The Delft Sand, Clay & Rock Cutting Model, 2019a

The Delft Sand, Clay & Rock Cutting Model.
6.8. Analytical/Numerical Water Pore Pressure Calculations.
As is shown in Figure 6-9, the water can flow from 4 directions to the shear zone where the dilatancy takes place.
Two of those directions go through the sand which has not yet been deformed and thus have a permeability of ki,
while the other two directions go through the deformed sand and thus have a permeability of kmax. Figure 6-12
shows that the flow lines in 3 of the 4 directions have a more or less circular shape, while the flow lines coming
from above the blade have the character of a straight line. If a point on the shear zone is considered, then the water
will flow to that point along the 4 flow lines as mentioned above. Along each flow line, the water will encounter
a certain resistance. One can reason that this resistance is proportional to the length of the flow line and reversibly
proportional to the permeability of the sand. Figure 6-20 shows a point on the shear zone and it shows the 4 flow
lines. The length of the flow lines can be determined with the equations (6-36), (6-37), (6-38) and (6-39). The
variable Lmax in these equations is the length of the shear zone, which is equal to hi/sin(), while the variable L
starts at the free surface with a value zero and ends at the blade tip with a value Lmax.
According to the law of Darcy, the specific flow q is related to the pressure difference Δp according to:
p
q k i k (6-34)
g s
w
The total specific flow coming through the 4 flow lines equals the total flow caused by the dilatation, so:
c
q v sin
p p p p
k k k k
max max i i
w gs1 w gs2 w gs3 w gs4
(6-35)
Figure 6-20: The flow lines used in the analytical method.
For the lengths of the 4 flow lines, where s2 and s3 have a correction factor of 0.8 based on calibration with the
experiments:
hb
s1 Lmax L
1
2
sin
With : 1
2
(6-36)
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