Numerical Modeling of Coupled Seabed Scour and Pipe Interaction ⇑ Hossein Fadaifard, John L

International Journal of Solids and Structures 51 (2014) 3449–3460

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International Journal of Solids and Structures

journal homepage: www.elsevier.com/locate/ijsolstr

Numerical modeling of coupled seabed scour and pipe interaction ⇑ Hossein Fadaifard, John L. Tassoulas

Department of Civil, Architectural, and Environmental Engineering, The University of Texas at Austin, Structural Engineering, Mechanics, and Materials, 301 E Dean Keeton St., Stop C1748, Austin, TX 78712-1068, USA article info abstract

Article history: Scouring of the seabed by ice masses poses great threat to structural integrity and safety of buried Received 8 December 2013 structures such as oil and gas pipes. Large ridge motions and large seabed deformations complicate study Received in revised form 20 April 2014 of the seabed scouring and pipe interaction using classical Lagrangian methods. We present a new Available online 21 June 2014 numerical approach for modeling the coupled seabed scour problem and its interaction with embedded marine pipes. In our work, we overcome the common issues associated with finite deformation inher- Keywords: ently present within the seabed scour problem using a rheological approach for soil flow. The seabed Seabed scouring is modeled as a viscous non-Newtonian fluid, and its interface described implicitly through a level-set Soil flow function. The history-independent model allows for large displacements and deformations of the ridge Soil–structure interaction Arbitrary Lagrangian Eulerian and seabed, respectively. Numerical examples are presented to demonstrate the capabilities of the method with regard to seabed scouring and the ensuing large soil deformation. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Wadhams (2000) were on average 2.5 m with maximum recorded gouge depths of 4.5 m. A more recent study of the Canadian Beau- The seabed scouring process (also referred to as seabed gouging, fort Sea reported gouges with average depths of 0.3 m (Héquette furrowing, plowing, scoring, scarping) describes the active physical et al., 1995), while in another (King et al., 2009), the statistical process that takes place when ice masses come in contact with the averaging of data collected during a mapping survey conducted seabed. Grounding of ice masses and subsequent scouring of the on the Grand Banks in 2004 revealed a mean trench depth of seabed occur when the draft exceeds the water depth. This interac- 0.3 m and widths of 31 m. According to these reports, trenches tion produces long gouges and basin-shaped depressions on the do not frequently exceed 1 m in depth, and gouges with depths seabed, which are common marked features in shallow marine exceeding 1 m are considered ‘‘extreme events’’ (Barrette, 2011). shelves with water as deep as 27 m (Carmack and Macdonald, The scouring of the seabed of the Arctic ocean has become of 2002; Miller et al., 2004). Seabed scouring is not limited to shallow particular interest in the past decade as a recent article by the US depth as evidence of scouring marks is visible in water depths of Geological Survey (USGS) estimated that the region is home to over over 100 m (Chari, 1980; Prasad and Chari, 1986; King et al., 10% of world’s undiscovered oil reserves, and 30% of world’s 2009). The initial contact between ice masses and the seabed untapped natural gas reserves (Gautier et al., 2009). Recovery of typically results in continual scouring of the seabed for several the oil and gas from reserves that lie beneath the sea requires con- kilometers with a preferred orientation (Woodworth-Lynas and struction of marine pipes which will be partly submerged beneath Guigné, 1990; Héquette et al., 1995; Wadhams, 2000), or, in other the sea surface with long spans supported on the seabed. Exposure instances, alternately impacting and rotating free of the seabed of the pipes to the environment leads to both man-made and geo- (Bass and Woodworth-Lynas, 1988), producing a series of craters. environmental hazards such as dragging of anchors and cables, First observed over a century ago (see e.g. Milne, 1876), seabed thermal buckling, strudel scouring of seabed, and scouring by ice scouring continues to affect one-third of the world’s coastlines masses. (Smale et al., 2008), with scouring rates as high as 8.2 In Arctic regions, seabed gouging by ice features is generally events km1 year1 (Lach, 1996). The depths of the gouge marks accepted to pose the most significant risk to buried structures such observed during early sonar scans of the Canadian Beaufort Sea as offshore gas and oil pipes (Barrette, 2011). The magnitude of the resultant forces due to plowing of the seabed by ice masses in an established scouring process is estimated to be 1000–10,000 tons ⇑ Corresponding author. Tel.: +1 5129835453. (Palmer et al., 1990), making marine pipes vulnerable to severe E-mail addresses: [email protected] (H. Fadaifard), [email protected] (J.L. Tassoulas). damage if they are to come in contact with the ice masses. The http://dx.doi.org/10.1016/j.ijsolstr.2014.06.013 0020-7683/Ó 2014 Elsevier Ltd. All rights reserved. 3450 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 burial of the marine pipes within the seabed is deemed to provide a hierarchy of meshes, which may possess the added disadvantage the ideal amount of protection against coming in contact with for- of deterioration, and in extreme cases, lack of convergence to a eign objects (Palmer and King, 2008). solution of the nonlinear equations. The contact problem and Although evidence of scouring was first observed over four dec- re-meshing requirements used in classical modeling approaches ades ago, there is limited amount of information available on the can be avoided by recasting the soil–object interaction (SOI) as a active scouring process in the open literature. The ice scouring fluid–object interaction (FOI) problem. The idea of approximating experiments performed in laboratory settings by Chari (1979) soil behavior as a highly viscous non-Newtonian fluid was success- and Barker and Timco (2002) and the in situ ice-ridge scour by fully tested by Raie and Tassoulas (2009) in numerical modeling of Liferov and Hyland (2004) are some of the few well-documented torpedo anchor installations in seabeds using a finite-volume and publicly available information on active scouring processes. based commercially available computational-fluid-dynamics A much sought-after quantity of interest in the design of marine (CFD) software. The