### Abstract

In geosciences, complex forward problems met in geophysics, petroleum system analysis, and reservoir engineering problems often require replacing these forward problems by proxies, and these proxies are used for optimizations problems. For instance, history matching of observed field data requires a so large number of reservoir simulation runs (especially when using geostatistical geological models) that it is often impossible to use the full reservoir simulator. Therefore, several techniques have been proposed to mimic the reservoir simulations using proxies. Due to the use of experimental approach, most authors propose to use second-order polynomials. In this paper, we demonstrate that (1) neural networks can also be second-order polynomials. Therefore, the use of a neural network as a proxy is much more flexible and adaptable to the nonlinearity of the problem to be solved; (2) first-order and second-order derivatives of the neural network can be obtained providing gradients and Hessian for optimizers. For inverse problems met in seismic inversion, well by well production data, optimal well locations, source rock generation, etc., most of the time, gradient methods are used for finding an optimal solution. The paper will describe how to calculate these gradients from a neural network built as a proxy. When needed, the Hessian can also be obtained from the neural network approach. On a real case study, the ability of neural networks to reproduce complex phenomena (water cuts, production rates, etc.) is shown. Comparisons with second polynomials (and kriging methods) will be done demonstrating the superiority of the neural network approach as soon as nonlinearity behaviors are present in the responses of the simulator. The gradients and the Hessian of the neural network will be compared to those of the real response function.

Original language | English |
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Pages (from-to) | 549-561 |

Number of pages | 13 |

Journal | Computational Geosciences |

Volume | 18 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - Aug 2014 |

Externally published | Yes |

### Keywords

- Basin modeling
- Gradient methods
- Hessian
- History matching
- Inversion
- Optimizers
- Proxies
- Seismic
- Uncertainty analysis

### ASJC Scopus subject areas

- Computer Science Applications
- Computers in Earth Sciences
- Computational Theory and Mathematics
- Computational Mathematics