04 Hydraulic Conductivity

8/13/2019 04 Hydraulic Conductivity

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Hydraulic Conductivity

Groundwater Hydraulics

Daene C. McKinney


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Summary Hydraulic Conductivity

Permeability Kozeny-Carman Equation Constant Head Permeameter Falling Head Permeameter

Heterogeneity and Anisotropy Layered Porous Media

Flow Nets Refraction of Streamlines Generalized Darcys Law


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Hydraulic Conductivity A combined property of the medium and the fluid Ease with which fluid moves through the medium

k = intrinsic permeability = density = dynamic viscosity

g = gravitational constant

g k K

Porous medium property

Fluid properties


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Hydraulic Conductivity Specific discharge ( q) per unit hydraulic gradient Ease with which fluid it transorted through porous medium Depends on both matrix and fluid properties

Fluid properties: Density , and Viscosity

Matrix properties Pore size distribution Pore shape Tortuosity Specific surface area Porosity

K AQ

q

flowVertical

g

k K

k = intrinsic permeability [L 2]


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Estimating ConductivityKozeny Carman Equation

A combined property of the medium and the fluid Ease with which fluid moves through the medium

k = intrinsic permeability = densityg = gravitational constant

= dynamic viscosityd = mean particle sizef = porosity

g k K

22

32

)1(180 d cd k

f

f

Kozeny Carman eq.


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Lab Measurement of ConductivityPermeameters

Darcys Law is useless unless we can measure theparameters

Set up a flow pattern such that We can derive a solution We can produce the flow pattern experimentally

Hydraulic Conductivity is measured in the lab with apermeameter

Steady or unsteady 1-D flow Small cylindrical sample of medium


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Lab Measurement of ConductivityConstant Head Permeameter

Flow is steady Sample: Right circular cylinder

Length, L Area, A

Constant head difference ( h) isapplied across the sampleproducing a flow rate Q

Darcys Law

ContinuousFlow

OutflowQ

Overflow

A

Q = KA b L


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Lab Measurement of ConductivityFalling Head Permeameter

Flow rate in the tube must equal that in the column

OutflowQ

Qcolumn = r column2

K h L

Qtube = r tube2 dh

dt

r tuber column

2 L K

dhh

= dt


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Heterogeneity and Anisotropy Homogeneous aquifer

Properties are the same atevery point

Heterogeneous aquifer Properties are different at

every point Isotropic aquifer

Properties are same in everydirection

Anisotropic aquifer Properties are different in

different directions Often results from stratification

during sedimentation

vertical horizontal K K

www.usgs.gov


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Layered Porous Media(Flow Parallel to Layers)

Q = Qii= 1

3

= ( bi K i h

x

)

i= 1

3

=h2 h1

W (bi K i )

i= 1

3

bi K ii= 1

3

= bK K Parallel =

1

bbi K i( )

i= 1

3

3 K

2 K

1 K

W

b Q

1b

2b

3b

1Q

2Q

3QQ =h2 h1

W bK


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Layered Porous Media(Flow Perpendicular to Layers)

b K

=bi

K i

i= 1

3

K Perpendicular =bbi

K i

i= 1

3

Q

3 K

2 K

1 K

W

b

1b

2b

3b

1h

2h

3h

Q


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Units

Hydraulic Conductivity K [L/T]

m/s gal/(day-ft 2)

Permeability k [L2]

m 2

ft 2 darcy

1 gal

day ft 2= 4.72 x10

7 ms


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Boundary Conditions

Specified Head Boundary

Specified Flow Boundary

No-flow boundary

h boundary = h(t )

qn boundary = q(t )

qn boundary = 0

Constant Head BC

Specified flow BCNo Flow BC

reservoir

dam

Constant Head BC


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Simple Flow Net Analysis Flow Line a line such that thevelocity vector is tangent to it

Flow net the set of Flow l ines and Equipotent ia l s intersect atright angles

Flow lines terminate onEquipotentials (delineatesboundaries of flow domain)

Discharge of any Flowtube (areabetween two Flow lines) per unitwidth is

q = K dm( ) dhds


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Simple Flow Net Analysis

q = K dm( ) dhds

ds dm

q = Kdh

dh =hn

q = K h

n

Q = mq = Kh m

n

n = number of head drops

m = number of flow tubes


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Flow Net Under a Dam

Flow happens Head above dam > head below dam

Bottom of reservoir Equipotential Flow is down

Impervious boundary, Streamline No-flow

Base of dam Streamline No flow

Water surface below dam Equipotential Constant head

FlowlineEquipotential

reservoir

dam


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Groundwater Flow Direction

Water levelmeasurements fromthree wells can be usedto determinegroundwater flowdirection


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Contour Map of Groundwater Levels

Contours ofgroundwater level(equipotential lines)and Flowlines(perpendicular toequipotiential lines)indicate areas ofrecharge and discharge


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Refraction of Streamlines Vertical component of

velocity must be the sameon both sides of interface

Head continuity alonginterface

So

2 K

1 K Upper Formation

12 K K

y

x

1

2

2q

1q

Lower Formation

q y1 = q y2

q1 cos

1=

q2 sin

2

h1

= h2

@ y = 0

K 1 K 2

=tan 1tan 2


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Darcys Law

Darcys Law 1-D expression When flow is not 1-D, q

is a vector with 3 components

Lh

K q

h K q

z y

x

q

q

q

q

1-D expression

vector with 3

components

3-D expression

q xq yq z

=

K xx K xy K xz K yx K yy K yz K zx K zy K zz

h x h y h z


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Darcys Law

Often we can align thecoordinate axes in theprincipale directions of

layering Horizontal conductivity

often order ofmagnitude larger thanvertical conductivity

q x = K xx h x

q y = K yy h y

q z = K zz h z

K xx = K yy >> K zz

q xq yq z

=

K xx 0 0

0 K yy 0

0 0 K zz

h x h y h z


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Summary Hydraulic Conductivity

Permeability Kozeny-Carman Equation Constant Head Permeameter Falling Head Permeameter

Heterogeneity and Anisotropy Layered Porous Media

Flow Nets Refraction of Streamlines Generalized Darcys Law

04 Hydraulic Conductivity

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