Physics-informed neural networks (PINNs) are an emerging technology in the scientific computing domain. Contrary to data-driven methods, PINNs have been shown to be able to approximate and generalize well a wide range of partial differential equations (PDEs) by imbedding the underlying physical laws describing the PDE. PINNs, however, can struggle with the modeling of hyperbolic conservation laws that develop shocks, and a classic example of this is the Buckley–Leverett problem for fluid flow in porous media. In this work, we explore specialized neural network architectures for modeling the Buckley–Leverett shock front. We present extensions of the standard multilayer perceptron (MLP) that are inspired by the attention mechanism. The attention-based model was, compared to the multilayer perceptron model, and the results show that the attention-based architecture is more robust in solving the hyperbolic Buckley–Leverett problem, more data-efficient, and more accurate. Moreover, by utilizing distance functions, we can obtain truly data-free solutions to the Buckley–Leverett problem. In this approach, the initial and boundary conditions (I/BCs) are imposed in a hard manner as opposed to a soft manner, where labeled data are provided on the I/BCs. This allows us to use a substantially smaller NN to approximate the solution to the PDE.

Modeling flow and transport in fractured porous media has been a topic of intensive research for a number of energy- and environment-related industries. The presence of multiscale fractures makes it an extremely challenging task to resolve accurately and efficiently the flow dynamics at both the local and global scales. To tackle this challenge, we developed a computational workflow that adopts a two-level hierarchical strategy based on fracture length partitioning. This was achieved by specifying a partition length to split the discrete fracture network (DFN) into small-scale fractures and large-scale fractures. Flow-based numerical upscaling was then employed to homogenize the small-scale fractures and the porous matrix into an equivalent/effective single medium, whereas the large-scale fractures were modeled explicitly. As the effective medium properties can be fully tensorial, the developed hierarchical framework constructed the discrete systems for the explicit fracture–matrix sub-domains using the nonlinear two-point flux approximation (NTPFA) scheme. This led to a significant reduction of grid orientation effects, thus developing a robust, applicable, and field-relevant framework. To assess the efficacy of the proposed hierarchical workflow, several numerical simulations were carried out to systematically analyze the effects of the homogenized explicit cutoff length scale, as well as the fracture length and orientation distributions. The effect of different boundary conditions, namely, the constant pressure drop boundary condition and the linear pressure boundary condition, for the numerical upscaling on the accuracy of the workflow was investigated. The results show that when the partition length is much larger than the characteristic length of the grid block, and when the DFN has a predominant orientation that is often the case in practical simulations, the workflow employing linear pressure boundary conditions for numerical upscaling give closer results to the full-model reference solutions. Our findings shed new light on the development of meaningful computational frameworks for highly fractured, heterogeneous geological media where fractures are present at multiple scales.