A Hybrid Probabilistic-Backpropagation Neural Network Solver for Nonlinear Systems in Reservoir Simulation

Abstrak

Reservoir simulation requires solving large, sparse systems of nonlinear equations, where iterative Krylov subspace solvers such as the conjugate gradient (CG), stabilized conjugate gradient (BiCG-STAB), and generalized minimal residual (GMRES) are widely applied. However, these methods often have limitations in terms of their stability and accuracy in nonlinear systems. This paper introduces a hybrid probabilistic backpropagation neural network (Prob-BPNN) solver that integrates neural-network-based initialization with probabilistic inference to improve robustness. The solver was benchmarked against CG, BiCG-STAB, and GMRES using two synthetic reservoir models with the GMRES solution at a tolerance of 10-10, serving as the reference solution. The results show that Prob-BPNN consistently achieved production profiles closely matching the reference solution, with errors of MAE ≤ 0.066, RMSE ≤ 0.071, MAPE ≤ 2.04%, and R2 ≥ 0.945. In contrast, CG and BiCG-STAB produced unstable and nonphysical results, with errors exceeding 292% and negative R2 values. In terms of computational performance, Prob- BPNN required 9.96 s in Case 1 and 45.90 s in Case 2, compared to 2.85 s and 1.53 s for GMRES, respectively. Although more computationally expensive, Prob-BPNN delivered convergence on the same residual order of magnitude (below 10-3) as GMRES while avoiding the severe instabilities observed in CG and BiCG-STAB. These findings indicate that the Prob-BPNN is preferable in applications where solver robustness and accuracy are critical, even at the expense of a higher execution time. Future research should focus on reducing computational overhead through parallelization and hybridization strategies to enhance the scalability of large-scale reservoir models.

Kata kunci

• nonlinear solver
• probabilistic
• neural networks
• backpropagation
• reservoir simulation