The depth-averaged (2D) lubrication theory is often adopted to simulate Newtonian flow in rough fractures. This approach, which is computationally much less expensive than using 3D CFD solvers, allows addressing large ensembles of stochastic fracture realizations. For creeping flow, the degree of approximation introduced is limited as long as the apertures vary relatively smoothly. We propose the first generalization of this approach addressing the flow of fluids whose rheology, described by the Ellis model, is shear-thinning (ST) above a crossover shear stress and Newtonian (of viscosity μ₀) below. The resulting nonlinear Reynolds equation for pressures is solved for a vast range of realistic rheological parameter values using a novel and specifically designed finite volume-based numerical model. The spatial discretization takes inspiration from the graph p-Laplacian to yield a symmetric Newton Jacobian, allowing for a highly efficient inexact implementation of the preconditioned conjugate gradient-based Newton-Krylov method. This is combined with a parameter continuation strategy to increase code robustness and ensure global convergence for flow indices as low as 0.1 with an excellent efficiency. This original solver is used to investigate realistic synthetic rough fracture geometries, which exhibits both self-affinity and a correlation length. The results show that the ST rheology mitigates the effects of aperture heterogeneities, increasing fracture transmissivity by several orders of magnitudes as compared to the Newtonian flow of viscosity μ₀ if the imposed macroscopic gradient is sufficiently large, and even rendering the rough fracture up to 10 times more permeable than a smooth fracture of identical mean aperture.

中文翻译：

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基于润滑的粗糙裂缝剪切稀化流动求解器

通常采用深度平均（2D）润滑理论来模拟粗糙裂缝中的牛顿流动。这种方法在计算上比使用 3D CFD 求解器便宜得多，允许解决随机裂缝实现的大型集合。对于蠕动流，只要孔径变化相对平稳，引入的近似程度就会受到限制。我们提出了这种方法的第一个推广，该方法解决了由 Ellis 模型描述的流体流动，其流变性在交叉剪切应力和牛顿（粘度μ ₀） 以下。使用新颖且专门设计的基于有限体积的数值模型，针对大量实际流变参数值求解所得的非线性雷诺压力方程。空间离散化的灵感来自于图p-拉普拉斯算子产生一个对称的牛顿雅可比行列，允许高效不精确地实现基于预条件共轭梯度的牛顿-克雷洛夫方法。这与参数延续策略相结合，以提高代码的鲁棒性并确保流指数低至 0.1 的全局收敛性具有出色的效率。这个原始求解器用于研究真实的合成粗糙裂缝几何形状，它表现出自亲和力和相关长度。结果表明，与粘度为μ ₀的牛顿流动相比，ST 流变学减轻了孔径不均匀性的影响，将裂缝透射率提高了几个数量级如果施加的宏观梯度足够大，甚至使粗糙裂缝的渗透性比相同平均孔径的光滑裂缝高 10 倍。