![Two types of Porosity
• Intergranular
o Between grains, mostly part of the effect of porosity, but also dead-
end pores
• Intragranular
o Within grains
o Usually not considered part of the effective porosity
o Incredible wide range of widths and length scales
• Simple dichotomous model - dual porosity
Darcy’s Law
In 1856 in Dijon, France, Henry Darcy conducted his now famous experiment of
pouring water through sediment-packed pipes to see how much would flow through
them in a given amount of time [volume of flow per unit time].
Flow through column is Q in L3
/T most important quantity
The flow per unit area is specific discharge
q =
Q
A
with units of velocity L/T –
called Darcian velocity or
Darcian flux, but not actual
velocity of the fluid
Q
∆h
h1
h2
∆l
Area, A
Flow, Q
Darcy showed that:
Q is in direction of decreasing head
q is proportional to h2 – h1 = ∆h, given ∆l fixed, q α (-∆h)
q is inversely proportional to ∆l, given ∆h fixed, q α (1/∆l)
The proportionality constant is K, and flow is from higher to lower hydraulic head.
1.72, Groundwater Hydrology Lecture 2
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![• Term 1 is loss – viscous friction against wall of solids and
• Term 2 is loss – dissipation of kinetic energy in pores – flow converges and
diverges.
Must have laminar flow within pores.
ρ d
v Inertial forces
R = ≡
e
µ Viscous forces
Laminar in pipes < 2,000 in rocks < 10
U
Darcy’s Law
Real Law
L0 L2 L1
Measures of hydraulic conductivity (L/T)
• Commonly cm/s, m/d, ft/d
• Older unit, gpd/ft2
, or meinzer
Measures of permeability, (L2
)
Often the Darcy Unit is used, recall
q −
=
dh
g
k
Q = 1 cm/s −
=
g
kρ dh
ρ
µ dl cp dl
1 darcy is the permeability that gives a specific discharge of 1 cm/s for a fluid with a
viscosity of 1 cp under a hydraulic gradient times density times g of 1 atm/cm.
2
• It equals about 10-8
cm
• About 0.831 m/d at 20o
Important – A typical aquifer measure of the transmission property of media for the
flow of water is given over a thickness, b
Transmissivity – T = Kb [L2
/T] 2D
• Very common quantity for site and regional studies
• Much more on this when we get to groundwater flow equation and well tests
1.72, Groundwater Hydrology Lecture 2
Prof. Charles Harvey Page 8 of 10](https://image.slidesharecdn.com/lecturegroundwaterhydrology-250624171339-ef5822dd/85/lecture_Groundwater-Hydrology-and-model-pdf-8-320.jpg)
![C
Relations Between Grain Size and Hydraulic Conductivity
2
10 )
(d
C
K =
Equation
Hazen (1911)
Reference
)
31
.
1
(
)
760
)(
04
66
.
9
(
2
g
g EXP
d
E
K σ
−
−
= Krumbein and Monk (1942)
( )2
3
2
1
180 φ
φ
µ
ρ
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
d
g
K
Kozeny-Carman (in Bear, 1972)
( )2
2
0
3
1 φ
φ
µ
ρ
−
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
sa
S
C
g
K
Kozeny-Carman (in de Marsily, 1986)
( )2
3
sa
T S
T
C
g
K
φ
µ
ρ
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
Kozeny Equation, modified by Collins
(1961)
C0 = factor reflecting pore shape and packing in the Kozeny-Carmen eqn. [-]
CT = factor reflecting pore shape and packing in Kozeny eqn, mod. By Collins [-]
= factor in the Hazen equation [T/L]
d10 = grain diameter for which 10% of particles are smaller [L]
dg = geometric mean grain diameter [L]
d = representative grain diameter [L]
g = gravitational acceleration [L/T2
]
K = hydraulic conductivity [L/T]
φ = total porosity, accounting for compaction [-]
µ = dynamic viscosity [M/LT]
ρ = density [M/L3
]
σg = geometric mean standard deviation [L]
Ssa = surface area exposed to fluid per unit volume of solid medium [1/L]
T = tortuosity [-]
1.72, Groundwater Hydrology Lecture 2
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![Remember Darcian Velocity is not an actual velocity; it is discharge per unit area
(area is TOTAL cross section)
Darcian Velocity
Total Area
Q
Area of
Pore Space
Acts like
this
Average Linear Velocity
Primary porosity – the original interstices
Secondary porosity – secondary dissolution or structural openings (fractures,
faults, and openings along bedding planes).
Computed porosity - n = 100[1-ρb/ρp]
ρb – bulk density, M/L3
-> mass of dry sample/original volume
ρp – particle density, M/L3
-> mass of dry sample/volume of mineral matter from
water-displacement test (2.65 g/cc)
Effective porosity – porosity available for flow, ne
Can have isolated water as dead-end pores or trapped gas. IMPORTANT to
transport!
