  A motivation to study Gr\"{o}bner theory for fields with valuations comes
from tropical geometry, for example, they can be used to compute
tropicalization of varieties \citep{maclagan2009introduction}. The
computational aspect of this theory was first studied in (Chen \& Maclagan,
2013). In this paper, we generalize this Gr\"obner basis theory to free modules
over polynomial rings over fields with valuation. As the valuation of
coefficients is also taken into account while defining the initial term, we do
not necessarily get a monomial order. To overcome this problem we have to
resort to other techniques like the use of ecart function where the codomain is
the well-ordered set $\mathbb{N}$, and thereby give a method to calculate the
Gr\"{o}bner basis for submodules generated by homogeneous elements.
