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Break-out resistance of offshore pipelines buried in inclined clayey seabed
Applied Ocean Research 94 (2020) 102007
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Break-out resistance of offshore pipelines buried in inclined clayey seabed a
a
b
b,⁎
Deqiong Kong , Jingshan Zhu , Leiye Wu , Bin Zhu a b
T
Center for Hypergravity Experimental and Interdisciplinary Research, Zhejiang University, Hangzhou 310058, China College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
ARTICLE INFO
ABSTRACT
Keywords: Bearing capacity Pipes and Pipelines Finite element limit analysis Clay
The application of marine pipelines prevails in offshore developments for the transportation of oil and gas products from offshore to onshore, and some of them are inevitably buried in continental slopes with moderate inclinations (mostly below 6°). Regrettably, very limited attention has been paid to the bearing capacity of these pipelines, which generates the very motivation of this study. A total number of 169 200 separate finite element limit analyses (FELA) were conducted on rigid plane-strain pipe sections in inclined clayey seabed, using a sound FELA code OxLim. The breakout resistances of the pipe along all possible directions are explored to determine the most vulnerable plane where a transverse displacement could most likely take place, and accordingly the minimum breakout resistance. Before reaching a full-flow state, this resistance was found to be considerably lower, mostly by 15% to 35%, than the uplift resistance. Design charts that can be immediately used for the modification of existing database are then provided, which promotes the safe design of pipelines regarding transverse displacement or buckling. The potential non-conservation of the most recent design approaches is highlighted through comparisons.
1. Introduction The development of marine energy resources is an important way to alleviate energy shortages worldwide, which involves the widely usage of submarine pipelines for the transportation of oil and gas products. Considering the long distance of transportation from offshore to onshore, some of these pipelines are inevitably buried in continental slopes that link the continental shelves and the deep sea plains, typically characterized by gentle inclination angles ranging from 3° to 6° at water depths from 200 m to 1500 m. Though marine pipelines in comparatively deep waters are laid directly on the seabed, there are possibilities that these pipelines can also be buried for the sake of higher on-bottom stability, subject to the availability of proper facilities. For instance, the Soil Machine Dynamics LtD built a deep-water plough (MD3-160) in the UK in 2017, which is capable of working at water depth of 2000 m and ploughing trenches of depth up to 3 m. As the exploitation of marine oil and gas resources continually moves to deeper sea, pipelines buried in inclined seabed will not be rare, and the stability of them will become a major concern in design. Offshore pipelines are prone to displace or buckle transversely due to the internal high pressure and the thermal-induced high axial loads, which is a common failure mode in practical engineering. As a result, the bearing capacity of pipelines in soil, especially in the upward
⁎
direction where the constraints are assumed to be the smallest, has attracted a long history of research interest. Solutions based on classic plasticity and numerical approaches have been widely pursued (Randolph and Houlsby [30]; [1, 28, 29]; Martin and Randolph [22]; [18, 34]), with the effects of unit weight, burial depth, interface roughness and tensile capacity and other relevant factors being investigated. For more complex soil properties, model tests have also been undertaken (Rowe and Davis [33]; [3, 5]) to provide useful insights. Recently, V–H failure envelops have also been obtained to provide more comprehensive understanding of the bearing behaviour of pipelines subject to the combination of horizontal and vertical loads [24–26, 37]. Therefore, there is little doubt that the current design approaches of pipelines in regard to on-bottom stability have rigorous and sound theoretical basis. However, little research evidence has been found on the bearing capacity of pipelines buried in inclined seabed, which deserves proper attention since the weakening effect of slope angles on the uplift resistance of shallow foundations has been highlighted in previous studies such as Kumar [15] and Banza-Zurita et al. [2]. Liu [17] studied the uplift bearing capacity characteristics of pipelines in a sandy slope by means of centrifuge tests, and found that the failing surface deflected towards the slope toe, exhibiting an asymmetric pattern. Gao et al. [10] discussed the on-bottom stability of a partially buried pipeline on sandy
Corresponding author. E-mail addresses: [email protected] (D. Kong), [email protected] (J. Zhu), [email protected] (L. Wu), [email protected] (B. Zhu).
https://doi.org/10.1016/j.apor.2019.102007 Received 28 July 2019; Received in revised form 18 October 2019; Accepted 25 November 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
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Notation D H k Pu Pu, b Pu, u Pu,local Pu,global
su T V w α β γ′ θ
pipe diameter, m horizontal load shear strength gradient, kPa/m breakout resistance calculated as the normal of (V, H) minimum breakout resistance uplift resistance undrained bearing capacity under a local failure mode undrained bearing capacity under a global failure mode
undrained shear strength, kPa tensile capacity at pipe–soil interface vertical load pipe embedment, m roughness factor of pipe–soil interface inclined angle of seabed submerged unit weight, kN/m3 angle between the most vulnerable plane and the upright plane of seabed
The soil was modeled as a rigid-plastic material obeying the Tresca yield criterion, with constant su or linearly varying undrained strength profile su = kz, where k is the strength gradient and z is the soil depth. Strength softening of the soil caused by installation was not accounted for, as the bearing behaviour of the pipelines was assessed a period of this process and strength recovery has been gained. The submerged soil unit weight was given in normalized forms γ′D/su and γ′/k for the two distinct profiles. Two extreme cases of pipe–soil interface roughness, namely completely smooth (α = 0) and completely rough (α = 1), were considered, with the former representing a unroughened pipe section and the latter coated with concrete or epoxy resin for anti-corrosion purpose. Regarding the tensile capacity T on the interface, unbonded conditions (T = 0) were assumed by a multitude of studies (Houlsby and Puzrin [11]; [24,25]), stating that the voids beneath the pipeline after trenching may prevent it from developing suctions, and this assumption has been adopted in the DNV design code [6]. However, it is also acknowledged that the high water pressure and the weight of the pipeline may extrude the water beneath its invert and promote reconsolidation. The breakout response associated with the development of tension / suction was reported even for pipelines directly laid on the seabed (e.g. [8,36]). As a result, analyses assuming bonded conditions (T = ∞) on the pipe–soil interface were widely conducted as well [3,4,24,26]. In this study, the values of T were determined as either zero or unlimited to consider both scenarios. It is noteworthy that in reality when a bonded condition is present, full mobilization of shear stress along the interface may not be likely, and the assumption made here is certainly a simplification. For each configuration of the model, the pipe was subjected to displacement-controlled probes at an interval of 2° to obtain both UB
slope and derived a pipe–soil interaction model that can yield predictions in good agreement with model test results. Apart from these limited work on sand, no research effort has been directed to clay, let alone the development of adequate assessment models. Above all, the existing studies are insufficient to guide the design of marine pipelines in inclined seabed concerning the uplift bearing capacity. Another important issue worth noting is the fact that the thermal-induced transverse displacement or buckling should take place along the most vulnerable direction, which does not necessarily coincide with the upright direction or the normal direction of the inclined seabed surface, as will be discussed later. Even in flat seabed, this issue has not been systematically investigated despite the derivation of abundant V-H failure envelopes as reported in Aubeny et al. [1], Merifield et al. [26], Martin and White [24], etc. To address these gaps, this paper presents an extensive numerical study on the bearing capacity of submarine pipelines buried in inclined clayey seabeds with different angles. The bearing capacities along all possible directions are carefully examined and compared, with reference to differing failure mechanisms, in an attempt to determine the minimum resistance against transverse displacement. Design charts to quantify the effect of inclination angle on the minimum breakout resistance of pipelines are then presented, taking into account the influence of various soil properties. Critical review of current industrystandard procedures is also presented to highlight the un-conservative shortcomings. 2. Methodology 2.1. Finite element limit analysis (FELA) code OxLim All analyses were performed using the FELA code OxLim [19–21], which obtains rigorous upper- and low-bound plastic solutions of complex geotechnical problems through implementing them as second order cone optimisation problems and solving them using MOSEK [27]. Discontinuities among the discrete elements were allowed to provide more degrees of freedom of the model and thus improve the computing accuracy. An adoptive remeshing technique introduced in Martin [23] was adopted for mesh generation and refinement, to obtain closelybracketed upper-bound (UB) and lower-bound (LB) solutions, with an allowed bracketing discrepancy of 1% in the present study. 2.2. Numerical details The object investigated in this paper is a plane-strain pipeline section fully buried in inclined soft clay seabed. The model configuration and the notations are illustrated in Fig. 1. The pipeline section was assumed to be rigid body with a diameter D, comprised of a 180-segments polygon, with its invert being buried at a depth of w (i.e. 1.5D to 6.0D at an interval of 0.5D). The width and depth (at the middle line) of the soil were 16D and 12D, respectively. Fixed boundaries were applied to the bottom and two sides of the soil domain, and free boundary conditions were assumed at the soil surface with an inclination angle of β, taking values of 0° (i.e. for a flat seabed), 1°, 3°, 5°, 7° and 10°.
