  In this paper, we study the Waldschmidt constant of a generalized fat point
subscheme $Z=m_1p_1+\cdots+m_rp_r$ of $\mathbb{P}^2$, where $p_1,\cdots,p_r$
are essentially distinct points on $\mathbb{P}^2$, satisfying the proximity
inequalities. Furthermore, we prove its lower semi-continuity for $r\le 8$.
Using this property, we also calculate the Waldschmidt constants of the fat
point subschemes $Z=p_1+\cdots+p_5$ giving weak del Pezzo surfaces of degree 4.
