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Effect of high hyperbaric pressure on rock cutting process

M. Alvarez Grima a,, S.A. Miedema b, R.G. van de Ketterij a, N.B. Yenigl a, C. van Rhee b

a MTI Holland B.V., Smitweg 6, 2963 AW Kinderdijk, The Netherlandsb Delft University of Technology, Section of Dredging Engineering, Mekelweg 2, 2628 CD Delft, The Netherlands

a b s t r a c ta r t i c l e i n f o

Article history:

Received 2 July 2014

Received in revised form 5 June 2015

Accepted 20 June 2015Available online 27 June 2015

Keywords:

Rock cutting

Hyperbaric pressure

Fracture propagation

Time dependence

When cuttingrockhyperbaric,two cases mayoccur. Therock mayencounter dilation or compaction dueto shear.

Dilation results in pore under pressures, while compaction results in pore overpressures. Dilation will increase

the cuttingforcesconsiderably,while compaction may decrease the cuttingforces. In both cases the cutting pro-

cess is supposed to be cataclastic.To dimension cutting tools fordeep seamining, theworst case should be inves-

tigated, which is the dilatant case. To understand the cutting mechanism experiments are carried out in a

pressure tank, simulating the hyperbaric conditions. Hyperbaric cutting appears to be very different from atmo-

spheric cutting due to the pore water pressures. The experiments have revealed that the cutting mechanism

changes from a chiptype mechanism under atmospheric conditions to a cataclastic (crushed) type under hyper-

baric conditions, resulting in higher cutting forces.

An analyticalmodel is presented to estimate the cutting forces under high hyperbaric conditions. The results ob-

tained with the analytical model agree rather well with the experimental data.

2015 Elsevier B.V. All rights reserved.

1. Introduction

When cutting rock hyperbaric, two cases may occur. The rock mayencounter dilation or compaction due to shear. Dilation results in pore

under pressure, while compaction results in pore over pressures. Dila-

tion will increase the cutting forces considerably, while compaction

maydecrease thecutting forces. In both cases the cuttingprocess is sup-

posed to be cataclastic. To dimension cutting tools for deep sea mining,

theworst case should be investigated, which is the dilatant case. To un-

derstand the cutting mechanism experiments are carried out in a pres-

sure tank, simulating the hyperbaric conditions.

Hyperbaric cutting appears to be very different from atmospheric

cutting due to the pore pressures. The experiments have revealed that

the cutting mechanism changes from a chip type mechanism under at-

mospheric conditions, to a cataclastic (crushed) type under hyperbaric

conditions.

In front of the chisel the rock is crushed and shearing of the crushed

rock results in dilation, resulting in pore under pressures. These under

pressures increase the effective stress and thus also the frictional

shear stress. These under pressures depend on the magnitude of the di-

lation or the magnitude of the dilation and the permeability of the

crushed rock and are limited by the water vapor pressure. Because of

the very low permeability of the crushed rock, cavitation is expected

to be in effect of low to very low cutting velocities. The experiments

were carried out at different velocities in order to quantify this effect.

From these experiments it was found that cavitation occurred already

at low velocities and thatthe forcescan be predictedwell withthe mod-ied sand cutting equations. Further it appeared that the cutting mech-

anism has changed in more than one way. Not only the mechanism

become cataclastic, but also the 3D chip pattern with a sideways

shape has become more a box cut, just followingthe shape of the chisel.

The experiments have proven that in the type of rock chosen, strong hy-

perbaric effects occur, which in terms of cuttingforces, can be described

for the cavitating case, with the theory given in the paper.

The increase in cutting forces can be explained by analyzing the

combined effect of cutting speed and hyperbaric pressure during the

rock cutting process.

According to previous studies reported in the literature, it appears

that the understanding of the cutting mechanism at high hyperbaric

pressures is rather limited. Most of the studies conducted are mainly

concerned with drill bits; cutting a very thin layer of rock (b1 mm).

An understanding of the mechanism of rock cutting at large water

depth is a requisite for proper design of rock cutting tools cutting a

layer of centimeters.

Several rock cutting theories have been published in the literature

such as (Evans, 1961, 1962, 1965; Goktan, 1995, 1997; Nishimatsu,

1972). These theories concern rock cutting under dry and/or atmo-

spheric conditions. They cannot be used to estimate the rock cutting

forces under hyperbaric conditions because the pore water pressure

and the effect of the cutting speed are not taken into account.

Kaitkay and Lei (2005)conducted lab experiments on the inuence

of hydrostatic pressure on rock cutting with drill bits on Carthage

Engineering Geology 196 (2015) 2436

Corresponding author.

E-mail address:[email protected](M. Alvarez Grima).

http://dx.doi.org/10.1016/j.enggeo.2015.06.016

0013-7952/ 2015 Elsevier B.V. All rights reserved.

Contents lists available at ScienceDirect

Engineering Geology

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marble. They foundthat the increase of conning pressure transformed

the cutting process from a brittle to a ductilebrittle failure mode. Lon-

ger chips were formed and the cutting forces increased with high hy-

drostatic pressure.

Van Kesteren (1995)examined the effect of pore water pressure in

rock by distinguishing two limiting conditions: drained and undrained.

In the drained condition, porewaterow dueto pore water pressuregra-

dient is possible without affecting the porous system itself. In the un-

drained condition pore water is not allowed to

ow through the poresand the pore water pressure will affect the stress state in the rock fabric.

