An absorbent article having a fluid-handling system includes a fluid permeable bodyside liner; a fluid impermeable outer cover; an absorbent core disposed between the liner and the outer cover, wherein the absorbent core includes superabsorbent material and optionally fluff pulp; and a core wrap at least partially encircling the absorbent core, wherein the core wrap includes a three-dimensionally patterned, wetlaid, cellulosic tissue nonwoven material. Fibers of the nonwoven material are entirely cellulose fibers, and the core wrap includes opposing core wrap surfaces each having a textured surface. Each surface includes an average material plane, a plurality of ridges extending in a z-direction from the average material plane, and a plurality of grooves alternating with the plurality of ridges, wherein the grooves depth extend in the opposite z-direction from the average material plane.