Appendix Module Measurement and mapping of hydraulic head Capsule summary by brucewayneishere

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									            Appendix 2

Module 2: Measurement and mapping of
           hydraulic head
   Capsule
 summary of
    basic
hydrogeology


 Michel Robin


                                                                                  1




   Groundwater Flow


                                Storage
       In                                                Out




Mass balance:
( Movement In – Movement Out = Change in Storage ) per unit time
                                                                                  2




   Movement vs Storage
      – Aquifer: BOTH Movement and Storage
        sufficient for purpose (definition relative to usage (purposely vague))
      – Aquiclude: NO movement
      – Aquitard: Some (very limited) movement

      – Confined Aquifer: sandwich – aquifer
        sandwiched between 2 aquitards(or aquicludes)
      – Unconfined Aquifer: open-face sandwich –
        aquifer bounded at the bottom by an aquitard
        and open at the top to the atmosphere
                                                                                  3




                                                                                      1
 Water storage
In pores, cracks, fractures etc
n = porosity = Volume Pores / Volume Total
         = [ L3Pores / L3Total]
         = [ L3Water / L3Total] at saturation

 Typical porosities - Unconsolidated
 Sand: 30-40%
 Clays: 35-60%                                                      Dagram
                                                                    unconsolidated
 Typical porosities - Rocks
 Sedimentary:                  < 10%
 Massive Ig. & Met:            1-2%                                   Dagram
 Fractured I&M: 3-8%                                                  consolidated


                                                                                            4




 Water release from storage
 Confined:
           Expansion of water and compression of aquifer;
           Storativity, S [ L3Water / L2Total / LHead ] = volume of water that can be
    removed from the aquifer per unit area of aquifer per unit drop in hydraulic head.
                      Specific Storage, Ss = S / b [ L3Water / L3Total              / LHead ]
                              b = thickness of aquifer [ LTotal ]
                      Range: S = 10-5 – 10-2
 Unconfined:
           Dewatering (replace water by air);
           Specific Yield, Sy [ L3Water / L2Total / LHead ]
                   Range: Sy = 0.2 – 0.4

                                                                                            5




 Storativity and Specific Yield
 Storativity (confined) and Specific Yield (unconfined) represent the
 same quantity:
 “The volume of water that can be removed from the aquifer per unit area
 of aquifer per unit drop in water level in the aquifer.”
 Or
 “The height of water that can be removed from the aquifer per unit drop
 in water level in the aquifer” (S or Sy = )w / )h)
                                                           - for S = 10- 3, removing 1
                          )w – height of water                cm of water from the
                          removed from aquifer                aquifer will result in a 10
                                                              m drop in water level.
                                                           - for Sy = 0.2, removing 1
                                                              cm of water from the
                         )h - Change in water level           aquifer will result in a 5
                                                              cm drop in water level.

                                                                                            6




                                                                                                2
    Storage measurements
    Confined:
       Can only be measured in the field using large-scale
       pump-tests (with at least 2 wells and $’s).



    Unconfined:
       Can be measured in the field using large-scale pump-
       tests (with at least 2 wells and $’s); some single well
       tests and lab tests.
                                                                                                          7




   Transient VS Steady state

                                              Storage
         In                                                                  Out




( Movement In – Movement Out = Change in Storage ) per unit time

   • Transient: if change in storage per unit time …0
   • Steady-state: if change in storage per unit time = 0; water may
   still be flowing but it is not accumulating or being lost. Kind of
   like a long-term average.
                                                                                                          8




    Water movement – Darcy’s Law
                    ∆h
 q = Q / A = −K        = − Ki
                    ∆l
q = Darcy Flux J [ L      3
                           Water   / L2Total / time ]


  = Specific discharge K
   = Darcy Velocity L
Q = Discharge [ L / time ]
                 3
                  Water


A = Cross-section area [ L ]         2
                                      Total


h = hydraulic head [ L ]   Water
                                                                               Source:
l = Distance between measurement points [ L                      Total   ]     Freeze and Cherry, 1979.
K = Hydraulic Conductivity [ L / L / time ]     2
                                                 Water   Total
                                                                               Groundwater.


 i = Hydraulic gradient [ L / L ]      Water    Total
                                                                                                          9




                                                                                                              3
Groundwater velocity VS
Darcy Flux
The Darcy Flux (sometimes called Darcy Velocity) is NOT a velocity!!
? To get gw velocity we must divide the Darcy Flux by the porosity:
                                                    v
                      q


      q [Lw3/Lt2/t]           q [Lw3/Lt 2/t] / n [Lw3/Lt3] = v [Lt/t]

                      ∆h
        q=K              = − Ki                                     v = q/n
                      ∆l
     v = Average linear groundwater velocity [ L                              Total   / time ]

     q = Darcy Flux [ L / L / time ]
                                   3
                                    Water
                                               2
                                                Total


     n = Porosity [ L / L ]
                          3
                           Water
                                      3
                                       Total
                                                                                                 10




Hydraulic Conductivity
• Expresses the ease with which water moves
  through the aquifer
• = 1 / (hydraulic resistance)
• Typical values
    Sand: 10-5 – 10-2 m/s                               Clay: 10-11 – 10-4 m/s
    Limestone: 10-6 – 10-2 m/s                          Shale: 10-13 – 10-9 m/s




                                                                                                 11




Hydraulic Conductivity
Measurements
• Lab measurements: permeameters
• Field measurements: pump tests, slug tests
    – Pump tests will also give an estimate of the storage coefficient (Specific
      storage)
    – Slug tests can only be used to estimate the hydraulic conductivity

• In this field camp we will be performing a
  bail test at Chalk River


                                                                                                 12




                                                                                                      4
 Hydraulic Head
• A measure of stored energy
• Energy per unit weight of water = (equivalent) Length of water
• Water level relative to an arbitrary reference (usually sea level)
                                           Observation well
                                                              Ground surface
h =ψ + z                                              Water
                                                      level
                                                                   L




                                                                       R
 h = hydraulic head [ L    Water      ]
                                                                                    h
 R = pressure head [ L    Water   ]

 z = elevation head [ L                              Measurement
                          Water ]
                                                     point
                                                                       z

                                           Reference level
                                           (sea level)
                                                                                                       13




  Water Table VS
  Potentiometric surface
By definition the water table is where the pressure
  head is equal to 0 in an unconfined aquifer.




