Capillary absorption by an initially dry porous sheet or surface is examized. The sheet absorbs moisture from one end while evaporation takes place across its surface. We are interested in the effect of evaporation on the progress and distribution of moisture in the sheet, including the equilibrium moisture profile. The process is modelled using a nonlinear diffusion equation with a linear sink. An algebraic analytical approximation is obtained relating the various physical parameters as well as an exact steady state solution for arbitrary properties. The complete problem is solved numerically. Good agreement is obtained between the analytical and numerical solutions.