ENV-2E1Y - Fluvial Geomorphology by p2Zi5wv


									        ENV-2E1Y - Fluvial Geomorphology

                             2004 - 2005

         Slope Stability and Related Topics

          Flownet of seepage of water through soil around an obstruction

                                Section 2

Seepage, Flow of Water, Pore Water Pressures
N. K. Tovey                  ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                  Section 2

                                                Slope Stability and Related Topics

                                      2. Seepage of Water in Soils/Permeability

2.1 Introduction
                                                                               Both the first an second are clearly important (the first arises
NOTE: FIRST A HEALTH WARNING!!!! There are                                     directly from the depth below the water table, whereas there is
some sections of this handout which are in shaded boxes.                       ample evidence (in the form of springs to indicate water flow
These are not essential parts, but complement the main                         through soils).
course. Thus for some of you who are doing ENV-2B31
(Mathematics I) you may object the simplistic approach                         What about the velocity head?
sometimes used in the handout. There is a more rigorous
approach for you in these boxes. In other cases, the boxes                     Typical velocities even in coarse sands will rarely exceed 10
show additional information which may be derived which may                     mm s-1, and the magnitude of the last term in this extreme
be of use in other courses (e.g. Hydrogeology). Please consult                 case will be (remember to convert to metres!!!!):-
the note before each box.
                                                                                    ( 0. 01)2
                                                                                               0. 000005 m   in terms of total head or 0.00005
                                                                                      2 x 10                  kPa in pressure terms.
For another set of notes for this section see the University
of West of England WEB Site on the topic.
                                                                               This is exceedingly small, and in most soils, the velocity will
http://fbe.uwe.ac.uk/public/geocal/SoilMech/water/water.htm                    be many orders of magnitude less than this so in future we can
A knowledge of the factors affecting the flow of water through                 conveniently neglect the velocity head term.
soils is important as the permeability of a soil affects the way
in which a soil consolidates which in turn affects its                         2.2 Hydraulic Gradient
mechanical properties. Equally water pressures may build up
within the soil and as seen the demonstrations can greatly                     Associated with water pressure is the hydraulic gradient
affect the ability of a soil to resist shearing. There are three               which is a measure how fast the water pressure is changing.
component parts to the water pressure:-                                        This in turn affects how fast water will flow and what the
                                                                               immediate pressure within the soil will be.
     i) that pressure arising from a static head
                                                                               Now consider flow between two points A and B (see Fig. 2.1).
            [ at any point at a height Z above the measuring
           datum, this pressure will be w Z where w is                       The hydraulic gradient is defined as the rate of change of head
           the unit weight of water.]                                          of water with distance (in the direction of measurement)

     ii) excess pore water pressure (i.e. a pressure head
           differential which actually causes water flow [ this
           has the symbol   u]
                                         w v2
     iii) a velocity head and equals        2g

The total pore water pressure (often abbreviated to pwp)
                              w v2
                 wZ  u 
 =                             2g                 .......2.1
For those who have done the Hydrology or Oceanography
options you may already be familiar with Bernoulli's equation                  Fig. 2.1 Flow of water in a channel of simple horizontal cross
of fluid flow i.e.                                                                      section
                               u                  v2
      H        Z                      
                              w                  2g        ........2.2        Since the flow is horizontal, there is no effect from the static
 total = position      +     pressure       +    velocity                      head of water.
 head     head                 head               head
                                                                               The head of water at A is h1
i.e. equation 2.1 is Bernoulli's equation expressed in terms of                             and at B h2
pressure rather than head of water.         [To convert from
                                                                               The excess pressure of the water at A (as the velocity is small)
equation 2.1 to 2.2 all we need do is divide by w.
                                                                               is                    w h1
How significant are these three terms in water flow in soils?                  and at B              w h2
N. K. Tovey                  ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                           Section 2

