Lateral instability and tunnel erosion of a submarine pipeline: competition mechanism

Abstract

In submarine geological and hydrodynamic environments, either tunnel erosion or lateral instability could be initiated where there is a shallowly embedded pipeline. Unlike previous studies on the tunnel erosion of sand and the lateral instability of pipelines, in this study we performed correlation analyses on the competition mechanism for these two physical processes. By correlating the critical flow velocities of these two processes, the instability envelope for the pipe–soil interaction system is established, which can be described using three key parameters: the embedment-to-diameter ratio, the dimensionless submerged weight of the pipe, and the corresponding critical flow velocity. The analysis procedure is further proposed to first determine the instability mechanism and then the critical velocity of ocean currents. Our parametric study indicates that tunnel erosion is more prone to emerging than lateral instability with small embedment-to-diameter ratio values. With increasing pipeline embedment, tunnel erosion can be suppressed and lateral instability therefore occurs more frequently. Moreover, for light pipelines, lateral instability is more likely to be triggered than tunnel erosion.

Appendix A: Pipe–soil interaction model for current-induced pipeline instability on a horizontal sandy seabed

The pipe–soil interaction model proposed by Gao et al. (2016) was derived by using a limit equilibrium approach for predicting ultimate lateral soil resistance to the partially embedded pipeline on a sloping sandy seabed in ocean currents. For the special case of a horizontal sandy seabed (i.e., the slope angle of the seabed is zero), the model can be simplified as follows.

As illustrated in Fig. 9a, the composite failure surface for the lateral pipe–soil interaction is comprised of the sliding-friction segment-DB and the passive-pressure segment-BC. A virtual retaining wall-AB is supposed to be perpendicular to the seabed surface. The carried soil wedge-ABD at failure (the shaded areas, see Fig. 9a) should be chosen as the analysis object. The main forces acting on the soil wedge-ABD include four components: the passive earth pressure (E ₁) on the virtual retaining wall-AB, the sliding-friction (E ₂) on the segment-DB, the submerged weight (W _b) of the wedge-ABD, and the pipe–soil interfacial force (P). The corresponding triangle of these forces is shown in Fig. 9b.

According to the geometric principle of the composition of forces, the pipe–soil interfacial force P is balanced by the resultant force of E ₁, E ₂, and W _b. As the horizontal component of the interfacial force P, the lateral soil resistance F _R can then be divided into the passive pressure (F _Rp) and the sliding friction (F _Rf) components, respectively:

(A1)

where φ ^' is the mobilized friction angle along the retaining wall-AB and φ is the internal friction of the sand. Based on the Coulomb’s theory of passive earth pressure, the passive earth pressure E ₁ can be obtained as follows:

$$ {E}_1=\frac{1}{2}{\gamma}^{\hbox{'}}{e}^2{K}_{\mathrm{p}} $$

(A2)

where γ ^' is the buoyant unit weight of sand, e is the initial embedment of pipeline, and K _p is the passive pressure coefficient:

$$ {K}_{\mathrm{p}}={\left[\frac{ \cos \varphi}{\sqrt{ \cos {\varphi}^{\hbox{'}}}-\sqrt{ \sin \left(\varphi +{\varphi}^{\hbox{'}}\right) \sin \varphi}}\right]}^2 $$

(A3)

In accordance with the 'law of sines' for the forces triangle (see Fig. 9b), the total sliding friction E ₂ in Eq. A1 can be determined as follows:

$$ {E}_2=\frac{ \cos {\varphi}^{\hbox{'}} \sin \left(\beta -\delta -\omega \right)}{ \cos \omega \cos \left(\beta -\delta +\varphi \right)}{E}_1 $$

(A4)

where β is the intersection angle between wall-AB and the line segment AD^' (Fig. 9a) and ω is the intersection angle between the direction of F _MN to the seabed surface (Fig. 9b):

$$ \beta =\frac{\pi}{2}-\frac{3}{4}\theta $$

(A5)

$$ \omega = \arctan \left(\frac{E_1 \sin {\varphi}^{\hbox{'}}-{W}_{\mathrm{b}}}{E_1 \cos {\varphi}^{\hbox{'}}}\right) $$

(A6)

The submerged weight W _b of the soil wedge in Eq. A6 can be calculated as follows:

$$ {W}_{\mathrm{b}}=\frac{\gamma^{\hbox{'}}}{8}\left[4{e}^2\frac{1+ \cos \theta}{ \sin \theta}-{D}^2\left(\theta - \sin \theta \right)\right] $$