large rigid-body displacement of the torpedo pipes is the optimum burial depth that maximizes safety while anchor, modeled as a rigid-object, was accommodated using minimizing the associated installation and maintenance costs. A frequent re-meshing of the computational domain. deterministic approach through numerical modeling of the seabed Other approaches which deviate from classical ones have been scour problem may be used for determining safety of a pipe during proposed for numerical analysis of complex problems subject to a scouring event. This type of numerical simulation can ultimately large deformations. Sayed and Timco (2009) for instance, utilized be used for finding optimum burial depths of pipes for given soil the Particle-in-Cell method (PIC) (Harlow, 1964) for flow-like types and scouring events. two-dimensional analysis of the large-displacement gouging Early efforts towards studying and gaining insight into the process. scouring process by ice masses have resorted to considering basic In this work, we use a simple constitutive model for soil in an failure modes and equilibrium of soil and ice ridges (see e.g. Chari, attempt to overcome shortcomings of classical methods for analy- 1979; Palmer et al., 1990). Recently, there has been increased sis of the seabed scour problem. We use a constitutive model that interest in employing advanced numerical methods for investigat- depends solely on the strain-rate of the soil, and re-write the gov- ing this complex phenomenon. Despite the maturity of computa- erning equations on a referential domain moving at an arbitrary tional solid mechanics (CSM) and the successfully proven velocity (ALE representation). The simplifying assumption of approaches of modeling soil as a porous medium, numerical mod- history-independent soil behavior allows the computational mesh eling of complex phenomena such as seabed scouring still faces to be moved independently from the soil particle motion, thereby difficulties. The two predominant difficulties in taking the conven- overcoming mesh distortion issues associated with classical tional approach are the interaction of a rigid object, such as an Lagrangian approaches. ice-mass, with the soil, and the subsequent large deformation We use a mesh-overlapping fictitious domain technique to (gouging) induced on and in the soil in the course of such allow for the arbitrarily large displacements of the ridge that are interaction. prescribed. Rather than following the conventional approach of The use of arbitrary Lagrangian–Eulerian (ALE) techniques (see tracking the seabed–ocean interface in a Lagrangian fashion, we e.g. Hirt et al., 1974; Hughes et al., 1981) has been extensively used adopt an Eulerian approach by describing it implicitly through a in the last four decades for solution of free-surface flow problems level-set function. The ridge-soil contact problem is therefore (see e.g. Souli and Zolesio, 2001; Lo and Young, 2004), fluid– naturally taken into account by appropriate boundary conditions. structure interaction (FSI) problems (see e.g. Bazilevs et al., 2008; This paper is organized as follows. In Section 2 we summarize Wick, 2011), and even in computational solid mechanics involving soil behavior under scouring, and in Section 3 we present how sim- large deformations (see e.g. Ghosh and Kikuchi, 1991; Rodríguez- ple constitutive model may be utilized for approximating soil Ferran et al., 2002). Within the context of fluid–structure behavior. The governing equations of the seabed-scour problem interaction problems, in ALE techniques, the structure is tracked and the constitutive model of soil are presented in Section 5.In in a Lagrangian fashion, while the fluid computational domain is Section 6 the variational formulation of the governing equations allowed to deform and warp so that conformity of meshes between and their discretization are presented. Numerical examples are the fluid and the object is maintained. Such procedures where a presented in Section 7. body-fitted mesh discretization evolves with the structure’s motion, suffer from the same issues present in Lagrangian tracking of the soil–object interface: in problems where the body is subject 2. Behavior of soil under scouring to large displacements and rotations, there is still need for computationally intense re-meshing of the domain of interest, Soon after the recognition of hazards ice features present by and inevitably remapping of variables between meshes. gouging, it was assumed that burial of marine pipes such that they Previous approaches in numerical modeling of seabed gouging do not come in direct contact with ice features is sufficient in pre- have been generally performed using ALE techniques or remeshing venting damage to the pipes. It was later determined that the soil procedures available within commercially available packages such with close proximity to the ridge undergoes large deformations, as LS-DYNA (see e.g. Konuk and Gracie, 2004; Konuk and Yu, 2007) making pipes susceptible to damage even in the absence of direct and ABAQUS (see e.g. Nobahar, 2005; Nobahar et al., 2007; Peek contact. Palmer et al. (1990) qualitatively identified three different and Nobahar, 2012). In these works, complex elastoplastic consti- modes of behavior for soil under scouring (Fig. 1): tutive models are used for the soil. Konuk and Gracie (2004) assumed undrained and isotropic soil response and used the CAP 1. Zone I: The region of seabed subjected to plowing by actively model (Simo et al., 1988) which includes nonlinear kinematic coming in contact with ice features is referred to as Zone I. hardening of the soil, while Peek and Nobahar (2012) utilized an Within this region whose depth is comparable to the scour elastoplastic material model with the von Mises yield criterion depth, the soil is subjected to large plastic deformations. Signif- and handled the large deformation using ALE methods combined icant upward and downward movement is present within this with re-meshing and re-mapping strategies. zone. Upward movement of soil particles is followed by subse- Re-meshing techniques are computationally demanding. In quent clearing to the sides of the ridge. Evidence of this mech- addition to the high computational cost, the regeneration of anism is seen in sonar scans of ocean floors which reveal scour computational domains requires remapping of variables between marks with raised shoulders on either side (Prasad and Chari, H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 3451

Fig. 1. Seabed zones distinguished by Palmer et al. (1990) (left) and schematic view of seabed scour from top (right).

