  We consider a continuum percolation model on $\R^d$, where $d\geq 4$.The
occupied set is given by the union of independent Wiener sausages with radius
$r$ running up to time $t$ and whoseinitial points are distributed according to
a homogeneous Poisson point process.It was established in a previous work by
Erhard, Mart\'{i}nez and Poisat~\cite{EMP13} that (1) if $r$ is small enough
there is a non-trivial percolation transitionin $t$ occuring at a critical time
$t\_c(r)$ and (2) in the supercritical regime the unbounded cluster is unique.
In this paper we investigate the asymptotic behaviour of the critical time when
the radius $r$ converges to $0$. The latter does not seem to be deducible from
simple scaling arguments. We prove that for $d\geq 4$, there is a positive
constant $c$ such that$c^{-1}\sqrt{\log(1/r)}\leq t\_c(r)\leq
c\sqrt{\log(1/r)}$ when $d=4$ and $c^{-1}r^{(4-d)/2}\leq t\_c(r) \leq c\
r^{(4-d)/2}$ when $d\geq 5$, as $r$ converges to $0$. We derive along the way
moment estimates on the capacity of Wiener sausages, which may be of
independent interest.