                                                                                         Source: Freeze and
                                                                                         Cherry, 1979. p48




                                                                                                       14




 Hydraulic Gradient
• THE driving Force
• i = Îh / Îx
 = Change in Energy per unit distance through which water moves
   Horizontal gradient
                                                               Vertical gradient
           L   h1                                                               L       h1
                                           Îh
               Îx                                                          z1                 Îh
                           L          h2
                                                              Îz            L
                                                                                        h2
      x1            x2                                                     z2

                                                                            iz = Îh / Îz
           ix = Îh / Îx                                                                                15




                                                                                                              5
Will the true Hydraulic Gradient stand up!
when measurement points are not aligned with the axes the gradient
measured between the points IS NOT the hydraulic gradient …
                                                     It is the vector projection of the gradient
                                                     on the line separating the points.
                             L       h1                   ix … Îh / Îx
             Îh                            L    h2
                                                          iz … Îh / Îz

                  p1                                      ip = Îh / L
                                     L
     Îz                                                     = projection of the
                                                            gradient i onto line p1-p2
                                      p2
                                                          To get the gradient…
                                 Îx
                                                                                           16




  Hydraulic Gradient (cont’d)
  … we need 3 measurement points in 2-D (and 4 in 3-D)

                                                     • Draw a line between 2 measurement
                                                       points and interpolate hydraulic head
                                                       along the line.

                                                     • Do the same for the other two points.

                                                     • Draw equipotential lines
                         h1 = 10
                             9                       • Gradient is perpendicular to
                                                       equipotential lines.
                                      8
              11
                                                     • Magnitude of gradient is the slope of the
                                           h2 = 7      potentiometric surface in the direction
                                      8
                                 9                     perpendicular to the equipotential lines.
                        10
                   11
                                                       i = Îh / L
   h3 = 12                                                                                 17
                                     L Îh = 2




  Hydraulic Gradient (cont’d)
 • This trick works in plan view or vertical cross-section, AS
   LONG AS the axes are on the same scale; in plan view this is
   usually no problem, but in vertical sections there is often a
   vertical exaggeration -> big problem.
 • Piezometers must be screened in the same aquifer: hydraulic
   head and gradients are discontinuous across aquifer boundaries.
 • When mapping a water table, only the shallowest piezometer in
   a multi-level installation should be used (especially if there are
   vertical gradients).
 • The next step is to use the gradient, the conductivity, and the
   porosity to estimate groundwater velocity; we can then
   calculate advective times.

                                                                                           18




                                                                                                   6
       Steady-state flow: Example
   For a sand:
   K = 10-5 m/s
   n = 0.3

qx = -Ki = -10-5 * 0.1 = -10-6 m/s
vx = qx / n = -10-6 / 0.3 = - 3.3x 10-6 m/s = 104 m/y




   qz = -Ki = -10-5 * (-0.4) = 4x10-6 m/s
   vz = qz / n = 4x10-6 / 0.3 = 13.3x10-6 m/s = 419 m/y   Source: Freeze and Cherry, 1979.                                                19




                   Field School
           3 components of Hydro Module
        • Bail Test: Well 21, Twin Lake
              – Hydraulic conductivity calculations
        • Water level survey: Twin Lake Site, CRNL
              – Water table map with approximate flow nets
              – Cross-section with approximate flow net
              – Velocity calculations and advective time calculations
        • Mass balance (water budget) at Lake Ogilvie
              – Seepage: meter installation and measurements
              – Piezometer installation
              – Mass balance calculation
                                                                                                                                          20




                                                                                                          Bail test recovery

                                                                                           5.5


        Bail Test                                                                           5
                                                                         Water level (m)




                                                                                           4.5


                                                                                            4


                                                                                           3.5
                                                                                                 1

                                                                                                     10
                                                                                                           19
                                                                                                                28

                                                                                                                      37

                                                                                                                           46
                                                                                                                                55
                                                                                                                                     64

                                                                                                                                          73
                                                                                                                                               82




                                                                                                                     time (sec)




   L         L                       L                               L


                                                                                                     L

                                              L




                                                                                                                                          21




                                                                                                                                                    7
  Water level survey, Chalk River
For a sand:
K = 10-5 m/s
n = 0.3

                             (b) )l = 60 m (a) )l = 50 m




                                                       Source: Fetter, C.W. 1994. Applied Hydrogeology p.115.

