- Email: [email protected]
Interaction forces between pipelines and submarine slides — A geotechnical viewpoint
Ocean Engineering 48 (2012) 32–37
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Interaction forces between pipelines and submarine slides — A geotechnical viewpoint Mark F. Randolph n, David J. White Centre for Offshore Foundation Systems, The University of Western Australia, Crawley, WA 6009, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 February 2011 Accepted 17 March 2012 Editor-in-Cheif: A.I. Incecik Available online 19 April 2012
Assessment of interaction forces between deep water pipelines and potential submarine slides, debris flows and turbidity currents is an important aspect of geohazard studies. Historically, interaction forces have tended to be expressed in terms of drag factors, within a traditional fluid mechanics framework, with the drag factors depending strongly on an equivalent Reynolds number for the non-Newtonian debris material. Here, we have followed a more geotechnical approach, allowing the interaction forces to be expressed in terms of a strain-rate dependent shear strength of the debris material, and with the inclusion of a drag term (with fixed drag coefficient) for high velocity, low strength, combinations. This superposition approach treats separately the interaction forces that arise from the strain-rate dependent strength and the inertia of the debris, rather than combining them into a single drag force. A failure envelope is proposed, allowing axial and normal interaction forces to be estimated for any angle of attack of the debris flow. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: Submarine slides Debris flow Pipelines Analysis Offshore engineering
1. Introduction Deep water oil and gas developments on the continental slope are prone to a number of geohazards, including submarine landslides, seismicity, mud volcanoes, shallow gas and gas hydrates (Kvalstad et al., 2001). Submarine landslides are perhaps the most significant of these, and assessment of the potential for a slide to be triggered, and to impact infrastructure, is a major component of geohazard assessment. Pipelines, particularly the export pipelines and trunklines that carry the hydrocarbon product to shore, are the most exposed to impact risk from submarine slides, because of their length and the varied terrain through which they must pass. Critical questions are therefore what impact forces are exerted on a pipeline by a submarine slide and the associated debris flow and turbidity current, and what will then be the response of the pipeline. This paper is concerned with the first of the above questions: evaluation of the impact forces from slide material (referred to here generically as debris flow) with given rheological properties and moving at a given velocity. The general situation is as indicated in Fig. 1. Attention is restricted to situations where the debris flow fully engulfs the pipeline. The effect of the incident impact angle on the resulting normal and axial (or frictional) tractions imposed on the pipeline will be considered. Studies of submarine slide runout have tended to be conducted within a fluid mechanics framework (e.g., Locat and Lee, 2002; Niedoroda et al., 2003; Gauer et al., 2005). This has led naturally to
the use of a similar framework to assess the potential ‘drag’ loading on a pipeline or other infrastructure in the path of a submarine slide. An alternative approach is followed here, whereby the tractions applied by the moving slide material are expressed (primarily) in terms of an appropriate shear strength of the material, but allowing for the effect of shear strain rate on that shear strength. The design approach developed in the paper is largely based on a re-analysis of numerical data from Zakeri (2009), who undertook multiphase analysis of a pipe suspended in water and impacted by slurry representing the debris flow. Zakeri (2009) noted generally good agreement between the numerical results and those from physical modelling of the problem. In both types of modelling it was found that the material surrounding the pipe was somewhat heterogeneous in its properties, with pockets of entrained water in addition to the slurry. However, here the situation has been idealised to that of a fully engulfed pipe with (rough) contact maintained with a homogeneous debris flow over its complete surface. Justification for the idealisation rests partly on the reasonable agreement obtained between the results from Zakeri (2009) and theoretical solutions for limiting cases of normal and tangential flow, and partly because the assumed boundary conditions should prove conservative.
2. Alternative approaches for estimating pipeline forces 2.1. Fluid mechanics approach
n
Corresponding author. Fax: þ61 8 6488 1044. E-mail address: [email protected] (M.F. Randolph).
The forces acting on a pipeline from low strength debris flow and turbidity currents have generally been estimated from the
0029-8018/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.03.014
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
33
and C f ¼ 0:08 þ
9:2 Re1:1 non-Newtonian
ð5Þ
His range of Reynolds numbers covered 1.5 to just over 300, with resulting coefficients ranging from about 1 to 15 (Cd) and 0 to 7 (Cf), as shown in Fig. 3 (using his notation of CD 90 and CD 0 for Cd and Cf). His numerical data are reinterpreted later in this paper. 2.2. Geotechnical approach Fig. 1. Schematic of debris flow interacting with pipeline.
