Hydraulic Conductivity & Porosity Does K change if we change the ...

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Hydraulic Conductivity & Porosity

• Today

– Hydraulic Conductivity

– Porosity

– Aquifer, Aquitard, etc

noise-to-signal photos

Does K change if we change the

porous medium?

• Yes

– Hydraulic conductivity is a property of theporous media

• It depends on the pore size, its distribution, and itsconnectivity

– In a clastic sediment this translates to a dependence on

» grain shape, size, and sorting

– K dependence on porous media isrepresented by a measurable property, the

intrinsic permeability, k [L2]

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Intrinsic Permeability, k

• k ! L2 , where we could define L in terms of acharacteristic distance, say grain size.

• For perfectly sorted (i.e., uniform diameter)spherical glass beads, k can be predicted on thebasis of diameter, d alone

k ! d2

• If the grain size varies then use, e.g., mediangrain size for d.

• What is the proportionality coeffcient?– Lots of empirical and some theoretical models …

• See text and other references

Empirical Intrinsic Permeability, k

• For real, mixed diameter and odd shaped grains,

a proportionality constant, C, must be included to

account for grain size distribution, grain shape,

and packing: k = C d2

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Relationship to Porosity

• Note that k typically cannot be correlatedwith porosity.

• For example, clay has a very high porositybut very low permeability,

– while well-sorted gravel, which also has highporosity, has a high permeability

• However, within a single lithologic type(such as sandstone), k typically increases

with increasing porosity, n.

Sorting and Size

• Large grains:

• Small grains:

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Sorting and Size

• Well sorted grains:

• Poorly sorted grains:

wikipedia

Sorting and Size

• Well sorted sandstone: “This is an example of well-rounded,

clean sandstone. The green areais open pore space. This rock hashigh porosity and probably highpermeability also.”

• Poorly sorted sandstone: “Poorly sorted coarse sandstone.

The spaces between the large,well-rounded grains are filled bysmall angular fragments in adark clay-rich matrix. This rockhas very low porosity andpermeability.”

www.geo.wvu.edu/~jtoro/Petroleum/Review%202.html

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Does K change if we change the

fluid in the porous medium?• Yes!

– Hydraulic conductivity is a property of the fluid

• Mainly the fluid dynamic viscosity, µ [M/LT]

• But also the fluid density, ! [M/L3]

• Often written instead in terms of

» Kinematic viscosity," = µ/! [L2/T]

» Specific weight, # = ! g [M/L2T2]

– Both ! and " depend on temperature and pressure

» Through an equation of state (EOS)

• How does K change with increasing µ & !?

• Decreases with µ

• Increases with !µ

1!K !"K

µ

! "# K

K is a property of both the fluid &

the porous medium

• We get:

and

• we can also now

express Darcy’s Law

in terms of these

quantities:

µ

! gkK

=

l

l

l

d

dhkg

d

dhgk

d

dhk

A

Qq

!

µ

"

µ

#

$=

$=

$==

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K is a property of both the fluid &

the porous medium

• For example, using

the empirical model,

k = C d2

• and

µ

! gCdK

2

=

ld

dhgCd

A

Qq

!

2

"==

The basic units for conductivity, K

• Units are [L/T]

• Commonly employed in current and

historical literature and reports:

– SI (preferable): m/s

– Meinzer (old USGS):

• gal per day per square foot

– = gal d-1 ft-2 = [L3 T-1 L-2] = [L/T]

– USGS (recent) and most consultants: ft/d

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The basic units for permeability, k

• Units are [L2]

• Commonly employed in current and historicalliterature and reports:– SI (preferable): m2, preferable

• but the numbers are very small !

– cm2, now commonly used

– ft2, now less common

– Darcy, common in oil, gas and deep basin work

• One darcy is the k which will permit

q=1 cm/s for µ = 1 cP at g(dh/dl) = 1 atm/cm

• 1 darcy " 10-12 m2 = 10-8 cm2

Natural Variation of K

• Its huge! Over 13 orders of magnitude!

