The two-dimensional dynamics of a thin film of viscous fluid spreading between a permeable horizontal plate and an overlying thin elastic sheet is explored. We use a lubrication model to describe the balance between the elastic stress, the hydrostatic pressure gradient and the viscous resistance of the flow, as fluid spreads laterally from a source and simultaneously drains through the plate. A family of asymptotic solutions are described in which the flow is dominated by either the hydrostatic pressure gradient or the elastic stress associated with the deformation of the sheet. In these solutions, although the deformation of the sheet above the porous plate arises from the fluid flow below the sheet, the fluid typically separates from the sheet a short distance upstream of the full extent of the draining zone, with the region of flow being driven purely by the hydrostatic pressure gradient. As a result, an air gap develops below the sheet up to the point where it touches back down onto the plate. With a very light or stiff elastic sheet, this touchdown point may extend far beyond the fluid draining zone, but otherwise it is similar to the extent of the draining zone.