  We present a new semi-analytic formalism for modeling metal absorption lines
that emerge from a clumpy galactic environment, ALPACA. We predict the
''down-the-barrel'' (DTB) metal absorption line profiles and the EW of
absorption at different impact parameters as a function of the properties of
the clumps, including the clump kinematics, the clump volume filling factor,
the clump number density profile and the clump ion column densities. With
ALPACA, we jointly model the stacked DTB CII$\lambda$1334 spectrum of a sample
of $z \sim$ 3 Lyman break galaxies and the EW v.s. $b$ profile of a sample of
$z \sim$ 2 star-forming galaxy-galaxy pairs. ALPACA successfully reproduced two
datasets simultaneously, and the best-fit prefers a low clump volume filling
factor ($\sim 3 \times 10^{-3}$). The radial velocities of the clumps are a
superposition of a rapidly accelerated outflow with a maximum velocity of $\sim
400\,\rm km\,s^{-1}$ and a velocity dispersion of $\sigma_{\rm cl} \sim\,120
\rm km\,s^{-1}$. The joint modeling reveals a physical scenario where the
absorption observed at a particular velocity is contributed by the clumps
distributed over a fairly broad range of radii. We also find that the commonly
adopted Sobolev approximation is at best only applicable within a narrow range
of radii where the clumps are undergoing rapid acceleration in a
non-volume-filling clumpy medium. Lastly, we find that the clump radial
velocity profile may not be fully constrained by the joint modeling and
spatially-resolved Ly$\alpha$ emission modeling may help break the degeneracy.
