step or step-pulse injection

The method of temporal moments is an efficient approach for analyzing breakthrough curves (BTCs). By matching the moments of the BTCs computed through parametric transfer-function models or one-dimensional transport models to those of the data, one can estimate the parameters characterizing the transfer function or apparent transport parameters. The classical method of moments presumes infinite duration. However, the measurement of BTCs is usually terminated prematurely, before the concentration has reached zero. Unless this truncation of the BTCs has been taken into account, the estimates of the parameters may be in error. Truncated measured BTCs are sometimes extrapolated assuming exponential decay. In this study, we use the concept of moments of the truncated impulse–response function [Jawitz JW. Moments of truncated continuous univariate distributions. Adv Water Res 2004;27:269–81] in the analysis of truncated BTCs corresponding to the commonly encountered step and step-pulse injection modes. The method is straightforward, based on the relation, which we derive, between truncated moments of the impulse–response function and the measured BTC. It is practical to apply and does not require the extrapolation of the measured BTC. The method is also accurate. In a numerical study we discuss how short a step-pulse injection may be so that we can approximate it as instantaneous. Finally, we apply the method to the analysis of a field-scale tracer test.