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Fluid Mechanics
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Chapter 9 Seepage Flow
§ 9-1 Introduction
§ 9–2 Section 1 Basic law of seepage flow
§ 9-3 Uniform and non-uniform flow of groundwater
§ 9-4 Well and catchment passage
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第九章 渗 流
§ 9-1 引 言
§ 9–2 渗流基本定律
§ 9-3 地下水均匀流和非均匀流
§ 9-4 井和集水廊道
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Chapter 9 Seepage Flow
Seepage flow occurs during fluid flows through porous medium
,Mechanics of seepage flow is the study of fluids movement law in
porous media,It is one of offsets of Mechanics of fluids,a subject
developed by mechanics of fluids,medium theory,physics and
chemistry,biology,
Seepage phenomenon is common in nature and artificial materials,
Such as seepage of groundwater,hot water and brine ;seepage of oil,
natural gas and coal seam gas;seepage of blood micro-circulation
and micro-bronchia in animals; transportation of water,gas and sugar
in plant; seepage of pottery,tile and so on artificial porous materials,
§ 9-1 Introduction
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第九章 渗 流
渗流是流体通过多孔介质的流动,渗流力学就是研究流体
在多孔介质中运动规律的科学。渗流力学是流体力学的一个分
支,是流体力学与介质理论、表面物理、物理化学以及生物学
交叉渗透而发展起来的一门边缘学科。
渗流现象普遍存在于自然界和人造材料中。如地下水、热
水和盐水的渗流;石油、天然气和煤层气的渗流;动物体内的
血液微循环和微细支气管的渗流;植物体内水分、气体和糖分
的输送;陶瓷、砖石、砂模、填充床等人造多孔材料中的渗流
等。
§ 9-1 引 言
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Mechanics of seepage is widely applied in application science and
project technology.Such as mechanics of soil,subterranean hydrology
,oil engineering,terrestrial heat engineering and micro-mechanism and
so on,Furthermore,it is also involved in national defense industry,such
as study of gas mask,
Characteristic of seepage flow:first,the area of unit volume space is
very large in porous medium,surface action is distinct,viscous action is
considered to any body and anytime ; second,the pressure is large in
subterranean seepage flow,so,the compressibility is usual considered;
third,the complexity of hole,large resistance,capillary effect and force
of molecular ; fourth,complicated process of physics and chemistry 。
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渗流力学在很多应用科学和工程技术领域有着广泛的应用。如
土壤力学、地下水文学、石油工程、地热工程、给水工程、环境工
程、化工和微机械等等。此外,在国防工业中,如航空航天工业中
的发热冷却,核废料的处理以及诸如防毒面罩的研制等都涉及渗流
力学问题。
渗流的特点在于:第一,多孔介质单位体积空隙的表面积较
大,表面作用明显,任何人和时候都必须考虑粘性作用;第二,
在地下渗流中往往压力较大,因而通常要考虑流体的压缩性;第
三,孔道形状复杂、阻力大、毛细管作用较普遍,有时还要考虑
分子力;第四,往往伴随有复杂的物理化学过程。
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Mechanics of seepage flow is not only long historical but also
young science,It has been past one and half centuries since Darcy law
The grow up of oil industry in 20th promoted the develop of seepage
flow,Along with correlative science develops,it is promote seepage
flow again,Recently mechanics of seepage flow developed into a new
stage along with develop of nonlinear mechanics; the establishment of
bifurcate theory and other theory,
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渗流力学是一门既有较长历史又年轻活跃的科学。从达西定
律的出现到现在已过去一个半世纪。 20世纪石油工业的崛起极大
地推动了渗流力学的发展。随着相关科学技术的发展,如高性能
计算机的出现,核磁共振,CT 扫描成像以及其他先进实验方法
用于渗流,又将渗流力学大大推进了一步。近年来,随着非线性
力学的发展,将分叉、混沌理论以及分形理论用于渗流,其他诸
如格气模型的建立等等,更使渗流力学的发展进入一个全新的阶
段。
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§ 9-2 Basic law of seepage flow
1,Model
Figure 9 — 1 is experiment set,cylinder is full of sand,interface
area is A,sand layer thick is l,thin net in the sand bottom。
Water flow through A to C
determine。 Neglecting kinetic energy,
difference of measure hydraulic head
H1- H2 is water head loss。
2,Darcy’s law for seepage flow
French engineer - Henri Darcy,did
many experiment utilize sandy soil in
1852-1855,achieved Darcy’s law
A
Q
l
B
C
1H 2H
Fig.9— 1 experiment set
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§ 9-2 渗流基本定律
一、渗流模型
图 9 — 1 为 渗流实验装置,圆筒内充填沙粒,界面积为 A,
沙 层厚为 l,沙底部有细网支撑。
水由稳压箱经阀 A 进入圆筒 C
测定。忽略动能,测压管水头差
H1- H2为渗流的水头损失。
二、达西渗流定律
法国工程师达西( Henri Darcy)
在 1852至 1855年利用沙质土壤进行了
大量实验,得到线性渗流定律。即达
西定律,
A
Q
l
B
C
1H 2H
图 9— 1 渗流实验装置
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3,coefficient of seepage flow
metrical method for coefficient,
( 1) empirical formula method:estimating according as soil
diameter,shape,structure,porosity,temperature,
( 2) experimental method,utilizing the setting like Fig.9— 1
and formula( 9— 1),
( 3) spot method:do experiment by pumping water from well,
calculate k according as well formula,
9 1
wwhhQ k A V k k J
ll
k
J
? ? ?or ( — )


Coefficient of seepage flow
Hydraulic gradient
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坡度,即水力坡度。流程范围内平均测压管—
具有流速的量纲;
流的一个系数,和流体性质综合影响渗渗流系数,是土壤性质—式中:
)—(或
J
k
kJ
l
h
kV
l
h
kAQ ww
19 ???
