Darcy's Law and the Field Equations of the Flow of Underground Fluids

Published in Petroleum Transactions, AIME, Vol. 207, 1956, pages 222–239. Paper prepared for presentation before Darcy Centennial Hydrology Symposium of the International Association of Hydrology held in Dijon, France. Sept. 20–26, 1956, and for dual publication as part of the Dijon Symposium and in the Darcy Centennial Issue of Journal of Petroleum Technology. ABSTRACT In 1856 Henry Darcy described in an appendix to his book, Les Fontaines Publiques de la Ville de Dijon, a series of experiments on the downward flow of water through filter sands, whereby it was established that the rate of flow is given by the equation: q = − K (h2 − h1) /l. in which q is the volume of water crossing unit area in unit time. l is the thickness of the sand, h1 and h2 the heights above a reference level of the water in manometers terminated above and below the sand, respectively, and K a factor of proportionality. This relationship, appropriately, soon became known as Darcy's law. Subsequently many separate attempts have been made to give Darcy's empirical expression a more general physical formulation, with the result that so many mutually inconsistent expressions of what is purported to be Darcy's law have appeared in published literature that sight has often been lost of Darcy's own work and of its significance. In the present paper, therefore. it shall be our purpose to reinform ourselves upon what Darcy himself did, and then to determine the meaning of his results when expressed explicitly in terms of the pertinent physical variables involved. This will be done first by the empirical method used by Darcy himself, and then by direct derivation from the Navier-Stokes equation of motion of viscous fluids.