Peter Permyakov，Georgy Popov，Stepan Varlamov

(1.俄罗斯科学院西伯利亚分院北极物理技术研究所，雅库茨克677010，俄罗斯;2.俄罗斯科学院西伯利亚分院 麦尔尼科夫冻土研究所，雅库茨克677010，俄罗斯)

0 Introduction

Construction and operation of large-scale linear engineering structures(oil and gas pipelines，railways，roads，power lines，etc.)are accompanied by undesirable cryogenic processes:heave，thermokarst，settlement，solifluction etc.

In this work［1］based on an approximate analytic solution of the problem of heat and moisture transfer in freezing soil developed the determination technique of heave.According to the authors，a comprehensive approach to the study of deformation of freezing and thawing soils allows to consider a common position of the development of heave deformation and settlement during phase transitions in soils that is heave is determined by a combination of development of deformations due to migration ice-accumulation，massive heaving out and shrinkage soils.Then to estimate the magnitude of de-formation it is applied the formula［1］:

where hvol——the magnitude soil heave due to volume growth，in 9%of pore water during freezing(heaving due to the massive heaving out)，m;hJw—the magnitude of heaving due to migrating moisture to the frozen zone，m;hsettl—the magnitude of settlement of the thawed part of soil，due to dehydration，m.This formula takes into account the joint process of heaving and shrinkage.The formula of frost heaving contains a lot of hard-defined empirical parameters being has not been brought to a final decision，it does not account for the main features of the freezing-thawing soil disperse systems，and therefore it reflects the process not enough adequately and completely.

Therefore in this paper，it is provided a mathematical model of frost heave in frozen soil and the results of the numerical experiment on forecast of frost heaving.

1 Materials and methods

Mathematical model of heave is based on the assumption that the expansion of the soil volume occurs in height(in the direction to the soil surface)due to an increase in pore substance due to the transition of water into ice，that is without the possibility of lateral expansion，as it is assumed in the problem of compression of compacted soil.

The magnitude of heaving S1，using the total volumetric moisture content θ，and porosity n can be described as follows:

Parameter θ，which is in the formula(2)，is determined from the solution of a system of simultaneous equations of heat and moisture exchange.

Mathematical model，taking into account the process of moisture transfer in soils，is described by the following system of equations［2］:

The system of equations(3)-(4)is closed by the equation of the amount of unfrozen water:

here c，cw—volumetric heat capacity of soil and water，J/(m3·K);T—temperature，K;τ—time，s;λ—the thermal conductivity of soil，W/(m·K);r，z—spatial coordinates(z—downward)，m;L—volume heat of phase transition，J/m3;W，Wi，Ww—total weight moisture in the form of ice and water;V=(Vr，Vz)—hydraulic rate，m/s;θ=θπ+θb— total volumetric moisture，the content of volumetric ice and water;kh—hydraulic coefficient，m/s;H=P－z—pressure，m; P—suction pressure，m;k—diffusion coefficient，m2/ s;v=0.1(v=0—decartesian and v=1－a cylindrical coordinate system);R，l—the width and depth of the area under consideration，m.

Conduction of heat equation contains convective summand that may be presented in non-divergence (non-conservative)and divergent(conservative) forms.In the numerical solution it is given the main consideration on approximation of the convective summand.It should be noted that schemes with directed differences of convective terms，taking into account the sign of the rate of filtration，are widely put into practice.Heaving of freezing rocks may occur in conditions of"open one"(with moisture inflow from the aquifer) and"closed"(without the inflow of moisture from the outside)systems.

On the surface(upper boundary)can be specified condition of infiltration snow water(effluent)or evaporation at the base of the boundary condition of the same type.On the left and right boundary of area sit is set conditions of impermeability.

2 Numerical experiment

The area of numerical simulation is two-dimensional-a vertical section of soil with coordinates(r，z).The initial parameters for the calculating experiments on heat and moisture transfer at the base of the pipeline are defined in relation to the climatic conditions of Central Yakutia.

The outer diameter of the pipe is 0.53 m，of thickness of the walls of the pipe －0.008，0.01，0.014 m，depending on the permafrost hydrogeological conditions.As anticorrosive coating pipeline it was made two layers of domestic materials"Polylen".To protect insulated pipeline against mechanical damage it was made solid lining by wooden antiseptic rack.Capacity of the pipeline－528 million m3/year and rated working pressure of 5.5 MPa.On the surface it is given the boundary condition of the third kind for the thermal conductivity with effective heat transfer coefficient，which takes into account the vegetation layer and the thickness of the snow cover.Maximum snow depth is 0.4 m.Average monthly ambient temperature and effective heat transfer coefficient，evaporation，atmospheric precipitation data are taken from the amended and rain gauge readings.Ground lithology and the linear part of the pipeline data are selected according to engineering research.

The numerical calculations(with and without the pipeline)in ten year in April show that only a change in the moisture content of a field，and the temperature is almost constant.The pipeline，buried at a depth of 1 m，takes the temperature of the surrounding soil and impacts minimal on the environment.

Fig.1 shows，calculated by the numerical method，the dynamics of heaving of seasonally thawed soil (soil)and pipeline(pipe)，laid at a depth of 1 m.

