• porous pavement;
  • highway drainage

[1] In the paper “Drainage hydraulics of permeable friction courses” by Randall J. Charbeneau and Michael E. Barrett (Water Resources Research, 44, W04417, doi:10.1029/2007WR006002, 2008), the model formulation for combining sheet flow and PFC flow used an incorrect “initial point” to integrate the governing equation. The original approach (section 3.6) uses an arbitrary “initial point” depth to develop solutions for the unsaturated part of the domain. A transition region, in which the Dupuit-Forchheimer assumptions are suspended, was used to join the PFC flow and sheet flow parts of the drainage profile.

[2] Further research toward development of numerical models for combined PFC and sheet flow [e.g., Eck, 2010] has suggested that the appropriate “initial point” is not arbitrary. By taking the initial point at the transition to sheet flow, the revised method maintains the Dupuit-Forchheimer assumptions throughout the domain. The revision changes the solution only in the region where sheet flow does not occur; the point where sheet flow begins (with respect to the drainage divide) is given by continuity and is unchanged between this correction and the original work. What is different is the choice of “initial point” used to develop solutions for the unsaturated part of the domain.

[3] A schematic view of combined PFC and sheet flow is shown in Figure 11. Figure 11 shows the water saturation thickness (h) within the PFC with total PFC thickness bp, the elevation of the water surface above a datum (H), the locations of a possible drainage divide (lateral coordinate origin, x = 0) and initiation of sheet flow (Ls) as measured from the upstream edge of pavement or drainage divide. The pavement slope (s) is shown, and the uniform rainfall intensity (r) is not shown. At the upstream edge of pavement is a potential seepage face which may function with an assumed seepage face height to allow outflow of upstream drainage from the groundwater divide, if one occurs. The possible function of the seepage face and location of a groundwater divide are determined through the solution procedure.


Figure 11. Schematic view of porous friction course with drainage divide and sheet flow.

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[4] At the location of “initial sheet flow” (x = Ls), the sheet flow discharge (Us) is zero and the distance from the drainage divide or upstream edge of pavement is found from

  • equation image

Substituting x = Ls and h = bp in equation (7) shows that at the location of initial sheet flow, dh/dx(x = Ls) = 0. Thus the relevant range of integration of equation (7) is from the origin where dh/dx = s to the location where dh/dx = 0. When a seepage divide occurs, equation (7) must also be integrated in the upstream direction, as described originally.

[5] Figure 12 shows an example of the revised solution. This solution provides a gradual rather than abrupt transition to saturated conditions and sheet flow because the Dupuit-Forchheimer assumptions are now applied throughout the flow domain. In both solutions, the region of downslope drainage without sheet flow is 270 cm long by the equation above. However, the location of this region shifts upslope by 37 cm. This shift results from changing the initial point for integrating equation (7). By integrating from Ls rather than L1 the PFC saturation thickness is lower at x = 0. The resulting thickness is matched by a smaller region of upslope drainage (L2) and so the drainage divide occurs further up the slope. The different location of the drainage divide causes more discharge to occur in the downstream direction and the sheet flow depth increases accordingly.


Figure 12. Original (black) and revised (gray) drainage profiles corresponding to L = 1000 cm, s = 0.03, r = 2.0 cm/h, bp = 5 cm, K = 1 cm/s, KDW = 900, and ν = 0.01 cm2/s.

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