Efficient Mitigation Measures for Reducing the Kinematic Distress of Offshore Pipelines Due to Seismic Fault Rupture

Abstract

Offshore high-pressure gas pipelines comprise critical infrastructure that often cross seismic regions for hundreds of kilometers. The intersection of such pipelines with seismic fault zones is frequently inevitable due to high costs or technical constraints of alternative routes. While typical mitigation measures, such as stronger materials or cross-sections and flexible joints, ensure pipeline integrity against earthquake-related geohazards, options for deep-water pipelines are more limited compared to onshore or even near-shore pipelines. This paper numerically investigates the efficiency of various mitigation approaches for surface-laid steel pipelines subjected to normal or reverse seismic faulting. Using ABAQUS finite-element software, the pipeline is simulated under realistic conditions for cohesive and non-cohesive seabed sediments. Critical fault displacements for different pipe steel materials, cross-sections, coatings, pressures, and orientations are calculated according to international standards. Results demonstrate that a 30° fault-pipe intersection angle is the most effective approach, increasing pipe’s capacity to fault dislocation by up to 90% for normal and 75% for reverse faults. Additionally, coating materials can increase a pipe’s resistance by up to 15%, whereas pressure difference variations may also have an impact. This study provides useful conclusions regarding the efficiency of the mitigation measures, the applicability of international standards, and the simulation of pipe–soil interaction.

1. Introduction

Offshore natural gas pipelines constitute large-scale critical infrastructure, often operating in depths of hundreds of meters under highly adverse and uncertain conditions and passing through areas characterized by the presence of earthquake-related geohazards. Due to their low mass, such pipelines are primarily vulnerable to kinematic distress caused by Permanent Ground Displacements (PGDs), such as seismic fault ruptures at the seabed surface, rather than inertial effects from ground shaking. Bypassing fault zones is the safest way to ensure pipeline integrity. However, operational constraints, such as high costs, technical difficulties, and geopolitical conflicts, often make this option unfeasible. In this context, implementing mitigation measures constitutes not only a technical necessity but also a core requirement for sustainable energy development, aligning with the European Green Deal and international energy security directives [1]. Ensuring pipeline integrity in geohazardous areas aligns with international energy safety directives and environmental protection programs, as it minimizes the environmental footprint associated with potential infrastructure failure.

Typical mitigation techniques—mainly used for onshore pipelines—include the optimal orientation of the pipe route, the use of stronger cross-sections and materials, and the reduction of pipe–soil interaction via the use of coatings, lighter backfill materials, draining systems or smaller burial depths [2]. The influence of backfill material, burial depth, pipe cross-sectional characteristics, and material properties has been investigated using experimental [3,4,5] and numerical models [6,7,8]. Karamitros et al. [9] studied the effects of pipe-fault intersection angle on pipelines crossing normal faults, while Joshi et al. [7] and Melisiannos et al. [10] studied the effects on pipelines crossing reverse faults. The impact of internal pressure and pipe material properties has been investigated by several researchers [11,12,13,14]. In addition, Karamitros et al. [15] and Vazouras and Karamanos [16] examined the effect of bends on pipe response due to fault rupture.

Yu et al. [17] studied the performance of buried offshore pipelines crossing active faults using an innovative non-linear vector form intrinsic finite element (VFIFE) method. Both operational conditions and empty waiting state were investigated, while different modes of collapse and damage propagation were classified based on the initial collapse location and buckling propagation direction. Also, the post-earthquake resumption and reconditioning of unpressurised and pressurized pipelines were evaluated, and the results indicated a significant effect of loading paths on structural responses [18,19]. Qiu et al. [20] investigated the distress of buried steel pipes subjected to fault-induced dislocation, with particular emphasis on fault offset, internal pressure and the diameter-to-thickness ratio.

Liu et al. [21] and Triantafyllaki et al. [22] examined surface-laid offshore pipelines, focusing on the effect of pipe mechanical and geometrical properties, soil properties, intersection angles, etc. In particular, Triantafyllaki et al. [22,23] presented a series of large-deformation parametric finite-element (FE) simulations of a partially embedded pipeline resting on a fine-grained seabed and subjected to different types of active tectonic faults (normal, reverse, and strike-slip). More recently, they explored pipeline design options against fault movements based on equations for predicting critical fault displacement [24].

More sophisticated techniques include the use of flexible joints, anchor points, innovative pipe materials, complex cross-sectional geometries, or the use of innovative backfill materials to reduce backfill weight. The influence of flexible joints on buried pipelines has been discussed in several studies [25,26,27,28], while the use of sliding supports has been successfully implemented on the aboveground part of the Trans Alaska Pipeline [29]. Expanded polystyrene (EPS) is a material that has been examined as a mitigation measure for buried pipelines [30,31,32], while the use of tire aggregate has also been investigated [33,34]. Berger et al. [35] and Guo et al. [36] investigated the impact of innovative cross-section geometries on pipe distress. Mokhtari and Alavi Nia [37] and Li et al. [38] examined the influence of carbon fiber-reinforced plastic (CFRP) wraps on buried pipelines. Rojhani et al. [39] conducted a series of centrifuge tests to investigate the beneficial impact of low-density gravel (LDG) with high porosity as backfill material. Yang et al. [40] studied the interaction between the pipeline and lightweight trench material during fault dislocations. Lastly, Gantes and Melissianos [41] compared the effect of different mitigation measures on buried pipelines subjected to strike-slip fault rupture.

While the seismic design of pipelines, particularly regarding fault-pipe intersection, has been extensively investigated for onshore and nearshore (i.e., shallow waters) pipelines, a significant knowledge gap persists for offshore pipelines in deep waters. At such depths, defined, for instance, in the case of the Trans Adriatic Pipeline (TAP) at depths exceeding 200 m [42], traditional mitigation measures remain technically and economically unfeasible, at least for the time being. This is primarily because deep-water pipelines are typically laid directly on the seabed at depths of hundreds up to thousands of meters.

This challenge is further intensified by the overgrowing pressure on global industries to continuously and safely transfer energy even in areas characterized by deep and ultra-deep waters, complex seabed topography and intense seismicity, such as the Southeastern Mediterranean Sea. In these areas, pipelines may cross depths up to 3000 m, inevitably intersecting active seismic fault zones. Indicatively, fault ruptures in offshore Crete, Greece, can result in PGDs reaching nearly 2 m within a 50-year pipeline life-cycle [43].