三、渗流系数
渗流系数测定方法,
( 1)经验公式法,根据土壤粒径、形状、结构、孔隙率,
影响流体粘性的温度等参数所组成的经验公式估算。
( 2)实验室方法,在实验室利用类似图 9— 1的装置及公式
( 9— 1)计算,方法简易,但不易取得未扰动土样。
( 3)现场方法,在现场利用钻井或原有井作抽水或灌水实
验,根据水井的计算公式求 k 值。
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§ 9-3 Uniform and non-uniform flow of groundwater
1,invariable uniform and non-uniform flow
1.1 invariable uniform seepage flow,hydraulic gradient is uniform in
any section,according to Darcy’s law u is equal in any point,
1.2 invariable non-uniform seepage flow
as Fig,9— 2
hydraulic pipe gradient of point in any section
? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
?
?
1
2
dH
J
ds
1
2
0 0
H
Fig,9— 2 hydraulic pipe
gradient formula Dupuit
2 9
constant
) — ( dS dH k kJ U
dS
dH J
- ? ?
? - ?
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§ 9-3 地下水均匀流和非均匀流
一、恒定均匀流和非均匀流
( 1)恒定均匀渗流,任一断面水力坡相同,由达西定律,任
意点流速 u 都相等。
( 2)恒定非均匀渗流,如图 9— 2 所示
任意过水断面上各点测压管坡度
)。此式称为裘皮幼公式(
)—(
常数
D u p u i t
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dS
dH
kkJU
dS
dH
J
-??
?-?
图 9— 2 测压管坡度
? ? ? ? ? ? ? ? ? ? ?
? ?
? ?
?
?
1
2
dH
J
ds
1
2
0 0
H
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2,Gradual change basic differential equation and soakage curve
In non-hydraulic seepage flow,surface of G-water is called soakage
surface,in plane is soakage curve,As in 9— 3,due to Dupuit formula
( ) 9 3
wher e hy dra ul i c g r a di e nt
0,sm ooth gra dient
0,up- gr a dient
0,down- gr a dient
9 4 i s s oakage curv e equat i on
dH dh
V k J k i
dS dS
i
i
i
i
w he
? ? - ? -
?
?
?
( — )
— ;;


22
12
22
12
0
2
2
o r 9 4
wh e r e,
ni
Q
l h h
kb
q
l h h
k
Q
q
b
?
?-
?-
?
( — )
Fig,9— 4
? ? ? ? ?? ? ? ? ? ? ?
J
0 0
H
Fig,9— 3
?? ? ? ? ? ? ?
h
?s
?
l
?? ??????????
0 0
2h
?? ? ? ? ? ? ?
1h
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二、渐变渗流基本微分方程和浸润曲线
在无压渗流中,重力水自由表面称浸润表面,平面问题中
为浸润曲线。如图 9— 3所示,由裘皮幼公式
即单宽流量。式中:
)—(或
。—的浸润曲线方程,见图下面仅给出
称为逆坡。
称为顺坡;
称为平坡;
渗流底坡;—式中
)—(
,
49
2
2
490
,0
,0
,0
39 )(
2
2
2
1
2
2
2
1
b
Q
q
hhl
k
q
hhl
kb
Q
i
i
i
i
i
dS
dh
ik
dS
dH
kJV
?
-?
-?
?
?
?
?
-?-??
? ? ? ? ?? ? ? ? ? ? ?
J
0 0
H
图 9— 3
?? ? ? ? ? ? ?
h
?s
?
图 9— 4
?? ??????????
0 0
2h
?? ? ? ? ? ? ?
1h
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§ 9-4 Well and catchment passage
1,Catchment corridor
A catchment corridor,section is rectangle,the bottom is impermeability,
i = 0,due to( 9— 3) get,
)0( dsdhbhkQ -?
)—(
),—(

)—(
69
2
)(
o b t a i n i n g
59 f o r m u l a i n t o,T a k i n g
c o r r i d o r, a t t a c h m e n t of s c o p e e f f e c t e d
,a f f e c tt i s n ' l e v e lt e r u n g r o u n d w a
e q u a t i o n, c u r v e s o a k a g e is
59
2
o b t a i n i n g a n d gi n t e g r a t i na f t e r
22
22
L
hHk
q
HzLx
is
LLx
T h i s
x
kb
Q
hz
-
?
