The animation below gives you a more quantitative idea about how different interaction energies contribute to the overall partitioning constant. These are only rough calculations but they give a good idea of the absolute strength of various interactions.
The partitioning between two liquid phases is typically dominated by H-bond interactions. But it would be erroneous to conclude from this that van der Waals interactions are weak compared to H-bond interactions. They are in fact much stronger for almost all molecules but in liquid/liquid partitioning these contributions largely cancel and thus do not become "visible". Therefore, the animation only shows air/solvent partitioning. Of course, you can easily deduce liquid/liquid partition data from combining the data from the two air/liquid partition systems for the respective liquids.
More details on the quantitative estimation of partition constants are given in the next chapter.