# Nonlinear Wave-Induced Uplift Force onto Pipelines Buried in Sloping Seabeds
## 1 Introduction  
Subsea pipelines are widely utilized for transporting gas and oil from offshore areas to land [1]. In order to maintain the stability of pipelines, some pipelines are buried in the seabed. However, this method cannot guarantee the stability of the pipeline completely due to the complexity of marine environments. One of the failure mechanisms is upheaval buckling due to temperature and pressure [2]. The hydrodynamic stability of pipelines buried in the seabed can be influenced by waves since waves can induce the sediment transport and pore pressure variation [3, 4]. This study is devoted to studying the latter. Under the cyclic wave load, the seabed may get liquefied [4], which may further threaten the stability of pipeline. Two wave-caused soil response mechanisms can be summarized from previous studies [5]. The first one relates to the formation process of pore pressure, which is called residual mechanism. Generally, the residual pore pressure occurs in the initial stage of the wave loading, which is induced by the contraction of soil particles. The other is an oscillatory mechanism and it is caused by the upward-directed vertical pressure gradient under the wave trough. Since lots of studies on wave-induced residual soil response have been performed [6, 7, 8, 9, 10, 11, 12], the present study focuses on the oscillatory soil response-caused uplift force applied to the pipeline. A number of studies involving wave-seabed interaction have been conducted for investigating the oscillatory seabed response in the past few decades. Among these,Yamamoto et al. [13] used Biot consolidation theory [14] to research the wave-caused response of the porous elastic seabed, and the result shows that the seabed permeability and the rigidity ratio of soil particle shear modulus to pore water bulk modulus are the main influencing factors for the seabed response. Based on the aforementioned study, [PERSON] and [PERSON] [15] carried out a study under the assumption of a finite thickness seabed and proposed an analytical solution to the seabed response caused by waves. [PERSON] and [PERSON] [16] studied the seabed liquefaction considering the combined current and wave load and found that the opposing current is able to relieve the liquefaction, while the following current is able to intensify it. [PERSON] et al. [17] carried out a 1D experiment in which the hydrostatic pressure was included to research the influences of the soil properties as well as the wave parameters on pore pressure and liquefaction. By carrying out some flume experiments, [PERSON] et al. [18] researched the wave-caused pore pressure response along the soil depth, in which the phase lag of oscillatory pore pressure was found. On the basis of the dynamic stiffness matrix, [PERSON] et al. [19] considered the combined load of current and wave to propose a semi-analytical solution for the dynamic response of multilayered transversely isotropic seabed and found an underestimated maximum liquefaction depth under the isotropic assumption. [PERSON] et al. [20] also proposed a semi-analytical solution using fully dynamic [PERSON]'s theory and second-order Stokes theory and found that the stratification can greatly impact the seabed response caused by waves and that it is easy to liquefy when the surface layer is hard. [PERSON] et al. [21] studied the seabed response around a dumbbell-shaped cofferdam under wave action using a three-dimensional numerical model built by OpenFOAM and found that the maximum liquefaction depth around the cofferdam occurred near the steel casing along the centerline of the cofferdam. [PERSON] et al. [22] investigated the wave-induced soil response during caisson installation using the PORO-FSSI-FOAM model and found that the sequence of caisson installation has a large effect on the momentary liquefaction, and a formula is proposed for describing the variation of the liquefaction depth with the caisson submerged depth.