Clay Blanket Calculations by iin12614

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									          Water flow in saturated soil

                          D A Cameron
                   Civil Engineering Practice 1




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   1
               SEEPAGE – water pressures
                      Water flows from points of high
                             to low TOTAL head

         WATER HEADS
         [“height of water”] x [w] = water pressure, u


         Total head = [elevation head + pressure head]

         i.e                     h = hT = [he + hp]

                      Kinetic head is ignored in soils

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                             Head of Water
       Pressure head = height water rises to in a
       standpipe above the point

                                                               No loss of
      Water table level                                        head, h, in soil
                                      hp                       mass,
                                                 h             so no flow
                                                               - Steady State
                                      he
                                                     Arbitrary
                                                     datum
      Element of soil
      within soil mass
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                          Confined Aquifer
       A water bearing layer, overlain and underlain
       by far less permeable soils.

        Water level in aquifer                                standpipe

                                                                  Clay, silt
                                                                  - no free water


                                                          x
                  Clay, silt


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    Steady flow in soils – Laminar flow

     Assumptions to theory:

     • Uniform soil, homogeneous & isotropic
     • Continuous soil media
     • Small seepage flow (non turbulent flow)
               Darcy’s Law of 1850 – a Frenchman


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                               Darcy’s Law

                                 q = kiA

   where               q         =         rate of flow (m3/s)
                       i         =         hydraulic gradient
                       A         =         area normal to flow
                                           direction (m2)
                       k         =         coefficient of
                                           permeability (m/s)


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                     Hydraulic Gradient, i

                                                h


                                                                      Area of
                                                                      flow, A



 Flow rate,
        q
                                            Length of flow, l


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                   Hydraulic Conductivity

    • Coefficient of permeability or just “permeability”

    • SATURATED soil permeability


  Hazen’s formula, for clean, almost uniform sands:

                                        2
                                       d10
                                    k
                                       100
   m/sec if particle size in mm

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         TYPICAL PERMEABILITIES

   Clean gravels                                           > 10-1 m/s

   Clean sands, sand-gravel 10-4 to 10-2 m/s

   Fine sands, silts                                 10-7 to 10-4 m/s

   Intact clays, clay-silts                          10-10 to 10-7 m/s




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                  Measuring Permeability
    [A] Laboratory                                 [A] Laboratory
    • Constant head test                             How good is the
    • Falling head test                              sample?
    • Other

    [B] Field                                       [B] Field
    • Pumping tests                                 Need to know soil
    • Borehole infiltration                         profile (incl. WT) &
                                                    boundary conditions
        tests

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     Lab Test 1: Constant head test

    • Cylinder of saturated coarse grained soil
    • Water fed under constant head
         ─ elevated water tank with overflow

    • Rate of outflow measured

       Repeat the above after raising the water tank



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   11
         1. Constant head permeameter
                 Water tank -
                  moveable



                                                                      ht

                     q                A
                                                                      hpB hpC
                                      B
                                                          l           he
                                       C
                                       D
                                                  soil
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT             12
                     Constant head test
             Suitable for clean sands and fine gravels

         EXAMPLE:
       • If the sample area is 4500 mm2,
       • the vertical distance between the 2
         standpipe points is 100 mm,
       • h is 75 mm
       • Outflow is 1 litre every minute

           What is the coefficient of permeability?

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   13
                                   Solution
       • 1000 cm3/min

           OR q = 16.7 cm3/sec = 16.7x10-6 m3/sec

       • i = 75/100 = 0.75

       • k = q/(iA)

              = (16.7x10-6)/(0.75x4500x10-6) m/sec

           k = 5 x 10-3 m/sec

              Typical permeability of a clean sand or gravel


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   14
      Test 2: Falling head permeameter

       For fine sands, silts, & maybe clays
    • Rate of water penetration into cylindrical
       sample from loss of head in feeder tube
    • Must ensure:
         − no evaporation
         − sufficient water passes through

                               A slow procedure


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   15
            2. Falling Head Permeameter
                                                   Level at time, t1
                                                 Level at time, t2
    Tube of cross-sectional
    area 'a'

                                                                       h1
                                                                  h2

      To permeameter cell



                                                       Level of cell outflow

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT            16
                    Falling head test
        • Soil sample length, L, & area, A

        • Flow in the tube = flow in the soil
             – tube has area “a”


                    a  L   h1 
                 k              ln 
                    A  (t 2 t1 )   h 2 
                                   


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   17
        3. Field testing – drawdown test
                   Pumping well                                       Water
           q
                                   r2                                 table

