Sparse digital antenna arrays constitute a promising detection technique for future large-scale cosmic-ray observatories. It has recently been shown that this kind of instrumentation can provide a resolution of the energy and of the shower maximum on the level of other cosmic-ray detection methods. Due to the dominant geomagnetic nature of the air-shower radio emission in the traditional frequency band of 30 to 80 MHz, the amplitude and polarization of the radio signal strongly depend on the azimuth and zenith angle of the arrival direction. Thus, the estimation of the efficiency and subsequently of the aperture of an antenna array is more complex than for particle or Cherenkov-light detectors. We have built a new efficiency model based on utilizing a lateral distribution function as a shower model, and a probabilistic treatment of the detection process. The model is compared to the data measured by the Tunka Radio Extension (Tunka-Rex), a digital antenna array with an area of about 1 km$^2$ located in Siberia at the Tunka Advanced Instrument for Cosmic rays and Gamma Ray Astronomy (TAIGA). Tunka-Rex detects radio emission of air showers using trigger from air-Cherenkov and particle detectors. The present study is an essential step towards the measurement of the cosmic-ray flux with Tunka-Rex, and is important for radio measurements of air showers in general.