The drained condition, undrained condition, and the transition zone are

determined by the Peclet number. It relates the cutting speed, cutting

depth, and the diffusivity coefcient, which in turns depends on the per-

meability, compressibility of the rock skeleton, compressibility of water,

compressibility of rock grains, and porosity. Additionally van Kesteren

(1995)explained that the pore pressures are of inuence on crack initi-

ation andpropagation. He arguesthat at thestrain rates occurringduring

dredging (order of 103 s1), the pore water is able to ow towards the

crack tips (drained condition), the transition towards undrained condi-

tion occurs at strain rate larger than 105 s1. This implies that crack ini-

tiation is not impeded by the resistance of the water pressure. At high

hyperbaric conditions, however,a different situationmight occur regard-

ing the speed of the process, strain rate, and ambient pressure. This in

fact, constitutes the main objective of this study. The magnitude of this

hyperbaric effect will probably depend on the ratio of the hydrostatic

pressure and the unconned compressive strength of the rock.

Detournay and Atkinson (2000)investigated the inuence of pore

pressure on the drilling response in low-permeability shear dilatant

rocks. Different pore pressure regimes were identied, which were con-

trolled by a dimensionless number. They found that in the high speed

regime, the rock in the shear zone is undrained and pressure drops in-

duced by shear-induced dilatancy, which leads to cavitation.

Huang et al. (1999)investigated the effect of cutting depth numeri-

cally by using DEM. They found that theductile failure mode in rockcut-

ting is characterized by a steadyow of a crushed material ahead of the

cutter. The brittle failure is characterized by the coalescence of micro

cracks and possibly formation of chips. They concluded that the transi-

tion between ductile and brittle failure mode in rock cutting dependson the depth of the cut.

Al-Shayea et al. (2000)investigated the effect of conning pressure

and temperature on mixed-mode (III) fracture toughness of a lime-

stone. Tests were conducted under an effective conning pressure of

28 MPa, and a temperature of up to 116 C. They found a substantial in-

crease in fracture toughness under conning pressure. The pure mode-I

fracture toughnessKICincreased by a factor of about 3.7 under a con-

ned pressure of 28 MPa compared to that under atmospheric pressure.

The pure mode-II fracture toughness KIICincreased bya factorof 2.4 for a

con

ning pressure of 28 MPa. The effect of temperature was only 25%more forKICat 116 C.

Sang et al. (2003) investigated the strain-rate dependency of the dy-

namic tensile strength of rock. The fracture processes were analyzed at

various strain-rates. They found that higher strain rates generated a

large number of micro cracks, which interfered with the formation of

the fracture plane. The observed increase in dynamic strength at high

strain rate was caused by crack arrests due to the generation of a large

number of micro cracks.

Funatsu et al. (2004)studied the combined effect of increasing tem-

perature and conning pressure on the fracture toughness of clay bear-

ing rocks. They found that the fracture toughness of sandstone

increased by approximately 470% at 9 MPa connement over its value

at atmospheric pressure.

Schmidt and Huddle (1997) investigated the effect of conningpres-

sure on fracture toughness of Indiana limestone. They found that KICin-

creased from 0.93 MN m3/2 at atmospheric pressure to 4.2 MN m3/2

at a conning pressure of 62 MPa.

Zijsling (1987)found that due to low permeability, cavitation will

occur in the crushed zone, even with very small layer thicknesses. The

result, combined with narrowing to a box cut, implies that the full

width of the cut has to be covered with chisels/pick points. So different

rows of chisels have to be staggered in contrary with atmospheric cut-

ting where there is overlap due to the 3D effect.

From the studies presented in the literature, it can be concluded that

high hyperbaric pressure affects the rock behavior and particularly the

fracture toughness, crack initiation and propagation. This of course

will dependon therock material properties such as porosity, permeabil-

ity, and elasticity (plasticity). It is expected that cutting rock at large

water depth will have a strong effect on the magnitude of the cuttingforces and energy required.

2. Time dependency of fracture initiation and propagation

This section discusses the impact of time and speed on the cutting process. The cutting process is divided into three different subprocesses (i)

forming of a crushed zone, (ii) fracturing by shear failure, and (iii) fracturing by tensile failure. Each of these sub-processes is inuenced by the speed

of the cutting process. A dimensional analysis has been carried out to estimate the impact of the different factors on the cutting process and estab-

lishing in this way the basis for the selection of the parameters that will be investigated in the laboratory (see Table 1). The dimensional analysis was

carried out by using the Buckingham theorem.

Fig. 1 schematizes the phenomena involved in the chip forming process during rock cutting for shallow (Verhoef, 1997), and for deep water con-

ditions. It is assumed thatunder high hyperbaric pressure (about 20 MPa) the failure mechanism will be predominantly shear, in contrast to a typical

shallow cutting process where thefailure mechanismwill be predominantly tensile. Theextension of the crushed zone in front and below the cutting

tooth is expected to be larger for high hyperbaric pressure.(i) Forming of a crushed zone: In this part of the process the cutting tool penetrates the rock and crushes it. The rock compressive strength will be

exceeded. Grains will be pulled outof the joints, pulverized, and pushed into the pores of the material further away from thecutting tool. The

water in the pores will be pushed further away into the material, resulting in high pore pressure. Water ow in the pores is governed by

Darcy's law. If the cutting speed is increased, the uid velocity in thepores will have to increase too. This can only occur if pressure difference

increases. The pore pressure near the tool tip has to decrease considering a pore pressure away from the tool tip approximately equal to the

hydrostatic pressure. This has an inevitable inuence on the required cutting force, which will increase linearly with pore pressure difference

near the tip.