                                    r1


                                                             h2
                                                h1


                                           Impermeable boundary
       Drawdown –
       phreatic or
       flow line
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT           18
                        Drawdown test
      Needs
      1. a well-defined water table and
      2. a confining boundary

      Must be able to
      1. pull down water table and
      2. create flow
            (phreatic line = uppermost flow line)



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   19
                                 Solution

                             Axi-symmetric problem

          By integration of Darcy’s Law,


                             q          r2 
                      k              ln  
                         π(h 2 h 2 )   r 
                             2   1   1 




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   20
                  TUTORIAL PROBLEMS
    A canal and a river run parallel, an average of 60 m
      apart. The elevation of water in the canal is 200 m
      and the river 193 m. A stratum of sand intersects both
      the river and canal below the water levels

    The sand is 1.5 m thick and is sandwiched between
      strata of impervious clay

    Compute the seepage loss from the canal in
     m3/s per km length of the canal, given the
     permeability of the sand is 0.65 mm/s

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   21
                           THE PROBLEM

   Sand
   seam


                       RL 200 m                          RL 193 m




                     canal
                                                           river
                                     60 m


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   22
                                 SOLUTION

              q = kiA
              k = 0.65 mm/s = 0.65 x 10-3 m/s
              h = 7 m

              q = 0.65 x 10-3 x 0.117 x 1.5 m2/m length
              q = 0.114 x 10-3         m3/sec /m length
              q = 0.114               m3/sec/km length



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   23
                                   Hydraulic
                               gradient, i = 0.117


                       RL 200 m                          RL 193 m

                                                h = 7 m




                                   l = 60 m


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   24
                                Flow Lines
           – shortest paths for water to exit

                                                Equipotential lines


                         hp1                                          h

           Flow
           tube                                                       hp2

                         h1e1                  l                     he2

                           Elevation head reference line
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT         25
                         The Flow Net
                         - FLOW LINES

         Run  parallel to impervious boundaries
         (impermeable walls or “cut-offs”) and the
         phreatic surface
         The “Phreatic surface” is the top flow line

               2 consecutive flow lines constitute
                                   a “flow tube”

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   26
                                                             h


      5 Flow Lines




                                  Impervious boundary

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   27
                              The Flow Net
                   - EQUIPOTENTIALS
           • Are lines of equal total head

           • The total head loss between consecutive
              equipotentials is constant

           • Equipotentials can be derived from
              boundary conditions and flow lines



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   28
                       Flownet Basics
       Water flow follows
       paths of maximum
       hydraulic gradient,
                                                                 h i
       imax                                             i m ax 
                                                                 b m in
     flow lines and
       equipotentials must
       cross at 90°, since:


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT       29
   Since q is the same, ratio of
   sides will be constant for all
   the “squares” along the flow
   tube

                                                               h




                                       M




Equi- potential lines
                                   Impervious boundary
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   30
                                                 h i 
             Flow
              q
                                          q  k      a  per m
                                                 b 
                                q  constant,if a : b  is constant

                                 hi  head lost between equipotent
                                                                  ials


        b
                           a
                                                 Common convention
                                                 draw “squares” with a = b

            “square, M”, a x b


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT          31
                                         Discharge in flow direction,
      Equipotentials                            = q per “flow tube”

                                                                      h3

                  90º

                                l
                                                            h2
  Flow lines

                                       h1


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT        32
                       Flownet Construction




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   33
                    Flow Net Calculations
          Nd equal potential drops along length of flow?
          Then the head loss from one line to another is:
                          h1-2 = (h) = h / Nd

          From Darcy’s Law, flow rate in a flow tube,
                                        Δh i 
                              ΔqkiA k      a1
                                        b 
                              OR
                                   Δh  a 
                                   N  b 
                              Δqk     
                                   d
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   34
                    Flow Net Calculations

       BUT              a=b
       AND              total flow for Nf “flow channels”,
                        per unit width is:


                                  Nf
                              qk    Δh
                                  Nd

                          But only for “squares”!


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   35
      Example: if k = 10-7 m/sec, what would be the
        flow per day over a 100 m length of wall?


                                           Dam
                                               cutoff
    50 m of water
                                                                5 m of water




                             Low permeability rock

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT            36
                               Calculations

         Nf = 5                             Answer:
         Nd = 14
         h = 45 m                          = 10-7(5/14)45 x 100 m length
         k = 10-7 m/sec                     = 0.000161 m3/sec
                                            = 13.9 m3/day




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT     37
           Example: what is the hydraulic gradient
                    in the “square” C?

