We investigate the different flow regimes in nonconsolidated porous media. The porous bulk is soaked with water, which is then pumped out of it, across the boundary defined by the particles at the edge of the bulk. Experiments are carried out on sand and glass beads soaked in distilled water and placed in a circular Hele-Shaw cell, the flow being radially convergent. We show, for a given value of flow velocity (the yield velocity), the existence of an unstable regime where the fluid-porous interface is deformed and branches upstream in the bulk. When this velocity is further increased, two cases arise depending on the value of the yield velocity: Either a second threshold is passed, global fluidization of the porous bulk sets in, and the flow becomes stable or the instability persists and the canal arborescence continues to grow. The driving mechanism of this instability is thus the permeability contrast across the edge of the porous bulk; when this contrast diminishes, the flow becomes stable. A force balance on the boundary particles predicts the threshold value for the fluid velocity, beyond which the flow is unstable. Using a Saffman-Taylor inspired linear perturbation analysis [Proc. R. Soc. London, Ser. A 245, 312 (1958)], a dispersion function is found (predicting the wavelength dependence of the instability growth amplitude), taking into account the particle arch formation in the porous bulk. We then find the velocity of propagation of the receding front, predicted to be proportional to the particle velocity beyond the front, itself described by a Bagnold concentrated suspension flow [Proc. R. Soc. London, Ser. A 225, 49 (1954)]. This front velocity is successfully confronted with experimental measurements. A screening effect characteristic of Laplacian growth phenomena is seen in the experiments as testified by flow rate conservation between the different branches of the arborescence and direct dye visualization. The topologies obtained are fractal and the measured dimension $D_f=1.6\mathrm{}-1.7$ compares favorably to the calculated dimension from the branching angle distribution.

• Received 5 February 1997

DOI:https://doi.org/10.1103/PhysRevE.58.6051