Quantum-well infrared photodetectors (QWIPs) have potential applicability in many remote-sensing applications, even in the space environment where low background fluxes are involved. In this environment, the detector arrays may need to be operated at temperatures lower than 77 K. At these temperatures, tunneling mechanisms such as Fowler-Nordheim and trap-assisted tunneling could dominate the dark current. The device resistance, which is bias-dependent, increases by several orders of magnitude at these temperatures and may pose a problem. We have seen offsets in the current-versus-voltage (I-V) characteristics (nonzero current at zero bias not associated with dopant migration) which could impair the compatibility of a QWIP array with a readout circuit. We propose that these offsets are due to an RC time- constant effect. We further propose that the resistance in this time constant is due to tunneling mechanisms (and not due to contact resistance), which in turn are structure- and bias-dependent. We discuss our observations and present a circuit model of a QWIP that explains these observations nearly completely.