At (a): q = Ki = 10-5 * 5 / 50 = 10-5 * 0.1 = 10-6 m/s
v = q / n = 10-6 / 0.3 = 3.3x 10-6 m/s = 104 m/y

 At (b): q = Ki = 10-5 * 5 / 60 = 10-5 * 0.083 = 8.3x10-7 m/s
                                                                                                           22
 v = q / n = 8.3x10-7 / 0.3 = 2.78x 10-6 m/s = 88 m/y




            Water table elevation horizontal flow
                         conditions
                                                                Ground Surface
                                               L                                L Water table




                                                                                                           23




          Water table elevation in a recharge area

                                                                Ground Surface

                                               Dry piezometer
                                                                                L Water table

                                                   L




                                               L




                                           L



                                                                                                           24




                                                                                                                8
   Water table elevation in a discharge area
                     L
                         L             L
                                           L   Ground Surface
                                                           L Water table




                                                    Moral of the story:
                                                    To plot water table elevations
                                                    from multi level piezometers,
                                                    use the highest piezometer
                                                    with water


                                                                                 25




                     Perched water tables

                                               Ground Surface
                                           L               L Perched water table




                                Dry piezometer


                                   L                       L Regional water table



                               L




                                                                                 26




Flow Nets
2-dimensional construction of equipotential lines and flow lines representing a
   flow domain at steady state.
• Equipotential Lines:
   Lines of equal hydraulic head - like contour lines on a contour map

• Flow Lines or Streamlines:
   Lines that are everywhere tangent to the flux vector. The mathematical
   functions that describe streamlines are called streamfunctions. A flow net
   gives a graphical presentation of flow lines and equipotential lines at the
   same time.
• Assumptions:
         - **** Steady-state flow ****
         - 2-D flow: No Flow in the third (ignored) dimension.

                                                                                 27




                                                                                      9
 Rules of the game
• Flow lines and equipotential lines are always perpendicular; except if one of the
      axes is exaggerated relative to the other.
• Equipotential lines are always perpendicular to impermeable boundaries
      (flow lines are always parallel to impermeable boundaries).
• Equipotential lines are always parallel to constant hydraulic head
  boundaries (flow lines are always perpendicular to constant hydraulic head boundaries).
• Symmetry boundaries that are groundwater divides can be represented by
  an impermeable boundary.
• Water table boundaries are NOT constant head boundaries; they are
  boundaries where h = z. An equipotential line will cross the w.t. at the
  elevation that corresponds to the equipotential line value (because h = z at
  the water table).
• Flow lines never intersect each other (there is no flow across a flow line, they are
      impermeable "boundaries").
• Equipotential lines and flow lines form curvilinear squares. The more
  closely spaced the equipotential lines are, the more closely spaced are the
  flow lines.
                                                                                                        28




     Flow net example, vertical
                                                   Hinge
                      Recharge zone                zone               Discharge zone
                                    o
                                < 90                                > 90o
                                                     = 90o
      Water table
         L                                                                  Groundwater
17
                                                                            Divide
16
                16
15
14
13                   15
12                                                                                                  L
11                                                                                      L
10
9                          14                                                  11
8
7
6                                          13
                                                             12
5
4
3                                                                                   Stagnant zone
2
1
                                                                                                        29




               Lake Ogilvie water budget




                                                                            Leakage
                          Seepage


                                                                                                        30




                                                                                                             10
    Seepage meters and piezometers



                         h2        L
L
                              )h
                         h1




                                       )z

          (a) Measure q directly from seepage meter
          (b) Measure i from piezometer (= )h / )z)
                                                       31
          (c) Calculate K from (a) and (b)   q = -Ki




                                                       32




                                                       33




                                                            11
34




35




36




     12
37




38




39




     13
            Hydro Module Schedule
               Day 1                                    Day 2
 AM                                       AM
 - Seepage bag installation               - Piezometer installation & surveying
 - Close-top seepage meter installation
 PM                                       PM
 - Water level survey Twin Lake,          - Piezometer installation & surveying
   CRNL                                     (cont’d)
 - Bail Test at Well 21, Twin Lake,       - Seepage Measurements Lake
   CRNL                                     Ogilvie
Evening                               Evening
Report: potentiometric surface        Team Final        Report
mapping (map view and cross-section),
bail test analysis.                   due 21:06
                                                                           40




                                                                                  14
Bail Test at Twin Lake:

Your hydro team will be given an assigned time to do a response test at Twin Lake Well 21.

1. Note down general information

In your field note book you should take note of the usual stuff (date, time, location, climatic
conditions, people with you, etc). You should also take note of the equipment that you are using
(instrument, model # , type of slug or bailer, its volume etc.).

2. Note down information about well configuration and aquifer

For the response test you should measure or find out the following information:

r       = (inside) radius of well casing [L]
R       = radius of well screen including sand-pack [L]
L       = length of well screen [L]
Lw      = distance between bottom of screen and static water level (i.e. the depth of water in the
        well at static equilibrium) [L]
Li      = distance between bottom of screen and impermeable layer below [L]

You should also find out what type of aquifer the well is located in (type of aquifer material,
confined vs unconfined, thicknesses of strata, etc); and also how the screen was constructed.
Much of this information can be found in the technical report by Inch, Killey and Munch (1989),
which was sent in the documentation before field school.

3. Response test

Using the ReelLogger TM instrument to record your data, perform your piezometer response test.
The idea is to produce an instantaneous change in hydraulic head in the piezometer and to
monitor the recovery of the hydraulic head to the initial level. The initial level is called the
“static level”; and the difference between the static level and the hydraulic head is the
“unrecovered drawdown”.

The ReelLogger TM instrument will give you readings of pressure head (depth of water above the
probe), p, and time (and possibly other things depending on the probes attached). To obtain
hydraulic head we need to know z, the elevation of the pressures probe; but this information is
unnecessary if the position of the pressure probe is not changed during the course of the response
test, in which case the unrecovered hydraulic head difference will be equal to the unrecovered
pressure head difference, because the (constant) elevation head cancels out:

H - h = (P + z) - (p + z) = P - p
where:
h      = hydraulic head [L]
H      = static water level (hydraulic head before slug or bail) [L]
z      = elevation head [L]
p      = pressure head [L] - straight out of the data logger
P      = static pressure head before slug or bail [L] - initial value in the data logger
H - h = unrecovered drawdown [L]
       = P-p (As we just saw)

This means that you can use your pressure probe data directly - provided the probe does not
move during the test. If the bailer or slug knocks the probe up or down during the test, you may
have to redo your test.

