The flow of a gas (saturated or dry) through a porous medium, which is partially occupied by a liquid phase, causes evaporation. The latter occurs, even if the inlet gas is fully saturated, as a result of volume expansion. This process, referred to as flow-through drying is important in a variety of natural and industrial and industrial applications, such as convective drying, fuel cells, and natural gas production, which is the context of this work. In this article, a mathematical model is developed to understand the process and to predict drying rates and the evolution of liquid saturation profiles. The model includes the effects of gas compressibility and capillarity.  Compressibility effects account for the evaporation into the saturated gas phase, whereas teh capillary pressure gradients cause the liquid flow that leads to spreading of the saturation profile. Two important parameters, a normalized viscous pressure drop across the medium and capillary "wicking" number, control the two respective regimes. Capillary-driven flow from regions of high saturation to regions of the low saturation leads to more uniform saturation profiles or spreading drying fronts. the results are compared against experimental data obtained by X-ray imageing (c) 2006 American Institute of Chemical Engineers ALChE J. 52: 2367-2380. 2006