An Optimization Framework for Automatic History Matching of Aquifer Drive Reservoir with Geological and Geostatistical Data Integration

Abstract Lately, we adapted two stochastic gradient algorithms, Adam and AdamW originally designed to train large deep learning models for automatic history matching of three-phase production data. In this paper, we extend their application by developing an optimization framework that systematically integrates well production data, geological models, and geostatistical constraints into reservoir simulation models. We theoretically show that the automatic history matching process can be formulated as a reaction–diffusion system. Solving this reaction–diffusion equation corresponds to a denoising diffusion process, where the reservoir parameters gradually evolve over iterations, reducing uncertainty while maintaining geological integrity. By adaptively adjusting governing coefficients, the proposed framework preserves geological structure while effectively reducing discrepancies between simulated and observed production data. A key innovation in this work is the introduction of two complementary nonlinear diffusion formulations to guide spatial regularization. The first suppresses smoothing in regions where the optimizer identifies significant parameter variations, thereby preserving geologically meaningful features such as faults and facies boundaries. The second encourages structural alignment with a reference model by promoting diffusion where the estimated model deviates from the prior, without enforcing its absolute values. Together, these formulations provide a physically interpretable mechanism to jointly minimize production data mismatch while regularizing toward geologically consistent spatial structures during history matching. Model prediction is conducted using a three-dimensional, three-phase black oil simulator on a corner-point grid. Gradients are computed using an embedded adjoint method. These include not only grid permeability and porosity, but also aquifer properties such as permeability, porosity, thickness, and radius—treated as unknowns and estimated automatically Our case studies demonstrate three key advantages. First, the framework preserves sharp geological features—such as facies boundaries and low-permeability barriers—by employing nonlinear diffusion strategies that either suppress smoothing in mismatched regions or extract structural alignment from prior models. Second, it integrates geostatistical constraints into the optimization, ensuring consistency with spatial priors. Third, it improves convergence stability by iteratively updating all parameters—including aquifer properties—within a reaction–diffusion formulation guided by adjoint gradients, allowing for efficient and robust history matching over long production periods. These results show that the proposed approach provides a physically interpretable mechanism to jointly minimize production data mismatch while regularizing toward geologically consistent spatial structures during history matching.