Estimating Drag Forces on Suspended
and Laid-on-Seafloor Pipelines Caused
by Clay-Rich Submarine Debris Flow Impact
A. Zakeri
Abstract Estimating the impact drag forces exerted by a submarine debris flow on
a pipeline is a challenge. The conventional geotechnical based methods available to
estimate drag forces on buried pipelines in unstable slopes are not applicable to a
debris flow impact situation as they ignore or significantly underestimate the shear
rate effects in the soil-structure interaction. The results of recent investigations
indicate that a fluid dynamics approach in conjunction with rheological principles
of non-Newtonian fluids provides a more appropriate way in the study of soil-pipe
interaction for submarine debris flow impact situations. To that extent, this paper
summarizes the results of a recent investigation on the impact of clay-rich submarine
debris flows on suspended (free-span) and laid-on-seafloor pipelines. It presents a
method to estimate the drag forces, longitudinal and normal to the pipe axis, for
various angles of impact. The investigation comprised experimental flume tests and
Computational Fluid Dynamics (CFD) numerical analyses.
Keywords Clay-rich submarine debris flow • impact • pipeline • drag force •
longitudinal • normal • Computational Fluid Dynamics (CFD)
1
Introduction
Submarine landslides and debris flows are amongst the most destructive geohazards,
economically and environmentally, for installations on the seafloor. Estimating the
drag forces caused by these geohazards is an important design consideration in
offshore engineering. Failure of a hydrocarbon pipeline may be economically and
environmentally devastating. With offshore oil and gas activities pushing into
deeper water, there is a need to better assess and quantify the risk associated with
geohazards. Research in the area was inspired in the wake of the 1969 Hurricane
A. Zakeri ()
Geotechnical Engineering Group, C-CORE, St. John’s, Newfoundland, Canada
e-mail: arash.zakeri@c-core.ca
D.C. Mosher et al. (eds.), Submarine Mass Movements and Their Consequences,
Advances in Natural and Technological Hazards Research, Vol 28,
© Springer Science + Business Media B.V. 2010
93
94
A. Zakeri
Camille in the Gulf of Mexico, during which three offshore platforms and the
associated network of pipelines suffered significant damage. Subsequent studies
concluded that the damage was mainly due to wave induced mass movement at the
seafloor and not the wind and surface wave action alone (Coleman and Prior 1978
Schapery and Dunlap 1978). As a result, a number of techniques were developed
between the mid 1970s and late 1980s to assess the drag forces arising from soilstructure interaction. The problem was investigated from two perspectives:
a geotechnical approach and a fluid dynamics approach. In the former, the drag
forces are directly linked to the soil shear strength either linearly or through a
power-law relationship including the rate of shear. The latter approach considers
the soil fully fluidized and applies fluid mechanics principles. These methods are
limited and provide a wide range of estimates for the drag forces. Further, the methods seem often to be misinterpreted and incorrectly applied to situations outside the
methods’ validity (Zakeri 2009a).
A moving failed mass from a submarine landslide undergoes a series of complex
processes from initial disintegration to glide blocks to fast moving fluid-like debris
flows and turbidity currents. It is this complex process that has led to considerable
confusion of nomenclature in the literature and inappropriate use of the methods in
assessing soil-structure interaction forces. For example, geotechnical-based methods
that were developed for the case of fully or partially buried pipeline in an unstable
slope are sometimes applied to the problem of debris flow impact. Mulder and
Alexander (2001) provide a clear and simple classification system for sedimentary
density flows based on their physical flow properties and grain support mechanisms
encompassing both cohesive and cohesionless soils. Clay-rich submarine debris
flows are often fully remoulded and in a fluid state of pseudoplastic type.
The method presented here to estimate the drag forces deals with the impact of
clay-rich submarine debris flow on suspended and laid-on-seafloor pipelines. It is
based on an experimental program consisting of physical flume experiments (Zakeri
et al. 2008) complemented by numerical analyses using the Computational Fluid
Dynamics (CFD) method (Zakeri 2009b; Zakeri et al. 2009). For the case of suspended pipelines, the method covers various angles of incidence making it possible to
estimate the impact drag forces parallel and normal to pipe axis. The paper begins
with a brief discussion on the experimental program.