                                           Dam
                                              cutoff
   50 m of water
                                                               5 m of water




                             Low permeability rock


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT           38
                        Calculations

            [h / Nd] = 45/14                         Answer: 1.1 and
                      = 3.2 m                           therefore
            head per drop                               dangerous!


            Average length of
            flow is about 3 m



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT     39
            Critical hydraulic gradient, ic
           The value of i for which the effective
           stress in the saturated system becomes
           ZERO!

           Consequences:

           no stress to hold granular soils together

            soil may flow 

                 “boiling” or “piping” = EROSION

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   40
    Seepage Condition – upward flow of water
                                               = satz = total stress
                                              u due to seepage,
           B
                                                   = i(z)(w)
                                     z
                                              (represents proportion of h
           A                                  occurring over length AB)
                                               =  - u
                                                  = (satz) – (wz + i(z)w)
                                               = z – i(z)w
          = 0, when           z = i(z)w         OR        i = (/ w)

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT            41
                  Likelihood of Erosion
               GRANULAR SOILS chiefly!
               When the effective stress becomes zero,
               no stress is carried by the soil grains

               Note: when flow is downwards, the effective
               stress is increased!

               So the erosion problem and ensuing
               instability is most likely for upward flow,
               i.e. water exit points through the foundations of
               dams and cut-off walls

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   42
DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   43
           Minimising the risk of erosion
                1. Add more weight at exit points



                                                        permeable concrete mats?




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT         44
                     Lengthen flow path?
                            1. Deeper cut-offs
                            2. Horizontal barriers
                            3. Impermeable blanket on exit surface




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   45
                    Simple cut-offs (FESEEP)


                                                      Nf = 5
                                                          Nf
                                                      Nd =10= 5
                                                          NN=115
                                                            d f=

                                                              Nd =13




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT    46
              “Impermeable” Clay Blanket




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   47
                  Summary: Key Points
              •   Heads in soil
              •   Darcy’s Law
              •   Coefficient of permeability
              •   Measurement of permeability
              •   Flownets
              •   Flownet rules
              •   Seepage from flownets
              •   Piping, boiling or erosion
              •   Critical hydraulic gradient

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   48
 Exercises
 a) Draw a flow net for seepage under a vertical sheet pile
    wall penetrating 10 m into a uniform stratum of sand
    20 m thick.


 b) If the water level on one side of the wall is 11 m above
    the sand and on the other side 1.5 m above the sand,
    compute the quantity of seepage per unit width of wall.
    [k = 3  10-5 m/s]


 c) What is the factor of safety against developing the
    “quick” condition on the outflow side of the wall?
    [sat= 21 kN/m3]

DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   49
               Finite Difference
             spreadsheet solution
        and other numerical approaches

         Authors:
                Mahes Rajakaruna (ex UniSA)
              & University of Sydney (FESEEP)



DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   50
          Finite Difference approach to flow nets
                      - flow line set up
  ROWCO   A     B   C   D   E    F   G   H   I   J   K   L   M   N   O   P   Q   R   S     T   U   V    W
  L

     1    100   Soil level                                                           104

     2    100                                                                        104

     3    100                                                                        104

     4    100
                                     Cell H5                                         104

     5    100                                                                        104

     6    100                                                                        104

     7    100                                                                        104

     8    100                                                                        104

     9    100                   Interior cell value =                    104 104 104 104 104 104 104

    10    100                   (H4+I5+H6+G5)/4
    11    100

    12    100
                                                         Impermeable boundary
    13    100

    14    100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100 100


DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT                                51
    Flow lines from finite difference
    program (spreadsheet)




                                                                      98-99
                                                                      99-100
                                                                      100-101
                                                                      101-102
                                                                      102-103




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT      52
    Equipotentials from finite difference
    program (spreadsheet)


                                                                      113-114
                                                                      112-113
                                                                      111-112
                                                                      110-111
                                                                      109-110
                                                                      108-109
                                                                      107-108
                                                                      106-107
                                                                      105-106
                                                                      104-105
                                                                      103-104
                                                                      102-103
                                                                      101-102
                                                                      100-101




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT    53
          FESEEP: University of Sydney
                                             cutoff




    Mesh of foundation soil




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   54
                       FESEEP Output
                     (University of Sydney)
    flownet




     pore pressures                       increasing




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT   55
                                            Dam
                                                cutoff
    50 m of water
                                                                 5 m of water




                              Low permeability rock




DIVISION OF INFORMATION TECHNOLOGY, ENGINEERING AND THE ENVIRONMENT             56

								
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