Self-intersection number of fibered surface

Question

Let $R$ be a complete discrete valuation ring with residue field $k$ (can assume algebraically closed), $f:X \to \mathrm{Spec}(R)$ a flat, proper family of projective curves (i.e., $f$ is also surjective and $X$ is a projective surface). Assume further that $X$ is non-singular. Denote by $Y$ the special fiber $f^{-1}(\mathrm{Spec}(k))$. Is there any way to compute the self-intersection of $Y$? Is there some literature which deals with similar questions? If necessary one can assume the generic fiber of $f$ is smooth.