These facts highlight the urgent necessity for the design of reliable, cost-effective and practically feasible mitigation measures for deep-water offshore pipelines. To bridge this gap, the present study provides a systematic, quantitative assessment of easily-implementable measures, such as steel grade, cross-sectional geometry, intersection angle, and coating materials for deep-water high-pressure steel pipelines intersecting with normal or reverse faults. The impact of different pipe orientations, cross-sections, steel materials, coatings, and pressures is examined for both clayey and sandy seabed. The pipeline is simulated via a numerical finite-element (FE) model, considering realistic mechanical and geotechnical parameters for deep-water seabed conditions, while pipe–soil interaction is calculated according to the methodology proposed by O’Rourke and Liu [2]. Critical fault displacements are calculated according to the limit states of contemporary international standards, and subsequently, they are compared in order to assess the effectiveness of each mitigation approach. By considering the adverse deep-sea seabed conditions, this work offers a practical framework, as well as preliminary screening and calibration tools for the efficient seismic design of offshore pipelines without needing to conduct for each case a series of computationally demanding numerical analyses.

Following the current introductory section, the rest of this paper is organized as follows: Section 2 outlines the analytical methodologies for the simulation of pipe–soil interaction in both cohesive and non-cohesive seabed sediments. This section also presents the existing limit state criteria for the seismic response of offshore pipelines and describes the numerical modeling of fault-pipe intersection, including the selected pipe material, cross-sectional geometry and service loads, as well as the geotechnical properties of the seabed sediments. Section 3 presents the numerical results derived from an extensive parametric investigation, considering different fault types and the impact of various key parameters, such as steel grade, cross-sectional geometry, intersection angle, pressure difference and coating material. Section 4 provides a comprehensive discussion of the findings, while Section 5 concludes this paper with insights, limitations, and recommendations for further research.

2. Materials and Methods

2.1. Pipe–Soil Interaction for Offshore Pipelines

Typical methodologies for simulating pipe–soil interaction for buried pipelines (e.g., American Lifeline Alliance—ALA [44] guidelines) are not applicable to surface-laid submarine pipelines, since they are usually partially embedded in the seabed. The simulation of pipe–soil interaction for surface-laid pipelines has attracted significant research interest [45]. The outcomes are summarized in both international standards [46] and design methodologies [2,47].

The present work adopts the methodology developed by O’Rourke and Liu [2], which uses efficient equations for calculating soil resistance along the axial, lateral and vertical directions. Two different types of seabed soils are examined: non-cohesive (e.g., sands) and cohesive (e.g., clays). Sandy seabed is found in shallow waters, while clayey seabed is more common in deeper waters. The adopted embedment depth is z_e = 0.3D and 0.5D for the respective seabed soils, where D is the outer pipe diameter.

2.1.1. Non-Cohesive Soils

For pipelines laid on non-cohesive soils, where soil response is drained, axial soil resistance can be—conservatively—considered bi-linear with ultimate value equal to:

$$T_u=w_s\mathit{tan}\left(k\phi \right)$$

(1)

where w_s is the submerged pipe weight per unit length, φ is the friction angle of the soil and k is the friction reduction factor. Typical values of k range between 0.5 for pipelines with coating and 0.9 for rough pipelines without coating [2,44,48]. These values can be increased by 10% for embedment depths z_e > 0.3D. According to the international standard DNVGL-RP-F114 [46] and Bai and Bai [47], the axial soil resistance can be computed with the following factor:

$$f_s={\displaystyle \frac{2\mathit{sin}\theta }{\theta +\mathit{sin}\theta \mathit{cos}\theta }}$$

(2)

where the angle θ = cos⁻¹(1–2z_e/D) is shown in Figure 1. The ultimate axial soil resistance, T_u, is mobilized for a relative displacement equal to x_u = 0.005 m.

O’Rourke and Liu [2] proposed the implementation of a trilinear model for calculating the lateral soil resistance. For large lateral movement, the pipeline exits the pseudo-trench created by the laying process and the seabed currents. Therefore, the berm resisting to the lateral movement becomes smaller, while the unconsolidated soil adjacent to the pipe also presents smaller frictional resistance.

Lateral breakout soil resistance can be calculated using the equation proposed by Verley and Sotberg [49]:

$$P_{brk}=w_s\tan (k\phi )+\gamma D^2\left(4.5-0.11{\displaystyle \frac{{\gamma }^{\prime }{D}^2}{w_s}}\right){\left({\displaystyle \frac{z_e}{D}}\right)}^{1.25}$$

(3)

where γ and γ′ denote the total and effective unit weight of the soil, respectively. The above breakout soil resistance is activated for relative displacement y_brk = 0.2D to 0.5D. Verley and Sotberg [49] also proposed a pair of equations based on the submerged pipe weight, w_s. These equations are adopted by DNVGL-RP-F114 [46] and Bai and Bai [47], with the expression for a light pipeline being similar to Equation (3).

Lateral residual soil resistance can be calculated using the first term of Equation (3) multiplied by a coefficient of 0.4 to 0.8 for smooth and rough pipelines, respectively. Herein, lateral residual soil resistance was conservatively set equal to:

$$P_{res}=w_s\tan \left(k\phi \right)$$

(4)

The relative displacement for the mobilization of residual soil resistance is taken as y_res = 1.5D, according to DNVGL-RP-F114 [46].

For vertical soil resistance, different values are proposed for upward and downward movements. Since the pipeline is laid directly on the seabed, with no soil above the pipeline to resist vertical movement, upward ultimate soil resistance, Q_up,u, is equal to the submerged pipe weight, w_s. This soil resistance can be mobilized for a very small relative movement, e.g., z_up,u = 0.001 m. Downward ultimate soil resistance can be calculated using the formula proposed in ALA [44] ignoring the effect of embedment as follows:

$$Q_{down,u}={\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\gamma N_{\gamma }{B}^2$$

(5)

where N_γ = e^{(0.18φ−2.5)} is the bearing capacity factor, and B (=D sinθ) is the contact width between the pipe and the seabed. The above-soil resistance can be mobilized for relative displacement z_down,u = 0.1D and 0.15D for dense and loose sand, respectively.

