In this paper, we first define various types of $k$-regularity of ordered semigroups and various types of $k$-Archimedness of ordered semigroups. Also, we define the relations $\tau^{(k)}$, $\tau^{(k)}_l$, $\tau^{(k)}_r$, $\tau^{(k)}_t$ and $\tau^{(k)}_b$ $(k\in Z^+)$ on an ordered semigroup. Using these notions, filter, and radical subsets of an ideal, left ideal and biideal of ordered semigroups we describe chains of $k$-Archimedean (left $k$-Archimedean, $t$-$k$-Archimedean) ordered subsemigroups.