Computational Mathematics Group, Physical & Computational Sciences Directorate, Pacific Northwest National Laboratory, Richland, Washington 99354, USA.
Phys Rev E. 2017 Sep;96(3-1):033314. doi: 10.1103/PhysRevE.96.033314. Epub 2017 Sep 28.
This work presents a method of model reduction that leads to models with three solutions of increasing fidelity (multifidelity models) for solute transport in a bounded layered porous media with random permeability. The model generalizes the Taylor-Aris dispersion theory to stochastic transport in random layered porous media with a known velocity covariance function. In the reduced model, we represent (random) concentration in terms of its cross-sectional average and a variation function. We derive a one-dimensional stochastic advection-dispersion-type equation for the average concentration and a stochastic Poisson equation for the variation function, as well as expressions for the effective velocity and dispersion coefficient. In contrast to the linear scaling with the correlation length and the mean velocity from macrodispersion theory, our model predicts a nonlinear and a quadratic dependence of the effective dispersion on the correlation length and the mean velocity, respectively. We observe that velocity fluctuations enhance dispersion in a nonmonotonic fashion (a stochastic spike phenomenon): The dispersion initially increases with correlation length λ, reaches a maximum, and decreases to zero at infinity (correlation). Maximum enhancement in dispersion can be obtained at a correlation length about 0.25 the size of the porous media perpendicular to flow. This information can be useful for engineering such random layered porous media. Numerical simulations are implemented to compare solutions with varying fidelity.
这项工作提出了一种模型降阶方法，该方法针对具有随机渗透率的有界层状多孔介质中溶质输运的三种保真度（多保真度模型）的模型。该模型将 Taylor-Aris 弥散理论推广到具有已知速度协方差函数的随机层状多孔介质中的随机输运。在降阶模型中，我们用其横截面平均值和变化函数表示（随机）浓度。我们推导出了平均浓度的一维随机对流-弥散型方程和变化函数的随机泊松方程，以及有效速度和弥散系数的表达式。与宏观弥散理论中与相关长度和平均速度呈线性比例的情况相比，我们的模型分别预测有效弥散与相关长度和平均速度呈非线性和二次依赖关系。我们观察到速度波动以非单调的方式增强弥散（随机尖峰现象）：弥散最初随相关长度 λ 增加，达到最大值，并在无穷远处（相关）减小到零。在大约垂直于流动的多孔介质尺寸的 0.25 处，可以获得最大的弥散增强。这些信息对于工程设计这种随机层状多孔介质可能是有用的。实施了数值模拟以比较不同保真度的解。