density and velocity of the flow, using drag coefficients (DNV, 1976). For steady state flow acting on a fully submerged pipeline, the two terms of most interest are those for normal loading (with velocity component, vn), expressed in terms of a drag coefficient, Cd, and for axial friction (with velocity component, va), based on a friction coefficient, Cf. The forces per unit length of pipeline of diameter, D are expressed as 1 2 rvn D Fn ¼ Cd 2 1 2 rva pD ð1Þ Fa ¼ Cf 2 where r is the density of the flowing material. Values of the drag coefficient are suggested as Cd ¼1.2 at Reynolds numbers, Re r4 105 and Cd ¼0.7 for Re Z6 105 with linear interpolation on a log–log scale between these limits; the corresponding friction coefficient is expressed as Cf ¼
and
0:075
ð2Þ
ðlog10 Re 2Þ2
For turbidity currents, DNV (1976) suggests that the Reynolds number should be based on that for water, expressed as Re ¼vnD/n, with the kinematic viscosity, n ¼1.6 10 6 m2/s. A similar formulation has been suggested for debris flows (Zakeri, 2009), with drag and friction coefficients expressed as power law functions of an equivalent Reynolds number for nonNewtonian fluids. The non-Newtonian Reynolds number is also referred to as a Johnson number, giving the relative magnitudes of a drag-related resistance, or stagnation pressure, to the mobilised shear stress, t, in the material, and is expressed as Re,non-Newtonian ¼
v2n
r t
ð3Þ
Zakeri (2009) reports results from both physical and numerical modelling. The latter was carried out using the CFD code, ANSYS CFX Version 11.0, considering pipelines placed at angles of 0, 30, 45, 60 and 901 to the flow direction (see Fig. 2). He provided a table of results for the normal and axial forces deduced on a short, typically 2 diameter, segment in the central part of the pipeline. Normal and frictional coefficients were obtained from the computed normal and frictional forces per unit area projected in the planes normal and parallel to the flow, treating all the data together, regardless of the flow angle. The coefficients were based on the full input flow velocity (i.e., not components of the flow velocity). The best fit expressions determined in terms of the nonNewtonian Reynolds numbers (also based on the full velocity, and with a mobilised shear stress, t, based on a strain rate of v/D) were given by C d ¼ 1:2þ
We propose an alternative approach for debris flows, which is more aligned with geotechnical methods for characterising the properties of seabed sediments and for assessing the capacity of foundation elements. This approach is to express the normal and frictional forces on the pipe in terms of a shear strength, su, adjusted for the high strain rates involved, and using appropriate bearing factors for normal load and an interface friction ratio for frictional loads, assuming full contact of the debris around the circumference of the pipe. For normal loading, it is also necessary to consider a conventional drag coefficient, which would become important for debris flow at high Johnson numbers (i.e., high ratios of rv2n/su). (Note that the nomenclature, su, is used for the shear strength to emphasise the geotechnical approach, although this is essentially identical to the mobilised shear stress, t, used to evaluate the non-Newtonian Reynolds number.) The normal and frictional forces per unit length of pipe are expressed as 1 2 Fn ¼ Cd rvn D þN p su,nom D ð6Þ 2
27:5 Re1:1 non-Newtonian
ð4Þ
F a ¼ f a su,nom pD
ð7Þ
where Np is the bearing factor for the force normal to the pipe, while fa is a frictional coefficient for the force parallel to the pipe. The relevant shear strengths are adjusted for the shear strain rate, taken as v/D (with v the total velocity of the debris flow). Thus if the shear strength is found to vary according to a power law relationship, the nominal shear strength is expressed as v=D b su,nom ¼ su,ref ð8Þ g_ ref where su,ref is a reference shear strength at a strain rate of g_ ref and v is the flow velocity. Dynamic finite element analyses using the techniques described by Wang et al. (2011) of flow normal to a cylinder, for rate independent soil under conditions where a gap is allowed to form behind the cylinder (in the wake of the flow) have been performed. These show a linear dependence of the normal force on v2n that is consistent with a drag coefficient of Cd ¼0.5. Complementary analyses in zero density soil, but with different forms of rate-dependent shear strength, have shown that it is sufficient to base su,nom on the nominal strain rate, v/D, provided that the coefficient b in Eq. (8) is less than about 0.3 (Zhu and Randolph, 2012). This restriction on b covers typical soil conditions under the range of water contents relevant for debris flows (Boukpeti et al., 2012). The bearing factor Np is then essentially independent of velocity, and consistent with values derived from rate-independent plasticity theory, ranging from 9.1 for a fully smooth pipe to 11.9 for a fully rough pipe (Randolph and Houlsby, 1984; Martin and Randolph, 2006). Based on these observations, the influence of strain rate on flow pattern can be neglected if the strength su relevant to the strain rate of v/D is adopted in Eq. (6), with the result that the
34
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
Fig. 2. Plan view of ANSYS CFX analyses (Zakeri, 2009).
values of bearing factor Np and drag factor Cd become independent of the flow velocity, v. For axial loading (i.e., flow parallel to the pipe), the friction force is expressed by Eq. (7), with the nominal shear strength given by Eq. (8). The coefficient fa reflects the enhanced shear strain rate immediately adjacent to the pipe (assuming a rough pipe-soil interface), in excess of the nominal strain rate of v/D. For a power law model, the coefficient fa may be derived analytically, and is given by (Einav & Randolph, 2006) b 1 1 ð9Þ fa ¼ 2
b
The value of fa lies in the range 1.2 to 1.4 for typical values of b between 0.05 and 0.15. Some comment is appropriate here regarding evaluation of the reference shear strength, su,ref, and the strain rate dependency of shear strength using the power law relationship in Eq. (6), or other forms of relationship such as Herschel–Bulkley (Huang and Garcia, 1998). Recent studies (Jeong et al., 2009; Boukpeti et al., 2009, 2012) have shown that the reference shear strength can be expressed as a function of water content with parameters that become increasingly independent of material type as the water content increases (and strength decreases). They also found that the rate parameter, b, showed no systematic variation with water content. In principle, therefore, the shear strength of the debris flow may be established from a known starting point (as provided by site investigation data from potential landslide zones) and a means of assessing the evolving water content with runout distance.
