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Development of approximate solutions for contaminant transport through fractured media

机译：开发污染物通过裂隙介质传输的近似解决方案

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Approximate solutions are sometimes very convenient and useful in engineering practices if the analytical solution is in a complicated form and difficult to evaluate accurately. This study develops four different approximate solutions for the problem of contaminant transport in fractured media presented in Tang et al. (1981). Their problem was solved analytically and the solutions of concentration distributions in the fracture and the rock expressed in infinite integrals had to rely on numerical approaches to obtain the results. The approximate solutions we develop herein include small-time solution, large-time solution, low-order approximate solution and high-order one based on the Pade approximation technique. The small-time solution gives very accurate concentrations at early times while the large-time solution yields excellent predictions at late times, as compared to Tang et al.'s solution (Tang et al., 1981). In contrast, the solution based on low-order Pade approximation with polynomials of degree one in the numerator and degree two in the denominator gives fairly good predictions over the entire time domain, especially in the intermediate period as compared with those of the small-time and large-time solutions. In addition, the solution based on high-order Pade approximation with polynomials of degree two in the numerator and degree three in the denominator is also developed and its predicted concentrations are also compared with Tang et al.'s solution (Tang et al., 1981). These results reveal that the Pade approximation has an advantage of being capable of producing more accurate results than the relationships of SPLT and LPST in the intermediate and late time periods.
机译：如果分析解决方案形式复杂且难以准确评估，则有时近似解在工程实践中会非常方便且有用。这项研究针对Tang等人提出的裂缝介质中污染物运移问题开发了四种不同的近似解。 （1981）。他们的问题通过解析得到解决，裂缝和岩石中浓度分布的解（用无穷积分表示）必须依靠数值方法来获得结果。我们在此开发的近似解包括基于Pade近似技术的小时间解，大时间解，低阶近似解和高阶解。与Tang等人的解决方案相比（Tang等人，1981），小型解决方案在早期提供了非常准确的浓度，而大型解决方案在后期提供了出色的预测。相比之下，基于低阶Pade逼近的解决方案具有分子一级的分母和分母二级的多项式，在整个时域内，特别是在中间时段，与小时间的相比，给出了相当好的预测和大型解决方案。此外，还开发了基于高阶Pade逼近的解决方案，其中分子的阶数为2，分母的阶数为3的多项式，并将其预测浓度与Tang等人的解决方案进行了比较（Tang等人， 1981）。这些结果表明，在中间和后期，Pade近似具有比SPLT和LPST的关系更精确的结果。

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