  We present a detailed study of the effective cones of Calabi-Yau threefolds
with $h^{1,1}=2$, including the possible types of walls bounding the K\"ahler
cone and a classification of the intersection forms arising in the geometrical
phases. For all three normal forms in the classification we explicitly solve
the geodesic equation and use this to study the evolution near K\"ahler cone
walls and across flop transitions in the context of M-theory compactifications.
In the case where the geometric regime ends at a wall beyond which the
effective cone continues, the geodesics "crash" into the wall, signaling a
breakdown of the M-theory supergravity approximation. For illustration, we
characterise the structure of the extended K\"ahler and effective cones of all
$h^{1,1}=2$ threefolds from the CICY and Kreuzer-Skarke lists, providing a rich
set of examples for studying topology change in string theory. These examples
show that all three cases of intersection form are realised and suggest that
isomorphic flops and infinite flop sequences are common phenomena.
