While local learning rules like equilibrium propagation have been theorized as efficient algorithms to implement with memristor technology, few works have simulated this algorithm with memristor nonidealities, such as cycle-to-cycle and device-to-device variations and nonlinear dynamics. Furthermore, a specific architecture and update scheme that can be implemented by peripheral circuitry needs to be outlined to move forward in this line of research. In this work, we propose an update scheme implementing equilibrium propagation that can be efficiently implemented in circuitry. We find that learning with memristor nonidealities achieves little or no accuracy degradation when compared to training without nonidealities.