  Gr\"obner bases have been generalized by replacing monomial orders with
constructions such as valuations and filtrations. We consider suitable
valuations on a rational valuation field $K(x,y)$ and analyze their behavior
when restricting to an underlying polynomial ring $K[x,y]$. In previous work,
the corresponding value groups were subsets of ${\mathbb Q}$, and in this paper
we consider the case when the value groups are isomorphic to ${\mathbb Z}
\oplus {\mathbb Z}$. Bounds on how the image of $K[x,y]$ grows with respect to
degree are given, and then a class a valuations that are suitable for use for
generalized Gr\"obner bases are described. We construct an example in which the
image of the underlying polynomial ring is non-negative, yet is not
well-ordered.
