Laser-accelerated electron beams have been created at a kHz repetition rate from the {it reflection} of intense ($sim10^{18}$ W/cm$^2$), $sim$40 fs laser pulses focused on a continuous water-jet in an experiment at the Air Force Research Laboratory. This paper investigates Particle-in-Cell (PIC) simulations of the laser-target interaction to identify the physical mechanisms of electron acceleration in this experiment. We find that the standing-wave pattern created by the overlap of the incident and reflected laser is particularly important because this standing wave can inject electrons into the reflected laser pulse where the electrons are further accelerated. We identify two regimes of standing wave acceleration: a highly relativistic case ($a_0~geq~1$), and a moderately relativistic case ($a_0~sim~0.5$) which operates over a larger fraction of the laser period. In previous studies, other groups have investigated the highly relativistic case for its usefulness in launching electrons in the forward direction. We extend this by investigating electron acceleration in the {it specular (back reflection) direction} and over a wide range of intensities ($10^{17}-10^{19}$ W cm$^{-2}$).
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