Hefetz, M{u}tze, and Schwartz conjectured that every connected undirected graph admits an antimagic orientation. In this paper we support the analogous question for distance magic labeling. Let $Gamma$ be an Abelian group of order $n$. A textit{directed $Gamma$-distance magic labeling} of an oriented graph $vec{G} = (V,A)$ of order $n$ is a bijection $vec{l}:V rightarrow Gamma$ with the property that there is a textit{magic constant} $mu in Gamma$ such that for every $x in V(G)$ $ w(x) = sum_{y in N^{+}(x)}vec{l}(y) - sum_{y in N^{-}(x)} vec{l}(y) = mu. $ In this paper we provide an infinite family of odd regular graphs possessing an orientable $mathbb{Z}_{n}$-distance magic labeling. Our results refer to lexicographic product of graphs. We also present a family of odd regular graphs that are not orientable $mathbb{Z}_{n}$-distance magic.
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