(A7)

where D is the diameter of pipeline and θ is the half angle of the pipeline penetration:

$$ \theta = \arccos \left(1-2 e/ D\right) $$

(A8)

In Eq. A4, δ is the mobilized pipe–soil interfacial friction angle, which is relative to the ocean current-induced drag force (F _D), lift force (F _L), the submerged weight of the pipeline (W _S), and the half angle of the pipeline penetration (θ):

$$ \delta = \arctan \left(\frac{F_{\mathrm{D}}}{W_{\mathrm{S}}-{F}_{\mathrm{L}}}\right)-\frac{3}{4}\theta $$

(A9)

Note that the absolute values of δ should be no larger than its critical value δ _crit (i.e., |δ| ≤ δ _crit, where δ _crit = arctan[(sinφ cos ν _s)/(1 − sin φ sin ν _s)], ν_s is the angle of soil dilation).

Submitting Eqs. A2 and A4 into Eq. A1, the lateral soil resistance F _R can be re-expressed as follows:

$$ {F}_{\mathrm{R}}=0.5{\gamma}^{\hbox{'}}{e}^2{K}_{\mathrm{p}} \cos {\varphi}^{\hbox{'}}\left(1+\frac{ \sin \varphi \sin \left(\beta -\delta -\omega \right)}{ \cos \omega \cos \left(\beta -\delta +\varphi \right)}\right) $$

(A10)

A _0.5, One half the area of a vertical cross-section of the soil displaced by the partially embedded pipe during penetration and oscillations (see Wagner et al. 1989); C _D, Drag force coefficient; C _L, Lift force coefficient; D, Pipe diameter; E, Initial embedment of pipe; (e/D)_T, Embedment-to-diameter ratio for transition of instability mechanisms; (e/D)_T ^', Trial value for the iteration calculations of (e/D)_T; E ₁, Passive earth passive in the pipe–soil model by Gao et al. (2016) (see Fig. 9); E ₂, Total sliding friction on a failure surface (see Fig. 9); F _L, Lift force on the pipe; F _D, Drag force on the pipe; F _R, Lateral soil resistance to pipe; F _Rp, Passive pressure component of lateral soil resistance; F _Rf, Sliding friction component of lateral soil resistance; F _Rw, Additional submerged weight component of lateral soil resistance; F _C, Pro force of seabed to pipe; Fr, Froude number, $U/\sqrt{gD}$; G, Gravitational acceleration;

G, Dimensionless submerged weight of pipe, W _s/(γ ^' D ²); I _R , Relative dilatancy index; i _cr0, Conventional critical hydraulic gradient for seepage failure (upward)i _cr0 = (1 − n)(s − 1); i _cr, Critical hydraulic gradient for the oblique seepage failure i _cr = (sin θ + cos  θ  tan  φ)(1 − n)(s − 1); K _p, Passive pressure coefficient; n, Porosity of soil;

p ^' , Mean effective stress (in kPa); P, Total pipe–soil interfacial force (see Fig. 9); Re, Reynolds number, UD/ν; r _emb, H, Reduction coefficients in the horizontal direction;

r _emb, L, Reduction coefficients in the vertical direction; s, Specific gravity of sand grains, ρ _s/ρ _w; U, Current velocity; U _CT, Critical velocity for tunnel erosion of sand; U _CL, Critical velocity for lateral instability of pipe; U _cr, Critical velocity for the instability of pipe–soil interaction system; (U _cr)_T, Critical velocity for transition of instability mechanisms;

W _s, Submerged weight of pipe per unit length; W _b, Submerged weight of the soil wedge (see Fig. 9); α, Slope angle of seabed surface; β ^', Empirical coefficient in the pipe–soil interaction model by Wagner et al. (1989); β, ω, Intersection angle (Fig. 9); γ ^', Effective unit weight of sand, γ ^' = (1 − n)(s − 1)ρ _w g; θ, Half angle of pipe penetration; θ _cr, Non-dimensional flow velocity for the onset of tunnel erosion; μ, Coefficient of sliding friction; ν, Kinematic viscosity of water; ν_s, Angle of soil dilation; ρ _s, Mass density of sand grains; ρ _w, Mass density of water; φ, Internal friction angle of sand; φ _crit, Critical state angle of shearing resistance of sands; φ ^', Mobilized friction angle along retaining wall (see Fig. 9); δ, Pipe–soil interfacial friction angle

Fig. 9

Lateral instability of a pipeline: (a) geometry of failure mechanism; (b) triangle of the forces on the wedge-ABD [shaded area in ( a )]