1. Groundwater Flow Equations and Flow Nets

2. Objective Need values of head, flow, and groundwater velocity
Resource use (well development)
environmental impact
contaminant transport
remediation
In many cases we need to make predictions
Approach: use models
Take into account aquifer and fluid properties (pore size, K, porosity, etc)
Account for boundary conditions: head, flow

3. Types of Models Physical models (scaled) Analog models
Electrical
Viscous flow
Mathematical models
Material, energy and momentum balance
Constitutive “laws” – e.g., Darcy’s Law
Input account aquifer and fluid properties (pore size, K, porosity, etc
Input boundary and initial conditions
Output head, flow, velocity

4. Groundwater Flow Equation Confined Flow

5. Homogeneous Isotropic
Hydraulic conductivity does not depend on position or direction

6. Steady State Conditions
Mass accumulation is equal to zero
THE
LaPLACE
EQUATION

7. Unconfined Flow
General Equation
Steady State
Chain rule

8. Unconfined Flow Assume changes in h are small Steady State
LaPlace Equation!

9. Model Solution
We have the model: head (h) is described in the equations in terms of x, y, z, and t
Need to solve the differential equation!
Solve model using integral calculus
Conditions of steady state (no change over time)
Simple boundary conditions
Solve model using techniques for PDEs
Conditions change with time
Laplace Transforms, combination of variables, perturbation
Solve model numerically
Write differential equations as algebraic equations
Solve model graphically

10. Flow Nets Graphical solution to LaPlace’s Equation
Graphical depiction of equipotential lines and flow lines
Valid for 2-D, incompressible media and fluid, steady state conditions

11. FLOW LINE - the average path a particle of water follows through the porous media (the vector mean path)
Here shown as a dye trace in flow through a dam
Rule: flow lines can never intersect or meet

12. EQUIPOTENTIAL LINE - line of equal potential (f) or head (f/g).
FLOW LINE Ho
EQUIPOTENTIAL LINE
EQUIPOTENTIAL LINE - line of equal potential (f) or head (f/g).
The same energy level exists everywhere along a given equipotential line
Equipotential lines can never intersect or meet

13. Flow Lines
Groundwater flows from high head (potential) to low head (potential)
More specifically, it follows the maximum gradient in head.
dh/dl is maximum for the shortest dl (l1)
Therefore, flow lines cross equipotential lines at right angles (isotropic medium)
If we know the distribution of equipotential lines in a homogeneous and isotropic aquifer, flow lines can easily be drawn. l4 l3 h1 l2 l1 h2 h1 h2 h3 h4

14. Flow and Equipotential Lines

15. Simple Flow Net
ho – 1DhT /4
ho – 3DhT /4
ho – 1DhT /2

16. Boundary Conditions
Impermeable Boundary (aka no-flow boundary) (for example, an ‘impermeable’ rock, concrete)
Constant Head Boundary (for example, a standing body of water)
Variable Head Boundary (for example, a water table)

17. Impermeable Boundary “no-flow boundary”
no-flow boundaries
Adjacent flowlines must parallel a no-flow boundary
Equipotential lines must be perpendicular to no-flow boundaries

18. Constant Head Boundary “equipotential surface”
boundaries
Flow lines must intersect a constant head boundary at right angles
Adjacent equipotential lines must ‘parallel’ the constant head boundary

19. What Type of Boundary ?

20. Variable Head Boundary (variable but known)
Neither a flowline nor an equipotential line
A line (surface) with known heads
provides known intersection points for equipotential lines WT
h = 93 92 91 90 92 91 90

21. Groundwater Mapping Hydrologic map
2-D representation of 3-D flow system
Measure water levels in piezometers at same point in time
Construct contours of equal head
Use interpolation
Equipotential lines are intersections of equipotential surfaces with the plane of the diagram
Isotropic aquifer: direction of flow at every point is perpendicular to contours
Hydraulic gradient computed from contour spacing

22. Contoured Equipotential Data

23. Groundwater Flow Draw flow lines perpendicular to equipotential lines
Identify sources and sinks
Identify impervious zones
Identify surface/gw interactions
Identify GW divides
Assess environmental impact
Evaluate contaminant transport

24. Intuitions from Flownets
Converging flowlines
Equipotential lines are more closely spaced
Groundwater velocities increase
Discharge areas (e.g. - springs, effluent streams, pumping wells, etc.)
Radiating (diverging) flowlines
Associated with recharge areas
Water is entering the ground - e.g. influent streams, some lakes, hilltops, injection wells, etc.

