N. K
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
26
N. K. Tovey ENV-2E1Y Fluvial Geomorphology 2004 – 2005 Section 2
Slope Stability and Related Topics
2. Seepage of Water in Soils/Permeability
26
N. K. Tovey ENV-2E1Y Fluvial Geomorphology 2004 – 2005 Section 2
2.1 Introduction
NOTE: FIRST A HEALTH WARNING!!!! There are some sections of this handout which are in shaded boxes. These are not essential parts, but complement the main course. Thus for some of you who are doing ENV-2B31 (Mathematics I) you may object the simplistic approach sometimes used in the handout. There is a more rigorous approach for you in these boxes. In other cases, the boxes show additional information which may be derived which may be of use in other courses (e.g. Hydrogeology). Please consult the note before each box.
For another set of notes for this section see the University of West of England WEB Site on the topic.
http://fbe.uwe.ac.uk/public/geocal/SoilMech/water/water.htm
A knowledge of the factors affecting the flow of water through soils is important as the permeability of a soil affects the way in which a soil consolidates which in turn affects its mechanical properties. Equally water pressures may build up within the soil and as seen the demonstrations can greatly affect the ability of a soil to resist shearing. There are three component parts to the water pressure:-
i) that pressure arising from a static head
[ at any point at a height Z above the measuring datum, this pressure will be gw Z where gw is the unit weight of water.]
ii) excess pore water pressure (i.e. a pressure head differential which actually causes water flow [ this has the symbol u ]
iii) a velocity head and equals
The total pore water pressure (often abbreviated to pwp)
= ...... 2.1
For those who have done the Hydrology or Oceanography options you may already be familiar with Bernoulli's equation of fluid flow i.e.
...... 2.2
total = position + pressure + velocity
head head head head
i.e. equation 2.1 is Bernoulli's equation expressed in terms of pressure rather than head of water. [To convert from equation 2.1 to 2.2 all we need do is divide by gw.
How significant are these three terms in water flow in soils?
Both the first an second are clearly important (the first arises directly from the depth below the water table, whereas there is ample evidence (in the form of springs to indicate water flow through soils).
What about the velocity head?
Typical velocities even in coarse sands will rarely exceed 10 mm s-1, and the magnitude of the last term in this extreme case will be (remember to convert to metres!!!!):-
in terms of total head or 0.00005 kPa in pressure terms.
This is exceedingly small, and in most soils, the velocity will be many orders of magnitude less than this so in future we can conveniently neglect the velocity head term.
2.2 Hydraulic Gradient
Associated with water pressure is the hydraulic gradient which is a measure how fast the water pressure is changing. This in turn affects how fast water will flow and what the immediate pressure within the soil will be.
Now consider flow between two points A and B (see Fig. 2.1).
The hydraulic gradient is defined as the rate of change of head of water with distance (in the direction of measurement)
Fig. 2.1 Flow of water in a channel of simple horizontal cross section
Since the flow is horizontal, there is no effect from the static head of water.
The head of water at A is h1
and at B h2
The excess pressure of the water at A (as the velocity is small) is gw h1
and at B gw h2
The gradient of the head drop (or pressure drop) is known as the hydraulic gradient (i). In this simple situation, the hydraulic gradient is:-
NOTE: The hydraulic gradient as defined above is dimensionless (i.e. has no units). In some other disciplines, it is defined in terms of pressure (rather than head) and thus is no longer dimensionless. In this case,
In this case, there are units associated with the hydraulic gradient (i.e. kN m-3). To keep things simple we shall use the first definition in this course.
The above is a simple description of what how to measure the hydraulic gradient. Strictly speaking we should be talking in terms of the differential coefficient:-
where s refers to a general direction of measurement.
For most of you it is sufficient to accept the above definition, but if you want the full derivation it is given in the following box (see WARNING given in introduction about these boxes).
2.3 The Permeameter
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 in the cylinder wall. Fine bore capillary tubes are inserted into sample and connected to manometers to measure water pressure at the two points, and thus the hydraulic gradient may be determined.
The experiment starts with the constant head reservoir at a given level. Water is allowed to pass through the sample for a few minutes (depending on the nature of the material under test) until a steady state is reached. The level of water in the pressure tappings is measured and water is allowed to flow into the measuring cylinder for a known period of time.
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) .
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:-
...... (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.