Fig. 1. Configuration of the numerical model. 2
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Fig. 3. Breakout resistances against probing angles in inclined seabed. From inside to outside, embedment w/D = 1.5 to 6 at intervals of 0.5. α = 1, γ′D/ su = 1.
Fig. 2. Breakout resistances against probing angles in inclined seabed. From inside to outside, embedment w/D = 1.5 to 6 at intervals of 0.5. α = 1, β = 3°.
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Fig. 4. Failure mechanisms corresponding to minimum breakout resistance Pu,b. T = 0, α = 1, w/D = 3.0, β = 3°.
Fig. 5. Comparison of failure mechanisms corresponding to Pu,u and Pu,b. T = ∞, α = 1, γ′D/su = 1, w/D = 3.0.
and LB failure loads at a full range of possible directions. Since the vector defined by (V, H) does not coincide with the displacement direction, where V and H are the load components along the vertical and horizontal planes respectively, the results presented in the form of V-H failure envelopes fail to provide sufficient information on the most vulnerable plane (MVP) along which transverse displacement is most likely to take place. Hence, the norm of (V, H), denoted by Pu, is plotted against the probing angle in polar coordinates, enabling the detection of the MVP as well as the minimum breakout resistance. Nonetheless, for the sake of simplicity, the term “failure envelope” is still used.
3. Results and discussions 3.1. Breakout resistances at differing probing directions Figs. 2 and 3 show the development of failure envelopes with burial depth of a rough pipe that can sustain zero or unlimited tension, with a combination of γ′D/su and β values being considered. Among all data points of each failure envelope, the one that has the shortest distance to the origin (i.e. denoting the smallest breakout load) is highlighted. Besides, the direction determined by the origin and this particular point coincides with the most vulnerable plane (MVP). When T = 0, all MVPs almost coincide with the outer normal planes (ONPs) of the seabed. Exceptions are achieved when the pipe is buried 4
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Fig. 6. Variation of Pu,u and Pu,b with embedment for totally rough pipelines.
deep enough to develop a full-flow state, i.e., the failure envelopes start to coincide and the MVPs shift to the upright plane, and in the cases with high values of γ′D/su and β (see Fig. 2(g)) that may induce a slope failure. The latter is confirmed in Fig. 4, where distinct failure mechanisms corresponding to breakout along MVPs for cases with γ′D/ su = 1 and 3 are illustrated. Apart from that, the failure mechanisms along MVPs and ONPs are almost identical in the unbonded cases, although not presented here. Through comparisons among the left five subplots in Fig. 2, it can be seen that the soil weight helps promote the development of full-flow mechanisms that are associated with higher resistances than global ones. This is evidenced by the considerably smaller sizes of failure envelopes in the weightless soil cases (see Fig. 2(a)) and the coincidence of failure envelopes at shallower depths as γ′D/su increases. Similar issue has been discussed in Martin and White [24]. In general, β appears to have limited effect on the failure envelopes for the default γ′D/su examined here, as shown in the left five subplots in Fig. 3. When T = ∞, the MVPs divert significantly from the ONPs for all cases examined, before the pipe reaching a full-flow state. This illustrates significantly more pronounced influence of the inclined soil surface profile on the pipe-soil response than in the unbonded cases. The bonded condition is also found to help facilitate the full-flow mechanism, as the failure envelopes coincide at much shallower depth than their counterparts in the unboned cases (compare the left and right halves in both Figs. 2 and 3). The angles between MVPs and ONPs show an overall decreasing trend with increasing γ′D/su (see Fig. 2(b), (d), (f), (h) and (j)), but exhibiting little dependence on the slope angles considered here (see Figs. 3(b), (d), (f), (h) and (j)). The most striking observation is that even in the flat seabed case, there is a huge discrepancy between MVPs and ONPs, implying the inadequacy of current design approach in which the transverse buckling was by default considered to occur in the upright direction. This issue is further explored
in Fig. 5, where the failure mechanisms corresponding to breakout along MVPs and ONPs are compared. The full-flow failure mechanism developed by the pipe when loaded uprightly can transition to a global one if loaded along the MVP, which denotes lower breakout resistance. The results of smooth pipes basically show similar patterns with those illustrated in Figs. 2–5 and the details are not discussed here. 3.2. Breakout resistances along MVPs Figs. 6 and 7 plot the breakout resistances evaluated along MVPs (denoted by Pu,b) in reference to the uplift resistances (denoted by Pu,u) for totally rough and smooth pipes respectively. When T = 0, the difference between the two is indiscernible for soils with low γ′D/su (see Figs. 6(a)–(b) and 7(a)–(b)), but becomes notable at larger values (see Figs. 6 (c) and 7(c))) . The inclination β has very little impact on Pu,u whilst its undermining effect on Pu,b is more pronounced, which can only be noted in soil cases with high γ′D/su. When bonded conditions (T = ∞) are assumed, β still has little impact on Pu,u but its influence on Pu,b becomes significantly more notable than in the unbonded cases (e.g. comparing subplots (b) and (e) in Fig. 6 and subplots (b) and (e) in Fig. 7). More importantly, remarkable discrepancies between Pu,b and Pu,u can be observed before a full-flow state has been achieved, for all parameters examined here, and this trend can be exacerbated by increasing γ′D/su and β. Another observation can be made is that the inclination of the seabed does not only reduce Pu,b, but also postpone the occurrence of a full-flow mechanism that can be inferred from the coincidences of the curves. Two important implications can be drawn here: (i) the assessment of breakout resistances based on uplift bearing capacities may greatly overestimate the on-bottom stability of pipelines buried in soft clay, and (ii) the pipelines may need to be embedded deeper than predicted using previous models to reach an optimal burial depth beyond which no / little 5
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Fig. 7. Variation of Pu,u and Pu,b with embedment for totally smooth pipelines.