2
2.1
Experimental Program
Flume Experiments
The physical flume experiments involved subaqueous gravity flow of various clay-rich
slurries impacting a model pipe in a direction normal to its axis. The slurries were a mixture of kaolin clay, sand and water. A total of 50 experiments were carried out in a 0.20 m
wide and 9.5 m long flume suspended inside a 0.6 m wide tank (Fig. 1). The flume slope
was adjustable (3° and 6°) and the bed was rough. The instrumentation consisted of:
Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines
0.45 m x 0.45 m x 0.85 m H
Head
Tank
GL2 Camera
(30 fps)
1.0 m
Instrumented
Pipe
5.9 m
6.2 m
High-speed
EPIX Cameras
0.20 m Wide Flume
Sloped at 3 and 6 Degrees GL2 Camera
(30 fps)
Gate
(0.2 m W x 0.075 m H)
2.3 m
3.0 m
Sonar
(min. 0.62m from Bed)
Flume Walls
- Clear
Plexiglass
- 6 mm
Thick
Ball Valve
Plug
150mm I.D.
P.V.C. Pipe
0.45 m
Rod, Supporting Sonar
at Tip (20 mm O.D.)
Slurry
Mixing
Tank
(190 Lit.)
0.3 m
Sonar Data
Acquisition
System
95
Chute
(0.2 m x 0.2 m x 0.3 m H)
10.0 m
Fig. 1 Experimental setup for flume experiments
• Two Canon GL2 cameras for measuring the slurry head velocities near the gate
and 5.9 m downstream − 720 W × 480 H pixels frame size at 30 frames/s (fps)
• Two high-speed EPIX cameras covering the area immediately upstream and
downstream of the pipe to investigate the impact and flow characteristics in the
wake − 1,280 W × 1,024 H pixels frame size at 30 fps and 550 W × 600 H pixels
at 96 fps
• Four load cells to measure the drag and vertical forces – sampling rate of
1,000 Hz
• One submersible sonar to measure slurry flow and overriding turbidity heights.
Transducer: A301S-SU, Olympus NDT and pulser/receiver: DPR300, JSR
Ultrasonics
Each time, 190 l of slurry was prepared in the mixing tank located some 6 m above
the flume and conveyed into the head tank. Two copper model pipe sizes, 22.2 and
28.6 mm O.D., were used. The high frequency sonar system consisted of a stationary 500 kHz submersible transceiver just below the mean water surface oriented
normal to the sloping bed surface, approximately 0.62 m above the bed surface. The
data collection protocol involved two sampling periods: the first period at 50 Hz for
60 s and the second period at 6 Hz for the next 30 min. For each ping, the system
sampled backscatter at a rate of 8 MHz for 10,000 samples (1.25 ms). The plexiglass flume walls are smooth and the shear stress induced on the slurries was
assumed to be negligible (Crowe et al. 2001). The experiments attempt to model
about 2 s of continuous flow, ideally under constant head condition. Prior to the
flume experiments, an extensive rheological study using laboratory rheometers was
carried out to determine the slurry properties and suitable mix design. Table 1
presents the different slurry compositions and material properties.
The Brookfield DV-III Ultra vane-in-cup rheometer was used to determine the
rheological properties of the slurries. The slurry preparation and rheology experiments
were carried out in accordance with the ASTM (D2196-05) procedures. The results of
the rheological experiments and mathematical models are presented on Fig. 2.
