Darcy Law by alicejenny

VIEWS: 22 PAGES: 20

									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
                                                              gradient




                                                       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)
  grad h = hydraulic gradient (L/L)
            h = head (L)
      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)
Scalar         Magnitude       Head, concentration,
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


          grad h                         q      grad h




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*
  Steady State Water Balance Equation
                        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.

								
To top