Title: Continual Learning of Neural Networks for Realtime Wireline Cable Position Inference. Abstract: In the oil fields, Wireline cable is spooled onto a drum where computer vision techniques based on convolutional neural networks (CNNs) are applied to estimate the cable position in real time for automated spooling control. However, as new training data keeps arriving to continuously improve the network, the re-training procedure faces challenges. Online learning fashion with no memory to historical data leads to catastrophic forgetting. Meanwhile, saving all data will cause the disk space and training time to increase without bounds. In this paper, we proposed a method called the modified-REMIND (mREMIND) network. It is a replay-based continual learning method with longer memory to historical data and no memory overflow issues. Information of old data are kept for multiple iterations using a new dictionary update rule. Additionally, by dynamically partitioning the dataset, the method can be applied on devices with limited memory. In our experiments, we compared the proposed method with multiple state-of-the-art continual learning methods and the mREMIND network outperformed others both in accuracy and in disk space usage.