Table 1lists the parameters inuencing the cutting process. When the hydrostatic pressure (Phyd), the volume of crushed zone (Vcr), and vis-

cosity () are chosen as running variables the following relation emerges by using the derived dimensionless numbers:

Encr

Phyd

Vcry p

Phyd;k3

V2

cr

;L3

Vcr;

VPhyd

V1=3

cr

;Phydt

;D3grain

Vcr;

EgrainPhyd

;EmatrixPhyd

;PhydV2=3cr

2 ;

11Phyd

;12Phyd

;13Phyd

;L3toolVcr" #: 1

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It becomes clear from the dimensionless numbertPhydt that time has an equivalent inuence on the crushing process as the hydrostaticpressure. Also the dimensionless number Encr EncrPhydVcr shows that the energy required for the crushing process also increases when eitherthe hydrostatic pressure,or the volume of the crushed zone increases. Thedimensionless numberLtool L

3tool

Vcrshows that an increase of oneof

the important dimensions of the cutting tool such as the width, or the contact area, has a signicant inuence on the volume of the crushed

zone and consequently on the energy involved in the crushing process.

(ii) Shear failure: Perpendicular to the crushed zone, approximately perpendicular to the upper surface of the tooth, a shear failure, which is in-

duced by shear stresses caused by the tool and thechanges in thecrushed zone will occur (see Fig. 1). This fracturing in sliding mode, refereed

as shear fracture (van Kesteren, 1995) occurs when the mode II stressintensity factor near the tip of the micro fracture in thematerial reaches

a critical value.

Crushed zone

Shear failure

Tensile FailureProduced chip

Shallow Water

Crushed zone

Shear failure

Tensile Failure

Produced chip

Deep Water

Fig. 1.Phenomenological description of the chip forming process.

Table 1

Parameters inuencing the cutting process.

Symbol Parameter Units

t Time s

Dgrain Diameter of the grains m

Phyd Hydrostatic pressure N/m2

Egrain Youngs' modulus of the grains N/m2

Ematrix Youngs' modulus of the material N/m2

Density water N s2 m4

11 Stress in the material N/m2



Vcr Volume crushed zone m3

Encr Energy crushed zone Nm

Ltool Length dimension of the tool M

k permeability m2

p Pressure difference N/m2

L Distance over which the pressure difference acts m

Viscosity N s/m2

V Speed of the water through the material m/s

Enf Energy fracture zone Nm

Lf Fracture length m

KIC Fracture toughness Nm3/2

pp Fluid pressure in fracture Nm2

Vx Fluid speed in fracture ms1

wf Fracture width m

vl Leak-off coef

cient Nm

1/2

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(iii) Tensile failure: Tensile stress increase as the horizontaldistance from the tooltip increases. At a certain moment, the ratio between the mode I

and mode II stress intensity factor near thetip of the propagating shear fracture will reach a critical level. At this pointthe shear fracture will

bifurcate into a tensile fracture. As soon as the fracture starts propagating, the pressure prole in the fracture inuences the fracture propa-

gation to a large extent. Thepressureprole in thefracture is determined by three factors: mass balance in the fracture, viscousuidow, and

elastic (or plastic) deformation of the fracture (Weijers, 1995). In view of the high pressures, the pressure dependency of the

uid densityneeds to be considered.

Fluid leak-off from the surrounding rock into the fracture (or from the fracture into the surrounding rock) typically follows the equation:

vlconstant Klffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffitti x

p 2where Kl is the leak-off coefcient, which is linearly dependent on the permeability of the material. The term (t ti) indicates the timeelapsed since

thefracture tip passed a certain part of the fracture wall. If inertia, compressibility and gravity are neglected the NavierStokes equation foruidow

in x-direction simplies to a linear relation between uid velocity and pressure gradient:

vx w2

12

pxx

3

wherewis the local fracture width andis the uid viscosity.

Finally the pressure will depend on the entrance width, fracture length, Young's modulus of the material, and over pressure distribution in the

fracture.

Once a dimensional analysis using the hydrostatic pressure (Phyd), fracture length (Lf) and uid viscosity () as running variables (see Table 1) is

performed, then the following dimensionless groups are obtained.

En f

PhydL3fy Phydt

;

KIC

Phydffiffiffiffiffi

Lfp ; PhydL2f

2 ;

ppPhyd

; vxPhydLf

;wfLf

;Lf

ffiffiffiffiffiffiffiffiffiffivl

pffiffiffiffiffiffiffiffiffiPhyd

p" #

: 4

Table 3

Rock properties at atmospheric conditions.

Test no. UCS MPa E (GPa) ()

BTS(MPa)

k liquid(m/s)

n(%)

s(Mg/m3)

1 7.92 5.95 0.31 0.88 3.1E06 37.86 2.78

2 7.92 5.95 0.31 0.88 3.1E06 37.86 2.78

3 7.92 5.95 0.31 0.88 3.1E06 37.86 2.78

4 8.75 7.53 0.25 1.09 8.5E07 34.64 2.76

5 8.75 7.53 0.25 1.09 8.5E07 34.64 2.76

6 8.75 7.53 0.25 1.09 8.5E07 34.64 2.76

7 8.75 7.53 0.25 1.09 8.5E07 34.64 2.76

8 9.29 5.89 0.27 1.15 1.4E07 33.17 2.76

9 10.62 8.32 0.23 1.05 2.8E07 31.66 2.78

10 10.64 9.01 0.27 1.13 2.2E08 33.92 2.79

11 8.86 8.20 0.31 0.86 1.5E07 35.12 2.77

12 8.86 8.20 0.31 0.86 1.5E07 35.12 2.77

13 8.86 8.20 0.31 0.86 1.5E07 35.12 2.77

14 10.54 9.98 0.33 x 3.4E09 35.89 2.80

15 10.54 9.98 0.33 x x x x

Table 2

Experimental program.