improvement on breakout resistances can be gained. The uplift resistances calculated using the approach recommend by the newly issued design code DNV RP-F114 [7] are provided for comparison, in which the tensile capacity on the pipe–soil interface was assumed to be zero. Inclination of the seabed was not accounted for in it and thus only flat seabed was considered. The uplift resistances were taken as the smaller values calculated from the following
Pu,local = Nc · s¯u·D
·Ap
Pu,global =
· D 2 ·(1/2
·H · D +
present numerical results than DNV [6] and Maitra et al. [18], even considering its simple form. A striking observation is that when unlimited tension was sustained, the numerical results could still be lower than the DNV predictions (see Figs. 6(e)–(f)). To summarize, the nonconservation of the DNV method is more notable when breakout of pipelines along MVPs is considered or an inclined seabed is involved, hence implying potential safety risks in design. To facilitate application of the presented numerical results, the ratio of Pu,b to Pu,u is introduced to account for the influence of the inclination of soil surface and the fact that loads lower than Pu,u could possibly induce transverse displacement of buried pipelines. Note that all values of Pu,u, even in inclined seabeds, were determined at β = 0°, owing to the facts that mostly β does not affect Pu,u and, more importantly, a multitude of solutions and data have already been developed regarding the uplift bearing capacity of buried pipelines [1,3,24,29]. The calculated Pu,b/Pu,u ratios corresponding to various combinations of parameters are provided in Table 1, and can be readily used to adjust the existing database. Since the calculated ratios for unboned interfaces are mostly close to one, only the results for bonded cases are provided. Due to the computing errors, some values slightly higher than one (with difference less than 0.02%) were obtained and have been modified as one. Fig. 8 explores the relationship between θ (the deflection angle from ONP to MVP) and γ′D/su for the cases with bonded pipe–soil interface. For the sake of simplicity, a flat surface profile is considered here and greater γ′D/su than those presented in Figs. 6 and 7 can thus be examined. A general decreasing trend can be seen for the totally rough and smooth pipes with burial depthless than 4D and 3D respectively, owing to the fact that soils with higher unit weight are more likely to trigger a full-flow failure mechanism. For a typical case with γ′D/su of 4 (e.g. D = 0.5 m, γ′ = 6.5 kN/m3 and su = 0.8 kPa), the MVPs for pipelines with burial depth of less than 3D deflect approximately 30°
(1)
/8) + 2· s¯u (H + D /2)
(2)
where Pu,local is the undrained bearing capacity under a local failure mode and Pu,global is that under a global failure mode, u is the average soil strength and equals su for the uniform soil considered here, Ap is the cross-sectional area of soil above the pipe and H is the cover height from pipe crown to the mudline. The bearing capacity factor Nc, which corresponds to a full-flow failure mode, is in the range between 9 (totally smooth) and 12 (totally rough) provided by Randolph and Houlsby [30]. The above model denotes a significant improvement over the previous version DNV RP-F110 [6] that the soil buoyancy effect is addressed more rationally. The predictions using DNV RP-F110 and the model proposed by Maitra et al. [18] are plotted here for comparison as well. In Maitra et al. [18], a critical embedment wlim, beyond which no separation could occur even under unbonded conditions, is calculated first. Then the uplift resistances for pipes with embedment below or above this critical value are determined separately and the lower value is chosen. It can be seen from Figs. 6 and 7 that DNV RP-F114 [7] yields overestimated results for a number of situations examined here, especially for soils with low γ′D/su (see Figs. 6(a) and 7(a)), and the overestimation is generally more severe for smooth pipelines than for rough ones. However, the estimates from DNV [7] is generally closer to the 6
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0.8305 0.8469 0.9128 0.9989 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8678 0.8849 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
β = 5° 0.8086 0.7933 0.7760 0.8243 0.8854 0.9350 0.9731 1.0000 1.0000 0.9996
0.9059 0.9206 0.9972 1.0000 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000
β = 3° 0.8654 0.8611 0.8593 0.9266 0.9966 0.9997 1.0000 0.9997 1.0000 1.0000
0.9192 0.9356 0.9992 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
So far, only soils with uniformly distributed strength are investigated. This section provides some further details of the other extreme case with su = kz. Two surface profiles are considered here, a flat seabed and an inclined seabed with β of 3° (default value). For an inclined surface profile, it is not practical to assume a strength varying linearly with depth. As a result, when a strength gradient with z was adopted (i.e. k), a strength gradient with x was also set, taking the value of tanβk. This would generate a soil strength profile equal at the surface and vary linearly with the distance normal to it. Since the inclination angle is very small, the strength profile of the inclined seabed did not vary much from the flat one. As discussed before, there is no major difference between the totally smooth and rough pipes, and α was taken as 0.5 by default here. Similar observations with the analyses using uniform soils can be seen in Fig. 9. The MVPs divert largely from the upright plane when T = ∞. The results obtained from the inclined and flat seabeds are very close. Two typical failure mechanisms corresponding to breakout along ONP (the upright plane in this case), i.e., a full-flow or localised one, and along MVP, i.e., a global one, are illustrated in Fig. 10. Even in the cases with very high γ′/k, the pipe is still prone to displace almost along the horizontal plane, as shown in Fig. 9(d). The angles between MVPs and ONPs show much greater values than those in Fig. 8, though not presented for clarity. The results in cases with T = 0 are not discussed here because the MVPs are observed to be coincide with the upright plane. Fig. 11 compares minimum breakout resistance Pu,b and the uplift resistance Pu,u for the two surface profiles. Higher values of γ′/k do slightly reduce Pu,u due to soil buoyancy for both surface profiles while only reduce Pu,b in the inclined case. Again, the comparison between Fig. 11(a) and 11(b) shows that the inclination angle of seabed has very little effect on the uplift resistance Pu,u. The effect of normalized unit weight on the discrepancy between Pu,b and Pu,u observed here is less pronounced than in the uniform soil cases, even when considering the more deflection from the upright plane to the MVPs. This discrepancy only appears to diminish when a burial depth of 6D has been reached, thus indicating considerably greater effect on the optimal burial depth than that in the uniform soil cases (i.e. 2.5D and 3D for the totally smooth and rough pipes respectively).
0.8479 0.8631 0.9694 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8646 0.8879 0.9972 1.0000 1.0000 1.0000 0.9998 1.0000 1.0000 1.0000
β = 5° 0.8442 0.8379 0.8347 0.8950 0.9644 1.0000 1.0000 1.0000 1.0000 0.9997
0.8911 0.9109 0.9968 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8207 0.8407 0.9517 1.0000 0.9992 0.9993 1.0000 1.0000 1.0000 0.9997
γ′D/su = 2
0.8853 0.9027 0.9975 1.0000 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000
β = 3° 0.8731 0.8707 0.8726 0.9423 0.9998 0.9998 1.0000 0.9998 1.0001 1.0000
0.8285 0.8523 0.9697 1.0000 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 0.8331 0.8634 0.9794 1.0000 1.0000 1.0000 0.9998 1.0000 0.9998 1.0000 0.8415 0.8675 0.9899 0.9997 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 0.7770 0.8084 0.9270 0.9788 0.9984 0.9989 0.9990 0.9989 0.9992 0.9990
γ′D/su = 1
0.8462 0.8701 0.9933 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
β = 1° 0.8993 0.8993 0.9024 0.9779 0.9998 1.0000 1.0000 0.9998 1.0000 1.0000 β = 0° 0.9112 0.9125 0.9160 0.9911 1.0000 0.9997 1.0000 0.9998 1.0000 1.0000 β = 7° 0.8322 0.8283 0.8307 0.8937 0.9715 0.9998 1.0000 1.0000 1.0000 0.9997 β = 5° 0.8443 0.8434 0.8480 0.9147 0.9938 1.0000 1.0000 1.0000 1.0000 0.9998 β = 3° 0.8558 0.8568 0.8627 0.9334 1.0000 1.0000 1.0000 1.0003 1.0000 1.0000 β = 1° 0.8668 0.8693 0.8766 0.9482 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 β = 0° 0.8721 0.8745 0.8810 0.9536 1.0000 0.9998 1.0000 0.9998 1.0000 1.0000
0.7777 0.8108 0.9291 0.9862 0.9989 0.9993 0.9992 0.9990 0.9994 0.9990
β = 7° 0.8073 0.8163 0.8298 0.8974 0.9783 0.9935 0.9993 0.9997 0.9995 0.9995
0.7764 0.8111 0.9271 0.9945 0.9987 0.9993 0.9992 0.9992 0.9988 1.0000
β = 5° 0.8092 0.8180 0.8309 0.8979 0.9796 0.9961 0.9994 0.9997 0.9994 0.9997
4. Conclusions This paper presents a programme of finite element limit analyses on the ultimate bearing capacity of a rigid plane strain pipe section buried in inclined clayey seabed. The main findings are summarised as follows.
γ′D/su = 0
0.7767 0.8093 0.9274 0.9861 0.9992 0.9993 0.9991 0.9990 0.9993 0.9991
smooth
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0.7783 0.8107 0.9271 0.9934 0.9991 0.9995 0.9994 0.9983 0.9993 0.9993
β = 3° 0.8088 0.8185 0.8310 0.8999 0.9802 0.9993 0.9993 1.0000 0.9995 0.9996 β = 1° 0.8102 0.8190 0.8331 0.9008 0.9806 0.9907 0.9995 1.0000 0.9994 0.9995 β = 0° 0.8094 0.8196 0.8326 0.9011 0.9809 0.9863 0.9995 0.9998 0.9995 0.9996 w/D 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
3.4. Effect of strength gradient
0.8291 0.8414 0.9171 0.9953 0.9994 0.9993 1.0000 1.0000 1.0000 0.9997
γ′D/su = 3
β = 1° 0.9130 0.9134 0.9155 0.9932 0.9997 1.0000 0.9998 0.9998 1.0000 0.9998 β = 0° 0.9349 0.9349 0.9374 1.0000 0.9998 0.9996 1.0000 0.9997 1.0000 1.0000 β = 7° 0.8132 0.7993 0.7870 0.8312 0.8855 0.9286 0.9622 0.9906 1.0000 0.9997
γ′D/su = 3 γ′D/su = 2 γ′D/su = 1 γ′D/su = 0 rough
Table 1 Breakout resistance ratio Pu,b/Pu,u,β=0 (T = ∞).
from the upright plane. This observation provides an interesting perspective to interpret some pipeline failures found in the Pinghu Oil & Gas Field, where the fractures always took place at the 5 o'clock direction (Prof. Xianghua Lai, personal communication). Also of interest to note is the sudden drop of θ in the simulations with w/D = 3.0. In this particular case, when γ′D/su increases from 3 to 4, the MVP that moderately deviates from the upright direction shifts to the upright plane. Also, the soil failure mechanism transitions from a global one to a localised / full-flow one, though not illustrated here. Such a transition is absent in analyses with deeper embedment (w/D ≥ 3.5) or takes place more gradually in cases with shallower embedment (w/D ≤ 2.5).
