A hydrodynamic model, based on a spectral multidomain penalty method (SMPM), is proposed to study the dissipation of the velocity fluctuations at the hyporheic zone. The model is based on the one dimensional Navier Stokes equations, assuming incompressibility and a hydrostatic approximation. The spatial discretization of the governing equations is done via the SMPM that is a multidomain collocation approach based on discontinuous non-overlapping subdomains that are connected by a penalty term that ensures stability of the solution by imposing weak continuity at the subdomain interfaces. The temporal discretization of the equations is handled with a high-order fractional time step technique. A spectral filter is used to stabilize the solution when spurious oscillations appear in the solution. A sinusoidal pulse is imposed at the water/sediment interface as a boundary condition, and the flow dynamics was evaluated under different conditions of kinematic viscosity. A progressive damping of the velocity fluctuations was observed in all cases, until full dissipation is reached. At that point, the flow can be governed by Darcy's law.