The gradient of the head drop (or pressure drop) is known as               Note the -ve sign as h decreases as we go in direction of
the hydraulic gradient (i). In this simple situation, the                  flow of water,
hydraulic gradient is:-                                                    i.e. from B to A.
                                        h1  h2
                                  i                                        the total pwp at A is         = uw      = w (Z + h)
         NOTE: The hydraulic gradient as defined above is                   while the total pwp at B is   = uw + duw
         dimensionless (i.e. has no units). In some other
         disciplines, it is defined in terms of pressure (rather                                          = w (Z + dZ + h + dh)
         than head) and thus is no longer dimensionless. In
         this case,                                                         i.e. the difference in head is    duw =           w (dZ +
                                        w (h1  h2 )                      dh)
         In this case, there are units associated with the                   w dZ is the pressure arising from the head difference
         hydraulic gradient (i.e. kN m-3). To keep things
         simple we shall use the first definition in this course.             wdh        is the pressure arising from the excess
                                                                           pressure in the standpipes which will cause flow of water
The above is a simple description of what how to measure the               from B to A.
hydraulic gradient. Strictly speaking we should be talking in
terms of the differential coefficient:-                                    In the alternative definition of hydraulic gradient used in
                                                                           some disciplines,
                                        dh                                                      du
                                 i                                                     i 
                                        ds                                                      ds and the units are kN.m-3
where s refers to a general direction of measurement.

For most of you it is sufficient to accept the above definition,         2.3 The Permeameter
but if you want the full derivation it is given in the following
box (see WARNING given in introduction about these boxes).               The permeability of a soil dictates how quickly water will flow
                                                                         within the soil, and more importantly how quickly excess
                                                                         water pressure will dissipate. The permeability of a soil may
                                                                         be measured with a Permeameter. For silts and sands it is
                                                                         common to use a constant head Permeameter (Fig. 2.3). For
                                                                         clays the permeability is so low and a falling head
                                                                         Permeameter is used.

                                                                         2.3.1 Constant Head Permeameter (Fig. 2.3)

                                                                         The apparatus consists of a vertical cylindrical tube in which
                                                                         is placed the sample. Below and above the sample are porous
                                                                         stones. Water is fed from a supply to a constant head
                                                                         reservoir and to the base of the sample. After passing through
                                                                         the sample the water flows into a measuring cylinder which is
                                                                         used to estimate the flow rate. In the more accurate
                                                                         permeameters there are two pressure at a fixed distance apart
 Fig. 2.2 Illustration of two types of water pressure.                   in the cylinder wall. Fine bore capillary tubes are inserted into
                                                                         sample and connected to manometers to measure water
 Let A and B be two points separated by a small distance                 pressure at the two points, and thus the hydraulic gradient
 ds, and let the excess p.w.p. be du = w dh                             may be determined.

 then the hydraulic gradient (denoted by i) is given by:-                The experiment starts with the constant head reservoir at a
                                                                         given level. Water is allowed to pass through the sample for a
                                     i                                 few minutes (depending on the nature of the material under
                                               s                        test) until a steady state is reached. The level of water in the
 In the limit as ds   --->   0                                           pressure tappings is measured and water is allowed to flow
                                                                         into the measuring cylinder for a known period of time.
             dh       or                        ……….(2.3)
      i                     i 
                                       1 du
             ds                        w ds                             The flow rate Q may be estimated (volume of water collected
                                                                         in cylinder divided by time). We also measure the total
                                                                         internal cross-sectional area of the cylinder (At) .

N. K. Tovey                  ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                   Section 2

                                                                                of water, the density of water, and a shape factor (i.e. relating
                                                                                to the shape of the voids). Since this is an introductory course,
                                                                                and the use of non-dimensional parameters for hydraulic
                                                                                gradient leads to a simpler set of units, this will be used in this
                                                                                course and is consistent with those used in the textbooks on
                                                                                the reading list..

                                                                                2.5 Experimental Results from Permeameter
                                                                                The results of a replication of Darcy's experiment are shown
                                                                                as line A. For many soils, the linearity of the line is good,
                                                                                confirming Darcy's Law, but in peats, the data is often very

Fig. 2.3 Constant Head Permeameter

The velocity va of the water as it flows in the part of the
cylinder above the sample may be obtained from:-

   va 
          At     ........................................(2.4)

  va is also known as the apparent velocity (i.e. the
velocity the water would have when passing through the soil if
the solids occupied zero volume. It is less than the actual
velocity.                                                                       Fig.   2.4   Experimental Permeability Results on Leighton
                                                                                             Buzzard Sand (after Schofield and Wroth, 1968).
2.4 Darcy's Law
                                                                                As the head is increased, the upward pressure gradient across
In 1856 Darcy, using equipment similar to that described                        the sample increases, and eventually at point C the upward
above and continued the experiment by raising the level of the                  force equals the downwards force from self-weight and
constant head reservoir while keeping the height of the sample                  LIQUEFACTION occurs (i.e. we have created a quicksand).
constant. Two points were noted. First, the flow rate                           The soil "boils" at this point. If we now reduce the pressure
increased, and secondly the hydraulic gradient also increased.                  gradient by lowering the constant head reservoir, we will find
More readings were taken with further increases in the height                   that the sample will settle again, but will occupy a greater
of the constant head reservoir.                                                 volume than previously.

