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Four-stage single-well push-pull (SWPP) tracer tests, including injection, chasing, resting, and pumping, were conducted in a fractured aquifer at Newark basin. An anomalous transport phenomenon observed in the SWPP tests is the linear decline of breakthrough curves (BTCs) at late time with slope of -1.8 in log-log plots. A time-dependent fractional model is developed to interpret the anomalous transport behavior. This model considers a time-dependent power law memory function and a time-dependent fractional advection-dispersion operator. The fractional advection-dispersion equations (fADE) are solved in a radial coordinate system using the implicit Euler method. A semi-analytical solution of the first-order rate-limited mobile-immobile model (FORMIM) is derived for comparison. It is found that both the nonlocal transport in time and space can produce the long-tailed BTC. A smaller time-fractional or space-fractional index leads to a lower peak concentration and a larger late-time slope. The mass distribution of the fractional-in-space (FS) model exhibits power law decline at the leading plume edge. Early breakthrough during pumping is not observed because the mobile mass at the start of pumping is nonzero and more concentrated near the wellbore. The capacity ratio is an important factor that affects the peak concentration. A larger capacity ratio leads to greater peak concentration. A smaller time-fractional index in the injection, chasing, or resting stage will move the BTC downward and the slope of the late time BTC is determined by the space-fractional index over all stages and the time-fractional index in the pumping stage. The capability of the existing models to recover the BTC of the SWPP test is discussed and some guidelines for how to choose the appropriate model to interpret the SWPP test data are proposed.