  This paper is dedicated to the effective viscosity of suspensions without
inertia, at low solid volume fraction $\phi$. The goal is to derive rigorously
a $o(\phi^2)$ formula for the effective viscosity. In previous works, such
formula was given for rigid spheres satisfying the strong separation assumption
$ d_{min} \ge c \phi^{-\frac13} r$, where $d_{min}$ is the minimal distance
between the spheres and $r$ their radius. It was then applied to both periodic
and random configurations with separation, to yield explicit values for the
$O(\phi^2)$ coefficient. We consider here complementary (and certainly more
realistic) random configurations, satisfying soft assumptions of separation and
long range decorrelation. We justify in this setting the famous Batchelor-Green
formula. Our result applies for instance to hardcore Poisson point process with
almost minimal hardcore assumption $d_{min} > (2+\epsilon) r$, $\epsilon > 0$.
