Geoscience Reference
In-Depth Information
crucial importance should be retained. A model is in fact an idealisation of reality.
With modelling, one looks for an appropriate idealisation.
Consider the consolidation process as the phenomenon of interest. The field
equation is, according to (6.4)
2
u/
z
2
=
)
[m
2
/s]
c
v
u/
t
with
c
v
= k
/(
(9.1)
w
By introducing scaling in space, time and the field variables, this equation is
rendered dimensionless, as follows
2
F/
Z
2
=
Y
with
F = u/u
0
,
Z = z/L
,
Y = t/T
and
C = c
v
T/L
2
(9.2)
C
F/
Here,
L
,
T
and
u
0
are arbitrary scaling factors, and
C
is a dimensionless
consolidation number. Note that the field variable scale
u
0
does not appear in
C
,
because (9.1) is a linear in
u
. For a non-linear process the field variable scale will
appear in the dimensionless number. The general solution of (9.2) in terms of
consolidation degree
U
as function of
Y
, according to (6.6) with
L = h
, becomes
2
) exp(-
CY
2
/4)
U =
1 - (8/
(9.3)
Now, physical similarity for a model (m) and the prototype (p) is satisfied, when
the number
C
is similar for both, i.e.
C
m
=
C
p
. If the same material is tested, i.e.
c
v
is identical, then obviously
T
m
/
T
p
= (
h
p
/h
m
)
2
must hold. The time in the prototype to
reach the same consolidation degree as in the model will be (
h
m
/h
p
)
2
faster. So, if
50% consolidation in an oedometer test of 2 cm high is reached in 0.5 day, then it
takes 125 days for a layer of 5 metres in the field to reach 50% consolidation. In
conclusion, it is sufficient to consider dimensionless numbers in order to achieve
physical similarity.
Model tests are usually performed for situations where the solutions are not
known. In most cases, the physical phenomena of interest can be described in terms
of differential equations based on conservation principles (conservation of mass,
momentum and/or energy) and constitutive laws. Making these dimensionless
results in the physical numbers related to the different phenomena involved.
Imposing similarity for all of these numbers, i.e. making them identical for model
and prototype, provides the necessary scaling rules.
In centrifuge testing the gravity is significantly increased. As an example, the
physical similarity for two phenomena, equilibrium and consolidation, can be
elaborated. Consider the fundamental equations for both processes (for simplicity
one spatial dimension is considered), as follows
'
u
equilibriu
m
g
(9.4a)
x
x
2
2
u
u
consolidat
ion
(9.4b)
x
c
t
v
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