## Abstract

The reservoir simulation is based on the solving of second-order nonlinear Partial Differential Equations(PDEs). Following the high-level of nonlinearity or irregular boundaries, analytical solutions are notapplicable to solve the supposed PDEs. To numerically solve the PDEs, applying nonlinear solvers arerecommended. Dependencies on derivatives and proper initial guesses are the main disadvantages of classicsolvers. To overcome the mentioned obstacles, solving supposed equations based on Adaptive NeuralNetwork (ANN) has been introduced. The algorithm starts by introducing an initial set into the Nonlinear Simultaneous Algebraic Equations(NSAE). The outputs are compared with the desired matrix of zeros to generate the required error. Thecalculated vectors of errors and its derivation are firstly employed to update the ANN weights throughapplying the adaption laws, and secondly, create the input vector to run the ANN. The outputs of the ANNare considered as corrections to be made to the initial set. Then, the corrected initial set is reintroducedinto equations. The procedure continues iteratively until the outputs of equations meet the required levelof accuracy. By taking advantages of the adaptive laws, the outputs of the presented algorithm have successfullybeen matched with answers of the classic solvers, but with less computational costs. The convergenceof the shown algorithm has practically been examined by assuming various mathematical types of initialsets. The implemented algorithm has been robust enough to converge for different forms of the initialsets, even for invalid values like minus numbers. However, records indicate that the convergence rates arestrongly dependent on the values of initial sets. Following the sensitivity analysis over the primary model ofANN lead to the optimized network, which could solve the supposed NSAE three times faster. It has beeninterpreted that the number of neurons (NN), the diagonal coefficient matrix of error , and the adaptivecoefficient (Fw) have the most significant impacts on the performance of the algorithm. In contrast to Newton's method as the most well-known nonlinear solver, the launched algorithm does notrequire any proper initial guesses. Moreover, the absolute independence of computing the partial derivative of the Jacobian matrix and its inversion, which causes a notable reduction of computational costs, is the otherremarkable advantage of the proposed approach. The represented algorithm can be taken as the platform todevelop the next generation of simulators working based on machine learning.

Original language | English |
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Title of host publication | Society of Petroleum Engineers - SPE Europec Featured at 82nd EAGE Conference and Exhibition |

Publisher | Society of Petroleum Engineers |

ISBN (Electronic) | 9781613997123 |

Publication status | Published - 2020 |

Event | SPE Europec Featured at 82nd EAGE Conference and Exhibition - Amsterdam, Netherlands Duration: 8 Dec 2020 → 11 Dec 2020 |

### Publication series

Name | Society of Petroleum Engineers - SPE Europec Featured at 82nd EAGE Conference and Exhibition |
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### Conference

Conference | SPE Europec Featured at 82nd EAGE Conference and Exhibition |
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Country/Territory | Netherlands |

City | Amsterdam |

Period | 8/12/20 → 11/12/20 |