Memristors are an electronic device whose resistance depends on the voltage history that has been applied to its two terminals. Despite its clear advantage as a computational element, a suitable transport model is lacking for the special class of interface-based memristors. Here, we adapt the widely-used Yakopcic compact model by including transport equations relevant to interface-type memristors. This model is able to reproduce the qualitative behaviour measured upon Nb-doped SrTiO$_3$ memristive devices. Our analysis demonstrates a direct correlation between the devices' characteristic parameters and those of our model. The model can clearly identify the charge transport mechanism in different resistive states thus facilitating evaluation of the relevant parameters pertaining to resistive switching in interface-based memristors. One clear application of our study is its ability to inform the design and fabrication of related memristive devices.