# Darcy Law by alicejenny

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```									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)
Figure from Hornberger et al. (1998)
Generalization of Darcy’s column

h/L = hydraulic

Q is proportional
to h/L

q = Q/A

Figure from Hornberger et al. (1998)
q = Q/A
q is a vector                   h
qx   K x
x
z

h
qy   K y
y
q
qz 2
z                                  h
qz   K z
x                             z
x
qx 1

In general: Kz < Kx, Ky
q = - K grad h
h
qx   K x
x

h
qy   K y
y

h
qz   K z
z
Vector Form of Darcy’s Law

q = - K grad h
q = specific discharge (L/T)
K = hydraulic conductivity (L/T)
q = - K grad h
q is a vector with 3 components

h is a scalar

K is a tensor with 9 components
(three of which are Kx, Ky, Kz)
1 component                    temperature
Vector         Magnitude and Specific discharge, (&
3 components   direction     velocity), mass flux,
heat flux

Tensor         Magnitude,      Hydraulic conductivity,
9 components   direction and   Dispersion coefficient,
magnitude       thermal conductivity
changing with
direction
Darcy’s law

q = - K grad h

q                equipotential line

Isotropic                                Anisotropic
Kx = Ky = Kz = K                         Kx, Ky, Kz
True flow paths

Linear flow
paths assumed
in Darcy’s law

Average linear velocity
Specific discharge
q = Q/A                  v = Q/An= q/n
n = effective porosity

Figure from Hornberger et al. (1998)
Representative Elementary Volume
(REV)

REV

q = - K grad h
Equivalent Porous Medium
(epm)
Law of Mass Balance + Darcy’s Law =
Governing Equation for Groundwater Flow
---------------------------------------------------------------

div   q = - Ss (h t) +R* (Law of Mass Balance)

h                 (Darcy’s Law)
q = - K grad Water balance equation

div (K grad h) = Ss (h t) –R*
Inflow = Outflow
Recharge

Discharge

Transient Water Balance Equation
Inflow = Outflow +/- Change in Storage
Outflow - Inflow = Change in Storage
Storage Terms

h
h

b

Unconfined aquifer                        Confined aquifer
Specific yield = Sy                         Storativity = S
S=V/Ah
S = Ss b
Ss = specific storage
Figures from Hornberger et al. (1998)
W
OUT – IN =
qx qy qz
(             W ) x y z
x   y   z
REV

= change in storage

= - V/ t
S=V/Ah
Ss = V / (x y z h)       Ss = S/b
here b =  z
V = Ss h (x y z)
t      t
OUT – IN =
qx qy qz
(             W )  Ss h
x   y   z             t
h
qx   Kx
x
h
qy   Ky
y
h
qz   Kz
z

      h         h      h      h
( Kx )     ( Ky )  ( Kz )  Ss    W
x     x   y     y  z   z      t
Law of Mass Balance + Darcy’s Law =
Governing Equation for Groundwater Flow
---------------------------------------------------------------

div   q = - Ss (h t) +W (Law of Mass Balance)

q = - K grad h                      (Darcy’s Law)

div (K grad h) = Ss (h t) –W
      h         h      h      h
( Kx )     ( Ky )  ( Kz )  Ss    W
x     x   y     y  z   z      t

     h        h     h
2D confined:        (Tx )     (Ty )  S    R
x    x   y    y     t
(S = Ss b & T = K b)

      h         h      h
2D unconfined:      (hKx )     (hKy )  Sy    R
x     x   y     y      t
Figures from:
Hornberger et al., 1998. Elements of Physical Hydrology,
The Johns Hopkins Press, Baltimore, 302 p.

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