25. Map Features
Recharge
Discharge

26. Map Features
Surface water interactions
Changes in T
Impervious zones

27. Map Features
Impermeable zones
Faults
GW divides

28. Effect of Symmetry Flow does not cross AB and DC
No-flow boundaries
Can use symmetry to isolate individual flow systems

29. Flow Net Construction General Rules
For homogeneous material, flow net construction is independent of K!!
Impermeable (no flow) boundaries
Flow lines parallel
Equipotential lines perpendicular
Constant head boundaries
Flow lines perpendicular
Equipotential lines parallel

30. Flow Net Construction Steps
Identify boundary conditions
No flow, constant head, variable but known head
Sketch boundaries to scale
Choose equipotential line contour interval and flowline spacing such that the grid constructed approximates squares
an enclosed circle is tangential to all 4 sides
ERASE AND REDRAW until all flow lines are perpendicular to equipotential lines, boundary conditions are satisfied, and grid is made up of squares
can have partial streamtube at edges
KEEP A SENSE OF HUMOR!

31. Simple Flow Net
ho – 1DhT /4
ho – 3DhT /4
ho – 1DhT /2

32. Simple Flow Net
ho – 1DhT /5
ho – 4DhT /5
ho – 1DhT /2

33. Constant head
boundary
No flow boundary
Squares?

34. Curvilinear squares
An enclosed circle is tangential to all 4 sides

35. Example Identify boundary conditions
No flow, constant head, variable but known
Choose equipotential line contour interval and flowline spacing to produce squares
ERASE AND REDRAW
all flow lines are perpendicular to equipotential lines
boundary conditions are satisfied
grid is made up of squares
can have partial streamtube at edges

36. Streamtubes
The region between adjacent flow lines is called a STREAMTUBE
Discharge through each stream tube is constant
flowlines cannot intersect
As flowlines converge, the cross-sectional area perpendicular to flow gets smaller
Flow stays constant within a streamtube,
Hydraulic gradient must increase
Equipotential lines get more closely spaced
low gradient
high
gradient

37. Flow in Streamtubes is Equal
dQ1
dQ2
dQ3
dQ1= dQ2= dQ3
dQ1+ dQ2+ dQ3 = QTOTAL
Total width, W
Flow lines are evenly spaced to make squares
Total flow is divided into equal increments
Therefore, every streamtube has an equal discharge

38. Darcy’s Law dq’1 dq’2 dq’3 dq’ = flow per streamtube per unit width
h - Dh
dq’1 Dl
dq’2 Dz
dq’3
dq’ = flow per streamtube per unit width
Dh = head drop btw equipotential lines
Dl = flow length between equipotential lines
Dz = height of streamtube
Unit width
dQ1 = – KDzDw(Dh/Dl)
dq’ = –KDz(1)(Dh/Dl)

39. Darcy’s Law dq’ = –KDz(1)(Dh/Dl)
h - Dh
dq’1 Dl
dq’2 Dz
dq’3
Flownet is made of squares so Dl = Dz
dq’ = –K(Dh)
q’ = –pK(Dh) where p = number of streamtubes
Dh = DhT/f where f = number of head drops
Unit width
q’ = –pKDhT/f
Q = q’W

40. Simple Flow Net Q = –KA(DhT/L) = –K(10)(5)(3/50) = –3K
DhT = 3 cm
W = 5 cm
B = 10 cm
L = 50 cm
Q = q’W = –pK(DhT/f)W = –2K(3/10)(5) = –3K

