We consider in this chapter (1) the average paths that fluid particles follow on their way through the soil and (2) the transport of solutes. The paths are average; deviations due to the discrete nature of the porous material are ignored. We refer to these average particle paths as path lines.
For steady flow the path lines coincide with the streamlines. For transient flow the pattern of streamlines changes with time; the path lines are different from the streamlines. Closed form expressions for streamlines and path lines exist only for a few cases; we usually resort to numerical methods. We first discuss the computation of points on streamlines and path lines, then cover an approximate technique for determining streamlines and path lines in three dimensions using the Dupuit- Forchheimer approximation.We finish this chapter with a brief discussion of solute transport.
Numerical Determination of Fluid Particle Paths
The equations for the streamlines usually are implicit in terms of the coordinates x and y. We can determine points on streamlines by the use of the stream function if the potential is harmonic. The application of a contouring procedure to a set of values of Ψ computed at the mesh points of a grid is one way of computing points of a set of streamlines. A disadvantage of this approach is that jumps of the stream function along branch cuts appear in the plot as heavy black lines. This problem can be avoided only by using a special contouring routine that processes the jumps across the branch cuts correctly. An alternative approach is to determine points of a given streamline in such a way that jumps in the stream function can be handled easily; we discuss such a procedure in this section.
If the potential is not harmonic, as in shallow flow with infiltration and in shallow transient flow, then the stream function cannot be used. We cover separately procedures for computing points on streamlines and path lines applicable to such cases.
A Procedure for Tracing Streamlines Using the Stream Function
We determine points on a streamline by a process that we refer to as tracing streamlines: we start at any given point of a streamline Ψ = Ψ j and proceed by calculating complex coordinates z of points on the streamline one by one.