The spheromak has a toroidal magnetic fusion geometry embedded in a simply connected flux conserver, which would simplify a reactor design if energy confinement were sufficiently good. Helicity injection is an efficient current-drive mechanism used in many spheromak experiments but has been shown theoretically and computationally to open the magnetic flux surfaces. Experiments are consistent with this: excellent confinement has been demonstrated in slowly decaying spheromaks with good surfaces, but it is poor in driven spheromaks. One option to overcome this limitation is to operate in a 'refluxing' mode in which the current drive and confinement phases of operation are separated in time. After the toroidal current and magnetic field have partially decayed, the current drive would be turned back on to rebuild the magnetic field. The physics of refluxing operation is explored in the resistive-MHD approximation using simulations and an integrated-physics model. The results suggest that sustaining the confinement phase for an extended time will require an auxiliary means of controlling the safety-factor profile.