Test no. Chisel width

(mm)

Wear angle

(o)

Hyperbaric pressure

(MPa)

Cutting velocity

(m/s)

Cutting depth

(mm)

Cutting angle

(o)

1 21 10 0 0.2 20 68

2 21 10 18 0.2 20 68

3 21 10 1.5 0.2 20 68

4 21 10 0 2.0 20 68

5 21 10 1.5 2.0 20 68

6 21 10 3 2.0 20 687 21 10 6 2.0 20 68

8 21 10 18 2.0 20 68

9 21 10 18 0.6 20 68

10 21 10 18 0.01 20 68

11 21 10 0 0.01 20 68

12 21 10 3 0.2 20 68

13 21 10 6 0.2 20 68

14 21 10 18 1.2 20 68

15 21 10 6 1.2 20 68


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The dimensionless number (tPhydt

) in Eq.(4)is the most interesting parameter for the purpose of this study. It suggests that time impact is

equivalent to the impact of thehydrostatic pressure, andinversely related to theuid viscosity. In other words, theinuence of timeon the fracturing

process is equivalent to the inuence of pressure on the fracturing process.

When fracture length and hydrostatic pressure are chosen as running variables, another interesting dimensionless numberarises:vx tvxLf . Thisdimensionless number shows that time is equally important as uid speed in the material, but more importantly, inversely proportional to fracture

length. As fracture length is proportional to the thickness of the layer cut during the cutting process, time is also inversely proportional to the layer

thickness.

Table 4

Results of executed tests.

Test

no.

Chisel

width

(mm)

Cutting

depth

(mm)

Hyperb.

pressure

(MPa)

Cutting

velocity

(m/s)

Actual cutting

velocity (m/s)

Average cutting

force

(kN)

Max cutting

force

(kN)

Min cutting

force

(kN)

Cuttin

cross

(mm2)2)

Specic

energy

(MJ/m3)

1 21 20 0 0.20 0.188 7.22 9.8 5.1 811 8.90

2 21 20 18 0.20 0.178 9.25 13.1 5.2 729 12.69

3 21 20 1.5 0.20 0.200 10.42 14.4 5.6 795 13.11

4 21 20 0 2.00 1.826 8.09 12.5 3.8 820 9.87

5 21 20 1.5 2.00 1.717 11.17 12.6 8.1 577 19.36

6 21 20 3 2.00 1.740 12.23 14.9 10.6 543 22.52

7 21 20 6 2.00 1.702 13.19 18.3 9.8 655 20.14

8 21 20 18 2.00 1.577 20.70 28.5 15.8 562 36.83

9 21 20 18 0.60 0.618 22.72 12.3 13.1 581 39.13

10 21 20 18 0.01 0.010 4.94 7.5 2.4 831 5.94

11 21 20 0 0.01 0.017 4.72 6.5 2.5 1070 4.41

12 2 1 20 3 0.20 0.202 11.36 16.2 7.5 675 16.83

13 2 1 20 6 0.20 0.207 11.29 15.3 7.4 833 13.55

14 21 20 18 1.20 1.238 12.74 17.4 8.2 541 23.54

15 2 1 20 6 1.20 1.188 10.90 12.6 7.9 596 18.28

Fig. 2.a) Experimental test set-up, hyperbaric tank, Ifremer, Brest. b) Cutting rig and hyperbaric tank Ifremer (Brest, France). c) Position of the sensors in the cutting rig.


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3. Experimental investigation

This section presents the laboratory cutting experiments performed

on Savonnieres limestone. The laboratory work focuses on investigating

the effect of the hyperbaric pressure on the magnitude of the cutting

forces and consequently the required power. The parameters that

were varied in the rst experimental series are the cutting speed and

the ambient pressure. The cutting depth, tooth geometry and cutting

angle were kept constant (seeTable 2).

Table 3lists the rock properties determined at atmospheric condi-

tions on the Savonnieres limestone samples used in the lab experi-

ments. The unconned compressive strengthUCS, Young's modulusE,

Poisson's ratio, Brazilian tensile strength, BTS, permeability k, porosity

n, and solid densitysare measured rock properties from the tests per-

formed by Delft University of Technology, Faculty of Civil and Geo Engi-

neering. The tests were done according to the ASTM (American Society

for Testing and Materials).

3.1. Rock cutting experiments

The rock cutting tests were executed in a hyperbaric tank having a

height of 2.2 m and a diameter of 1 m. The tests were done by Deltares

at Ifremer, Brest in France (Fig. 2a) (van Kesteren, 2009). The pressure

rating is 1000 bar. The experiments were conducted at scale 1:1. This al-

lows investigating the effect of the hyperbaric pressure on the cutting

process regardless of the scaling effects. Therefore, the formation of a

realistic crushed zoneas wellas a proper crack initiation and propagation

could be achieved without masking the effect of the ambient pressure.

