POAC09­79

THERMODYNAMIC EVOLUTION OF ICE RIDGES UNDER WEATHER CONDITIONS IN THE BAYDARATSKAYA BAY

Shestov A. S. 1,2 , Marchenko A. V. 1,2

1 The University Centre in Svalbard, Longyearbyen, Norway 2 State Oceanographic Institute, Moscow, Russia

ABSTRACT Results of numerical simulations of thermodynamic consolidation of floating ridge are performed for the weather conditions in the region of the Baydaratskaya Bay in winter season 2002­2003. It is shown that consolidated layer of model ice ridge is extended to the depth 5 m. Below this level freeze bonds are formed in narrow gaps between ice blocks inside the ridge keel. The destruction of the ridge keel due to its impact into the seabed is studied by numerical simulations using elastic model of the aggregate of solid ice blocks filling the ridge keel. Strength properties of freeze bonds are introduced in the model using the data of field experiments. It is shown that unconsolidated rubble of the ridge keel should be destroyed due to the interaction of the keel with the seabed before the formation of deep gouges in the seabed.

1. INTRODUCTION Impact of icebergs keels into the seabed can cause damage of underwater pipelines and communications. In­situ observations of seabed gouging by icebergs demonstrated significant destruction of seabed by iceberg keels and the incorporation of ice blocks broken from the iceberg into the soil along the gouge (see, e.g., Woodworth­Lynas et al., 1991). Formation of deep gouges by icebergs is more or less evident because of their big mass and high strength of iceberg ice. Direct observations of ice gouging by ridge keels are absent because of harsh conditions for the underwater observation in­situ. Field studies and numerical simulations show that consolidated parts of ice ridges keels can extend in 1.5­2 times deeper than surrounded level ice when macroscopic porosity of the ridge keel is greater 0.2, while local peculiarities of ice blocks packing inside ridge keels can cause the formation of consolidated protrusions extended to bigger depths (Marchenko, 2008).

Numerous observations of seabed relief forms in the Baydartskya Bay of the Kara Sea excited the study of ice gouging phenomenon as one of most dangerous natural phenomena for the construction and exploitation of underwater gas pipeline from Yamal to Central Russia (Odisharia et al., 1997). Possibility of iceberg drift to the Baydaratskaya Bay from glaciers on North­East of Novaya Zemlya was discussed by Elisov (1995) (Fig. 1b). He also performed numerical simulations of icebergs drift, mentioned about icebergs events in the region in 1930­ the years and assumed that wide and deep gouges observed on the seabed at depths below 18 m can be formed by icebergs. Grounded ice similar to an iceberg was pictured in the region of the pipeline cross­section in May 2006 (Fig. 1a). Thus although iceberg events are very rare in the

POAC 09 Luleå, Sweden

Proceedings of the 20th International Conference on Port and Ocean Engineering under Arctic Conditions

June 9­12, 2009 Luleå, Sweden

region they however can explain the formation of deep and wide gouges. Field observations show significant sedimentation of the seabed gouges in 14 years from 1992 to 2006 reducing their depth calculated from the level of unperturbed bottom from 1 m to 0.6 m (Marchenko etl., 2007; Ogorodov et al., 2008) (Fig. 1c). Therefore icebergs occurrence can be estimated roughly as few times per 30 years if observed gouges were created by icebergs in the region.

Numerous field observations show the existence of grounded ice ridges (stamukhas) at depths smaller 15 m in the region under the consideration. Drilling studies of floating ice ridges in the middle of the Baydaratskaya Bay in 2007 have shown that maximal drafts of ridge keels are smaller 12 m. Therefore it is difficult to explain the formation of deep and wide gouges at depths below 18 m by the impact of ridge keels into the seabed. Nevertheless ridge keels with drafts deeper 25 m were observed in the Arctic (see, e.g., Vinje et al, 1998). In the present paper we analyse the destruction of ridge keels caused by their impact into the seabed for the weather conditions and seabed soils of the Baydaratskaya Bay.

Figure 1. Grounded ice picture in the Baydaratskaya Bay in May 2006 (a), scheme of icebergs drift from Novaya Zemlya glaciers to the Baydaratskaya Bay (b), sedimentation of seabed gouge

from 1992 to 2006 (c).

2. MODEL OF THERMODYNAMIC CONSOLIDATION OF ICE RIDGE CONSISTING OF ICE BLOCKS AND SLUSH

Computational domain 2D model ridge consisting of solid ice blocks and slush­filled caves in between these ice blocks in the ridge keel is shown in Fig. 1. The mass exchange between the brine in the slush and sea water surrounding the ridge keel is absent. The bottom and lateral boundaries of the computational domain coincide with straight lines of b z z = ( m 20 − = b z ) and ± = x x ( m 50 = + x and m 0 = − x ), and the upper boundary of the computational domain coincides with the surface of the ridge sail. The ridge was constructed from unfrozen ice blocks with a thickness of 0.5 m and had not initial consolidation. The initial porosity of the slush inside caves in between ice blocks is shown in Figure 1 as 0.4, 0.5, 0.6, 0.7, 0.8 or 0.9, respectively. The ridge keel is surrounded by sea water with a constant freezing point C T f ° − = 8 . 1 . The total area of solid ice shown in Fig. 1 is equal to 348.9 m 2 , and the total area of slush in between these ice blocks is equal to 142.4 m 2 . The initial area of pure ice in the slush was assumed to be 56.2 m 2 . The ridge was formed from a level ice sheet of about 700 m length and with a thickness of 0.5 m. Thus, the macroscopic porosity of the ridge keel, equaling the ratio of the areas filled by slush to the total area of the keel, is 29 . 0 = P .