2.1.2. Cohesive Soils

The response of cohesive soils can be drained or undrained, depending on the soil state and the rate of drainage relative to movement. Axial soil resistance for cohesive soils can be modeled as either bi-linear or tri-linear, depending on whether the conditions are drained or undrained, respectively [46]. According to O’Rourke and Liu [2], axial breakout soil resistance can be calculated by the expression proposed by ALA [44] as follows:

$$T_{brk}=\alpha s_u{A}_{contact}$$

(6)

where α is the adhesion factor, s_u is the undrained shear strength of the intact soil, and A_contact = Dθ is the contact area per unit length of the partially buried pipeline (see Figure 1).

According to Bai and Bai [47], the residual soil resistance, T_res, can be calculated using the undrained shear strength of the remolded soil, defined as the ratio of intact shear strength to soil sensitivity (s_u,r = s_u/S_t). For the relative soil displacements for the mobilization of axial breakout and residual resistance, the values of DNVGL-RP-F114 [46] were used, namely x_brk = min (0.005 m; 0.01D) and x_res = min (0.03 m; 0.06D), respectively.

Equation (6) assumes that pipe–soil interaction is dominated by the undrained shear strength and ignores the influence of the submerged pipe weight. DNVGL-RP-F114 [46] proposes a different equation where axial soil resistance is a function of the submerged pipe weight. For typical values of pipe weight and undrained shear strength, Equation (6) results in higher values of axial soil resistance compared to the expression given in DNVGL-RP-F114 [46]. Therefore, Equation (6) was used in the current study as a more conservative approach.

Regarding lateral soil resistance, O’Rourke and Liu [2] proposed the implementation of a trilinear model using the equations developed by Bruton et al. [50]. Equation (7) gives the lateral breakout soil resistance:

$$P_{brk}=0.2w_s+3z_e\sqrt{D{s}_u{\gamma }^{\prime }}$$

(7)

This is mobilized for a relative displacement equal to y_brk = 0.1D. Lateral residual soil resistance is a function of submerged pipe weight and can be calculated as follows:

$${\displaystyle \frac{P_{res}}{w_s}}=1-0.65\left[1-exp\left(-{\displaystyle \frac{1}{2}}{\displaystyle \frac{s_u}{{\gamma }^{\prime }D}}\right)\right]\le 1$$

(8)

According to DNVGL-RP-F114 [46], residual soil resistance is mobilized for relative displacements, i.e., y_res = 1.5D.

Concerning the vertical soil resistance, different values are proposed for upward and downward movements. For undrained soil response, upward breakout soil resistance needs to consider the effect of adhesion between the pipe and seabed, in addition to the submerged pipe weight, as follows:

$$Q_{up,brk}=w_s+\alpha s_{u}B$$

(9)

The above-soil resistance can be mobilized for a very small relative movement, i.e., z_up,brk = 0.001 m. After the complete separation of pipe and seabed, i.e., for relative movement equal to the embedment depth (z_up,res = z_e), only the submerged pipe weight is exerted on the pipe (Q_up,res = w_s).

Ignoring the effect of embedment, downward ultimate soil resistance can be calculated using Equation (10) proposed in ALA [44]:

$$Q_{down,u}=s_u{N}_{c}B$$

(10)

where N_c is a bearing capacity factor expressed as a function of friction angle, φ, and for φ = 0, N_c = 5.1. Bai and Bai [47] proposed smaller values of N_c; however, the conservative value of 5.1 was selected herein. The above-soil resistance can be mobilized for relative displacement z_down,u = 0.1D and 0.15D for stiff and soft clay, respectively.

2.2. Limit-State Criteria

Limit-state criteria can be either stress-based or strain-based. Stress-based criteria are usually used for load-controlled conditions, where structural response is primarily governed by the imposed loads. Conversely, strain-based criteria are usually used for displacement-controlled conditions, where structural response is primarily governed by imposed displacements. The kinematic distress of pipelines due to normal or reverse fault rupture is a combination of bending and axial strains, tensile or compressive, respectively. The integrity of offshore pipelines under such loading conditions has to be examined against tensile rupture, local buckling, global buckling, as well as service loads, such as pressure difference and temperature difference.

2.2.1. Tensile Rupture

Tensile rupture depends on the quality of girth welds. According to the international standard DNVGL-ST-F101 [51], when the total nominal strain is anticipated to be higher than 0.4%, extra requirements need to be satisfied regarding the material properties and the welding quality in order to avoid their fracture. Similarly, Eurocode 3 [52] suggests a plastic tensile strain limit equal to 0.5% for plate materials and welds. Analogously, the American standard API 1104 [53] can be applied in pipe girth welds when the maximum axial strain does not exceed 0.5%.

When girth welds are properly constructed, failure occurs on the pipeline. This is a more favorable situation allowing for higher strain limits of ε_t,cr = 2% to 4%. DNVGL-ST-F101 [51] notes that extra measures should be taken when the plastic strain is anticipated to be higher than 2%. ALA [44] guidelines suggest a strain limit of 2% as a serviceability limit state and 4% as an ultimate limit state. Eurocode 8 [54] suggests an allowable tensile strain of 3%, while the Canadian standard CSA-Z662 [55] suggests an allowable strain of 2.5%.

2.2.2. Local Buckling

Local buckling appears on pipelines under compression and is characterized by the rapid development of wrinkles on the pipe wall. It is a critical failure mode for steel pipelines, since failure appears for lower strains compared to tensile rupture, i.e., ε_c,cr = 1% to 2%. ALA [44] guidelines suggest two characteristic equations for the calculation of strain limits:

$${\epsilon }_{c,cr}=0.5{\displaystyle \frac{t}{D}}-0.0025+3000{\left({\displaystyle \frac{dPD}{2Et}}\right)}^2$$

(11)

$${\epsilon }_{c,cr}=1.76{\displaystyle \frac{t}{D}}$$

(12)

Equation (11) refers to the serviceability limit state, while Equation (12) corresponds to the ultimate limit state; while D is the outer pipe diameter, t is the wall thickness, dP is the pressure difference and E denotes the Young’s Modulus of the pipe material. Equation (11) accounts for both the cross-sectional characteristics and the pressure difference. Similar equations are suggested by the Canadian standard CSA-Z662 [55], Eurocode 3 [52] and DNVGL-ST-F101 [51]. Eurocode 8 [54] suggests an approach similar to Equation (12), where the maximum compressive strain should be equal to ε_c,cr = min (0.01; 0.4t/D).

