  $G$-operators, a class of differential operators containing the differential
operators of minimal order annihilating Siegel's $G$-functions, satisfy a
condition of moderate growth called Galochkin condition, encoded by a $p$-adic
quantity, the size. Previous works of Chudnovsky, Andr\'e and Dwork have
provided inequalities between the size of a $G$ -operator and certain
computable constants depending among others on its solutions. First, we recall
Andr\'e's idea to attach a notion of size to differential modules and detail
his results on the behavior of the size relatively to the standard algebraic
operations on the modules. This is the corner stone to prove a quantitative
version of Andr\'e's generalization of Chudnovsky's Theorem: for
$f(z)=\sum_{\alpha, k,\ell} c_{\alpha, k,\ell} z^{\alpha} \log(z)^k f_{\alpha,
k,\ell}(z)$, where $f_{\alpha, k,\ell}(z)$ are $G$-functions, we can determine
an upper bound on the size of the minimal operator $L$ over
$\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the
$f_{\alpha, k,\ell}(z)$, the rationals $\alpha$ and the integers $k$. We give
two applications of this result: we estimate the size of a product of two
$G$-operators in function of the size of each operator; we also compute a
constant appearing in a Diophantine problem encountered by the author.