0.7809 0.7750 0.7834 0.8327 0.8765 0.8567 0.8458 0.8499 0.8805 0.9276
β = 7° 0.7207 0.6818 0.6554 0.6338 0.6969 0.6742 0.6576 0.6622 0.6834 0.7244
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(1) The most vulnerable plane (MVP) along which a transverse displacement of pipeline could most likely take place diverts greatly from the outer normal plane (ONP) of the soil surface. A full-flow failure mechanism developed by an uplift force can easily shift to a global one when the pipe was loaded along the MVP. This carries an important implication that the minimum breakout resistance Pu,b of 7
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Fig. 8. Angle between the most vulnerable plane and the upright plane. T = ∞, β = 0°.
Fig. 9. Breakout resistance against probing angle in seabeds with linearly varying strength (T = ∞). Subplots (a)-(d) show results with β = 0° and (e)-(h) show results with β = 3°.
a pipeline is greatly overestimated by the assessment based on uplift bearing capacities, and an overestimation of approximately 15%-35% can be observed even for soils with moderate unit weight and inclined angle. (2) The inclination angle β of the soil surface has limited effect on the uplift bearing capacity Pu,u of a pipeline, but this effect on Pu,b is notable. In addition, the existence of inclination angles also postpones the occurrence of a full-flow mechanism by 0.5-1D, suggesting deeper optimal burial depth beyond which little improvement on bearing capacities can be gained. (3) The DNV RP-F114 approach [7] yields moderate overestimation of the breakout resistance regarding transverse displacement or buckling (along MVPs), but the predictions are in general closer to the present numerical results than that from other methods. It is worth noting that the non-conservation of the DNV approach, which was developed based on unbonded interface assumption, exists even in some cases with bonded interfaces. (4) Design charts are developed to modify the database on uplift resistance of pipelines to account for the factors described above,
which are expected to benefit the safe design of marine pipelines regarding on-bottom stability. For all analyses presented, the redistribution and softening of soil strength caused by the pipeline installation process was not accounted for, which inevitably constitutes a limitation of the present study. Progress on this issue may be achieved through further exploration of a number of recently developed pipeline-soil interaction models (e.g. [13, 35], [14]; [32]). Moreover, only total stress analyses were conducted in this paper, without any attempt to look into the pore pressure response. Some recent work (e.g. [12, 16]; Randolph et al. [31]; [9]) could help shed some interesting light on this issue. Data availability statement All data generated or used during the study are available from the corresponding author by request.
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Fig. 10. Comparison of failure mechanisms corresponding to Pu,u and Pu,b for pipes buried in seabed with linearly varying strength (T = ∞). Subplots (a) and (b) show results with β = 0°, and (c) and (d) show results with β = 3°.
Fig. 11. Variation of Pu,u and Pu,b with embedment of pipelines buried in seabeds with linearly varying strength. T = ∞, γ′ /k = 1. (a) β = 0°, (b) β = 3°.
Declaration of Competing Interest
acknowledge the fund from the National Natural Science Foundation of China (research grant: 51809232) and the Zhejiang Provincial Natural Science Foundation of China (research grant: LCD19E090001).
The authors declar that they have no conflict of interest related to this project. Acknowledgement
Supplementary materials
The authors gratefully thank Professor Chris Martin at the University of Oxford for providing the FELA code OxLim, and
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apor.2019.102007. 9
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References
(2006) 604–634. [20] A. Makrodimopoulos, C.M. Martin, Upper bound limit analysis using simplex strain elements and second-order cone programming, Int. J. Numer. Anal. Methods Geomech. 31 (6) (2007) 835–865. [21] A. Makrodimopoulos, C.M. Martin, Upper bound limit analysis using discontinuous quadratic displacement fields, Commun. Numer. Methods Eng. 24 (11) (2008) 911–927. [22] C.M. Martin, M.F. Randolph, Upper bound analysis of lateral pile capacity in cohesive soil, Géotechnique 56 (2) (2006) 141–145. [23] C.M. Martin, The use of adaptive finite-element limit analysis to reveal slip-line fields, Géotech. Lett. 1 (4–6) (2011) 23–29. [24] C.M. Martin, D.J. White, Limit analysis of the undrained bearing capacity of offshore pipelines, Géotechnique 62 (9) (2012) 847–863. [25] R.S. Merifield, D.J White, M.F. Randolph, The effect of pipe–soil interface conditions on the undrained breakout resistance of partially-embedded pipelines, Proceedings of the Twelfth International Conference on Computer Methods and Advances in Geomechanics, 2008, pp. 4249–4256. [26] R. Merifield, D.J. White, M.F. Randolph, The ultimate undrained resistance of partially embedded pipelines, Géotechnique 58 (6) (2008) 461–470. [27] MOSEK ApS (2010). The MOSEK optimization tools manual, Version 5. Available online atwww.mosek.com. [28] J.D. Murff, D.A. Wagner, M.F. Randolph, Pipe penetration in cohesive soil, Géotechnique 39 (2) (1989) 213–229. [29] A.C. Palmer, C.P. Ellinas, D.M. Richards, J. Guijt, Design of pipelines against upheaval buckling, Proceedings of the Offshore Technology Conference, Houston, 1990 Paper OTC 6335. [30] M.F. Randolph, G.T. Houlsby, The limiting pressure on a circular pile loaded laterally in cohesive soil, Géotechnique 34 (4) (1984) 613–623. [31] M.F. Randolph, C. Gaudin, S.M. Gourvenec, D.J. White, N. Boylan, M.J. Cassidy, Recent advances in offshore geotechnics for deep water oil and gas developments, Ocean Eng. 38 (7) (2011) 818–834, https://doi.org/10.1016/j.oceaneng.2010.10.021. [32] A. Rismanchian, D.J. White, M.F. Randolph, C.M. Martin, Shear strength of soil berm during lateral buckling of subsea pipelines, Appl. Ocean Res. 90 (2019) (2019) 101864. [33] R.K. Rowe, E.H. Davis, Behaviour of anchor plates in clay, Géotechnique 32 (1) (1982) 9–23. [34] K. Roy, B. Hawlader, S. Kenny, I. Moore, Uplift failure mechanisms of pipes buried in dense sand, Int. J. Geomech. 18 (8) (2018) 04018087. [35] Y. Tian, M.J. Cassidy, C.K. Chang, Assessment of offshore pipelines using dynamic lateral stability analysis, Appl. Ocean Res. 50 (2015) (2015) 47–57. [36] D.J. White, H.R.C. Dingle, The mechanism of steady friction between seabed pipelines and clay soils, Géotechnique 61 (12) (2011) 1035–1041. [37] T. Zhou, Y.H. Tian, M.J. Cassidy, Effect of tension on the combined loading failure envelope of a pipeline on soft clay seabed, Int. J. Geomech. 18 (10) (2018) 4018131.
