The dynamic on-bottom stability analysis of a subsea pipeline is performed by using finite element-based advanced simulation software, which is named Flexcom [46]. Flexcom is a highly versatile software package, which is commonly used in the structural analysis of conventional and unconventional offshore structures. The pipeline is modeled as a line with an automatic mesh generation, depending on the specified element length. The structural geometric properties (i.e., internal and external diameters, wall thickness, Young’s modulus, shear modulus, mass density, etc.) are assigned to the pipeline. The hydrodynamic loads in the form of drag, inertia, and lift force coefficients are allocated to the pipeline. The external and internal coatings, such as corrosion and concrete coatings, are added by deciding the thickness of each coating layer. The sea state consists of two-dimensional irregular waves accompanied by a steady current, and both are perpendicular to the pipeline.

Figure 1 shows the external forces (including the water-induced hydrodynamic forces, the self-weight of the pipeline, and the soil resistance) which affect the subsea pipeline segment.

The time-domain simulation is performed for a duration of 3 h (10,800 s), as recommended in RP-F109 [16]. To perform full dynamic stability analysis for the pipeline, three different types of analyses are performed and then subsequently aggregated into one dynamic analysis. These three types of analyses include static analysis, quasi-static analysis, and time-domain dynamic analysis.

The static analysis is performed to study the effect of the time-independent loadings such as water current and temperature loadings. In such analysis, the full model is designed to include the geometrical layout of the pipeline and its material and hydrodynamic properties, the environmental properties, as well as the geomechanical properties of the soil (shear strength, friction, adhesion, compressibility, abrasion, corrosion, permeability, seepage, lateral earth pressure, consolidation, bearing capacity, slope stability), and geotechnical properties of the soil (grain-size distribution, weight–volume relationships, relative density, Atterberg/consistency limits, soil texture, soil phases, etc.).

In the quasi-static analysis, all vertical constraints along the length of the pipeline are removed, and the pipeline is allowed to fall onto the seabed, under the combined effect of the gravity and buoyancy loads. Settling the pipeline quickly over the seabed under the combined effect of the gravity and buoyancy loads may be performed by specifying a significant mass damping to minimize initial transients. It is worth mentioning that the static and quasi-static analyses merely facilitate contact initiation between the pipeline and the seabed for penetration, due to the mutual action of pipeline self-weight and the seabed soil stiffness.

The dynamic analysis is performed in the time domain, to investigate the effect of the time-dependent loads (e.g., the wave loads, etc.), and the nonlinear response of the concrete-coated pipeline. For a given design condition, in which the pipeline is required to withstand combined static and dynamic loads, Flexcom [46] builds a fully dynamic solution for the combined loads by superposing the static, quasi-static, and dynamic analyses and then integrating all loads through all stages to get the full response of the pipeline at the end of the dynamic simulation.

### 2.1 Pipe–soil interaction

The interaction between the concrete-coated pipeline and the soil plays a significant role in the overall pipeline response and its contribution to the on-bottom stability as a resisting force to the surrounding hydrodynamic loads. Pipe–soil interaction is a very complex process because of the mutual influence between several parameters occurring throughout the pipeline installation and operation conditions. These parameters include the hydrodynamic cyclic loading on the pipeline and the pipe–soil interaction. Different hydrodynamic models have been developed based on the Morison's equations [29] to accurately predict the cyclic loadings on the pipeline from waves and currents. These models were developed in Ref. [11, 17, 22, 25, 35, 36, 42].

The seabed soil resistance consists mainly of two components: the friction between the pipeline and the seabed soil (i.e., pure friction term) and the resistance due to the embedment of the pipeline into the seabed soil (i.e., passive resistance term). Lyons [27] experimentally concluded that the Coulomb friction model was not appropriate to describe the pipe–soil interaction, especially when the pipeline is placed on a soil of soft clay or loose sand. Coulomb friction model does not function in the properties of wave, pipe, and soil.