96
A. Zakeri
Table 1 Slurry composition and material properties
Sand gradation
Slurry
Percentage material by mass
Clay
Water
Sand
Density (kg/m3)
Mesh Size (mm)
0.425
% Passing
100
10% Clay
15% Clay
20% Clay
25% Clay
30% Clay
35% Clay
10
15
20
25
30
35
1,681.0
1,685.7
1,687.7
1,689.6
1,691.6
1,694.0
0.300
0.212
0.150
0.106
0.075
0.053
99.5
95.5
77.5
33.5
8.5
0.5
35
35
35
35
35
35
55
50
45
40
35
30
Notes:
Sand properties: Specific Gravity = 2.65, Uniformity coefficient (Cu) = 1.7 defined as the ratio of
the maximum particle size of the smallest 60% (d60) over that of the smallest 10% (d10) of the
granular sample. Cu = 1 for a single-sized soil, Cu < 3 a fairly uniform grading and Cu > 5 a wellgraded (Whitlow 2001).
About 5% of the mass of sand was replaced by black diamond coal slag for visual purposes. The
black diamond slag had the same specific gravity and grain size distribution as the sand.
Shear Stress (Pa)
300
10% Clay
15% Clay
20% Clay
25% Clay
30% Clay
35% Clay
Herschel-Bulkley
Power-Law
200
100
0
0
10 20 30 40 50 60
Shear Rate (1/s)
Fig. 2 Results of the rheological experiments and Herschel-Bulkley and Power-Law mathematical model fits. Shear stresses are in Pascal
All the results of the rheological experiments were repeatable within ± 5%. Both the
Herschel-Bulkley and Power-Law models had a confidence of fit of 98% or greater.
2.2
Numerical Analyses
The situations tested in the experiments were numerically analyzed using Computational
Fluid Dynamics (CFD) method. The analyses were carried out using the CFD
software, ANSYS CFX 11.0, which is based on the Finite Volume (FV) method for unstructured grids. The FV method uses the integral form of the conservation equations.
Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines
97
With tetrahedra or hexahedra Control Volumes (CVs), unstructured girds are best
adapted to the FV approach for complex 3D geometries (Ferziger and Perić 2002).
The flume experiments constitute an incompressible two-phase, water and slurry, flow
regime. The inhomogeneous two-phase separated Eulerian–Eulerian multiphase
flow model of the CFX program was used to simulate the experiments. A general
description of the theory and the associated differential form formulations used to
analyze the flume experiments are included in Appendix A.
The computational procedures consisted of first setting up and calibrating a
numerical model to simulate the flume experiments using the rheological models
given in Fig. 2. The numerical analyses were quite successful in closely simulating
the subaqueous slurry flow characteristics (e.g. slurry head velocities, hydroplaning,
slurry flow and overriding turbidity heights) as well as calculating the impact forces
(normal drag and vertical) on the pipe models. Further, the flow parameters such as
slurry velocity and shear rate profiles (upstream and around the pipe) computed in
the CFD numerical analyses together with the high-speed camera images indicated
that the experimental setup for suspended pipeline appropriately modelled the prototype situation. The model was then used to complement the flume data by running
additional simulations with different slurry velocities and pipe diameters for both
the suspended and laid-on-seafloor cases. Later, the numerical model was extended
to cover all angles of incidence for the suspended pipeline case. As a result, both
the normal and longitudinal (with respect to the pipe axis) impact drag forces on a
suspended pipeline were investigated (Zakeri 2009b).
3
Method Developed to Estimate the Impact Drag Forces
The drag forces measured in the physical experiments and calculated in the simulations were correlated to the slurry head velocities measured upstream of the model
pipe within a distance between 5 to 10 times the pipe diameter. It was observed in
the experiments that in this range, the slurry flow is not affected by the presence of
the pipe and therefore, the slurry head velocities could be considered as the free
upstream flow velocity, U∞. The fluid flow characteristics around an object of a
given shape strongly depend on parameters such as the object size and orientation,
relative velocity between object and fluid, and fluid properties. For the drag force,
it is customary to use the inertia type of definition (White 2006), and define it by
using a drag coefficient, CD, through dimensional analysis by Eq. 1:
CD =
FD
1
r ⋅ U ∞2 ⋅ A
2
(1)
where, FD and A are the total drag force and the projected slurry-pipe contact area
(i.e. pipe diameter times the contact length), respectively, and r is fluid density.