Darcy found that the apparent velocity         (v) was proportional             We may now repeat the experiment, but this time, although
                                                                                still displaying a linear trend, the points lie on a line with a
to the hydraulic gradient   (i)                                                 higher gradient. As might be expected, the rate of flow of
                            dh                                                  water in the loose sample is greater than for the dense sample
i.e.       va  ki   k              .                   .........(2.5)        and hence the coefficient of permeability (k) has increased.
                                                                                The results shown in Fig. 2.4 were obtained for the same
k is known as the coefficient of permeability                                   Leighton Buzzard Sand as used in most of the demonstrations
                                                                                described in section 1.
NOTE: k has the units of velocity (i.e. m.s-1) because the
hydraulic gradient in non-dimensional. In some branches of                      The cylinder containing the sample has a uniform cross-
Science, the hydraulic gradient is measured in terms of the                     section and thus there is a uniform hydraulic gradient within
pressure (rather than head). In such cases, permeability is                     the soil sample and equals up the cylinder and it is equal to
measured in units of m3.s-1.kN-1. The two sets of readings                                                   h
differ only by a factor of the unit weight of water. There is                                          i
further confusion in that the term hydraulic conductivity is
sometimes used instead of permeability.                                         where dh is the difference in the level of the pressure
This latter term, used in Hydrogeology, differs from                               and ds is the distance between the tappings.
permeability in that it also attempts to allow for the viscosity
N. K. Tovey                   ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                Section 2

                                                                          The volume of the reduction in water in the capillary tube in a
The initial voids ratio for the (initially medium dense) sand             time t will equal the flow of water through the sample in the
may be calculated as follows:-                                            same period.

Let m be the total mass of sand                                                            dh khA
       its length                                                                  a       
and                                                                                        dt L
then volume occupied = A                                                                    h           t
                                                                                               dh A
                                                                                                 1           1

                                   m                                               i .e.  a    k  dt
                                                                                               h   L0
and volume of sand grains       = Gs  w                                                     h   2

                                                                                             aL         h
                                                                                   k  2.3        log 10 0 ...............(2.7 )
where Gs is the specific gravity of the solid particles.     Hence                           At 1       h1

   A  m G      AGs  w                                                Thus to measure the permeability using a falling head
e          s w
                          1                                             Permeameter, the sample is enclosed in the sample tube and
     m              m
       Gs  w                 ...........(2.6)
                                                                          the capillary tube is filled so that the water level is nearly at
                                                                          the top. A valve remains closed preventing the water flowing
                                                                          through the sample until the experiment is ready to begin. The
                                                                          height of the water in the capillary tube is measured and a stop
2.6 Falling Head Permeameter (Fig. 2.5).                                  clock is started as the valve is opened. The time for the level
                                                                          in the capillary tube to fall to a second level is noted. The
When the permeability of fine grained silts/clays is to be                measurements needed are thus:- two heights on the capillary
determined it is found that the flow rate is so low that it cannot        tube, the diameter of the capillary tube, the length and
be measured accurately. A falling head Permeameter is then                diameter of the sample.
                                                                          2.7 Formation of a Quicksand - Piping
                     Let a be cross-section of capillary tube
                         A be cross-section of sample tube                Referring back to graph in section 2.5. When point C was
                         L be length of sample tube                       reached the sand appeared to boil as in a quicksand and piping
                         ho be initial height of water at t = 0           occurred.
                         h be height at time t
                     and h1 be height at time t1                          The seepage force is that force exerted by the water seeping
                                                                          through the soil.

                                                                          In the situation above, at point C, the seepage force equals the
                                                                          submerged weight of the sand.