[1] C.P. Aubeny, H. Shi, J.D. Murff, Collapse load for cylinder embedded in trench in cohesive soil, Int. J, Geomech. 5 (4) (2005) 320–325. [2] E. Bazán-Zurita, D.M. Williams, J.K. Bledsoe, A.D Pugh, AEP 765 kV Transmission Line: Uplift Capacity of Shallow Foundations on Sloping Ground, Electrical Transmission Line and Substation Structures: Structural Reliability in a Changing World, 2006, pp. 215–226. [3] C.Y. Cheuk, W.A. Take, M.D. Bolton, J.R.M.S. Oliveira, Soil restraint on buckling oil and gas pipelines buried in lumpy clay fill, Eng. Struct. 29 (6) (2007) 973–982. [4] C.Y. Cheuk, D.J. White, M.D. Bolton, Large-scale modelling of soil–pipe interaction during large amplitude cyclic movements of partially embedded pipelines, Can. Geotech. J. 44 (8) (2007) 977–996. [5] C.Y. Cheuk, D.J. White, M.D. Bolton, Uplift mechanisms of pipes buried in sand, J. Geotech Geoenviron. Eng. 134 (2) (2008) 154–163. [6] Det Norske Veritas (DNV) (2007). Global buckling of submarine pipelines: structural design due to high temperature / high pressure. Recommended Practice F110. [7] Det Norske Veritas (DNV) (2017). Pipe–soil interaction for submarine pipelines. DNVGL Recommended Practice DNVGL–RP–Fl14. [8] H.R.C. Dingle, D.J. White, C. Gaudin, Mechanisms of pipe embedment and lateral breakout on soft clay, Can. Geotech. J. 45 (5) (2008) 636–652. [9] J. Fredsøe, Pipeline–seabed interaction, J. Waterw. Port Coast. Ocean Eng. 142 (6) (2016) 3116002, , https://doi.org/10.1061/(ASCE)WW.1943-5460.0000352. [10] F.P. Gao, N. Wang, J. Li, X.T. Han, Pipe–soil interaction model for current-induced pipeline instability on a sloping sandy seabed, Can. Geotech. J. 53 (11) (2016) 1822–1830. [11] G.T. Houlsby, A.M. Puzrin, The bearing capacity of a strip footing on clay under combined loading, Proc. R. Soc. A Math. Phys. Eng. Sci. 455 (1983) (1999) 893–916. [12] D.S. Jeng, L. Cheng, Wave-induced seabed instability around a buried pipeline in a poroelastic seabed, Ocean Eng. 27 (2) (2000) 127–146. [13] D. Kong, C. Martin, B. Byrne, Sequential limit analysis of pipe–soil interaction during large-amplitude cyclic lateral displacements, Géotechnique 68 (1) (2018) 64–75. [14] D. Kong, L. Feng, B. Zhu, Assessment model of pipe–soil interaction during large-amplitude lateral displacements for deep-water pipelines, Comput. Geotech. 117 (2020) 103220, , https://doi.org/10.1016/j.compgeo.2019.103220. [15] Jyant Kumar, Upper bound solution for pullout capacity of anchors on sandy slopes, Int. J. Numer. Anal. Methods Geomech. 21 (7) (1997) 477–484. [16] Z. Lin, Y. Guo, D.S. Jeng, C. Liao, N. Rey, An integrated numerical model for wave–soil–pipeline interactions, Coast. Eng. 108 (2016) 25–35. [17] J.W. Liu, Centrifuge modeling on uplift behavior of shallow buried pipe in inclined sand, PhD thesis Zhejiang University, 2017. [18] S. Maitra, S. Chatterjee, D. Choudhury, Generalized framework to predict undrained uplift capacity of buried offshore pipelines, Can. Geotech. J. 53 (11) (2016) 1841–1852. [19] A. Makrodimopoulos, C.M. Martin, Lower bound limit analysis of cohesive frictional materials using second-order cone programming, Int. J. Numer. Methods Eng. 66 (4)
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Break-out resistance of offshore pipelines buried in inclined clayey seabed a
a
b
b,⁎
Deqiong Kong , Jingshan Zhu , Leiye Wu , Bin Zhu a b
T
Center for Hypergravity Experimental and Interdisciplinary Research, Zhejiang University, Hangzhou 310058, China College of Civil Engineering and Architecture, Zhejiang University, Hangzhou 310058, China
ARTICLE INFO
ABSTRACT
Keywords: Bearing capacity Pipes and Pipelines Finite element limit analysis Clay
The application of marine pipelines prevails in offshore developments for the transportation of oil and gas products from offshore to onshore, and some of them are inevitably buried in continental slopes with moderate inclinations (mostly below 6°). Regrettably, very limited attention has been paid to the bearing capacity of these pipelines, which generates the very motivation of this study. A total number of 169 200 separate finite element limit analyses (FELA) were conducted on rigid plane-strain pipe sections in inclined clayey seabed, using a sound FELA code OxLim. The breakout resistances of the pipe along all possible directions are explored to determine the most vulnerable plane where a transverse displacement could most likely take place, and accordingly the minimum breakout resistance. Before reaching a full-flow state, this resistance was found to be considerably lower, mostly by 15% to 35%, than the uplift resistance. Design charts that can be immediately used for the modification of existing database are then provided, which promotes the safe design of pipelines regarding transverse displacement or buckling. The potential non-conservation of the most recent design approaches is highlighted through comparisons.
1. Introduction The development of marine energy resources is an important way to alleviate energy shortages worldwide, which involves the widely usage of submarine pipelines for the transportation of oil and gas products. Considering the long distance of transportation from offshore to onshore, some of these pipelines are inevitably buried in continental slopes that link the continental shelves and the deep sea plains, typically characterized by gentle inclination angles ranging from 3° to 6° at water depths from 200 m to 1500 m. Though marine pipelines in comparatively deep waters are laid directly on the seabed, there are possibilities that these pipelines can also be buried for the sake of higher on-bottom stability, subject to the availability of proper facilities. For instance, the Soil Machine Dynamics LtD built a deep-water plough (MD3-160) in the UK in 2017, which is capable of working at water depth of 2000 m and ploughing trenches of depth up to 3 m. As the exploitation of marine oil and gas resources continually moves to deeper sea, pipelines buried in inclined seabed will not be rare, and the stability of them will become a major concern in design. Offshore pipelines are prone to displace or buckle transversely due to the internal high pressure and the thermal-induced high axial loads, which is a common failure mode in practical engineering. As a result, the bearing capacity of pipelines in soil, especially in the upward
⁎
direction where the constraints are assumed to be the smallest, has attracted a long history of research interest. Solutions based on classic plasticity and numerical approaches have been widely pursued (Randolph and Houlsby [30]; [1, 28, 29]; Martin and Randolph [22]; [18, 34]), with the effects of unit weight, burial depth, interface roughness and tensile capacity and other relevant factors being investigated. For more complex soil properties, model tests have also been undertaken (Rowe and Davis [33]; [3, 5]) to provide useful insights. Recently, V–H failure envelops have also been obtained to provide more comprehensive understanding of the bearing behaviour of pipelines subject to the combination of horizontal and vertical loads [24–26, 37]. Therefore, there is little doubt that the current design approaches of pipelines in regard to on-bottom stability have rigorous and sound theoretical basis. However, little research evidence has been found on the bearing capacity of pipelines buried in inclined seabed, which deserves proper attention since the weakening effect of slope angles on the uplift resistance of shallow foundations has been highlighted in previous studies such as Kumar [15] and Banza-Zurita et al. [2]. Liu [17] studied the uplift bearing capacity characteristics of pipelines in a sandy slope by means of centrifuge tests, and found that the failing surface deflected towards the slope toe, exhibiting an asymmetric pattern. Gao et al. [10] discussed the on-bottom stability of a partially buried pipeline on sandy
Corresponding author. E-mail addresses: [email protected] (D. Kong), [email protected] (J. Zhu), [email protected] (L. Wu), [email protected] (B. Zhu).
https://doi.org/10.1016/j.apor.2019.102007 Received 28 July 2019; Received in revised form 18 October 2019; Accepted 25 November 2019 0141-1187/ © 2019 Elsevier Ltd. All rights reserved.
