A residual-accelerated Jacobian method for rapid convergence in reservoir simulation

Abstract Reservoir simulation requires efficient algorithms for complex, nonlinear systems. We introduce a Residual–Accelerated Jacobian (RAJ) for fully implicit reservoir simulation. RAJ assembles the Jacobian by finite differences with a residual–adaptive step $$h_{\textrm{res}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>h</mml:mi> <mml:mtext>res</mml:mtext> </mml:msub> </mml:math> that enlarges far from convergence and shrinks near tolerance; saturation probes are projected to physical bounds. The governing residual is unchanged, so accuracy is preserved while Jacobian columns remain stable across nonlinear episodes. This adaptive mechanism ensures that the RAJ method remains responsive and can effectively adjust 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. We evaluate RAJ on SPE10 (top five layers) and the Norne field (corner-point with NNCs), reporting CPU time, Newton/linear iterations, and robustness indicators (wasted steps/iterations), alongside accuracy parity. On SPE10 water injection, RAJ runs in 226.73 s vs FD 254.42 s ( –10.9% ); all methods use 43 Newton iterations and $$\approx $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>≈</mml:mo> </mml:math> 2215 linear iterations, with no wasted steps. On PE10 gas injection, RAJ completes in 1600.55 s vs FD 1772.12 s ( –9.7% ) and lowers wasted work (wasted time-step fraction 31.98% vs 36.38% ; wasted linear iterations 5385 vs 6644 ). On Norne, RAJ takes 276.81 s vs FD 313.85 s ( –11.8% ) with 51 vs 53 Newton iterations and zero wasted steps. As expected, analytical derivatives are fastest where available (water 154.816 s , gas 1177.69 s , Norne 223.935 s ). Overall, RAJ delivers comparable or better CPU times than fixed-step FD while preserving accuracy and reducing wasted work, offering a practical, drop-in alternative when analytical Jacobians are unavailable.