For the normal drag force, FD–90, A is the pipe cross-sectional area projected onto
98
A. Zakeri
the plane normal to the flow direction and for the longitudinal drag force, FD–0, onto
the plane parallel to the flow direction. CD is a function of both the Froude number
(Fr) and the Reynolds number (Re), which are the most important dimensionless
parameters for studying incompressible fluid flow around an object. For many
flows the gravitational effects are unimportant such as for the flow around a body
or an airfoil where gravity waves are not generated. In that case, the Froude number
is irrelevant and the drag coefficient becomes only a function of the Reynolds
number (Kundu and Kohen 2004). The classical definition of the Reynolds number
for a Newtonian fluid is:
Re Newtonian =
rU ∞ D
m
(2)
where, m is the absolute (dynamic) viscosity, and D is the length characteristic –
here, the pipe diameter. This definition is not directly applicable to the problem of
non-Newtonian fluid flow past a circular cylinder. Hence, an ad hoc Reynolds
number was proposed for shear-thinning, non-Newtonian fluids described by the
Power-law or Herschel-Bulkley rheological models. It was based on the apparent
viscosity as opposed to the absolute viscosity. The apparent viscosity is defined as
the ratio of shear stress to the rate of shear of a non-Newtonian fluid. The apparent
viscosity changes with changing rates of shear and must, therefore, be reported as
the value at a given shear rate. Here, for the impact situations the shear strain rate
immediately outside the boundary layer is defined as:
. U
g= ∞
D
(3)
where, U∞ is the approaching debris head velocity. The pipe diameter is taken as the
length scale. In shear-thinning fluids of the Power-law or Herschel-Bulkley models,
the apparent viscosity, mapp, is defined as:
Power-law fluid:
.
.
t = a . g n thus : mapp = a .g n −1
(4)
.
.
t
t = t C + K . g n thus : m app = .C + K . g n −1
g
(5)
Herschel-Bulkley fluid:
In the above equations, t and tC are the fluid shear stress and fluid yield stress,
respectively, and the parameters a, n and K are model parameters which are determined from rheology testing. The Bingham model is a special case of the HerschelBulkley model where the fluid parameter, n, is equal to unity and the consistency,
K, is the same as the Bingham viscosity, mB. The behaviour of most clay-rich debris
flows can be described by the Herschel-Bulkley model (Locat 1997). The results of
the numerical analyses indicated that Eq. 3 provides a reasonable approximation for
the rate of shear induced on the slurry upon impact with the pipe as the magnitude
Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines
99
of the shear rate induced on the slurry drops significantly away from the pipe
surface – an order of magnitude within about a millimeter away from the pipe surface
and two orders of magnitude within about 3 mm. This relatively small distance
from the pipe surface basically constitutes the boundary layer thickness. Hence, the
use of Eq. 3 is also appropriate for the field situation of submarine debris flow
impact on pipelines. Using the shear rate defined by Eq. 3 and the apparent viscosity, defined by Eq. 4 or 5, the following form of the Reynolds number is proposed
for the problem of debris flow impact on pipelines:
Re Newtonian =
r . U ∞2
r . U ∞2
r . U ∞2
. thus :Re non − Newtonian =
.=
m .g
m app ⋅ g
t
(6)
The drag coefficient, CD, was obtained using the total drag force (i.e. the sum of the
viscous and inertia forces) measured from the experiments and calculated in the
simulations using Eq. 1, and its dependency on the Reynolds number defined by
Eq. 6 was then investigated. Ultimately, the following relationships were proposed
for estimating the normal drag force on a pipeline for design purposes:
Suspended Pipeline:
C D − 90 = 1.4 +
17.5
(7)
1.25
non − Newtonian
Re
Laid-on-seafloor Pipeline:
C D − 90 = 1.25 +
11.0
(8)
1.15
non − Newtonian
Re
The above proposed relationships (dashed line for Eq. 7 and bold solid line for Eq. 8)
are shown on Fig. 3 along with the results of the physical and numerical experiments.