The difficult part of this test is to produce an “instantaneous” change in water level - especially in
highly conductive aquifers (such as the Twin Lake Aquifer). If a slug or a bailer is used the time
required is on the order of one or two seconds (depending on the length of the slug or bailer); but
if the change of water level is produced by pouring a known volume of water into the well, then
the time required may be several seconds, which may be too long. Time permitting, you should
repeat your test, and try different ways of producing the initial change in water level. The data
will be downloaded upon returning to camp.

4. Calculations

Import your data into a separate Excel sheet, call it “Raw Data”. Don’t modify the raw data in
case you have to return to it later. Copy this sheet to a new sheet, say, “Clean Data”, and clean
up all unwanted stuff so that you have only time and pressure head. Now copy your clean data to
a new sheet, say, “Plots”.

For each response test that you did, prepare a plot of ln(|H-h|) as a function of elapsed time.
Make sure that your units are consistent - use SI units. Look for a straight-line portion of the
plot in the middle of the range. Do a regression of this segment of the plot and obtain the slope
of the resulting line. Put the line on the same plot with your data. From the slope, calculate T0 =
( - 1 / slope). Using this value of T0 calculate the hydraulic conductivity of the formation using
both the Hvorslev Method and the Bouwer and Rice Method. - the theory follows.

5. Report

Your report should be concise, but complete. Prepare your report by hand (except for the plots).
You can use point form where appropriate. Prepare one report per team.

- Give the general information and the aquifer and well information (items 1 and 2 above).

- Hand in a copy of your semi-log plots properly labelled and titled (title should include your
team members’ names and ID #, the well ID and the site, and the test number (if you did more
than one).
- Give a sample calculation of the hydraulic conductivity.

- Prepare a table containing your hydraulic conductivity values: one row for each test that you
conducted; and one column for

- Compare the results with the Hvorslev Method and the B & R Method with the value(s)
reported by Inch et al.

- Prepare a list of the sources of error in the methods. Discuss (one or two sentences) each
source and give an estimate of its magnitude.

- The probe and its cable that you used in these tests are normally encased in a 1-inch diameter
polyvinyl tubing. Why was it removed for your tests?
                                                                                                   4


Piezometer Response Tests (slug / bail tests)
•     Procedure is the same for both slug- and bail-tests:
      <     Record, H, the original hydraulic head in the piezometer.
      <     Add a slug of known volume of water to the well, raising the water level in the
            well to H0 instantaneously, or remove a known volume (bail) to lower the water
            level to H0 . H0 can be measured directly in the well (sometimes difficult) or it
            can be estimated from the volume of the slug and the cross-section area of the
            piezometer.
      <     As the water re-enters the formation record the hydraulic head, h(t).

•     Used in low to moderately high conductivity material

•     Requires no pumping apparatus

•     Can be done in small-diameter piezometers

•     For formation of low to medium permeabilities; the higher the permeability, the larger the
      volume that must be injected or withdrawn, and the f-f--f---faster one must be at
      measuring the recovery.

•     There are several methods for analysing the data; the most commonly used is Hvorslev’s
      in confined aquifers, and.




    Sciences de la Terre                                              Piezometer Response Tests
    Earth Sciences              M.J.L. Robin                          EVS 4910 EVS Field Course
                                                                                                      5


Hvorslev(1951)
•         Point piezometers (not fully-penetrating)
•         Confined aquifers




Source: Freeze and Cherry, 1979. Groundwater


Observed that flow in or out of the well will be proportional to the hydraulic conductivity and the
unrecovered head difference (H - h):




Where r is the radius of the piezometer, F is a factor that depends on the shape and dimension of
the piezometer intake (including sand- or gravel pack), K is the hydraulic conductivity, and (H-
h) is the unrecovered head difference.

This differential equation can be solved, subject to the initial condition: h = H0 at t = 0. The
solution is:




     Sciences de la Terre                                                 Piezometer Response Tests
     Earth Sciences                            M.J.L. Robin               EVS 4910 EVS Field Course
                                                                                                        6




The left-hand side of the equation is called the normalized unrecovered head difference and T0 is
the “basic time lag” defined as:




This value is obtained graphically. We plot                    as a function of t and read the value


of t that corresponds to                        . The reason we do this is that when t = T0, the

right hand side of the equation is equal to exp(-t / T0) = exp(-1) = 0.37; and therefore




when



Therefore T0 is the value of t when                           . Although we can use a linear plot

(which will give an exponential decay curve), it is easier to read values off of a straight line, and
so the normalized unrecovered head difference is usually plotted on a log scale and the time on a
linear scale.

For a piezometer intake of length L and radius R (including the sand or gravel pack), and when
L/R > 8 the hydraulic conductivity, K, is given by:




                                                                                      Hvorslev


where: K       = hydraulic conductivity [L T-1]
               r      = radius of well casing [L]
               R      = radius of well screen including sand-pack [L]
               L      = length of well screen [L] L/R > 8




    Sciences de la Terre                                                   Piezometer Response Tests
    Earth Sciences                 M.J.L. Robin                            EVS 4910 EVS Field Course
                                                                                                     7


               T0     = value of t when                          .


If H0 is unknown (because of the inability to make the measurement) then it can be estimated
instead from a plot of H-h as a function of t. T0 is then simply the slope of the line:




or, if you happen to be using semi-log paper, you can convert this to base 10:




Simple, yet elegant, I’m shore yawl agrea ....

But the catch is that we only get the hydraulic conductivity. An other method called the Cooper-
Bredehoeft-Papadopoulos Method will give an estimate of the storativity, using a curve-matching
technique but the results are error-prone (plus the name of the method is hard to pronounce) so
the technique is not commonly used.



Bouwer and Rice (1976)
(This discussion is taken from Bouwer, Ground Water, 1989, 27:304-309.)

