Flux Type Boundary Conditions Research Articles

On the diffusion wave model for overland flow: 2. Steady state analysis

Following our study in the first paper (Govindaraju et al., this issue) on steep slopes, we are now presenting some results on analysis of the steady state phase of the solution. It is physically obvious that a steady state is reached for constant rainfall, since the depth of overland flow at the outflow section increases until equilibrium is achieved and continuity is satisfied. Numerical and analytical steady state results for flux type boundary conditions are presented as new material in this paper. The upstream boundary condition is one of zero inflow. Both the zero‐depth gradient and the critical depth down‐stream boundary conditions are investigated here. For steep slope situations, the upstream boundary condition of zero depth adopted in paper 1 of these companion papers is found to be justified, A method of finding the complete solution (i.e., including the time‐dependent part) by superimposing the steady state profile and a transient component is discussed for the zero‐depth gradient lower end condition. The steady state results clearly show that the critical depth condition at the downstream boundary is a stringent requirement and therefore is likely to pose problems. This helps us to understand why some of the numerical techniques used by previous researchers failed for certain parameters.

A generalized solution for solute flow in soils with mobile and immobile water

An analytical solution is presented for the movement of a solute through a porous medium which is leached at a constant rate. The solution takes into account transfer into an immobile water fraction. The generalized solution is valid for both finite or semi‐infinite media and for concentration or flux type boundary conditions. The solution consists of a convolution integral of two functions. The first function is the derivative with respect to injected pore volume of the corresponding analytical solution of the solute transfer equation without transfer into immobile water. The second function is the J function, for which series approximations are presented. The influence of the boundary conditions on the solution is discussed.

Mass transfer in porous media with immobile water

An analytical solution is presented for the movement of a solute through a porous medium, which is leached at a constant rate. The solution takes into account diffusion into an immobile water fraction. The generalized solution is valid for both finite or semi-infinite media and for concentration or flux-type boundary conditions. The solution consists of a convolution integral of two functions. The first function is the derivative vs. time of the corresponding solutions of the solute transfer equation without lateral diffusion. The second function is the J-function, for which a series approximation was developed. The use of the solution is illustrated with displacement studies in a 30 cm long column, filled with glass beads. It is shown that effluent concentration distributions from, and solute concentration distributions inside the column, when unsaturated, cannot be explained without taking into account an immobile water fraction.

Étude numérique de l'influence des conditions aux limites sur les solutions de l'équation de la dispersion pour un écoulement unidimensionnel

At low Péclet's numbers, solutions of the dispersion equation for a linear flow in porous columns are greatly affected by the boundary condition at the entrance of the column. Including an entrance chamber which precedes the porous bed in mathematical models yields a more realistic simulation of the physical processes involved. Hence, the concentration distributions are also depending upon the ratio of concentration gradients on both sides of the entrance section of the bed. Mathematical simulations showed that the standard flux-type boundary condition leads to better predictions as the porous medium is more dispersive and the entrance chamber shorter.