− K ∆h − K dh
V = =
n ∆l n dl
e e
V is the average linear pore water velocity. Measure of the rate of advection
of a slug of water. V is larger than the Darcian Velocity. -> q = ne V
1.72, Groundwater Hydrology Lecture 2
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lecture_Groundwater Hydrology and model.pdf
- 1. 1.72, Groundwater Hydrology Prof. Charles Harvey Lecture Packet #2: Aquifers, Porosity, and Darcy’s Law Lake River zone Saturated Infiltration (Exposed Water Table) Precipitation Unsaturated Water table zone - Aquifer Unsaturated Zone, Vadose Zone, Soil Moisture Zone, Zone of Aeration – rock, water and air Capillary Fringe – region above water table where water rises due to capillary forces in the porous medium. Saturated Zone, Phreatic Zone – rock and water Water Table • Top of saturated zone • Depressed version of topography • Surface waters are manifestations of the water table – exposed water table Aquifer - a geologic unit that stores and transmits water Recharge Area Unconfined Aquifer Confined or Artesian Aquifer Water Table 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 1 of 10
- 2. Unconfined Aquifer – water is in contact with atmospheric pressure – drill and well hit the water table Confined Aquifer – recharge upgradient forces water to flow down and get trapped under an aquiclude. Water is under pressure due to the weight of the upgradient water and the confinement of the water between “impermeable” layers. Water flows to surface under artesian pressure in an Artesian Well. Aquifer contamination Contaminant Source Unconfined Aquifer Confined or Artesian Aquifer Water Table In confined vs. unconfined aquifers • Although unconfined aquifers are used for water supply, they are often contaminated by wastes and chemicals at the surface. • Confined aquifers are less likely to be contaminated and thereby provide supplies of good quality. • Mechanisms of transport are advection and dispersion. • There can be chemical interactions in aqueous phase or between the water and solid media • Covered in Contaminant Hydrology course (1.72 is a prerequisite) 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 2 of 10
- 3. In A.D. 1126, in the province of Artois: Unconfined Aquifer Unconfined Aquifer Water Table Confined or Artesian Aquifer Water Table Confined or Artesian Aquifer Key Aquifer Properties Porosity – Percentage volume occupied by voids. Porosity is independent of scale. For example, a pile of marbles and a pile of beach balls have spherical shape and differing sizes; the porosities are identical due to the similar shaping. Permeability – Measures the transmission property of the media and the interconnection of the pores. Related to hydraulic conductivity and transmissivity. (more later) Good aquifer – High porosity + High permeability • Sand and gravel, sandstone, limestone, fractured rock, basalt Aqiuiclude, Confining bed, Aquitard – “impermeable” unit forming a barrier to groundwater flow. • Granite, shale, clay 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 3 of 10
- 4. Porosity Poorly Sorted Fracture Intragranular Well Sorted Decreased Porosity Dissolution by Diogenesis Diogenesis – The formation of rock; pores fill up with precipitations of mineral and reduce porosity 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 4 of 10
- 5. Two Origins of Porosity Primary • A function of grain size distribution, also packing • Decreases with depth – compaction and pressure solution 0 2 3 4 5 6 7 1 Shales Porosity vs. Depth Curves Depth (kilometers) (Athy, 1930) Sandstones (Blatt, 1979) 10 20 30 40 50 Porosity Porosity increases as depth decreases. This is on account of the weight on top of the deeper materials. Porosity also tends to increase with grainsize. Why? Secondary • Dissolution • Fracture Lithology Q Calcite Fracture Number uartz SS Cemented SS Limestone 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 5 of 10
- 6. Two types of Porosity • Intergranular o Between grains, mostly part of the effect of porosity, but also dead- end pores • Intragranular o Within grains o Usually not considered part of the effective porosity o Incredible wide range of widths and length scales • Simple dichotomous model - dual porosity Darcy’s Law In 1856 in Dijon, France, Henry Darcy conducted his now famous experiment of pouring water through sediment-packed pipes to see how much would flow through them in a given amount of time [volume of flow per unit time]. Flow through column is Q in L3 /T most important quantity The flow per unit area is specific discharge q = Q A with units of velocity L/T – called Darcian velocity or Darcian flux, but not actual velocity of the fluid Q ∆h h1 h2 ∆l Area, A Flow, Q Darcy showed that: Q is in direction of decreasing head q is proportional to h2 – h1 = ∆h, given ∆l fixed, q α (-∆h) q is inversely proportional to ∆l, given ∆h fixed, q α (1/∆l) The proportionality constant is K, and flow is from higher to lower hydraulic head. 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 6 of 10
