Graphical Solutions for Hillslopes: Discharge, Head, and Velocity Diagrams

Graphical solutions for discharge $\left(Q\right)$ , head $\left(h\right)$ , and velocity are presented for general recharge conditions on hillslopes underlain by a horizontal impermeable layer and with a constant-head boundary at the toe and a no-flow boundary upslope. Recharge $\left(R\right)$ may be defined for different sections of the hillslope having constant, but not necessarily equal, hydraulic conductivities. $Q$ diagrams are determined using conservation of mass, which requires that the change in $Q$ between two points on a hillslope be equal to the area of the $R$ diagram between them. $h$ diagrams have “exact” (with Dupuit’s assumptions) values for head. Darcy velocity, $q$ , diagrams may be made by dividing the value of the $Q$ diagram at any point by the value of the $h$ diagram at that point. Pore velocity diagrams may be made by dividing values from the $q$ diagrams by the effective porosity. Three examples illustrate the graphical solutions for various scenarios.