In the last decade, the interest towards fluids characterized by a complex rheology has increased in the scientific community. Physicochemical, rheological and fluid mechanical approaches are adopted to characterize the peculiarities of non-Newtonian fluids. These fluids show remarkable properties that can be exploited to improve remediation techniques or optimize industrial operations. While the existence of actual yield stress is still debated, the presence of a plug or pseudo-plug zone is frequent in emulsions, soft glassy materials, jammed non-colloidal suspensions or colloidal gels. Even if simple yield stress models are often faulted because of their acknowledged deficiencies, they allow to study important phenomena without introducing excessive complexity. In this study, the randomness describing the aperture field of a natural rock fracture is coupled with a bi-viscous fluid rheology of parameter ϵ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon$$\end{document}, representing the viscosity ratio; the cases ϵ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0$$\end{document} and ϵ→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow {1}$$\end{document} represent the Bingham and Newtonian behaviour, respectively. The conceptual model proposed describes the flow of such fluids through a fracture with aperture variable along a single direction, the aperture being constant along the other. The aperture variation is modelled via a generic probability distribution function of assigned mean and variance. Two limit flow cases are considered: (1) parallel arrangement (PA), representing the case of maximum conductance, with the fluid flowing in the direction of channels of constant aperture; and (2) series arrangement (SA), the case of minimum conductance, with flow directed orthogonally to the constant aperture side of the fracture. Results are illustrated for log normal and gamma aperture distributions. The influence of aperture variability and applied pressure gradient on flow rate is investigated for both arrangements. The pressure gradient affects in a nonlinear fashion the flow rate, with a marked increase around a threshold value for both PA and SA. The channel flow rate exhibits a direct dependency upon aperture variability for PA, an inverse one for SA. The shape of the distribution has an impact on model responses: for the PA, the influence is significant but limited to an intermediate threshold range of pressure gradients, while results for the SA are affected in the whole range of pressure gradient. An example application in dimensional form is included.

中文翻译：

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裂缝中双粘性流体流动的通道模型

在过去的十年中，科学界对具有复杂流变学特征的流体越来越感兴趣。采用物理化学、流变学和流体力学方法来表征非牛顿流体的特性。这些流体显示出卓越的特性，可用于改进修复技术或优化工业运营。虽然实际屈服应力的存在仍有争议，但在乳液、软玻璃材料、堵塞的非胶体悬浮液或胶体凝胶中，塞或假塞区的存在很常见。即使简单的屈服应力模型由于其公认的缺陷而经常出错，它们也允许研究重要现象而不会引入过多的复杂性。在这项研究中，ϵ=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength {\oddsidemargin}{-69pt} \begin{document}$$\epsilon =0$$\end{document} 和 ϵ→1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage {amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\epsilon \rightarrow {1}$$\ end{document} 分别代表宾厄姆和牛顿行为。所提出的概念模型描述了此类流体通过裂缝的流动，裂缝的孔径沿单个方向可变，而孔径沿另一个方向不变。孔径变化通过指定均值和方差的通用概率分布函数建模。考虑了两种极限流动情况： (1) 平行排列 (PA)，代表最大电导的情况，流体沿恒定孔径的通道方向流动；(2) 串联排列 (SA)，在最小电导的情况下，流动方向垂直于裂缝的恒定孔径侧。说明了对数正态分布和伽马孔径分布的结果。研究了两种布置的孔径可变性和施加的压力梯度对流速的影响。压力梯度以非线性方式影响流速，PA 和 SA 在阈值附近显着增加。通道流速表现出对 PA 的孔径可变性的直接依赖，而对 SA 则相反。分布的形状对模型响应有影响：对于 PA，影响是显着的，但仅限于压力梯度的中间阈值范围，而 SA 的结果在整个压力梯度范围内都会受到影响。包括尺寸形式的示例应用程序。