Chugunov Vladimir, Fomin Sergei
Moscow City University, Moscow, Russia.
California State University, Chico, CA, USA.
Philos Trans A Math Phys Eng Sci. 2020 May 29;378(2172):20190283. doi: 10.1098/rsta.2019.0283. Epub 2020 May 11.
Reservoir contamination by various contaminants including radioactive elements is an actual environmental problem for all developed countries. Analysis of mass transport in a complex environment shows that the conventional diffusion equation based on Fick's Law fails to model the anomalous character of the diffusive mass transport observed in the field and laboratory experiments. These complex processes can be modelled by non-local advection-diffusion equations with temporal and spatial fractional derivatives. In the present paper, fractional differential equations are used for modelling the transport of radioactive materials in a fracture surrounded by the porous matrix of fractal structure. A new form of fractional differential equation for modelling migration of the radioactive contaminant in the fracture is derived and justified. Solutions of particular boundary value problems for this equation were found by application of the Laplace transform. Through the use of fractional derivatives, the model accounts for contaminant exchange between fracture and surrounding porous matrix of fractal geometry. For the case of an arbitrary time-dependent source of radioactive contamination located at the inlet of the fracture, the exact solutions for solute concentration in the fracture and surrounding porous medium are obtained. Using the concept of a short memory, an approximate solution of the problem of radioactive contaminant transport along the fracture surrounded by the fractal type porous medium is also obtained and compared with the exact solution. This article is part of the theme issue 'Advanced materials modelling via fractional calculus: challenges and perspectives'.
包括放射性元素在内的各种污染物对储层的污染，是所有发达国家都面临的一个现实环境问题。对复杂环境中质量传输的分析表明，基于菲克定律的传统扩散方程无法对在现场和实验室实验中观察到的扩散质量传输的异常特性进行建模。这些复杂过程可以用具有时间和空间分数阶导数的非局部对流扩散方程来建模。在本文中，分数阶微分方程被用于模拟放射性物质在由分形结构的多孔基质包围的裂缝中的传输。推导并论证了一种用于模拟裂缝中放射性污染物迁移的新形式的分数阶微分方程。通过应用拉普拉斯变换求出了该方程特定边值问题的解。通过使用分数阶导数，该模型考虑了裂缝与周围分形几何多孔基质之间的污染物交换。对于位于裂缝入口处的任意随时间变化的放射性污染源的情况，得到了裂缝和周围多孔介质中溶质浓度的精确解。利用短记忆概念，还得到了放射性污染物在由分形类型多孔介质包围的裂缝中传输问题的近似解，并与精确解进行了比较。本文是主题为“通过分数阶微积分进行先进材料建模：挑战与展望”的一部分。
