  Distributions of potential and lithium content inside lithium ion batteries
highly affects their performance and durability. An increased heterogeneity of
the lithium distribution is expected in thick electrodes with high energy
densities or cycling at high currents. To optimize electrodes and cells
designs, it is crucial to probe lithium concentration gradients across the
depth of the electrode, but also to predict their occurrence and magnitude as a
function of materials properties. Here, we follow the lithium distribution
across a $80~\mu m$ thick porous graphite electrode using a $1~\mu m$ focused
synchrotron X-ray beam. The sequential formation of the individual Li$_x$C$_6$
phases during lithium de-insertion is extracted from X-ray diffraction
patterns, allowing the quantification of lithium concentration across the
electrode thickness. Analyzing the evolution of heterogeneities as a function
of time, we recover the striking features we predicted with a porous electrode
model, including the succession of homogeneous and heterogeneous distributions
of lithium. However, a clear difference is obtained at high stoichiometry, with
a much more homogeneous distribution than initially predicted. Revisiting the
interplay between transport and kinetic transfers limitations in the porous
electrode model, we suggest that the kinetics of lithium (de)-insertion is
highly reduced during the LiC$_6$/LiC$_{12}$ phase transition.
