Neural Networks and their Derivatives for History Matching and other Seismic, Basin and Reservoir Optimization Problems

Description: In geosciences, complex forward problems met in geophysics, petroleum system analysis and reservoir engineering problems often requires 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 of 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 non linearity 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 the first point, a complete description of a neural network equivalent to a second order polynomial will be given. 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. Application: On a real case study, the ability of neural networks to reproduce complex phenomena (water-cuts, production rates. etc.) is showed. Comparisons with second polynomials (and kriging methods) will be done demonstrating the superiority of the neural network approach as soon as non linearity 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. Results and conclusions: (1) Neural Network can replace advantageously polynomial and kriging approaches as proxies for inverse problems and uncertainty analysis, (2) A neural network giving a bilinear polynomial will be explicitly given, (3) Gradients and Hessian of neural network can be calculated and use by optimizers. Keywords: Proxies, History Matching, Gradient Methods, Optimizers, Basin Modelling, Seismic Inversion, Uncertainty Analysis, Hessian