2.2.3. Global Buckling

Global buckling is the phenomenon where an excessive lateral (or upward) movement occurs along a part of a pipeline, e.g., due to high PGDs, and is related to the development of compressive pipe distress. Surface-laid offshore pipelines are more susceptible to global buckling because they are less restrained compared to onshore or offshore pipelines that are buried. When the submerged pipe weight exceeds lateral soil resistance, lateral buckling occurs; otherwise, the pipeline tends to move upward, causing upheaval buckling. It is noted that the pipeline does not necessarily fail under such global buckling phenomena, since the developed distress may be within the tolerance limits [2,47,56].

International guidelines [56] provide design procedures and limit-state criteria for submarine pipelines against global buckling. In particular, when the pipeline is susceptible to global buckling it should be tested for failure modes, such as local buckling, tensile rupture and fatigue. For tensile rupture, strain-based limit states similar to those given in Section 2.2.1 are suggested, while both stress-based and strain-based criteria are proposed for local buckling. The stress-based limit state for combined axial force, bending moment and pressure is a rather conservative criterion and can be calculated as follows:

$${\left(n_1{\displaystyle \frac{\left|M_{sd}\right|}{M_p}}+{\left(n_1{\displaystyle \frac{S_{sd}}{S_p}}\right)}^2\right)}^2+{\left(n_2{\displaystyle \frac{dP(D-t)}{2t{\sigma }_y}}\right)}^2\le 1 for 15\le {\displaystyle \frac{D}{t}}\le 45 and {\displaystyle \frac{S_{sd}}{S_p}}<4$$

(13)

where n₁ and n₂ are safety factors, M_sd (=n₃M) and S_sd (=n₃S) are the design bending moment and axial force, respectively, which are equal to the calculated loads multiplied by another safety factor n₃; M_p and S_p are the plastic bending moment and axial force capacities, respectively; dP is the pressure difference; and σ_y is the yield strength of the steel material.

2.2.4. Pressure Integrity and On-Bottom Stability

Deep-water submarine pipelines are characterized by high internal pressure caused by the natural gas, and high external pressure existing at great water depths. The difference between the internal pressure, P_i, and the external pressure, P_e, (dP = P_i − P_e) causes hoop stress on the pipe wall, which has to be smaller than the material yield strength, σ_y, to ensure pipe integrity against leakage. Equivalent stress-based criteria are proposed by several guidelines [44,51,57]. According to the international standard DNVGL-ST-F101 [51], the maximum allowable pressure during operation is given by

$$P_{max}=n_4{\displaystyle \frac{2t}{D-t}}{\sigma }_y$$

(14)

where n₄ is a safety factor depending on the safety class and the location.

An additional problem considered for deep-water natural gas pipelines is the on-bottom stability, since the absence of cement coating and the low weight of the pressurized natural gas make the pipeline susceptible to floating. For instance, the deep-water part of the TAP pipeline is covered only with a 3 mm thick anti-corrosive polyethylene (PE) coating [42]. Therefore, pipelines with a big outer diameter and a small cross-sectional width tend to float underwater. The international standard DNVGL-RP-F109 [58] proposes the following limit to avoid floatation:

$$n_5{\displaystyle \frac{b}{w_s+b}}\le 1$$

(15)

where w_s is the submerged pipe weight, b is the pipe buoyancy and n₅ = 1.1 is a design factor to increase safety.

2.3. Problem Description

In the present study, a deep-water pipeline under kinematic distress due to normal and reverse fault rupture is investigated. As presented in Figure 1, the pipeline is partially embedded in the seabed surface. The applied PGDs are depicted in Figure 2, for the case of normal fault rupture, and are equal to the following:

$$\begin{array}{c}{d}_v=d\sin \alpha \\ d_a=d\cos \alpha \sin \beta \\ d_l=d\cos \alpha \cos \beta \end{array}$$

(16)

where d is the fault displacement; d_v, d_a and d_l are the vertical, axial and lateral displacements, respectively; α is the fault dip angle (Figure 2a); and β is the pipe-fault intersection angle (Figure 2b).

2.3.1. Numerical Model

The investigation is conducted utilizing the static-standard analysis of ABAQUS software [59]. As seen in Figure 3, a 10.3 km long model with 9200 elements is created. Pipe elements are used for the simulation of the pipeline and PSI elements for the simulation of the pipe–soil interaction. The intersection between the pipeline and the fault occurs in the middle of the model. Fine mesh is adopted in the middle of the model along L_cr = 300 m ≈ 500D, where the element length is equal to d_FE = 0.5 m. Adjacent to the above part, two 5 km long parts are created so that the boundary conditions do not influence the pipe response. The element length of the above parts increases gradually from d_FE = 0.5 m near the middle of the model to d_FE ≈ 5 m near the two ends of the model.

The analysis is conducted in two steps. During the first step, internal and external pressure are applied. The “far-field” nodes of the PSI elements are pinned and the edges of the pipeline are fixed. During the second step, fault movement is applied by imposing the corresponding PGDs on the “far-field” nodes of the PSI elements and the pipe end towards the hanging wall, as seen in Figure 3.

The pipeline is modeled with PIPE31 elements, which are 2-noded linear Timoshenko pipe elements that allow transverse shear deformation. Pipe–soil interaction is modeled with PSI34 elements, which are 4-noded elements with two nodes attached to the pipe and two representing “far field” conditions. The above elements are more sophisticated compared to conventional springs, since they simulate pipe–soil interaction per unit length in three directions simultaneously, while adjusting the direction of pipe–soil interaction according to the pipe rotation. Material and geometrical non-linearities are considered in a computationally efficient manner.

2.3.2. Material Properties

The selected materials, cross-sectional dimensions and pressures of the pipeline, as well as the seabed geotechnical properties, are suitably chosen to present realistic deep-water conditions. Concerning the pipeline, the outer diameter is selected equal to D = 0.61 m (24″) and three different wall thicknesses were investigated: t = 25 mm, 32 mm and 38 mm, or 1″, 1.25″ and 1.5″. The unit weight of the pressurized natural gas is set equal to γ_gas = 1.5 kN/m³ [22], while a 5 mm thick PE coating is assumed for some of the examined cases. The above values are selected to meet the criterion of on-bottom stability, as described in Equation (15).

