Martin Šanda, B673martin.sanda@fsv.cvut.cz

Martina Sobotková, B609martina.sobotkova@fsv.cvut.cz

CTU in Prague, FCE, Dept. of Irrigation, Drainage and landscape management

Groundwater hydraulics – lecture6

dam seepage, confined aquifer

Based on Darcy’s equation and Dupuit assumptions (planar horizontal flow)

➢For transient nonhomogeneous anisotropic environment

➢For steady state flow in homogeneous isotropic environment

t

H S

y

H K

y

x

H K

xyx

0

02

2

2

2

= y

H +

x

H

Kx,y,z=fce(x,y,z)

Flow equation

By Valentová, J - Hydraulika podzemní vody

kde So is specific storativity (sink or source)

DUPUIT ASSUMPTIONS – TO REPEAT

The hydraulic height H (x, y, z) is equal to the groundwater level h (x, y),

the flow lines are horizontal lines, the vertical are equipotentials. The

gradient of the potential is given by the slope of the plane and is constant

after the vertical.)y (x ,

dx

dh )z y,(x ,

dx

dH

REALITY HYDRAULIC APPROACH

WATER TABLEWATER TABLE

θ

K sd

z d K

sd

H d Kvs sin ) (x h h ,

dx

dh Kvx

Pavlovsky method for seepage through the trapezoidal damContrasting Dupuit assumptions, it considers real shape of the earth dam, allows to determine seepage face

D

β

d

HK

α

D H

H ) D (h K qdz )(z v q

K

KI

D

0

I

lntg1

At second part, delineated by Z=0 and Z=λ, Dupuit assumption is used :

2

22 dDKqII

Third part has to be split at two parts horizontally, since right side of the dam follows different

boundary conditions.

Flow at Z = λ is determined as summation of two flows.

The dam is divided vertically into three parts, flow must be equal at all of them.

Assumption is adopted for the flow, that flowline are approximately same length, as if they were

horizontal and intersection same point at Z axis.

Flow at X=0 can be written as follows:

q2

q1h2

d

H = h2

H = z

At lower part q1 followsz d

dz ) h d ( K qh

0

2

tg21

Upper part, q2 follows:

dz K q

d

h2

tg2

Total flow at vertical at third section is:

h d

d ) h d ( Kqq qIII

2

221 ln1tg

Three equations show three flows qI, qII a qIII with four unknowns (D,d,λ,q). Therefore fourth

equation has to be added.

L

dtg

Solving set of equations leads to value of seepage, partly shape of the water table and point of

seepages

Pavlovsky method can be adopted for the earth dams with drains

foot drain

In this case, no seepage face occurs

x = λ h = h2; flows for parts and II are solved

only

2

222hD

KqII

No seepage face as well,

Flows for I and II, while x = λ h = 0

2

2DKqII

Inner drain

Correctness of seepage according to Dupuit assumptions

d

h(x)

While shape of groundwater adopting Dupuit assumptions is determined incorrectly – neglecting

seepage face, it can be proved that value of seepage with vertical walls is determined exactly

Specific flow at arbitrary vertical can be written exactly as follows

dzx

zxHKq

xh

0

,

Leibnitz rule is used for the right hand side

dzx

zxH

dx

dhhxH

dx

dhhxHdzzxH

dx

dxh

xh

xh

xh

2

1

2

1

,,,, 1

12

2

Whre in our applicatoin h1 = 0 a h2 = h(x)

xhxh

dx

dxH

dx

dhhxHdzzxH

dx

ddz

x

zxH

00

00,,,

,

0Hydraulic head at

groundwater table = level

of water h

xhxh

dx

dhhxhdzzxH

dx

ddz

x

zxH

00

,,,

Can be condensed into:

xhh

dzzxHdx

d

K

q

0

2

2,

It can be integrated 0 do x:

xhh

dzzxHddxK

q

0

2

2,

Ch

dzzxHK

qxxh

0

2

2,

Integration constant can be determined from x = L, hydraulic head H(x,z) = h2 :

Ch

dzhK

qLh

2

2

2

0

2

2 2C

2

2h

K

qL

Left side of the dame follows these boundary conditions :

d<z h zz H

h <z hz Hx

1

11

pro,0

0pro,00

Substitution integration constant C and using second boundary condition:

22

,002

2

0

2 h

K

qLddzz H

d

for

for

Splitting right hand side integral into two parts:22

02

2

2

0

1

1

1

h

K

qLddzzdzh

d

h

h

Integrating results in :

2220

2

2

22

1

22

1

h

K

qLdhdh

After reorganizing: Dupuit - Forchheimer equation for seepage is adopted:

L

hhKq

2

2

1

2

2

Steady flow in the confined aquifer

Detail of sampling point

Flow is determined by such equation and boundary conditions

0

dx

dHxT

dx

d

BA HHpro;HH0pro Lxxfor for

Hydraulic head development within aquifer is described

L

x

AB

A

xT

xd

xT

xd

HH

HxH

0

0

Specific flow is determind from the equation :

L

BA

xT

xd

HHq

0

Lets solve flow in this aquifer assuming that at distance A, horizontal sink gallery is present,

subtracting flow qp per unit meter. Hydraulic head at point of sink is Ha and sink is equal to

summation of inflows from both sides.