The tooth used has a width of 21 mm, a clearance angle of 10 and a

cutting angle of 68. The cutting forces were measured with strain

gauges with maximum range of 50 kN in tension. The sensor was tailor

made by Deltares to accommodate the requirements of the Ifremer

tank. There is one horizontal sensor and two vertical sensors, which

measures the torque as well. The horizontal sensor is located behind

thechisel and thevertical sensors are located at thetwo sides of thecut-

ting rig. This enables the measurement of the position of the cutting

force onthe chisel(Fig.2b, Fig.2c).For each test thecutting velocity, dis-

placement, cutting forces, and ambient pressure were recorded. After

completion of each test, the cutting debris were collected and analyzed

in order to relate the effect of the cutting speed and pressure with the

size of the cutting debris and tooth production.

The particle sizedistribution of coarse samples (63 m31mm)was

determined by sieve analysis according to NEN 5753. Particles largerthan 31 mm were weighed individually whereas the distribution of

ne particles, less than 63m were determined on a sub-sample of

3 gram dry soil and measured with Malvern Mastersizer 2000. A laser

device was used to scan the linear cutting grooves and to measure the

groove prole cross section. This allows getting more insight into the

combined effect of the cutting speed and pressure on the side break-

out angle.

3.2. Results of experiments

Table 4lists the results of the cutting experiments average hori-

zontal cutting forces, minimum and maximum cutting forces, cut cross

section as determined with a laser device, and the specic energy. The

specic cutting energy SE (MJ/m3) is dened as the amount of energy

required per volume of excavated rock. Table 5lists the particle sizes

of the cutting debris in terms of gravel, sand and nes fraction as

percentage.

3.2.1. Effect of hyperbaric pressure and cutting speed on cutting forces

Fig. 3and Table 4show that in general the cutting forces increase as

the hyperbaric pressure increases. Under high hyperbaric pressure the

cutting process changes into an apparent ductile (cataclastic) mode

and failure of the rock will be predominately shear. In brittle cutting

process, the chip formation is dominated by tension cracks as common-

ly encountered in shallow rock cutting. However, when ductile behavior

is prevailing, the crack formation is mostly along shear planes and more

force will be required to create a chip. Besides, high hyperbaric pressure

0

5

10

15

20

25

0 5 10 15 20

Cutting

forces,

Fh

(kN)

Hyperbaric pressure (MPa)

Fig. 3.Cutting forces versus hyperbaric pressure.

Table 5

Particle size distribution of cutting debris.

Test no. Chisel width (mm) Cutting depth (mm) Hyperbaric pressure (bar) Cutting velocity (m/s) Gravel

N2 mm (%)

Sand

b2 mm

N63m (%)

Fines

b63m

N20m

(%)

1 21 20 0 0.20 83.55 16.43 0.02

2 21 20 180 0.20 76.65 23.35 0.01

3 21 20 15 0.20 76.68 23.32 0.00

4 21 20 0 2.00 80.30 19.69 0.015 21 20 15 2.00 54.55 45.45 0.00

6 21 20 30 2.00 54.76 45.24 0.00

7 21 20 60 2.00 51.12 48.86 0.01

8 21 20 180 2.00 57.17 42.78 0.05

9 21 20 180 0.60 68.98 30.89 0.13

10 21 20 180 0.01 88.84 11.14 0.01

11 21 20 0 0.01 85.05 14.90 0.05

12 21 20 30 0.20 78.77 21.23 0.00

13 21 20 60 0.20 66.32 33.67 0.01

14 21 20 180 1.20 66.86 33.14 0.00

15 21 20 60 1.20 73.07 26.93 0.00


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will lead to high friction forces between the material cut and the top

surface of the cutting tool (i.e., apparent ow mechanism). The crushed

zone below and in front of the tooth will increase.

Thescatter in the magnitude of the cutting forcesat 18 MPapressure

inFig. 3is attributed to the effect of the cutting speed in combination

with the drained and/or undrained behavior of the material in relation

to its permeability and porosity. It was observed that at lower cutting

speeds the effect of the pressure is counteracted. The experimental re-

sults indicate that for low cutting speeds (v b1 m/s) despite the ductile

behavior of the rock, much less changes in the magnitude of cutting

forces are observed. The results showed that the increase in cutting

forces varies from 4.7 kN at atmospheric conditions up to 22.7 kN at

18 MPa hyperbaric pressures for a cutting speed equal to 2 m/s. The in-

crease in cutting forces is approximatelyve times higherthan the mag-

nitude of the cutting forces at atmospheric conditions.Fig. 4shows an

overview of complete cut, where the differences in side-break out

angle between atmospheric and hyperbaric pressures can be seen for

different cutting speeds.

As an example two selected tests are presented showing the cutting

forces versus timefor both atmospheric conditions and hyperbaric con-ditions (Figs. 5 and 6). They correspond with test 1 and test 8 as listed in

Table 4. Clearly the gures show the combined effect of pressure and

cutting speed on the magnitude of the cutting forces.

