  We consider the problem of the vortex contribution to thermal properties of
dirty d-wave superconductors. In the clean limit, Volovik has argued that the
main contribution to the density of states in a d-wave superconductor arises
from extended quasiparticle states which may be treated semiclassically, giving
rise to a specific heat contribution \delta C(H)\ltsim H^{1/2} . We show that
the extended states continue to dominate the dirty limit, but lead to a H \log
H behavior at the lowest fields, H_{c1}\ll H\ll H_{c2}. This crossover may
explain recent discrepancies in specicific heat measurements at low
temperatures and fields in the cuprate superconductors. We further discuss the
field dependence of transport properties within the same model.