•      Open boreholes or screened wells
•      Point piezometers or fully-penetrating
•      Unconfined (but can be used for confined aquifers if the top of the well screen is far
       enough below the bottom of the confining layer.)

The equations are identical to the Hvorslev method, giving a straight line on a semi-log scale:




    Sciences de la Terre                                                Piezometer Response Tests
    Earth Sciences                M.J.L. Robin                           EVS 4910 EVS Field Course
                                                                                                          8


except that the constant T0 is now:




which we can quickly rearrange to obtain the conductivity:




                                                                                      Bouwer and Rice



where:

K      = hydraulic conductivity [L T-1]
r      = radius of well casing [L]
t      = time since slug or bail was applied [T]
h      = hydraulic head [L]
H      = static water level (head before slug or bail)
       [L]
H0     = head produced by slug or bail [L]
H - h = unrecovered draw-down [L]
H - H0 = initial unrecovered draw-down [L]
R      = radius of well screen including sand-pack [L]
Re     = effective radial distance over which head is
       dissipated (= distance over which K is being
       measured) [L]
L      = length of well screen [L]
Lw     = distance between bottom of screen and static
       water level [L]
Li     = distance between bottom of screen and
       impermeable layer below [L]




Note that this diagram shows a slug test, wherein a volume is added to the well which produces
an initial head, H0, higher than the static level, H. In the bail test, water is bailed out of the well
thereby reducing the initial head relative to the static level. The analysis is exactly the same for
the bail or the slug test.


    Sciences de la Terre                                                     Piezometer Response Tests
    Earth Sciences                  M.J.L. Robin                             EVS 4910 EVS Field Course
                                                                                                   9

Also note that the Bouwer and Rice Method is almost identical to the Hvorslev Method, except
for the quantity ln (Re/R), which can be calculated in two ways depending on whether or not the
well extends down to an impermeable boundary. If the well does not reach the boundary, i.e. Li
> 0 in the diagram, then:




otherwise, Li = 0, and:




The constants A,B, and C are dimensionless parameters that depend on , L / R, the ratio of the
screen length to the screen + sand-pack radius. They can be obtained from the following graph:




Source: Bouwer, 1989 (Ground Water 27;306)




     Sciences de la Terre                                              Piezometer Response Tests
     Earth Sciences                          M.J.L. Robin              EVS 4910 EVS Field Course
                                                                                                        10




A plot of the unrecovered head drop (H-h) on a log
scale as a function of time, t, on a linear scale will
give a straight line, at first, but as time progresses and
“drawdown of the ground-water table becomes
increasingly significant as the test progresses, the
points [...] begin to deviate from the straight line [...]”
Bouwer, 1989 (Ground Water 27;306)


In this and the following figures Yt is equivalent to
(H-h)




                                                               Source: Bouwer, 1989 (Ground Water 27)
But field data often deviates from the straight
line at early times giving a “double straight line
effect”. Bouwer (1989) explains the early
behaviour as “probably due to a highly
permeable zone around the well (gravel pack or
developed zone), which quickly sends water into
the well...” Early times are also error-prone,
especially in highly conductive materials,
because of difficulties in achieving an
instantaneous pulse: it takes a certain time to
introduce or remove the slug or bailer.

The remedy: only use the middle segment of the
curve, [B-C].


                      Source: Bouwer, 1989 (Ground Water 27)




    Sciences de la Terre                                           Piezometer Response Tests
    Earth Sciences                      M.J.L. Robin                EVS 4910 EVS Field Course
                                                                                                      11


Summary of the Bouwer & Rice Method:

1-     Find the value of L /R, the ratio of screen length to screen + sand-pack radius
2-     Obtain A and B from the graph if the screen does not reach the underlying impermeable
       boundary or C if it does.
3-     Plug these values into the formula and obtain ln(Re/R)
4-     Prepare a semi-log plot of Log (H-h) VS t. Examine the plot and find a segment that is
       linear in the middle of the range. Using regression, obtain the slope of this linear
       segment. Calculate T0 from the slope.
5-     Obtain K by substituting (3) and (4) into the formula given above.




Disadvantages of Bail / Slug tests:
•      Bail / slug tests do not give storage coefficients.
•      The piezometer intake has to be of high quality and in good condition. A clogged intake
       or screening that is too fine will interfere with flow, underestimating the conductivity. If
       the intake has been over-developed the measured conductivity will be too high.
•      This type of response test is difficult to do in highly conductive formations; a pressure
       transducer and data logging device may help in these situations.




     Sciences de la Terre                                                Piezometer Response Tests
     Earth Sciences               M.J.L. Robin                            EVS 4910 EVS Field Course
                                                                                                       1


Lake Ogilvie Seepage
•      The purpose of this exercise is to measure seepage in and out of lake Ogilvie, and to
       estimate a water budget for the lake.

•      Read the material on seepage measurements by Dave Lee (one article in this package and
       workshop notes attached).

•      I have attached a sketch of the lake with a preliminary survey of electrical conductivity of
       the lake sediment. This information can sometimes be used to find possible areas of
       groundwater discharge.

Procedure:

On Day 1 there will be a demonstration of the various methods. The group will be divided up
into smaller teams to install piezometers, and seepagemetres. Each team will be assigned a
different location in and/or around Lake Ogilvie to measure seepage and hydraulic gradient.

