Dupuit_Forchheimer_Unconfined_Aquifer_Recharge_Presentation.ppt

The Dupuit-Forchheimer theory simplifies
groundwater flow in unconfined aquifers by
assuming horizontal flow and neglecting
vertical components.
For unconfined ground water flow Dupuit
developed a theory that allows for a simple
solution based on the following assumptions:
1) The water table or free surface is only
slightly inclined
2) Streamlines may be considered horizontal
and equipotential lines, vertical
3) Slopes of the free surface and hydraulic
gradient are equal
The Dupuit-Forchheimer theory provides a
simplified model of flow in unconfined
aquifers with recharge. It's useful in
approximations but has limitations under
certain conditions.
• Limitations:
• 1. Assumes negligible vertical flow, suitable
for gentle gradients only.
• 2. Neglects aquifer heterogeneity and
extreme gradients.

1. Dupuit-Forchheimer Theory for
Unconfined Aquifers with
Recharge Flow
Understanding Flow Dynamics in
Unconfined Aquifers with Recharge
Presented By:
I20MA061,I20MA060,I20MA056

2. Introduction to Unconfined
Aquifers
•An aquifer whose upper water surface
(water table) is at atmospheric pressure, and
thus is able to rise and fall.

3. Basics of the Dupuit-
Forchheimer Theory
•The Dupuit-Forchheimer theory simplifies
groundwater flow in unconfined aquifers by
assuming horizontal flow and neglecting
vertical components.

4. Dupuit Assumptions
For unconfined ground water flow Dupuit
developed a theory that allows for a simple solution
based on the following assumptions:
1) The water table or free surface is only
slightly inclined
2) Streamlines may be considered horizontal
and equipotential lines, vertical
3) Slopes of the free surface and hydraulic
gradient are equal

5. Derivation of the Dupuit
Equation - Unconfined Flow

6. Derivation of the Dupuit
Equation
Darcy’s law gives one-dimensional flow per unit
width as:
q = -Kh dh/dx
At steady state, the rate of change of q with
distance is zero, or
d/dx(-Kh dh/dx) = 0
OR (-K/2) d
2
h
2
/dx
2
= 0
Which implies that,
d
2
h
2
/dx
2
= 0

7. Dupuit Equation
Integration of d
2
h
2
/dx
2
= 0 yields
h
2
= ax + b
Where a and b are constants. Setting the boundary
 
    
condition h = h
o at x = 0, we can solve for b
b = h
o
2
Differentiation of h
2
= ax + b allows us to solve for a,
a = 2h dh/dx
And from Darcy’s law,
hdh/dx = -q/K

8. Dupuit Equation
So, by substitution
h
2
= h
0
2
– 2qx/K
Setting h = h
L
2
= h
0
2
– 2qL/K
Rearrangement gives
q = K/2L (h
0
2
- h
L
2
) Dupuit Equation
Then the general equation for the shape of the parabola is
h
2
= h
0
2
– x/L(h
0
2
- h
L
2
) Dupuit Parabola
However, this example does not consider recharge to the aquifer.

9. Cross Section of Flow
q

10. Adding Recharge W -
Causes a Mound to Form
Divide

11. Dupuit Example
Example:
2 rivers 1000 m apart
K is 0.5 m/day
average rainfall is 15 cm/yr
evaporation is 10 cm/yr
water elevation in river 1 is 20 m
water elevation in river 2 is 18 m
Determine the daily discharge per meter width into each
River.

12. Example
Dupuit equation with recharge becomes
h
2
= h
0
2
+ (h
L
2
- h
0
2
) + W(x - L/2)
If W = 0, this equation will reduce to the parabolic
Equation found in the previous example, and
q = K/2L (h
0
2
- h
L
2
) + W(x-L/2)
Given:
L = 1000 m
K = 0.5 m/day
h
0
= 20 m
h
L= 28 m
W = 5 cm/yr = 1.369 x 10
-4
m/day

13. Example
For discharge into River 1, set x = 0 m
q = K/2L (h
0
2
- h
L
2
) + W(0-L/2)
= [(0.5 m/day)/(2)(1000 m)] (20
2
m
2
– 18 m
2
) +
(1.369 x 10
-4
m/day)(-1000 m / 2)
q = – 0.05 m
2
/day
The negative sign indicates that flow is in the opposite direction
From the x direction. Therefore,
q = 0.05 m
2
/day into river 1

14. Example
For discharge into River 2, set x = L = 1000 m:
q = K/2L (h
0
2
- h
L
2
) + W(L-L/2)
= [(0.5 m/day)/(2)(1000 m)] (20
2
m
2
– 18 m
2
) +
(1.369 x 10
-4
m/day)(1000 m –(1000 m / 2))
q = 0.087 m
2
/day into River 2
By setting q = 0 at the divide and solving for x
d, the
water divide is located 361.2 m from the edge of
River 1 and is 20.9 m high

15. Summary
•The Dupuit-Forchheimer theory provides a
simplified model of flow in unconfined
aquifers with recharge. It's useful in
approximations but has limitations under
certain conditions.
•Limitations:
•1. Assumes negligible vertical flow, suitable
for gentle gradients only.
•2. Neglects aquifer heterogeneity and extreme
gradients.

16. Thank You !!