3. Re-analysis of data from Zakeri (2009) Given that Reynolds number is proportional to v2, and that the drag coefficients proposed by Zakeri (2009) are close to varying
inversely with Reynolds number, it is instructive to replot his numerical data as a function of the inverse of Reynolds number. The results for the extreme conditions of flow either normal or parallel to the pipe are shown in Fig. 4. The data are closely fitted by linear relationships. Substituting these fits into Eq. (1), making use of Eq. (3) (and replacing the mobilised shear stress, t, by the nominal shear strength, su,nom based on the flow velocity) leads to F n ¼ 0:4 12 rv2n þ13su,nom D ð10Þ F a ¼ 1:5su,nom pD These relationships are essentially identical to Eqs. (6) and (7), with coefficients of Cd ¼0.4, Np ¼13 and fa ¼1.5. These compare with the theoretical values of Np ¼11.9 for a rough pipe and fa ¼1.4 (using Eq. (9) with power law exponent, b, in the range 0.11–0.14 as adopted by Zakeri, 2009). The drag coefficient of Cd ¼0.4 compares with the value of 0.5 estimated from numerical modelling of Wang et al. (2011) at high values of rv2n/su. The results for flow at different angles to the pipe have been replotted similarly in Fig. 5, taking the relevant components of velocity (normal or parallel to the pipe) to evaluate an equivalent drag coefficients and Reynolds numbers. Two adjustments have been made: (1) For convenience the horizontal axes are plotted as 2/Re rather than 1/Re, so that the gradients represent equivalent Np and fa coefficients directly. (2) The drag coefficients for flow normal to the axis have been shifted by the limiting drag coefficient for very high Reynolds number (i.e., the intercept for 2/Re ¼ 0); it was found that the best overall adjustment was by a value of 0.6/siny, where y is the flow direction relative to the pipe axis. Although the degree of scatter increases slightly, linear relationships through the origin are then obtained for both components of
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
35
Drag Coefficient Normal to Pipe Axis, CD - 90
20 2007 Inv. - Num. Analyses 2007 Inv. - Exp. Data Angle of Attack Analysis Proposed for Design Fit to Angle of Attack Analysis Data
16
12
8
4
0 1
10
100
Reynolds Number, Re non-Newtonian
Drag Coefficient Parallel to Pipe Axis, CD - 0
10 Angle of Attack Analysis Proposed for Design
8
6
Fig. 4. Drag coefficient data replotted against inverse of Reynolds number.
4 decreases in proportion to sin2y, while the drag coefficient only increases as 1/siny. The variation of bearing and friction factors with the angle of the flow relative to the pipeline axis may be interpreted in terms of a failure envelope, as shown in Fig. 6. The dashed curve shows a fit through the deduced numerical data points, expressed as
2
0
1
10
100
Reynolds Number, Re non-Newtonian Fig. 3. Drag coefficients deduced from numerical analysis (Zakeri, 2009).
flow. For the extremes of normal and parallel flow, the resulting drag coefficient of 0.6 (for y ¼901), and Np and fa factors of 12.8 and 1.5 are close to the theoretical values of 11.9 and 1.4 as previously noted. For intermediate flow angles, the Np and fa values decrease relative to the values for the extremes relevant for each factor, as summarised in Table 1. For the flow component normal to the pipe, the apparent increase in drag coefficient (expressed as 0.6/siny) as the flow deviates from perpendicular to the pipe is surprising, and does not appear to have any particular logic. It should be noted, however, that the drag force for a given resultant velocity still decreases, since the normal component of velocity (squared)
fa f a,0
3
þ
Np Np,90
1
¼1
ð11Þ
where, consistent with the data, fa,0 has been taken as 1.54 and Np,90 has been taken as 12.8. Although the envelope provides a reasonable fit to the data points for 45 and 601, the data point for 301 lies well within the envelope, and indeed implies some concavity in the envelope. The reason for this is unclear, and would require further detailed numerical simulation for flow angles lying between 0 and 451 to resolve. A design envelope is proposed that is constrained to the theoretical limiting bearing factors of fa,0 ¼1.4 and Np,90 ¼ 11.9 (see full curve in Fig. 6). In addition to the failure envelope, a relationship is required to determine the position on this envelope that corresponds to a particular angle of flow, and hence provide the resulting bearing factors Np (and fa). The resultant force on the pipe, which is the vector sum of Fn and Fa, is not parallel to the flow direction. It is proposed that the bearing
36
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
Fig. 6. Failure envelope for varying flow angle relative to pipe axis.
Table 2 Input data for example application.
Table 1 Summary of bearing and axial resistance factors.
0 30 45 60 90
Values from numerical data of Zakeri (2009)
Pipeline diameter (m) Density of flow material (t/m3) Reference shear strength, su,ref (kPa) Reference shear strain rate, g_ ref (s 1) Power law coefficient, b
0.8 1.5 0.3 1 10 5 0.13
4. Illustrative example application Values from proposed design approach
Axial coefficient, fa
Axial Normal coefficient, coefficient, Np fa
Normal coefficient, Np
1.54 0.96 0.88 0.70 0
0 7.5 10.6 11.7 12.8
0 7.3 9.3 10.8 11.9
1.4 1.02 0.84 0.64 0
factor, Np, may be expressed as Np ¼ Np,90 ðsin yÞ0:7 ¼ 11:9ðsin yÞ0:7
Value
numerical data. Although most of the numerical data lie outside the proposed design curve, and as such would provide a more conservative assessment of interaction forces, the design curve is constrained by theoretical values (for the given boundary conditions assumed in the numerical analysis) for flow parallel and normal to the pipe. For those cases the numerical data slightly overestimates the interaction forces (as is common with displacement-based finite element computations). The inner proposed curve is therefore considered to be a reasonable basis for design.
