A longitudinal analysis of transient flow between a river and an underlying aquifer is developed to calculate flow rates between the river and the aquifer and the location of groundwater seepage into the river as it changes over time. Two flow domains are defined in the analysis: an upstream domain of fluvial recharge, where water flows vertically from the river into the unsaturated portion of the aquifer and horizontally in saturated parts of the aquifer, and a downstream domain of groundwater seepage to the river, where groundwater flows parallel to the underlying impermeable base. The river does not necessarily penetrate completely through the aquifer. A one-dimensional, unsteady flow equation is derived from mass conservation, Darcy's law, and the geometry of the river-aquifer system to calculate the water table position and the groundwater seepage rate into the river. Models based on numerical and analytical solutions of the flow equation were applied to a reach of the Methow River in north central Washington. The calibrated models simulated groundwater seepage with a root-mean-square error less than 5% of the mean groundwater seepage rates for three low-flow evaluation periods. The analytical model provides a theoretical basis for a nonlinear exponential base flow recession generated by a draining aquifer, but not an explicit functional form for the recession. Unlike cross-sectional approaches, the longitudinal approach allows the analysis of the length and location of groundwater seepage to a river, which have important ecological implications in many rivers. In the numerical simulations, the length of the groundwater seepage varied seasonally by about 4 km and the upstream boundary of groundwater seepage was within 689 m of its location at a stream gage on 9 September 2001 and within 91 m of its location on 6 October 2002. To demonstrate its utility in ecological applications, the numerical model was used to calculate differences in length of groundwater seepage to the Methow River under an early runoff scenario and the timing of those differences with respect to life stages of chinook salmon.