Rainfall soaking into soil, plant roots taking up water, a paper tissue mopping up a spill, or even filling a baby's nappy, are all commonly encountered examples of imbibition. This is the reverse process to drainage, where now it is the wetting phase that invades the non-wetting phase. Primary imbibition is the invasion of a wetting phase into a porous medium initially completely saturated with the non-wetting phase. It describes, for instance, water advancing into dry sand. This process is complicated by the formation of wetting films and layers in advance of water movement through the centres of the pore space (Dussan, 1979; de Gennes, 1985; Cazabat et al., 1997; de Gennes et al., 2003), although in many cases, as discussed below, wetting layer flow can be ignored. It is more normal to encounter secondary imbibition, where the wetting phase enters a porous medium in which some wetting phase is already present in layers and occupying the smaller regions of the pore space established after primary drainage. This is always the case for waterflooding an oil reservoir, the migration of CO 2 in an aquifer or the rise of the water table through damp soil.

We will describe two distinct processes, whose competition controls the nature of the displacement. The first is the swelling of wetting layers, leading to snap-off. The non-wetting phase can then become trapped in the larger pores, surrounded by the wetting phase in narrower regions. This has a major impact on oil recovery, the security of CO 2 storage and the amount of trapped air near the water table. The second process is piston-like advance, or the direct filling of pores and throats from adjacent pores filled with the wetting phase, which is the reverse of the displacement in drainage. However, there is a subtlety associated with the filling of pores, discussed below. Piston-like advance alone leads to a rather flat advance of fluid and very little trapping.

We will assume that the wetting phase is water, while the non-wetting phase is oil: this helps focus attention on applications in oil recovery and renders the text less abstract. We will also discuss CO 2 storage, where CO 2 is the non-wetting phase, and the infiltration of water into soil.