Fig. 5. Replotted drag coefficient data for general flow conditions.
Flow angle relative to pipe axis (1)
Property
ð12Þ
with the corresponding value of fa then being found from Eq. (11). Eq. (12) is a purely empirical relationship, which appears to give reasonable predictions of the numerical data. The proposed failure envelope, together with spot values of the factors for the same angles as examined numerically by Zakeri (2009), are shown in Fig. 6 and the bearing and friction factors are also summarised in Table 1. The factors all lie within 10% of those deduced from the
An example application of the design approach derived above is provided here, the input data for which is summarised in Table 2. Relatively low water content material, with density 1.5 t/m3 and reference shear strength of 0.3 kPa is assumed to impact a pipeline of diameter 0.8 m with velocity of 2 m/s. Adopting a reference shear strain rate of 1 10 5 s 1 (corresponding to a typical laboratory shearing rate for soil of about 3%/h) and a power law coefficient of b ¼0.13, the nominal shear strength is calculated as su,nom ¼1.51 kPa. This gives a Reynolds (or Johnson) number, rv2/su,nom ¼4.0, suggesting that the geotechnical bearing resistance will dominate in comparison with the drag force. The resulting normal and frictional forces for varying angles of attack are shown in Fig. 7. The maximum (total) normal force is just under 16 kN/m (average pressure of 20 kPa across the pipe diameter), and comprises 90% geotechnical resistance and 10% inertial drag. The maximum frictional resistance is 34% of the maximum normal force, and corresponds to an average shear stress around the pipe circumference of 2.1 kPa. The normal component of force develops rapidly as the flow angle (relative to the pipeline axis) increases from zero. The angle of the resultant force reaches 451 for a flow angle of only 101 (Fig. 7(b)).
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
37
angled flow impacting a segment of pipe. The data have been reprocessed in order to evaluate bearing and frictional factors (from which the resulting forces may be evaluated) for flow at different angles to the pipe axis. Separately, drag coefficients, independent of the strength of the material, have been identified; these become increasingly important at non-Newtonian Reynolds numbers exceeding about 10. The design approach is expressed as a failure envelope that links the bearing and frictional factors, together with an algorithm that expresses the normal bearing factor as a function of the flow angle. An example application is included to illustrate how the different components of force may be evaluated for a particular debris flow or turbidity current impacting a pipeline at different angles.
Acknowledgements This work forms part of the activities of the Centre for Offshore Foundation Systems (COFS), established under the Australian Research Council’s Research Centres Program and now supported by Centre of Excellence funding from the State Government of Western Australia and by the Australian Research Council’s Federation and Future Fellowship programs. The research described here is part of a Joint Industry Project administered and supported by the Mineral and Energy Research Institute of Western Australia, and by BP, BHP Billiton, Chevron, Petrobras, Shell and Woodside. The financial support of all the participants is gratefully acknowledged. References
Fig. 7. Resulting force variations with flow angle for illustrative example.
5. Conclusions This paper has attempted to develop a more rational basis for assessing the impact forces of debris flows and turbidity currents on offshore pipelines. Any submarine landslide is likely to lead to mass transport of the slide material, with the strength and consistency of the material gradually changing from what may be classed as ‘soil’ to a fluid with suspended solids. Models for estimating impact forces have tended to be couched in a fluid mechanics framework throughout this material regime, with drag and frictional coefficients expressed as functions of Reynolds number and matched to experimental data. The proposed design approach is based on a single model for the nominal shear strength of the flow material, expressed as a power law function of a nominal strain rate that is linked to the flow velocity and pipe diameter. Analytical solutions for the geotechnical bearing resistance (for flow normal to the pipe) and frictional resistance (for flow parallel to the pipe) are used to constrain data from an extensive numerical study by Zakeri (2009) that investigated normal and frictional forces arising from
Boukpeti, N., White, D.J., Randolph, M.F., Low, H.E., 2012. Strength of fine-grained soils at the solid–fluid transition. Ge´otechnique 62 (3), 213–226. Boukpeti N., White D.J., Randolph M.F., (2009). Characterization of the solid–liquid transition of fine-grained sediments. Proc. Conf. on Offshore Mechanics and Arctic Engineering, Honolulu. OMAE2009-79738. DNV, 1976. Rules for the Design, Construction and Inspection of Submarine Pipelines and Pipeline Risers. Recommended Practice. Det Norske Veritas, Hovik, Norway. Einav, I., Randolph, M.F., 2006. Effect of strain rate on mobilised strength and thickness of curved shear bands. Ge´otechnique 56 (7), 501–504. Gauer, P., Kvalstad, T.J., Forsberg, C.F., Bryn, P., Berg, K., 2005. The last phase of the Storregga slide: simulation of retrogressive slide dynamics and comparison with slide-scar morphology. Mar. Pet. Geol. 22, 171–178. Huang, X., Garcia, M.H., 1998. A Herschel–Bulkley model for mud flow down a slope. J. Fluid Mech. 374, 305–333. Jeong, S.W., Leroueil, S., Locat, J., 2009. Applicability of power law for describing the rheology of soils of different origins and characteristics. Can. Geotech. J. 46, 1011–1023. Kvalstad, T.J., Nadim, F., Harbitz, C.B., (2001). Deepwater geohazards: geotechnical concerns and solutions. Proc. Offshore Tech. Conf., Houston, Paper OTC 12958. Locat, J., Lee, H.J., 2002. Submarine landslides: advances and challenges. Can. Geotech. J. 39 (1), 193–212. Martin, C.M., Randolph, M.F., 2006. Upper bound analysis of lateral pile capacity in cohesive soil. Ge´otechnique 56 (2), 141–145. Niedoroda, A., Reed, C., Hatchett, L., Young, A., Kasch, V., (2003). Analysis of past and future debris flows and turbidity currents generated by slope failures along the Sigsbee escarpment. Proc. Offshore Technology Conference, Houston, Paper OTC 15162. Randolph, M.F., Houlsby, G.T., 1984. The limiting pressure on a circular pile loaded laterally in cohesive soil. Ge´otechnique 34 (4), 613–623. Wang, D., Randolph, M.F., White, D.J., 2011. A dynamic large deformation finite element method and element addition technique. Int. J. Geomech., ASCE. revised September 2010. Zakeri, A., 2009. Submarine debris flow impact on suspended (free-span) pipelines: normal and longitudinal drag forces. Ocean Eng. 36 (6–7), 489–499. Zhu, H., Randolph, M.F., 2011. Numerical analysis of a cylinder moving through rate-dependent soils. Ocean Eng. 38 (7), 943–953.
