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

The Delft Sand, Clay & Rock Cutting Model.
the blade is then limited to a mixed boundary conditions problem. The potential theory can be used to solve this
problem. For the determination of the water vacuum pressures it is necessary to have a proper formulation of the
boundary condition in the shear zone. Miedema (1985A) derived the basic equation for this boundary condition.
In later publications a more extensive derivation is published. If it is assumed that no deformations take place
outside the deformation zone, then:
2 2
p p
2 2
x
y
0
(12-15)
Making the boundary condition in the shear plane dimensionless is similar to that of the breach equation of Meijer
and Van Os (1976). In the breach problem the length dimensions are normalized by dividing them by the breach
height, while in the cutting of sand they are normalized by dividing them by the cut layer thickness. Equation
(12-15) is the same as the equation without a wedge. In the shear plane A-B the following equation is valid:
ki
p p w g vc hi
sin( )
n
with: n'
k n' n' k h
max 1 2
max i
(12-16)
This equation is made dimensionless with:
p
'
p
n'
n g v h /k
w c i max
(12-17)
The accent indicates that a certain variable or partial derivative is dimensionless. The next dimensionless equation
is now valid as a boundary condition in the deformation zone:
max 1 2
'
ki
p p
sin( )
k n n
'
(12-18)
The storage equation also has to be made dimensionless, which results in the next equation:
'
2 2
x
p
p
y
2 2
'
0
(12-19)
Because this equation equals zero, it is similar to equation (12-15). The water under-pressures distribution in the
sand package can now be determined using the storage equation and the boundary conditions. Because the
calculation of the water under-pressures is dimensionless the next transformation has to be performed to determine
the real water under-pressures. The real water under-pressures can be determined by integrating the derivative of
the water under-pressures in the direction of a flow line, along a flow line, so:
P
calc
s
'
'
p
ds'
(12-20)
s
This is illustrated in Figure 12-7 and Figure 12-8. Using equation (12-20) this is written as:
real
'
p w g vc hi
p s
s (12-21)
' k
s
s max s hi
P ds ds' with: s'
This gives the next relation between the real emerging water under pressures and the calculated water under
pressures:
P
real
w g vc hi
Pcalc
(12-22)
k
max
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