On the theoretical derivation of Darcy and Forchheimer formulas

Abstract

Darcy's law q = KJ(q = vector of specific discharge; J = — grad E = hydraulic gradient; E — energy of flow; K — hydraulic conductivity of porous medium) is historically reviewed as well as its numerous extensions. The author starts from the Navier‐Stokes hydrodynamic equations of viscous flow, and using statistical methods (space averaging) shows that: The head (or energy) used in Darcy law is different from that used in viscous flow; which recommends large piezometric intakes. The formula J = aq + bqq + c∂q/∂t is obtained (a, b, c = coefficients in isotropic media). Usually the third term may be neglected. At low values of Reynolds numbers, Darcy's law is obtained with a = 1/K in the Kozeny‐Carman form. It is found that a = the hydraulic resistivity is more important than K. At larger values of Reynolds number, the Forchheimer formula is obtained, with b depending on the soil texture (grain diameter d) and porosity, not on temperature nor viscosity. The Laplace equation is replaced by a non linear Poisson‐type equation with K = f(J). Darcy's law then becomes physically meaningless. A minimum principle is found: Darcy flow is such that the kinetic energy or the dissipation becomes minimum. The application of the Navier‐Stokes equations to rocks with seams or cracks gives Darcy's law in its tensorial form.