The passive soil resistance models have initially been developed through the PIPESTAB project [10, 45] and then the AGA project [1, 9]. Both models were appropriate for use in predicting the dynamic pipeline response and for soft clay and loose sand soils, and both are implemented in PONDUS [21] and AGA software [33], respectively. By revisiting the PIPESTAB [10, 45] and AGA [1, 9] test data and other test data sources [32], an empirical approach was proposed by Verley and Sotberg [44] and Verley and Lund [43] to evaluate embedment for sand and clay soils, respectively. Both empirical approaches are considered the basis for the current design methodologies represented in RP-F109 [16].

Neglecting the passive soil resistance results in decreasing its total soil resistance (compared to its actual resistance); especially when the pipeline is placed on soil of soft clay or loose sand, the m utual interaction between the pipeline and the seabed soil may not be modeled accurately [49], and the pipeline may also large ly displace in the lat eral direction.

In the present case study, only the pure friction term is considered, due to the present capabilities of the software, and it is represented herein by the Coulomb friction model. The Coulomb friction model is considered as the simplest method for modeling the interaction between the pipeline and the seabed soil, and it can be used in both the static and dynamic analysis [49]. The Coulomb friction model assumes pure steady plastic frict ional resistance between the pipeline and the seabed soil, and it does not consider any embedment-based cyclic loads or passive soil resistance.

To ensure the stability of the pipeline on the seabed in the presence of the Coulomb friction model, Eq. (1) must be satisfied.

$$F_{{\text{f}}} \le \mu F_{{\text{c}}}$$

(1)

where \({F}_{\mathrm{f}}\) is the friction force induced by the wave and current between the pipeline and the seabed in a direction parallel to the seabed, \(\mu\) is the coefficient of friction between the pipeline and the seabed, and \({F}_{\mathrm{c}}\) is the contact force induced by the wave and current between the pipeline and the seabed in a direction perpendicular to the seabed: \({F}_{\mathrm{c}}={W}_{\mathrm{s}}-{F}_{\mathrm{L}}\), in which \({W}_{\mathrm{s}}\) is the pipeline submerged weight in a direction perpendicular to its span and \({F}_{\mathrm{L}}\) is the lift force induced by the wave and current in a direction perpendicular to the pipeline.

The seabed is modeled as an elastic flat surface having longitudinal and transverse friction coefficients and a constant linear stiffness value. For the concrete-coated pipelines (as per the present case study), the RP-F109 [16] recommends a friction coefficient (\(\mu\)) of 0.6 for sand and rock soil and 0.2 for clay soil.

At each iteration step of the time-domain simulation, the software surveys all the nodes of the structural model of the pipeline, to check whether they are in contact with the seabed or not (as long as either a rigid or elastic seabed has been specified as part of the model). If any node registered such contact, and if either or both of the seabed friction coefficients are nonzero, it is assumed as attached to the seabed (in the plane of the seabed) using a friction coefficient-based nonlinear spring approach.

In an ideal Coulomb friction model, each of these nonlinear springs would have a force–deflection relationship in which its stiffness (in the region corresponding to zero-deflection point) is infinite. Assume, momentarily, that such infinite force–deflection relationship refers to the longitudinal direction. If there is no longitudinal force on the node, the node does not move (corresponding to zero deflection in the infinite force–deflection relationship). Indeed, the node should remain in the same location until the total nodal force exceeds the limiting friction force (\(\mu {F}_{\mathrm{c}}\)) at which point the node may move with this movement resisted by a constant force equal to the limiting friction force.

Flexcom [46] uses the modified Coulomb friction model as shown in Fig. 2, in which the infinite stiffness of the ideal Coulomb friction model (in the region corresponding to zero-deflection point) is replaced by a very high (but not infinite) stiffness (\(k\)) around the zero-deflection point. Such modification is performed by employing a slightly modified nonlinear spring characteristic.