The pipeline is assumed to consist of stainless-steel grades API 5L X60, X65 and X70 with Young’s Modulus E = 210 GPa and Ramberg–Osgood plasticity with coefficients α_r = 1 and n = 20:

$$\epsilon ={\displaystyle \frac{\sigma }{E}}+{\alpha }_r{\displaystyle \frac{{\sigma }_y}{E}}{\left({\displaystyle \frac{\sigma }{{\sigma }_y}}\right)}^n$$

(17)

In Equation (17), the yield strength, σ_y, and ultimate strength, σ_u, of the above steel grades are presented in

Table 1. Material properties of API 5L steel grades.

Grade	σy (MPa)	σu (MPa)
X60	414	517
X65	448	531
X70	486	565

. The maximum allowable pressure, P_max, is calculated from Equation (14) with n₄ = 0.72, similar to Triantafyllaki et al. [22]. Five different scenarios are examined concerning pressure difference: dP/P_max = 0%, 25%, 50%, 75%, and 100%; dP/P_max = 0% corresponds to equal internal and external pressure (P_i = P_e), while for the other cases the internal pressure is always assumed to be higher (P_i > P_e).

Offshore surface-laid pipelines frequently develop “snaked” routes due to the strains that are developed by the service loads (pressures and temperatures). In contrast, buried pipelines are restricted by the surrounding soil and, therefore, they develop stresses, instead of strains. After a sensitivity analysis, it was concluded that the difference between the above scenarios is negligible in terms of critical fault displacement. In the present study, the pipeline is assumed to present zero initial strains, due to the simplicity in the simulation.

Concerning the seabed properties, four different soils are selected: soft clay, stiff clay, loose sand, and dense sand. The soil characteristics are taken from the geotechnical report of TAP pipeline [60] to ensure realistic conditions. The material properties are presented in

Table 2. Seabed material properties.

	φ (ο)	su (kPa)	γ (kN/m3)	St
Soft Clay	-	20	17	1.5
Stiff Clay	-	50	19	3
Loose Sand	30	-	18	-
Dense Sand	40	-	20	-

, where φ is the friction angle, s_u is the undrained shear strength, γ is the total unit weight, and S_t is the soil sensitivity. It is noted that undrained shear strength usually increases with depth, with very small values near the surface of the seabed. However, the pipe weight consolidates the clay around the pipeline, increasing the undrained shear strength. Herein, the constant values presented in Table 2 are adopted, as a conservative approach.

The embedment depth of the pipe is assumed equal to z_e = 0.5D, which is a rather conservative value taking into account the laying process and seabed currents [2]. Four different friction reduction factors are selected: k = 0.9, 0.7, 0.5, and 0.3. Values of k = 0.9 and 0.7 correspond to rough and smooth steel surfaces, respectively, whereas the value of k = 0.5 corresponds to PE coating. According to relevant studies [2,44,48], the value of k = 0.3 is very small and it was adopted in order to simulate the impact of a material that could lead to extreme reduction of pipe–soil interaction. For instance, O’Rourke et al. [61] reported that friction reduction factors k ≈ 0.3–0.4 could be achieved using Plexiglas or hard epoxy coatings, while Subba Rao et al. [62] stated that friction reduction factors could be as low as k = 0.2.

The above values are used for the calculation of pipe–soil interaction along the axial, lateral and vertical (upward and downward) directions, according to the methodology proposed by O’Rourke and Liu [2]. The adopted models are either bi-linear or tri-linear and are presented in Figure 4, together with the corresponding equations. It has to be stressed that for upward pipe movement greater than the embedment depth, axial and lateral soil resistance decrease. However, the influence of the above phenomenon on the pipe distress was insignificant and hence neglected in the cases studied.

3. Numerical Results

The numerical study is conducted for normal and reverse fault ruptures and three different dip angles: α = 60°, 45° and 30° (Figure 2a). A clayey seabed is adopted for a normal fault, while a sandy seabed for a reverse fault, to examine soil–pipe interaction mechanisms for both fault types and seabed conditions. A total of 162 FE analyses were conducted in the current parametric investigation: more specifically, 90 for normal fault ruptures and 72 for reverse fault ruptures.

3.1. Normal Fault

For normal fault ruptures, the pipeline is assumed to be laid on soft and stiff clay, having the material properties listed in Table 2. The impact of different mitigation approaches is investigated and corresponding main parameter values are presented in

Table 3. Main parameter values for the examined mitigation approaches for normal fault rupture.

Parameter	Value
Steel grade	X60, X65, X70
D/t	24, 19.2, 16
β (°)	90°, 60°, 45°, 30°
dP/Pmax (%)	0, 25, 50, 75, 100

, where X60, X65 and X70 are steel grades (Table 1); D/t is the diameter-to-thickness ratio; β is the pipe-fault intersection angle (Figure 2b); and dP/P_max is the pressure difference ratio. It is noted that the effect of pipe coating could be examined via the reduction of the adhesion factor in Equation (6). However, the adhesion factor, apart from pipe–soil roughness, reflects the variation in the undrained shear strength (s_u) around the pipe–soil interface; thus, it is difficult to be determined accurately [47]. Hence, the conservative approach of ALA [44] is adopted in this study, neglecting the effect of coating in the cases examined.

The “base model” of the investigation consists of a pipeline with outer diameter D = 0.61 m (24″), wall thickness t = 25 mm (1″) and X65 steel grade. The pressure difference of the pipeline is dP = 14 MPa, or 50% of the maximum allowable pressure, which is equal to P_max = 28 MPa, according to Equation (14). The pipeline intersects with the fault perpendicularly, i.e., β = 90°. The values of soil resistance forces and relative displacements are presented in

Table 4. Soil resistance forces and relative displacements.

	Axial		Lateral		Upward		Downward
	Tbrk (kN/m)	Tres (kN/m)	Pbrk (kN/m)	Pres (kN/m)	Qup,brk (kN/m)	ws	Qdown,u (kN/m)
Soft Clay	19.3	12.9	8.8	0.6	13.5	1.2	62.9
Stiff Clay	45.3	15.1	15.6		30		157.2
	xbrk (mm)	xres (mm)	ybrk (mm)	yres (mm)	zup,brk (mm)	zup,res (mm)	zdown,u (mm)
Soft Clay	5	30	61	915	10	305	92
Stiff Clay							61

. Figure 5 and Figure 6 present the maximum (ε_max) and minimum strains (ε_min) of the above pipeline for soft and stiff clay, respectively; dip angles α = 60°, 45° and 30°; and fault displacement d = 4 m.

