The characterization of porous media via digital testing usually relies on intensive numerical computations that can be parallelized in GPUs. For absolute permeability estimation, Stokes flow simulations are carried out at the micro-structure to recover velocity fields that are used in upscaling with Darcy’s law. Digital models of samples can be obtained via micro-computed tomography (μCT) scans. As μCT data is three-dimensional, meshes grow cubically with image dimensions, causing the numerical problem at hand to become compute- and memory-bound as either resolution improves or larger fields-of-view are considered. While the usual focus is on accelerating solvers, memory usage continues to be a significant limitation for analyses of representative volumes in relatively accessible hardware. In this work, we explore the possibility of implementing MINRES solvers in GPU that favor a reduction in memory allocation. These solvers are applied to matrix-free FEM-based permeability characterization of μCT images. Our goal is to enable the study of 10003 voxel images in single GPU machines. Implementations that only require five, three, or two n-sized vectors of variables are presented, with n being the number of unknowns. Further, we employ a mesh numbering strategy that enables node-by-node massively parallel operations within a non-monolithic voxel-based pore space without storing connectivity tables. The proposed solvers, available through the open-source chfem software, are verified against analytical models for simple three-dimensional micro-structures, then are validated against numerical Digital Petrophysics benchmarks. A consumer-grade graphics card with 12GB of RAM is employed for the characterization of images with up to roughly 540 million voxels in a matter of tens of minutes. Stokes flow FEM-based simulations in meshes with 449 million degrees-of-freedom (DOFs) are carried out in 9 to 15 min, allocating less than 10GB in global memory. Finally, simulations on three 10003 carbon fiber domains, amounting to more than 3.7 billion DOFs, were run on a high-end GPU with 80GB of RAM in under 2.5 h, achieving very close agreement with flow-tube permeability experiments.

中文翻译：

---

使用单个 GPU 在千兆体素多孔介质中实现基于 FEM 的绝对渗透率估计


通过数字测试对多孔介质进行表征通常依赖于可以在 GPU 中并行化的密集数值计算。对于绝对磁导率估计，在微观结构处进行斯托克斯流模拟，以恢复用于使用达西定律放大的速度场。样本的数字模型可以通过显微计算机断层扫描 （μCT） 扫描获得。由于 μCT 数据是三维的，网格随图像尺寸呈立方增长，导致当分辨率提高或考虑更大的视场时，手头的数值问题会受到计算和内存的限制。虽然通常的重点是加速求解器，但内存使用仍然是分析相对可访问硬件中的代表性卷的重大限制。在这项工作中，我们探索了在 GPU 中实现有利于减少内存分配的 MINRES 求解器的可能性。这些求解器应用于 μCT 图像的无基质 FEM 渗透性表征。我们的目标是在单个 GPU 机器中研究 10003 个体素图像。给出了只需要 5 个、3 个或 2 个 n 大小的变量向量的实现，其中 n 是未知数的数量。此外，我们采用了网格编号策略，可以在非整体体素的孔隙空间内实现逐节点大规模并行操作，而无需存储连接表。通过开源 chfem 软件提供的所提出的求解器，根据简单三维微结构的解析模型进行验证，然后根据数值数字岩石物理基准进行验证。 具有 12GB RAM 的消费级显卡用于在数十分钟内表征多达大约 5.4 亿个体素的图像。在具有 4.49 亿个自由度 （DOF） 的网格中，基于 Stokes 流 FEM 的仿真可在 9 到 15 分钟内完成，分配的全局内存不到 10GB。最后，在具有 80GB RAM 的高端 GPU 上，在不到 2.5 小时内对三个 10003 碳纤维域（总计超过 37 亿个自由度）进行模拟，与流管渗透率实验非常接近。