Abstract This paper is concerned with the estimation of the effective transport characteristics of fluid-saturated porous media via rigorous microstructure-property relations. We are particularly interested in predicting the formation factor F , mean survival time τ, principal NMR (diffusion) relaxation time T1, principal viscous relaxation time Θ1, and fluid permeability k. To do so, we employ rigorous methods to estimate the fluid permeability and these other transport properties of “hyperuniform” and nonhyperuniform models of porous media from microstructural information. Disordered hyperuniform materials are exotic amorphous states of matter that have attracted great attention in the physical, mathematical and biological science but little is known about their fluid transport characteristics. In carrying out this investigation, we not only draw from ideas and results of the emerging field of hyperuniformity, but from homogenization theory, statistical geometry, differential equations (spectrum of Laplace and Stokes operators), and the covering and quantizer problems of discrete geometry. Among other results, we derive a Fourier representation of a classic rigorous upper bound on the fluid permeability that depends on the spectral density to infer how the permeabilities of hyperuniform porous media perform relative to those of nonhyperuniform ones. We find that the velocity fields in nonhyperuniform porous media are generally much more localized over the pore space compared to those in their hyperuniform counterparts, which has implications for their permeabilities. Rigorous bounds on transport properties suggest a new approximate formula for the fluid permeability that provides reasonably accurate permeability predictions of a certain class of hyperuniform and nonhyperuniform porous media. These comparative studies shed new light on the microstructural characteristics that determine the transport properties of general porous media. Our findings also have implications for the design of porous materials with desirable transport properties.

中文翻译：

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通过严格的微观结构-性质关系预测超均匀多孔介质的输运特性

摘要 本文涉及通过严格的微观结构-性能关系估计流体饱和多孔介质的有效输运特性。我们特别感兴趣的是预测地层因子 F 、平均生存时间 τ、主要 NMR（扩散）弛豫时间 T1、主要粘性弛豫时间 Θ1 和流体渗透率 k。为此，我们采用严格的方法从微观结构信息估计多孔介质的“超均匀”和非超均匀模型的流体渗透率和这些其他传输特性。无序超均匀材料是奇异的非晶态物质，在物理、数学和生物科学中引起了极大的关注，但对其流体传输特性知之甚少。在进行这项调查时，我们不仅借鉴了超均匀性新兴领域的思想和结果，还借鉴了同质化理论、统计几何、微分方程（拉普拉斯和斯托克斯算子的频谱）以及离散几何的覆盖和量化问题。在其他结果中，我们推导出了流体渗透率的经典严格上限的傅立叶表示，该上限取决于谱密度，以推断超均匀多孔介质的渗透率相对于非超均匀多孔介质的渗透率如何。我们发现，与超均匀多孔介质中的速度场相比，非超均匀多孔介质中的速度场通常更集中于孔隙空间，这对其渗透率有影响。输运特性的严格界限提出了一种新的流体渗透率近似公式，该公式为某类超均匀和非超均匀多孔介质提供了相当准确的渗透率预测。这些比较研究揭示了决定一般多孔介质传输特性的微观结构特征。我们的发现也对设计具有理想传输特性的多孔材料有影响。