Heavy transport vessels (HTVs) are widely used in long-distanced dry-tow transport. Currently, two end operations, loading and discharge, are executed in harbours or protected waters where waves are virtually non-existent. There is potentially a huge market demand to execute these operations at offshore sites. To expand such operations to the offshore open seas, a systematic investigation of vertical motions between HTV and cargo is needed. The main challenge brought by these offshore discharge and loading operations is the small gap between HTV and cargo, both subject to environment conditions (waves, current, wind etc). Non-linearity due to gap flow is found from previous forced oscillation tests and CFD calculations. This thesis seeks to tackle this non-linearity in the time-domain framework. The thesis report starts with the literature study based on Molin where the external forced due to forced-oscillation is described for a deeply-submerged circular disc extremely close to seabed. It has been found that inertial and cushioning forces are dominant for the small gap problem. Essentially, both forces can be expressed by added mass. The evaluations cases in the thesis are circular cylinders with certain drafts, for which the free surface effect must also be addressed by employing separation of frequency technique. The problem is divided into two parts for time domain equations in convolution form: gap part and free surface part. Gap part is frequency independent whereas free surface part is frequency dependent due to radiated waves generated by oscillating cargo. For the latter, it has been concluded that free surface part is largely independent of gap part, thus keeping radiation force by convolution integration untouched. After separation of frequency, the main task is to find the proper formulations to quantify the gap part force. Two important variables have been identified: gap height and inclination angle. The quantification starts with verification for two numerical tools: AQWA and New method. Both tools are sufficiently accurate to quantify the influence from gap height and inclination angle on added mass of vertical motions. Thus, the external force is obtained and validated against measurements from forced oscillation tests and analytical results from Molin. With constructed formulations for added mass, the DLL for external force can be implemented into time domain solver. The implementation can be done via two approaches: interpolation of database approach and direct formulation approach. Moreover, a systematic scheme has been proposed to implement time domain simulations for the small gap problem. Preliminary time domain simulations of regular waves at natural frequency show that the conventional linear results over-predict the heave response compared to non-linear results. This systematic methodology to implement time domain simulations will throw light on further analysis for workability.