Much of this work was done as part of a Senior Thesis project by Nickolay Neshkov ('96).
There are a variety of models available to simulate the motion of an air/water interface in a porous medium. Among the key factors to consider are: how rapidly the interface is pushed, the degree of disorder in the system, and how much the local interface motion depends on whether neighboring sites are wet or dry.
This figure is from a model with a very low driving force. The water invades from the top and heads downward. The different intensities correspond to different times in the simulation. For very low driving force, note that the interface is very irregular and tends to snake its way through the path of least resistance.
This figure is from the same model with a much higher driving force. The water invades from the top and heads downward. The different intensities correspond to different times in the simulation. Note that, at least on average, the interface proceeds smoothly ahead. The larger driving force tends to overwhelm small-scale irregularities in the random porous medium.
The actual experimental images tend to look a lot more like the higher driving force model, even when the experiment is run very slowly. (In this image, the fluid is dark and is entering from the bottom.)