3.1.1. Impact of Steel Grade

Figure 7 presents the critical fault displacements, d_cr, for X60, X65 and X70 steel grades, soft and stiff clay seabed and three different dip angles, while Figure 8 presents the variation in pipe strength through a column chart depicting the differences in critical fault displacements, Δd_cr.

3.1.2. Impact of Cross-Sectional Geometry

Figure 9 presents the critical fault displacements of three different cross-sectional geometries. For diameter-to-thickness ratios D/t = 24, 19.2, and 16, the diameter is equal to D = 0.61 m for every case, while the wall thickness is equal to t = 25, 32, and 38 mm, respectively. Figure 10 depicts the differences in critical fault displacements, Δd_cr.

3.1.3. Impact of Intersection Angle

Figure 11 displays the critical fault displacements for four different pipe-fault intersection angles, β = 90°, 60°, 45° and 30° (Figure 2b), and Figure 12 presents the differences of critical fault displacements, Δd_cr.

3.1.4. Impact of Pressure Difference

Figure 13 presents the critical fault displacements for five different pressure difference ratios dP/P_max = 0, 25, 50, 75 and 100%, as well as for maximum allowable pressure P_max = 28 MPa, dP = 0, 7, 14, 21 and 28 MPa, respectively. Figure 14 shows the differences of critical fault displacements, Δd_cr.

3.2. Reverse Fault

For reverse fault rupture, the pipeline is assumed to be laid on loose and dense sand having the material properties presented in Table 2. The impact of main parameter values of the examined mitigation approaches is investigated, as presented in

Table 5. Main parameter values for the examined mitigation approaches for reverse fault rupture.

Parameter	Value
Steel grade	X60, X65, X70
D/t	24, 19.2, 16
β (°)	90°, 60°, 45°, 30°
k	0.9, 0.7, 0.5, 0.3
dP/Pmax (%)	0, 25, 50, 75, 100

, where X60, X65, and X70 are steel grades (Table 1); D/t is the diameter-to-thickness ratio; β is the pipe-fault intersection angle (Figure 2b); k is the friction reduction factor; and dP/P_max is the pressure difference ratio.

The “base model” of the investigation consists of a pipeline with outer diameter D = 0.61 m (24″), wall thickness t = 25 mm (1″) and X65 steel grade. The pressure difference of the pipeline is dP = 14 MPa, or 50% of the maximum allowable pressure, which is equal to P_max = 28 MPa according to Equation (14). The pipeline intersects with the fault perpendicularly, i.e., β = 90°. The values of soil resistance forces and relative displacements are presented in

Table 6. Soil resistance forces and relative displacements.

	Axial	Lateral		Upward	Downward
	Tu (kN/m)	Pbrk (kN/m)	Pres (kN/m)	ws (kN/m)	Qdown,u (kN/m)
Loose Sand	0.6	5.8	0.4	1.2	60.9
Dense Sand	0.8	7.2	0.6		409.1
	xu (mm)	ybrk (mm)	yres (mm)	zup,u (mm)	zdown,u (mm)
Loose Sand	5	305	915	10	92
Dense Sand		122			61

. Figure 15 and Figure 16 present the vertical pipe displacement, as well as the maximum and minimum strains of the above pipeline for loose and dense sand; dip angles α = 60°, 45° and 30°; and fault displacement d = 1 m.

3.2.1. Impact of Steel Grade

Figure 17 presents the critical fault displacements, d_cr, for X60, X65 and X70 steel grades, loose and dense sand and three different dip angles, while Figure 18 shows the differences of critical fault displacements, Δd_cr.

3.2.2. Impact of Cross-Section

Figure 19 depicts the critical fault displacements for diameter-to-thickness ratios D/t = 24, 19.2 and 16; loose and dense sand; and three different dip angles, while Figure 20 shows the differences of critical fault displacements, Δd_cr.

3.2.3. Impact of Intersection Angle

Figure 21 illustrates the critical fault displacements for four different pipe-fault intersection angles, β = 90°, 60°, 45° and 30° (Figure 2); loose and dense sand; and three different dip angles. Figure 22 presents the differences of critical fault displacements, Δd_cr.

3.2.4. Impact of Coating

Figure 23 illustrates the critical fault displacements for four different friction reduction factors k = 0.9, 0.7, 0.5 and 0.3; loose and dense sand seabed; and three different dip angles. Figure 24 presents the differences of critical fault displacements, Δd_cr.

3.2.5. Impact of Pressure Difference

Figure 25 depicts the critical fault displacements for four different pressure difference ratios dP/P_max = 0, 25, 50, 75 and 100%; loose and dense sand seabed; and three different dip angles. Figure 26 presents the differences of critical fault displacements, Δd_cr.

4. Discussion

4.1. Normal Fault

The extensive parametric investigation conducted in the current study has initially examined the case of normal fault rupture. As seen in Figure 5 and Figure 6, both tensile and compressive strains are developed due to the presence of bending strains. Under such conditions, pipe integrity should be tested for tensile rupture or local buckling, as described in Section 2. In this work, a rather conservative tensile strain limit is adopted, i.e., ε_t,cr = 0.5%, which is lower than the compressive strain limits against local buckling. Similar strain limits are proposed in several guidelines and correspond to the failure of girth welds [51,52,53].

Figure 7 shows the impact of steel grade, indicating that the use of stronger steel improves pipe integrity, as expected. This can be attributed to the increased values of PGDs along the axial direction for the same fault displacement, which results in bigger axial tension. It is noted that the pipe capacity increases linearly in all the examined cases.

Figure 8 clearly demonstrates that the variation in pipe strength is slightly affected by seabed properties and fault dip angle. The use of X65 steel grade, instead of X60 steel grade, increases pipe strength by a mean value of 14.2% for soft clay and 14.6% for stiff clay. The use of X70 steel grade increases pipe strength by a mean value of 29.6% for soft clay and 29.7% for stiff clay.

