  We generalise the notion of Gr\"obner fan to ideals in R[[t]][x_1,...,x_n]
for certain classes of coefficient rings R and give a constructive proof that
the Gr\"obner fan is a rational polyhedral fan. For this we introduce the
notion of initially reduced standard bases and show how these can be computed
in finite time. We deduce algorithms for computing the Gr\"obner fan,
implemented in the computer algebra system Singular. The problem is motivated
by the wish to compute tropical varieties over the p-adic numbers, which are
the intersection of a subfan of a Gr\"obner fan as studied in this paper by
some affine hyperplane, as shown in a forthcoming paper.
