Fast interplanetary coronal mass ejections (ICMEs) are the drivers of strong space weather storms such as solar energetic particle events and geomagnetic storms. The connection between the space-weather-impacting solar wind disturbances associated with fast ICMEs at Earth and the characteristics of causative energetic CMEs observed near the Sun is a key question in the study of space weather storms, as well as in the development of practical space weather prediction. Such shock-driving fast ICMEs usually expand at supersonic speeds during the propagation, resulting in the continuous accumulation of shocked sheath plasma ahead. In this paper, we propose a "sheath-accumulating propagation" (SAP) model that describes the coevolution of the interplanetary sheath and decelerating ICME ejecta by taking into account the process of upstream solar wind plasma accumulation within the sheath region. Based on the SAP model, we discuss (1) ICME deceleration characteristics; (2) the fundamental condition for fast ICMEs at Earth; (3) the thickness of interplanetary sheaths; (4) arrival time prediction; and (5) the super-intense geomagnetic storms associated with huge solar flares. We quantitatively show that not only the speed but also the mass of the CME are crucial for discussing the above five points. The similarities and differences between the SAP model, the drag-based model, and the"snow-plow" model proposed by Tappin are also discussed.