                                                                          Let critical value of dh at which the quicksand occurs be
                                                                          dhcrit. the corresponding critical hydraulic gradient (icrit)
                                                                          will be given by:-
                                                                                                     va 

                                                                              the upwards force       = Atw dhcrit =          At du

                                                                           whilst the downwards force            = At'dz

                                                                          thus the critical hydraulic gradient occurs when

                                                                                           h crit  A '     Gs  1
                                                                              i crit              t     
                                                                                           z       At  w    1e             .........(2.8)
Fig. 2.5 Falling Head Permeameter
                                                                          During piping the volume occupied by the sand increases and
From Darcy                                                                consequently the voids ratio. The second line (loose in Fig.
                 Q = kiA                                                  2.4) was obtained by repeating the experiment at the new
                                                                          voids ratio.
N. K. Tovey                 ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                  Section 2

At a condition of piping, the sand becomes completely                           1) estimations of pump capacities in water resources, and
buoyant and it is as though the effect of gravity had been                      2) stability calculations on slopes.
reduced to zero.
                                                                                The flow of water through soils is directly analogous to two
If flow of water had been downwards, the effect of gravity                      other processes namely HEAT FLOW and the flow of
would be increased. This effect can be used in model analysis                   electricity. Those of you who have done the Energy
to study slope stability using models of a slope..                              Conservation (ENV-2D02) course will already be familiar
                                                                                with the equations defining heat flow through a component of
2.8 Typical Values of k                                                         a building (such as the walls). Those of you who did Physics
                                                                                at school or who have done Geophysics will be familiar with
                  gravels                      k > 10 mms -1                    electrical conductivity.

                   sands     10 mms-1 > k > 10-2 mms-1                          These analogies are helpful, as we can model both heat flow
                                                                                and water flow using an electrical analogue model which is
                    silts   10-2 mms-1 > k > 10-5 mms-1                         easier to change than is either a brick wall or a layer of
                   clays                        k < 10 -5 mms-1
                                                                                In HEAT FLOW

2.9 Actual Seepage Velocity                                                            Q    ( 1   2 )
                                                                                where Q is the heat flow rate
The actual velocity of seepage through the pores must be
                                                                                        1 is the internal temperature
greater than the apparent velocity as calculated by equation
2.4.                                                                                     2 is the external temperature
                                                                                        A is the cross  section area
                Q                                                                        is the thicknessof wall
           va 
                At          ........................................(2.4           and k is the thermal conductivity
                                                                                In the FLOW of ELECTRICITY

 If vs is the actual velocity, and      Av     is the actual cross                       kA
                                                                                       I    ( E1  E2 )
section of the voids                                                                       
                                                                                where I is the heat flow
then Q = vsAv = vaAt - i.e. the water flowing through                                   E1 is the inletpotential
the soil pores must equal the apparent flow mentioned earlier.
This is the continuity equation..                                                       E2 is the outletpotential
                                                                                        A is the cross  section area
                         A      V   v       1e
                  vs  va t  va t  a  va                                              is the thicknessof wall
i.e.                     Av     Vv   n       e                                     and k is the electrical conductivity
                                                                                In the FLOW of WATER in SOILS
 where Vt and Vv are the volume of the total sample and
that of the voids respectively.                                                        Q    ( h1  h2 )
for dense Leighton Buzzard Sand      vs is approximately 2.6va                  where Q is the flow rate
                                                                                        h1 is the intlet head
NOTE: In the symbols, upper case V is used for volumes,                                 h2 is the outlethead
but lower case v is used for velocities.                                                A is the cross  section area
                                                                                         is the thicknessof wall
2.10 Flow of Water in Soils
                                                                                   and k is the permeability
In the permeameter the shape of the sample is cylindrical and
analysis of the flow of water through the soil is simple. In                    The solution of the problem through a cylinder is
nature, however, the boundary conditions are never as simple                    straightforward and can become quite complex in more
and ways must be found to enable estimates to be made of                        general situations. The flow of water (or flow of heat, or flow
seepage.                                                                        of electricity) is governed by Laplace's equations and suitable
                                                                                solutions to these equations. There are five ways in which
This is important for two reasons:                                              solutions may be achieved:-
N. K. Tovey                ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                 Section 2

