1. Hydrology and Fluid Flow
GLE/CEE 330 Lecture Notes
Soil Mechanics
William J. Likos, Ph.D. Department of Civil and Environmental Engineering University of Wisconsin-Madison

2. The Hydrologic Cycle

3. Water volume, in cubic miles
Where is the Water?
Water source
Water volume, in cubic miles
Percent of total water
Oceans
317,000,000
97.24%
Icecaps, Glaciers
7,000,000
2.14%
Ground water
2,000,000
0.61%
Fresh-water lakes
30,000
0.009%
Inland seas
25,000
0.008%
Soil moisture
16,000
0.005%
Atmosphere
3,100
0.001%
Rivers
300
0.0001%
Total water volume
326,000,000
100%
Source: Nace, U.S. Geological Survey, 1967 and The Hydrologic Cycle (Pamphlet), U.S. Geological Survey, 1984

4. Water Table Fluctuations
(USGS)

5. Saturated and Unsaturated Zones
Low hydraulic conductivity; k = f(S)

6. Aquifers
(Water Encyclopedia)

7. Perched Groundwater Table

8. Why do Geotechs Care about Water?
1) Predict quantity of flow
Earth dam
Typical Section

9. Why do Geotechs Care about Water?
2) Predict rate of flow (e.g., contaminant transport)

10. Why do Geotechs Care about Water?
3) Predict pore water pressure!
Terzaghi’s Effective Stress
s’ = effective stress
s = total stress
uw = pore water pressure sv uw
Total stress (self weight, external loads) is carried by the
soil skeleton and the pore pressure
= s’+uw
The stress carried by the soil skeleton governs engineering behavior (strength, volume change). If the pore pressure changes, the effective stress changes, and changes in behavior occur

11. 1D Fluid Flow – Darcy’s Experiment
Qin
Qout
Influent water reservoir
Effluent water reservoir
Henry Darcy
( ) h1 L h2
datum A
Saturated
Sand
Flow

12. Influent water reservoir Effluent water reservoir
Qin
Qout
Influent water reservoir
Effluent water reservoir h1 L h2
datum A
Saturated
Sand
Flow
Q = volumetric flow rate (volume/time), e.g., cm3/sec
At steady state, Q = Qin = Qout
Darcy found that
Dh = total head loss
i = Dh/DL = “hydraulic gradient”

13. Darcy’s Law For steady state saturated flow in porous media
k = “hydraulic conductivity” (length/time)
Dht = total head loss (length)
DL = distance over which head loss occurs (length)
A = cross-sectional area of flow (length squared) L A
datum h1 h2
Qin
Qout

14. q = v = “Darcy velocity”
or “discharge velocity” v k
So, k is a property of the soil (and fluid) that
quantifies conductivity to water flow.
Sand – high k
Clay – low k i

16. Typical Values of Hydraulic Conductivity

17. Hydraulic Conductivity vs. Intrinsic Permeability
Hydraulic conductivity (k), units = length/time (cm/s)
depends on both soil and fluid properties
soil properties → void ratio (e), grain size distribution (D10)
fluid properties → density (r), viscosity (m)
Intrinsic permeability (K), units = length2 (m2)
depends only on soil properties
for water, K is about 5 orders less than k
for example, if k = 10-5 cm/s then K = cm2

18. Hydraulic Conductivity vs. Intrinsic Permeability
Example:

19. Discharge Velocity vs. Seepage Velocity
Discharge velocity (v), aka “Darcy” velocity
gross flow velocity out of a cross-section
Seepage velocity (vs)
actual fluid velocity around soil grains
governs “piping” (internal erosion processes)
vs > v
For mass conservation at discharge:
Q1 = vsAv
Q2 = vAt
At = Av + As Av As
Since n < 1, vs > v

20. Total Hydraulic Head Bernoulli’s Equation:
ht = total hydraulic head: (m)
z = elevation head: (m)
uw/gw = pressure head: (N/m2)/(N/m3) = (m)
v2/2g = velocity head: (m2/s2)(m/s2) = (m)
Daniel Bernoulli
( )
We may also write:
For most flow processes in soil v is small; so we usually assume:

21. Total Hydraulic Head, ht
Total hydraulic head (“total head”) is the governing variable for fluid flow.
There MUST be a difference in ht for flow to occur between any two points.
“Head Loss” Dht =ht1 - ht2 ≠ 0
Consider a beaker of water
Will there be flow between 1 and 2?
(Is there a “head loss” between 1 and 2?) z
ht1 = ht2
so Dht = 0
so NO FLOW he 1
ht1 hp
z (+) 2
ht2
Datum (z = 0)
he or hp or ht

22. (flow will stop when water level reaches Point 3)
Can water flow uphill?
YES! – but only if Dht ≠ 0 3
hp1
FLOW
hp2 2
he2
z (+) 1
he1
Datum
ht1 > ht2
So FLOW from 1 to 2
(flow will stop when water level reaches Point 3)

23. ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day
Qin
ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day
Qout 3’ 6’
25’ 2’
10’ A B C 5’
datum
Point
he (ft)
hp (ft)
ht (ft) A 5 20 25 B ? C 10
(Need to find total head at B by interpolation)

24. ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day
Qin
ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day
Qout 3’ 6’
25’ 2’
10’ A B C 5’
datum ht
25’
htb
assume head loss is
linear in the soil
10’ 3’ 6’ x A B C

25. ksilt= 10-5 cm/s = 2.8 X 10-2 ft/day
Qin
Qout 3’ 6’
25’ 2’
10’ A B C 5’
datum
Point
he (ft)
hp (ft)
ht (ft) A 5 20 25 B 15 C 10

26. Drawing Head Diagrams
General approach (for steady state systems with homogeneous soil):
Establish elevation datum (arbitrary)
Compute and draw he profile relative to elevation datum.
Compute hp for points located “outside” the soil. Recall that hp is the height from the point to the nearest peizometer (water level).
Compute ht from ht = he + hp.
Pressure head at any point “inside” the soil can then be calculated graphically from the pressure head profile or analytically using the fact that the head loss is linear within the soil.

