Flow through Soils II (ch8)
2D flow
Vel. vectors are confined to a single plane
1D flow vel vectors parallel
2D flow vel vectors not necessarily parallel
wall
soil
Laplace’s Equation
dxx
vv xx
zv
xv
1D flow - Darcy’s law 2D flow - Laplace’s equation
dzz
vv zz
x
z
dz
dx L
Rate of in vel in z-dir
Laplace’s Equation
Laplace’s equation
“represents energy loss through any resistive medium”
Assumptions
Darcy is valid
S = 100%
is of constant volume
Isotropic kx = kz
Homogeneous k = same throughout
Remember main point = get Q
Examples
Q beneath a dam Q in excavations
Solving
Mathematics
02
2
2
2
z
h
x
h
Graphical
Flownets
Flownets
Flowline
wall
impervious
Equipotential line
Flowpath – “channel” between two flowlines
Equipotential line – along any eqpl, the total head is the same
Flownets
vel
Flowline
Flowline
node
eqpl
Requirements:
Perpendicular crossings at nodes
Maintain “squareosity”
Flownets - Example
h = 14 – 2 = 12 m
h at first eq = 24 m
Q = h L k (NFP/NED)
Q = 0.3 m3/sec
NFP = 3
NED = 12
h at last eq = 12 m
Flownets - Example
h at first eq = 24 m
Head loss per drop = h/NED = 12 / 12 = 1 m
he at A = 8 m
h at A = 24 – 8 = 16 m
hp at A = h – he = 16 – 8 = 8 m
uA = (hp)(w) = 78.5 kPa
Flownets - Example
h at first eq = 24 m
Head loss per drop = h/NED = 12 / 12 = 1 m
he at B = 10 m
h at B = 24 – 9 = 15 m
hp at B = h – he = 15 – 10 = 5 m
uB = (hp)(w) = 49 kPa
Uplift Pressures
Case 1 (no flow)
u = (hp)(w) = uhydrostatic
Draw flownet
Compute h at several points along the structure’s base
Determine hp at these points
Case 2 (with flow)
Find u at these points by u = (hp)(w)
Filters
Problem
- water pressure
- soil filter
- Soil particles migrate out
Sol.
- drains
Problem
Sol.
Filters
2 filter purposes
Allow adequate drainage
Soil retained is called base soil
Disallow particle migration
Filter soil
Filters
2 filter failure types
Clogging: base clogs filter pores – (k decreases)
Piping: base migrates through filter
D
dmax
D
dmax
Simple Cubic Packing
D/dmax = 2.4
Tetrahedral Packing
D/dmax = 6.5
Interstice SizesFilters - principles
To disallow particle migration
To allow adequate drainage (maintain proper k)
Terzaghi’s Filter criteria
5485
15 tod
D 54
15
15 tod
D