Geoscience Reference
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shear plane, following (9.17), under the maximum strength and above the critical
strength may occur. There is no preference for any specific orientation in the
corresponding range. The fact that in numerical simulations a certain shear band is
found, is purely a result of calculation coincidence (numerical localisation). For a
dilatation angle
between 0 and
, the range of R cr appears to be: sin
< R cr <
tan
. Hence, a conservative approach would be to adopt a safe friction angle
safe
= atan(sin
.
Localisation depends on the geometry and the deformation conditions. In
practice, a physical trigger (e.g. a local weakness) may cause a sudden collapse
(bifurcation). Hence, even with certainty about the essential parameters, the
residual strength is usually uncertain. Non-associative constitutive models should
be handled with care. There is need for a new comprehensive constitutive model
that better incorporates stress rotations and induced anisotropy, which arise with
localisation in dilatant materials.
). For
= 34
=
it yields
safe = 29.2
=
Numerical singularities
The calculation result of a numerical model may be very accurate but not
realistic. This is the case at singularity points, i.e. at locations where the
discretisation does not cover the reality properly. As an example, a point well in a
standard porous flow field is considered (after Kono). The real solution of the
induced potential head
for a well with a constant Q discharge is logarithmic.
Hence the potential head
w and the head at a small
distance d from the well and the corresponding discharge become
between the well head
r
ln(
)
d
(
)
(9.18)
d
d
w
r
ln(
w
)
d
(
)
d
w
with
Q
2
kHr
2
kH
(9.19)
real
r
ln(
r
/
d
)
w
Here, kH is the transmissivity (permeability times layer height), r is the radial
coordinate and r w the actual well diameter. If in a numerical finite difference
approach a linear head distribution is used, it will lead to a value
N in the well
point of the square grid (grid size d ) and the corresponding discharge becomes (4
neighbouring grid points contribute equally)
Q
4
kH
(
)
(9.20)
FD
d
N
In order to achieve the same discharge, i.e. Q real = Q FD , the following must hold
2
r w
(
)(
ln
)
(9.21)
l
w
d
N
d
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