Human brain homeostasis requires carefully regulated transport of nutrients and waste, but its governing mechanisms are hard to quantify. By combining multi-modal gMRI data over 48 hours with high-fidelity inverse computational modelling, we here identify several potential transport mechanisms that best explain the clinical observations. Our findings support the combined roles of local, vascular and glymphatic-type clearance pathways within human brain tissue with tissue flow velocities on the order of $\mu$m/min, local clearance rates at the order of 10**(-3)/min, and enhanced diffusion by a factor 3.5, in sleeping and sleep-deprived subjects. Reduced advection fully explains reduced tracer clearance after sleep-deprivation, supporting the role of sleep and sleep deprivation on human brain clearance.