Contents lists available at SciVerse ScienceDirect
Ocean Engineering journal homepage: www.elsevier.com/locate/oceaneng
Interaction forces between pipelines and submarine slides — A geotechnical viewpoint Mark F. Randolph n, David J. White Centre for Offshore Foundation Systems, The University of Western Australia, Crawley, WA 6009, Australia
a r t i c l e i n f o
a b s t r a c t
Article history: Received 25 February 2011 Accepted 17 March 2012 Editor-in-Cheif: A.I. Incecik Available online 19 April 2012
Assessment of interaction forces between deep water pipelines and potential submarine slides, debris flows and turbidity currents is an important aspect of geohazard studies. Historically, interaction forces have tended to be expressed in terms of drag factors, within a traditional fluid mechanics framework, with the drag factors depending strongly on an equivalent Reynolds number for the non-Newtonian debris material. Here, we have followed a more geotechnical approach, allowing the interaction forces to be expressed in terms of a strain-rate dependent shear strength of the debris material, and with the inclusion of a drag term (with fixed drag coefficient) for high velocity, low strength, combinations. This superposition approach treats separately the interaction forces that arise from the strain-rate dependent strength and the inertia of the debris, rather than combining them into a single drag force. A failure envelope is proposed, allowing axial and normal interaction forces to be estimated for any angle of attack of the debris flow. Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved.
Keywords: Submarine slides Debris flow Pipelines Analysis Offshore engineering
1. Introduction Deep water oil and gas developments on the continental slope are prone to a number of geohazards, including submarine landslides, seismicity, mud volcanoes, shallow gas and gas hydrates (Kvalstad et al., 2001). Submarine landslides are perhaps the most significant of these, and assessment of the potential for a slide to be triggered, and to impact infrastructure, is a major component of geohazard assessment. Pipelines, particularly the export pipelines and trunklines that carry the hydrocarbon product to shore, are the most exposed to impact risk from submarine slides, because of their length and the varied terrain through which they must pass. Critical questions are therefore what impact forces are exerted on a pipeline by a submarine slide and the associated debris flow and turbidity current, and what will then be the response of the pipeline. This paper is concerned with the first of the above questions: evaluation of the impact forces from slide material (referred to here generically as debris flow) with given rheological properties and moving at a given velocity. The general situation is as indicated in Fig. 1. Attention is restricted to situations where the debris flow fully engulfs the pipeline. The effect of the incident impact angle on the resulting normal and axial (or frictional) tractions imposed on the pipeline will be considered. Studies of submarine slide runout have tended to be conducted within a fluid mechanics framework (e.g., Locat and Lee, 2002; Niedoroda et al., 2003; Gauer et al., 2005). This has led naturally to
the use of a similar framework to assess the potential ‘drag’ loading on a pipeline or other infrastructure in the path of a submarine slide. An alternative approach is followed here, whereby the tractions applied by the moving slide material are expressed (primarily) in terms of an appropriate shear strength of the material, but allowing for the effect of shear strain rate on that shear strength. The design approach developed in the paper is largely based on a re-analysis of numerical data from Zakeri (2009), who undertook multiphase analysis of a pipe suspended in water and impacted by slurry representing the debris flow. Zakeri (2009) noted generally good agreement between the numerical results and those from physical modelling of the problem. In both types of modelling it was found that the material surrounding the pipe was somewhat heterogeneous in its properties, with pockets of entrained water in addition to the slurry. However, here the situation has been idealised to that of a fully engulfed pipe with (rough) contact maintained with a homogeneous debris flow over its complete surface. Justification for the idealisation rests partly on the reasonable agreement obtained between the results from Zakeri (2009) and theoretical solutions for limiting cases of normal and tangential flow, and partly because the assumed boundary conditions should prove conservative.
2. Alternative approaches for estimating pipeline forces 2.1. Fluid mechanics approach
n
Corresponding author. Fax: þ61 8 6488 1044. E-mail address: [email protected] (M.F. Randolph).
