Sediment undulations observed on the continental shelf in muddy prodeltas are often marked by the presence of gas in the upslope location. The growth and migration of bubbles within fine-grained muddy sediments are associated with sediment fracturing that damages the sediment and changes its effective mechanical properties. The presented theoretical investigation demonstrates by modeling that undulations may be induced by creep deformations downslope from the region with gas presence. Comparison of the dimensions of the modeled undulations with the observed ones suggests that the undulations may be enhanced over time by different processes, e.g., by the different types of the sediment deposition.
 Sediment undulations are often observed on the continental shelf in muddy prodeltas. They are interpreted as (1) depositional and transport features developed from the multiple passages of turbidity currents [Lee et al., 2002], internal waves [Puig et al., 2007], hyperpycnal flows [Wheatcroft et al., 2006], sea waves, and tides [Palanques et al., 2002]; (2) sediment deformation features, e.g., slump waves [Gardner et al., 1999], resulting from seismic, storm, and other loading [Correggiari et al., 2001], and creep waves [Hill et al., 1982]; or (3) features resulting from a combination of deformation and depositional processes [Faugères et al., 2002; Cattaneo et al., 2004]. Established diagnostics does not allow one to distinguish unambiguously between the suggested patterns [Gardner et al., 1999; Lee et al., 2002].
 Sediment undulations in the mud wedge are often marked by the presence of gas in the upslope location [Correggiari et al., 2001; Trincardi et al., 2004; Cattaneo et al., 2004; Berndt et al., 2006; Puig et al., 2007; Urgeles et al., 2011]. Specifically, acoustic turbidity interpreted as gas bubbles overlaid with layers of relatively undeformed soft muddy sediment with no gas presence terminates downslope at the onset of the sediment waving. Although a gas presence was clearly identified in the studied location, the nature of the coupling between gas and sediment undulations located in the adjacent regions has not yet been explored, and depositional processes were often suggested as being responsible for the undulations origination [Puig et al., 2007; Urgeles et al., 2007, 2011].
 Previously, gas observed within aquatic sediments at a relatively shallow water depth was associated with sediment deformation initiation via its introduction into pores through diagenesis of organic material. This causes an elevation in the pore pressure, a drop in the frictional resistance, and the formation of a “weak” detachment induced by reduced sedimentary cohesion [e.g., Sultan et al., 2004; Kvalstad et al., 2005; Urgeles et al., 2007] below the region experiencing the deformation. This effect is potentially hazardous when a low-permeability (e.g., muddy) sediment is unloaded, particularly in relatively shallow water depths [Vanoudheusden et al., 2004].
 Alternatively, in this paper, I show by modeling that creep deformation activated by the upslope free gas may be responsible for the sediment weakening and for the origination of the downslope sediment undulations. The study does not preclude other processes from playing a role in the undulation evolution.
2 Background and Suggested Hypothesis
2.1 Mechanics of Gas Migration in Muddy Aquatic Sediments
 Biogenic methane (CH4) sources are found in abundance on the continental shelf worldwide in marine sediments [e.g., Best et al., 2006]. When the concentration of the dissolved CH4 in pore waters exceeds the solubility of the gas (affected, in turn, by temperature, pressure, salinity, and other factors), methane bubbles form and thereafter try to escape the sediment through ebullition driven by buoyancy [Rothfuss and Conrad, 1998]. The strong subbottom methanotrophy in the marine environment usually prevents the escape of the methane to the water column. This leads to the formation of a well-defined gas horizon at some distance below the seafloor [Whiticar, 2002].
 The perceived pliability of soft muddy sediments and the observed fluidization patterns (e.g., gravity flow) erroneously suggest that such sediments could act fluidly or plastically in response to stress induced by the growth and migration of bubbles. However, recent laboratory simulations have shown that these sediments respond mechanically as a fracturing elastic solid [Boudreau et al., 2005; Boudreau, 2012]. The importance of grain size in determining the behavior of gassy sediments has recently been demonstrated by Jain and Juanes , suggesting that gas migration in fine-grained (muddy) sediments is governed by a fracture-dominated regime, while in coarse-grained (sandy) sediments, it occurs by capillary invasion through the sediment framework.