1986) with reported heights as high as 6 m (Woodworth-Lynes soil resistance. The water surrounding the ice ridge may be et al., 1996). This fanning or clearing mechanism of soil is assumed to act as a lubricant, potentially resulting in reduced fric- assumed to prevent continuous build up of soil ahead of the tional resistance from the soil (Chari, 1975, 1979; Prasad and Chari, ridge, resulting in what is sometimes referred to as a steady- 1986). state scouring event. Plowing of saturated soil by mechanical tools has been reported 2. Zone II: Directly below Zone I lies a layer of soil subject to large to have significant speed effects as the force required for cutting plastic deformation during a scouring event despite not coming increases significantly with cutting speeds (Palmer, 1999). Such in direct contact within the path of ridge. Plowing of soil in Zone significant increase in forces may be explained by reduction in I results in indirect transfer of large shearing forces in this layer, pore-water pressure and the subsequent increase in shear resis- with significant forward displacement of soil particles. Centri- tance. In their numerical models, Sayed and Timco (2009) also fuge tests, geological evidence and numerical modeling have observed significant increase in ridge stresses and forces with been utilized to gain insight into the extent and depth of this increasing scouring rates, and hypothesized that the inertia of layer. the moving soil plays a role in the increase in the stresses with 3. Zone III: The region in which the soil starts to behave elastically increased loading rates. is referred to as Zone III. Soil deformations and stresses are minimum in this region, making it the safest region for burial of 3. Soil as a non-Newtonian viscous fluid marine pipes. Following the discussion presented in Section 2, the rapid load- Current design approaches assume failure of pipes in Zone I. The ing rates the soil experiences under scouring by ice masses suggest average depth of this zone in a given location can be estimated by the possibility of using simpler constitutive laws than those com- studying existing scour marks of the seabed. While this informa- monly used in soil mechanics. Under these rapid loading rates, it tion is useful in reducing risk of pipes coming in direct contact with can be assumed that the interaction between individual soil ice masses, it does not guarantee that pipes will not be over- particles is limited and hence friction is not prevalent. The soil con- stressed by the loads transmitted to them by the soil. Although stitutive model using a general fluid mechanistic approach can be pipes can be designed to safely accommodate plastic deformations, written as (see e.g. Tardos, 1997; Srivastava and Sundaresan, 2003) excessive deformations can lead to wrinkling or local buckling of f ðp0; mÞ the pipe wall, or fracture, e.g. associated with a girth weld defect. r0 ¼ _ p0I; ð1Þ _ Excessively deep trenching for pipes is cost-prohibitive, making it jj preferable instead to trench and bury pipes in Zone II. A main s f 1 f quantity of interest in design of marine pipes is thus the optimum _ ¼ r v rx v I; ð2Þ x 3 burial depth that minimizes operating costs while ensuring their structural integrity and safety during a scouring event. where r0 is the effective stress tensor, _ is the strain-rate tensor, j_ j The typical speed of ice masses is estimated to be 0.1 m/s is a measure of strain-rate, I is the identity tensor, and f is an appro- (Palmer, 1997; Yang and Poorooshasb, 1997), with occurrence den- priate function which generally depends on the effective pressure p0 sity of 2:2 108 m2 s1 (King et al., 2009). Most surveyed regions and soil volume fraction m. of terrains marked by scour marks are almost horizontal with Various forms have been previously described for the structure slopes of about 0.001 (Chari, 1979; Yang and Poorooshasb, 1997). of f ðp0; mÞ in particle and powder flows (see e.g. Schaeffer, 1987; Ocean scour records indicate continual gouging of the seabed once Tardos, 1997; Srivastava and Sundaresan, 2003). The particular scouring is initiated, and it is general practice to assume the pro- form of the constitutive law described in Schaeffer (1987) ffiffiffi cess is displacement driven. Due to the large loading rates, the soil p f ¼ 2p0 sin /; ð3Þ is assumed to be loaded in an undrained condition and the contin- ffiffiffiffiffiffiffiffiffiffi ued shear deformation takes place without any change in void ratio p j_ j¼ _ : _ ; ð4Þ (see e.g. Lach, 1996; Palmer, 1997). The loading of the soil in an undrained condition further results in a near-incompressible where p0 is the critical state pressure (Srivastava and Sundaresan, behavior of the soil. 2003), represents a simple representation of stresses in the critical A large set of different values for the undrained shear strength state. To ensure zero soil pressure at a minimum prescribed volume of soil has been reported ranging from almost zero up to about fraction mmin while allowing for rapid pressure increase as a maxi-