3.2.2. Effect of hyperbaric pressure and speed on cutting debris size

Fig. 7shows the average cross sectional area, Acrof a groove. The

cross sectional area is normalized with the applied cutting area of the

tooth, which equals to the product of the tooth width and the cutting

depth. Ascan be seenin Fig. 7a (speed = 0.2 m/s, pressure equals to at-

mospheric) a large side-break out angle occurred indicating a short

shear path reaching a bifurcation point from shear to tension earlier

than what is shown inFig. 7d (speed = 2 m/s, hyperbaric pressure =

18 MPa).

Fig. 8shows the ratio between the average cutting cross sectional

area and the cutting area of the tooth versus the cutting velocity. Fig. 9

shows the ratio between the average cutting cross sectional area versus

pressure times speed to illustrate the combined effect on tooth produc-

tion. As can be seen inFigs. 8 and 9, at low cutting speed the area is

about 2.5 times the applied cutting area of the tooth, which is caused

by the small crushed zone and large chips that are formed (Figs. 10a

and10b). This still holds for the tests conducted at atmospheric condi-

tion and high cutting speed (Fig. 10c), where the area is about 2 times

the applied cutting area of the tooth. At high cutting velocity, however,

when the hyperbaric pressure increases (i.e., p = 18 MPa) the produc-tion ratio drops to 1.3. In this situation ratherthanchips,lumps with the

appearance of clay are formed (Fig. 10d). This is a clear evidence of a

more ductile process with increasing cutting speed and pressure.

Fig. 5.Cutting forces versus time (Test 1, atmospheric conditions). Black solid line horizontal cutting force, red solid line vertical cutting force, blue solid line cutting speed.

(a) (b) (c) (d)

Fig. 4. Overview of complete cut a) at atmospheric pressure with low cutting velocity (0.2 m/s) b) at high hyperbaric pressure (180 bar) with low cutting velocity (0.2 m/s), c) at atmo-

spheric pressure with high cutting velocity (2 m/s) and d) at high hyperbaric pressure (180 bar) with high cutting velocity (2 m/s).


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In general the grain size of the debris becomes neras the hyperbar-

ic pressure and cutting speed increases. The gravel fraction decreases

and the sand, andne fractions increase with increasing cutting speed

(Table 5).

4. Analytical models

4.1. Analytical model for atmospheric conditions

The model for rock cutting under atmospheric conditions presented

hereis based on the ow type of cutting mechanism. Although in gener-

al rock will encounter a more brittle failure mechanism and the ow

type considered represents the shear failure mechanism, the ow type

mechanismformsthe basis for all cutting processes. In the case of brittle

failure an equivalent shear strengthcis determined, which is based on

the tensile strength of the rock.

Fig. 11 illustrates the forceson the layer of rock cut. Theforces acting

on this layer are:

A normal force acting on theshear surface N1 resulting from the grain

stresses.

A shear forceS1as a result of internal ctionN1 tan().

A shear force Cas a result of the shear strength (cohesion) c. This force

can be calculated by multiplying the cohesive shear strength cwith

the area of the shear plane.

A force normal to the toothN2resulting from the grain stresses.

A shear force S2 as a result of the soil/steel friction N2 tan() or exter-

nal friction.

(a) (b)

(c) (d)

Fig. 7. Composition of laser scan cutgeometry a) at atmospheric pressurewith lowcutting velocity (0.2m/s) b) at highhyperbaric pressure(180 bar)with lowcutting velocity(0.2 m/s),

c) at atmospheric pressure with high cutting velocity (2 m/s) and d) at high hyperbaric pressure (180 bar) with high cutting velocity (2 m/s).

Fig. 6.Cutting forces versus time (Test 8, 18 MPa). Black solid line horizontal cutting force, red solid line vertical cutting force, blue solid line cutting speed.


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The normal forceN1and the shear force S1can be combined to a

resulting grain forceK1.

Theforces actingon a straight tooth when cutting rock, canbe distin-guished as:

A force normal to the bladeN2resulting from the grain stresses.

A shear force S2 as a result of the soil/steel friction N2 tan() or exter-

nal friction.

Combining the forces N2 and S2 will lead to a resulting force K2,

which is the unknown force on the blade. By taking the horizontal and

vertical equilibrium of forces, an expression for the force K2 on the

blade can be derived.

The forceCdue to the cohesive shear strength cis equal to:

C chiw

sin 5

where cis the shear strength (cohesion), hi is the cutting depth, w is the

tooth width, andis the shear angle.

The factor in Eq.(5) is the velocity strengthening factor, which

causes an increase of the cohesive shear strength. In clay (Miedema,

1992, 2010) this factor has a value of about 2 under normal cutting con-

ditions. In rock the strengthening effect is not reported, so a value of 1

should be used. On the blade a force component in the direction of cut-

ting velocity Fh and a force perpendicular to this direction Fvcan be dis-

tinguished.

FhK2 sin 6

FvK2cos : 7

The following equations for the horizontal Fhand verticalFvcuttingforces are found.

Fh chiw cos sin

sin sin 8

Fvchiw cos cos sin sin 9

where is the cutting angle, is the external friction angle, andisthe

internal friction angle.

The cohesioncis assumed to be about 50% of the UCS value, when

the internal friction angle is small or not taken into account.

To determine the shear angle where the horizontal forceFhis at a

minimum, the denominator of Eq.(8)has to be at a maximum. This oc-curs when the rst derivative ofFhwith respect toequals to zero, and

the second derivative is negative.

sin sin

sin 2 0 10

2

2 : 11

This gives for the cutting forces:

Fh2 chiw cos sin

1 cos H Fchiw 12

Fv2chiw cos cos

1 cos V Fchiw: 13

Fig. 12shows the values of the horizontal cutting force coefcient

HFas a function of the blade angle and the internal friction angle

of the rock. (SeeFig. 11.)

