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Numerical resolution of moderately high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality for the classical numerical methods including finite difference, finite element and spectral methods. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by tracking the deterministic motion, random jump, resampling and reweighting of particles. Real-valued weighted particles are adopted by SPM to approximate the high-dimensional solution, which automatically adjusts the point distribution to intimate the relevant feature of the solution. A piecewise constant reconstruction with virtual uniform grid is employed to evaluate the nonlinear terms, which fully exploits the intrinsic adaptive characteristic of SPM. Combining both, SPM can achieve the goal of adaptive sampling in time. Numerical experiments on the 6-D Allen-Cahn equation and the 7- D Hamiltonian-Jacobi-Bellman equation demonstrate the potential of SPM in solving moderately high-dimensional nonlinear PDEs efficiently while maintaining an acceptable accuracy
The Hydraulic Fracturing Test Site 1 (HFTS-1) was a field study performed in the Wolfcamp Formation in the West Texas Permian (Midland) Basin, USA, with a focus on improving the efficiency of hydraulic fracturing. Investigating site-specific rock-fluid geochemical interactions during hydraulic fracturing is an important step to understanding the impact on formation shale porosity, permeability, and long-term shale gas production. During field operations in this region, hydraulic fracturing fluid (HFF) injection usually starts with a concentrated acid spearhead for rapid rock dissolution, followed by the injection of near-neutral pH slickwater containing chemicals and proppants. A multistep sequential injection approach was used to investigate different stages of rock-fluid interactions. The carbonate content in the host rock is important when acid spearhead is considered, as carbonate mineral dissolution is rapid and can result in porosity and permeability changes in the shale matrix. In this study, we designed flow-through experiments using fractured carbonate-rich and clay-rich Wolfcamp shale cores with (1) a short-time acid soaking step and (2) a long-term slickwater flow-through step to simulate the injection method used at HFTS-1. The fluid chemistry was analyzed. A thorough mineralogical progression [e.g., Calcium (Ca) dissolution and iron (Fe) redox progression] in the cores during HFF injection was also characterized and imaged by synchrotron microprobe. Reactive transport modeling was performed based on the experimental setup. The results showed that the acid spearhead is a crucial step in creating a reaction front by mineral dissolution, especially in carbonate-rich shales. A slight layer of ferrihydrite precipitated during the slickwater flow-through period. This study provides insights into potential geochemical impact due to hydraulic fracturing operations in the Permian Basin.
This paper addresses the methodological challenges of comparing the iron industries in the Qin-Han and Roman empires by creating "modeling domains" as a pragmatic and utilitarian approach. These domains, built from literary and archaeological evidence, represent generalized rules and frameworks, paired with diachronic, fragmented landscapes that depict the progressive acquisition and integration of lands with established metallurgical traditions. The paper argues that simply reaching this step is not enough, as each domain should be understood as part of a larger aggregative set, with an "external" dimension. The paper further discusses the distancing effect and the need for caution in cross-domain discussions, emphasizing the importance of historical and social specificity. The Roman-Parthian and Han-Nanyue examples are used to illustrate these challenges and opportunities. The paper concludes that the comparative approach should be ever-expanding, leading to a continual dialogue between domains and a deeper understanding of the dynamics of control, trade, and technological exchange in different historical and social contexts.