groundwater hydraulics water flow in porous media...

Groundwater hydraulics

Water flow

in porous media,

hydraulic conductivity,

Darcy’s law

Saturated flow

Darcy, H., 1856. Les Fountaines de la Ville de Dijon

Henri Darcy (1856) filtration of water for fountains in

Dijon

After many experiments he found that water flow

through the soil column depends on:

• directly proportional to pressure drop

• inversely proportional to the length

• directly proportional to the crossectional area

• dependent on coefficient which is specific for

each media

Henry Darcy

Darcy’s lawHenri Darcy 1856

Darcy law

Q = flow [L3.T-1]

A = crossectional area [L2]

Ks = saturated hydraulic conductivity [L.T-1]

rH = H1 – H2 (hydraulic head drop) [L]

L= sample lenght [L]

valid in fully

saturated porous

media

For example: under

the ground water

level

L

ΑΔΗΚQ S

LH2

H1

datum

z1

z2

h1

h2

Hi = hi+zi

Sn

Q

S

Q v

n

p

Darcian velocity Porous velocity

S

Q v

n

v vp Where n is porosity

z

HKv

y

HKv

x

HKv

HgradK K

z

y

x

neboJv

for:

Transforms to the:

More gereneral form:

Pro 1D vertical flow

kde:

q ... Volume flux [L.T-1]

Q ... Flow rate [L3.T-1]

A ... Crossectional area [L2]

L

HKq s

A

Qq

dl

dHKq s

HKdl

dHKq Ss

note: negative sign

due to the fact grad H

aims against flow

direction

For 3D flow

Coefficient or the saturated hydraulic

conductivity KsAlso called (sometimes) filtration coefficient, darcy’s coefficient or

permeability (incorrect)

commonly used units Ks (m.s-1), (cm.d-1), (cm.s-1)

Ks is property of water-solid interaction. Parameter which is related only to

the porous media (independently on flowing liquid)

is:

k = Ksn/g [ L2]

where n kinematic viscosity when m is dynamic viscosity

Permeability k

m

gk K

Ks for different media

k (

cm

2)

Ks

(cm

.s-1

)

Ks

(m.s

-1)

source: Císlerová a Vogel, 1998

cla

y

sil

t sa

nd

y l

oa

m sa

nd

gra

ve

l

Permeability

Permeability reflects porous medium only - k [m2].

Is used for transport of solutes in porous space

Permeability can be estimated from number of formulas by combination of porosity,

grain distribution and shapes of particles

2dk = cEmpirical formulas: where d is diameter of effective grain (d10),

c=45 for clayey sand, c=140 clear sand.

m

m

m d

P

α

n

) - n(

βk

100

112

3

21

Where β=5, α is grain shape parameter, α=6 for spherical grains,

α=7.7 for sharp grains, Pm is procents of grains found on the

sieve with dm diameter

Physically based equation Carmana - Kozeny:

22

3

S

0 M) n- 1 (

n Ck Where C0 is empiric constant, MS is specific surfaceof unit volume of porous space

Formula of Carman – Kozeny for hydraulic conductivity

72

d e gn K e

n

222/3

Where n is porosity, e is number of porosity, de is effective grain

g is gravity acceleration, ν is kinematic viscosity of water

Example 1 :

Vertical column: q=?

b=

10cm

q

const. head

seepage face

1) datum definition

H = 0

H2

= H1 = z1

2

1

2) points 1 and 2 with known hydraulic

heads

= z2

+z

3) Darcy’s law

1

12

.110100

)0110(100

dcm

L

HHK

L

HKq ss

WATER

SOIL

Ks=100cm/dL=

100cm

dH=10 cm

Water

head inflow

Free

outflow

100 cm

Ks

H1

H2

Example 2

horizontal column: q = ?

1) Step 1, definition of datum and coordination system

2) Definition of points 1 and 2). Then x1 = 0 and h1 = 10 cm,

x2=100cm, h2 = 0, z1 = z2 = 0, L = x2 - x1 = 100 cm

3) Hydraulics heads are then H1 = h1 + z1 = 10 cm, H2 = h2 + z2 =

0 cm

5) Darcy’s law

112 .10100

)100(100

dcm

L

HHK

L

HKq ss

1) Measurements of Ks using constant head permeameter

VODA

PŮDA

b

L

q

Const. head

Seepage face

Soil column

H1 = 0 + 0 (lower end)

H2 = b + L (upper end)

H = (b+L) - 0

then:

Ks measurements

LbqL

H

qLKs

LbAtVL

HAt

VLKs

In practice we measure Q, resp. V/t, then:

WATER

SOIL

Q

A

Ks= ?

sample

po

rou

s d

iscs

Q

measurement

overflow

inflowconstant head

b

L

Fig. : http://www.utdallas.edu/~brikowi/Teaching/Geohydrology/LectureNotes/Darcy_Law/Permeameters.html

Constant head permeameter

soil sample must be carefully saturated with water to

measure “real” Ks.

