What is Wrong with the Boundary Conditions in Column Tracer Tests
Abstract
Solute transport in a column is probably one of the most fundamental problems investigated in contaminant hydrology and soil physics because it serves as a benchmark for testing transport theories, for measuring dispersivities, etc. Despite its importance, there are still dispute and inconsistency on how to deal with the boundary conditions involved in such problems. The boundary condition could impose great influence upon transport in a column, particularly when the length of the column is relatively short, or the socalled Peclet number is not large. There are three types of boundary conditions to choose for transport in a column. Among these three types of boundary conditions, only the thirdtype boundary satisfies the mass balance requirement rigorously. The first type boundary, despite its frequent use in previous studies, could lead to serious mass balance problems. The most serious problem is on how to deal with the outlet boundary. Some studies have used a zero concentration gradient at the outlet (the socalled Danckwerts' boundary condition). This is named the model A. Another idea is to treat the finite length column as a part of an infinitely long column and to calculate the concentration at the outlet based on a formula developed for an infinitely long column. This is named the model B. The model A satisfies the mass balance requirement but was found to fit with the experimental data poorly. The model B does not satisfy the mass balance requirement, but usually agree well with the experimental data. So, the dilemma is: which model to choose? At present, most investigators prefer to choose the model B because of its close agreement with the experimental data, despite of its violation of the mass balance requirement. But the question is: why the model A, which satisfies the mass balance requirement, does not fit with the experimental data? It turns out that the advectiondispersion equation (ADE) that uses the Fick's first law to describe the hydrodynamic dispersion has some problems, particularly in the regions near the two boundaries. Taylor (1921) has pointed out that the dispersion coefficient varies linearly with time at the beginning and tends to its asymptotic, Fickian value after a travel time of a few correlation scales. Dagan and Bresler (1985) have further pointed out that the constant dispersivity is attained after the solute body has traveled tens of conductivity integral scales. For transport in a homogeneous column, the integral scale of the conductivity is probably around the pore scale or equivalent to the dispersivity value. Therefore, for a finite column whose length is not much greater than the dispersivity value, the transition zones in which solute transport is nonFickian could consist of a significant portion of the column length. It is such nonFickian transport in the column that is responsible for the failure of the model A. But still, why does the model B yield the right solution? There is no answer to this question based on a rigorous quantitative analysis yet. To resolve the dilemma, one must carry out a nonFickian transport study to deal with the transition zones. It is my hypothesis that if the nonFickian transport analysis succeeds, one will find that the mass balance requirement is indeed satisfied in the model B. Dagan and Bresler (1985) have pointed to the right direction, but a rigorous analysis has not followed. This is something interesting and worthwhile to investigate. REFERENCES CITED Dagan, G., and Bresler, E., 1985. Comment on ¡°Fluxaveraged and volumeaveraged concentration in continuum approaches to solute transport¡± by J.C. Parker and M.Th. van Genuchten. Water Resources Research, 21: 1299 1300. Taylor, G.I., 1921. Diffusion by continuous movements. Proc. London Math Soc. 2: 196212.
 Publication:

AGU Fall Meeting Abstracts
 Pub Date:
 December 2007
 Bibcode:
 2007AGUFM.H12C..06Z
 Keywords:

 1829 Groundwater hydrology;
 1831 Groundwater quality;
 1832 Groundwater transport