                                                                         This is the two-dimensional version of Laplace's equation,
         1) Mathematical solutions                                       and solution to seepage problems from which we can
                a) exact solutions for certain simple                    obtain the water pressure at any point require solution of
                     situations                                          this fundamental equation within appropriate boundary
                b) solutions by successive approximate -                 conditions. In this course we shall be using graphical
                     e.g. relaxation methods                             solutions,      but if you are doing ENV-2B31
         2) Graphical solutions                                          (Mathematics), you may like to attempt a mathematical
         3) Solutions using the electrical analogue                      solution of the problem to simple boundary value
         4) Solutions using models                                       conditions. The example shown in section 2.14 may also
                                                                         be solved provided that you transform the co-ordinates
Mathematical solutions are beyond the scope of this course               using conformal transformation beforehand.
(although anyone doing the Maths options may wish to follow
section 2.11, and then attempt some solutions as applications
                                                                         There must be continuity: i.e. the water flowing in across
for the Maths Course).
                                                                         the bottom and left hand border must equal the water
                                                                         flowing out across the other two borders.
The Relaxation Method requires the development of computer
software. An example is already in use in dynamic heat flow
                                                                                                         v x                    v z
computations in the Energy Conservation Course as a self                    v x  z  v z x  ( v x         . x )z  ( v z       . z)x
contained computer package, and it is hope to adapt this for                                             x                      z
use in water flow in soils for this course in the next few years.                v x   v z
                                                                         i. e.              0                     ..........( 2. 10)
Graphical Solutions are the method we shall adopt in this
                                                                                 x     z
                                                                         for convenience we will make the substitution           = ki
Electrical Analogue method are nice to use in practicals, and
were in fact used in the past in this course when more time                               i                             i
                                                                         i. e.          k                    and        k
was devoted to the topic.
                                                                                     x    x                         z    z
Solutions using models. These are expensive to construct, but                                                                 
nevertheless can give useful information particularly in                 thus         vx                  and v z  
                                                                                                x                              z
complex situations when it becomes difficult to define the
conditions easily in mathematical terms. A model of water
                                                                         Substituting for vx in Equation (2.10) we get
flow around an obstruction was constructed many years ago
and is now used in the Hydrogeology option.
                                                                                    2  2
                                                                                             0 ............(2.11)
2.11 Flow Equations - 2-D case                                                     x 2 z 2

This section contains the Mathematical derivation of the basic           Note the analogy with flow of electricity:
equations governing the flow of water in two-dimensions.
This is strictly for those who are mathematically inclined - for
                                                                                  2V    2V
those who are not - skip the following highlighted box.                                                              0
                                                                                 x 2   z 2
                                                                         where V is the electrical potential or voltage

                                                                         or heat flow

                                                                                  2    2
                                                                                                                    0
                                                                                 x 2   z 2
                                                                         where  = temperature

                                                                         By analogy  is termed the potential in the flow
                                                                         of water.
 Fig. 2.5 Schematic of two-dimensional flow of water

                                                                         As solutions to the above equations already
                                                                         exist for heat/electricty flow, we may use these
                                                                         soltuions to solve flow of water problems.

                                                                         2.12 Graphical Solutions - Flow Nets (Fig.
N. K. Tovey               ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                           Section 2

2.12 Graphical Solutions - Flow Nets (Fig. 2.6).

In this course we shall only concern ourselves with the
graphical solution to Laplace's Equation. While this may at
first sight seem rather crude, it is nonetheless very effective
and can be used for all shapes of flow channel provided that a
little care is used when drawing. We do this by drawing
flownets - i.e. we draw lines parallel to the line of flow of
water (flowlines), and lines at right angles which are lines of
equal pressure or equipotentials. Experience certainly helps,
and you may wish to practice by drawing arbitrary shapes to
define boundaries and attempt to fill in the appropriate flow

In the permeameter we had a situation where flowlines are
vertical and lines of equal pressure (equi-potentials) are

                                                                       Fig. 2.7 Asymmetric Flownet

                                                                               Note: equipotentials are still orthogonal to flow
                                                                               lines and it can be shown that all regions bounded by
                                                                               two adjacent flowlines and two adjacent equipotentials
                                                                               are curvilinear rectangles of similar proportion (or in
                                                                               most cases squares - See figure 2.8). The following is
                                                                               merely a proof of the this, you may skip the following
                                                                               box unless you are particularly interested.

