# 105328616-3-GW-Flow-Equations

```					Equations of Groundwater Flow

Description of ground water flow is based on:

Darcy’s Law

Continuity Equation – describes conservation of fluid
mass during flow through a porous medium; results in a
partial differential equation of flow.

Laplace’s Eqn - most important in math
Derivation of 3-D GW Flow
Equation from Darcy’s Law
z

ρx                               ∂
V                           ρ x + ( V)
V      ρx
∂x

y
Mass In - Mass Out = Change in Storage

∂     ∂    ∂
ρ ) ρz) 0
− (V − ( y − (V =
ρx)   V
∂
x     ∂
y    ∂
z
Derivation of 3-D GW Flow
Equation from Darcy’s Law
Replace Vx, Vy, and Vz with Darcy using Kx, Ky, and Kz

∂  ∂ ∂
h     ∂ ∂
h    ∂
h
ρy + ρz
ρx +  K   K  0
K               =
∂
x   ∂ ∂ ∂ ∂ ∂
x  y   y  z  z

Divide out constant ρ, and assume Kx= Ky= Kz = K

∂h ∂h ∂h
2    2   2
+ 2 + 2 =0
∂x ∂
2
y   ∂z
∇h=0cldaa E.
2
le p e n
a Llc q
Transient Saturated Flow
∂     ∂    ∂    ∂
ρ ) ρz)
− (V − ( y − (V = (n
ρx)   V         ρ)
∂
x     ∂
y    ∂
z    ∂
t
A change in h will produce change in ρ and n, replaced
with specific storage Ss = ρg(α + nΒ). Note, α is the
compressibility of aquifer and B is comp of water,
therefore,

∂ ∂ ∂
h    ∂ ∂ ∂
h    h    ∂h
x 
K   + y 
K          =s
+ z  S
K
∂ ∂ ∂ ∂ ∂ ∂
x   x  y   y  z   z    ∂
t
Solutions to GW Flow Eqns.
Solutions for only a few simple problems can be obtained
directly
Generally need to apply numerical methods to address
complex boundary conditions.
∂h ∂h ∂h
2    2   2
+ 2 + 2 =0
∂x ∂
2
y   ∂z
∇h=0cldaa E.
2
le p e n
a Llc q

h0                                h1
Transient Saturated Flow
Simplifying by assuming K = constant in all dimensions
And assuming that S = Ssb, and that T = Kb yields

h
∂ ∂  ∂ ∂  ∂ ∂  S∂
h         h    s h
 +   +   =
x ∂  y ∂  z ∂
∂  x ∂  y ∂ z K t∂

∂2h ∂2h ∂2h S ∂  s h
+ 2+ 2 =
∂x 2
∂y   ∂z   K∂ t
S∂h
∇ h=
2
rm o,
a
fo Jcb     hs
e
Ti
T∂t
Simplifying by assuming K = constant in all dimensions
and assuming that Transmissivity T = Kb and
Q = flow rate to well at point (x,y) yields

∂h ∂h
2
+ 2=−
2
( )
x
Q ,y
∂x ∂
2
y    T

```
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 views: 7 posted: 10/7/2012 language: English pages: 7
Description: Drainage Engineering