DISCRETE TIME- AND LENGTH-AVERAGED SOLUTIONS OF THE ADVECTION-DISPERSION EQUATION

Authors: LEIJ, FEIKE - U.C. RIVERSIDE; TORIDE, NOBUO - U.C. RIVERSIDE

Submitted to: Water Resources Research
Publication Type: Peer Reviewed Journal
Publication Acceptance Date: 2/14/1995
Publication Date: N/A
Citation: N/A

Interpretive Summary: The study of chemical movement in the subsurface frequently involves the analysis of experimental solute concentrations obtained in the laboratory or in the field. These concentrations typically represent values that are averages for a finite sampling interval (time required to fill a test tube or the length of a soil core) whereas most expressions to describe such data are based on continuous mathematical models for much smaller scales. This paper presents discrete mathematical solutions that can correctly describe experimental averaged concentrations for an arbitrary sampling time or length. The paper also compares the use of (shifted) continuous and discrete solutions; shifting the continuous solution by half the sampling interval generally yields similar results to those obtained with the (discrete) time- or length- averaged analysis except for extreme cases with low disperson coefficients and large sampling intervals. 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.

Technical Abstract: Solute concentrations obtained from displacement experiments in porous media often represent values that are averaged over a finite sampling interval. These averaged concentration values represent discrete values that are often used implicitly as "point" values in continuous descriptions for solute transport such as the Advection- Dispersion Equation (ADE). 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 similar results 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.