??
?
?-
Fig,9— 5
Catchment corridor
?? ???????????? ? ? ? ? ? ?
L
??
x
o
x
H
h
z
N
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§ 9-4 井和集水廊道
一、集水廊道
某集水廊道,横断面为矩形,底为不透水层,见图 10— 8,
底坡 i = 0,由式( 9— 3)得 )0(
ds
dhbhkQ -?
积分得
)—(
)式,可得单侧渗流量—代入(
。将为集水廊道的影响范围
,称后,地下水位不受影响
此式为浸润曲线方程。
)—(
69
2
)(
59
,
59
2
22
22
L
hHk
q
HzLx
LLx
x
kb
Q
hz
-
?
??
?
?-
图 9— 5 集水廊道
?? ???????????? ? ? ? ? ? ?
L
??
x
o
x
H
h
z
N
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2,Diving well( non-hydraulic well)
Diving water has free surface,the well constructed in it called
diving well,The bottom of well is impermeability,as 9— 6
22
0
22
0
D ue to D up uit f or m ul a
2
l n 9 7
w he n R,H,is c a l l e d w e l l e f f e c te d dia m e te r
()
9 8
ln
the m a x de sc e nt o f u nd e r gr ou nd w a te r
c a l l e d
dz
Q A U rz k
dr
Qr
zh
kr
r z R
k H h
Q
R
r
S H h is
?
?
?
? ? ?
-?
??
-
?
?-
( — )
( — )
w a te r l e v e l d e pth
Fig,9— 6 Diving well
? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ???
r
Hh N
0r
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二、潜水井(无压井)
具有自由水面的地下水称无压水或潜水,在其中建的
井称潜水井。井底打到不透水层称完全井,如图 9— 6
称水位降深地下水面的最大降落
)—(
为井的影响半径称时,在
)—(
应用裘皮幼公式
hHS
r
R
hHk
Q
RHzRr
r
r
k
Q
hz
dr
dz
krzAUQ
-?
-
?
??
?-
???
89
ln
)(
,
79 ln
2
0
22
0
22
?
?
?
图 9— 6 潜水井
? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ???
r
Hh N
0r
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3,Artesian well
Aquifer is between two impermeable layer,pressure is larger
than air pressure,called artesian,the well is served by it called
artesian well,as in fig.9— 7,for entirety well
0
0
2,i s
hy dr a ul i c he a d f or c or r e sponding point
a f t e r i nte gr a t i ng a nd ge t t i ng
hy dr a ul i c c ond uit he a d c ur v e e qua t i on
l n 9 9
t a ke i nto,
()
ln
dz
Q AV rt k z
dr
r
Qr
zh
k t r
z H r R ge t
H h k t
Q
R
r
?? ? ?
-?
??
-
?
( — )
9 10
9 10 f or m ul a i s t r y t o de duc e e qua t i on w h e n h t h t??
( — )
( — ), 。
Fig,9— 7
Artesian entire well
? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ???
r
H
h N
0r
x
s
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三、自流井
含水层位于两不透水层之间,压强大于大气压,称为自流
层,由其供水的井称自流井,见图 9— 7,对完全井
的情况。时公式,试推导)式是—(
)—(
得流量公式代入
)—(
方程积分得测压管水头曲线
点的测压管水头为对应
thth
r
R
kthH
Q
RrHz
r
r
kt
Q
hz
rz
dr
dz
krtAVQ
??
-
?
??
?-
???
109
109
ln
)(
,
99 ln
,2
0
0
?
图 9— 7 自流完全井
? ? ? ? ? ? ? ? ? ? ? ?? ? ? ? ? ? ? ???
r
H
h N
0r
x
s
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4,Big orifice well and delve drain
The diameter of big orifice well is above 2 m,most is incomplete
well,the bottom water quantum is part of gross。 Supposing a big
well,well wall is impermeability,bottom is hemisphere,water supply
is only through bottom,the section is concentric with bottom
0
2
2
0
2
2
2 9 1 1
o r if ic e o r a w e l l F x ih a e i th in k
th e s e c tio n is e l l ip a e,s tr e a m l in e is h y p e r b o l a ;
y ie l d in g f o r m u l a is
4 9 1 2 )
RH
r H S
dz
Q A U r k
dr
dr
Q r d z
r
Q k R S
to
Q k r S
?
?
?
-
??
?
?
?
??
( — )

( —
?S
H r
0r N
Fig.9— 8
Big orifice well
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四、大口井和基坑排水
大口井一般直径在 2 m 以上,多为不完全井,井底水量是
总量一部分。设一大口井,井壁四周不透水,井底为半球形,
供水仅能通过井底。过水断面是与井底同心的半球面,则
12)9 4
119 2
2
2
0
2
2
0
—(
产水量公式为
是双曲线,水断面是椭圆形,流线
梅认为过对平底大口井,福希海
)—(
SkrQ
k R SQ
dzr
r
dr
Q
dr
dz
krAUQ
H
SH
R
r
?
?
?
??
??
-
?
?
?
图 9— 8 大口井
?S
H r
0r N
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