Figure 1. The ice ridge model and computational domains.

Governing equations The heat transfer equation describing temperature distribution inside the solid ice blocks and sea water, and the motion of phase boundaries between water and ice inside is written as follows (Meirmanov, 1992):

( ) ( ) ( ) T k t T T T l c f i ∇ ∇ =

∂ ∂

− + δ ρ ρ ,

∂ ∂

∂ ∂

= ∇ z x

, (1)

where t is the time, x and z are horizontal and vertical coordinates, T is the temperature, l is the latent heat, and ) ( f T T − δ is the Dirac delta function. Symbols ρ , c and k denote density, specific heat capacity and heat conductivity, respectively. Equation (1) describes the heat transfer in ice when f T T < and in water when f T T > . Therefore, it is assumed that ( ) ( ) i i i k c k c , , , , ρ ρ = when f T T < , and ( ) ( ) w w w k c k c , , , , ρ ρ = when f T T > . Subscripts i and w indicate values for ice and water correspondingly. Specific heat capacities i c and w c are determined as temperature function as in (Marchenko, 2008), turbulent thermal conductivity of sea water is assumed to be bigger molecular heat conductivity in 1000 times.

Thermodynamic evolution of the slush is described by equation (Marchenko, 2008)

( ) T k t T

T T p

l c i ∇ ∇ = ∂ ∂

− 2 0 0 ρ ρ , (2)

where 0 p and 0 T is the initial porosity and temperature in domains. The coefficients in equation (2) are defined as follows

fi i w w c p c p c ρ ρ ρ ) 1 ( − + = , i w k p pk k ) 1 ( − + = (3)

where p is the slush porosity. It is assumed that in the case of ice blocks melting in the bottom part of the keel the released slush is turned into sea water immediately.

It is assumed that the slush transform into solid ice when its microscopic porosity is smaller 0.3 since upper bound of micro­porosity of ice samples tested for uniaxial compressive strength in small scale experiments can be estimated at this level (Høyland, 2007).

Boundary and initial conditions Energy is taken away through the top surface, where exchange with the surrounding air consists of turbulent fluxes (sensible (H) and latent (LE)) and radiation balance (R). In the model, radiation is treated as a surface flux, thus the boundary condition on the top surface of the ridge is formulated as follows:

R LE H F z + + = + , , ) (x h z + = , (4)

where n T k F i z ∂ ∂ − = + / , is the energy flux into the atmosphere and n T ∂ ∂ / denotes the normal derivative of the temperature on the surface of the ridge. Fluxes H, LE and R are expressed in terms of the air temperature, the wind speed, the specific heat capacity of air, the relative

humidity of the air, the density of air, the albedo and the surface temperature of the ridge. The formulas for these fluxes were taken from (Makshtas, 1991). Solar radiation was not considered in the simulations for the ridge and wind velocity with temperature variation was carried out with full set of meteorological data. Data were obtained from the weather station Ust’­Kara located on the east cost of the Baydaratskaya Bay. Period of winter season for calculations was chosen from the appearing of first negative temperatures in October 2002 until the first positive temperatures in June 2003. Variations of the air­temperature and absolute wind velocity are shown in Fig. 2.

a) b)

Figure 2. Temperature (a) and absolute wind velocity (b) data used in the simulations

The bottom of the computational domain is held by a constant temperature:

b T T = , b z z = , (5)

and heat fluxes through lateral boundaries of the domain are equal to zero:

0 , = ± x F , (6)

where x T k F x ∂ ∂ = ± / , m , calculated with ± = x x . In simulations temperature at the bottom of the computational domain is equal to ­1.75°C.

Initial conditions determine the spatial distribution of the initial temperature ( 0 T ) and porosity ( 0 p ) over the computational domain as follows:

) , ( 0 z x T T = , ) , ( 0 0 z x p p = , 0 = t . (7)

Initial distribution of the porosity used in the simulations is shown in Fig. 1 in domains with slush. Initial temperatures of water, slush and solid ice domains are equal to ­1.75°C, ­2°C and ­ 8°C respectively.