Fig.1 Seasonal dynamics of heaving of the active layer of the soil and the pipe

Consider the annual cycle of the dynamics of the surface active layer and the pipeline with frost heave.In the autumn and winter months(November-March) it occurs freezing of the top active layer，which is accompanied by migration of pore water.Winter low temperatures give birth ice crystals in free water of wet ground.When it enters the whole ice under the action of forces of crystallization it attracted loosely coherent first，and then some portion of the water film.Thus on mineral particles coated with a thin layer of water film it occurs surface unrealized energy through which water comes close to strongly bound thin film of water from the downstream wet ground.The process of moving the film and capillary water to the freezing front is called by the suggestion of M.I.Sumgin［3］，the migration of moisture during seasonal freezing of ground.The magnitude of heaving is due to increase in volume in 9%of pore water freezing.This process plays a major role in the formation of frost heave.The magnitude of heaving during the winter increases monotonically due to the migration of pore water to the freezing front.In May when snow water enters there is a sharp increase in volume(heaving)of the upper layers of the active layer.In the summer months(June-August)，when there is an intensive evaporation due to drying of the upper ground layers it shrinks the active layer.It should be noted that in Central Yakutia in average annual balance the evaporation dominates over precipitation.Autumn rains(September)stop the process of settlement from drying out.All of the above process is repeated cyclically every year.

"Seasonal shaking"of gas pipeline in comparison to the surface of the ground is a little late.The amplitude of the seasonal fluctuations of the pipeline is 3.8 cm，and the ground surface is 5 cm.Peak value of loosening is observed in late May-the maximum，and in early November-the minimum.The general course of this numerical experiment agrees well with the field observations.Perennial cyclical"seasonal shaking"pushes up gas pipeline in winter，resulting in emergency condition.

Fig.2 Dynamics of heaving

The study examined the distribution of temperature and total humidity in the section of the two-dimensional area of model problem for 50 years.The depth and width of the area is respectively 16 and 36 m.The ground consists of three layers:0～4.5 m-loam;4.5～8 m-clay，8～16 m-sand.In the lower right corner on the depth of 16 m ground water flows with positive temperature 0.8 degrees by Celsius(open system).

Top is the usual cyclical seasonal freezing and thawing of the ground based on precipitation and evaporation.Availability of ground water is warm influence of the temperature regime of rock massif.Cyclic freezing-thawing on the surface causes the migration of groundwater，the formation of injection ice on the depth of 8～15 m.This process is known as permafrost heaving mounds(bulgunnyakhi-Yakut name).

Fig.2 shows the dynamics of heaving on the right end for 50 years.Over time，the process of frost heaving increases，and the growth rate gradually decays.The magnitude of heaving on the right end of the field are，respectively，0.56(in a year)，1.49(in 20 years)and 1.72 m(in 50 years).

If the difference of the maximum and minimum values is more than 0.5 m heave，the ground belongs to highly dangerous heaving and creates unfavorable conditions on the stability of engineering structures.Similar results were obtained during the field observation experiment on gas pipelines，which are given in the works［4-5］.

Fig.3 Coefficient's dynamics of change of irregularity factor for different values of the diffusion coefficient

Fig.3 shows the irregularity factor of heaving wet soil kirreg.The numerical value of this ratio is expressed by following formula: Where hmax，hmin—the maximum and minimum values of heaving，m;L—the calculated distance between the points of maximum and minimum heaving，m.

The figure shows that each year the irregularity factor heave of the gas pipeline increases monotonically and reaches a maximum value and then gradually damps.This is due to the fact that the zone extends along the length of the heaving.The maximum value of the coefficient of irregularity equals 0.24 m/m in 35 years by the diffusion coefficient 100·k.Growth factor affects negatively the stress-strain state of engineering structures.

3 Conclusions

Thus，the numerical prediction can draw the following conclusions:

1)Within ten years around the gas pipeline with underground pro-laying it occurs changes in humidity field，although the transported gas takes the temperature of the surrounding soil;

2)It was considered a"seasonal shaking"of gas pipeline.The magnitude of heaving increases during the winter monotonically due to the migration of pore water to the front of the freezing.In May admission snow water there is a sharp increase in volume(heaving)of the upper layers of the active layer.In the summer months(June-August)，when there is an intensive evaporation due to drying of the upper soil layers it shrinks the active layer.

3)Availability of ground water in the lower horizons has a warm influence of the temperature regime rock massif.Cyclic freezing-thawing from the surface causes the migration of groundwater，the formation of ice injection.Depending on the degree of heaving ground there is an increase in the difference between the maximum and minimum values of heaving，which affects negatively the tension-strain state but linear engineering structures.

［1］ Shesternev D M，Shesternev D D.Heaving rocks under conditions of cryolithozone degradation［M］.Yakutsk:Per-mafrost Institute，2012.

［2］ Permyakov P P，Ammosov A P.Mathematical simulation of technogenic pollution in cryolithozone［M］.Novosibirsk:Nauka，2003.

［3］ Sumgin M I.Physical and mechanical processes in wet and frozen soils in connection with the formation of the heave on the roads［M］.Moscow:Transpechat NKPS，1929.

［4］ Akagawa S，Huang S，Ono T，et al.Sudden up-lift of buried child gas pipeline monitored at the boundary of permafrost and non-permafrost［J］.Permafrost engineering.Fifth international symposium.Proceeding.Yakutsk，2002，(1):125-129.

［5］ Pazinyak V V，Kutvitskaya N B，Minkin M A.Experimental studies of stability of pipelines on large-scale groundwater models［J］.Earth's Cryosphere，2006，(1):51-55.