Groundwater Hydraulics

1. Darcy’s law
Groundwater Hydraulics
Daene C. McKinney

2. Outline
• Properties – Aquifer Storage
• Darcy’s Law
• Hydraulic Conductivity
• Heterogeneity and Anisotropy
• Refraction of Streamlines
• Generalized Darcy’s Law

3. Aquifer Storage
• Storativity (S) - ability of
an aquifer to store water
• Change in volume of
stored water due to
change in piezometric
head.
• Volume of water released
(taken up) from aquifer
per unit decline (rise) in
piezometric head.
Unit area
Unit decline
in head
Released
water

4. Aquifer Storage
• Fluid Compressibility (b)
• Aquifer Compressibility (a)
• Confined Aquifer
– Water produced by 2
mechanisms
1. Aquifer compaction due to
increasing effective stress
2. Water expansion due to
decreasing pressure
• Unconfined aquifer
– Water produced by draining
pores
gV a
S = rg(a +fb)
S = Sy

5. Unconfined Aquifer Storage
• Storativity of an
unconfined aquifer (Sy,
specific yield) depends
on pore space drainage.
• Some water will remain
in the pores - specific
retention, Sr
• Sy = f – Sr
Unit area
Unit decline
in head
Released
water

6. Porosity, Specific Yield, & Specific Retention
yr SS f

7. Confined Aquifer Storage
• Storativity of a confined
aquifer (Ss) depends on
both the compressibility
of the water (b) and the
compressibility of the
porous medium itself
(a).
Unit area
Unit decline
in head
Released
water

8. Example
• Storage in a sandstone aqufier
• f = 0.1, a = 4x10-7 ft2/lb, b = 2.8x10-8 ft2/lb, g = 62.4 lb/ft3
• ga  2.5x10-5 ft-1 and gbf  1.4x10-7 ft-1
• Solid  Fluid
• 2 orders of magnitude more storage in solid
• b = 100 ft, A = 10 mi2 = 279,000,000 ft2
S = Ss*b = 2.51x10-3
• If head in the aquifer is lowered 3 ft, what volume is released?
V = SAh = 2.1x10-6 ft3

9. Darcy
http://biosystems.okstate.edu/Darcy/English/index.htm

10. hL
z1
z2
P1/g
P2/g
Q
Q
L
v
Sand
column
Datum
plane
Area,A
h1 h2
Darcy’s Experiments
• Discharge is
Proportional to
– Area
– Head difference
Inversely proportional to
– Length
• Coefficient of
proportionality is
K = hydraulic conductivity
L
hh
AQ 21 
 Q = -KA
h2 -h1
L
Q = -KA
Dh
L

11. Darcy’s Data
0
5
10
15
20
25
30
35
0 5 10 15 20
Flow,Q(l/min)
Gradient (m/m)
Set 1, Series 1
Set 1, Series 2
Set 1, Series 3
Set 1, Series 4
Set 2

12. Hydraulic Conductivity
• Has dimensions of velocity [L/T]
• A combined property of the medium and the fluid
• Ease with which fluid moves through the medium
k = cd2 intrinsic permeability
ρ = density
µ = dynamic viscosity
g = specific weight
Porous medium property
Fluid properties

13. Hydraulic Conductivity

14. Groundwater Velocity
• q - Specific discharge
Discharge from a unit cross-
section area of aquifer
formation normal to the
direction of flow.
• v - Average velocity
Average velocity of fluid
flowing per unit cross-
sectional area where flow is
ONLY in pores. A
Q
q 
ff A
Qq
v 

15. dh = (h2 - h1) = (10 m – 12 m) = -2 m
J = dh/dx = (-2 m)/100 m = -0.02 m/m
q = -KJ = -(1x10-5 m/s) x (-0.02 m/m) = 2x10-7 m/s
Q = qA = (2x10-7 m/s) x 50 m2 = 1x10-5 m3/s
v = q/f = 2x10-7 m/s / 0.3 = 6.6x10-7 m/s
/”
h1 = 12m h2 = 12m
L = 100m
10m
5 m
FlowPorous medium
Example
K = 1x10-5 m/s
f = 0.3
Find q, Q, and v

16. Hydraulic Gradient
Gradient vector points in the direction of greatest rate of increase of h
Specific discharge vector points in the opposite direction of h

17. Well Pumping in an Aquifer
Aquifer (plan view)
y
h1 < h2 < h3
x
h1
h2 h3
Well, Q
q
h
Circular hydraulic
head contours
K, conductivity,
Is constant
Hydraulic gradient
Specific discharge

18. Validity of Darcy’s Law
• We ignored kinetic energy (low velocity)
• We assumed laminar flow
• We can calculate a Reynolds Number for the flow
q = Specific discharge
d10 = effective grain size diameter
• Darcy’s Law is valid for NR < 1 (maybe up to 10)
NR =
rqd10
m

19. Specific Discharge vs Head Gradient
q
Re = 10
Re = 1
Experiment
shows this
a
tan-1(a)= (1/K)
Darcy Law
predicts this

20. Estimating Conductivity
Kozeny – Carman Equation
• Kozeny used bundle of capillary tubes model to derive an
expression for permeability in terms of a constant (c) and
the grain size (d)
• So how do we get the parameters we need for this
equation?
2
2
3
2
)1(180
dcdk










f
f
Kozeny – Carman eq.