2.4 Darcy's Law
In 1856 Darcy, using equipment similar to that described above and continued the experiment by raising the level of the constant head reservoir while keeping the height of the sample constant. Two points were noted. First, the flow rate increased, and secondly the hydraulic gradient also increased. More readings were taken with further increases in the height of the constant head reservoir.
Darcy found that the apparent velocity (v) was proportional to the hydraulic gradient (i)
i.e...... (2.5)
k is known as the coefficient of permeability
NOTE: k has the units of velocity (i.e. m.s-1) because the hydraulic gradient in non-dimensional. In some branches of Science, the hydraulic gradient is measured in terms of the pressure (rather than head). In such cases, permeability is measured in units of m3.s-1.kN-1. The two sets of readings differ only by a factor of the unit weight of water. There is further confusion in that the term hydraulic conductivity is sometimes used instead of permeability.
This latter term, used in Hydrogeology, differs from permeability in that it also attempts to allow for the viscosity 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 non-linear.
Fig. 2.4 Experimental Permeability Results on Leighton Buzzard Sand (after Schofield and Wroth, 1968).
As the head is increased, the upward pressure gradient across the sample increases, and eventually at point C the upward force equals the downwards force from self-weight and LIQUEFACTION occurs (i.e. we have created a quicksand). The soil "boils" at this point. If we now reduce the pressure gradient by lowering the constant head reservoir, we will find that the sample will settle again, but will occupy a greater volume than previously.
We may now repeat the experiment, but this time, although still displaying a linear trend, the points lie on a line with a higher gradient. As might be expected, the rate of flow of water in the loose sample is greater than for the dense sample and hence the coefficient of permeability (k) has increased. The results shown in Fig. 2.4 were obtained for the same Leighton Buzzard Sand as used in most of the demonstrations described in section 1.
The cylinder containing the sample has a uniform cross-section and thus there is a uniform hydraulic gradient within the soil sample and equals up the cylinder and it is equal to
where dh is the difference in the level of the pressure tappings,
and ds is the distance between the tappings.
The initial voids ratio for the (initially medium dense) sand may be calculated as follows:-
Let m be the total mass of sand
and its length
then volume occupied = A
and volume of sand grains =
where Gs is the specific gravity of the solid particles. Hence
...... (2.6)
2.6 Falling Head Permeameter (Fig. 2.5).
When the permeability of fine grained silts/clays is to be determined it is found that the flow rate is so low that it cannot be measured accurately. A falling head Permeameter is then employed.
Let a be cross-section of capillary tube
A be cross-section of sample tube
L be length of sample tube
ho be initial height of water at t = 0
h be height at time t
and h1 be height at time t1
Fig. 2.5 Falling Head Permeameter
From Darcy
Q = k i A
The volume of the reduction in water in the capillary tube in a time t will equal the flow of water through the sample in the same period.
Thus to measure the permeability using a falling head Permeameter, the sample is enclosed in the sample tube and 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 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 measurements needed are thus:- two heights on the capillary tube, the diameter of the capillary tube, the length and diameter of the sample.
2.7 Formation of a Quicksand - Piping
Referring back to graph in section 2.5. When point C was reached the sand appeared to boil as in a quicksand and piping occurred.
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:-
the upwards force = Atgw dhcrit = At du
whilst the downwards force = Atg'dz
thus the critical hydraulic gradient occurs when
...... (2.8)
During piping the volume occupied by the sand increases and consequently the voids ratio. The second line (loose in Fig. 2.4) was obtained by repeating the experiment at the new voids ratio.
At a condition of piping, the sand becomes completely buoyant and it is as though the effect of gravity had been reduced to zero.
If flow of water had been downwards, the effect of gravity would be increased. This effect can be used in model analysis to study slope stability using models of a slope..
2.8 Typical Values of k
gravels k > 10 mms-1
sands 10 mms-1 > k > 10-2 mms-1
silts 10-2 mms-1 > k > 10-5 mms-1
clays k < 10-5 mms-1
2.9 Actual Seepage Velocity
The actual velocity of seepage through the pores must be greater than the apparent velocity as calculated by equation 2.4.
...... (2.4 repeated)
If vs is the actual velocity, and Av is the actual cross section of the voids
then Q = vsAv = vaAt - i.e. the water flowing through the soil pores must equal the apparent flow mentioned earlier. This is the continuity equation..