Constant head permeameter

2) Measurement of Ks falling-head permeameter

Ks= ?

b(t)

L

q

Falling head

Seepage face

Experiment is done on soil sample in the

laboratory. Initial water level is equal to b0

H1 = 0, H2(t) = L+b(t), H(t) = [b(t) + L] - 0

becomes:

b0

L

LbK

dt

dbq s

dtL

K

Lb

db s

Lb

LbLb

Lb

db bb

b

b

01lnln

1

0

1

0

Integration of the left side

b1WATER

SOIL

11

0

1

0

t

ss

t

s

L

tKdt

L

Kdt

L

K

Right side integration

dtL

K

Lb

db s

L

tK

Lb

Lb s 1

0

1ln

then:

Lb

Lb

t

LKs

1

0

1

ln

Zdroj: http://www.utdallas.edu/~brikowi/Teaching/Geohydrology/LectureNotes/Darcy_Law/Permeameters.html

Falling-head permeameter

sample

Po

rou

s

dis

cs

Q

wa

ter

leve

l d

rop

b0

L

b1

Lb

Lb

t

L

A

AK bs

1

0

1

ln

Consider different

crossectional areas of

burette and column:

db

d

where:

t1…. time

Ab … cross. area of burette

A ... cross. area of the sample

VODA

PŮDA

b

L

h0

h2 = b, z2=L

h1 = 0, z1 = 0

Ks is constant, h(z) = ?

Darcy’s law:

z

zhK

L

LbK

L

HKq sss

zh

zL

bz

L

hhh

12

In this case, pressure head

profile is linear .

Example 3 :

Pressure head along the soil column h(z)=?

q

q12 q1z

WATER

SOIL

Homogeneity and inhomogeneity of the environment

with regards to hydraulic conductivity

The environment is homogeneous: hydraulic conductivity is same at all points

of the environment

The environment is inhomogeneous = heterogeneous: environment where

hydraulic conductivity varies according to position in the area

Heterogeneity with gradual change Heterogeneity with sudden change

distance

Example of inhomogeneous environment, accepted as homogeneous

Practical situations lead us to considerations whether accept or neglect inhomogeneity

Calculating flow velocity

z

HKv

y

HKv

x

HKv

HgradK K

z

y

x

neboJv

Where K is the function of x,y and z.

clay lens

recursive pattern

or

Isotropy and anisotropy of the environment

The environment is isotropic when hydraulic conductivity is same at all directions

The environment is anisotropic when hydraulic conductivity is direction dependent, e.g.

Hydraulic conductivity in vertical direction is lower than in horizontal direction

This property is due to the formation of the structure (e.g. sediments)

Hydraulic conductivity of anisotropic environment is defined by tensor of hydraulic conductivity

3D

2D

yyyx

xyxx

zzzyzx

yzyyyx

xzxyxx

KK

KK

KKK

KKK

KKK

KK

Component Kxy defines part of the velocity in x direction caused

by unit hydraulic gradient in y direction

Vector of velocity is tilted from the direction of the hydraulic

gradient in anisotropic environment

Sedimentary rock – horizontal structure

https://commons.wikimedia.org/w/index.php?curid=8912695


Eolic sediment – vertical structure



Main axes of anisotropy are the direction where hydraulic conductivity

reaches maximal or minimal value

If coordinate system x,y,z is parallel to main axes of anisotropy,

tensor of hydraulic conductivity is

yy

xx

zz

yy

xx

K

K

K

K

K

0

0

00

00

00

KK

3D2D

Components of flow in anisotropic environemnt are following

z

H K

y

H K

x

H K v

z

H K

y

H K

x

H K v

z

H K

y

H K

x

H K v

zzyzxzz

yzyyyxy

xzxyxxx

z

H Kv

y

H Kv

x

H Kv

zz

yy

xx

neboor

Combination of homogeneity (1) and isotropy (2) in the in environment

homogeneous, isotropic homogeneous, anisotropic

inhomogeneous, anisotropicinhomogeneous, isotropic

Limits of Darcy’s law validityF

LO

W V

EL

OC

IT

HYDRAULIC GRADIENT

Darcy’s law is valid in certain interval of hydraulic gradient and flow velocities only

Laminar flow

Turbulent flow

Prelinear zone Linear zone Postlinear zone

Prelinear flow

Happens in very fine materials (clay and silt), flow velocity is

calculated by Swartzendruber:

. 3

Jln

J

1 ,J)exp( J

3

4 J K v 0

0

0

kde

J0 is limit value of hydraulic gradient, where flow is actuvated.

J0 < 0.5 for silt, J0= 0.5 - 1.0 for clay.

Postlinear flow

Happens in coarse materials (gravel or sandy gravel), where higher velocities occur

Type of flow: laminar vs. turbulent is indicated by Reynolds number:

ν

d Re

v

Critical value of Re is dependent on given material. Typically 10 - 100 (1 - 10).

D is grain size and n is kinematic viscosity

where

For postlienar zone, flow can be calculated by combined method

2 m 1.6 , B A grad B A gradm

vvnebovv2

A, B are constants, characterizing permeability of the environment, d or de is effective grain:

,11 2tKBKA

engdedgn

K 933.4log4Ka72

10

45

t

223

n

Pavlovskij equation

) (grad K 5.0

v

e

ed

14 d 20 n K

In most of the cases groundwater flow meets linear zone where Darcy’s law

is applicable. Upper limit of Darcy’s law might be overcome in carstic, dolomitic or

volcanic areas with cavities.

or

Transmissivity

Transmissivity – is ability of aquifer to transfer water in one horizontal

meter of aquifer of thickness B and hydraulic conductivity K

T = KB

Transmissivity units [m2/s]

Transmissivity can be applied if vertical flow component is not considered

x

H W B K Q

x

y

H W B K Q

y

groundwater hydraulics water flow in porous media...

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