Discrete Time‐ and Length‐Averaged Solutions of the Advection‐Dispersion Equation

Abstract

Solute concentrations obtained from displacement experiments in porous media frequently represent discrete values as a result of averaging over a finite sampling interval. For example, effluent curves are made up of time‐averaged concentrations while volume‐averaged concentrations are obtained from core samples. The discrete concentrations are often described by continuous solutions of macroscopic solute transport equations such as the advection‐dispersion equation (ADE). The continuous solution is often shifted to describe the average concentration. This paper compares continuous and time‐ or length‐averaged solutions of the one‐dimensional ADE cast in terms of flux‐averaged and resident concentrations. Expressions for the time‐ and length‐averaged concentrations are presented for solute applications described by Dirac delta or Heaviside functions (instantaneous and continuous releases of the solute) using four different combinations of solute application and detection modes. A temporal and spatial moment analysis was conducted to compare the traditional continuous description with the discrete time‐ or length‐averaged approach. Graphical and tabular data are presented to evaluate the accuracy of continuous solutions of the ADE for determining transport parameters. Although significant errors may occur for extreme cases with low dispersion coefficients and large sampling intervals, shifting the continuous solution by half the sampling interval generally yields results similar to those obtained with the time‐ or length‐averaged analysis. An advantage of averaged concentrations is that they permit greater flexibility to conduct experiments, since averaged concentrations provide an exact description of the data regardless of the sampling interval.