Drainage condition (seepage)

A special boundary condition is needed if the phreatic surface reaches an open, freely draining surface. In such a case the pore fluid can drain freely down the face of the dam, so at all points on this surface below its intersection with the phreatic surface. Above this point negative pore water pressures occur, with their particular value depending on the solution. Above this point negative pore water pressures occur, with their particular value depending on the solution. This drainage-only flow condition consists of prescribing the flow velocity on the freely draining surface in a way that approximately satisfies the requirement of \(p^w=0\) on the completely saturated portion of this surface:

\[ q^w = \begin{cases} \tilde{k}_s \left(p^w - p^w_{target} \right) & \text{if}~~ p^w > p^w_{target} \\ 0 & \text{else}\end{cases} \]

where \(p^w_{target}\) is the target pore water pressure at the considered surface and \(\tilde{k}_s\) is the proportionality coefficient:

\[ \tilde{k}_s \approx f \dfrac{k^{sat}c}{\gamma_w} \]

Therein, \(f\) is a scaling factor usually in the range of \(10^3 \leq f \leq 10^5\), \(k^{sat}\) is the saturated hydraulic conductivity, \(\gamma_w\) is the unit weight of water and \(c\) the characteristic element size of the underlying finite element.