 As a consequence of the bubbles' propagation pattern, the seabed gassy mud should be riddled with subvertical fractures, some open and some closed at any given moment [Boudreau et al., 2005], providing a lowered-resistance conduit for the movement of other bubbles [Best et al., 2006]. This sometimes facilitates the exchange of solutes with overlaying pore water and is capable of even resuspending the sediments [Klein, 2006]. Such fracturing may affect the structural integrity of the muddy sediment and change its effective mechanical properties.
2.2 Sediment Fracturing and its Effective Mechanical Properties
 The direction and amount of the fracturing are found to significantly influence the mechanical behavior of the material. During the material deformation, flaws or cracks develop and new ones may be generated. Changes in Young's modulus and sometimes also in the Poisson ratio were suggested to be a linear function of a scalar damage variable, 0 ≤ α ≤ 1, describing the sensitivity of the macroscopic elastic properties to the distributed damage [Kachanov, 1994]. Here α = 0 corresponds to a totally undamaged material and α = 1 to a totally destroyed material. In general, the evolution of the damage may include two opposite processes: the fracturing and the recovery of the material associated with microcracks healing. Although no consensus has yet been achieved with respect to the damage evolution rule, in many studies, the damage variable α relates to inelastic strain [Martin and Chandler, 1994; Alves et al., 2000].
 In this paper I hypothesize that the upslope part of the sediment, initially fractured and therefore damaged by the migrating gas, is characterized by a smaller Young's modulus than the undamaged sediment parts. This inhomogeneity [Eshelby, 1957] generates a nonuniform stress field on the slope. The local plastic deformations and macroscopic damage of the sediment may evolve over time, inducing creep folding in the downslope part of the structure with no gas presence.
 The simple numerical model is designed to verify the suggested hypothesis. Considering a low stress applied onto the mud on the slope under the specified conditions, the mud would behave like a viscoelastoplastic [Liu and Mei, 1989] or an elastoplastic [Roscoe and Burland, 1968] solid. In this paper the latter approach is used.
 The generalized geometry of the model is composed of three inclined rectangular segments placed above a rigid base (Figure 1a). There is no gas in the downslope Segment A, whereas gas is present in the upslope Segment C and is absent in the upslope Segment B. The motivation for this subdivision of the upslope sediment part into two Segments (B and C) is that in the marine environment, gas bubbles do not usually reach the seafloor due to the intensive subbottom methanotrophy that forms a well-defined gas horizon at some distance below the seafloor [Whiticar, 2002]. This generalized geometry does not reflect the entire variety of the configurations observed in the field, including a specific mud wedge geometry [e.g., Cattaneo et al., 2007], a variable gas window location, or settings with intercalated intervals of undulated and nonundulated surfaces [e.g., Cattaneo et al., 2004]. The suggested simplified geometry is designed to verify the hypothesis that under the modeled conditions, the upslope gas is capable of inducing the downslope undulations.
3.2 Model Formulation
 Force equilibrium equation is modeled for 2-D plane strain conditions [Malvern, 1969]:
where c is the stiffness tensor, u are displacements, ρm is the material bulk density, and g is the gravity acceleration. The total strain rate () is decomposed into elastic (e) and plastic (p) parts: . For the plasticity component, the yield function is described by F = ϕ(σ) − σys, where ϕ(σ) is an effective stress chosen here to be von Mises stress and σys is the yield stress level (σ is a stress tensor). The material behaves elastically if F < 0. Isotropic softening material behavior is modeled here by
where is an effective plastic strain, Ks is a positive isotropic hardening modulus, and σys0 is an initial yield stress.
 An effective plastic strain, , at any time t accumulated since the beginning of the deformations is defined as
where is a plastic strain rate and is an effective plastic strain rate. The damage evolution is modeled by changing the local Young's modulus according to E = E0(1 − α) [e.g., Kachanov, 1994], where E0 is an undamaged Young's modulus and α a scalar damage variable. For the sake of simplicity, the evolution of the scalar damage variable is modeled as a linear function of the effective plastic strain, (k is a constant coefficient). No fracture healing is incorporated into the model. The model is purely mechanical with no transport component allowing sediment deposition. It thus does not allow assessing the role of the transport processes in the development of undulations.
3.3 Boundary Conditions
 The base boundary is fixed (u = 0) (Figure 1a); the external vertical boundaries of the sloped segments (A, B, and C) are rolled along the vertical plane (i.e., reflecting zero displacements normal to the boundary, n ⋅ u = 0, where n is a unit vector normal to the boundary). This boundary condition is chosen to simulate a continuity of the sediment cover on the slope. The inclined external boundaries of Segments A and B are subjected to the hydrostatic (water loading) boundary condition (σ ⋅ n = Pw, where σ is the stress tensor, n is normal to the boundary, and Pw is the hydrostatic pressure, Pw = ρwgh, where ρw is the water density, g is the gravity acceleration, and h is the distance from a sea surface). Continuity of displacement is prescribed in all the internal boundaries of the system. Initial damage is generated at t = 0.