The modified Coulomb friction model is performed to overcome the convergence difficulty inside the iterative scheme of the software while searching for the correct solution of the deflections (as in the case of most FE programs). This point is crucial to the operation of the seabed friction model. As the displacement at any arbitrary node depends, principally, on the stiffness of this section of the force–deflection curve, the stiffness is given by Eq. (2).

$$k = \frac{{\mu F_{{\text{c}}} }}{{L_{{\text{m}}} }}$$

(2)

where \(k\) is the stiffness of the Coulomb friction model and \({L}_{\mathrm{m}}\) is the mobilization length which is taken as 5% of steel pipeline outer diameter.

The value of the mobilization length (\({L}_{\mathrm{m}}\)) affects the stiffness of the nonlinear spring, i.e., the smaller is this value, the greater is the stiffness and the closer the friction model is to an ideal model. However, reducing the mobilization length of the pipeline makes it harder for the program t o converge on a correct solution of the deflections. Note also that separate mobilization lengths may be used in both the longitudinal and transverse directions, with the longitudinal value typically being shorter than the transverse one.

### 2.2 Environmental loads

The environmental loads on the subsea pipelines, for an arbitrary-selected sea state, are represented by a combination of two-dimensional irregular waves and current loadings. Such loads are assumed to be dependent on the total velocity of water particles at the pipeline level and pipeline velocity. The total velocity of water particles is calculated as the summation of the wave-induced particle velocity and the steady current velocity. The wave-induced particle velocity is generated by the transformation of the wave spectrum at the sea surface. JONSWAP wave spectrum model [19] as given by Eq. (3) is selected to represent the two-dimensional irregular sea waves.

$$S_{\eta } \left( f \right) = \alpha \cdot g^{2} \cdot \left( {2\pi } \right)^{ - 4} \cdot f^{ - 5} \cdot {\text{e}}^{{\left[ { - \frac{5}{4}\left( {\frac{f}{{f_{{\text{p}}} }}} \right)^{ - 4} } \right]}} \cdot \gamma^{{e^{{\left[ { - 0.5\left( {\frac{{f - f_{{\text{p}}} }}{{\sigma \cdot f_{{\text{p}}} }}} \right)^{2} } \right]}} }}$$

(3)

where \(\alpha\) is the Phillip’s constant, \(g\) is the gravitational acceleration, \(f\) is the computational wave frequency, \(f_{{\text{p}}}\) is the spectral peak frequency, \(f_{{\text{p}}} = \frac{1}{{T_{{\text{p}}} }}\), \(\gamma\) is the peakedness parameter of the wave spectrum, and σ is the spectral width parameter given by Equation (4).

$$\sigma = \left\{ {\begin{array}{*{20}c} {0.07, } & { {\text{if}} \;f < f_{{\text{p}}} } \\ {0.09,} & { {\text{else}}} \\ \end{array} } \right.$$

(4)

Based on the value of \(\varphi = \left( {T_{{\text{p}}} /\sqrt {H_{{\text{s}}} } } \right)\), the value of Phillip’s constant (\(\alpha\)) and peakedness parameter (\(\gamma\)) of the wave spectrum are calculated according to Eq. (5) and Eq. (6), respectively.

$$\alpha = \left\{ {\begin{array}{*{20}l} {2.73H_{{\text{s}}}^{2} /T_{{\text{p}}}^{4} } \hfill & {\varphi \le 3.6} \hfill \\ {0.036 - 0.0056\varphi } \hfill & {3.6 < \varphi < 5.0} \hfill \\ {5.07H_{{\text{s}}}^{2} /T_{{\text{p}}}^{4} } \hfill & {\varphi \ge 5.0} \hfill \\ \end{array} } \right.$$

(5)

$$\gamma = \left\{ {\begin{array}{*{20}l} {5.0} \hfill & {\varphi \le 3.6} \hfill \\ {{\text{e}}^{{\left( {5.75 - 1.15\varphi } \right)}} } \hfill & {3.6 < \varphi < 5.0} \hfill \\ {1.0} \hfill & {\varphi \ge 5.0} \hfill \\ \end{array} } \right.$$

(6)

Different random seed numbers are used to get different time series of wave heights, by assigning different phases to wave components during discretizing the wave spectrum.