You will set up the seepagemeters at the beginning of Day 1 and make the measurements at the
end of Day 2

Open-top seepage metres:
             1.     Take a water level outside the meter and inside (take down these
                    numbers);
             2.     If the water level is higher inside, install a syphon, let equilibrate for a
                    minute, attach an empty seepage bag, and note the time;
             3.     If the water level is lower inside, install a syphon, let equilibrate for a
                    minute, attach a bag with about 500mL in it, note the time, and the exact
                    amount of water in the bag;
             4.     If the water level is the same, go to the next seepage meter.
             5.     Let the seepage metre do its thing for a day or so (you can check up on it
                    to make sure that it is not bursting with water or completely emptied out;
             6.     To take the seepage reading after the equilibration time, take a water level
                    outside the metre and inside (take em down) - the two levels should be the
                    same, otherwise the syphon did / is not working properly - and you have to
                    start from scratch;
             7.     Measure the contents of the bag and take note of the time;
             8.     Pull up the syphon from the meter enough to un-prime the syphon and
                    leave the tube there for the next group (if you are the last group, pull out
                    the syphon AND the seepage meter.

Close-top seepage metres and piezometres:




    Sciences de la Terre                                                        Lake Ogilvie Seepage
    Earth Sciences                M.J.L. Robin                           EVS 4910 EVS Field Course
                                                                                                      2

Follow directions in the D. Lee articles.

•      During the afternoon of Day 2 compile the seepage and piezometric measurements and
       prepare the report.

Report:

•      Calculate hydraulic conductivities from your measurements, and give the average K that
       you obtained.

•      Calculate the seepage flux at each measurement location - don’t forget to make
       appropriate corrections for the open-top meters if necessary.

•      Write the seepage fluxes that were obtained directly on the electrical conductivity map.
       Use positive values for groundwater discharge (seepage) and negative values for
       groundwater recharge (leakage). Qualitatively describe the spatial correlation between
       seepage and electrical conductivity.

•      Discuss the pros and cons of each seepage measurement method: open-top; close-top; EC

•      Based on your measurements and the given EC map draw an approximate map of seepage
       in and out of Lake Ogilvie. Calculate the total seepage in an out of the lake (give a range
       of possible values).

•      Draw a cross-section of the Lake Ogilvie Site indicating the zones of groundwater
       recharge and discharge.

•      Water Budget: write a ledger of all the inputs and outputs to Lake Ogilvie, with estimated
       errors.




    Sciences de la Terre                                                       Lake Ogilvie Seepage
    Earth Sciences                 M.J.L. Robin                         EVS 4910 EVS Field Course
                                                                  3




Sciences de la Terre                       Lake Ogilvie Seepage
Earth Sciences         M.J.L. Robin   EVS 4910 EVS Field Course
                                                                                                        1


Water level survey at Twin Lake (Water Table
Map & Piezometric x-section)
For this afternoon in the field, you will need a lunch, plenty of drinking water, sun screen lotion,
insect repellent, etc. You should wear clothing suitable for walking in the woods (proper
footwear, long pants, safety glasses, hat, etc). And of course, you should have your trusty field
note book.

For this exercise you will be working in the same 4 teams of 3-4 individuals (because we have
only 4 water level tapes). Each team will be assigned a portion of the Twin Lake Site territory,
and you will measure water levels in wells and multi-level piezometers in your territory. At one
point during the day (at your assigned time) you will have to interrupt your water level survey to
do your piezometer response test. At the end of the day the TA will compile all water level
measurements from all the groups, and distribute them.

Your team will prepare a report for this exercise that includes a potentiometric surface map (map
of water table elevation) and a potentiometric surface in cross-section, from which you will
calculate average linear groundwater flow velocities, and recharge/discharge rates. Blank maps
and cross-sections will be handed out at field camp.

1. Field measurements

Before you begin your survey, take note of the particular water level tape that you are using:
brand, length, AND serial number (if available). Take note of any possible correction to the
readings on the tape (some tapes that have been cut and repaired will require a correction).

Complete your survey of water levels: at each well, measure in meters (S.I. units):

•       Well number
•       depth to water: distance from the top of the standpipe to the water;
•       stickup: height of the standpipe from the ground surface to the top of the standpipe;
•       take note of any condition that could invalidate the measurements - plugged up wells
        damaged casing etc.
If the well location is a multi-level piezometer (most are), identify the level of the piezometer and
measure its depth to water.
When taking a measurement, especially in the narrow-diameter piezometers, try not to submerge
the tip of the water level meter too deeply, or to move it up and down, to pin point the exact
location of the air-water interface. This action can cause some smearing of the water against the
piezometer wall or water level tape, which can produce an intermittent and weak signal from the
water level meter. It is also a good idea to dry the water level meter between measurements. At
sites where cross-contamination from one well to another may be an issue, the water level meter
should also be decontaminated (cross-contamination will not be an issue at Twin-Lake).




    Sciences de la Terre                                Water Table Map & Piezometeric x-section
    Earth Sciences                 M.J.L. Robin                           EVS 4910 EVS Field Course
                                                                                                       2

Transform your water level measurements so that they are expressed as meters above sea level -
ground surface elevations are provided in a spreadsheet.

At the end of the day the TA will compile the data from all teams into a single spreadsheet and
distribute them. Your team will then complete the exercise with the data from all teams.


2. Report

On the title page of this report give the title, “Water Table Map and Piezometric Cross-section”
and the names of the members of the team, their affiliation and student ID (if applicable). The
report consists of two items: A water table map and a piezometric cross-section. Arm yourself
with a sharp pencil, a good eraser, and a lot of patience.

A) Water Table Map:

Officially, the water table is defined as the surface where the hydraulic head is equal to the
elevation head; or the surface where the pore pressure is equal to atmospheric pressure. In
English, it is the water level in the soil. The water table elevation can be measured with a water
table well, which is just a well with a long screen that spans the unsaturated and the saturated
zones. At the Twin Lake Site there are no water table wells, so we will use the next best thing:
the shallowest piezometer at each location. The error introduced by this approximation will
depend on the depth of the screen below the water table and on the magnitude of the vertical
component of the hydraulic gradient (“vertical gradient”, for short): The deeper the screen
location relative to the water table and the larger the gradient, the larger the error. In the grand
scheme of things the error is usually fairly small.