 The model is solved using a Comsol Multiphysics simulation environment based on the finite element method. An unstructured triangular grid is used with 2 m maximum element size in the bulk, subsiding to 1 m element size at the upper structure boundary. The model is solved implicitly with built-in adaptive time stepping.
3.4 Material Properties
 Properties of the muddy sediments are approximated here from the known data. At the water depth h ≅ 60 m, Young's modulus in the undamaged Segments A and B is E0 ≅ 5 ⋅ 106 Pa [see Boudreau, 2012, and references therein]. In Segment C, Young's modulus is initially reduced due to fractures caused by the vertical bubbles migration, with the initial scalar damage variable α0 = 0.2. The initial yield stress level is taken as σys0 ≅ 4 ⋅ 105 Pa [Broek, 1986] and the Poisson ratio as ν = 0.45 [Boudreau, 2012, and references therein]. A small isotropic hardening modulus is Ks = 8000 Pa, the average sediment bulk density is ρm = 1600 kg/m3 extrapolated from Silva and Brandes , and the water density is ρw = 1000 kg/m3.
4 Simulations of Sediment Undulations Initiation on the Slope
 Creep is defined here as a gravity-driven phenomenon evolving under the constant loading and preserving a very low deformation rate [e.g., Mulder and Cochonat, 1996; Shillington et al., 2012]. Upslope, gas-induced sediment fracturing does not appear instantaneously but rather develops continuously as bubbles migrate. The calculations presented in the paper demonstrate the origination of the downslope undulations with an initially defined amount of the upslope damage. The fracturing initially reduces the Young's modulus in the damaged Segment C (Figure 1b) and thus creates an inhomogeneity [Eshelby, 1957] defined by the nonuniform stress field within and around it under the loading. In the beginning of the simulations when the inhomogeneity is introduced (Figure 1b), Segment C becomes significantly thinned due to the decreased Young's modulus. These deformations induce enhanced von Mises stresses (Figure 1c) and significant yielding in Segment C and at its tip (Figure 1d). The first undulation appears on the sediment surface above the Segment C tip.
 Under gravity, the undulations reach the slope toe (Figure 2). Enhanced von Mises stress is discerned below the undulations troughs (Figure 2a). Significant plastic yielding and the damage accumulation develop in Segment C and in Segment A at its toe (Figure 2b), which correspond to the locations with reduced Young's modulus (not shown in the picture). The deformations in the entire structure grow with respect to the initial undeformed configuration (Figure 1). The vertical displacements and the total displacements grow toward the slope toe (Figures 2c and 3a), indicating the mass movement in this direction. Undulations with a wavelength of ~80 m and an amplitude of ~ 25 cm develop at the sediment surface (Figure 2c). The entire structure thins toward the toe due to the differential hydrostatic loading. Below the undulations troughs, the areas of the enhanced shear stresses appear (Figure 3b). This stage may be defined as an initial phase of the undulation evolution.
 The simulated results demonstrate that the following two processes are important in creep folding generation under the given conditions: (1) gravity-induced downslope mass movement associated with a relative extension in the upper part of the downslope segment and accommodated further at the slope toe by the enhanced compression [e.g., Gardner et al., 1999; Lee and Chough, 2001] and (2) localization of the deformations along the slope in the form of the creep folding triggered by the upslope gas-induced inhomogeneity that generates stress concentrations and promotes shear planes localization in the downslope region evolving from its top. It was verified that when no upslope inhomogeneity is present, no folding appears downslope, although the general gravity-driven downslope mass movement remains.
 Recent studies suggested a dominant role of the depositional and transport processes (hyperpycnal flows, internal waves, and bottom currents) in the origination of sediment undulations on the continental shelves [e.g., Urgeles et al., 2007; Puig et al., 2007; Urgeles et al., 2011]. Despite its importance, creep has less frequently been addressed in the literature as an important mass movement process occurring on submarine slopes. Preserving a very low rate of deformation that is difficult to record, in many cases, creep leaves no faults and headwall scars [Shillington et al., 2012]. This often discards the creep from the potential mechanisms inducing the undulations on the slopes, in favor of the depositional processes, as suggested fairly by Cattaneo et al. . Despite this, creep was demonstrated to be a proven alternative mechanism to explain sediment undulations [Hill et al., 1982; Piper et al., 1985; Mulder and Cochonat, 1996; Lee and Chough, 2001; Shillington et al., 2012].
