We investigate through numerical simulations of the Navier-Stokes equations the influence of the surface roughness on the fluid flow through fracture joints. Using the Hurst exponent $H$ to characterize the roughness of the self-affine surfaces that constitute the fracture, our analysis reveal the important interplay between geometry and inertia on the flow. Precisely, for low values of Reynolds numbers Re, we use Darcy’s law to quantify the hydraulic resistance $G$ of the fracture and show that its dependence on $H$ can be explained in terms of a simple geometrical model for the geometric tortuosity $\tau$ of the channel. At sufficiently high values of Re, when inertial effects become relevant, our results reveal that nonlinear corrections up to third-order to Darcy’s law are aproximately proportional to $H$. These results imply that the resistance $G$ to the flow follows a universal behavior by simply rescaling it in terms of the fracture resistivity and using an effective Reynolds number, namely, Re/$H$. Our results also reveal the presence of quasi-one-dimensional channeling, even considering the absence of shear displacement between upper and lower surfaces of the self-affine fracture.

中文翻译：

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流过三维自仿射裂缝

通过Navier-Stokes方程的数值模拟，我们研究了表面粗糙度对流体流过断裂缝的影响。使用赫斯特指数$H$为了表征构成裂缝的自仿射表面的粗糙度，我们的分析揭示了几何形状和流动惯性之间的重要相互作用。精确地，对于雷诺数Re的低值，我们使用达西定律来量化水力阻力$G$ 并显示其对 $H$ 可以用简单的几何曲折几何模型来解释 $\tau$频道的 在足够高的Re值下，当惯性效应变得相关时，我们的结果表明，达西定律的三阶非线性校正近似成比例于$H$。这些结果表明抗性$G$ 通过简单地根据裂缝的电阻率重新缩放比例并使用有效的雷诺数（即Re /$H$。我们的结果还揭示了准一维通道的存在，即使考虑到自仿射裂缝的上下表面之间没有剪切位移。