IIIP qqq

Total amount of water taken by the gallery is:

L

a

aB

a

aAIIIP

xT

xd

HH

xT

xd

HHqqq

0

Flow in confined aquifer with overflow

Semipermeable

layer

Main aquifer

Driving equation

00

xHH

dx

dHT

dx

dModified to:

T

xHH

dx

Hd

2

2

0

2

2

kde,0 where

H(x) is general solution of such equation

021 expexp HxCxCxH

Integration constants C1 a C2 are determined by boundary conditions, as follows

BA HHpro;HH0pro Lxx

Integration constants are given by equations

LL

LHHHHC

CHHC

AB

A

expexp

exp002

201

Substitution we receive for H(x)

xHH

xLHH

LHxH BA sinhsinh

sinh

1000

Where hyperbolic sinus is following:

xxx expexp2

1sinh

for for

Flow can be determined bydx

dHTq When H(x) is derived

Equation can be solved for x se if hydraulic head in main collector is near H0.

Integration constants follow

012 a0 HHCC A

Hydraulic head is described by equation:

xHHHxH A exp00

Flow is described by

00 exp

HxHTxHH

T

dx

dHTxq A

and

Free groundwater

semipermeable

impervious

Balanced flow: 0022 xqNdx

dqxxqxxqxNxxq vv

q and qv – i.e. overflow velocity is substituted

1

11 kde,aK

BBxhq

dx

dhxKhq v

datum.

Flow in free aquifer with overflow

and where

Final driving equation 022 1

2

22

BxhN

dx

hdK

K

NBD

KADxAh

dx

hd 1

2

22 1022

this equation can be modified with constants A a D into

Boundary conditions for x = 0, h = h0 and for x = L , h = 0. Ananytical solution for the shape of

groundwater

xDAhAx

hxh

21

0

2

03

2

6

Specific flow can be described based on the equation

2/1

3

2

D

xAhxKh

dx

dhxKhxq

Same flow eq as in previous case 0 xqNdx

dqv

Overflow velocity

001

xh

B

xhKqv

Flow in free aquifer with overflow

Final equation 022 0

2

22

xhN

dx

hdK

Linearization for final solution by transmissivity of aquifer T = H hs, kde hs is averaged

Height of groundwater.

0nebo,0

2

0

2

2

0

2

2

T

Nxh

dx

hdN

xh

dx

hdT

General solution of the equation is function h(x):

TNxCxCxh 2

210 expexp

Integration constants are defined by boundary conditions x=0, h=h0, pro x=L, h=hL

or

Variable with no physical meaning. Derived based on Dupuit assumptions and can be

used in isotropic and homogenenous environment or homogeneous in horizontal

plane – values of hydraulic conductivity may change over vertical axis.

Definition for free head aquifer

yxh

dzzKyxhzyxG

,

0

,,

Girinsky potential is defined in such a way that its derivation in arbitrary direction equals

to specific flow in that direction:

yxy

Gyxqyx

x

Gyxq yx ,,a,,

yxG

yxq ,,

Or in general

In homogeneous aquifer with free head Girinsky potential is deifned as

yxhK

yxG ,2

, 2

Girinsky potential

Steady flow in aquifer with free head is defined as

0dy

dq

dx

dq yx

Whre using Girinsky potential can be rewritten

0,,2

2

2

2

yx

y

Gyx

x

G

Girinsky potential for the confined aquifer

B

dzzKyxHzyxG0

,,

Confined and homogeneous

H

BKBG

2

dx

dGxq Used property of Girinsky potential

Boundary conditions x = 0, G = G(0) and x = L, G = G(L) solution received

L

GLGq

0

Calculating seepage using Girinsky potential

Case study

Simulation of unsteady flow in levees

Císlerová a Zumr (Reserach UDRŽITELNÁ VÝSTAVBA – MSM

6840770005)

• Inputs

• Geometry of levee and

bedrock

• Material characteristics

• Design flood

• Outputs

• Pore pressure

• Moisture

• Velocity field

• 2D network of final elements – program ARGUS ONE

• Matematical model S2D_DUAL – flow and transport in

heterogeneous porous space (prof. Vogel, CTU in Prague)

1)(

z

hhK

zt

Water regime is levee

Pressure field

Horizontal velocity

Water regime is levee

Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir

Water regime is levee

Horizontal velocity

Pressure field Flood height, volume in reservoir