500 kPa (Prasad and Chari, 1986). Higher soil strength results in mum volume fraction mmax is reached, Johnson and Jackson (1987) lower expected scour depth (Prasad and Chari, 1986) and higher proposed an expression of the form 3452 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 ( r ðmmminÞ s > elastic-perfectly-plastic material with no strain-rate effects. The 0 k ðm mÞ m mmin p ¼ max ; ð5Þ effect of this parameter on the stress level is highlighted in Fig. 2. 0 m 6 m min Despite our assumption of incompressibility of the seabed, we where k is a parameter representing the stiffness of the bulk soil, chose to retain the deviatoric portion of the strain-rate tensor since and r and s are constants. Sayed and Timco (2009) utilized these it is not identically zero on the discrete level. This simple rheologi- pffiffiffiffiffiffiffi cal model of soil behavior allows use of an Eulerian method without expressions while taking j_ j¼maxð 2I2; c0Þ in seabed scour simulations to allow for plastic deformation of soil, where I2 is the sec- the need to project history of plastic stresses. The parameter c_ 0 2ð0; 1Þ reflects the threshold strain-rate at ond strain-rate invariant, and c0 is a specified yield strain-rate. Rapid scouring of saturated clay by ice masses results in the which soil response starts to show dependence on the strain-rate. seabed being loaded in an undrained condition and continued Some of the published results for the threshold strain-rate shear deformation of the seabed takes place without any change include (see Díaz-Rodríguez et al. (2009) and references therein) 7 1 7 1 in void ratio or any increase in pore-water pressure (Palmer, 1:39 10 s for Drammen clay, 4:17 10 s for Haney clay, 8 1 10 1 1997; Lach, 1996). Thus the behavior of clay may be described 2:78 10 s for various postglacier clays, 2:78 10 s for 7 1 solely in terms of its undrained shear strength and the saturated Mexico City soil, and 1 10 s for Kaolin clay (Rattley et al., mass density. The undrained shear strength of soils is dependent 2008). on the strain-rate past a critical strain-rate value, sometimes Raie and Tassoulas (2009) selected the softening coefficient khb referred to as the threshold strain-rate (Díaz-Rodríguez et al., at 0.1 on the basis of observed 10% increase in undrained strength 2009). Past the threshold strain-rate, a 5–15% increase in the with 10-fold increase in strain-rate in laboratory anchor installa- strength for every 10-fold increase in the strain-rate is observed tion tests performed in True (1976). The threshold strain-rate of 1 (see e.g. Vaid et al., 1979; Casacrande and Wilson, 1951; Lehane the soil was assumed to be 0.024 s , based on the yield shear et al., 2009). strain-rate in a vane shear test used for determining the undrained For numerical modeling of torpedo anchor installations in strength of the soil. seabeds, Raie and Tassoulas (2009) used a simpler rheological We wish to point out the following regarding the model model for approximating soil behavior under such conditions by described by Eq. (6): removing the dependence of f on the critical pressure and soil volume fraction. Following the work of Raie and Tassoulas (2009), the 1. Based on earlier discussion, a more realistic model would spec- Herschel–Bulkley model which combines the Bingham plastic ify constant undrained shear strength for strain-rates below the model for modeling the plastic forces along with the power-law threshold strain-rate, c_ 0. In the model considered here, we uti- model can be obtained by selecting lize a linearly increasing stress with strain-rate, thereby avoid- ing a discontinuity in the equivalent viscosity. f r ¼ _ pI; ð6Þ 2. While the model includes reduction in the shear strength of soil, j_ j it is not history-dependent, and cannot predict local hardening ( or softening of soil as predicted by soil plasticity models. s0 _ 6 _ f c_ c c0; ¼ 0 ð7Þ _ s _ _ 4. Effective soil stiffness in two-dimensions jj c_ c > c0; The Herschel–Bulkley model described in the previous section c_ s ¼ s 1 þ k log : ð8Þ aims to reproduce the viscous behavior of the soil, but, in two 0 hb 10 _ c0 dimensions, fails to account for the resistance exerted on the pipe by the out-of-plane soil in response to pipe displacement and In the above, r andpffiffiffiffiffiffiffiffiffiffiffiffiffip are the total stress and total pressure, respectively, c_ ¼ 2_ : _ is a measure of the strain-rate, deformation. For two-dimensional simulations, we may choose to approximate the stiffness provided in response to deformation s0 2ð0; 1Þ is the maximum stress the soil experiences in the linear for embedded pipe by employing empirical force–displacement regime, and khb 2½0; 1Þ is a dimensionless softening parameter highlighting strain-rate dependence of undrained shear strength relations (also referred to as p–y curves, see e.g. Reese and Van Impe, 2001) proposed to correlate the lateral capacity of piles to past the threshold yield strain-rate, with khb ¼ 0 representing an soil strength. Consider an elastic beam with stiffness EI of infinite length (approximating the embedded pipe in the context of our study) supported on an elastic two-way (linear) foundation (approximating the soil) subjected to a distributed load w of uniform magnitude over a length of 2a (Fig. 3). Then, the beam’s displacement, u, can be analytically computed to be ( w x4 þ A x2 þ A 0 6 x 6 a 24EI 1 2 uðxÞ¼ 2b ; ð9Þ P1Dbx þ bM1Cbx a 6 x < 1 ks where, k 1=4 b ¼ s ; ð10Þ 4EI

bðxaÞ Dbx ¼ e cosðbðx aÞÞ; ð11Þ

bðxaÞ Cbx ¼ e ½cosðbðx aÞÞ sinðbðx aÞÞ ; ð12Þ

P1 ¼wa; ð13Þ Fig. 2. Herschel–Bulkley model with different values of khb. H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 3453

Fig. 3. Distributed load of magnitude w on an inﬁnite elastic foundation.