The forces acting on a pipeline from low strength debris flow and turbidity currents have generally been estimated from the
0029-8018/$ - see front matter Crown Copyright & 2012 Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.oceaneng.2012.03.014
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
33
and C f ¼ 0:08 þ
9:2 Re1:1 non-Newtonian
ð5Þ
His range of Reynolds numbers covered 1.5 to just over 300, with resulting coefficients ranging from about 1 to 15 (Cd) and 0 to 7 (Cf), as shown in Fig. 3 (using his notation of CD 90 and CD 0 for Cd and Cf). His numerical data are reinterpreted later in this paper. 2.2. Geotechnical approach Fig. 1. Schematic of debris flow interacting with pipeline.
density and velocity of the flow, using drag coefficients (DNV, 1976). For steady state flow acting on a fully submerged pipeline, the two terms of most interest are those for normal loading (with velocity component, vn), expressed in terms of a drag coefficient, Cd, and for axial friction (with velocity component, va), based on a friction coefficient, Cf. The forces per unit length of pipeline of diameter, D are expressed as 1 2 rvn D Fn ¼ Cd 2 1 2 rva pD ð1Þ Fa ¼ Cf 2 where r is the density of the flowing material. Values of the drag coefficient are suggested as Cd ¼1.2 at Reynolds numbers, Re r4 105 and Cd ¼0.7 for Re Z6 105 with linear interpolation on a log–log scale between these limits; the corresponding friction coefficient is expressed as Cf ¼
and
0:075
ð2Þ
ðlog10 Re 2Þ2
For turbidity currents, DNV (1976) suggests that the Reynolds number should be based on that for water, expressed as Re ¼vnD/n, with the kinematic viscosity, n ¼1.6 10 6 m2/s. A similar formulation has been suggested for debris flows (Zakeri, 2009), with drag and friction coefficients expressed as power law functions of an equivalent Reynolds number for nonNewtonian fluids. The non-Newtonian Reynolds number is also referred to as a Johnson number, giving the relative magnitudes of a drag-related resistance, or stagnation pressure, to the mobilised shear stress, t, in the material, and is expressed as Re,non-Newtonian ¼
v2n
r t
ð3Þ
Zakeri (2009) reports results from both physical and numerical modelling. The latter was carried out using the CFD code, ANSYS CFX Version 11.0, considering pipelines placed at angles of 0, 30, 45, 60 and 901 to the flow direction (see Fig. 2). He provided a table of results for the normal and axial forces deduced on a short, typically 2 diameter, segment in the central part of the pipeline. Normal and frictional coefficients were obtained from the computed normal and frictional forces per unit area projected in the planes normal and parallel to the flow, treating all the data together, regardless of the flow angle. The coefficients were based on the full input flow velocity (i.e., not components of the flow velocity). The best fit expressions determined in terms of the nonNewtonian Reynolds numbers (also based on the full velocity, and with a mobilised shear stress, t, based on a strain rate of v/D) were given by C d ¼ 1:2þ
We propose an alternative approach for debris flows, which is more aligned with geotechnical methods for characterising the properties of seabed sediments and for assessing the capacity of foundation elements. This approach is to express the normal and frictional forces on the pipe in terms of a shear strength, su, adjusted for the high strain rates involved, and using appropriate bearing factors for normal load and an interface friction ratio for frictional loads, assuming full contact of the debris around the circumference of the pipe. For normal loading, it is also necessary to consider a conventional drag coefficient, which would become important for debris flow at high Johnson numbers (i.e., high ratios of rv2n/su). (Note that the nomenclature, su, is used for the shear strength to emphasise the geotechnical approach, although this is essentially identical to the mobilised shear stress, t, used to evaluate the non-Newtonian Reynolds number.) The normal and frictional forces per unit length of pipe are expressed as 1 2 Fn ¼ Cd rvn D þN p su,nom D ð6Þ 2
27:5 Re1:1 non-Newtonian
ð4Þ
F a ¼ f a su,nom pD
ð7Þ
where Np is the bearing factor for the force normal to the pipe, while fa is a frictional coefficient for the force parallel to the pipe. The relevant shear strengths are adjusted for the shear strain rate, taken as v/D (with v the total velocity of the debris flow). Thus if the shear strength is found to vary according to a power law relationship, the nominal shear strength is expressed as v=D b su,nom ¼ su,ref ð8Þ g_ ref where su,ref is a reference shear strength at a strain rate of g_ ref and v is the flow velocity. Dynamic finite element analyses using the techniques described by Wang et al. (2011) of flow normal to a cylinder, for rate independent soil under conditions where a gap is allowed to form behind the cylinder (in the wake of the flow) have been performed. These show a linear dependence of the normal force on v2n that is consistent with a drag coefficient of Cd ¼0.5. Complementary analyses in zero density soil, but with different forms of rate-dependent shear strength, have shown that it is sufficient to base su,nom on the nominal strain rate, v/D, provided that the coefficient b in Eq. (8) is less than about 0.3 (Zhu and Randolph, 2012). This restriction on b covers typical soil conditions under the range of water contents relevant for debris flows (Boukpeti et al., 2012). The bearing factor Np is then essentially independent of velocity, and consistent with values derived from rate-independent plasticity theory, ranging from 9.1 for a fully smooth pipe to 11.9 for a fully rough pipe (Randolph and Houlsby, 1984; Martin and Randolph, 2006). Based on these observations, the influence of strain rate on flow pattern can be neglected if the strength su relevant to the strain rate of v/D is adopted in Eq. (6), with the result that the
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M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
Fig. 2. Plan view of ANSYS CFX analyses (Zakeri, 2009).
