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 • Increases with K K K µ µ

K is a property of both the fluid & the porous medium • We get: k g K = µ and Q k dh q = = • we can also now A µ dl express Darcy’s Law kg dh in terms of these = quantities: µ dl kg dh = dl

5 K is a property of both the fluid & the porous medium • For example, using the empirical model, Cd 2 g 2 K = k = C d µ

2 • and Q Cd g dh q = = A dl

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: 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

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!

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) Good aquifers: 10-5 < K < 10-3 m/s : -11 -7 Aquitards 10 < K < 10 m/s Typical 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 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

Natural Variation of Porosity, n • Its varies much less, but the variation is still important.

• In nature, SZ 2005 – 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 (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 Property R constant O with P I direction C

A N I S Property O changes T with R O direction P I C 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

b 1 K1 h A hB A B b2 K2

b3 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 h – h = -h Q1 = K1 (b1w) A B L Width (into page): w h Q2 = K 2 (b2 w) “Area” of layer i: wbi, i=1,2,3 L h Q3 = K 3 (b3 w) L

b1 K 1 or, generalizing for layer i, hA h b K B 2 2 !h Q = "K (b w) b K i i i 3 3 !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 Discharge in each layer [L3/T]: over all three layers? !h Q = "K (b w) Use conservation of mass or continuity: 1 1 1 !L Total discharge = sum of layer discharges 3 !h Q Q Q Q Q Q2 = "K2 (b2 w) total = i = 1 + 2 + 3 !L i=1 !h Q = "K (b w) 3 3 3 !L

b1 K 1 or, generalizing for layer i, hA h b K B 2 2 !h Qi = "Ki (bi w) b K 3 3 !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 !h Q Q Q Q Q or Q w (K b K b K b ) total = ! i = 1 + 2 + 3 total = " 1 1 + 2 2 + 3 3 i=1 !L

b Later, we'll introduce another term for flow parallel to layers : 1 K1 "h "h Q = #w (T + T + T ) = #w T hA h total "L 1 2 3 "L b K B 2 2 where, in this example layer transmissivity T = K b b K i i i 3 3 3 3 total transmissivity T T K b = ! i =! i i 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? Discharge in layer i [L3/T]: Specific discharge in layer i [L/T]: Qi !h !h qi = = "Ki depends on Ki Qi = "Ki (bi w) b w !L !L i where i=1,2,3: Seepage velocity in layer i [L/T]:

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

b q K !h 1 v = i = " i K1 i n n L h e,i e,i ! Travel time A to B in layer i [T]: A hB b2 K2 x B 1 !L b K t = dx = 3 3 i " v v xA i i x x (1 A B 1 (1 (1 2 1 (1 ' !h $ 1 = (ne,i Ki !h !L = (ne,i Ki % " !L x 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 !h Recall: Q = "K (b w) total eff total !L Total Discharge b b b b over all three layers total = 1 + 2 + 3 From conservation of mass or continuity: Total discharge = sum of layer discharges 3 !h Q Q Q Q Q or Q w (K b K b K b ) total = ! i = 1 + 2 + 3 total = " 1 1 + 2 2 + 3 3 i=1 !L

Equate the Qtotal’s b 1 K1 h A hB b2 K2

b3 K3

L

16 Effective Conductivity: flow parallel to layers !h !h Equate the two terms: " w (K b + K b + K b ) = "w K (b ) !L 1 1 2 2 3 3 !L eff total

Eliminate common elements: (K1b1 + K2b2 + K3b3 ) = Keff (btotal )

K1b1 + K2b2 + K3b3 Solve for the effective conductivity : Keff = btotal K b + K b + K b = 1 1 2 2 3 3 b1 + b2 + b3

The “effective conductivity” is the equivalent homogenous K that results in the same discharge. It’s ! Kb the result of “upscaling” or “spatial averaging” over the Keff = heterogeneities. In general, for flow parallel to layers: !b

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