  In this paper we consider the space $\mathbb{R}^2$ with the river metric
$d^*$ and different types of convexity of this space. We define $W$-convex
structure in $(\mathbb{R}^2,d^*)$ and we give the complete characterization of
the convex sets in this space. We consider some measures of noncompactness and
we give the moduli of noncompactness for considered measures on this space.
