In this paper, a multi-pair amplify-and-forward (AF) massive MIMO system is investigated, where multiple source users communicate with multiple destination users via a single relay. The achievable rate of the system and its analytical expression are studied for power-scaling laws when ZFT/ZFR algorithm is adopted at the relay. The imperfect channel state information (CSI) is considered and the MMSE estimator is utilized. Analytical consequences reveal that with the number of antennas growing into infinity, the effect of small-scale fading, the inter-user interference and the noise could be eliminated. And the achievable rate will be affected by channel errors, the number of users, and the power-scaling law. Simulation results effectively verify our analytical expressions.