1. Effects of Boundary Condition on Shape of Flow Nets
ERTH403/HYD503 Lecture 6
Effects of Boundary Condition on Shape of Flow Nets ? ? ? ? ?
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

2. Seepage and Dams Flow nets for seepage through earthen dams
ERTH403/HYD503 Lecture 6
Seepage and Dams
Flow nets for seepage through earthen dams
Seepage under concrete dams
Uses boundary conditions (L & R)
Requires curvilinear square grids for solution
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

3. ERTH403/HYD503 Lecture 6
Flow Net in a Corner:
Streamlines Y are at right angles to equipotential F lines
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

4. Flow to Pumping Center:
ERTH403/HYD503 Lecture 6
Flow to Pumping Center:
Contour map of the piezometric surface near Savannah, Georgia, 1957, showing closed contours resulting from heavy local groundwater pumping (after USGS Water-Supply Paper 1611).
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

5. Example: Travel Time h = 100 m h = 92 m Orphanage Luthy Lake
ERTH403/HYD503 Lecture 6
The deadly radionuclide jaronium-121 was released
in Luthy Lake one year ago. Half-life = 365 days
Concentration in Luthy Lake was 100 g/L.
If orphans receive jaronium over 50 g/L, they
will poop their diapers constantly. How long
will it take for the contaminated water to get
to Peacock Pond and into the orphanage water
supply? Will jaronium be over the limit?
Example: Travel Time
h = 92 m
Luthy Lake
h = 100 m
Peacock Pond
Orphanage
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

6. Travel time Contamination in Luthy Lake Travel time is only 131 days!
ERTH403/HYD503 Lecture 6
Travel time
h = 92 m
Luthy Lake
h = 100 m
Peacock Pond
Contamination in Luthy Lake 1 2 3
Note: because l terms are squared, accuracy in estimating l is important. 4
Travel time is only 131 days!
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

7. Travel time Contamination in Luthy Lake Travel time is only 131 days!
ERTH403/HYD503 Lecture 6
Travel time
Supposed we used averages?
h = 92 m
Luthy Lake
h = 100 m
Peacock Pond
Contamination in Luthy Lake 1 2 3 4
Travel time is only 131 days!
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

8. Two Layer Flow System with Sand Below
ERTH403/HYD503 Lecture 6
Two Layer Flow System with Sand Below
Ku / Kl = 1 / 50
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

9. Two Layer Flow System with Tight Silt Below
ERTH403/HYD503 Lecture 6
Two Layer Flow System with Tight Silt Below
Flow nets for seepage from one side of a channel through two different anisotropic two-layer systems. (a) Ku / Kl = 1/50. (b) Ku / Kl = 50. Source: Todd & Bear, 1961.
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

10. Flow nets in anisotropic media
ERTH403/HYD503 Lecture 6
Flow nets in anisotropic media
SZ2005 Fig. 5.11
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

11. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
So far we have only talked about flownets in
isotropic material. Can we draw flownets for
anisotropic circumstances?
For steady-state anisotropic media, with x and y
aligned with Kx and Ky, we can write the flow equation:
dividing both sides by Ky:
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

12. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Next, we perform an extremely cool transformation
of the coordinates:
This transforms our governing equation to:
Laplace’s Eqn!
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

13. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Steps in drawing an anisotropic flownet:
Determine directions of max/min K. Rotate axes
so that x aligns with Kmax and y with Kmin
2. Multiply the dimension in the x direction by
(Ky/Kx)1/2 and draw flownet.
3. Project flownet back to the original dimension by
dividing the x axis by (Ky/Kx)1/2
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

14. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Example: Ky Kx
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

15. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media Ky Kx
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

16. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media Kx Ky
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

17. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

18. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

19. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

20. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
25%
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

21. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
25%
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

22. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

23. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

24. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

25. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

26. Flownets in Anisotropic Media
ERTH403/HYD503 Lecture 6
Flownets in Anisotropic Media
Kx = 15Ky
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

27. ERTH403/HYD503 Lecture 6
Flow Nets: an example
A dam is constructed on a permeable stratum underlain by an impermeable rock. A row of sheet pile is installed at the upstream face. If the permeable soil has a hydraulic conductivity of 150 ft/day, determine the rate of flow or seepage under the dam.
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

28. ERTH403/HYD503 Lecture 6
Flow Nets: an example
The flow net is drawn with: m = head drops = 17
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

29. Flow Nets: the solution
ERTH403/HYD503 Lecture 6
Flow Nets: the solution
Solve for the flow per unit width:
q = m K
= (5)(150)(35/17)
= 1544 ft3/day per ft
total change in head, H
number of head drops
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