4.1.1. Validation of experiments under atmospheric conditions

As shown inTable 3, the rock used in the experiments had a UCS

value between 7.92 and 10.64 MPa, resulting in a shear strength cbe-

tween 4 and 5.3 MPa, when the angle of internal friction is not taken

into account. The tensile strength of the rock was between 0.86 and

1.15 MPa (BTSseeTable 3). According toMiedema (2014)the failure

mechanism of such a rock will be shear failure. The internal frictionangle of the rock is unknown, but an estimated value between 15 and

30 gives a reasonable range. According to Fig. 12the value ofHFis be-

tween 2 and 3.5, resulting in horizontal cutting forces between 3.4 kN

and 6 kN with a shear strength of 4 MPa, and between 4.5 kN and

7.9 kN with a shear strength of 5.3 MPa. This gives a total range from

3.4 kN up to 7.9 kN. On the one hand the atmospheric experiments

showed a strong 3D failure pattern (Fig. 7) with a cross section much

larger than the box cut of the tooth resulting in underestimation of

thecutting forces, on theother hand thetheorycalculates themaximum

cutting force at the start of the shear failure resulting in an overestima-

tion of the cutting forces. Assuming that these two effects more or less

compensate one to each other, the range of the theoretical atmospheric

cutting forces is 3.4 kN to 7.9 kN, matching the measured atmospheric

cutting forces ranging from 4.72 kN to 8.09 kN (seeTable 4).

Fig. 9. Ratio ofthe averagecut cross sectional areato thecutting area of toothas a function

of the product cutting speed and hyperbaric pressure.

Fig. 8. Ratio of the averagecut cross sectional areato the cutting areaof chisel versus cut-

ting velocity.


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4.2. Analytical model for hyperbaric conditions

The differences between rock cutting under atmospheric conditionsand under hyperbaric conditions is concerned with the extra pore pres-

sure forces W1 and W2 on the shear plane and on theblade as explained

below.Fig. 13illustrates the forces on the layer of rock cut. The forces

acting on the layer are:

A normal force acting on the shearsurface N1 resulting from the grain

stresses.

A shear forceS1as a result of internal ction angleN1 tan().

A forceW1as a result of water under pressure in the shear zone.

A shear forceCas a result of the cohesive shear strength c. This force

can be calculated by multiplying the cohesive shear strengthcwith

the area of the shear plane.

A force normal to the tooth N2resulting from the grain stresses.

A shear forceS2as a result of the external friction angleN2 tan().

A shear force A as a result of pure adhesion between the rockand the tooth a. This force can be calculated by multiplying

the adhesive shear strength a of the rock with the contact

area between the rock and the tooth. In most rocks this force

will be absent.

A forceW2as a result of water under pressure on the tooth.

The normal force N1 andthe shear force S1 on the shear plane can be

combined to a resulting grain forceK1

K1ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi

N21S21q

: 14

Theforces acting on a straight tooth when cuttingrock,can be distin-

guished as:

A force normal to the toothN2resulting from the grain stresses.

A shear forceS2as a result of the external friction angleN2 tan().

A shear forceAas a result of pure adhesion between the rock and the

tooth. This force can be calculated by multiplying the adhesive shear

strengtha of the rock with the contact area between the rock and

the tooth. In most rocks this force will be absent.

A forceW2as a result of water under pressure on the tooth.

Fig. 14 shows the abovementioned forces. When the forces N2 and S2are combined to a resulting force K2, and the adhesive force and the

water under pressures are known, then the resulting force K2is the un-

known force on the tooth. By taking the horizontal and vertical equilib-

rium of forcesan expression forthe force K2 on thetooth can be derived.

K2ffiffiffiffiffiffiffiffiffiffiffi ffiffiffiffiffiffi

N22S22q

: 15

The forceK2on the tooth is:

K2W2sin W1sin Ccos A cos sin :

16

The forces on the tooth can be derived from Eq. (16). On

the tooth a force component in the direction of the cutting

velocity Fh and a force perpendicular to this direction Fv can be

distinguished.

FhW2 sin K2 sin 17

FvW2cos K2cos : 18

The pore pressure forces can be determined in the case of full-

cavitation or in the case of no cavitation according to:

W1

wg z 10 hiw

sin or W1

P1mhiw

sin 19

Fig. 11.The forces on the layer cut in rock (atmospheric).

(a) (b) (c) (d)

Fig. 10. Productionof thecut a) at atmospheric pressure with lowcutting velocity(0.2 m/s) b) at high hyperbaricpressure (180 bar) with lowcutting velocity (0.2m/s),c) at atmospheric

pressure with high cutting velocity (2 m/s) and d) at high hyperbaric pressure (180 bar) with high cutting velocity (2 m/s).


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W2wg z 10 hbw

sin or W2P2mhbw

sin 20

whereP1mis the pore water pressure on the shear zone and P2mis the

pore water pressure on the tooth, hbis the tooth blade height,hiis the

cutting depth,zis the water depth, wwidth of the tooth,wwater den-

sity,ggravitational acceleration, andcutting angle.

The forcesCandAare determined by the cohesive shear strength c

and the adhesive shear strengthaaccording to:

Cchiwsin 21

Aahbwsin : 22

4.2.1. Validation of experiments under hyperbaric conditions

First of all it isassumed thatthe adhesive shear strength ofthe rockis

zero. According toMiedema (2014)the shear angle is about 2025

for the case considered here. Taking an internal friction angle of 20

and a shear angle of 22, together with a shear strength of 5.3 MPa,

the cutting forces can be determined as shown inFig. 15.