Typical ranges of values: K (m/s)

Gravel 10-3 to 101

Sand 10-7 to 10-2

Silt 10-9 to 10-5

Clay & Shale 10-12 to 10-9

Karst limestone 10-5 to 10-1

Sandstone 10-10 to 10-5

Igneous & Metamorphic rocks 10-13 to 10-10

(unfractured)

Use values in your text and cite them

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• Its huge! Over 13

orders of magnitude!

Miocene alluvial fan sediments

in Southern California.

Mainly a mixture of debris flow and

channel/ sheetflood deposits.

Peter Mozley.

Notice: The wide variation in grain

size and in sorting suggesting wide

spatial variation of conductivity in

just this one outcrop.


• Its huge! Over 13 orders of magnitude!


Gravel 10-3 to 101

Sand 10-7 to 10-2

Silt 10-9 to 10-5





(unfractured)

Good aquifers: 10-5 < K < 10-3 m/s

Aquitards: 10-11 < K < 10-7 m/s Typ

ical

val

ues

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Aquifers, Aquitards, and Aquicludes

• Aquifer: a saturated permeable geologic unitthat can store & transmit significant quantities ofgroundwater under ordinary hydraulic gradients&/or can yield economic quantities of water towells (i.e., store and transmit water)

• Aquitard: permeable geologic unit capable oftransmitting geologically significant amounts ofwater, but not economic quantities

• Aquiclude: a geologic unit that cannot transmitgeologically significant amounts of water

These are relative terms;

depend on “local” or “regional” conditions.

Review: What is a

• Confined aquifer?

• Phreatic aquifer?

• Perched aquifer?

• Water table?

• Capillary fringe?

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Natural Variation of Conductivity, K

• Its huge!

• In nature, over 13



Gravel 10-3 to 101

Sand 10-7 to 10-2

Silt 10-9 to 10-5





(unfractured)

Use values in your text and cite them

SZ 2005

Natural Variation of Porosity, n• Its varies much less,

but the variation is

still important.

• In nature,

– n varies over 3 orders of magnitude

– while ne varies more.Porosity n (%)

Gravel 25 - 40

Clay 40 - 70

Karst Limestone 5 - 50

Sandstone 5 – 30

Crystalline Rock 0 – 5

Normally, well-sorted sedimentary materials have a larger porosity than poorly

sorted ones, due to filling of the voids between larger grains by smaller ones.

(Fetter, 2001)

SZ 2005

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Natural Variation of conductivity, K,

in a particular deposit

• Its still huge!

• In a particular deposit

not unusual to be 7


Miocene alluvial fan sediments

in Southern California.

Mainly a mixture of debris flow and

channel/ sheetflood deposits.

Peter Mozley.

Notice: The wide variation in grain size and in

sorting suggesting wide spatial variation of

conductivity (& porosity) in just this one

outcrop.

K measures hydraulic properties at a point, not necessarily for a

whole system. If K is the same at all points, the system is

uniform or homogeneous. If not, it is heterogeneous.

heterogeneoushomogeneous

Examples in natural systems:

sand with clay lenses connected fractures

Homogeneous/Heterogeneous deposits

Lesson today: averaging or upscaling heterogeneity leads to (upscaled) anisotopy

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Property Types

• Scalar Properties– Have no directional component

– Examples• Porosity, Density, Compressibility, Viscosity

• States: Pressure, Heads, Concentrations

• Vector or Tensor Properties– Have directional component

– Isotropic v. anisotropic• Isotropic: same in all directions

• Anisotropic: property varies with direction

– Examples:• Permeability, Hydraulic Conductivity (later: Transmissivity)

• States: specific discharge, seepage velocity, solute flux density

Properties as functions of

location and directionHOMOGENEOUS HETEROGENEOUS

Property changes with locationProperty constant with location

Property

constant

with

direction

Property

changes

with

direction

ISOTROPIC

ANISOTROPIC

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How does averaging or upscaling

heterogeneity leads to (upscaled) anisotopy?

first upscaled volume

second upscaled volume

original volume

Spatially average the heterogeneities reduces heterogeneity creates anisotropy(smooths)

K1

K2

K3b3

b2

b1

hA hB

#L

Heterogeneity:

flow parallel to layers

AB

How much water flow from the reservoir at A to the reservoir at B?