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Notation D H k Pu Pu, b Pu, u Pu,local Pu,global
su T V w α β γ′ θ
pipe diameter, m horizontal load shear strength gradient, kPa/m breakout resistance calculated as the normal of (V, H) minimum breakout resistance uplift resistance undrained bearing capacity under a local failure mode undrained bearing capacity under a global failure mode
undrained shear strength, kPa tensile capacity at pipe–soil interface vertical load pipe embedment, m roughness factor of pipe–soil interface inclined angle of seabed submerged unit weight, kN/m3 angle between the most vulnerable plane and the upright plane of seabed
The soil was modeled as a rigid-plastic material obeying the Tresca yield criterion, with constant su or linearly varying undrained strength profile su = kz, where k is the strength gradient and z is the soil depth. Strength softening of the soil caused by installation was not accounted for, as the bearing behaviour of the pipelines was assessed a period of this process and strength recovery has been gained. The submerged soil unit weight was given in normalized forms γ′D/su and γ′/k for the two distinct profiles. Two extreme cases of pipe–soil interface roughness, namely completely smooth (α = 0) and completely rough (α = 1), were considered, with the former representing a unroughened pipe section and the latter coated with concrete or epoxy resin for anti-corrosion purpose. Regarding the tensile capacity T on the interface, unbonded conditions (T = 0) were assumed by a multitude of studies (Houlsby and Puzrin [11]; [24,25]), stating that the voids beneath the pipeline after trenching may prevent it from developing suctions, and this assumption has been adopted in the DNV design code [6]. However, it is also acknowledged that the high water pressure and the weight of the pipeline may extrude the water beneath its invert and promote reconsolidation. The breakout response associated with the development of tension / suction was reported even for pipelines directly laid on the seabed (e.g. [8,36]). As a result, analyses assuming bonded conditions (T = ∞) on the pipe–soil interface were widely conducted as well [3,4,24,26]. In this study, the values of T were determined as either zero or unlimited to consider both scenarios. It is noteworthy that in reality when a bonded condition is present, full mobilization of shear stress along the interface may not be likely, and the assumption made here is certainly a simplification. For each configuration of the model, the pipe was subjected to displacement-controlled probes at an interval of 2° to obtain both UB
slope and derived a pipe–soil interaction model that can yield predictions in good agreement with model test results. Apart from these limited work on sand, no research effort has been directed to clay, let alone the development of adequate assessment models. Above all, the existing studies are insufficient to guide the design of marine pipelines in inclined seabed concerning the uplift bearing capacity. Another important issue worth noting is the fact that the thermal-induced transverse displacement or buckling should take place along the most vulnerable direction, which does not necessarily coincide with the upright direction or the normal direction of the inclined seabed surface, as will be discussed later. Even in flat seabed, this issue has not been systematically investigated despite the derivation of abundant V-H failure envelopes as reported in Aubeny et al. [1], Merifield et al. [26], Martin and White [24], etc. To address these gaps, this paper presents an extensive numerical study on the bearing capacity of submarine pipelines buried in inclined clayey seabeds with different angles. The bearing capacities along all possible directions are carefully examined and compared, with reference to differing failure mechanisms, in an attempt to determine the minimum resistance against transverse displacement. Design charts to quantify the effect of inclination angle on the minimum breakout resistance of pipelines are then presented, taking into account the influence of various soil properties. Critical review of current industrystandard procedures is also presented to highlight the un-conservative shortcomings. 2. Methodology 2.1. Finite element limit analysis (FELA) code OxLim All analyses were performed using the FELA code OxLim [19–21], which obtains rigorous upper- and low-bound plastic solutions of complex geotechnical problems through implementing them as second order cone optimisation problems and solving them using MOSEK [27]. Discontinuities among the discrete elements were allowed to provide more degrees of freedom of the model and thus improve the computing accuracy. An adoptive remeshing technique introduced in Martin [23] was adopted for mesh generation and refinement, to obtain closelybracketed upper-bound (UB) and lower-bound (LB) solutions, with an allowed bracketing discrepancy of 1% in the present study. 2.2. Numerical details The object investigated in this paper is a plane-strain pipeline section fully buried in inclined soft clay seabed. The model configuration and the notations are illustrated in Fig. 1. The pipeline section was assumed to be rigid body with a diameter D, comprised of a 180-segments polygon, with its invert being buried at a depth of w (i.e. 1.5D to 6.0D at an interval of 0.5D). The width and depth (at the middle line) of the soil were 16D and 12D, respectively. Fixed boundaries were applied to the bottom and two sides of the soil domain, and free boundary conditions were assumed at the soil surface with an inclination angle of β, taking values of 0° (i.e. for a flat seabed), 1°, 3°, 5°, 7° and 10°.
Fig. 1. Configuration of the numerical model. 2
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Fig. 3. Breakout resistances against probing angles in inclined seabed. From inside to outside, embedment w/D = 1.5 to 6 at intervals of 0.5. α = 1, γ′D/ su = 1.
Fig. 2. Breakout resistances against probing angles in inclined seabed. From inside to outside, embedment w/D = 1.5 to 6 at intervals of 0.5. α = 1, β = 3°.
3
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Fig. 4. Failure mechanisms corresponding to minimum breakout resistance Pu,b. T = 0, α = 1, w/D = 3.0, β = 3°.
Fig. 5. Comparison of failure mechanisms corresponding to Pu,u and Pu,b. T = ∞, α = 1, γ′D/su = 1, w/D = 3.0.
and LB failure loads at a full range of possible directions. Since the vector defined by (V, H) does not coincide with the displacement direction, where V and H are the load components along the vertical and horizontal planes respectively, the results presented in the form of V-H failure envelopes fail to provide sufficient information on the most vulnerable plane (MVP) along which transverse displacement is most likely to take place. Hence, the norm of (V, H), denoted by Pu, is plotted against the probing angle in polar coordinates, enabling the detection of the MVP as well as the minimum breakout resistance. Nonetheless, for the sake of simplicity, the term “failure envelope” is still used.
3. Results and discussions 3.1. Breakout resistances at differing probing directions Figs. 2 and 3 show the development of failure envelopes with burial depth of a rough pipe that can sustain zero or unlimited tension, with a combination of γ′D/su and β values being considered. Among all data points of each failure envelope, the one that has the shortest distance to the origin (i.e. denoting the smallest breakout load) is highlighted. Besides, the direction determined by the origin and this particular point coincides with the most vulnerable plane (MVP). When T = 0, all MVPs almost coincide with the outer normal planes (ONPs) of the seabed. Exceptions are achieved when the pipe is buried 4
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Fig. 6. Variation of Pu,u and Pu,b with embedment for totally rough pipelines.
deep enough to develop a full-flow state, i.e., the failure envelopes start to coincide and the MVPs shift to the upright plane, and in the cases with high values of γ′D/su and β (see Fig. 2(g)) that may induce a slope failure. The latter is confirmed in Fig. 4, where distinct failure mechanisms corresponding to breakout along MVPs for cases with γ′D/ su = 1 and 3 are illustrated. Apart from that, the failure mechanisms along MVPs and ONPs are almost identical in the unbonded cases, although not presented here. Through comparisons among the left five subplots in Fig. 2, it can be seen that the soil weight helps promote the development of full-flow mechanisms that are associated with higher resistances than global ones. This is evidenced by the considerably smaller sizes of failure envelopes in the weightless soil cases (see Fig. 2(a)) and the coincidence of failure envelopes at shallower depths as γ′D/su increases. Similar issue has been discussed in Martin and White [24]. In general, β appears to have limited effect on the failure envelopes for the default γ′D/su examined here, as shown in the left five subplots in Fig. 3. When T = ∞, the MVPs divert significantly from the ONPs for all cases examined, before the pipe reaching a full-flow state. This illustrates significantly more pronounced influence of the inclined soil surface profile on the pipe-soil response than in the unbonded cases. The bonded condition is also found to help facilitate the full-flow mechanism, as the failure envelopes coincide at much shallower depth than their counterparts in the unboned cases (compare the left and right halves in both Figs. 2 and 3). The angles between MVPs and ONPs show an overall decreasing trend with increasing γ′D/su (see Fig. 2(b), (d), (f), (h) and (j)), but exhibiting little dependence on the slope angles considered here (see Figs. 3(b), (d), (f), (h) and (j)). The most striking observation is that even in the flat seabed case, there is a huge discrepancy between MVPs and ONPs, implying the inadequacy of current design approach in which the transverse buckling was by default considered to occur in the upright direction. This issue is further explored
in Fig. 5, where the failure mechanisms corresponding to breakout along MVPs and ONPs are compared. The full-flow failure mechanism developed by the pipe when loaded uprightly can transition to a global one if loaded along the MVP, which denotes lower breakout resistance. The results of smooth pipes basically show similar patterns with those illustrated in Figs. 2–5 and the details are not discussed here. 3.2. Breakout resistances along MVPs Figs. 6 and 7 plot the breakout resistances evaluated along MVPs (denoted by Pu,b) in reference to the uplift resistances (denoted by Pu,u) for totally rough and smooth pipes respectively. When T = 0, the difference between the two is indiscernible for soils with low γ′D/su (see Figs. 6(a)–(b) and 7(a)–(b)), but becomes notable at larger values (see Figs. 6 (c) and 7(c))) . The inclination β has very little impact on Pu,u whilst its undermining effect on Pu,b is more pronounced, which can only be noted in soil cases with high γ′D/su. When bonded conditions (T = ∞) are assumed, β still has little impact on Pu,u but its influence on Pu,b becomes significantly more notable than in the unbonded cases (e.g. comparing subplots (b) and (e) in Fig. 6 and subplots (b) and (e) in Fig. 7). More importantly, remarkable discrepancies between Pu,b and Pu,u can be observed before a full-flow state has been achieved, for all parameters examined here, and this trend can be exacerbated by increasing γ′D/su and β. Another observation can be made is that the inclination of the seabed does not only reduce Pu,b, but also postpone the occurrence of a full-flow mechanism that can be inferred from the coincidences of the curves. Two important implications can be drawn here: (i) the assessment of breakout resistances based on uplift bearing capacities may greatly overestimate the on-bottom stability of pipelines buried in soft clay, and (ii) the pipelines may need to be embedded deeper than predicted using previous models to reach an optimal burial depth beyond which no / little 5
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Fig. 7. Variation of Pu,u and Pu,b with embedment for totally smooth pipelines.
