# faculty kfupm edu sa ES shaibani Basic Princip porosity

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```					Groundwater flow

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Basic principles of GW Flow
Porosity and effective porosity
   Total porosity is defined as the part of rock that's void space
nT = Vv/VT = (VT – Vs)/VT
Vv :     void volume,
VT :     total volume
Vs:      solid volume

   void ratio
e = Vv/Vs
 Primary porosity: interstitial porosity (original in the rock)

   Secondary porosity: fracture or solution porosity
   Primary porosity range from 26% to 47% (using different
arrangements and packing of ideal spheres).

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Porosity of Sediments and Rocks

   Depending on grain size, particle shape and arrangement,
diagenetic features, actual values of porosity can range from
zero or near zero to more than 60%.
   In general, for sedimentary rocks, the smaller the particle
size, the higher the porosity.
   Total porosity amount of pore space (does not require
pore connection)
   Effective porosity: percentage of interconnected pore
space available for groundwater flow.
   Effective porosity can be one order of magnitude smaller
than total porosity (difference greatest in fractured rocks).

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Measurement of porosity

   In the lab, porosity is measured by taking a
sample of known volume (V),
    sample is dried in an oven at 105oC until it
reaches a constant weight (expelling moisture).
   Dried sample is then submerged in a known
volume of water and allowed to remain in a sealed
chamber until saturated
   Volume of voids is equal to original water volume
minus volume in the chamber after saturated
sample is removed. Result is effective porosity.

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   Total porosity is found from:

n  100 [1  (  b /            s ]
   bulk density is mass of sample after dried
divided by original sample volume, particle
density is oven-dried mass divided by
volume of solid determined from water-
displacement test.
   In most rocks and soils, particle density is

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n = 47.65 %         n = 25.95%

(a) Cubic packing of spheres
(b) Rhombohedral packing
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Darcy’s Experimental Law

  Darcy's Experimental Law :
 water was passed through a sand column and the volumetric
flow rate Q was measured at the outlet
 The cross-sectional area of the sand column was known, as
was the length of the sand in the column. During the
experiment, Darcy measured the distance between the water
levels in the two manometers at various flow rates.
He tabulated Q, A, L , and (h1 - h2). He calculated Q/A and (h1-h2)/L.
Q/A is a volumetric flow rate per unit surface area and is termed
specific discharge.

Q      K (h1  h2 )      h
q               K
A           l            l

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Darcy’s Experimental Law (contd.)
   Darcy's law is stated as:
The velocity of flow is proportional to hydraulic gradient
 Darcy's law is valid for flow through most granular material.
 The law holds as long as flow is laminar.
 In turbulent flow, water particles take more circuitous paths.
Darcy's velocity (q)
 Darcy's q is a "superficial" velocity.
 Actual velocity v is the volumetric flow rate per unit area of
connected pore space.
 Therefore, v = q/ne = (Ki/ne) , where neA is the effective area of
flow and ne is the effective porosity. v is the linear velocity of
groundwater. v is always larger than the superficial velocity and
increases with decreasing effective porosity.

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Henry Darcy

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Other forms of Darcy's law

q = Ki
Q = KiA

v = Ki/ne

v = q/ne
v and q are both vector quantities (with
magnitude and direction).
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Darcy’s experiment

Q      h
qK
A      l

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Darcy’s Law, contd.
   Actual velocity (linear gw velocity)
= v = q / ne
ne = effective porosity
   Applicability of Darcy’s Law:
   Laminar flow

–Turbulent flow

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Applications of Darcy’s Law

 Predict   groundwater flow to a
well
 Predict rate and direction of
contaminants movement
different locations in an aquifer
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   wl's are measured with reference to a common datum, taken
arbitrarily at the base of the sample. Absolute values of wl
elevations were of no concern to Darcy (only the difference
between them). We are concerned here with the actual
water level elevations; and what they mean.

   Manometers:
   devices to measure wl elevations in the lab
   Piezometers:
    a tube or a pipe to measure wl elevations in field. It's open at
top where measurements are taken, and open at bottom to
facilitate entrance of water.
   A common datum is sea level (elevation zero).

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   Total head is a function of:
   Bernoulli equation.
under conditions of steady flow, total
energy of an incompressible fluid is
constant at all positions along a flow path in
a closed system. This may be written as

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(total energy contained in the water):
P   v2
gz         constant
w 2
g:    acceleration due to gravity
z:    elevation of base of piezometer
P:    pressure exerted by water column,
:    fluid (water) density
v:           velocity

Divide through by g to get (next slide):

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P    v2
z          constant
g w 2 g

Equation above describes total energy
contained by the fluid.
   1st term: energy of position

   2nd term: energy due to sustained fluid pressure

   3rd term: energy due to fluid movement

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   velocity head is ignored (slow movement)

P  v2
h z      
g w 2 g
   Stated simply:
total head (h) is the sum of elevation of the base
of piezometer and length of water column in
piezometer

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Example 3.1
   gs elevation: 1000 m
   DTW:            25 m
   peizometer: 50 m
   Water density: 1000 kg/m3

Find:
(c) pressure

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Example 3.1
Solution:

(c)   pressure

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   Total head =          h = hp + z
distance
i = dh / dx
   in vector form, gradient may be written as:

h    h  h
grad h = h=      i       j k
x    y  z

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   Equipotential lines =             h = 120m

Lines of equal hh         h = 100m

h = 80m

interval/horizontal      Direction of
groundwater         Angle = 90
distance                 flow

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3-point problem

B

A             E

C

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Hydraulic conductivity and
permeability
   Hydraulic conductivity (K) is constant of
proportionality in Darcy’s Experimnet
Q      h
qK
A      l
 HC: ease with which groundwater flows
through the porous medium
 Sands& gravels:      high K
 Clay& shales:        low K
 Units:   [L/T] e.g. m/d, ft/d, gpd/ft2
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Example 3.3:
     Groundwater flows through a buried
valley aquifer with a cross sectional area
of 1 x 106 ft2,and a length of 2x 104 ft.
     Hydraulic head at gw entry = 1000 ft
      Hydraulic head at gw exit = 960 ft
     Groundwater discharge = 1 x 105 ft3/day
(1)   what’s HC of aquifer in ft/day, m/d.
(2)   If effective porosity = 0.3, find the
linear groundwater velocity.

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Intrinsic Permeability
k w g
K 

w:   density of water         [kg/m3]
g:    accelration of gravity   [m/sec2]
:    viscosity                [kg/(m.sec)]
k:    intrinsic permeability   [m2]
K:    hydraulic conductivity   [m/sec]

1 m2 = 104 cm2    = 1.013x1012 Darcy

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Intrinsic Permeability

k w g
K 
Example 3.4               

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Empirical Approaches for estimation

K  Cd
   Hazen                      2
10

4)
   Harleman
k  (6.54x 10 d       2
10

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Grain-Size Distribution

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Laboratory Measurements of K

 Field Tests
 Empirical relations
 Lab measurements

   What’s the most reliable?

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Laboratory Measurements of K

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QL
K 
Ah

aL               h0
K  2.3                log10
A (t 1  t 0 )       h1

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Mapping Flow in Geological Systems

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Mapping Flow in Geological Systems

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Mapping Flow in Geological Systems

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Mapping Flow in Geological Systems

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QL
K 
aL Ah            h0
K  2.3                log10
A (t 1  t 0 )       h1

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