Equation for one dimensional vertical flow of groundwater. 1. The rigorous theory

A new mathematical derivation of the one‐dimensional flow equation in an elastic, saturated, porous medium is presented. The approach involving the consideration of a fixed elemental volume in fixed coordinates is developed by starting from both Lagrangian and Eulerian definitions of the position vector. The confusion existing about these two fundamental points of view has so far led to theoretically erroneous results. The Lagrangian and Eulerian formulations prove to be equivalent and provide the same outcome if they are correctly interpreted and consistently applied. The rigorous equation, compared with Cooper's (1966) equation, turns out to contain an additional nonlinearity resulting from the correct expansion of the partial spatial derivative of the grain velocity. It is also shown that an approach based on a deforming element in fixed coordinates is simple and straightforward, since it does not introduce the grain velocity into the development. However, it needs a particular definition for the compressibility different from the classical one. It is proved that these two compressibilities are not equal; their mathematical link is derived. Copyright 1973 by the American Geophysical Union.