Development of approximate solutions for contaminant transport through fractured media

Abstract

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) [1]. 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 Padé 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) [1]. In contrast, the solution based on low-order Padé 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 Padé 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) [1]. These results reveal that the Padé 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.