  We study the hydrodynamic behaviour of the asymmetric simple exclusion
process on the lattice of size $n$. In the bulk, the exclusion dynamics
performs rightward flux. At the boundaries, the dynamics is attached to
reservoirs. We investigate two types of reservoirs: (1) the reservoirs that are
weakened by $n^\theta$ for some $\theta<0$ and (2) the reservoirs that create
particles only at the right boundary and annihilate particles only at the left
boundary. We prove that the spatial density of particles, under the hyperbolic
time scale, evolves with the entropy solution to a scalar conservation law on
$[0,1]$ with boundary conditions. The boundary conditions are characterised by
the boundary traces at $x=0$ and $x=1$ which take values from $\{0,1\}$.
