Polynomial approximate solutions of an unconfined Forchheimer groundwater flow equation

We consider a one-dimensional, unconfined groundwater flow equation for the horizontal propagation of water. This equation was derived by using a particular form of the Forchheimer equation in place of Darcy's Law. Such equations can model turbulent flows in coarse and fractured porous media. For power-law head, exponential head, power-law flux and exponential flux boundary conditions at the inlet, the problems can be reduced, using similarity transformations, to boundary-value problems for a nonlinear ordinary differential equation. We construct quadratic and cubic approximate solutions of these problems. We also numerically compute solutions using a new modification of a method of Shampine, which exploits scaling properties of the governing equation. The polynomial approximate solutions replicate well the numerical solutions and they are easy to use. Last, we compare the predicted wetting front positions from our quadratic and cubic polynomials to predictions based on Adomian polynomials of the same degrees. The work demonstrates the value of polynomial approximate solutions for validating numerical solutions and for obtaining good approximations for water profiles and the extent of water propagation. This work also presents a new application of Shampine's method for this type of groundwater flow equation. We note that this paper introduces additional classes of approximate solutions for the Forchheimer equation. Up to this date, not many solutions are known, especially for the transient cases considered here.