# Shukla

## Die Bodenkultur - Journal for Land Management, Food and Environment

M. K. Shukla, S. Klepsch and W. Loiskandl:

## Summary

Mathematical conceptualization of the solute transport phenomenon at the microscopic level i. e. at an individual pore, is difficult because of the complex pore geometry of soils. Hence most models are developed at macroscopic level using an average pore water velocity for convective flow and a diffusion-dispersion coefficient.

The insights into separate dispersion of a solute within a capillary of constant diameter caused by molecular diffusion from that by velocity distribution was first provided by TAYLOR (1953). Using statistical concepts, SCHEIDEGGER (1954) and others have massumed a simple random walk stochastic process to describe transport in a fluid saturated homogeneous, isotropic porous medium. Using the representative elementary volume concept, the random capillary models were made physically more realistic.

Assuming fluid is a continuous medium and each point has a flow path RIFAI et al. (1956) and EINSTEIN (1937) developed a stochastic model which accounts for displacement for motion and rest phases. The classical convective dispersion equation was proposed by LAPIDUS and AMUNDSON (1952) which was later extended to include several complicated equilibrium and nonequilibrium processes (NIELSEN et al., 1986; VAN GENUCHTEN and WIERENGA, 1976; SELIM et al., 1976; CAMERON and KLUTE; 1977, KRUPP et al., 1972 etc.).

In this paper several equilibrium and nonequilibrium mathematical models are described along with their analytical solutions for various initialand boundary conditions for both continuous and pulse type tracer application.

The nondimensional parameters are also given in this paper. The nonequilibrium two site, one site, two region and anion exclusion models have also been combined into one nondimensional equation taking into account all the nonequilibrium processes.

The paper further describes some of the commercially available software describing solute transport. The software described in the paper include one, two and three dimensional solute transport through porous media using either analytical or numerical solution of the problem. The various adsorption isotherms which could be employed in these models are also included in the paper. At the end an iterative solution is presented which is helpful to verify some of the models in case source code is not available.

Key words:  Miscible displacement, breakthrough curves, porous media, pore water velocity, equilibrium, nonequilibrium, adsorption.