 Although an upslope gas location was often distinguished in the vicinity of the sediment undulations in the muddy prodeltas and recent experimental studies do indicate that the presence of gas in muddy sediments modifies their effective mechanical properties [e.g., Sultan et al., 2012], the role of gas in triggering the deformations remained unresolved [Puig et al., 2007; Sultan et al., 2008; Urgeles et al., 2011; Correggiari et al., 2001; Trincardi et al., 2004; Cattaneo et al., 2004; Berndt et al., 2006]. The presented modeling demonstrates that fracturing induced by the migration of gas bubbles in the upslope region of the muddy sediment is capable of triggering deformations in the downslope region with no gas presence. This new mechanism of sediment destabilization is an alternative to the mechanism of the weak layer formation caused by gas inclusion into the sediment pores [Kvalstad et al., 2005], where gas is supposed to underlie the deformed region in order to trigger the deformation. Moreover, it was numerously demonstrated that the mature gas bubbles residing in the muddy sediment are much bigger than its pores size [e.g., Boudreau, 2012, and references therein]. Therefore, this mechanism of gas bubbles inclusion in pores of the muddy sediment seems questionable.
 A modeled concave upward morphology and enhanced shear zones between the undulations indicate a similarity to the field observations [e.g., Correggiari et al., 2001; Cattaneo et al., 2004]. Moreover, these localized shear stresses grow with time, and finally, undulations troughs may be transformed into faults, in agreement with the predictions of Lee and Chough  and Shillington et al. . In addition, evolving plastic deformations amplified toward the toe may lead to the formation of a decollement zone [e.g., Lee and Chough, 2001].
 At a similar water depth (60 m), sediment undulations summarized by Urgeles et al.  that developed in Mediterranean prodeltas in a similar sediment type reached about 100 m–200 m in their wavelength and about 0.55 m–1 m in their amplitude. Undulations originated initially in the current study are characterized by the smaller amplitude and somewhat smaller wavelength. Therefore, the important role of other processes in further evolution of the undulations cannot be discarded.
 Once formed initially (e.g., by creep), the undulations may be permanently enhanced by the differential sediment deposition [e.g., Correggiari et al., 2001; Urgeles et al., 2011]. In this context, the preceding numerical modeling demonstrates [e.g., Lee et al., 2002] that sediment undulations amplified by turbidity currents or hyperpycnal flows require an initial step-like topography or a preexistent seafloor roughness that can be generated by any kind of the deformation. However, in addition to the suggested short-term deformations generated by the discrete events of the earthquakes and subsequently enhanced by the long-term sediment depositions [e.g., Sultan et al., 2008] creating the intercalated intervals of the undulated and nonundulated surfaces [Cattaneo et al., 2004], a permanent coupling may exist between the evolving time-dependent creep deformation and time-dependent sediment deposition [e.g., Faugères et al., 2002].
 Moreover, additional fracturing in the upslope segments induced by the upward migration of the newly formed bubbles may cause an additional reduction in the local sediment stiffness, and amplification of the sediment undulations downslope. This study demonstrates that the role of creep on the seafloor and its triggers is important and should be studied more extensively.
 The modeling conducted suggests the following.
 Upslope fracturing of a muddy sediment by the migrating upward gas bubbles changes the effective mechanical properties of the sediment and creates stress concentrations under the loading. This mechanism may trigger sediment creeping, resulting in the formation of downslope undulations in muddy sediments.
 A comparison of the dimensions of the modeled undulations with the observed ones indicated that the undulations may be enhanced over time by other processes, e.g., by the different types of the sediment deposition.
 The study was funded by the Ministry of Energy and Water Resources, Israel (grant ES-50-2011). I would like to thank Yizhaq Makovsky for his important comments, the Editor, and reviewers Øyvind Hammer, Roger Urgeles, and Antonio Cattaneo for their substantial contributions in enhancing this paper.
 The Editor thanks Antonio Cattaneo, Roger Urgeles, and Øyvind Hammer reviewer for their assistance in evaluating this paper.