improvement on breakout resistances can be gained. The uplift resistances calculated using the approach recommend by the newly issued design code DNV RP-F114 [7] are provided for comparison, in which the tensile capacity on the pipe–soil interface was assumed to be zero. Inclination of the seabed was not accounted for in it and thus only flat seabed was considered. The uplift resistances were taken as the smaller values calculated from the following
Pu,local = Nc · s¯u·D
·Ap
Pu,global =
· D 2 ·(1/2
·H · D +
present numerical results than DNV [6] and Maitra et al. [18], even considering its simple form. A striking observation is that when unlimited tension was sustained, the numerical results could still be lower than the DNV predictions (see Figs. 6(e)–(f)). To summarize, the nonconservation of the DNV method is more notable when breakout of pipelines along MVPs is considered or an inclined seabed is involved, hence implying potential safety risks in design. To facilitate application of the presented numerical results, the ratio of Pu,b to Pu,u is introduced to account for the influence of the inclination of soil surface and the fact that loads lower than Pu,u could possibly induce transverse displacement of buried pipelines. Note that all values of Pu,u, even in inclined seabeds, were determined at β = 0°, owing to the facts that mostly β does not affect Pu,u and, more importantly, a multitude of solutions and data have already been developed regarding the uplift bearing capacity of buried pipelines [1,3,24,29]. The calculated Pu,b/Pu,u ratios corresponding to various combinations of parameters are provided in Table 1, and can be readily used to adjust the existing database. Since the calculated ratios for unboned interfaces are mostly close to one, only the results for bonded cases are provided. Due to the computing errors, some values slightly higher than one (with difference less than 0.02%) were obtained and have been modified as one. Fig. 8 explores the relationship between θ (the deflection angle from ONP to MVP) and γ′D/su for the cases with bonded pipe–soil interface. For the sake of simplicity, a flat surface profile is considered here and greater γ′D/su than those presented in Figs. 6 and 7 can thus be examined. A general decreasing trend can be seen for the totally rough and smooth pipes with burial depthless than 4D and 3D respectively, owing to the fact that soils with higher unit weight are more likely to trigger a full-flow failure mechanism. For a typical case with γ′D/su of 4 (e.g. D = 0.5 m, γ′ = 6.5 kN/m3 and su = 0.8 kPa), the MVPs for pipelines with burial depth of less than 3D deflect approximately 30°
(1)
/8) + 2· s¯u (H + D /2)
(2)
where Pu,local is the undrained bearing capacity under a local failure mode and Pu,global is that under a global failure mode, u is the average soil strength and equals su for the uniform soil considered here, Ap is the cross-sectional area of soil above the pipe and H is the cover height from pipe crown to the mudline. The bearing capacity factor Nc, which corresponds to a full-flow failure mode, is in the range between 9 (totally smooth) and 12 (totally rough) provided by Randolph and Houlsby [30]. The above model denotes a significant improvement over the previous version DNV RP-F110 [6] that the soil buoyancy effect is addressed more rationally. The predictions using DNV RP-F110 and the model proposed by Maitra et al. [18] are plotted here for comparison as well. In Maitra et al. [18], a critical embedment wlim, beyond which no separation could occur even under unbonded conditions, is calculated first. Then the uplift resistances for pipes with embedment below or above this critical value are determined separately and the lower value is chosen. It can be seen from Figs. 6 and 7 that DNV RP-F114 [7] yields overestimated results for a number of situations examined here, especially for soils with low γ′D/su (see Figs. 6(a) and 7(a)), and the overestimation is generally more severe for smooth pipelines than for rough ones. However, the estimates from DNV [7] is generally closer to the 6
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0.8305 0.8469 0.9128 0.9989 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8678 0.8849 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
β = 5° 0.8086 0.7933 0.7760 0.8243 0.8854 0.9350 0.9731 1.0000 1.0000 0.9996
0.9059 0.9206 0.9972 1.0000 0.9998 1.0000 1.0000 1.0000 1.0000 1.0000
β = 3° 0.8654 0.8611 0.8593 0.9266 0.9966 0.9997 1.0000 0.9997 1.0000 1.0000
0.9192 0.9356 0.9992 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
So far, only soils with uniformly distributed strength are investigated. This section provides some further details of the other extreme case with su = kz. Two surface profiles are considered here, a flat seabed and an inclined seabed with β of 3° (default value). For an inclined surface profile, it is not practical to assume a strength varying linearly with depth. As a result, when a strength gradient with z was adopted (i.e. k), a strength gradient with x was also set, taking the value of tanβk. This would generate a soil strength profile equal at the surface and vary linearly with the distance normal to it. Since the inclination angle is very small, the strength profile of the inclined seabed did not vary much from the flat one. As discussed before, there is no major difference between the totally smooth and rough pipes, and α was taken as 0.5 by default here. Similar observations with the analyses using uniform soils can be seen in Fig. 9. The MVPs divert largely from the upright plane when T = ∞. The results obtained from the inclined and flat seabeds are very close. Two typical failure mechanisms corresponding to breakout along ONP (the upright plane in this case), i.e., a full-flow or localised one, and along MVP, i.e., a global one, are illustrated in Fig. 10. Even in the cases with very high γ′/k, the pipe is still prone to displace almost along the horizontal plane, as shown in Fig. 9(d). The angles between MVPs and ONPs show much greater values than those in Fig. 8, though not presented for clarity. The results in cases with T = 0 are not discussed here because the MVPs are observed to be coincide with the upright plane. Fig. 11 compares minimum breakout resistance Pu,b and the uplift resistance Pu,u for the two surface profiles. Higher values of γ′/k do slightly reduce Pu,u due to soil buoyancy for both surface profiles while only reduce Pu,b in the inclined case. Again, the comparison between Fig. 11(a) and 11(b) shows that the inclination angle of seabed has very little effect on the uplift resistance Pu,u. The effect of normalized unit weight on the discrepancy between Pu,b and Pu,u observed here is less pronounced than in the uniform soil cases, even when considering the more deflection from the upright plane to the MVPs. This discrepancy only appears to diminish when a burial depth of 6D has been reached, thus indicating considerably greater effect on the optimal burial depth than that in the uniform soil cases (i.e. 2.5D and 3D for the totally smooth and rough pipes respectively).
