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Development of approximate solutions for contaminant transport through fractured media. (English) Zbl 1432.74064
Summary: 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 [D. H. Tang et al., “Contaminant transport in fractured porous media: analytical solution for a single fracture”, Water Resources Res. 17, No. 3, 555–564 (1981; doi:10.1029/WR017i003p00555)]. 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 [loc. cit.]. 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 [loc. cit.]. 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.
MSC:
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
44A10 Laplace transform
65R20 Numerical methods for integral equations
74G10 Analytic approximation of solutions (perturbation methods, asymptotic methods, series, etc.) of equilibrium problems in solid mechanics
74R10 Brittle fracture
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