41. Total width = 80 m
K = 10-3 cm/sec
z = 60 m
Data: p = 2; f = 7; DhT = 60 m
q’ = –pK(DhT/f)
= –2(10–5 m/s)(–60 m/7) = 1.714x10–4 m3/sec/m
Q = q’*W
= (1.714x10–4 m3/sec/m) x 80 m = 1.371x10-2 m3/sec
Q = 1185 m3/day; gal/d, gpm; 0.48 cfs

42. Pipe Flow
What size pipe would be required to carry the quantity of seepage flow through the dam?
Mannings Eq:
Pipes flowing full:
A = pD2/4
R = A/Wp = pD2/4/pD = D/4
6” concrete pipe laid at 1% grade (small pipe, flat grade)
n = 0.013
A = 3.14 x 0.52/4 =
R = 0.5/4 = 0.125
Q = 1.486/0.013 x x 0.125(2/3) x 0.01(1/2) = 0.56 cfs

43. p/f = Shape factor
There are an infinite number of possible flownets that can be constructed for any set of boundary conditions.
However, the shape factor p/f will be constant (if the flownet is constructed with squares)
p = 3, f = 6, p/f = 0.5
p = 6, f = 12, p/f = 0.5

44. Refraction of Flow Lines Heterogeneous aquifersK1 q1 K2 q2
Where q1 and q2 are the angles of the flowlines with respect to the perpendicular to the conductivity boundary

45. Equipotential Lines K1 K2
Draw equipotential lines as squares in material 1
Equipotential lines must be continuous across boundary
Isotropic – flow lines and equipotential lines form right angles in material 2
Rectangles are formed in material 2 K1 q1 K2 q2

46. Relate K1 and K2 by Continuity & Geometry
Flow in streamtube is constant
Head drop between equipotential lines is same in both materials
Flow area
Flow length Q1 l1 K1 a q1 b c K2 l2 Q2 q2

47. Tangent Law Eliminate Dh Substitute Cancel b, w Q1 l1 K1 a q1 b c K2

48. Flow from High K to Low K K1 K1 > K2 K2
Flow lines refracted towards the perpendicular
Flow net squares shorten to rectangles
Flow path btw equipotential lines, l, becomes smaller, increasing Dh/l
Steeper gradient and greater areas needed to produce the same Q

49. Flow from Low K to High K K1 K1 < K2 K2
Flow lines refracted away from the perpendicular
Flow net squares elongate to rectangles
Flow path btw equipotential lines, l, becomes greater, decreasing Dh/l
Smaller gradient and flow area needed to produce the same Q

50. Flow Net Shape Write continuity Eq. Substitute RecognizeK1 K2 l1 l2 Q1 Q2 q1 q2 a b c
Flow net squares elongate (or shorten) by the ratio of hydraulic conductivities

51. Path of Least Resistance
GW flows obliquely across heterogeneous systems
Flow lines ‘prefer’ to use high K formations as conduits; flow lines ‘prefer’ to cut across low K units by the shortest route
In flow systems with aquifer/aquitard conductivity contrasts of 2 orders of magnitude or more
(K1 > 100K2)
Flow lines tend to become ‘parallel’ to high K units, and ‘perpendicular’ to low K units

52. Effect of Lenses – High K
Low K
FLOW LINE REFRACTION NEAR HIGH K LENS

53. Effects of Lenses – Low K
High K
FLOW LINE REFRACTION NEAR LOW K LENS

54. Flow Nets in Anisotropic Systems
Directional variation in K
Flow lines no longer meet equipotential lines at right angles
We can still construct the flow net by transforming the coordinates
no longer 90o

55. Coordinate Transformation
Anisotropic System Flownet construction
For example, due to sedimentary layering
Generally
Expand vertical coordinate by the ratio
Construct flow net as usual using square elements
Collapse back to original dimensions KZ KX
KX > KZ

56. Coordinate Transformation Example
Transformed section