Saturated conditions generally exist below a water table, either as part of the permanent groundwater system (aquifer) or in the vadose zone as perched water. For isotropic and steady-state conditions, such systems can be modeled by Laplace’s equation. Because it is linear, Laplace’s equation is much easier to solve than the variably saturated forms of Richards’ equation and, hence, provides a convenient place to begin. Analyses of water flow for drainage and groundwater systems borrow heavily from the classical (and old!) work in heat conduction, hydrodynamics, and electrostatics. This section presents analytical solutions for subsurface drainage and well discharge in fully penetrating confined aquifers (the solutions are the same). Included are the definition of stream functions and demonstrations of the Cauchy–Riemann relations. A comparable numerical solution is presented, and also for the ponded drainage and well discharge, and the results compared with the analytical solutions. A more complex example is then presented concerning drainage below a curved water table. These results are followed by travel-time calculations relevant to solute movement from the soil surface to a drainage system. A short section covering analytical techniques with three-dimensional images is then given, followed by a section covering additional topics, which includes a complex image example (two dimensional) and some relationships for Fourier series. Consider a point source in a two-dimensional x—y plane, as in figure 3-1. The origin corresponds to a source that is assumed to be an infinite line perpendicular to the x—y plane. If the steady flow rate is Q, then the conservation of mass results in . . . Q = Jr(2πr) (3-1) . . . where Jr is the Darcian flow in the r direction and evaluated at a polar radius r. The dimensions of Q are [L2T-1] corresponding to a volume of flow per unit time from a unit length of the line perpendicular to the x—y plane. Values of Q are taken to be positive for water entering the system.
Approximate analytic solution of the Dirichlet problems for Laplace’s equation in planar domains by a perturbation method