0.8479 0.8631 0.9694 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8646 0.8879 0.9972 1.0000 1.0000 1.0000 0.9998 1.0000 1.0000 1.0000
β = 5° 0.8442 0.8379 0.8347 0.8950 0.9644 1.0000 1.0000 1.0000 1.0000 0.9997
0.8911 0.9109 0.9968 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.8207 0.8407 0.9517 1.0000 0.9992 0.9993 1.0000 1.0000 1.0000 0.9997
γ′D/su = 2
0.8853 0.9027 0.9975 1.0000 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000
β = 3° 0.8731 0.8707 0.8726 0.9423 0.9998 0.9998 1.0000 0.9998 1.0001 1.0000
0.8285 0.8523 0.9697 1.0000 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 0.8331 0.8634 0.9794 1.0000 1.0000 1.0000 0.9998 1.0000 0.9998 1.0000 0.8415 0.8675 0.9899 0.9997 0.9997 1.0000 1.0000 1.0000 1.0000 1.0000 0.7770 0.8084 0.9270 0.9788 0.9984 0.9989 0.9990 0.9989 0.9992 0.9990
γ′D/su = 1
0.8462 0.8701 0.9933 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
β = 1° 0.8993 0.8993 0.9024 0.9779 0.9998 1.0000 1.0000 0.9998 1.0000 1.0000 β = 0° 0.9112 0.9125 0.9160 0.9911 1.0000 0.9997 1.0000 0.9998 1.0000 1.0000 β = 7° 0.8322 0.8283 0.8307 0.8937 0.9715 0.9998 1.0000 1.0000 1.0000 0.9997 β = 5° 0.8443 0.8434 0.8480 0.9147 0.9938 1.0000 1.0000 1.0000 1.0000 0.9998 β = 3° 0.8558 0.8568 0.8627 0.9334 1.0000 1.0000 1.0000 1.0003 1.0000 1.0000 β = 1° 0.8668 0.8693 0.8766 0.9482 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 β = 0° 0.8721 0.8745 0.8810 0.9536 1.0000 0.9998 1.0000 0.9998 1.0000 1.0000
0.7777 0.8108 0.9291 0.9862 0.9989 0.9993 0.9992 0.9990 0.9994 0.9990
β = 7° 0.8073 0.8163 0.8298 0.8974 0.9783 0.9935 0.9993 0.9997 0.9995 0.9995
0.7764 0.8111 0.9271 0.9945 0.9987 0.9993 0.9992 0.9992 0.9988 1.0000
β = 5° 0.8092 0.8180 0.8309 0.8979 0.9796 0.9961 0.9994 0.9997 0.9994 0.9997
4. Conclusions This paper presents a programme of finite element limit analyses on the ultimate bearing capacity of a rigid plane strain pipe section buried in inclined clayey seabed. The main findings are summarised as follows.
γ′D/su = 0
0.7767 0.8093 0.9274 0.9861 0.9992 0.9993 0.9991 0.9990 0.9993 0.9991
smooth
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
0.7783 0.8107 0.9271 0.9934 0.9991 0.9995 0.9994 0.9983 0.9993 0.9993
β = 3° 0.8088 0.8185 0.8310 0.8999 0.9802 0.9993 0.9993 1.0000 0.9995 0.9996 β = 1° 0.8102 0.8190 0.8331 0.9008 0.9806 0.9907 0.9995 1.0000 0.9994 0.9995 β = 0° 0.8094 0.8196 0.8326 0.9011 0.9809 0.9863 0.9995 0.9998 0.9995 0.9996 w/D 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6
3.4. Effect of strength gradient
0.8291 0.8414 0.9171 0.9953 0.9994 0.9993 1.0000 1.0000 1.0000 0.9997
γ′D/su = 3
β = 1° 0.9130 0.9134 0.9155 0.9932 0.9997 1.0000 0.9998 0.9998 1.0000 0.9998 β = 0° 0.9349 0.9349 0.9374 1.0000 0.9998 0.9996 1.0000 0.9997 1.0000 1.0000 β = 7° 0.8132 0.7993 0.7870 0.8312 0.8855 0.9286 0.9622 0.9906 1.0000 0.9997
γ′D/su = 3 γ′D/su = 2 γ′D/su = 1 γ′D/su = 0 rough
Table 1 Breakout resistance ratio Pu,b/Pu,u,β=0 (T = ∞).
from the upright plane. This observation provides an interesting perspective to interpret some pipeline failures found in the Pinghu Oil & Gas Field, where the fractures always took place at the 5 o'clock direction (Prof. Xianghua Lai, personal communication). Also of interest to note is the sudden drop of θ in the simulations with w/D = 3.0. In this particular case, when γ′D/su increases from 3 to 4, the MVP that moderately deviates from the upright direction shifts to the upright plane. Also, the soil failure mechanism transitions from a global one to a localised / full-flow one, though not illustrated here. Such a transition is absent in analyses with deeper embedment (w/D ≥ 3.5) or takes place more gradually in cases with shallower embedment (w/D ≤ 2.5).
0.7809 0.7750 0.7834 0.8327 0.8765 0.8567 0.8458 0.8499 0.8805 0.9276
β = 7° 0.7207 0.6818 0.6554 0.6338 0.6969 0.6742 0.6576 0.6622 0.6834 0.7244
D. Kong, et al.
(1) The most vulnerable plane (MVP) along which a transverse displacement of pipeline could most likely take place diverts greatly from the outer normal plane (ONP) of the soil surface. A full-flow failure mechanism developed by an uplift force can easily shift to a global one when the pipe was loaded along the MVP. This carries an important implication that the minimum breakout resistance Pu,b of 7
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Fig. 8. Angle between the most vulnerable plane and the upright plane. T = ∞, β = 0°.
Fig. 9. Breakout resistance against probing angle in seabeds with linearly varying strength (T = ∞). Subplots (a)-(d) show results with β = 0° and (e)-(h) show results with β = 3°.
a pipeline is greatly overestimated by the assessment based on uplift bearing capacities, and an overestimation of approximately 15%-35% can be observed even for soils with moderate unit weight and inclined angle. (2) The inclination angle β of the soil surface has limited effect on the uplift bearing capacity Pu,u of a pipeline, but this effect on Pu,b is notable. In addition, the existence of inclination angles also postpones the occurrence of a full-flow mechanism by 0.5-1D, suggesting deeper optimal burial depth beyond which little improvement on bearing capacities can be gained. (3) The DNV RP-F114 approach [7] yields moderate overestimation of the breakout resistance regarding transverse displacement or buckling (along MVPs), but the predictions are in general closer to the present numerical results than that from other methods. It is worth noting that the non-conservation of the DNV approach, which was developed based on unbonded interface assumption, exists even in some cases with bonded interfaces. (4) Design charts are developed to modify the database on uplift resistance of pipelines to account for the factors described above,
which are expected to benefit the safe design of marine pipelines regarding on-bottom stability. For all analyses presented, the redistribution and softening of soil strength caused by the pipeline installation process was not accounted for, which inevitably constitutes a limitation of the present study. Progress on this issue may be achieved through further exploration of a number of recently developed pipeline-soil interaction models (e.g. [13, 35], [14]; [32]). Moreover, only total stress analyses were conducted in this paper, without any attempt to look into the pore pressure response. Some recent work (e.g. [12, 16]; Randolph et al. [31]; [9]) could help shed some interesting light on this issue. Data availability statement All data generated or used during the study are available from the corresponding author by request.
8
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Fig. 10. Comparison of failure mechanisms corresponding to Pu,u and Pu,b for pipes buried in seabed with linearly varying strength (T = ∞). Subplots (a) and (b) show results with β = 0°, and (c) and (d) show results with β = 3°.
Fig. 11. Variation of Pu,u and Pu,b with embedment of pipelines buried in seabeds with linearly varying strength. T = ∞, γ′ /k = 1. (a) β = 0°, (b) β = 3°.
Declaration of Competing Interest
acknowledge the fund from the National Natural Science Foundation of China (research grant: 51809232) and the Zhejiang Provincial Natural Science Foundation of China (research grant: LCD19E090001).
The authors declar that they have no conflict of interest related to this project. Acknowledgement
Supplementary materials
The authors gratefully thank Professor Chris Martin at the University of Oxford for providing the FELA code OxLim, and
Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.apor.2019.102007. 9
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