The measured horizontal cutting forces seem to increase rapidly as

the water depth increases starting at zero water depth (atmospheric

conditions), but having a less steep increase for water depths larger

than 60 m. This may be explained by two effects: The rst effect is thefact that at zero water depth the failure mechanism is brittle shear fail-

ure, meaning thatthe average cutting force will be smaller than the the-

oretically calculated cutting force. This failure mechanism transits to

ductile shear failure at larger water depths. This transition takes place

at a water depth of about 50100 m. The second effect is the 3D side

way cross section of the rock cut, being bigger than the box cut of the

Fig. 14.The forces on the blade in rock (hyperbaric).

Fig. 12.The ductile (shear failure) horizontal force coefcient.

Fig. 13.The forces on the layer cut in rock (hyperbaric).


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chisel. Figs. 4 and 7 show that this second effect decreases with increas-

ing water depth (hyperbaric pressure). The second effect does increase

the cutting forces since the cross section cut is larger than the chisel

width times the layer thickness. The following empirical equation

(Eq. (23)) takes both effects into account. The rst term between

brackets gives the rst effect; the transition from brittle shear failure

to ductile shear failure. Thesecond term between brackets gives thesec-

ond effect; the decreasing 3D cross section at large water depth. The co-

efcients1,2, and3in Eq.(23)depend on the rock properties. The

ones used in this study are 1= 3.33,2= 200, and3= 400.

Fh; cF h 1 1z 10

1 2

z 10 3

: 23

It is important to mention that Eq.(23)is an empirical equation based

upon a limited set of experiments. For theother types of rocks the coef-

cients1,2, and 3might be different than the ones used in this

study.

4.2.2. Inuence of the cutting velocity

From the experiments it is clear that there is an inuence of the cut-

ting velocity on the cutting process. In general a higher cutting velocity

gives a higher cutting force, but there is a lot of scatter. The theoretical

cutting forces in the previous section are determined assuming full cav-itation, but the question is Was there full cavitation during all experi-

ments?.Miedema (1987, 2013)derived an equation to determine the

transition cutting velocityVcbetween the non-cavitating regime and

the fully cavitating regime based on pore water ow according to

Darcy equation.

Vcd1 z 10 kmc1h1

: 24

The proportionality coefcients c1 and d1 have values close to 0.6 and

4, respectively (Miedema, 1987). As can be seen inTable 3, the perme-

abilities of the rocks show a lot of scatter. The permeabilities measured

(km) differ by a factor of 1000. So it is not possible to determine the tran-

sition velocities exactly, but it is stillpossible to see if Eq. (24) may explain

the velocity effect. The answer to this question is: at the lowest perme-

ability the transition velocity is about 0.2 m/s (at z = 1800 m) with a di-

latation (pore volume increase) of 0.01, and 0.02 m/s with a dilatation of

0.1. At the highest permeability the transition velocity is 187 m/s (at z =

1800 m) with a dilatation of 0.01 and 18.7 m/s with a dilatation of 0.1. The

cutting velocities used in the experiments are within this range, so the

only conclusion that can be drawn is that some tests had full cavitation

where the cutting forces do not depend on the cutting velocity, while

with other tests this was not the case and the cutting forces depended

on the cutting velocity. More detailed information has to be available togive a better prediction about the drained or undrained behavior of the

rock during the cutting process. For now, the conclusion is that the pre-

diction based on the equations for full cavitation gives a good upper

limit for the cutting forces and thus the required cutting energy and

power.

5. Discussion and conclusions

The purpose of this study wasto investigate the effect of high hyper-

baric pressures on rock cutting performance. The following conclusions

are drawn from the analysis of the results:

1. In general the cutting forces increase as the hyperbaric pressure in-

creases. This can be explained by taken into account that underhigh hyperbaric pressures the brittle behavior of the material and

the brittle cutting process changes into an apparent ductile mode.

This effect, however, is more noticeable for the combined effect of

high cutting speed and high pressure.

2. The experiments showed that the cutting forces at large hydrostatic

pressure (18 MPa) can be aboutfourto six times higherthan the cut-

ting forces at shallow ambient pressure or atmospheric conditions.

3. It was observed that when cutting rock at high hyperbaric pressures,

the side-break outangle is much narrow (i.e., box cut) than theside-

break out angle as commonly found when cutting rock at atmospher-

ic conditions or dry conditions. This results in a decrease in tooth

production.

4. The experiments reveal that contrary to dry and or atmospheric rock

cutting the effect of speed in combination with the hyperbaric

Fig. 15.The theoretical and corrected cutting forces and the measured horizontal cutting forces.


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pressure is signicant on the magnitude of the cutting forces and re-

quired energy. This is due to the crushed rock created in front of the

tooth resulting in dilation and pore under pressures.

5. The analytical models presented in this paper are an extension of the

models developed byMiedema (1987). The calculations show that

the analytical models can reproduce the measured values rather

well. It is important to mention, however, that the calculations

done with the hyperbaric cutting model assume full cavitation. The

results show a good upper limit for thecutting forcesand thus the re-quired cutting energy and power.

Acknowledgments

This research has been sponsored by the Royal IHC(IHC). We would

like to thank Mr. W.G.M. van Kesteren and Mr. J. Pennekamp from

Deltares for the execution of the rst series lab experiments of this

study. We also would like to thank Mr. Y Le Guen from the laboratory

of Ifremer, Brest, France.

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