The current loading is represented by a uniform current velocity distribution considering the effect of the boundary layer at the pipeline level and seabed roughness. Current velocity is given by Eq. (7) according to RP-F109 [16].

$$U_{{\text{c}}} = U_{{\text{c}}} \left( {z_{{\text{r}}} } \right) \cdot \left( {\frac{{\left( {1 + \frac{{z_{0} }}{D}} \right) \cdot \ln \left( {\frac{D}{{z_{0} }} + 1} \right) - 1}}{{\ln \left( {\frac{{z_{{\text{r}}} }}{{z_{0} }} + 1} \right)}}} \right) \cdot \sin \left( {\theta_{{\text{c}}} } \right)$$

(7)

where \(U_{{\text{c}}} \left( {z_{{\text{r}}} } \right)\) is the current velocity at reference measurement height \(z_{{\text{r}}}\) in a direction perpendicular to the pipeline, \(D\) is the external pipeline diameter including all coatings, \(z_{{\text{r}}}\) is the current reference measurement height above the seabed, \(z_{0}\) is the seabed roughness parameter, and \(\theta_{{\text{c}}}\) is the angle between current velocity direction and pipeline.

### 2.3 Hydrodynamic forces

The hydrodynamic forces which result from the wave and current loadings on the subsea pipeline are divided into horizontal and vertical forces. The horizontal forces result from the drag force (\(F_{{\text{D}}}\)) and the inertia force (\(F_{{\text{M}}}\)) and are calculated using Morison’s equations [29] as given by Eqs. (8) and (9), respectively. The vertical forces result from the lift force (\(F_{{\text{L}}}\)) and are calculated as given by Eq. (10). The drag, inertia, and lift forces are directly applied in a direction perpendicular to the pipeline segment as distributed loads.

$$F_{{\text{D}}} = \frac{1}{2}\rho_{{\text{w}}} DC_{{\text{D}}} U_{{\text{r}}} \left| {U_{{\text{r}}} } \right|$$

(8)

where \(\rho_{{\text{w}}}\) is the mass density of the seawater, \(D\) is the pipeline external diameter, \(C_{{\text{D}}}\) is the coefficient of the drag force associated with the ambient flow passing the pipeline, and \(U_{{\text{r}}}\) is the relative velocity between the fluid and the pipeline in a direction perpendicular to the pipeline, \(U_{{\text{r}}} = (U_{{\text{w}}} + U_{{\text{c}}} - U_{{\text{p}}} )\), in which \(U\) is the total velocity of the water-particles contributed by the wave and current in a direction perpendicular to the pipeline: \(U = (U_{{\text{w}}} + U_{{\text{c}}} )\), and \(U_{{\text{p}}}\) is the pipeline velocity in a direction perpendicular to its span.

$$F_{{\text{M}}} = \frac{1}{4}\rho_{{\text{w}}} \pi D^{2} \left( {C_{{\text{M}}} \frac{\partial U}{{\partial t}} - C_{{\text{a}}} \frac{{\partial U_{{\text{p}}} }}{\partial t}} \right)$$

(9)

where \(C_{{\text{M}}}\) is the coefficient of the inertia force associated with the ambient flow passing the pipeline, \(C_{{\text{a}}}\) is the coefficient of the added mass associated with the ambient flow passing the pipeline, \(C_{{\text{a}}} = \left( {C_{{\text{M}}} - 1} \right)\), \(\frac{\partial U}{{\partial t}}\) is the acceleration of the water particles contributed by the wave and water-current, and \(t\) is the computational time.

$$F_{{\text{L}}} = \frac{1}{2}\rho_{{\text{w}}} DC_{{\text{L}}} \left( {U - U_{{\text{p}}} } \right)^{2}$$

(10)

where \(C_{{\text{L}}}\) is the coefficient of the lift force associated with the ambient flow passing the pipeline.