A Mathematical Model of the Rainwater Flows in a Green Roof
  A model is presented for the gravity-driven flow of rainwater descending through the soil layer of a green roof, treated as a porous medium on a flat permeable surface representing an efficient drainage layer. A fully saturated zone is shown to occur. It is typically a thin layer, relative to the total soil thickness, and lies at the bottom of the soil layer. This provides a bottom boundary condition for the partially saturated upper zone. It is shown that after the onset of rainfall, well-defined fronts of water can descend through the soil layer. Also the rainwater flow is relatively quick compared with the moisture uptake by the roots of the plants in the roof. In separate models the exchanges of water are described between the (smaller-scale) porous granules of soil, the roots and the rainwater in the inter-granule pores. 
Catherine Adley, Mark Cooker, Gemma Fay, Ian Hewitt, Andrew Lacey,   Niklas Mellgren, Marguerite Robinson and Michael Vynnycky
physics.geo-ph