Stream functions in three‐dimensional groundwater flow

Abstract

Development of the partial differential equation of stream function ψ in two‐dimensional groundwater flow is based on the assumption that the curl of hydraulic gradient vector (∇×∇ϕ) is equal to zero. The equation of ψ is expressed in terms of hydraulic conductivity. This equation is valid for steady state groundwater flow only. In this paper, two partial differential equations of stream functions χ and λ in three‐dimensional groundwater flow are developed. In development of these equations, it is assumed that the curl of pseudopotential gradient (∇×∇β) is equal to zero. Pseudopotential gradient is the hydraulic gradient in the direction of flow. The equations of χ and λ are expressed in terms of pseudopotential conductivity. Pseudopotential conductivity is the hydraulic conductivity in the direction of flow. It is a scalar function and can be defined in terms of hydraulic gradient and hydraulic conductivity. Therefore besides being valid for steady state flow, the equations of χ and λ can be applied to evaluate stream functions in anisotropic porous media at a given point in time, provided that the hydraulic gradient field is known at the point in time. Application of the χ and λ equations is demonstrated by considering steady state groundwater flow under the Borden landfill in Ontario, Canada. The two equations will be valuable in visualization of groundwater flow and contaminant transport, design of numerical solution grids for contaminant transport modeling, and design of contaminant remediation well systems.