We use simple models of the spatial structure of the quasar broad-line region (BLR) to investigate the properties of so-called ghostly damped Ly α (DLA) systems detected in Sloan Digital Sky Survey (SDSS) data. These absorbers are characterized by the presence of strong metal lines but no H i Ly α trough is seen in the quasar spectrum indicating that, although the region emitting the quasar continuum is covered by an absorbing cloud, the BLR is only partially covered. One of the models has a spherical geometry, another one is the combination of two wind flows, whereas the third model is a Keplerian disc. The models can reproduce the typical shape of the quasar Ly α emission and different ghostly configurations. We show that the DLA H i column density can be recovered precisely independently of the BLR model used. The size of the absorbing cloud and its distance to the centre of the AGN are correlated. However, it may be possible to disentangle the two using an independent estimate of the radius from the determination of the particle density. Comparison of the model outputs with SDSS data shows that the wind and disc models are more versatile than the spherical one and can be more easily adapted to the observations. For all the systems, we derive log N(H i)(cm−2) > 20.5. With higher quality data, it may be possible to distinguish between the models.