21. Measuring Conductivity
Permeameter Lab Measurements
• Darcy’s Law is useless unless we can measure the
parameters
• Set up a flow pattern such that
– We can derive a solution
– We can produce the flow pattern experimentally
• Hydraulic Conductivity is measured in the lab with a
permeameter
– Steady or unsteady 1-D flow
– Small cylindrical sample of medium

22. Measuring Conductivity
Constant Head Permeameter
• Flow is steady
• Sample: Right circular cylinder
– Length, L
– Area, A
• Constant head difference (h) is
applied across the sample
producing a flow rate Q
• Darcy’s Law
Continuous Flow
Outflow
Q
Overflow
A
Q = KA
b
L
Sample
head difference
flow

23. Measuring Conductivity
Falling Head Permeameter
• Flow rate in the tube must equal that in the column
Outflow
Q
Qcolumn = prcolumn
2
K
h
L
Qtube = prtube
2 dh
dt
rtube
rcolumn
æ
è
ç
ö
ø
÷
2
L
K
æ
è
ç
ö
ø
÷
dh
h
= dt
Sample
flow
Initial head
Final head

24. Heterogeneity and Anisotropy
• Homogeneous
– Properties same at every
point
• Heterogeneous
– Properties different at every
point
• Isotropic
– Properties same in every
direction
• Anisotropic
– Properties different in different
directions
• Often results from stratification
during sedimentation
verticalhorizontal KK 
www.usgs.gov

25. Layered Porous Media
(Flow Parallel to Layers)
3K
2K
1K
W
b
1b
2b
3b
1Q
2Q
3Q
h
h2
h1
Piezometric surface
Q
datum

26. Layered Porous Media
(Flow Perpendicular to Layers)
Q
3K2K1K
b
Q
L
L3L2L1
h1
Piezometric surface
h2
h3
h

27. Anisotrpoic Porous Media
• General relationship between specific
discharge and hydraulic gradient
K is symmetric, i.e., Kij = Kji.

28. Principal Directions
• Often we can align the
coordinate axes in the
principal directions of
layering
• Horizontal conductivity
often order of
magnitude larger than
vertical conductivity
qx = -Kxx
¶h
¶x
qy = -Kyy
¶h
¶y
qz = -Kzz
¶h
¶z
Kxx = Kyy >> Kzz
qx
qy
qz
é
ë
ê
ê
ê
ù
û
ú
ú
ú
= -
Kxx 0 0
0 Kyy 0
0 0 Kzz
é
ë
ê
ê
ê
ù
û
ú
ú
ú
¶h
¶x
¶h
¶y
¶h
¶z
é
ë
ê
ê
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
ú
ú

29. Groundwater Flow Direction
• Water level
measurements from
three wells can be used
to determine
groundwater flow
direction

30. Groundwater Flow Direction
Set of linear equations can be solved for a,
b and c given (xi, hi, i=1, 2, 3)
3 points can be used to define a plane
Equation of a plane in 2D

31. Groundwater Flow Direction
Negative of head gradient in x direction
Negative of head gradient in y direction
Magnitude of head gradient
Direction of flow

32. x
q = -5.3 deg
Well 2
(200, 340)
55.11 m
Well 1
(0,0)
57.79 m
Well 3
(190, -150)
52.80 m
Example

33. Contour Map of Groundwater Levels
• Contours of
groundwater level
(equipotential lines)
and Flowlines
(perpendicular to
equipotiential lines)
indicate areas of
recharge and discharge

34. Refraction of Streamlines
• Vertical component of
velocity must be the same
on both sides of interface
• Head continuity along
interface
• So
2K
1K
Upper Formation
12 KK 
y
x
1q
2q
2q
1q
Lower Formation
qy1
= qy2
q1 cosq1 = q2 sinq2
h1 = h2 @ y = 0
K1
K2
=
tanq1
tanq2

35. Summary
• Properties – Aquifer Storage
• Darcy’s Law
– Darcy’s Experiment
– Specific Discharge
– Average Velocity
– Validity of Darcy’s Law
• Hydraulic Conductivity
– Permeability
– Kozeny-Carman Equation
– Constant Head Permeameter
– Falling Head Permeameter
• Heterogeneity and Anisotropy
– Layered Porous Media
• Refraction of Streamlines
• Generalized Darcy’s Law