20
Pipe on Seafloor Model
Exp. 10% Clay
2007 Inv. - Num. Analyses
2007 Inv. - Exp. Data
Exp. 15% Clay
Angle of Attack Analysis
16
Exp. 20% Clay
16
Proposed for Design
Exp. 25% Clay
Fit to Angle of Attack Analysis Data
Drag Coefficient, CD-90
Drag Coefficient Normal to Pipe Axis, CD - 90
20
12
8
Exp. 30% Clay
Exp. 35% Clay
Fit to Exp. Data
12
Num. 10% Clay
Num. 15% Clay
Num. 20% Clay
Num. 25% Clay
Num. 30% Clay
8
Num. 35% Clay
Fit to Num. Data
Proposed
4
4
0
0
1
10
Reynolds Number, Re
100
non-Newtonian
1
10
Reynolds Number, Re
100
non-Newtonian
Fig. 3 Drag coefficient versus Reynolds number curves: (left) suspended pipe model and (right)
pipe on seafloor model. The angle of attack is normal to the pipe axis
100
A. Zakeri
10
Drag coefficient Parallel to Pipe Axis, CD - 0
Fig. 4 Drag coefficient versus
Reynolds number: suspended
pipeline, longitudinal impact
drag force
Angle of Attack Analysis
Proposed for Design
8
6
4
2
0
1
10
Reynolds Number, Re
100
non-Newtonian
For the longitudinal drag force, the following CD–0-Re relationship was proposed:
C D − 0 = 0.08 +
9.2
1.1
non − Newtonian
Re
(9)
The above relationship which is based on CFD numerical analysis is shown on
Fig. 4. The drag force normal to pipe axis is developed as a result of both dynamic
pressures and viscous forces around the pipe whereas, the longitudinal drag force
is due to the shear stress on the pipe surface. As such, the drag coefficients
(i.e. CD–0) computed from the CFD model are believed to be representative of the
prototype.
4
Discussion
The experiments together with the simulations covered Reynolds numbers between
about 2 and 320 (i.e. more than two log cycles), and therefore, the results are considered appropriate for practical applications. From the results, it was noted that the
numerical analysis predicts a slightly higher drag forces than those measured in the
flume experiments when the Reynolds number is less than about 10 – see Fig. 3.
This difference was explained as follows. In the prototype the fluid-pipe interaction
in an impact with a submarine debris flow takes place in a relatively short duration
after which, the debris has either passed the pipeline or slowed down in the
upstream velocity. This was also the case for the flume experiments. In the CFD
Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines
101
simulations of the flume experiments however, the slurry was given a slightly
longer time to develop its flow around the pipe which in turn, also results in a larger
contact area between the pipe surface and the slurry flow and therefore, larger drag
forces. Further, small pockets of water may have become trapped between the
slurry flow and copper pipe surface in the flume test (i.e. causing partial slippage),
which in turn, results in lower drag force measurements for the low velocity and
highly viscous slurry experiments. This, together with the no-slip boundary condition,
explains why the drag coefficients computed by the CFD simulations are higher than
those estimated from the flume experiments for the Reynolds numbers of about 10
and less. As such, slightly lower curves than the ones predicted by the CFD analyses
were proposed for the design in the region of low Reynolds numbers.
It should be noted that the method assumes that the pipeline is moored to the
seafloor and does not deform when hit by the debris flow. Two conceptual measures
to control and mitigate the impact drag forces have been discussed by Zakeri et al.
(2009). These measures consist of an upstream berm to protect a laid-on-seafloor
pipeline in shallow waters or fjords crossings and a cable-controlled system for pipelines
installed in deep waters.
5
Conclusions
A combined experimental and numerical approach was used to develop a simple
method for estimating the drag forces on suspended and laid-on-seafloor pipelines
caused by a clay-rich debris flow impact. The method was based on fluid dynamics
principles where the drag force is presented in the non-dimensional form – drag
coefficient. An ad-hoc Reynolds number was defined to describe flow characteristics of clay-rich debris flows of shear-thinning non-Newtonian fluid behaviour
described by Bingham or Herschel-Buckley rheological models. Drag coefficients
were calculated for situation where the impact causes drag forces both normal and
longitudinal to the pipe axes.