                                                                        Consider a single flow channel initially of width a1 and a
                                                                        spacing between equipotentials of b1. The potential drop
                                                                        between all equipotentials is Dh. At a second point in
                                                                        the flow channel where its width is now a2 the distance
                                                                        between the equipotentials is b2. By continuity we must
                                                                        have the same amount of water flowing in both parts of
                                                                        the flow channel.

Fig. 2.6 Flowlines and Equipotentials in a simple situation
         where all flow lines are parallel

Two points to note:-

     1) flowlines and equipotentials are at right angles to
        one another.
     2) the cylinder walls are also flowlines.

The distances  between the equipotentials are equal and thus
the head drops between the equipotentials are also equal.               Fig. 2.8

Each flow channel is defined by two adjacent flow lines define           Let a1 be the width of channel when velocity is v1
the region in which water moves, i.e. water will not move from
                                                                        and a2 be the width of channel when velocity is v2
one channel to another (apart from a minor effect from
                                                                        then q = v1a1 = v2a2 = v3a3
What happens if cylinder is not of constant cross-section? (see
Fig. 2.7)                                                                                        kh                   kh
                                                                        but v1 = k i1 =             and v2 = k i2 = 
                                                                                                  b1                    b2
By symmetry, flow lines must diverge, but since the package
of water A must still remain within the flowlines c - c, d - d                                a1        a
its velocity will change and become less. If the dashed lines           thus    q   kh          kh 2
represent equipotentials, then they must become progressively                                 b1        b2
further apart as one proceeds towards the right.
N. K. Tovey                   ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                           Section 2

                                                    a1   a                                                nf k H
Hence for equal pressure drops we must have             2                                     nf q f 
                                                    b1   b2             the total seepage =                nd
                                                                                                           ..............       (2.12)
   i.e. the ratio is constant.
Thus we must draw rectangles of constant proportion.
                                                                        In other words if we require to determine the total volume of
However this is difficult when we have curvilinear figures
                                                                        water seeping we need only draw a flow net and count the
unless a = b (i.e. we have squares - a special case of the above
                                                                        number of pressure drops and flowlines.
                                                                        To work out the pressure at any point, which is what we really
General Flownets solution               - (See figure 2.9)              want, the procedure is equally simple and will be illustrated
                                                                        with reference to a particular example (Section 2.13).
Solutions are relatively straightforward. We need:-

1) draw the appropriate flownet
2) count the number of pressure drops in the flow net (over
                                                                        2.13 Seepage around an obstruction - (See figure
   the relevant distance)
3) count the number of flow lines
4) do a simple calculation as given by equation 2.12 below.             The example shown in Fig. 2.10 has a vertical obstruction
                                                                        which is preventing the normal flow of water. It also
                                                                        represents the model rig in the Soil Mechanics Lab which was
Let total pressure drop between AB and CD be           H and let
                                                                        formerly used as a practical in the fore-runner to this course,
there be nd pressure drops and nf flow lines.                           but recently has been used in the Hydrology courses.

Assuming uniform permeability k and h to be pressure drop               On one side of the obstruction, the water level is maintained
across one square of side.                                              at a high level while the water level on the other side is at
  h    H                                                                approximately the level of the soil. Water will flow in around
                                        where qf is the flow           the obstruction and upwards on the downward side of it. Thus
     nd 
                                         per unit cross-section         there will be an upward flow of water immediately
ByDarcy' s Law v  ki                   and
                                                                        downstream of the obstruction, and if the upward force of the
                                 nd                                    water exceeds the downward submerged weight of the column
                       kHa               a x 1 is the cross-            of soil, then piping will occur (i.e. a quicksand will form).
  and q f  kia 
                       nd               section between flow
                                         lines.                          We first draw the flownet and note that piping will occur if
  but a                                                               seepage force at A >= buoyant weight of column of sand.
Hence the total flow in a single flow channel is given by:-
                                                                        If the difference in the height of the water on the two sides of
                                                                        the obstruction is H, then the pressure drop between two
                       kH                                               equipotential lines will be wH / nd.
               qf 
and if the number of flowlines is              nf
  this space is left blank for notes:

