A Residual-Accelerated Jacobian Method for Rapid Convergence in Reservoir Simulation

In the realm of reservoir simulation, efficiently solving complex, non-linear systems is paramount. This paper introduces a novel Residual-Accelerated Jacobian (RAJ) method, designed to significantly enhance the convergence rate in reservoir simulation problems. Central to this novel method is the use of residual feedback for dynamically adjusting the step size in finite difference partial derivatives for Jacobian approximations. This adjustment enables faster and more stable convergence towards the solution, especially in scenarios marked by steep gradients and pronounced non-linear behaviors typical in reservoir models. In the RAJ method, the Jacobian matrix, crucial for Newton-Raphson iterations in reservoir simulations, is computed using a finite difference approach with an adaptively adjusted step size (h). This adaptation is guided by the local residual of each cell, which measures the deviation of the current solution from the tolerated solution. When the residual is large, suggesting a significant deviation, the RAJ method increases h, allowing broader exploration of the solution space. Conversely, as the solution approaches convergence, indicated by a smaller residual, h is reduced, allowing for finer and more precise iterations. This adaptive mechanism ensures that the RAJ method is responsive to the changing dynamics of the simulation, enabling faster convergence without compromising the accuracy of the solution, a crucial aspect in handling complex, non-linear systems in reservoir simulations. Numerical evaluations conducted using a spectrum of test cases from simple benchmark problems to intricate, realistic reservoir models, demonstrate the method's versatility and efficiency. Our findings indicate a reduction in the number of iterations required to achieve convergence compared to traditional methods, without any compromise in accuracy. The method is particularly effective in models characterized by high non-linearity, showcasing a noticeable enhancement in performance. In conclusion, the RAJ method stands as a significant advancement in the field of reservoir simulation. Offering a robust and efficient solution for addressing the inherently complex nonlinear problems in this domain, this method promises to revolutionize current practices and methodologies, paving the way for more precise, reliable, and efficient reservoir management strategies.