Reis Fabio D A Aarão, Voller Vaughan R
Instituto de Física, Universidade Federal Fluminense, Avenida Litorânea s/n, 24210-340 Niterói RJ, Brazil.
Department of Civil, Environmental, and Geo- Engineering, Saint Anthony Falls Laboratory, University of Minnesota, 500 Pillsbury Drive SE, Minneapolis, Minnesota 55455, USA.
Phys Rev E. 2019 Apr;99(4-1):042111. doi: 10.1103/PhysRevE.99.042111.
We study a random-walk infiltration (RWI) model, in homogeneous and in fractal media, with small localized sources at their boundaries. In this model, particles released at a source, maintained at a constant density value, execute unbiased random walks over a lattice; a model that represents solute infiltration by diffusion into a medium in contact with a reservoir of fixed concentration. A scaling approach shows that the infiltrated length, area, or volume evolves in time as the number of distinct sites visited by a single random walker in the same medium. This is consistent with numerical simulations of the lattice model and exact and numerical solutions of the corresponding diffusion equation. In a Sierpinski carpet, the infiltrated area is expected to evolve as t^{D_{F}/D_{W}} (Alexander-Orbach relation), where D_{F} is the fractal dimension of the medium and D_{W} is the random-walk dimension; the numerical integration of the diffusion equation supports this result with very good accuracy and improves results of lattice random-walk simulations. In a Menger sponge in which D_{F}>D_{W} (i.e., a fractal with a dimension close to 3), a linear time increase of the infiltrated volume is theoretically predicted and confirmed numerically. Thus, no evidence of fractality can be observed in measurements of infiltrated volumes or masses in media where random walks are not recurrent, although the tracer diffusion is anomalous. We compare our findings with results for a fluid infiltration model in which the pressure head is constant at the source and the front displacement is driven by the local gradient of that head. Exact solutions in two and three dimensions and numerical results in a carpet show that this type of fluid infiltration is in the same universality class of RWI, with an equivalence between the head and the particle concentration. These results set a relation between different infiltration processes with localized sources and the recurrence properties of random walks in the same media.
我们研究了一种随机游走渗透（RWI）模型，该模型适用于均匀介质和分形介质，且在其边界处有小的局部源。在这个模型中，从源释放的粒子保持恒定的密度值，在晶格上执行无偏随机游走；该模型表示溶质通过扩散渗透到与固定浓度储库接触的介质中。一种标度方法表明，渗透长度、面积或体积随时间的演化与单个随机游走者在同一介质中访问的不同位点数量有关。这与晶格模型的数值模拟以及相应扩散方程的精确解和数值解一致。在谢尔宾斯基地毯中，预计渗透面积将按(t^{D_{F}/D_{W}})（亚历山大 - 奥尔巴赫关系）演化，其中(D_{F})是介质的分形维数，(D_{W})是随机游走维数；扩散方程的数值积分以非常高的精度支持了这一结果，并改进了晶格随机游走模拟的结果。在(D_{F}>D_{W})的门格尔海绵（即维数接近3的分形）中，理论上预测并通过数值证实了渗透体积随时间呈线性增加。因此，在随机游走不常返的介质中，尽管示踪剂扩散是反常的，但在渗透体积或质量的测量中没有观察到分形性的证据。我们将我们的发现与一种流体渗透模型的结果进行比较，在该模型中，源处的压头恒定，前沿位移由该压头的局部梯度驱动。二维和三维的精确解以及地毯中的数值结果表明，这种类型的流体渗透与RWI属于同一普适类，压头和粒子浓度之间存在等效性。这些结果建立了具有局部源的不同渗透过程与同一介质中随机游走的常返性质之间的关系。
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