What is the rate of specific discharge and seepage velocity in each layer?

How long would it take a non-reactive tracer to move from A to B in each layer?

steady flow

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Heterogeneity:


K1

K2

K3b3

b2

b1

hA hB

#L

L

hwbKQ

L

hwbKQ

L

hwbKQ

!

!"=

!

!"=

!

!"=

)(

)(

)(

333

222

111

Head differencein each layer (same):

hA – h

B = -#h

Width (into page): w

“Area” of layer i: wbi, i=1,2,3

Discharge in each layer [L3/T]:

L

hwbKQ iii

!

!"= )(

or, generalizing for layer i,


where i=1,2,3

L

hwbKQ

L

hwbKQ

L

hwbKQ

!

!"=

!

!"=

!

!"=

)(

)(

)(

333

222

111

Heterogeneity:


K1

K2

K3b3

b2

b1

hA hB

#L

Discharge in each layer [L3/T]:

or, generalizing for layer i,


where i=1,2,3:

Total Discharge

over all three layers?

321

3

1

QQQQQi

itotal ++==!=

L

hwbKQ iii

!

!"= )(

Use conservation of mass or continuity:

Total discharge = sum of layer discharges

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Heterogeneity:


)( 332211 bKbKbKL

hwQtotal ++!

!"=

K1

K2

K3b3

b2

b1

hA hB

#L


Total Discharge

over all three layers? Use conservation of mass or continuity:


321

3

1

QQQQQi

itotal ++==!=

or

!!==

==

=

"

"#=++

"

"#=

3

1

3

1

321

ivity transmiss total

smissivitylayer tran

example thisin where,

)(

:layers toparallelflow for rmanother te introduce ll we'Later,

i

ii

i

i

iii

total

bKTT

bKT

TL

hwTTT

L

hwQ

Heterogeneity:


L

hwbKQ iii

!

!"= )( L

hK

wb

Qq i

i

ii

!

!"==

L

h

n

K

n

qv

ie

i

ie

ii

!

!"==

,,

K1

K2

K3b3

b2

b1

hA hB

#L

Discharge in layer i [L3/T]:

where i=1,2,3:

What is the rate of specific discharge and seepage velocity in each layer?

Specific discharge in layer i [L/T]:

Seepage velocity in layer i [L/T]:

How long would it take a non-reactive tracer to move from A to B in each layer?

Travel time A to B in layer i [T]:

L

h

n

K

n

qv

ie

i

ie

ii

!

!"==

,,

i

x

x i

i

v

Ldx

vt

B

A

!== "

1

1

1

11

,

2111

,L

L

hKnLhKniieiie

!"#

$%&

'

!

!(=!!(=

(

(((xA xB

depends on Ki

depends on Ki & ne,i

x

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Effective Conductivity:

• Replace a heterogeneous porous media with anupscaled “equivalent” homogeneous porous media

• Find the effective property of that equivalent mediathat preserves the upscaled metric of interest– Here that is total discharge

– But there are other metrics (eg, solute flux) leading toother effective properties (eg, macrodispersion)

• Effective hydraulic conductivity

– Is the equivalent homogeneous Keff that preserves the

total discharge, Qtotal



L

hwbKQ totalefftotal

!

!"= )(

)( 332211 bKbKbKL

hwQtotal ++!

!"=

K1

K2

K3b3

b2

b1

hA hB

#L

Total Discharge

over all three layers From conservation of mass or continuity:


321

3

1

QQQQQi

itotal ++==!=

or

Recall:

321bbbb

total++=

Equate the Qtotal’s

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)()( 332211 totaleff bKL

hwbKbKbK

L

hw

!

!"=++

!

!"

)()( 332211 totaleff bKbKbKbK =++

total

effb

bKbKbKK 332211

++=

321

332211

bbb

bKbKbK

++

++=

!!

=b

KbKeff

Equate the two terms:

Eliminate common elements:

Solve for the effective conductivity :

The “effective conductivity” is the equivalent

homogenous K that results in the same discharge. It’s

the result of “upscaling” or “spatial averaging” over the

heterogeneities. In general, for flow parallel to layers:

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