  By means of digital image correlation, we experimentally characterize the
deformation of a dry granular medium confined inside a Hele-Shaw cell due to
air injection at a constant overpressure high enough to deform it (from 50 to
250 kPa). Air injection at these overpressures leads to the formation of so
called pneumatic fractures, i.e. channels empty of beads, and we discuss the
typical deformations of the medium surrounding these structures. In addition we
simulate the diffusion of the fluid overpressure into the medium, comparing it
with the Laplacian solution over time, and relating pressure gradients with
corresponding granular displacements. In the compacting medium we show that the
diffusing pressure field becomes similar to the Laplace solution on the order
of a characteristic time given by properties of the pore fluid, granular medium
and the system size. However, before the diffusing pressure approaches the
Laplace solution on system scale, we find that it resembles the Laplacian field
near the channels, with the highest pressure gradients on the most advanced
channel tips and a screened pressure gradient behind them. We show that the
granular displacements more or less always move in the direction against the
local pressure gradients, and when comparing granular velocities with pressure
gradients in the zone ahead of channels, we observe a Bingham type of rheology
for the granular paste (the mix of air and beads), with an effective viscosity
$\mu\_B$ and displacement thresholds $\nabla P\_c$ evolving during mobilization
and compaction of the medium. Such a rheology, with disorder in the
displacement thresholds, could be responsible for placing the pattern growth at
moderate injection pressures in a universality class like the Dielectric
Breakdown Model with $\eta=2$, where fractal dimensions are found between 1.5
and 1.6 for the patterns.
