In this paper, we propose a methodology that combines finite-element modeling with neural networks in the numerical modeling of systems with behavior that involves a wide span of spatial scales.

The method starts by constructing a high-resolution model of the subsurface, including its elastic mechanical properties and pore pressures. A second model is also constructed by scaling up mechanical properties and pressures into a coarse spatial resolution. Inexpensive finite-element solutions for stress are then obtained in the coarse model. These stress solutions aim at capturing regional trends and large-scale stress correlations. Finite-element solutions for stress are also obtained in high resolution, but only in a small subvolume of the 3D model. These stress solutions aim at estimating fine-grained details of the stress field introduced by the heterogeneity of rock properties at the fine scale.

A neural network is then trained to infer the transformation rules that map stress solutions between different scales. The inputs to the training are pressure and mechanical properties in high and low resolutions. The output is the fine-scale stress computed in the subvolume of the high-resolutionmodel.

Once trained, the neural network can be used to approximate a high-resolution stress field in the entire 3D volume using the coarse-scale solution and only providing high-resolution material properties and pressures.

The results obtained indicate that when the coarse finite-element solutions are combined with the neural-network estimates, the results are within a 2 to 4% error of the results that would be computed with high-resolutionfinite-element models, but at a fraction of the cost in time and computational resources. This paper discusses the benefits and drawbacks of the method and illustrates its applicability by means of a worked example.

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