The motion of a capillary interface across a deformable granular material in a confined geometry shows the complex interplay between viscous forces, solid friction, and capillary forces. In a horizontal quasi-one-dimensional geometrical confinement, a millifluidic tube, the displacement of a three-phase flow consisting of two fluids and a mobile granular phase exhibits viscous or frictional displacement regimes, as shown in [Phys. Rev. Lett. 117, 028002 (2016).]. In the present paper we explore in detail the dynamics in both regimes by making use of a new set of data. The viscous displacement regime which is characterized by a fluidization of the immersed granular material dragged by the flow driving the displacement of the capillary interface is interpreted from a rheological point of view. The frictional displacement regime which displays a self-structuring of the granular material left in the tube behind the invading capillary interface is interpreted by using a model based on the Janssen law to predict the typical size of the plugs obtained.