N. K. Tovey             ENV-2E1Y Fluvial Geomorphology 2004 – 2005        Section 2

Fig. 2.9 Generalised Flownet
N. K. Tovey              ENV-2E1Y Fluvial Geomorphology 2004 – 2005        Section 2

Fig. 2.10 Flow of water around an obstruction

N. K. Tovey                     ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                                 Section 2

NOTE: we must include w as we are now dealing with                               if we know the voids ratio, then from as this may indicate
                                                                                  whether fine material is washed out from the soil because of
pressure, and not merely head of water.
                                                                                  high actual seepage velocities.
 If the number of equipotential drops from the base of the                                                          vs  va
obstruction to the surface of the soil on the downstream side is                  From equation (2.9)                          e
                                               N ab w H                          At any point within our flow net we can estimate the hydraulic
                                                  nd                              gradient by dividing the head drop between two equipotentials
Then the upward seepage force                =                                    by the distance between them

                                                  ' 
                                                 where  is
 and the downward force of the soil =                                                i.e.   nd
the depth of penetration of the obstruction into the soil. A
quicksand will occur if                                                           Once again the following is not essential to the course, but
                                                                                  may be considered if you wish.
                 N ab w H
                            '
                    nd                                                              Alternatively we may measure the actual seepage
                                                                                    velocity on a model determine permeability by
   hence                                                                            rearranging equation 2.16.
                 Nab H '
                       ..................(2.14)
                  nd  w                                                            i.e.
                                                                                                   v s nd e
As an example in the use of a factor of safety, we may predict                               k           .
such a factor (Fs) as follows                                                                         H 1e
                                                                                    and by Darcy
                           actual downward force of the soil
Factor of safety =        ------------------------------------------------                     kH                      1  e kH
                         force required to just resist seepage force                    va             i. e. v s          .
                                                                                               nd                       e nd  …(2.16)
            '  nd
   Fs          .               .........................( 2. 15)                 2.14 Flow nets (Summary)
           w H N ab
                                                                                  Rules for drawing flow nets:-
    In the above example,           nd = 10 and          Nab 3.5                 1) All impervious boundaries are flow lines.
                                                                                  2) All permeable boundaries are equipotentials
and noting that very approximately                      ' = w                   3) Phreatic surface - pressure is atmospheric, i.e. excess
                                                                                     pressure is zero.
                                10                                                       A further requirement is that the change in head
                     Fs                                                                  between adjacent equipotentials equals the vertical
                               3. 5H
                                                                                          distance between the points on the phreatic surface.
i.e. the distance       must exceed 0.35 times the difference in                 4) All equipotentials are at right angles to flow lines
head of water.                                                                    5) All parts of the flow net must have the same geometric
                                                                                     proportions (e.g. square or similarly shaped rectangles).
The following box is not essential to the course, however it is                   6) Good approximations can be obtained with 4 - 6 flow
complementary to the course and to aspects of the Hydrology                          channels. More accurate results are possible with higher
courses.                                                                             numbers of flow channels, but the time taken goes up in
We are sometimes interested in the actual seepage velocity vs                        proportion to the number of          channels. The extra
                                                                                     precision is usually not worth the extra effort.
 If we wished to work out the quantity of water flowing
 around the obstruction then from equation 2.12, and noting
 that nd = 10 and the number of flow channels is
 approximately 4.25.                                                              · a difficulty in flownet analysis is the determination of the
                                                                                      top flow line in flow down a slope or through an earth
                          nf                                                          dam.
 Q  n f q f  kH                 0. 425 kH                                      · It is difficult to give hard and fast rules - best try and see.
                                                                                      However, a good approximation is a parabola.
 If, for example, the permeability       ( k ) = 1 mms-1 and if a                 · In regions of sharply varying flow channels, there are
 field situation, H = 10 m                                                            more advanced methods which allow the subdivision of
                                                                                      squares into quarter size squares just at the key points,
 Q = 0.425 x 10-3 x 10 = 4.25 litres.s-1.                                             but these techniques are beyond the scope of this course.
N. K. Tovey               ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                           Section 2

