Let $X$ be a smooth, projective, integral surface over a field $k$ of characteristic $0$. Then one can define the invertible sheaf of differential forms $\Omega^2_X$ (which is indeed the canonical sheaf). Let $k(X)$ be the function field of $X$, then the $k(X)$-vector space of Kahler differentials $\Omega^2_{k(X)|k}$ is the vector space of the rational sections of the invertible sheaf $\Omega^2_X$.
Now let's change the notations and let's deal with the arithmetic case. Let $K$ be a number field, and let $X\to S=\operatorname{Spec }O_K$ be an arithmetic surface (smooth, integral, flat...). Also in this case one can define the sheaf of relative differential forms $\Omega^1_{X/S}$ (note that now we care about the relative dimension of $X$). One can show that $\Omega^1_{X/S}$ coincides with the Grothendieck dualising sheaf.
My question is the following:
Is there any explicit description of the rational sections of $\Omega^1_{X/S}$? What is the relationship with the Kahler differentials $\Omega^1_{K(X)|K}$?