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
751 downloads 0 Views 2MB Size
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]

1

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

2

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:

3

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

4

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=

5

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

6

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

7

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:

8

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?

9

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.

10

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

11

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

12

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?

13

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

Head difference

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

14

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 #

15

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

16

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

17