A vertical dispersion model for solute exchange induced by underflow and periodic hyporheic flow in a stream gravel bed

Interstitial flows in stream gravel beds are driven by stream slope and controlled by hydraulic conductivity (underflows) or induced by pressure differentials on the streambed surface (hyporheic flows). They enhance solute exchange between surface water and a streambed. To study the solute transport in a stream gravel bed, a 2-D transient advection-dispersion mass transfer model was formulated. The velocity field includes an underflow and a spatially periodic hyporheic flow, e.g., due to standing surface waves or bed forms. Two dimensionless scaling parameters emerged: R measures the relative strength of hyporheic flow to underflow in the streambed and λ is the ratio of dispersivity of the gravel bed to the pressure wavelength along the streambed. In the analysis of mass transfer of nonconservative substances into a streambed, an explicit 2-D analysis of interstitial flow is often undesirable. Therefore the numerical solutions for the 2-D concentration fields under periodic boundary conditions were reduced to 1-D vertical concentration profiles (by streamwise averaging of the solute concentrations). The profiles were matched to the solution of an unsteady vertical 1-D dispersion equation that introduces a depth variable "enhanced dispersion coefficient D_E(y)" that lumps all forms of interstitial advective and dispersive transport in the streambed. Functions D_E(y) were determined for many combinations of independent parameters by inverse modeling, and the dependence of D _E(y) on the dimensionless parameters R and λ was determined from these results. Analytical relationships for D_E(y, R, λ) have been proposed and validated against available experimental data. Knowledge of D_E(y) allows the estimation of solute/mass transfer rates in streambeds under wavy boundary conditions, without explicit analysis of the interstitial flow. This is a distinct advantage for applications in stream water quality and/or pore water quality studies.