Some Mathematical Problems Associated with Hyporheic Flow

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1715504Wang The supply of fresh water is under pressure for many reasons, including natural variability, population growth, expansion of business activities, rapid urbanization, climate change, depletion of aquifers, and pollution. Therefore, there is an urgent need to study water resources, especially unfrozen fresh water in terms of surface water and groundwater. Surface water and groundwater interact with each other. A prime example is the hyporheic zone, a portion of the bed and bank of a river or stream where surface water and groundwater mix, exchanging solutes and water at various scales. The hyporheic zone is critical to the ecology of river corridors. In particular, it is important in controlling the flux and location of water exchange between stream and subsurface; providing a habitat for benthic and interstitial organisms; providing a spawning ground and refuge for certain species of fish; providing a rooting zone for aquatic plants; serving as an important zone for cycling of carbon, energy, and nutrients; providing a natural attenuation zone for certain pollutants by biodegradation, sorption, and mixing; and moderating river water temperature. The purpose of this project is to develop and analyze better models of flows in the hyporheic zone in order to improve our understanding of the fundamental physical processes associated with hyporheic flows. Graduate students participate in the work of the project. The investigator and colleagues study the Navier-Stokes-Darcy-heat system that models coupled surface water-groundwater interaction together with thermal effects. The mathematical model presents several challenges: the strong nonlinearity associated with the Navier-Stokes flow in the river, the complications arising from including physically and biologically important thermal effects, the substantial disparity of spatial and temporal scales between flows in the river and in surrounding porous media (small Darcy number), the uncertainty associated with the geometric form and the permeability of the riverbed and riverbank, and the different physics in different parts of the physical domain. Although these difficult issues have been studied separately before for some subsystems of the models under consideration here, the need for a better integration of the physical and biological factors relevant to flow and transport in the hyporheic zone requires a more comprehensive model. The work of the investigator proceeds in steps. First is mathematical analysis of the model in terms of asymptotic behavior in the physically important small Darcy number regime. Second, he designs, analyzes, and implements accurate and efficient decoupled numerical methods for the model so that the results can be compared to experiments performed by collaborators. Third, he uses the model to assess the validity of various simplifications utilized by the water resources community in hyporheic flow studies. Graduate students participate in the work of the project.