## Hydraulic Conductivity & Porosity. Does K change if we change the porous medium?

Hydraulic Conductivity & Porosity • Today – Hydraulic Conductivity – Porosity – Aquifer, Aquitard, etc noise-to-signal photos Does K change if we ch...
Author: Lawrence Dawson
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 the porous media • It depends on the pore size, its distribution, and its connectivity – In a clastic sediment this translates to a dependence on » grain shape, size, and sorting

– K dependence on porous media is

represented 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 a

characteristic distance, say grain size. • For perfectly sorted (i.e., uniform diameter) spherical glass beads, k can be predicted on the basis of diameter, d alone k ! d2 • If the grain size varies then use, e.g., median grain 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 correlated with porosity. • For example, clay has a very high porosity but very low permeability, – while well-sorted gravel, which also has high porosity, 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 area is open pore space. This rock has high porosity and probably high permeability also.”

• Poorly sorted sandstone: “Poorly sorted coarse sandstone. The spaces between the large, well-rounded grains are filled by small angular fragments in a dark clay-rich matrix. This rock has very low porosity and permeability.” 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 1 K! #K" K"! • Increases with !

µ

! µ

K is a property of both the fluid & the porous medium • We get:

K=

k!g µ

and • we can also now express Darcy’s Law in terms of these quantities:

Q k# dh =\$ A µ dl k"g dh =\$ µ dl kg dh =\$ ! dl

q=

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K is a property of both the fluid & the porous medium • For example, using the empirical model, k = C d2

Cd 2 ! g K= µ

Q Cd 2 g dh q= =" A ! dl

• and

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 historical literature 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: Gravel Sand Silt Clay & Shale Karst limestone Sandstone Igneous & Metamorphic rocks (unfractured)

K (m/s) 10-3 to 101 10-7 to 10-2 10-9 to 10-5 10-12 to 10-9 10-5 to 10-1 10-10 to 10-5 10-13 to 10-10

Use values in your text and cite them

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Natural Variation of K • Its huge! Over 13 orders of magnitude!

Miocene alluvial fan sediments in Southern California. Mainly a mixture of debris flow and channel/ sheetflood deposits.

Notice: The wide variation in grain size and in sorting suggesting wide spatial variation of conductivity in just this one outcrop.

Peter Mozley.

Natural Variation of K • Its huge! Over 13 orders of magnitude! K (m/s)

Gravel Sand Silt Clay & Shale Karst limestone Sandstone Igneous & Metamorphic rocks (unfractured) Good aquifers: 10-5

Aquitards:

10-3 to 101 10-7 to 10-2 10-9 to 10-5 10-12 to 10-9 10-5 to 10-1 10-10 to 10-5 10-13 to 10-10

< K < 10-3 m/s 10-11 < K < 10-7 m/s

Ty pic al va lue s

Typical ranges of values:

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Aquifers, Aquitards, and Aquicludes • Aquifer: a saturated permeable geologic unit that can store & transmit significant quantities of groundwater under ordinary hydraulic gradients &/or can yield economic quantities of water to wells (i.e., store and transmit water) • Aquitard: permeable geologic unit capable of transmitting geologically significant amounts of water, but not economic quantities • Aquiclude: a geologic unit that cannot transmit geologically 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 orders of magnitude! SZ 2005

Typical ranges of values:

K (m/s)

Gravel Sand Silt Clay & Shale Karst limestone Sandstone Igneous & Metamorphic rocks (unfractured)

10-3 to 101 10-7 to 10-2 10-9 to 10-5 10-12 to 10-9 10-5 to 10-1 10-10 to 10-5 10-13 to 10-10 Use values in your text and cite them

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. Gravel Clay Karst Limestone Sandstone Crystalline Rock

Porosity n (%) 25 - 40 40 - 70 5 - 50 5 – 30 0–5

SZ 2005

(Fetter, 2001)

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.

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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 orders of magnitude! Miocene alluvial fan sediments in Southern California. Mainly a mixture of debris flow and channel/ sheetflood deposits. Notice: The wide variation in grain size and in sorting suggesting wide spatial variation of conductivity (& porosity) in just this one outcrop.

Peter Mozley.

