第二章 土的渗透性 Chapter 2 Permeability of Soil
第一节 简介
2.1 Introduction
In this lecture, we’ll learn
•
Darcy’s Law (达西定律)
•
Determination of Coefficient of Permeability (渗透系数的测定)
•
Seepage and Flow Net (渗流和流网)
•
Effective stress (有效应力)
Arch dam
(Courtesy of H. Chanson)
Aix-en-Provence, France
(1854)
Drac river, France (1962)
Height = 155 m
第二节 总水头
2.2 Total head
地下水
(i) Hydrostatic Groundwater Condition
静水条件
Pore-water pressure (孔隙水压力 ) at a depth z
z below the water table
u  w  z
where
w (kN/m3) is unit weight of the water (水的重度)
z (m) is depth (深度) of the water below water table
u
(ii) Total head 总水头
u
h
z
w
地下水
uA/w
A
where h is total head, u/w is pressure
head and z is elevation head
zA
Two piezometers ( 测 压 计 ) are
installed at A and B. At hydrostatic
groundwater condition, levels in
both piezometers will be the same,
i.e. hA = hB.
uB/w
B
zB
Datum
Water will flow from A to B if there
is a difference between the water
levels in the piezometers (in this
case hA > hB).
hA
uA

 zA
w
hB
uB

 zB
w
uA/w
A
zA
B
uB/w
zB
Datum
第三节 达西定律
2.2 Darcy’s Law
(i) Darcy’s Law 达西定律
h
q  k   A  k i  A
L
h
where
q = flow rate (m3/s) 渗流量
h = total head difference (m) 水头差
L = length of flow (m) 渗径长度
i = hydraulic gradient (-) 水力梯度
k = coefficient of permeability (m/s) 渗透系数
A = cross-sectional area of the specimen (m2) 横截面积
L
(ii) Validity of Darcy's law 达西定律的有效性 P63
Darcy’s law is valid for laminar flow (层流) condition
where Reynolds number is smaller than or equal to 1.
Reynolds number (雷诺数) is defined as follows:
vd
Re 
1

where  is density of water (水的密度), v is velocity of
water (流速),  is viscosity of water (水的粘滞系数) and
d is average diameter of soil grains (土粒子平均粒径).
第四节 渗透系数的测定
Determination of Coefficient of Permeability
(i) Constant head test 常水头测试
Q
v At
QL
k 

h
i
A  h  t
L
where Q is volume of water
collected, L is length of the
specimen, A is cross-sectional
area of the specimen, h is total
head difference and t is duration
of the test.
(ii) Falling head test 变水头测试
A 2  L  h1 
k
 ln 
A1  T  h 2 
2-19 -P68
where L is length of the
specimen, A1 is cross-sectional
area of the specimen, A2 is crosssectional area of the standpipe,
h1 is total head at t = 0 and h2 is
total head at t = T
(iii) Coefficient of permeability of soil
土的渗透系数的范围
砾石
砂,砾石与
砂混合物
细砂,粉土,粉土
与粘土混合物
粘土,粉土与
粘土混合物
第五节 二维渗流和流网 2-D Seepage and Flow Net
(i) Laplace Equation 拉普拉斯方程

Considering a two-dimensional element of soil of sizes dx
and dz in the x and z directions, respectively.

It is assumed that the soil is homogeneous and isotropic
with respect to permeability.

The governing differential equation for groundwater flow
is obtained by equating the flow rates into and out of the
element.
 h
 h
k x  2  k z  2  0 ---x
z
2

2
 h  h
 2 0
2
x
z
2
2
For most practical geotechnical problems, the Laplace’s
equation for 2-D seepage is solved graphically by
drawing flow nets.
(ii-a) Flow Net 流网

A flow net consists of two sets of curves – equipotential
lines (等势线) and flow lines (流线) – that intersect each
other at 90°.

Along an equipotential, the total head is constant.

