Figure from Hornberger et al. (1998) Darcy’s data for two different sands

Figure from Hornberger et al. (1998) Range in hydraulic conductivity, K 13 orders of magnitude

Figure from Hornberger et al. (1998)

Generalization of Darcy’s column  h/L = hydraulic gradient q = Q/A Q is proportional to  h/L

q is a vector q x z qx 1qx 1 qz 2qz 2 zz xx q = Q/A In general: K z < K x, K y

q = - K grad h

Vector Form of Darcy’s Law q = - K grad h q = specific discharge (L/T) K = hydraulic conductivity (L/T) grad h = hydraulic gradient (L/L) h = head (L)

q = - K grad h K is a tensor with 9 components (three of which are K x, K y, K z ) q is a vector with 3 components h is a scalar

Scalar 1 component MagnitudeHead, concentration, temperature Vector 3 components Magnitude and direction Specific discharge, (& velocity), mass flux, heat flux Tensor 9 components Magnitude, direction and magnitude changing with direction Hydraulic conductivity, Dispersion coefficient, thermal conductivity

q = - K grad h Darcy’s law grad h qequipotential line grad hq Isotropic Kx = Ky = Kz = K Anisotropic Kx, Ky, Kz

Figure from Hornberger et al. (1998) Linear flow paths assumed in Darcy’s law True flow paths Average linear velocity v = Q/An= q/n n = effective porosity Specific discharge q = Q/A

Representative Elementary Volume (REV) REV Equivalent Porous Medium (epm) q = - K grad h

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) +R* (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) – R * Water balance equation

Inflow = Outflow Recharge Discharge Steady State Water Balance Equation Transient Water Balance Equation Inflow = Outflow +/- Change in Storage Outflow - Inflow = Change in Storage

Figures from Hornberger et al. (1998) Unconfined aquifer Specific yield = S y Confined aquifer Storativity = S b hh hh Storage Terms S =  V / A  h S = S s b S s = specific storage

 x  y  z = change in storage OUT – IN = = -  V/  t S s =  V / (  x  y  z  h)  V = S s  h (  x  y  z) tt tt W REV S =  V / A  h S s = S/b here b =  z

OUT – IN =

Law of Mass Balance + Darcy’s Law = Governing Equation for Groundwater Flow div q = - S s (  h  t) +W (Law of Mass Balance) q = - K grad h (Darcy’s Law) div (K grad h) = S s (  h  t) – W

2D unconfined: 2D confined: (S = S s b & T = K b)

Figures from: Hornberger et al., Elements of Physical Hydrology, The Johns Hopkins Press, Baltimore, 302 p.