w M ¼2EIA a2; ð14Þ 1 1 2 "# 2 wa 2b 1 2 A1 ¼ 3 ðÞþba þ 1 a ; ð15Þ 2a þ 8b EI ks 6EI ks

2b 2 w 4 A2 ¼ ðÞP1 þ bM1 A1a a ð16Þ ks 24EI and the local coordinate system is placed at the projection of the midpoint of the ridge down on the pipe when it is directly above it (Fig. 3). The midpoint displacement of beam then is

2b 2 w 4 u0 :¼ uð0Þ¼A2 ¼ ðÞP1 þ bM1 A1a a ð17Þ Fig. 4. Seabed–ocean–pipe–ridge system. ks 24EI with bending moment of the fluid and the soil domains, and denote the open and bounded r p M0 ¼ 2EIA1 ð18Þ ice-ridge and the pipe domains by X and X , respectively. Our starting point is modeling both the soil and the ocean as at the origin. Assuming that the ice-ridge produces a uniformly dis- incompressible fluids, thereby working with a single governing dif- tributed load along the length of the pipe equal in size to the out-of- ferential equation to represent both the soil and the ocean. This plane dimension of the ridge, the equivalent soil resistance due to motivates identifying the ocean and seabed domains with aid of a displacement (as opposed to viscous flow) is approximated as dynamic implicit level-set function (Osher et al., 2004) / 2 X~ f wL ð0; TÞ!R in the open time interval t 2ð0; TÞ of length T > 0 such Ksoil ¼ ð19Þ s o u0 that X ¼fxj/ > 0g, ocean domain X ¼fxj/ < 0g, and the ocean–seabed interface @Xos ¼fxj/ ¼ 0g (see Fig. 5). The behaviors and ks is determined from p–y curves. of the seabed and the ocean are thus implicitly defined and distinguished with aid of the level-set function and different consti- 5. Governing equations tutive models. Following the discussion in Section 2 and consistently with pre- Consider the system shown in Fig. 4 composed of ocean, seabed, vious numerical (see e.g. Nobahar et al., 2007; Konuk and Yu, 2007; marine pipe, and an ice-feature herein referred to as an ice ridge Sayed and Timco, 2009; Peek and Nobahar, 2012) and experimen- (or simply ridge). Let Xo; Xs Rn be open and bounded domains tal (see e.g. Lach, 1996; Barker and Timco, 2002) studies, we denoting the ocean and seabed domains, respectively, where n is assume that the ridge is sufficiently strong relative to the seabed the spatial dimension (either 2 or 3). Let X~ f denote the union of to be treated as a rigid body and undergoes a prescribed motion. 3454 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460

velocities of the ridge, respectively, r denoting the vector pointing from the center of mass of the ridge to a given point on the ridge’s body. Then, using an overlapping mesh for the ridge the variational form for the coupled seabed–pipe interaction problem using a fictitious domain approach (Glowinski et al., 1999) reduces to finding the fluid velocity–pressure pair fvf ; pg2Sf , pipe velocity vp 2Sp, level-set function / 2S/, and Lagrange multipliers k 2Sr such that Z ! f f f @v f m f f w q þ v v rxðq v Þ dx f @t X Z v Z Z f f f f f þ rxw : r dx w q g dx w kdx ZXf Z Xf Z Xr p f f p @v p p w t dx þ qrx v dx w þ rxw : r dx Z@Xsp Z@Xsp Xp @t Fig. 5. Implicit declaration of seabed and ocean through level-set function /. þ wp tp dx þ wr ðvf ðvr x rÞÞdx p r Z@X X ! The governing equations for combined fluids (seabed and ocean), / @/ f m pipe, and the evolution of the seabed in time can then be written as þ w þðv v Þrx/ dx ¼ 0 ð24Þ Xf @t v f f @v f m f f f f q ð/Þ j þ v v rxðq v Þrx r ð/Þq ð/Þg f f p p / / r r @t v for all fw ; qg2W ; w 2W ; w 2W , and w 2W. Note that in ¼ 0 in X~ f ð0; TÞ; ð20Þ the above, the domain of integration of the INS equations for the fluids is extended into the overlapping region occupied by the object using the fictitious domain method (Glowinski et al., 1999) f f ~ f q ð/Þrx v ¼ 0; in X ð0; TÞ; ð21Þ assuming that the object domain is laid out such that it overlaps