The next two sections (i.e. 2.15 and 2.16) of this handout
are not essential for the course, but you should appreciate              2.16 Flow through soil having differing
in general terms what is going on. You will not be                              permeabilities in x and z directions
expected to do numeric problems based on the following.
                                                                         The above discussion has centred on the assumption that
 2.15 Flow between regions of different permeability                     permeability is uniform in all directions. Not
                                                                         infrequently there is a difference, through lamination,
 Flowlines and equipotentials in Fig. 2.11 in Region A form              between the horizontal and vertical permeabilites.
 squares of side a. In region b, these become rectangles of
 size b x  . Let permeability be ka and kb then by                      Let these permeabilities be kx and kz respectively.
 continuity flow in channel must be same in both A and B.
 Hence:-                                                                 We may drawn an equivalent flow net if we first adjust
                                                                         the scale of the original diagram so that we multiply all
                                                                         horizontal distances by a factor of
                                                                                                      (See figure 2.12)
                                                                         If we do this, we can then still use square (or rather
                                                                         curvilnear square) flow nets on the transformed diagramd
                                                                         diagram provided that we recognise that the effective
                                                                         permeability is also modified as follows:-

                                                                         The equivalent permeability is then k = (kxkz)

 Fig. 2.11 Water flow across a boundary between regions of               The geometric modification is a form of conformal
           different permeability                                        transformation (which those doing ENV-2B31 may have
                                                                         come across). Indeed the problem of the seepage around
      a         b                       ka   b
 q     k a h  k b h           so                                    an obstruction may also be solved by transforming the
      a                                kb                              actual scale into a square net using an appropriate
                a         b                a                            conformal transformation, although in this case it does
 and AB                       and AC                                 require elliptical functions in the transformation.
              sin  a   sin  b          cos  a   cos  b
             ka      b      tan  b                                      For details see pages 59-64 of Critical State Soil
       so               
             kb             tan  a                                     Mechanics by Schofield and Wroth (1968) - publisher
  Thus in passing between two zones flowlines are refracted              McGraw Hill - It is in the Library. and I also have a copy
 such that the ratio of the tangents of angles of incidence are          for those interested
 proportional to the permeabilities. Note: when drawing flow
 nets at boundaries, squares in one zone will become
 rectangles in the second zone.

Fig. 2.11 Example of transformation to allow for anisotropic conditions

2.17 Uplift on Obstructions                                            This pressure arises from both the static and excess heads.
                                                                       For an obstruction which has a flat base (which is horizontal,
When water seeps under a large obstruction such as a boulder           the static head will be constant along its length and equal to
which is partly buried in the soil, the obstruction experiences        the depth of the base below the water level on the down stream
an uplift from the total water pressure exerted on the base.           end. The excess head will vary along the underside of the
N. K. Tovey                  ENV-2E1Y Fluvial Geomorphology 2004 – 2005                                           Section 2

Fig. 2.13 Uplift on a rectangular boulder by water passing underneath

If the total uplift force exceeds the downward force from the            Because the base of the obstruction is 2m below the surface
self weight of the obstruction, then the object will be displaced        the uplift force from the static head is 2w multiplied by
downstream. To assess the likelihood of this happening, a
                                                                         width (i.e. 6w kN per metre length). From the excess head
flownet is drawn for the water seeping through the soil. A
                                                                         (see graph), the upward force is the area under the curve
graph is plotted with the x-axis as the distance along the base
from the upstream face, and the water pressure as gauged                 multiplied by w. In this example (since the line is nearly
from the equipotentials (+ the static head) as the y-axis. The           linear), the upward force = 6w kN per metre length, i.e. in
total force is then evaluated by working out the area under the          this case it equals the static head uplift. The total uplift will
curve. If this exceeds the with of the obstruction then it will          then be the sum of these two components, i.e. 12w kN m-1.
be displaced
                                                                         This uplift will considerably reduce the ability of the
The pressure distribution shows the excess head an has been              obstruction to resist movement through the pressure of water
drawn from position of intersection of equipotentials along the          (and has significance both as potential boulder blockages in a
base of the obstruction. The equipotentials are nearly equi-             river and also as man-made drop structure built in river
distant from each other in this region and the distribution is           engineering works to dissipate energy (see RDH's part of the
nearly linear. By counting equipotentials, the head at the               Course).     There may also be erosion from the possible
upstream head is 0.75 of total head, and at the down stream              quicksand which may form at the down stream end of the
end it is 0.25 of the total head.                                        obstruction.

  space for further notes.


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