Number for the rising salt dome would range
from 1.9 ϫ 10
Ϫ6 to 9.7 ϫ 10
Ϫ20 , and the Froude
Number from 6.1 ϫ 10
Ϫ18 to 3.1 ϫ 10
Ϫ24 . These very
small ratios justify ignoring the effects of inertial
forces when scaling models of salt dome development. A laboratory model of salt dome development is illustrated in Fig. 4.13a (Ramberg, 1967,
p. 123).
Many experimental observations of viscous
flow in conduits have demonstrated that the
Reynolds Number must be greater than approximately 1 ϫ 10
3 for the flow regime to become
turbulent (Bird et al., 1960, pp. 183–8). For
Reynolds Numbers less than this value, flow in
conduits is laminar (Fig. 4.8b). Because the range
of Reynolds Numbers estimated for the rising salt
dome (and many other tectonic processes involving flow) is many orders of magnitude less than
this transition value, the flow regime is laminar.
The precise scaling of inertial forces between the
model and prototype is not necessary as long as
the model is well within the laminar flow regime.
The condition for strict dynamic similarity in
terms of inertial forces between model and prototype can be relaxed without compromising the
usefulness of the experimental results.
4.4.3 Gravitational forces: a problem for
model similarity?
It was noted in the preceding section that all corresponding lengths, masses, and times in the
model and the prototype must adhere to the given
model ratios to maintain strict similarity. In
general these are thought of as independent
ratios, to be chosen at the convenience of the
experimenter, or as constrained by the available
materials. However, models of tectonic processes
often are carried out on a lab bench where the
acceleration of gravity is approximately 9.8 m s
Ϫ2 ,
and this value is not significantly different for the
prototype. Thus, the model ratio for accelerations
is approximately one. Examining the model ratio
for accelerations we find:
(4.108)
Thus, the model length and time ratios are not
independent. Solving for the model time:
a r ϭ L r T Ϫ2
r ϭ 1, so L r ϭ T 2
r
(4.109)
Using a reasonable range of model ratios for
length (10
Ϫ6 Ͻ L r Ͻ 1) and for the time scales for
tectonic processes (10
2 s Ͻ T p Ͻ 10
15 s), we calculate
a range of model time scales from T m ϭ 10
Ϫ1 s to T m
ϭ 10
15 s. At the lower end of this range the experimenter would have few problems, but clearly the
upper end is unattainable. For example, a
common model length scale is ten centimeters
(0.1 m), corresponding to a prototype length scale
of one kilometer (1000 m). A common time scale
for the duration of tectonic processes is one
million years (10
6 years). Given these values, the
model time scale is ten thousand years (10
4 years),
not a practical duration for experiments designed
to be observed by humans!
The constraint on the model time scale
imposed by similar accelerations of gravity acting
on the prototype and the model is not fatal to all
model experiments in tectonics because the inertial forces associated with accelerations in the
prototype may be insignificant compared to other
forces. Recall, for example, that flow of many geological materials is in the realm of low Reynolds
Number and low Froude Number flow. With very
small ratios of inertial to viscous forces (4.55) and
inertial to gravitational forces (4.102), the inertial
forces can be ignored.
There is, however, a constraint imposed on the
strength of model materials that must deform
under their own weight to simulate the deformation of very large masses of rock. Considering the model ratio for gravitational forces we
have:
(4.110)
Here the ratio of gravitational acceleration in the
model and prototype is approximately one. Given
this ratio of gravitational forces, the corresponding ratio of stresses and strengths is:
(4.111)
mg
pg
ϭ
m L m
p L p
Ϸ
L m
L p
F mg
F pg
ϭ
m g m L 3 m
p g p L 3
p
Ϸ
m L 3 m
p L 3
p
L m
L p
ϭ
T m
T p
2
, so T m ϭ T p √
L m
L p
4.4 SCALED LABORATORY MODELS
149
from 1.9 ϫ 10
Ϫ6 to 9.7 ϫ 10
Ϫ20 , and the Froude
Number from 6.1 ϫ 10
Ϫ18 to 3.1 ϫ 10
Ϫ24 . These very
small ratios justify ignoring the effects of inertial
forces when scaling models of salt dome development. A laboratory model of salt dome development is illustrated in Fig. 4.13a (Ramberg, 1967,
p. 123).
Many experimental observations of viscous
flow in conduits have demonstrated that the
Reynolds Number must be greater than approximately 1 ϫ 10
3 for the flow regime to become
turbulent (Bird et al., 1960, pp. 183–8). For
Reynolds Numbers less than this value, flow in
conduits is laminar (Fig. 4.8b). Because the range
of Reynolds Numbers estimated for the rising salt
dome (and many other tectonic processes involving flow) is many orders of magnitude less than
this transition value, the flow regime is laminar.
The precise scaling of inertial forces between the
model and prototype is not necessary as long as
the model is well within the laminar flow regime.
The condition for strict dynamic similarity in
terms of inertial forces between model and prototype can be relaxed without compromising the
usefulness of the experimental results.
4.4.3 Gravitational forces: a problem for
model similarity?
It was noted in the preceding section that all corresponding lengths, masses, and times in the
model and the prototype must adhere to the given
model ratios to maintain strict similarity. In
general these are thought of as independent
ratios, to be chosen at the convenience of the
experimenter, or as constrained by the available
materials. However, models of tectonic processes
often are carried out on a lab bench where the
acceleration of gravity is approximately 9.8 m s
Ϫ2 ,
and this value is not significantly different for the
prototype. Thus, the model ratio for accelerations
is approximately one. Examining the model ratio
for accelerations we find:
(4.108)
Thus, the model length and time ratios are not
independent. Solving for the model time:
a r ϭ L r T Ϫ2
r ϭ 1, so L r ϭ T 2
r
(4.109)
Using a reasonable range of model ratios for
length (10
Ϫ6 Ͻ L r Ͻ 1) and for the time scales for
tectonic processes (10
2 s Ͻ T p Ͻ 10
15 s), we calculate
a range of model time scales from T m ϭ 10
Ϫ1 s to T m
ϭ 10
15 s. At the lower end of this range the experimenter would have few problems, but clearly the
upper end is unattainable. For example, a
common model length scale is ten centimeters
(0.1 m), corresponding to a prototype length scale
of one kilometer (1000 m). A common time scale
for the duration of tectonic processes is one
million years (10
6 years). Given these values, the
model time scale is ten thousand years (10
4 years),
not a practical duration for experiments designed
to be observed by humans!
The constraint on the model time scale
imposed by similar accelerations of gravity acting
on the prototype and the model is not fatal to all
model experiments in tectonics because the inertial forces associated with accelerations in the
prototype may be insignificant compared to other
forces. Recall, for example, that flow of many geological materials is in the realm of low Reynolds
Number and low Froude Number flow. With very
small ratios of inertial to viscous forces (4.55) and
inertial to gravitational forces (4.102), the inertial
forces can be ignored.
There is, however, a constraint imposed on the
strength of model materials that must deform
under their own weight to simulate the deformation of very large masses of rock. Considering the model ratio for gravitational forces we
have:
(4.110)
Here the ratio of gravitational acceleration in the
model and prototype is approximately one. Given
this ratio of gravitational forces, the corresponding ratio of stresses and strengths is:
(4.111)
mg
pg
ϭ
m L m
p L p
Ϸ
L m
L p
F mg
F pg
ϭ
m g m L 3 m
p g p L 3
p
Ϸ
m L 3 m
p L 3
p
L m
L p
ϭ
T m
T p
2
, so T m ϭ T p √
L m
L p
4.4 SCALED LABORATORY MODELS
149