Two commonly suggested forms of the equation linking head loss and velocity for flow of water through coarse granular media are the Forchheimer and exponential relations. These have been combined with the continuity expression to give the differential equations applicable, within the limits of validity of the parent relations, to actual regions of flow. The resultant nonlinear elliptic partial differential equations have been solved by numerical methods including the direct finite difference and finite element methods.
Experimental results and associated analytical work were carried out to determine the accuracy of the nonlinear relations as compared to the linear Darcy Law, when applied over an extended Reynolds number range. Solutions have been obtained for some examples of unconfined flow with boundary conditions similar to those likely to be encountered in practical applications. The experimental work in a circular tank and an open flume has shown that good agreement between observed and calculated values of discharge and piezometric head can be obtained when the coefficients in the nonlinear head loss equations are accurately known. The results indicate that while the flow patterns from the Darcy and the nonlinear solutions are only significantly different for a high degree of curvature of the phreatic line, a nonlinear solution will usually be necessary for accurate predictions of discharge.