Liquids that flow in unsaturated rock fractures exhibit a range of behavior distinct from flow within the porous matrix. Flow patterns observed in laboratory experiments include aperture-spanning droplets, snapping and continuous capillary threads or rivulets, islands of local saturation, and fingering. Knowledge of the contact area and flow rate for each mode is necessary to achieve successful modeling of the fracture–matrix solute transfer. A geometric model for a rivulet cross section that includes the shape of the curved meniscus between the two walls of a fracture is used in this paper to calculate (i) the flow rate within the menisci corners, (ii) a critical flow rate that determines whether a rivulet will snap or be continuous, and (iii) the width of the wetted contact against a fracture wall. For low flow rates that are near the critical value between snapping and continuous conditions, the contact area was found to be up to 18 times greater than that predicted using a simpler rectangular geometry for the rivulet cross section. A wider contact area also suggests a slower mean flow and larger time constant for solute transfer.