Mishra M, Martin M, De Wit A
Nonlinear Physical Chemistry Unit and Center for Nonlinear Phenomena and Complex Systems, Faculté des Sciences, Université Libre de Bruxelles (ULB), CP 231 Campus Plaine, 1050 Brussels, Belgium.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Dec;78(6 Pt 2):066306. doi: 10.1103/PhysRevE.78.066306. Epub 2008 Dec 11.
When a sample of fluid of finite size is displaced in a porous medium by another miscible fluid, viscous fingering may occur when the two fluids have different viscosities. Depending whether the sample is more or less viscous than the carrier fluid, the log-mobility ratio R [defined as R=ln(mu_{2}mu_{1}) where mu_{2} and mu_{1} are the viscosities of the sample and of the carrier] is respectively positive or negative. In the case of a linear displacement of a finite slice of fluid, R>0 leads to fingering of the rear interface of the sample where the less viscous carrier invades the more viscous sample. If R<0 , it is on the contrary the frontal interface of the sample that develops fingers where the less viscous sample displaces the more viscous bulk solution. We investigate here numerically the differences in fingering dynamics between the positive and negative log-mobility ratio cases leading to the growth of fingers against or along the direction of the flow, respectively. To do so, we integrate Darcy's law coupled to a convection-diffusion equation for the concentration of a solute ruling the viscosity of the finite-size sample. The statistical moments of the solute's concentration distribution are analyzed as a function of dimensionless parameters of the problem such as the length of the slice l , the log-mobility ratio R , and the ratio between transverse and axial dispersion coefficients . We find that, on average, the mixing zones and the width of the sample broadening due to fingering are larger for negative R than for positive R . This is due to the fact that fingers travel quicker in the flow direction than against the flow. Relevance of our results are discussed for interpretation of experimental results obtained in chromatographic separation and for understanding conditions of enhanced spreading of contaminants in aquifers.
当有限大小的流体样本在多孔介质中被另一种可混溶流体驱替时，如果两种流体具有不同的粘度，就可能会出现粘性指进现象。根据样本的粘度相对于驱替流体是更高还是更低，对数迁移率比R[定义为R = ln(μ₂/μ₁)，其中μ₂和μ₁分别是样本和驱替流体的粘度]分别为正或负。在有限厚度流体切片的线性驱替情况下，R>0会导致样本后界面出现指进，此时粘度较低的驱替流体侵入粘度较高的样本。相反，如果R<0，则是样本的前界面出现指进，即粘度较低的样本驱替粘度较高的主体溶液。在此，我们通过数值方法研究正、负对数迁移率比情况下指进动力学的差异，这两种情况分别导致指状物朝着或逆着流动方向生长。为此，我们将达西定律与一个对流扩散方程联立，该对流扩散方程用于描述控制有限大小样本粘度的溶质浓度。分析溶质浓度分布的统计矩与问题的无量纲参数的函数关系，这些无量纲参数包括切片长度l、对数迁移率比R以及横向和轴向扩散系数之比。我们发现，平均而言，负R情况下因指进而导致的混合区和样本展宽宽度比正R情况下更大。这是因为指状物在流动方向上的移动速度比逆着流动方向更快。我们讨论了研究结果对于解释色谱分离实验结果以及理解含水层中污染物增强扩散条件的相关性。