Homogeneous/Heterogeneous deposits 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.

homogeneous

heterogeneous

Examples in natural systems:

sand with clay lenses

connected fractures

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 direction HOMOGENEOUS

HETEROGENEOUS

I S O T R O P I C

Property constant with direction

A N I S O T R O P I C

Property changes with direction

Property constant with location

Property changes with location

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How does averaging or upscaling heterogeneity leads to (upscaled) anisotopy?

original volume

first upscaled volume

second upscaled volume

Spatially average the heterogeneities

reduces heterogeneity

creates anisotropy

(smooths)

Heterogeneity: flow parallel to layers steady flow

hA

b1 A

K1 b2

K2 b3

hB B

K3

#L 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?

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Heterogeneity: flow parallel to layers How much water flow from the reservoir at A to the reservoir at B?

Discharge in each layer [L3/T]:

in each layer (same):

!h !L !h Q2 = " K 2 (b2 w) !L !h Q3 = " K 3 (b3 w) !L Q1 = " K 1 (b1 w)

hA – hB = -#h Width (into page): w “Area” of layer i: wbi, i=1,2,3

b1

hA

K1 b2

K2 b3

hB

or, generalizing for layer i,

Qi = " K i (bi w)

K3

!h !L

where i=1,2,3 #L

Heterogeneity: flow parallel to layers How much water flow from the reservoir at A to the reservoir at B?

Total Discharge over all three layers?

Discharge in each layer [L3/T]:

Use conservation of mass or continuity: Total discharge = sum of layer discharges 3

Qtotal = ! Qi = Q1 + Q2 + Q3 i =1

b1

hA

K1 b2

K2 b3

K3

hB

!h !L !h Q2 = " K 2 (b2 w) !L !h Q3 = " K 3 (b3 w) !L Q1 = " K1 (b1 w)

or, generalizing for layer i,

Qi = " K i (bi w)

!h !L

where i=1,2,3: #L

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Heterogeneity: flow parallel to layers How much water flow from the reservoir at A to the reservoir at B?

Total Discharge over all three layers? Use conservation of mass or continuity: Total discharge = sum of layer discharges 3

Qtotal = ! Qi = Q1 + Q2 + Q3

or

Qtotal = " w

i =1

b1

hA

Later, we' ll introduce another term for flow parallel to layers :

K1 b2

hB

K2 b3

!h ( K1b1 + K 2 b2 + K 3b3 ) !L

K3

"h "h (T1 + T2 + T3 ) = # w T "L "L where, in this example layer transmissivity Ti = K i bi Qtotal = # w

3

3

total transmissivity T = ! Ti = ! K i bi i =1

i =1

#L

Heterogeneity: flow parallel to layers 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?

Specific discharge in layer i [L/T]:

Discharge in layer i [L3/T]:

Qi = " K i (bi w)

!h !L

qi =

vi = b1

vi =

qi K !h = " i K1 ne ,i ne ,i !L

b2

hB

K2 b3

xB x

#L

qi K !h depends on Ki & ne,i =" i ne ,i ne ,i !L

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

ti =

K3

xA

depends on Ki

Seepage velocity in layer i [L/T]:

where i=1,2,3:

hA

Qi !h = "Ki bi w !L

1

" v dx =

xA

i

!L vi (1

' !h \$ 1 = (ne1,i K i(1 !h (1 !L2 = (ne1,i K i(1 % " !L & !L #

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Effective Conductivity: • Replace a heterogeneous porous media with an upscaled “equivalent” homogeneous porous media

• Find the effective property of that equivalent media that preserves the upscaled metric of interest – Here that is total discharge – But there are other metrics (eg, solute flux) leading to other effective properties (eg, macrodispersion)

• Effective hydraulic conductivity – Is the equivalent homogeneous Keff that preserves the total discharge, Qtotal

Effective Conductivity: flow parallel to layers Recall:

Qtotal = " K eff (btotal w)

Total Discharge over all three layers

!h !L

btotal = b1 + b2 + b3

From conservation of mass or continuity: Total discharge = sum of layer discharges 3

Qtotal = ! Qi = Q1 + Q2 + Q3

or

i =1

!h ( K1b1 + K 2 b2 + K 3b3 ) !L

Equate the Qtotal’s

b1

hA

Qtotal = " w

K1 b2

K2 b3

hB

K3

#L

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Effective Conductivity: flow parallel to layers Equate the two terms:

"w

!h !h ( K1b1 + K 2b2 + K 3b3 ) = " w K eff (btotal ) !L !L

Eliminate common elements:

( K1b1 + K 2b2 + K 3b3 ) = K eff (btotal )

Solve for the effective conductivity :

K eff = =

K1b1 + K 2b2 + K 3b3 btotal K1b1 + K 2 b2 + K 3b3 b1 + b2 + b3

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:

K eff =

! Kb !b

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