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Network lifetime is a crucial performance metric to evaluate data-gathering wireless sensor networks (WSNs) where battery-powered sensor nodes periodically sense the environment and forward collected samples to a sink node. In this paper, we propose an analytic model to estimate the entire network lifetime from network initialization until it is completely disabled, and determine the boundary of energy hole in a data-gathering WSN. Specifically, we theoretically estimate the traffic load, energy consumption, and lifetime of sensor nodes during the entire network lifetime. Furthermore, we investigate the temporal and spatial evolution of energy hole, and apply our analytical results to WSN routing in order to balance the energy consumption and improve the network lifetime. Extensive simulation results are provided to demonstrate the validity of the proposed analytic model in estimating the network lifetime and energy hole evolution process.Index Terms-wireless sensor network, network lifetime, energy hole, energy efficiency, routing.