  We construct an explicit sequence $V_{k_n,a_n}$ of crystalline
representations of exceptional weights converging to a given irreducible
two-dimensional semi-stable representation $V_{k,{\mathcal{L}}}$ of
$\mathrm{Gal}({\overline{\mathbb{Q}}}_p/{\mathbb{Q}}_p)$. The convergence takes
place in the blow-up space of two-dimensional trianguline representations
studied by Colmez and Chenevier. The process of blow-up is described in detail
in the rigid analytic setting and may be of independent interest. Also, we
recover a formula of Stevens expressing the ${\mathcal{L}}$-invariant as a
logarithmic derivative.
  Our result can be used to compute the reduction of $V_{k,{\mathcal{L}}}$ in
terms of the reductions of the $V_{k_n,a_n}$. For instance, using the zig-zag
conjecture we recover (resp. extend) the work of Breuil-M\'ezard and
Guerberoff-Park computing the reductions of the $V_{k,{\mathcal{L}}}$ for
weights at most $p-1$ (resp. $p+1$), at least on the inertia subgroup. In the
cases where zig-zag is known, we are further able to obtain some new
information about the reductions for small odd weights. Finally, we explain
some apparent violations to local constancy in the weight of the reductions of
crystalline representations of small weight.