In the handout, you are provided with an areal map that you can use as a base map to draw on.
On the map, indicate the elevation of the water table at all piezometer locations. Use the
shallowest piezometer at each location.

1- Equipotential lines:
       Draw equipotential lines, that is, lines of equal water table elevations, using regular
       intervals - you should end up with something that looks like a topographic map of the
       water table.
2- Flow lines:
       Draw approximate flow lines on your water table map. The lines should be at right angle
       to the equipotential lines.
3- Recharge / Discharge:
       At sites where there are multi-level piezometers, determine the direction of groundwater
       flow based on the vertical gradient. Label the areas of recharge (downward flow) and
       discharge (upward) directly on your map.
4- Gradient, Darcy Flux, Velocity:
       Using your water table map, estimate an approximate hydraulic gradient between Lake



    Sciences de la Terre                                 Water Table Map & Piezometeric x-section
    Earth Sciences                 M.J.L. Robin                            EVS 4910 EVS Field Course
                                                                                                         3

       233 and Twin Lake. Calculate the Darcy Flux between the two lakes (you’ll need an
       estimate of the hydraulic conductivity for this) and the average linear velocity (you’ll
       need and estimate of the porosity for this). Estimate the travel time for a non-reactive
       contaminant to travel from one lake to the other. Write your calculation in one of the
       corners of your water table map.




B) Piezometric Cross-section

In the handout, you are provided with a cross-section, Cross-Section C, which you will use for
this exercise. Identify all the wells located on the section. Most of these wells are multi-level
wells, and you should have a water level for each screen elevation. You will need to write down
on the cross section the water level measured at each screen elevation.

1- Equipotential lines:
       Using the water levels that you just wrote on your section, draw contour lines of equal
       water level elevations. These are also called equipotential lines. Again, use regular
       intervals. The water level surface that you obtain is called a piezometric surface. Draw
       the position of the water table on your section. The water table IS NOT an equipotential
       line; it is the line that joins all the points where the water level is equal to the elevation.
2- Flow lines:
       Draw approximate flow lines on your piezometric surace. The flow lines should be at
       right angle to the equipotential lines ONLY if there is no vertical exaggeration, otherwise
       they will cross at an angle.
3- Recharge / Discharge:
       Label the areas of recharge and dischargedirectly on your map. You can identify
       recharge areas where flowlines cross the water table downward, and discharge areas
       where they cross upward. Areas where the flow lines follow the water table are called
       “Hinge lines” - also indicate them on your diagram.




    Sciences de la Terre                                 Water Table Map & Piezometeric x-section
    Earth Sciences                  M.J.L. Robin                            EVS 4910 EVS Field Course
           TL-series borehole                                                                                                             TR-467
           C-series borehole
           Lake                                                                                                                           Use this map
           Wetland                                                                                                                        for your W.T.
           Waste M anagement Area                                                                                                         map
           Overgrown road (OGR)
           M ain dirt road
           Piezometer plowed or
           not there




                                                                 N
    A = Fig 22
                                                    0            200          400 ft
    B = Fig 24
    C = Fig 23                                          0        50         100 m                                                                       177




                                                                                                                                           183
                                                                                                                                                  182

                                                                                                                                                            181
                                                                                                                                                180
                                                                                48                      150
                                                                                                                                                                 138
                                                                                              47                7                                     139
                                          50                                                                               40               140               B
                                                                                                                    41           8
                                                                 49                                                                                   142         141
                                                                                                               A
                                                                                                                                                            25
                                 51                                                                                              5          143
                                                                                                         42 39
                            53                                   6
                                                                                    54                                                    144
                                                                                                          1                 9
                                    52          19                                                                   C                                           21
                                                                                                                            3
                                                                                                               65                                     145
                                                                                                                                     69                           32
                                                                                                   59
                                                                     4                                              66                                      31
                                                            21                                                              2
              46                                                                                        Twin
                                         50
                                                                                              60
                                                                                                                                67
                                                                                                    12
             (Hill)                                                                 11
                                                                                                                    Lake
                                    43  23 30 13                       20
     44                               h)
                                   itc24 2225 29
                                 (D
  Sa             57
                              18
                                  28 27
                                            26
                                                                                                                                      M ain dirt
    nd
       pit                                   14
                                                                                         15                                           road
           /du                                                       10
              nes                                                                                             Helpful Hints:
                                                                     17                                       1.       #28 is hidden behind trees to the left
                                                                                                                       of the overgrown road going uphill
                                               16
                   45                                                                                         2.       #56 is at the end of the OGR
                                                                                                              3.       #55 is up on the hill but broken at the
                                    56              55 (Hill)                                                          base (small trees may still be flagged)
               S an                                                                                           4.       Follow the ditch to find #23, 24 and
                   d pi
                          t/du                                                                                         18.
                              nes
                                                                                                              5.       To find #44, follow the OGR or at
                                                                                                                       #46, turn and go up steep hill, #44 is
Fe-oxide seep                                                                                                          at the top!
                                                                             Wall &                           6.       Follow the flagged trees from the
                                                                                                                       Main dirt road to find #15 and the
                                                                             Curtain                                   others.
Basics and More: Henry Darcy and His Law                                                      Page 1 of 6




                    Darcy's Law Basics and More
                                                   by

                                            Glenn Brown

                                    Oklahoma State University

Back to Groundwater

      Introduction
      One-Dimensional Flow
      Simple Discrete Form
      Differential Form
      Flow Variables
      Darcy Flux
      Seepage Velocity
      One Dimensional Flow at an angle to the coordinate axis
      Special 1-D Flows
      Horizontal flow
      Vertical Flow
      Unit Gradient Flow
      Other Measures of the Flow Proportionality
      Transmissivity
      Permeability

Introduction

Darcy's Law is a generalized relationship for flow in porous media. It shows the volumetric flow rate is
a function of the flow area, elevation, fluid pressure and a proportionality constant. It may be stated in
several different forms depending on the flow conditions. Since its discovery, it has been found valid for
any Newtonian fluid. Likewise, while it was established under saturated flow conditions, it may be
adjusted to account for unsaturated and multiphase flow. The following outlines its common forms and
assumes water is the working fluid unless otherwise stated.