only one fluid initially. With this extension, as noted above, we @/ f m ~ f denote the combined seabed-ocean domain as Xf . Extension of þðv v Þrx/ ¼ 0; in X ð0; TÞ; ð22Þ @t v the formulation to a force-driven ridge involves additional nonlin- earity but is otherwise straightforward. @vp Prior to arriving at the semi-discrete form of the coupled prob- qp r rp qpg ¼ 0; in X~ p ð0; TÞ: ð23Þ @t x lem, it is worth noting that the fluid density qf and the viscosity lf (embedded in rf ) fields are discontinuous across the seabed inter- where v is the position in a referential domain, vm ¼ @x j is the @t v face. The resulting discontinuous integrands can be integrated by f velocity of the referential frame, q is the (discontinuous) fluid localizing the interface and computing the discontinuous integrals f density-field, r is the fluid Cauchy stress, and g is the gravitational as in the work by Marchandise and Remacle (2006). We take the constant. alternative approach of the discontinuous properties across an inter- The first two equations above are a particularly convenient face by means of regularizing the level-set function / (see e.g. form of the incompressible Navier–Stokes (INS) equations for Sussman et al., 1994; Peng et al., 1999) as a signed-distance function. two-phase flows in primal variables written on a reference frame The semi-discretized form of the equation is obtained by replac- (ALE form) (see e.g. Hughes et al., 1981; Takashi and Hughes, ing the infinite-dimensional trial and weighing function spaces by 1992). Eq. (22) is the standard scalar advection equation, or the f f their finite-dimensional approximations. Let rmom, and rcont denote level-set equation (see e.g. Sussman et al., 1994, 1998; Osher the discrete residuals of the momentum and continuity, et al., 2004) written in a non-conservative form, describing the respectively, and let r/ denote the discrete residual of the level-set evolution of the seabed interface in time. Finally, Eq. (23) expresses equation, i.e. the conservation of linear momentum of the pipe. 0 1 The above equations are coupled through appropriate boundary f conditions. We assume that the soil is fully saturated at the time of f f @@vh f m f A f r ¼ q ð/ Þ þ v v rxðv Þ rx r ð/ Þ mom h @t h h h h h scouring and there is no mass transfer between the seabed and the v ocean, yielding continuity of velocity along the interface as well as f / g k 0; 25 constant density within each fluid. Finally, we use no-slip condi- q ð hÞ h ¼ ð Þ tions between the pipe and the soil, and the ridge and the soil.

f f rcont ¼ q ð/hÞrx vh; ð26Þ 6. Variational formulation and discretization

@/h f m In this section we present the coupled ridge-seabed–pipe inter- r/ ¼ þðv v Þrx/h: ð27Þ @t v action problem. We assume that the clay is saturated, and hence there is no transfer of mass between the seabed and the ocean. Fur- Under spatial discretization, let each sub-domain be discretized 6 thermore, we restrict this formulation to a ridge with prescribed into nel elements such that X ¼[eXe; 0 e < nel where denotes motion. the appropriate sub-domain, and nel is the number of elements of Let Sf ; Sp; S/, and Sr be appropriate trial solution function that given sub-domain. Working now with finite-dimensional spaces for the fluid velocity–pressure pairs, pipe velocity, level- approximations to trial and weighing function spaces and aug- set function, and ridge displacements, respectively. Similarly, let menting the discretized equations with stabilization terms to allow Wf ; Wp; W/, and Wr denote the appropriate weighing function for equal-order interpolation of fluid velocity and pressure fields as spaces for the fluid velocity and pressure fields, pipe velocity, well as minimizing the oscillation of the velocities near boundary level-set function, and Lagrange multipliers, respectively. Further- layers leads to the following semi-discrete variational formulation: r f f p p / o more, let v and x be the prescribed translational and angular find fvh; phg2Sh; v 2Sh; /h 2Sh ; kh 2Sh such that H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 3455

(a) (b)

Fig. 6. Proﬁle of seabed at various indicated times for (a) Test 4. (b) Test 5. Extent of the yielding region of soil is indicated in dashed lines in both cases. 0 1 Z p Z Z f Z p @vh p p p p f @v f f f f @ f h m f A þ wh þ rxwh : rh dx þ wh t dx wh q þ vh vh rxðq vhÞ dx þ rxwh : rh dx p @t p f @t f X @X X v X Z Z Z Z nXel1 f m f f f f f f sp þ smomðvh vh Þrxwh rmom dx w q g dx w kh dx w t dx f h h h e¼0 Xe ZXf XoZ @Xsp Z Z nXel1 nXel1 f o f o f f smom f þ qh rx vh dx þ lh ðvh ðvh xh rÞÞdx þ scontrx w r dx þ rxq r dx sp r h cont f h mom @X X Xf Xf q Z ! e¼0 e e¼0 e Z / @/h f m nXel1 þ wh þðvh vh Þrx/h dx f m / f f @t þ s/ðv vh Þrxw r/ dx ¼ 0 ð28Þ X v f h h e¼0 Xe 3456 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460

(a)

(b)

Fig. 7. Comparison of the horizontal (f1) and vertical (f2) components of forces acting on the ridge from this study with experimental and numerical work of Lach (1996) and numerical results of Yang and Poorooshasb (1997). (a) Test 4 (b) Test 5.

f f p p / o o for all fwh; qhg2Wh; wh 2Wh; /h 2Wh , and lh 2Wh. In the of the level-set function and hence prevent artificial numerical above, the last four summations are residual-based stabilization perturbation of the seabed–ocean interface. terms added. Specifically, the first summation is the streamline We employ widely used definitions (see e.g. Bazilevs et al., upwind Petrov–Galerkin (SUPG) stabilization term (see e.g. 2007) of the stabilization parameters (s’s) given by ! Tezduyar, 1991; Tezduyar and Sathe, 2003) added to diffuse numer- 2 1=2 C lf ical oscillations of the velocity field near boundary layers at high t f m f m : ; smom ¼ 2 þðvh vh ÞGðvh vh ÞþCI f G G ð29Þ Reynold numbers, the second summation is the least-squares sta- Dt q bilization term (see e.g. Aliabadi and Tezduyar, 2000) based on 1=2 the incompressibility constraint, and the third summation repre- scont ¼ ðÞsmg g ; ð30Þ sents the pressure stabilizing Petrov–Galerkin (PSPG) (Tezduyar, 1=2 1991) for the INS equations. The last term is a stabilization term Ct f m f m s/ ¼ þðv v ÞGðv v Þ ; ð31Þ of the level-set equation to mitigate artificial numerical oscillations Dt2 h h h h H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 3457 P @nk @nk n1 @nj where Gij ¼ and g ¼ . In the above, CI is a positive con- outer boundaries. A non-reflecting boundary condition is @xi @xj i j¼0 @xi stant derived from the element-wise inverse estimate (Harari and employed for the level-set function.