- 7. ∆h dh Hydraulic gradient q − = K − = K ∆l dl K is hydraulic conductivity and has units of velocity (L/T). It is a function of both media and fluid. Q is a flow per unit cross section and is not the actual velocity of groundwater flow. ∆h represents the frictional energy loss due to flow through media. Darcy’s law is a macroscopic law. It doesn’t tell you about the flow through individual pores. The discharge is Q in L3 /T ∆h q − = K − = KiA ∆l i is commonly used for gradient. Note the difference between Q and q! What is hydraulic conductivity? K is a property of both media and fluid. g kρ Experiments show: K = µ • K is the intrinsic permeability (L2 ), a property of media only • ρ is the mass density (M/L3 ) • µ is the dynamic viscosity (M/LT) and measures the resistance of fluid to shearing that is necessary for flow Range of Applicability of Darcy’s Law At extreme gradients some have questioned the applicability of Darcy’s Law (controversial) Low Gradients • Compacted clays and low gradients • Threshold gradient to get flow • Below a certain gradient – nonlinear High Gradients dh = q c + q c 2 dl 1 2 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 7 of 10
- 8. • Term 1 is loss – viscous friction against wall of solids and • Term 2 is loss – dissipation of kinetic energy in pores – flow converges and diverges. Must have laminar flow within pores. ρ d v Inertial forces R = ≡ e µ Viscous forces Laminar in pipes < 2,000 in rocks < 10 U Darcy’s Law Real Law L0 L2 L1 Measures of hydraulic conductivity (L/T) • Commonly cm/s, m/d, ft/d • Older unit, gpd/ft2 , or meinzer Measures of permeability, (L2 ) Often the Darcy Unit is used, recall q − = dh g k Q = 1 cm/s − = g kρ dh ρ µ dl cp dl 1 darcy is the permeability that gives a specific discharge of 1 cm/s for a fluid with a viscosity of 1 cp under a hydraulic gradient times density times g of 1 atm/cm. 2 • It equals about 10-8 cm • About 0.831 m/d at 20o Important – A typical aquifer measure of the transmission property of media for the flow of water is given over a thickness, b Transmissivity – T = Kb [L2 /T] 2D • Very common quantity for site and regional studies • Much more on this when we get to groundwater flow equation and well tests 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 8 of 10
- 9. C Relations Between Grain Size and Hydraulic Conductivity 2 10 ) (d C K = Equation Hazen (1911) Reference ) 31 . 1 ( ) 760 )( 04 66 . 9 ( 2 g g EXP d E K σ − − = Krumbein and Monk (1942) ( )2 3 2 1 180 φ φ µ ρ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = d g K Kozeny-Carman (in Bear, 1972) ( )2 2 0 3 1 φ φ µ ρ − ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = sa S C g K Kozeny-Carman (in de Marsily, 1986) ( )2 3 sa T S T C g K φ µ ρ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = Kozeny Equation, modified by Collins (1961) C0 = factor reflecting pore shape and packing in the Kozeny-Carmen eqn. [-] CT = factor reflecting pore shape and packing in Kozeny eqn, mod. By Collins [-] = factor in the Hazen equation [T/L] d10 = grain diameter for which 10% of particles are smaller [L] dg = geometric mean grain diameter [L] d = representative grain diameter [L] g = gravitational acceleration [L/T2 ] K = hydraulic conductivity [L/T] φ = total porosity, accounting for compaction [-] µ = dynamic viscosity [M/LT] ρ = density [M/L3 ] σg = geometric mean standard deviation [L] Ssa = surface area exposed to fluid per unit volume of solid medium [1/L] T = tortuosity [-] 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 9 of 10
- 10. Remember Darcian Velocity is not an actual velocity; it is discharge per unit area (area is TOTAL cross section) Darcian Velocity Total Area Q Area of Pore Space Acts like this Average Linear Velocity Primary porosity – the original interstices Secondary porosity – secondary dissolution or structural openings (fractures, faults, and openings along bedding planes). Computed porosity - n = 100[1-ρb/ρp] ρb – bulk density, M/L3 -> mass of dry sample/original volume ρp – particle density, M/L3 -> mass of dry sample/volume of mineral matter from water-displacement test (2.65 g/cc) Effective porosity – porosity available for flow, ne Can have isolated water as dead-end pores or trapped gas. IMPORTANT to transport! − K ∆h − K dh V = = n ∆l n dl e e V is the average linear pore water velocity. Measure of the rate of advection of a slug of water. V is larger than the Darcian Velocity. -> q = ne V 1.72, Groundwater Hydrology Lecture 2 Prof. Charles Harvey Page 10 of 10