Described are absorbent members useful in the containment of body liquids such as urine. The storage members have a high capillary suction capacity. For purposes of the present disclosure, capillary suction capacity is measured in terms of the member's ability to uptake liquid at relatively high capillary pressures, which are generally encountered when the member is positioned in an absorbent article. In particular, capillary suction capacity is measured in terms of a member's capillary sorption absorbent capacity, which is measured in accordance with the Capillary Sorption method described in the Test Methods section of the disclosure. The absorbent members may include an osmotic absorbent (preferably a hydrogel-forming absorbent polymer) or a high surface area material, or a combination thereof.