Hughes, 1992), Ct is a constant introduced to reflect the time- In the experimental setup, Lach (1996) estimated the undrained dependence of the stabilization parameters for transient simula- shear-strength of the soil to vary from 10 kPa at the surface to tions, @nk is the inverse Jacobian mapping of the finite elements 25 kPa at the bottom of the domain. For our numerical simulation, @xi between physical and parent domains. Repeated indices imply we use a linear profile varied from 10 kPa at the initial seabed summation. surface to 25 kPa at the bottom of the domain: For the motion of the mesh, we solve an additional equation ( 15 6 motivated from conservation of momentum for a linearized solid 10 þ H ðH x2Þ x2 H; m p s0ðx2Þ¼ ð33Þ continuum. The variational problem reads: find v 2S such that 10 x2 > H: Z nXel1 @ m m vh m m 1 In the above, H is the initial seabed depth, and x2 is the soil location wh þ rxwh : rh bðdet GÞ dn ¼ 0 ð32Þ f @t e¼0 Xe measured from the bottom of the domain. The mass density of the seabed is assumed to be 1400 kg/m3, consistent with average m p m m reported saturated clay densities. Our results indicate strong sensi- for all w 2W . In the above, rh ðuh Þ is a (discrete) linear pseudo- stress tensor for the mesh motion, and b is a positive scalar chosen tivity on the threshold strain-rate c_ 0. For the results presented 1 from experience to preserve mesh quality. herein, we use a constant value of c_ 0 ¼ 0:0005 s based on a cali- Finally, Eqs. (28) and (32) are discretized in time using the bration against a single experimental test and remained fixed second-order generalized-a method (Chung and Hulbert, 1993). throughout all other studies. This value lies in between the value chosen by Raie and Tassoulas (2009) for numerical modeling of torpedo anchor installation, and the experimental data reported by

7. Numerical studies Rattley et al. (2008) for Kaolin clay. The softening coefficient khb is selected to be 0.1, corresponding to 10% increase in the undrained 7.1. Scour simulation with every ten-fold increase in the strain-rate. The ridge is advanced 30 m horizontally with a prescribed hor- Lach (1996) performed a series of model centrifuge tests of izontal velocity of 0:073 m/s while its motion is restrained in the scouring of saturated Speswhite Kaolin clay by a rigid indentor in vertical direction. The Newton–Raphson method is used to linear- the absence of a pipe. Two-dimensional numerical modeling of ize Eq. (28), and we use a strongly-coupled solution methodology some of these centrifuge tests was carried out by Lach (1996) by solving the entirety of the system of equations simultaneously and Yang and Poorooshasb (1997). at each time-step. In this section we consider tests designated as Test 4 and Test 5 Snapshots of the seabed profile at various time instances, sepa- in Lach (1996) using the approach highlighted herein. Two of the rated approximately by 100 s, of scouring process are shown in main differences between the two tests are the initial scour height, Fig. 6. Dashed lines indicating regions in soil exceeding the yield and the initial effective vertical preconsolidation stress. The soil in strain-rate (referred to as yielded regions) along with contour plots both tests was overconsolidated, and the effective vertical precon- of strain-rate normalized against the threshold strain-rate are plot- solidation stress in Test 5 was 30 kPa higher than that of Test 4. ted in these figures. The extent and overall shape of the yielding A computational domain of size ½85 40 m2 with a structured region shown these figures qualitatively agree well with the con- quadrilateral mesh and element sizes of 0:5 0:5m2 is utilized for tour extent of the plastic region shown in Nobahar (2005). both cases. The ridge is discretized with an unstructured triangular Consistent with the experimental setup, an out-of-plane ridge- mesh with elements of typical side dimensions of 0.6 m. We width of 10 m was assumed for computing total forces. Computed assume no-slip boundary conditions along the boundary of the components of the resultant force acting on the ridge are shown in ridge and at the bottom of the domain, and slip conditions on other Fig. 7. For comparison with other studies, both experimental and (a) (b)

Fig. 8. Subscour horizontal soil displacements. (a) Test 4 (b) Test 5. 3458 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460

Fig. 9. Profile of seabed at various indicated times for (a) No pipe. (b) Embedded pipe. Extent of the yielding region of soil is indicated in dashed lines in both cases. numerical results of Lach (1996) and numerical ones of Yang and the vertical component. A stable configuration of the front-mound Poorooshasb (1997) are presented in the same figures. could not be reached without a clearing mechanism in two- Both Lach (1996) and Yang and Poorooshasb (1997) encoun- dimensions. tered severe numerical difficulties associated with mesh distortion The advantage of our approach over classical Lagrangian meth- due to large deformations. Yang and Poorooshasb (1997) reached a ods is evident here where we are able to move the ridge by over steady-state value, though displaced the ridge by less than 8 m 30 m without any numerical difficulties. There is excellent agree- from its original position. Lach (1996) successfully moved the ridge ment in both the predicted horizontal and vertical force compo- by 15 m before encountering numerical difficulties associated with nents acting on the ridge for Test 4. Although in the centrifuge excessive element distortion while matching the horizontal com- tests a steady-state behavior for the mound was observed, our ponent of the ridge force very closely, though underpredicting simulations predict a rapidly increasing mound forming ahead of H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 3459