One-Dimensional Flow

Simple Discrete Form

A one-dimensional flow column is shown in Figure 1.




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Basics and More: Henry Darcy and His Law                                                          Page 2 of 6




Figure 1. Simple column.

For a finite 1-D flow, it may be stated as

                   _____[1]


where,
Q = volumetric flow rate (m3/s or ft3/s),
A = flow area perpendicular to L (m2 or ft2),
K = hydraulic conductivity (m/s or ft/s),
l = flow path length (m or ft),
h = hydraulic head (m or ft), and
   = denotes the change in h over the path L.

The hydraulic head at a specific point, h is the sum of the pressure head and the elevation, or

      h = (p/ g + z)_____[2a]
      h = (p/ + z)_____[2b]

where,
p = water pressure (N/m2, lb/ft2),
  = water density (kg/m3),
  = water specific weight (lb/ft3),
g = acceleration of gravity (m/s2 or ft/s2), and
z = elevation (m or ft).

Equation [2a] is the normal SI form of the equation, while [2b] is the usual form used with English units.
The hydraulic head is the height that water would rise in a peizometer. Thus, h is simply the difference
in height of water in peizometers placed at the inlet and the outlet ( h = hin-hout). Substituting [2a] into
[1] yields,




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Basics and More: Henry Darcy and His Law                                                         Page 3 of 6



                                [3]


Equation [3] is approximately the form Darcy used to analyze his experimental data. Note that the flow
is not a function of the absolute pressure or the elevation. It is only a function of the change in hydraulic
head.

Differential Form

A more general form of the equation results when the limit of h with respect to the flow direction l, as
the flow path L goes to zero. Applying that step to equations [1] and [3] yields,

                                            _____[4]


The minus signs on the right hand terms reflects that the hydraulic head always decreases in the
direction of flow.

Flow Variables

Darcy Flux

The Darcy flux is defined as,

      q = Q /A_____[5]

where q = Darcy flux (m/s or ft/s).

The Darcy flux is the volumetric flow per unit area. Substitution of equation [5] into [4] yields,

                                         _____[6]


Seepage Velocity

While the Darcy flux has the units of velocity, it is not the velocity of the water in the pores. The solid
matrix takes up some of the flow area. The average pore water velocity is termed the seepage velocity, v,
and is given by

      v = Q/A = q/ _____[7]

where is the porosity of the porous media. The maximum pore velocity is a function of the pore
geometry and cannot be easily predicted except for simple shaped. In circular tubes the maximum
velocity is twice v.

One Dimensional Flow at an Angle to the Coordinate Axis

Darcy's Law is not a function of the flow direction in a homogeneous material. However, the gradient of




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Basics and More: Henry Darcy and His Law                                                         Page 4 of 6



h is calculated along the flow path, l, and the flow area, A is measured normal to l. Therefore, the
geometry of flow must be accounted for if the flow is measured relative to a different direction. Figure 2
shows the simple column tilted up.




Figure 2. Flow at an angle to the horizontal.

Assuming a 2-D space,

      z = x tan( )_____[8]
      dl = dx / cos( )_____[9]
      dl = dz / sin( )_____[10]

where,
  = angle to horizontal, and
x = horizontal distance (m or ft).

Substitution of equation [8] and [9] into [4] produces a relation relative to the x direction.

                                                  _____[11]


Simplifying produces,


                                                _____[12]


If the area of flow is measured normal to the x axis, Ax will be larger than the area normal to l. The two
areas are related by,




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Basics and More: Henry Darcy and His Law                                                      Page 5 of 6



      A = cos( )Ax [13]

Substitution of equation [13] into [12] produces

                                            _____[14]


By similar methods the flow may be expressed relative to the vertical direction by substitution of
equation [10] into [4]


                                        _____[15]


where Az is the area of flow normal to the vertical axis.

Special 1-D Flows

Horizontal flow

In horizontal flow,   = 0 and equation [14] reduces to

                           _____[16]


Vertical Flow

In vertical flow up, sin( ) = 1 and equation [15] reduces to


                                _____[17]


Unit Gradient Flow

In vertical downward flow, if dp/dz = 0, equation [15] reduces to the unit gradient form.

      Q = AzK (down)_____ [18]

Other Measures of the Flow Proportionality

Transmissivity

In saturated groundwater analysis with nearly horizontal flow, it is common practice to combine the
hydraulic conductivity and the thickness of the aquifer, b into a single variable,

      T = bK_____ [19]




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Basics and More: Henry Darcy and His Law                                                         Page 6 of 6



where T = transmissivity (m2/s, ft2/s).

Permeability

When the fluid is other than water at standard conditions, the conductivity is replaced by the
permeability of the media. The two properties are related by,

      K=k g/       = kg / _____ [20]

where,
k = permeability, (m2 or ft2),
  = fluid absolute viscosity, (N s/m2 or lb s/ft2) and
  = fluid kinematic viscosity, (m2/s or ft2/s).

Ideally, the permeability of a porous media is the same to different fluids. Thus, you may predict the
flow of one fluid, from the measurement of a second with equation [20]. However in practice, the solid
matrix may swell or sink with different fluids and produce different values of k. Substitution of equation
[20] into [4] yields,

                                           _____[21]


Likewise, substitution into equation [6] produces,

                                       _____[22]




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