  Large-scale or high-resolution geologic models usually comprise a huge number
of grid blocks, which can be computationally demanding and time-consuming to
solve with numerical simulators. Therefore, it is advantageous to upscale
geologic models (e.g., hydraulic conductivity) from fine-scale (high-resolution
grids) to coarse-scale systems. Numerical upscaling methods have been proven to
be effective and robust for coarsening geologic models, but their efficiency
remains to be improved. In this work, a deep-learning-based method is proposed
to upscale the fine-scale geologic models, which can assist to improve
upscaling efficiency significantly. In the deep learning method, a deep
convolutional neural network (CNN) is trained to approximate the relationship
between the coarse grid of hydraulic conductivity fields and the hydraulic
heads, which can then be utilized to replace the numerical solvers while
solving the flow equations for each coarse block. In addition, physical laws
(e.g., governing equations and periodic boundary conditions) can also be
incorporated into the training process of the deep CNN model, which is termed
the theory-guided convolutional neural network (TgCNN). With the physical
information considered, dependence on the data volume of training the deep
learning models can be reduced greatly. Several subsurface flow cases are
introduced to test the performance of the proposed deep-learning-based
upscaling method, including 2D and 3D cases, and isotropic and anisotropic
cases. The results show that the deep learning method can provide equivalent
upscaling accuracy to the numerical method, and efficiency can be improved
significantly compared to numerical upscaling.