(a) (b)

Fig. 10. Average pipe and soil displacements at center-line of pipe. (a) Horizontal displacements (b) Vertical displacements.

the ridge as shown in Fig. 6, consistent with the slowly increasing such that the ridge is initially exposed to soil in the direction of its vertical resistance of the ridge shown in Fig. 7. The rapidly increas- motion. ing mound height ahead of the ridge may be attributed to lack of Snapshots of the seabed profiles in presence and absence of a existence of a clearing mechanism for the soil in two-dimensional pipe at various times are shown in Fig. 9(a) and (b), respectively. analyses. There is some discrepancy in the resultant ridge forces The average pipe and soil displacements at the center-line of the computed for Test 5 (Fig. 7(b)). Such discrepancy between numer- pipe are shown in Fig. 10. Although there is an initial difference ical and experimental result was also present in the work of Lach between our results and those reported by Nobahar (2005), there (1996) (indicated as in Fig. 7(b)), and we suspect it is due to the is good agreement in the overall behavior and the final displace- effects of soil over-consolidation in this work. ments in both cases. We speculate that the difference is due to our The experimental subscour soil displacements in the steady- treatment of soil as a viscous (strain-rate dependent) fluid in con- state region of the scouring measured in the centrifuge tests are trast to the elastoplastic (strain dependent) model used by marked in Fig. 8. The resultant horizontal subscour soil displace- Nobahar (2005) and, furthermore, the increase in the soil yield stress ment below the ridge as obtained from the analysis is shown in following the initial embedment step performed by Nobahar (2005). the same figure. Since a steady-state behavior of the mound was Nobahar et al. (2007) and Peek and Nobahar (2012) performed never achieved in the analysis, the subscour soil displacements comparable scouring simulations in three-dimensions. The uncou- are shown at two different locations (20 m and 30 m marks, refer pled analyses showed lower pipe deformations, consistent with the to Fig. 6). The subscour displacements at different marks both behavior observed in our simulations. In addition, the estimated show a rapidly decreasing pattern with depth and are in good pipe deformations in these analyses were less than 1.5 m, suggest- agreement with overall observed behavior measured in the tests. ing the necessity of inclusion of the effective springs discussed in Section 4.

7.2. Coupled ridge-soil–pipe interaction 8. Summary and conclusions Nobahar (2005) performed two-dimensional simulations of the seabed scouring problem using the commercial finite element In this paper, we presented a new numerical approach for package ABAQUS/Explicit. Scouring in presence and absence of a numerical modeling of the seabed-scour problem and its interac- pipe was considered to investigate the necessity of monolithic cou- tion with marine pipes. In our formulation, a rheological simula- pled versus staggered analyses in ridge-soil–pipe interaction. tion was used for the seabed. A simple viscous non-Newtonian In this section we also present the results for the seabed scour model with a constitutive model in terms of the strain-rate of in presence and absence of a pipeline using the procedure high- the soil was utilized. The ridge was modeled as a rigid object using lighted herein. For both cases, we use a rectangular computational a fictitious domain method to allow for arbitrarily large ridge domain of size ½85 40. In absence of a pipe, a structured quadri- motion, and an interface capturing technique was utilized for the lateral mesh with element sizes of 0:5 0:5m2 is utilized. For seabed–ocean interface, thereby removing the need for computa- scouring in presence of a pipeline, we use an unstructured quadri- tionally expensive remeshing and remapping of state variables lateral mesh within the vicinity of the pipeline, and a structured between a hierarchy of meshes. quadrilateral mesh with element sizes of 0:5 0:5 away from the In the first numerical example, two 2-dimensional seabed scour pipe. Same ridge geometry and discretization as in the previous problems in the absence of a pipe were considered. There was example is used for both examples in this section. excellent agreement between the resistance forces offered by soil Slip conditions are imposed on the lateral and the top walls, and during scouring with experimental results. Yielding of soil local- no-slip condition is imposed on the bottom wall of the domain. A ized around the ridge was predicted. In addition, there was excel- non-reflecting boundary condition is employed for the level-set lent agreement between experimental and numerical subscour soil function along all boundaries of the domain, including that of the displacements. In these analyses, there was a continual increase in pipe. Consistent with Nobahar (2005), a constant undrained shear mound height ahead of the ridge, speculated to be due to lack of a strength of 25 kPa is assumed for the soil. As in the previous exam- clearing mechanism for soil in front of the ridge in two-dimen- ple, a constant strain-rate threshold of c_ 0 ¼ 0:0005 is used for the sional computations. fluid model representing the soil. The second numerical example considered the seabed scour Nobahar (2005) had an initial embedment step to reach the problem and its interaction with an embedded pipe. The pipe- required scour depth. Similar to the example considered above, deformation history was found to be comparable to the results of we instead initialize the gouge profile and embedment of the ridge other two-dimensional computations. Coupled pipe–soil interaction 3460 H. Fadaifard, J.L. Tassoulas / International Journal of Solids and Structures 51 (2014) 3449–3460 predicted smaller pipe displacements compared with soil displace- Lehane, B., O’Loughlin, C., Gaudin, C., Randolph, M., 2009. Rate effects on ments in the absence of a pipe, consistent with three dimensional penetrometer resistance in kaolin. 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