Let $S$ be an Enriques surface over complex numbers. It is known that $S$ admits an elliptic fibration over $\mathbb{P}^1$ with $12$ nodal singular fibers and $2$ double fibers. How can I see this wellknown fact? Is it possible to explicitly construct such a fibration via the Enriques lattice $NS(S)\cong U\oplus E_8$?
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A complete and detailed treatment of elliptic pencils on Enriques surfaces (included the result you are quoting) can be found in the book by BarthHulekPetersVan de Ven Compact Complex Surfaces (Ergebnisse der Mathematik und ihrer Grenzgebiete 4, Springer). See in particular Chapter VIII, Section 17 "Elliptic pencils". 