27. Lambe and Whitman (1969)

28. Lambe and Whitman (1969)

29. Example Problem:
Determine the components of head at A, B, and C
What is the hydraulic conductivity if you measure v = 2 cm/hr?

30. Point
he (cm)
hp (cm)
ht (cm) A 80 20
100 B 10 90 C 40 45 85 D 50 30

31. Determining Hydraulic Conductivity
Varies over ~13 orders of magnitude!
tight shale ~ cm/s to clean gravel ~ 102 cm/s
No one method is ideal for all soil types
Laboratory Methods:
Constant Head
Falling Head
Constant Flow
Field Methods:
Point Infiltrometers (small scale)
Pumping Tests (large scale)
Challenges:
Disturbance!
Scale?
Anisotropy
Advantages:
Control of boundary conditions
Control of stress state [k = f(e)]
Cost effective?
(varved clay; kh ≠ kv)

32. Constant Head Method Qin Dh A = (k)(Dh/DL)(A) Qin = Qout s1 s3 Qout DL
Steady-state flow established under constant gradient (constant head)
Measure Q for applied gradient to determine k from Darcy’s Law
Want to keep i < 30 to avoid changes to soil fabric
Qin
Requires undisturbed specimen
Flexible wall or rigid wall systems
Very common
Best for relatively high k (e.g., sand) Dh A
= (k)(Dh/DL)(A)
Qin
= Qout s1 s3
Qout DL

33. Qin = 892 ml in 112 sec Dh = 60cm d=18 cm Qout DL=16.7 cm
Constant Head Example Problem
Qin = 892 ml
in 112 sec
Dh = 60cm
d=18 cm
Qout
DL=16.7 cm
= (k)(Dh/DL)(A)
Qin
= Qout

34. Falling Head Method graduated Dh@ t0 standpipe (a) t0 t1 falling head
Hydraulic gradient changes with time
Works for relatively low k (e.g., clay)
graduated
standpipe (a) t0 t0 t1
falling
head
boundary a t1 A
Qout DL
constant
head
boundary

35. Constant Flow Method Q Syringe Pump A Dh DL Q = (k)(Dh/DL)(A)
Apply Q rather than measure it
Measure Dh rather than apply it
Q = -0.1 cc/min
Q = 0.1
Q = -0.2
Q = 0.2
Q = -0.3
Q = 0.3
Q = (k)(Dh/DL)(A)

36. Constant Flow Method Can apply extremely small flow rate Q
Thus, we can measure extremely small k, and still keep low gradient (i<30)
k as low as cm/s
Very useful for low perm materials (clay, shale, concrete!)

37. Empirical Correlations
“Empirical” = Relying on or derived from observation or experiment
Often desirable to estimate k from other (easily measured) properties
Hazen’s correlation
% Finer
k = hydraulic cond. (cm/s)
C = empirical coefficient (0.8 to 1.2; commonly 1.0)
D10 = 10% finer diameter from sieve analysis (mm)
0.1mm < D10 < 3mm and Cu < 5
D10
Log D
Makes sense that the
smaller grains govern
hydraulic conductivity

38. Anisotropic Flow k4 H4 kx k3 H3 k2 H2 kz k1 H1 (varved clay; kh ≠ kv)
Generally kx > kz

39. Anisotropic Flow Example
Silt (5 mm)
Analysis area
(repeating structure) kz kx
Clay (20 mm)
ks = 3 × 10-4 cm/s
kc = 6 × 10-7 cm/s

40. Capillarity
Sandstone, Golden Colorado
hc ~ 9’ (275 cm)

41. Capillarity

42. Height of Capillary Rise
(Lu and Likos, 2004)

43. 2D Fluid Flow – Flow Net Analysis
What is the distribution of total head and flow velocity in the domain? y x z dz
S = 1.0 dx

44. Conservation of Mass: dz S = 1.0 dx
(any velocity change in one direction must be negated by an equal and opposite velocity change in the other direction)

45. 3) From mass conservation
1) From Darcy’s Law:
3) From mass conservation
Laplace Equation
2) Define “velocity potential” f
Partial diff. equation defining distribution of f (total head) in 2 dimensions at steady state

46. Solution to Laplace eq. is a “flow net”
Equipotential lines – same total head (like contour lines)
Streamlines (Flow lines) lines along which flow occurs
Equipotential
Lines
Flow
Lines

47. Procedures for drawing flow nets:
Draw 2-D cross-section of problem to scale
Draw boundaries in ink
Take advantage of symmetry
Select integer number of flow “tubes” (Nf ~4 to 6)
Sketch flow lines
Sketch equipotential lines
Follow b/a = 1 rule
EL and FL must intersect at right angles
No FL or El may intersect
Lines are smooth b a EL EL FL
ht1 FL
ht2

50. Using Flow Nets – Flow Prediction
Nf = 3
Nd = 12
b/a = 1
Dh = 10 m
k = 1× 10-3 cm/s
10m

51. Cut-Off Walls
(Terzaghi, Peck, Mesri)

52. Impervious Blankets, Graded Filters
(Terzaghi, Peck, Mesri)