The impact of a pipe cross-section is investigated. Figure 9 illustrates that the use of a thicker pipe wall increases pipe integrity. Finally, it is noted that the pipe capacity increases linearly in all the examined cases. Figure 10 clearly demonstrates that the variation in pipe strength is slightly affected by seabed properties and fault dip angle. The use of a pipe with wall thickness equal to t = 32 mm, instead of t = 25 mm, increases pipe strength by a mean value of 27.9% for soft clay and 26% for stiff clay. The use of bigger wall thickness (t = 38 mm) increases pipe strength by a mean value of 50.5% for soft clay and 49.4% for stiff clay. By investigating the impact of different intersection angles, it can be clearly observed from Figure 11 that—as expected—crossing fault area with a smaller intersection angle increases pipe integrity.

As seen in Figure 12, the variation in pipe strength is slightly affected by seabed properties and fault dip angle. Pipe crossing at an angle β = 60°, instead of perpendicular crossing (β = 90°), increases pipe strength by a mean value of 15% for soft clay and 13.7% for stiff clay. In addition, pipe crossing at an angle β = 45° increases pipe strength by a mean value of 39.8% for soft clay and 38.5% for stiff clay. Lastly, pipe crossing at an angle β = 30° increases pipe strength by a mean value of 92.6% for soft clay and 88.1% for stiff clay. The favorable contribution of smaller intersection angles for normal fault rupture is consistent with the findings of Karamitros et al. [9] for buried pipelines.

In contrast to the previous mitigation schemes, the capacity of the pipe increases exponentially in all the examined cases, i.e., a small reduction in the intersection angle may be very beneficial for the pipe integrity. Nonetheless, it has to be stressed that the relative PGDs between the footwall and the hanging wall may create submarine landslides. Surface-laid offshore pipelines are vulnerable to submarine landslides, and a smaller intersection angle increases the exposed pipe length within a potential landslide zone.

Regarding the impact of pressure difference, results from Figure 13 demonstrate that critical fault displacement initially increases for pressure difference ratios dP/P_max = 25 and 50 and subsequently decreases for bigger values. This behavior is in compliance with previous results reported by Trifonov and Cherniy [13] and can be attributed to the contribution of hoop stress to the von Mises yield criterion.

Figure 14 shows that the variation in pipe strength seems to be slightly affected by seabed properties and fault dip angle. For soft clay and pressure difference ratios dP/P_max = 25, 50, 75 and 100%, the mean differences of critical fault displacement are Δd_cr = 4, 4.3, 0.6 and −7.3%, respectively, compared to dP/P_max = 0%. For stiff clay and pressure difference ratios dP/P_max = 25, 50, 75 and 100%, the mean differences of critical fault displacement are Δd_cr = 3.6, 3.7, −0.6 and −9.4%, respectively, compared to dP/P_max = 0%. Τhe maximum improvement of pipe strength is observed between pressure difference ratios dP/P_max = 100 and 50% and is equal to 11.6% for soft clay and 13.1% for stiff clay.

Conclusively, results from Figure 7, Figure 9, Figure 11 and Figure 13 clearly demonstrate that, on the one hand, critical fault displacement is generally smaller for stiffer clay, since soil resistance is greater. On the other hand, a smaller dip angle results in smaller critical fault displacements due to the increased amount of PGDs along the axial direction for the same fault displacement.

4.2. Reverse Fault

For reverse faulting, Figure 15 shows that the pipeline is under global upheaval buckling, since the vertical upward movement of the pipeline is much bigger than the fault displacement. Specifically, the maximum pipe movement is dy_max = 4.6, 5.7 and 6 m for loose sand and dy_max = 4.8, 5.9 and 6.2 m for dense sand and α = 60°, 45° and 30°, respectively. Hence, the pipeline is tested for failure modes corresponding to local buckling or tensile rupture. As seen in Figure 16, compressive pipe strains present higher values than the corresponding tensile strains, indicating that local buckling criteria should be met to maintain pipe integrity.

Under such conditions, pipe integrity is examined against local buckling utilizing the stress-based criterion of Equation (13), according to the international standard DNVGL-RP-F110 [56]. It is noted that the above criterion is rather conservative compared to the strain-based criteria of Section 2. Moreover, it has to be stressed that under realistic submarine conditions, the presence of underwater currents and/or curved pipe parts may result in the development of lateral instead of upheaval buckling. However, the failure mode of the pipeline is not expected to differentiate. As far as the impact of steel grade is concerned, Figure 17 displays that -as expected- the use of stronger material increases pipe integrity.

As shown in Figure 18, the variation in critical fault displacement is moderately affected by seabed properties. On the other hand, fault dip angle affects the variation in fault displacement, with smaller dip angles being less beneficial. The above behavior can be attributed to the increased amount of PGDs along the axial direction for the same fault displacement. The use of X65 steel grade, instead of X60, increases pipe strength by Δd_cr = 5.9, 2.8 and 0.7% for loose sand and Δd_cr = 5.6, 3.5 and 2.2% for dense sand, with α = 60°, 45° and 30°, respectively. The use of X70 steel grade increases pipe strength by Δd_cr = 15.8, 10.5, and 7.1% for loose sand and Δd_cr = 18.7, 12.9 and 9.8% for dense sand, with α = 60°, 45° and 30°, respectively.

However, Figure 19 reveals that the use of thicker pipe wall does not necessarily increase pipe integrity. More specifically, for dip angle α = 60°, a thicker pipe results in reduction in critical fault displacement, while for α = 30° critical fault displacement slightly increases. For α = 45° critical fault displacement remains constant for loose sand and presents a small reduction for dense sand.

The above results differ from the corresponding results for normal fault, where thicker pipe is beneficial for the pipe integrity. The difference can be attributed to the calculation of axial soil resistance of cohesive and non-cohesive soils. For reverse fault rupture axial soil resistance was calculated from Equation (1), which accounts for the submerged pipe weight. Since thicker pipelines are heavier, axial soil resistance increases, thus, any benefit due to thicker pipe wall may be counterbalanced. The phenomenon seems to be more intense for larger fault dip angles, due to the development of higher bending strains which contribute to pipe failure.

As seen in Figure 20, the variation in pipe strength cannot be correlated to the seabed properties and fault dip angle. For ratio D/t = 19.2, instead of 24, the difference of critical fault displacement is Δd_cr = −6.2, 0.2 and 11.6% for loose sand and Δd_cr = −9.7, −4.7, and 1.9% for dense sand, with α = 60°, 45° and 30°, respectively. For D/t = 16 the difference in critical fault displacement is Δd_cr = −6.4, 0.2 and 11.2% for loose sand and Δd_cr = −10.3, −5.6, and 1.3% for dense sand, with α = 60°, 45° and 30°, respectively. In general, a smaller dip angle and a looser sand have a more beneficial impact for the pipeline.