A pair of adjacent flow lines define a flow channel
through which the rate of flow of pore fluid is constant.

The loss of head between two successive equipotentials is
called the equipotential drop.
(ii-b) Properties of Flow Net – sketching rules 流网特性

A flow lines cross the equipotentials at right angles.

A flow line cannot cross other flow lines.

An equipotential line cannot cross other equipotential
lines.
The flow net must be constructed so
that each element is a curvilinear
square such that a circle may be
inscribed within it that touches all
four of its sides as shown in the
figure on the right.


Impermeable boundaries and
lines of symmetry are flow
lines, e.g. lines EF and FG in
the figure on the top right are
flow lines.

Bodies of water, such as
reservoirs behind a dam, are
equipotentials, e.g. line AB
in the figure on the bottom
right is an equipotential line.
(ii-c) Typical Flow Nets 流网类型
(ii-d) Flow Net - Flow Rate Calculations 流网计算

Consider water flows through
the flow element shown in the
figure on the right.

The flow rate through this
element is given by:
b=l
h
q  k i  A  k 
 b  k  h
l

If NF is number of flow channels, Nd is number of
equipotential drops and the total head difference is H,
the total flow rate is
H
NF
qT  q  NF  k 
 NF  k  H 
Nd
Nd
第六节 有效应力 Effective Stress
(i) Effective Stress Principle 有效应力原理

The total stress () carried by a saturated soil is the sum
of effective stress (’) carried by the soil particles and
the pressure carried by the pore water (u).
  ' u

Deformation of soil is a function of the change in
effective stress and not total stress.

Effective stress is not the contact stress between two soil
particles but is the average stress on a plane through the
soil mass as shown in the following figure.
P
Ns
P  N s A  A s 


u
A
A
A
X
X
  ' u
P
(ii) Effective Stress under Hydrostatic Condition
静水条件下的有效应力

地下水
Consider a soil element at a depth z
below ground surface with water
table at the ground surface as
shown in the figure on the right:
'    u   sat  z   w  z
'  'z
z
(iii-a) Effective Stress under Downward Seepage Condition
向下渗流时的有效应力

地下水
'    u   sat  z   w  z  h 
'  'z   w  h
h
z
Water flow
Consider a soil element at a depth z
below ground surface with water
table at the ground surface and
water flows downwards as shown
in the figure on the right:
(iii-b) Effective Stress under Upward Seepage Condition
向上渗流时的有效应力

'    u   sat  z   w  z  h 
'  'z   w  h
地下水
h
z
Water flow
Consider a soil element at a depth z
below ground surface with water
table at the ground surface and
water flows upwards as shown in
the figure on the right:
第七节 渗透力和临界梯度
Seepage force and critical hydraulic gradient
7.1 seepage force 渗透力
As water flows through soil it exerts a drag on the soil paricles
resulting in head losses.if the head losses over a flow distance
L,is h,the seepage force is
jw
h w

 i w
L
In seepage condition the effective stress contain two parts
seepage downwards
 z   ' z  js z   ' z  iz w
seepage upwards
 z   ' z  js z   ' z  iz w
'
'
7.2 Types of seepage failure 渗透破坏形式
When flow is upward,with the increase of gradient i，the
vertical effective stress become zero:
 z   ' z  js z   ' z  iz w  0
'
The soil loses its strength and behaves like a viscous fluid.
when the upward seepage forces exceeds the downward force
of the silt,s "boiling" occurs.
when the seepage force push the bottom of an excavation
upward,We call this "Heaving" .
If the upward seepage forces exceed the submerged
weight,the particles may be carried upwards to be deposited
at the ground sueface and a "pipe" is formed in the soil near
the surface.
pipe
boiling
7.3 Critical Hydraulic Gradient 临界水力梯度
There exist a critical head difference (hc) such that ’ = 0.

'  'z   w  h c  0
h
 ' G  1
i 


 G  1  1  n 
z

1e
c
s
cr
s
w