Results of numerical simulations Numerical simulations were carried out by Comsol Multiphysics 3.5. Freeze bonds and consolidated layer are shown in Fig. 4 after 10 days (a) and 200 days (b) of the consolidation by

colors. Grey color shows sea water. Red line at the ice bottom shows the isotherm at the freezing point. One can see that freeze bonds formed in the middle part of the ridge keel after 10 days due to initial cold reserves of the ice blocks and melted later due to the influence of heat fluxes from the sea. The consolidated layer extended to the depth 5 m during the cold season. Ice blocks in the bottom part of the ridge keel melted and caves filled by the sea water inside the keel.

Figure 4. Freeze bonds in the ridge keel after 10 days (a) and to the end of the winter season (b).

3. MODELLING OF RIDGE KEEL DESTRUCTION

Ridge keel model The aggregate of solid ice blocks in the ridge is considered as a linear elastic material with stress­ strain relationship el Dε σ = where D is the elasticity matrix (6x6). Stresses with strains are defined by column vector

[ ] T xz yz xy z y x τ τ τ σ σ σ σ = , [ ] T xz yz xy z y x γ γ γ ε ε ε ε = (8)

In calculations Young’s modulus GPa 4 = E , Poisson ratio 33 . 0 = ν and density 3 kg/m 920 = ρ were used to describe solid ice behavior under the loading.

The ridge keel was tested for the destruction at the end of the model season. Fracture condition for the freeze bonds is considered in the Coulomb­Mohr form

c tg n n + ≤ ϕ σ τ , (9)

where the cohesion and the angle of internal friction are equal to kPa 40 = c and ° = 45 ϕ (Timco et al., 2000; Shafrova and Hoyland, 2007). In terms of principle values of the stresses ( x σ and

z σ ) the yield curve corresponding to the condition (8) is described by equation

0 2 2

) , ( = + −

− +

≡ c f z x z x z x

σ σ σ σ σ σ (10)

Forces applied to the keel

Forces applied to the ridge keel by seabed soil depend on the soil properties. In the Baydaratskaya Bay seabed seabed soils are represented by sands with the cohesion smaller 4 kPa and the angle of internal friction about 30 o , and by clays with mean cohesion 20 kPa and mean angle of internal friction 17 o . Lower bound of forces created by a soil on solid indentor imitating ridge keel (Fig. 5a) are estimated with the formula describing passive pressure of the soil on vertical wall without friction (Sokolovskii, 1990)

γ γ γ σ

sin 1 sin 1 cot

− +

− = wz k h , (11)

where k is the soil cohesion, γ is the angle of internal friction and w is the specific weight of

the soil in the water. Fig. 5b) shows mean stresses h dy h h h / 0

∫ − = σ σ calculated with formula

(10) for 19 types of seabed soils from the region of the pipeline cross­section in the Baydaratskaya Bay. Thick line shows the stress 20 kPa, which can be considered as lower bound of the stresses. The same level of stresses was estimated from the load applied to the several

working knifes of plough (The qualification trials of plough PBS­7/9, 2005). Evaluating horizontal force applied per unit area of the knife is about 20 kPa.

Figure 5. Scheme of the indentation of ridge keel in the seabed (a). Mean stresses over the ice­ soil contact surface (b).

The stress ( v σ ) at the ridge bottom caused by its grounding due to tidally induced water level depression is determined as follows

L dz z gS w h

w v / ) ( 0

∫ −

= ρ σ , (12)

where w ρ is water density, g is the gravity acceleration, ) (z S is the area of horizontal cross­ section of solid ice in the ridge keel at the level z , 0 > h is the drop of the water level, and L is the length of contact line between the ridge keel and seabed. Simulations were performed with

cm 30 = w h . For the model ridge it means that kPa 50 = v σ (Fig. 6a).

Results of numerical simulations Results of destruction tests are represented in Fig. 6 for the end of the winter season when consolidated layer has maximal sizes. Destructed areas where the condition 0 ) , ( < z x f σ σ is satisfied are shown by white color. The ridge keel was destructed under both types of loading in the vertical (Fig. 6a) and horizontal (Fig. 6b) directions. It means that the impact of the ridge keel in the seabed causes significant displacements of ice blocks filling the keel before the seabed soil will get significant deformations.

4. CONCLUSIONS Numerical simulations have demonstrated that consolidated layer of the model ice ridge constructed from ice blocks with thickness 0.5 m and maximal draft of the ridge keel 20 m extends to 5 m depth under the weather conditions in the Baydaratskaya Bay in cold season 2002­ 2003. Freeze bonds in the middle part of the ridge keel formed due to initial cold reserves of solid ice blocks on initial stage of the consolidation and then melted. Ice blocks at the bottom of ridge keel melted over all winter season.

The destruction of ridge keel was simulated with elastic model for the aggregate of solid ice blocks filling ridge keel and Coulomb­Mohr fracture condition of freeze bonds between the

blocks. Numerical simulations have shown that the impact of unconsolidated part of the ridge keel into seabed soil causes the destruction of freeze bonds between ice blocks. It was found for the impact of the ridge keel into the seabed in horizontal and vertical directions both.

a)

b)

Figure 6. Simulated destructions of the ridge keel under its vertical (a) and horizontal (b) loading by the seabed soil.

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