We performed a systematic analysis of all INTEGRAL observations from 2003 to 2009 of 14 supergiant fast X-ray transients (SFXTs), implying a net exposure time of about 30 Ms. For each source we obtained light curves and spectra (3–100 keV), discovering several new outbursts. We discuss the X-ray behaviour of SFXTs emerging from our analysis in the framework of the clumpy wind accretion mechanism we proposed. We discuss the effect of X-ray photoionization on accretion in close binary systems such as IGR J16479−4514 and IGR J17544−2619. We show that, because of X-ray photoionization, there is a high probability of an accretion disc forming from the capture of angular momentum in IGR J16479−4514, and we suggest that the formation of transient accretion discs could be partly responsible for the flaring activity in SFXTs with narrow orbits. We also propose an alternative way to explain the origin of flares with peculiar shapes observed in our analysis applying the model of Lamb et al., which is based on accretion via the Rayleigh–Taylor instability and was originally proposed to explain Type II bursts.