The experimental setup, instrumentation and testing procedures in the flume as
well as the CFD simulations worked very satisfactorily. In practice, submarine pipe
diameters range between 0.1 to 1.0 m. Assuming a debris flow velocity between 1 to
10 m/s, density of 1,600 kg/m3 and shear stress between 0.5 and 2.0 kPa, the shear
rate upon the impact with a pipe would be in the range of 1 s−1 and 100 s−1, and one
would find the corresponding Reynolds number to be between 0.8 and 320. The
experiments covered the Reynolds numbers between about 2 and 320 (i.e. more than
two log cycles), and therefore, are considered appropriate for practical purposes.
Acknowledgements The work presented here (ICG Contribution No. 258) was supported by the
Research Council of Norway through the International Centre for Geohazards (ICG) and the LeifEiriksson stipend awarded to the author. Their support is gratefully acknowledged. Further, the
author is thankful to Prof. David White and Prof. Christopher Baxter for their review efforts and
constructive comments.
102
A. Zakeri
References
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Appendix A – Theory for CFD Numerical Analysis
A general description of the theory and the constitutive equations used in the CFD
analyses are briefly presented here. In the formulation, the different phases are
denoted by lowercase Greek letters, a and b, and the total number of phases is NP.
In the inhomogeneous model, each phase has its own velocity and other relevant
flow fields while the pressure field is shared between the incompressible fluid
phases (CFX 2007). In this model, the fluids interact via the inter-phase mass and
momentum transfer terms. The phase continuity equation is expressed by:
P
∂
ra ra )+ ∇ • ra ra U a = SMS a + ∑ Gab
(
∂t
b =1
(
)
N
(A.1)
where, ra, ra, and Ua are the phase volume fraction, density and velocity, respectively,
and SMSa is the user specified mass sources. Gab is the mass flow rate per unit
Estimating Drag Forces on Suspended and Laid-on-Seafloor Pipelines
103
NP
volume from phase b to phase a, which must obey the rule: G ba = − G ba ⇒ ∑ G a = 0 .
a =1
It is important to define the direction of the mass transfer in the conservative equa+
+
tions. A convenient method is to express Gab by: G ab = G ab
. The term G ab > 0
− G ba
represents a positive mass flow Nrate per unit volume from phase b into phase a. The
P
volume fraction is bound by:
ra = 1 . The momentum equation for a continuous
∑
fluid phase is:
a =1
∂
(ra ra Ua )+ ∇• ra (ra Ua ⊗ Ua )
∂t
(
( (
)
= − ra ∇Pa + ∇• ra ma ∇U a + (∇U a )
Np
(
T
))
(A.2)
)
+
+
+ ∑ G ab
U b − G ba
Ua + SM a + Ma
b =1
where, Pa and ma are the pressure and viscosity, respectively, and SMa is the user
defined momentum sources due to external body forces. T is the matrix transpose
operation. Ma is the sum of interfacial forces acting on phase a due to the presence
of other phases and is obtained from:
VM
TD
Ma = ∑ Ma b = MaDb + MaLUB
b + M a b + M a b + ...
(A.3)
b ≠a
where, the terms indicated above, in order, represent the inter-phase drag force, lift
force, wall lubrication force, virtual mass force and turbulence dispersion force.
+
+
Finally, the term G ab
U b − G ba
Ua represents the momentum transfer induced by
the inter-phase mass transfer. The governing transport equations result in 4 × NP + 1
equations with 5 × NP unknowns that correspond to (u, v, w, r, P)a for a = 1 to NP,
where u, v, and w and the velocity components in the x, y and z directions, respectively. Given that the fluids in the inhomogeneous multiphase flow share the same
pressure field, the transport equations are solved by imposing the constraint of
Pa = P for all a = 1 to NP.
(
)