Compared to normal fault rupture, Figure 21 displays that crossing with a smaller intersection angle increases pipe integrity in all the examined cases. As presented in Figure 22, the variation in pipe strength is moderately affected by seabed properties and fault dip angle. Pipe crossing at an angle β = 60°, instead of perpendicular crossing (β = 90°), increases pipe strength by a mean value of 14.4% for loose sand and 13.9% for dense sand. Pipe crossing at an angle β = 45° increases pipe strength by a mean value of 28.3% for loose sand and 32.2% for dense sand. Pipe crossing at an angle β = 30° increases pipe strength by a mean value of 71.5% for loose sand and 75.4% for dense sand. The favorable contribution of small intersection angles to pipe behavior during reverse faulting is consistent with the results reported by Joshi et al. [7].

Moreover, similarly to the case of normal fault rupture, the mean value of pipe strength (solid line) increases exponentially for both loose and dense sand seabed. However, the improvement in pipe strength is smaller, compared to the case of normal faults, for the corresponding intersection angles. Finally, it has to be stressed that the relative PGDs between the footwall and the hanging wall may create submarine landslides. Surface-laid pipelines are vulnerable to submarine landslides, and a smaller intersection angle increases the exposed pipe length within the potential landslide zone. Regarding the impact of coating, results in Figure 23 demonstrate that–as expected–for smaller coating factors, pipe integrity increases for all the examined cases.

As it can be observed in Figure 24, the variation in critical fault displacement seems to be moderately affected by seabed properties. On the other hand, fault dip angle affects the variation in fault displacement, with smaller dip angles being more beneficial. This finding can be attributed to the fact that smaller friction reduction factor results in smaller axial soil resistance on the pipeline. For smaller dip angles, the affected length of the pipeline is bigger. Friction reduction factor k = 0.7 (smooth steel), instead of 0.9 (rough steel), increases pipes strength by Δd_cr = 2.2, 2.6 and 4.4% for loose sand and Δd_cr = 2.6, 5.3 and 5.7% for dense sand, with α = 60°, 45° and 30°, respectively. Friction reduction factor k = 0.5 (PE coating) increases pipes strength by Δd_cr = 5.1, 7.9 and 11.4% for loose sand and Δd_cr = 7.6, 13.2 and 13.1% for dense sand, with α = 60°, 45° and 30°, respectively. The extreme reduction factor k = 0.3 increases pipes strength by Δd_cr = 9.8, 11.8 and 16.3% for loose sand and Δd_cr = 14, 17.5 and 20.9% for dense sand, with α = 60°, 45° and 30°, respectively.

Figure 25 shows that critical fault displacement gradually decreases for larger pressure difference ratios. This behavior can be attributed to the stress-based failure criterion of Equation (13), since for a bigger pressure difference the corresponding part of the equation dominates pipe failure. It can be easily observed that when the pressure difference, dP, is equal to the maximum allowable pressure, P_max, (i.e., dP/P_max = 100%), the capacity of the pipe is practically reached; thus, its failure occurs even for very small fault displacements.

It is evident from Figure 26 that the variation in pipe strength seems to be slightly affected by seabed properties and fault dip angle. For dense sand and pressure difference ratios dP/P_max = 25, 50, 75 and 100%, the mean differences of critical fault displacement are Δd_cr = −22.2%, −48.4%, −73.2% and –98.5%, respectively. For dense seabeds, the mean differences of critical fault displacement are Δd_cr = −20.4%, −45.4%, −71.2% and −95.3%, respectively.

Finally, results from Figure 17, Figure 19, Figure 21, Figure 23 and Figure 25 clearly demonstrate that critical fault displacement is generally smaller for denser sand and for smaller dip angle, due to the increased amount of PGDs along the axial direction for the same fault displacement.

5. Conclusions

The present study investigates the efficiency of easily-applicable mitigation measures for deep-water high-pressure pipelines under seismic faulting. Pipe ruptures for normal and reverse faults with different dip angles have been investigated, while the pipeline is assumed to be embedded in cohesive and non-cohesive soils under realistic seabed conditions. The main aim of this investigation, i.e., the quantitative assessment of the effect of different steel grades, cross-sectional geometries, coatings, pressures and pipe-fault intersection angles, has been successfully addressed in terms of critical fault displacements, according to recent international standards. An efficient numerical FE model has been developed for the simulation of pipe distress, while pipe–soil interaction is simulated according to the methodology proposed by O’Rourke and Liu [2]. The main findings of this investigation can be summarized as follows:

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Crossing the fault with a small intersection angle seems to be the most efficient mitigation technique for both normal and reverse faulting. Pipe strength increases exponentially up to 90% and 75% for an intersection angle β = 30° and for normal and reverse faults, respectively. However, the potential occurrence of a submarine landslide due to faulting at the seabed surface may counteract the above favorable situation since the exposed pipe length is increased compared to perpendicular intersection.

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Standard polyethylene-based pipe coating (k = 0.5) may increase pipe strength up to 10% compared to a rough pipe surface (k = 0.9). The use of more sophisticated coating materials can further improve the friction reduction factor to k = 0.3 and may increase pipe strength up to 15%.

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The influence of pipe pressure is minor compared to the other mitigation techniques. A pressure difference of approximately half of the allowable pressure (dP ≈ 0.5P_max) seems to be the most favorable operational condition. In addition, the stress-based criterion seems to overestimate the effect of pipe pressure resulting in insignificant critical fault displacements.

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According to the stress-based failure criterion, pipe strength seems to be affected by seabed properties and fault dip angle during reverse faulting, especially for different steel grades, cross-sections and coatings. On the other hand, the strain-based criterion of tensional rupture is slightly affected.

The present study has been focused on submarine pipelines under normal and reverse faulting for certain pipe and seabed conditions. Future work may investigate strike-slip faults, more complex seabed conditions, etc. Moreover, the use of more sophisticated mitigation techniques, such as flexible joints, could be examined either individually or in conjunction with conventional mitigation measures, considering also cost and technical implementation issues, especially in adverse deep-water conditions.