In this study, we introduce a new type of surface curves called $D$-type curve. This curve is defined by the property that the unit Darboux vector $vec{W}_{0} $ of a surface curve $vec{r}(s)$ and unit surface normal $vec{n} $ along the curve $vec{r}(s)$ satisfy the condition $leftlangle vec{n} ,vec{W}_{0} rightrangle =text{constant}$. We point out that a $D$-type curve is a geodesic curve or an asymptotic curve in some special cases. Then, by using the Frenet vectors and parametric representation of a surface pencil as a linear combination of the Frenet vectors, we investigate necessary and sufficient condition for a curve to be a $D$-type curve on a surface pencil. Moreover, we introduce some corollaries by considering the $D$-type curve as a helix, a Salkowski curve or a planar curve. Finally, we give some examples for the obtained results.