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首页> 外文期刊>Applied Mathematical Modelling >New Stochastic Model For Dispersion In Heterogeneous Porous Media: 1. Application To Unbounded Domains
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New Stochastic Model For Dispersion In Heterogeneous Porous Media: 1. Application To Unbounded Domains

机译：异质多孔介质中分散的新随机模型：1.在无界域中的应用

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摘要

A new model of solute dispersion in porous media that avoids Fickian assumptions and that can be applied to variable drift velocities as in non-homogeneous or geometrically constricted aquifers, is presented. A key feature is the recognition that because drift velocity acts as a driving coefficient in the kinematical equation that describes random fluid displacements at the pore scale, the use of Ito calculus and related tools from stochastic differential equation theory (SPDE) is required to properly model interaction between pore scale randomness and macroscopic change of the drift velocity. Solute transport is described by formulating an integral version of the solute mass conservation equations, using a probability density. By appropriate linking of this to the related but distinct probability density arising from the kinematical SPDE, it is shown that the evolution of a Gaussian solute plume can be calculated, and in particular its time-dependent variance and hence dispersivity. Exact analytical solutions of the differential and integral equations that this procedure involves, are presented for the case of a constant drift velocity, as well as for a constant velocity gradient. In the former case, diffusive dispersion as familiar from the advection-dispersion equation is recovered. However, in the latter case, it is shown that there are not only reversible kinematical dispersion effects, but also irreversible, intrinsically stochastic contributions in excess of that predicted by diffusive dispersion. Moreover, this intrinsic contribution has a non-linear time dependence and hence opens up the way for an explanation of the strong observed scale dependence of dispersivity.
机译：提出了一种新的溶质在多孔介质中分散的模型，该模型避免了Fickian的假设，并且可以像在非均质或受几何约束的含水层中那样应用于可变的漂移速度。一个关键特征是认识到，由于漂移速度在描述孔隙尺度上随机流体位移的运动方程中起着驱动系数的作用，因此需要使用Ito演算和随机微分方程理论（SPDE）的相​​关工具来正确建模孔尺度随机性与漂移速度宏观变化之间的相互作用。通过使用概率密度公式化溶质守恒方程的积分形式来描述溶质的运移。通过将其适当地链接到由运动学SPDE产生的相关但独特的概率密度，可以看出，可以计算出高斯溶质羽流的演化，特别是其随时间变化的方差，从而可以计算出分散性。对于恒定的漂移速度以及恒定的速度梯度，给出了此过程涉及的微分方程和积分方程的精确解析解。在前一种情况下，恢复了对流扩散方程所熟悉的扩散扩散。然而，在后一种情况下，显示出不仅存在可逆的运动色散效应，而且还具有不可逆的，固有的随机贡献，其超过了扩散色散所预测的贡献。此外，这种内在的贡献具有非线性的时间依赖性，因此为解释所观察到的强分散性的比例依赖性开辟了道路。

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