Contingency research to find optimal operations and post-contingency recovery plans in distribution networks has gained a major attention in recent years. To this end, we consider a multi-period optimal power flow (OPF) problem in distribution networks, subject to the N-1 contingency where a line or distributed energy resource fails. The contingency can be modeled as a stochastic disruption, an event with random magnitude and timing. Assuming a specific recovery time, we formulate a multi-stage stochastic convex program and develop a decomposition algorithm based on stochastic dual dynamic programming (SDDP). Realistic modeling features such as linearized AC power flow physics, engineering limits and battery devices with realistic efficiencies curves are incorporated. We present extensive computational tests to show efficiency of our decomposition algorithm and out-of-sample performance of our solution compared to its deterministic counterpart. Operational insights on battery utilization, component hardening, and length of recovery phase are obtained by performing analyses from stochastic disruption-aware solutions.