A tubular graft for use in a stent graft. The tubular graft may include a first woven layer that forms a first side of the tubular graft, where the first woven layer has a set of first warp ends. A second woven layer may forma a second side of the tubular graft, where the second woven layer has a set of second warp ends, and where the second warp ends are distinct from the first warp ends. A woven pocket flap may extend from the first woven layer, where a pocket opening is defined between the woven pocket flap and the first woven layer, and where the woven pocket flap includes at least one common weft yarn with the first woven layer.