Automatic History Matching of Three-Phase Production Data with Adaptive Moment Estimation (Adam)

Abstract Traditional gradient-based methods often struggle with slow convergence rates and may fail to meet convergence criteria because gradient descent can become trapped in a local minimum. In contrast, sensitivity-based methods achieve much faster convergence rates but incur high computational costs for sensitivity calculations and large memory resources to manage the dense matrices inherent in normal equations. This paper presents the application of the Adaptive Moment Estimation (Adam) algorithm, originally designed for training large-scale deep learning models, to achieve the task of automatic history matching. Adam adaptively adjusts the learning rate for each model parameter using both current and historical gradient information to optimize updates during each iteration. The implementation of the Adam algorithm involves three key stages: model prediction, gradient calculation, and iterative optimization. Model prediction is conducted using a three-dimensional, three-phase black oil simulator developed based on a corner-point grid, which effectively captures reservoir geological features. Gradients are derived through an adjoint formulation in this simulator, and the development is described in detail. The grid porosity, along with horizontal and vertical permeability, is computed by minimizing an objective function that incorporates water cut and gas-oil ratio (GOR) To validate our methodology, we employed the PUNQ-S3 reservoir model through three numerical experiments: individual matches for GOR and water cut, along with an analysis where both were matched simultaneously. Our case studies demonstrate that the Adam algorithm successfully matched a full 16.5 years of well production data. Adam achieved convergence at rates that were previously unachievable by conventional methods, particularly when simultaneously matching water cut and GOR. Furthermore, we implemented warm restarts and optimal step size decay—techniques proven effective in deep learning—to boost the convergence rate in our specific tests of the Adam algorithm for matching water cut and GOR. Our numerical experiments suggest that a class of adaptive gradient-based optimization techniques provides an effective solution of automatic history matching.