Transactions American Geophysical Union · 1940 · 110 citations · 0 references
EngineeringEarth SciencesFluid DynamicsFluid MechanicsHydraulicsDarcy FlowFluid PropertiesHydraulic ConductivityPotential FlowDarcy's LawDifferential Equation FormulationHydrodynamicsHydrodynamics Of Moving WaterPotential Flow Theory
Existing analytical treatments of groundwater flow are based on the limited‑validity concept that motion derives from a velocity potential, which aligns with conservation of matter but not energy. The paper demonstrates that adopting a potential defined as the work required to transform unit mass of fluid from a standard state to the point’s state yields a more exception‑less analytical theory of groundwater motion. The authors define this potential as the work needed to transform unit mass of fluid from a standard state to the point’s state. The derived potential Φ leads to the differential equation q = -σ grad Φ, expressing Darcy’s law, which is mathematically analogous to Ohm’s law and yields equivalent deductions in analogous situations.
The existing analytical treatments of ground‐water flow have mostly been founded upon the conception of limited validity, borrowed from the theory of the flow of the ideal frictionless fluids of classical hydrodynamics, that ground‐water motion is derivable from a velocity‐potential. This conception is in conformity with the principle of the conservation of matter but not with that of the conservation of energy. In the present paper it is shown that a more exception‐less analytical theory results if a potential whose value at a given point is defined to be equal to the work required to transform unit‐mass of fluid from an arbitrary standard state to the state at the point in question, is employed. Denoting this function by Φ, it is shown that the differential equation of fluid flow in an isotropic medium is given by q = ‐ σ grad Φ, where q is the flow‐vector whose magnitude is equal to the volume of fluid crossing a unit of area normal to the flow‐direction in unit‐time, and σ a specific conductivity‐parameter depending upon the properties both of the fluid and of the medium. This is an expression of Darcy's law and is physically, as well as mathematically, analogous to Ohm's law in electricity and leads to the same deductions in analogous situations.