Supplementary data to the article: Upscaling reactive transport and clogging in shale microcracks by deep learning

Fracture networks in shales exhibit multiscale features. A rock system may contain a few main fractures and thousands of microcracks, whose length and aperture are orders of magnitude smaller than the former. It is computationally prohibitive to resolve all the fractures explicitly for such multiscale fracture networks. One traditional approach is to model the small-scale features (e.g. microcracks in shales) as an effective medium. Although this fracture-matrix conceptualization significantly reduces the problem complexity, there are classes of physical processes that cannot be accurately upscaled by effective medium approximations, e.g. microcrack clogging during mineral reactions. In this work, we employ deep learning in place of effective medium theory to upscale physical processes in small-scale features. Specifically, we consider reactive transport in a fracture-microcrack network where microcracks can be clogged by precipitation. A deep learning multiscale algorithm is developed, in which main fractures are explicitly resolved while reactive transport and clogging in microcracks are upscaled as a wall boundary condition of the main fractures. The wall boundary condition is constructed by recurrent neural networks, which take concentration histories as input and predict the solute transport from main fractures to microcracks. The deep learning multiscale algorithm is firstly employed in specific scenarios, then a general model is developed which can work under various conditions. The new approach is validated against fully resolved simulations and an analytical solution, providing a reliable and efficient solution for problems that cannot be upscaled by effective medium models.

页岩中的裂隙网络具有多尺度特性。岩石体系可包含少量主裂隙与数千条微裂纹，后者的长度与开度较前者相差数个数量级。对于此类多尺度裂隙网络，显式解析所有裂隙的计算成本极高，难以实现。传统方法之一是将小尺度特征（如页岩中的微裂纹）建模为有效介质（effective medium）。尽管该裂隙-基质概念框架大幅降低了问题复杂度，但存在一类物理过程无法通过有效介质近似（effective medium approximation）进行准确的尺度上推，例如矿物反应过程中的微裂纹堵塞。本研究采用深度学习替代有效介质理论，实现小尺度特征下物理过程的尺度上推。具体而言，我们研究了裂隙-微裂纹网络中的反应输运问题，其中微裂纹可因矿物沉淀发生堵塞。本研究开发了深度学习多尺度算法，显式解析主裂隙，同时将微裂纹内的反应输运与堵塞过程尺度上推为主裂隙的壁面边界条件。该壁面边界条件由循环神经网络（recurrent neural network, RNN）构建，以浓度历史序列作为输入，预测溶质从主裂隙向微裂纹的输运过程。该深度学习多尺度算法首先在特定场景下开展应用与验证，随后开发出可适配多种工况的通用模型。新方法通过全解析模拟与解析解进行验证，为无法通过有效介质模型实现尺度上推的问题提供了可靠且高效的解决方案。