values of bearing factor Np and drag factor Cd become independent of the flow velocity, v. For axial loading (i.e., flow parallel to the pipe), the friction force is expressed by Eq. (7), with the nominal shear strength given by Eq. (8). The coefficient fa reflects the enhanced shear strain rate immediately adjacent to the pipe (assuming a rough pipe-soil interface), in excess of the nominal strain rate of v/D. For a power law model, the coefficient fa may be derived analytically, and is given by (Einav & Randolph, 2006) b 1 1 ð9Þ fa ¼ 2
b
The value of fa lies in the range 1.2 to 1.4 for typical values of b between 0.05 and 0.15. Some comment is appropriate here regarding evaluation of the reference shear strength, su,ref, and the strain rate dependency of shear strength using the power law relationship in Eq. (6), or other forms of relationship such as Herschel–Bulkley (Huang and Garcia, 1998). Recent studies (Jeong et al., 2009; Boukpeti et al., 2009, 2012) have shown that the reference shear strength can be expressed as a function of water content with parameters that become increasingly independent of material type as the water content increases (and strength decreases). They also found that the rate parameter, b, showed no systematic variation with water content. In principle, therefore, the shear strength of the debris flow may be established from a known starting point (as provided by site investigation data from potential landslide zones) and a means of assessing the evolving water content with runout distance.
3. Re-analysis of data from Zakeri (2009) Given that Reynolds number is proportional to v2, and that the drag coefficients proposed by Zakeri (2009) are close to varying
inversely with Reynolds number, it is instructive to replot his numerical data as a function of the inverse of Reynolds number. The results for the extreme conditions of flow either normal or parallel to the pipe are shown in Fig. 4. The data are closely fitted by linear relationships. Substituting these fits into Eq. (1), making use of Eq. (3) (and replacing the mobilised shear stress, t, by the nominal shear strength, su,nom based on the flow velocity) leads to F n ¼ 0:4 12 rv2n þ13su,nom D ð10Þ F a ¼ 1:5su,nom pD These relationships are essentially identical to Eqs. (6) and (7), with coefficients of Cd ¼0.4, Np ¼13 and fa ¼1.5. These compare with the theoretical values of Np ¼11.9 for a rough pipe and fa ¼1.4 (using Eq. (9) with power law exponent, b, in the range 0.11–0.14 as adopted by Zakeri, 2009). The drag coefficient of Cd ¼0.4 compares with the value of 0.5 estimated from numerical modelling of Wang et al. (2011) at high values of rv2n/su. The results for flow at different angles to the pipe have been replotted similarly in Fig. 5, taking the relevant components of velocity (normal or parallel to the pipe) to evaluate an equivalent drag coefficients and Reynolds numbers. Two adjustments have been made: (1) For convenience the horizontal axes are plotted as 2/Re rather than 1/Re, so that the gradients represent equivalent Np and fa coefficients directly. (2) The drag coefficients for flow normal to the axis have been shifted by the limiting drag coefficient for very high Reynolds number (i.e., the intercept for 2/Re ¼ 0); it was found that the best overall adjustment was by a value of 0.6/siny, where y is the flow direction relative to the pipe axis. Although the degree of scatter increases slightly, linear relationships through the origin are then obtained for both components of
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
35
Drag Coefficient Normal to Pipe Axis, CD - 90
20 2007 Inv. - Num. Analyses 2007 Inv. - Exp. Data Angle of Attack Analysis Proposed for Design Fit to Angle of Attack Analysis Data
16
12
8
4
0 1
10
100
Reynolds Number, Re non-Newtonian
Drag Coefficient Parallel to Pipe Axis, CD - 0
10 Angle of Attack Analysis Proposed for Design
8
6
Fig. 4. Drag coefficient data replotted against inverse of Reynolds number.
4 decreases in proportion to sin2y, while the drag coefficient only increases as 1/siny. The variation of bearing and friction factors with the angle of the flow relative to the pipeline axis may be interpreted in terms of a failure envelope, as shown in Fig. 6. The dashed curve shows a fit through the deduced numerical data points, expressed as
2
0
1
10
100
Reynolds Number, Re non-Newtonian Fig. 3. Drag coefficients deduced from numerical analysis (Zakeri, 2009).
flow. For the extremes of normal and parallel flow, the resulting drag coefficient of 0.6 (for y ¼901), and Np and fa factors of 12.8 and 1.5 are close to the theoretical values of 11.9 and 1.4 as previously noted. For intermediate flow angles, the Np and fa values decrease relative to the values for the extremes relevant for each factor, as summarised in Table 1. For the flow component normal to the pipe, the apparent increase in drag coefficient (expressed as 0.6/siny) as the flow deviates from perpendicular to the pipe is surprising, and does not appear to have any particular logic. It should be noted, however, that the drag force for a given resultant velocity still decreases, since the normal component of velocity (squared)
fa f a,0
3
þ
Np Np,90
1
¼1
ð11Þ
where, consistent with the data, fa,0 has been taken as 1.54 and Np,90 has been taken as 12.8. Although the envelope provides a reasonable fit to the data points for 45 and 601, the data point for 301 lies well within the envelope, and indeed implies some concavity in the envelope. The reason for this is unclear, and would require further detailed numerical simulation for flow angles lying between 0 and 451 to resolve. A design envelope is proposed that is constrained to the theoretical limiting bearing factors of fa,0 ¼1.4 and Np,90 ¼ 11.9 (see full curve in Fig. 6). In addition to the failure envelope, a relationship is required to determine the position on this envelope that corresponds to a particular angle of flow, and hence provide the resulting bearing factors Np (and fa). The resultant force on the pipe, which is the vector sum of Fn and Fa, is not parallel to the flow direction. It is proposed that the bearing
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M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
Fig. 6. Failure envelope for varying flow angle relative to pipe axis.