30. ERTH403/HYD503 Lecture 6
Flow Nets: An Example
There is an earthen dam 13 meters across and 7.5 meters high.The Impounded water is 6.2 meters deep, while the tailwater is 2.2 meters deep. The dam is 72 meters long. If the hydraulic conductivity is 6.1 x 10-4 centimeter per second, what is the seepage through the dam if the number of head drops is = K = 6.1 x 10-4cm/sec
= m/day
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

31. Flow Nets: the solution
ERTH403/HYD503 Lecture 6
From the flow net, the total head loss, H, is = 4.0 meters.
There are (m=) 6 flow channels and 21 head drops along each flow path: Q = (mKH/number of head drops) x dam length = (6 x m/day x 4m / 21) x (dam length) = 0.60 m3/day per m of dam
= 43.4 m3/day for the entire 72-meter length of the dam
After Philip Bedient
Rice University
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

32. Aquifer Pumping Tests Why do we need to know T and S (or K and Ss)?
ERTH403/HYD503 Lecture 6
Aquifer Pumping Tests
Why do we need to know T and S (or K and Ss)?
-To determine well placement and yield
-To predict future drawdowns
-To understand regional flow
-Numerical model input
-Contaminant transport
How can we find this information?
-Flow net or other Darcy’s Law calculation
-Permeameter tests on core samples
-Tracer tests
-Inverse solutions of numerical models
-Aquifer pumping tests
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

33. Steady Radial Flow Toward a Well
ERTH403/HYD503 Lecture 6
Steady Radial Flow Toward a Well
Aquifer Equation, based on assumptions becomes an ODE for h(r) :
steady flow in a homogeneous, isotropic aquifer
fully penetrating pumping well & horizontal,
confined aquifer of uniform thickness, thus
essentially horizontal groundwater flow
flow symmetry: radially symmetric flow
(Laplace’s equation)
(Laplace’s equation in x,y 2D)
(Laplace’s equation in radial coordinates)
Boundary conditions
2nd order ODE for h(r) , need two BC’s at two r’s, say r1 and r2
Could be
one Dirichlet BC and one Neumann, say at well radius, rw , if we know the pumping rate, Qw
or two Dirichlet BCs, e.g., two observation wells.
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

34. Steady Radial Flow Toward a Well
ERTH403/HYD503 Lecture 6
Steady Radial Flow Toward a Well
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

35. Island with a confined aquifer in a lake
ERTH403/HYD503 Lecture 6
Island with a confined aquifer in a lake b h1 h2 Q
We need to have a confining layer for our 1-D (radial) flow equation to apply—if the aquifer were unconfined, the water table would slope down to the well, and we would have flow in two dimensions.
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

36. Steady Radial Flow Toward a Well
ERTH403/HYD503 Lecture 6
Steady Radial Flow Toward a Well
Constant C1:
ODE
Multiply both sides by r dr
Separate variables:
Integrate
Integrate again:
Evaluate constant C1 using continuity.
The Qw pumped by the well must equal
the total Q along the circumference for every
radius r. We will solve Darcys Law for r(dh/dr).
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

37. Steady-State Pumping Tests
ERTH403/HYD503 Lecture 6
Steady-State Pumping Tests b h1 h2 Q r2 r1
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

38. Thiem Equation Integrate ERTH403/HYD503 Lecture 6
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

39. Steady-State Pumping Tests
ERTH403/HYD503 Lecture 6
Steady-State Pumping Tests
Why is the equation logarithmic?
This came about during the switch to radial coordinates. h r
What is the physical rationale for the shape of this curve (steep at small r, flat at high r)?
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

40. Think of water as flowing toward the well through a series of rings.
ERTH403/HYD503 Lecture 6
Think of water as flowing
toward the well through a series
of rings.
As we approach the well the rings get smaller. A is smaller but Q is the
same, so must increase.
(Driscoll, 1986)
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

41. ERTH403/HYD503 Lecture 6
The Thiem equation tells us there is a logarithmic relationship between head and distance from the pumping well.
hlc
one log cycle
(Schwartz and Zhang, 2003)
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

42. ERTH403/HYD503 Lecture 6
The Thiem equation tells us there is a logarithmic relationship between head and distance from the pumping well.
hlc
one log cycle
If T decreases, slope increases
(Schwartz and Zhang, 2003)
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006

43. Steady-State Pumping Tests
ERTH403/HYD503 Lecture 6
Steady-State Pumping Tests
Can we determine S from a Thiem analysis?
No—head isn’t changing!
Hydrology Program, New Mexico Tech, Prof. J. Wilson, Fall 2006