54
Geotechnical News December 2011
GEO-INTEREST
tion of the Seepage Force for the soil
(solid + water volume) associated with
the same particle.
So the problem comes down to find-
ing the factor by which the velocity
term in the F
D
equation must be mul-
tiplied to make the F
D
force associated
with a single particle equal to the S
F
force for a single particle.
Theoretical/Idealized Approach
In order to give mathematical
expression to the relationship between
Seepage Force and Drag Force we
must limit ourselves to dealing with
spherical particles of uniform size.
By looking at a single particle and
the volume occupied by that single par-
ticle we can write:
S
F
=
i
γ
w
(1+e) D
3
π/6
F
D
= C
D
ρ
(v
V
2
/2) D
2
π/4
In this particular instance I have
chosen to temporarily revert to using
C
D
rather than using the component F
B
and F
P
, and this is simply for conve-
nience: More mutual terms cancel out.
Now, setting S
F
= F
D
and recalling
that v
A
=
i k
, we get:
v
V
2
= 4 g D (1+e) v
A
÷ 3
k
C
D
which gives,
v
V
÷ v
A
= K =
2 √ ( g D (1+e) ÷ 3
k
C
D
v
A
)
This equation for K, it should be
noted, requires an iterative process to
recognize the fact that C
D
, and in non-
laminar flow situations,
k
, are both
functions of relative velocity. Such nu-
merical awkwardness is avoided in the
alternative approach outlined below.
The implication of the above math-
ematical derivation is that a value can
be given to the Crowding Factor once
the permeability of the soil has been
established. Although I offer a theo-
retical solution for evaluating saturated
soil permeability in the next article, it
must be said that such solutions are at
best approximations, and lab testing of
good specimens is really the only way
to go if there is any hope for accuracy
in subsequent computations.
The above theoretical approach is
useful inasmuch as it provides math-
ematical continuity to the overall hy-
pothesis, however, the following ap-
proach is likely to be more useful in
practice.
Empirical/Practical Approach
Earlier in this article I used the
permeameter to help explain the
Seepage Force. Now it would make
sense to look again at this standard
piece of laboratory equipment for
an empirical solution to our current
problem. What we can get from this
tool is not only the permeability [
k
]
needed to solve the above equation,
but furthermore, we get a direct
measurement of the actual Seepage
Force exerted on the volume of soil
comprising the specimen. And in fact,
this is all we need to know in order to
determine the value of K for whatever
real soil, and degree of compaction,
used to make the specimen.
How this is accomplished for a soil
containing a range of particle sizes re-
quires some explanation. Full details
of this procedure, and a computer pro-
gram to facilitate the calculations will
be given in the next article. Suffice to
say at this time, that what is involved is
finding, by iteration, the unique value
of v
V
which will achieve the criterion
that the summation of the individual
Drag Forces on the particles within the
mass should equal the Seepage Force
for that volume of soil.
Although the permeameter is a stan-
dard piece of equipment in geotechni-
cal labs, my preference for this particu-
lar investigation is for using the triaxial
apparatus instead. There are four rea-
sons for this choice:
1. Triaxial technicians are familiar
with constructing specimens to ex-
plicit specifications and they know
how to saturate and de-air soils. Air
entrained in an otherwise saturated
soil would artificially decrease
the measured permeability and in-
crease the Seepage Force.
2. The flexible membrane in which
the specimen is enclosed provides
a good boundary for the outer soil
particles once the cell pressure ex-
ceeds the pore water pressure. A
rigid (metal or glass cylinder) en-
casement of soil results in signifi-
cantly higher void spaces around
the specimen perimeter and this
leads to artificially high values of
permeability and lower Seepage
Forces. This is particularly impor-
tant in coarse uniformly graded
materials such as can be tested in
the large diameter setups available
to us nowadays.
3. After the permeability and Seepage
Force have been determined in the
drained-mode the specimen can
then be strained to see whether the
soil tends to contract or dilate. This
tells us whether deformation of the
soil modeled in the test specimen
will lead to increases or decreases
in pore water pressure.
4. It is a simple matter at this stage to
perform a routine drained or und-
rained compression test at the de-
formation rate of interest.
So Where Are We Now?
Fluid Mechanics and Hunter Rouse
have given us access to hydrodynamic
aspects of water flow at various
velocities around spherical particles of
various diameters, and that allows us to
separate such energy flow losses into
those which create water pressure and
those (viscous) which do not. The visit
to Fluid Mechanics also gave us a way
of looking at liquefaction and the idea
that the structural collapse/fall came
before the pore pressure rise. Following
this valuable excursion into Fluid
Mechanics, it is appropriate to return
to Soil Mechanics once it comes down
to non-discrete particles in crowded
assemblies, and to those aspects of
soil-structure and agglomerations
which geotechnical engineering is all
about. I believe the combination of
these sister disciplines gives us the best
of both worlds.
In the Next Article
The next article, Part 6, will be the
last in this series. The details of how
to calculate the pore water pressure
generated in any gradation of a saturated
soil-structure under deformation will
be explained.
I will make some general statements
about what I believe to be the most im-
portant facts about the water in the soil.
W. E. Hodge, P.Eng, M.ASCE
(778) 997-4505
