  Let $\mathcal{O}_K$ be a mixed characteristic complete discrete valuation
ring with perfect residue field. We study $\mathbb{B}_\mathrm{dR}^+$-crystals
on the (log-) prismatic site of $\mathcal{O}_K$, which are crystals defined
over the de Rham period sheaf. We first classify these crystals using certain
log connections. By constructing a Sen--Fontaine theory for
$\mathbf{B}_{\mathrm{dR}}^+$-representations over a Kummer tower, we further
classify these crystals by (log-) nearly de Rham representations. In addition,
we compare (log-) prismatic cohomology of these crystals with the corresponding
Sen--Fontaine cohomology and Galois cohomology.
