The Theory of Groundwater Motion


M. King Hubbert


The existing analytical treatments of ground-water flow have mostly been founded upon the erroneous conception, 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 exceptionless analytical theory results if a potential whose value at a given point is defined to be equal to the work required to transform a unit mass of fluid from an arbitrary standard state to the state at the point in question is employed. Denoting this function by $\Phi$, it is shown that the differential equation of fluid flow in an isotropic medium is given by $q = – \sigma$ grad $\Phi$, 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 $\sigma$ a specific conductivity parameter depending upon both the properties of the fluid and 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. It is shown that $\sigma = \kappa\rho/\eta$, where $\kappa$ is the permeability parameter depending upon the geometrical properties of the medium only and $\rho$ and $\eta$ are the density and viscosity, respectively, of the fluid. The remainder of the paper is devoted to deducing the consequences of Darcy’s law as just expressed, with particular regard for the practical problems of ground-water hydrology.