Why is there water flowing in a river when it isn’t raining?

In the absence of rain, most of the flow in a river is water that drains slowly from the ground.

As the groundwater is depleted, the flow rate in a river gradually recedes. The rate of recession is determined primarily by the underlying geology of the river basin. Some rivers, for example in the eastern Wairarapa where there is a tertiary mudstone geology, recede quickly and dry up in summer. Others, for example in the volcanic ash areas of the central North Island, have remarkably well sustained low flows. Some South Island rivers that drain the higher mountains of the Southern Alps are sustained in spring and summer by snowmelt, and experience lowest flows in winter when most of the precipitation is snow.

April 2011 river flows. [NIWA]