Table 2 Input data for example application.
Table 1 Summary of bearing and axial resistance factors.
0 30 45 60 90
Values from numerical data of Zakeri (2009)
Pipeline diameter (m) Density of flow material (t/m3) Reference shear strength, su,ref (kPa) Reference shear strain rate, g_ ref (s 1) Power law coefficient, b
0.8 1.5 0.3 1 10 5 0.13
4. Illustrative example application Values from proposed design approach
Axial coefficient, fa
Axial Normal coefficient, coefficient, Np fa
Normal coefficient, Np
1.54 0.96 0.88 0.70 0
0 7.5 10.6 11.7 12.8
0 7.3 9.3 10.8 11.9
1.4 1.02 0.84 0.64 0
factor, Np, may be expressed as Np ¼ Np,90 ðsin yÞ0:7 ¼ 11:9ðsin yÞ0:7
Value
numerical data. Although most of the numerical data lie outside the proposed design curve, and as such would provide a more conservative assessment of interaction forces, the design curve is constrained by theoretical values (for the given boundary conditions assumed in the numerical analysis) for flow parallel and normal to the pipe. For those cases the numerical data slightly overestimates the interaction forces (as is common with displacement-based finite element computations). The inner proposed curve is therefore considered to be a reasonable basis for design.
Fig. 5. Replotted drag coefficient data for general flow conditions.
Flow angle relative to pipe axis (1)
Property
ð12Þ
with the corresponding value of fa then being found from Eq. (11). Eq. (12) is a purely empirical relationship, which appears to give reasonable predictions of the numerical data. The proposed failure envelope, together with spot values of the factors for the same angles as examined numerically by Zakeri (2009), are shown in Fig. 6 and the bearing and friction factors are also summarised in Table 1. The factors all lie within 10% of those deduced from the
An example application of the design approach derived above is provided here, the input data for which is summarised in Table 2. Relatively low water content material, with density 1.5 t/m3 and reference shear strength of 0.3 kPa is assumed to impact a pipeline of diameter 0.8 m with velocity of 2 m/s. Adopting a reference shear strain rate of 1 10 5 s 1 (corresponding to a typical laboratory shearing rate for soil of about 3%/h) and a power law coefficient of b ¼0.13, the nominal shear strength is calculated as su,nom ¼1.51 kPa. This gives a Reynolds (or Johnson) number, rv2/su,nom ¼4.0, suggesting that the geotechnical bearing resistance will dominate in comparison with the drag force. The resulting normal and frictional forces for varying angles of attack are shown in Fig. 7. The maximum (total) normal force is just under 16 kN/m (average pressure of 20 kPa across the pipe diameter), and comprises 90% geotechnical resistance and 10% inertial drag. The maximum frictional resistance is 34% of the maximum normal force, and corresponds to an average shear stress around the pipe circumference of 2.1 kPa. The normal component of force develops rapidly as the flow angle (relative to the pipeline axis) increases from zero. The angle of the resultant force reaches 451 for a flow angle of only 101 (Fig. 7(b)).
M.F. Randolph, D.J. White / Ocean Engineering 48 (2012) 32–37
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angled flow impacting a segment of pipe. The data have been reprocessed in order to evaluate bearing and frictional factors (from which the resulting forces may be evaluated) for flow at different angles to the pipe axis. Separately, drag coefficients, independent of the strength of the material, have been identified; these become increasingly important at non-Newtonian Reynolds numbers exceeding about 10. The design approach is expressed as a failure envelope that links the bearing and frictional factors, together with an algorithm that expresses the normal bearing factor as a function of the flow angle. An example application is included to illustrate how the different components of force may be evaluated for a particular debris flow or turbidity current impacting a pipeline at different angles.
Acknowledgements This work forms part of the activities of the Centre for Offshore Foundation Systems (COFS), established under the Australian Research Council’s Research Centres Program and now supported by Centre of Excellence funding from the State Government of Western Australia and by the Australian Research Council’s Federation and Future Fellowship programs. The research described here is part of a Joint Industry Project administered and supported by the Mineral and Energy Research Institute of Western Australia, and by BP, BHP Billiton, Chevron, Petrobras, Shell and Woodside. The financial support of all the participants is gratefully acknowledged. References
Fig. 7. Resulting force variations with flow angle for illustrative example.
5. Conclusions This paper has attempted to develop a more rational basis for assessing the impact forces of debris flows and turbidity currents on offshore pipelines. Any submarine landslide is likely to lead to mass transport of the slide material, with the strength and consistency of the material gradually changing from what may be classed as ‘soil’ to a fluid with suspended solids. Models for estimating impact forces have tended to be couched in a fluid mechanics framework throughout this material regime, with drag and frictional coefficients expressed as functions of Reynolds number and matched to experimental data. The proposed design approach is based on a single model for the nominal shear strength of the flow material, expressed as a power law function of a nominal strain rate that is linked to the flow velocity and pipe diameter. Analytical solutions for the geotechnical bearing resistance (for flow normal to the pipe) and frictional resistance (for flow parallel to the pipe) are used to constrain data from an extensive numerical study by Zakeri (2009) that investigated normal and frictional forces arising from
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