New analytical models are introduced to describe the motion of a Herschel–Bulkley fluid slumping under gravity in a narrow fracture and in a porous medium. A useful self-similar solution can be derived for a fluid injection rate that scales as time $t$ ; an expansion technique is adopted for a generic injection rate that is power law in time. Experiments in a Hele-Shaw cell and in a narrow channel filled with glass ballotini confirm the theoretical model within the experimental uncertainty.