  We study the uniqueness of solutions of the initial-boundary value problem in
the quarter-plane for the chemical flood conservation law system in the class
of piece-wise $\mathcal C^1$-smooth functions under certain restrictions. The
vanishing viscosity method is used locally on the discontinuities of the
solution to determine admissible and inadmissible shocks. The Lagrange
coordinate transformation is utilized in order to split the equations. The
proof of uniqueness is based on an entropy inequality similar to the one used
in the classical Kru\v{z}kov's theorem.
