外文翻译

外文翻译：

模拟永冻土层下的凝析气井井喷

共轭气体凝析气井的井喷问题被认为是一种新的问题提法。因此，在普通条件下和井喷条件下，研究和模拟井下周围岩石的温度和井下永冻岩层再解冻过程中对井下流体的影响，是非常有意义的。

井喷在凝析气领域，特别是在遥远的北方，可能会导致严重的环境后果。为了选择一种好的手段，来防止井喷事故的发生，因此搞清楚井下液体流速、压力分布和温度场的分布是非常有必要的。不同的人对于井喷流体的计算有着不同的特点和见解。但是，他们通常都是不考虑凝析气混合物的成飞和相变，而只是考虑温度方面的影响。井喷可能发生在油井开始运转的时候，也可能发生在油井的生产过程中。

在北方等领域，有一个很明显的特点，就是永冻土层的存在。由于采油井的运转，导致永冻层的解冻，因此可能进一步导致严重的井喷事故的发生。为了进一步了解计算解冻方面的因素，因此了解温度场的分布和其随时间是非常有必要的。因此就要求多年冻土区的流体在钻井方向的议案调查必须包括流体的实际流量（天然气、凝析气混合物、油）和围岩温度场的计算一般初步认为，流体的流动比较稳定，因此可以初步的估计特征时间差（流体通过沿井方向的时间和和使岩石上的温度改变一度之间的时间差）。

1、在构建凝析气的数学模型议案中，有两相多组分混合物沿井方向运动，我们将有以下假设：油井的运转时稳定的和一维的压力和阶段性的温度在每一个横截面的条件是相同的，在井截面上单位时间内通过的混合物满足局部热力学平衡。因此就可以用大量的连续扩散运动来近似的描述多组分混合物的状况。

在这些假设的情况下，系统方程就是描述了这种两相多组分混合物沿井垂直方向的作业形式【2-4】：

coast G G G g m =+=1 (1.1)

]})1([1{])1([21

2221222ρ??ρρ??ρλρ-+--++=l s g l s g m m G G F dz d G G DF g dz dp (1.2) )()]2

()2([22T T DK g W i G W i G dz d c m l l l g g g --=+++πρ (1.3) 其中l G 、m G 和 g G 分别是混合物和气体的质量流率和液相，P 和T 是混合物的压力和绝对温度，D 和F 是通道截面的直径和面积，?是真中的气体含量， m ρ、 l ρ 和 s ρ 是混合物的密度、气体和液体的密度；在达西魏兹巴赫公式中m λ是液压阻力系数；g i 和 l i 是气体和液体的焓；l w 和 g w 是他们的速度，g 为重力加速度；k 是传热系数；c T =)(?c T 是井壁岩石的温度，?是沿井轴方向测量的坐标。

为了使系统（1.1）-（1.3）更准确，我们有了上面的两个阶段的状态方程。 ),(T P g g ρρ=;),(T P l l ρρ= (1.4)

),Re ,,,,(m m m l g b l We Fr ρρμμβ??= ，),Re ,,,,,(m m m l

g b l m m We Fr ερρμμ?λλ= (1.5)

gD w Fr m m 2= ，σρρg l m m Dw We -=22，l

b g l m m Dw μββμ?βρ?βρ)1()1()1(Re 1212-++--=-- 方程式1.5用混合物的速度（确定的）说明了不同流体的不同形式。除此之外，也可以由气相体积浓度体现，四个系统可以区分为：泡沫、环形的、分散的环形的、分散的。

此外还有：

1)(,)1(,--=-==l

l g g g g l l l g g g G G G w F G w F G ρρρβρ??ρ (1.6)

)(1,)1(l

l g g m l g m G G F w ρρρβ?ρρ+=-+= 在这里，是流体的含量，l μμ，b 和 g μ是水的液相粘度和气体阶段的粘度； 是管壁的相对粗糙度；是液体的张力；m w 是混合物的体积平均流速。

假设本系统的温度是已知的。在初始的瞬间，它对应的是体热梯度，随后由外部温度场计算来确定其具体值。由前面给定的条件，从1.1-1.6式可得给定的流体g G ，l G ，m w ,g w ,l w , m ρ，l ρ，g ρ，P ,T,m λ，?，g i ，l i ，g μ，l μ，β和σ。的函数。

这里我们引入新的变换式。

g D T P G G F T P m m l

l g g ρφλρ??ρ+=Φ=Φ-+=Φ=Φ122222112),(];)1([1),( (1.7)

)2

()2(),(2233l l l g

g g w i G w i G T P +++=Φ=Φ; )(),(44T T DK g T P c m --=Φ=Φπρ ])1([1)(132141432T

p dz dT T T dz dp ?Φ?-?Φ?+Φ?=?Φ?Φ-?Φ?Φ?=φ； (1.8) T p T p ?Φ??Φ?-?Φ??Φ?+=?1331)1( (1.9)

为了从p(H)=*p , T(H)=*T 中整合（1.8）已知的初始条件，其中H 是井的深度， *p 和 *T 是井底的压力和温度。

在井喷条件下，*p 和 m G 的数量是未知的，因此必须指定两个条件，用来整合系统1.8。

其中一个是表示灌入井中流体的关系式【9】

222)(m

m k G G B +A =P -P * (1.10) 在这里，是已知的油藏压力；A 和B 是滤波器系数，其取决于被调查井的稳定情况。这里有井喷条件的临界条件（Z=0）；从这点上讲，则可以确定井喷的第二个条件：0→∞→dz dT dz dp ，当Z=0时，或如下：1.8

0=?； 0=z (1.11)

2、为了确定孔壁的外部温度，有必要去解决就和其周围环境的传热问题。 如上所述，良好的流动性被认为是稳定的，土壤中的热场是不稳定的。因此，问题解决如下，在最初的瞬间温度由地温梯度和1.8式决定。P(z)和T(z)的值由 和 给定。然后假设在某个时间间隔T(z)是恒定的，由外部的温度场计算新的T （z ）的值。直到T(z)为给定的值的范围。P(z)和T(z)之间的关系可以再次由T(z)的计算方法还有**T ，p 和 m G ，然后继续外部温度场的计算，进一步确定P(z)和T(z)的关系。

在描述围岩温度场方程的推导过程中，我们将作如下假设：假设在T=0℃时，岩层中的水发生相变，与径向热通量相比较垂直井的热影响区比较下一点，且在井中由于热传导产生的热通量小于由对流产生的热通量。

我们假设最初0 z h 土壤为永久冻结，而h < z H 土壤解冻。我们把多年

冻土区区分成两个阶段。

在开始阶段，假设土壤是在温度为T< 0°C 下冻结。第二个阶段的开始时期对应的是，土壤中空袭的温度对应的是零度的时刻。第二个阶段的土壤可分为两个区：解冻和冻结区，区分于其相变从井的中心开始的时刻。

在所有这些情况下，土壤的温度分布，可以用一下无量纲形式的热传导方程式来描述：

20

00022)1(r a x r r t r r r x t i

i i i i i i i τζττθθθθ===T T =??+??=??，，，， (2.1) 在这里，1=i 对应的是解冻的土壤, 2=i 对应的是冻土，3=i 对应的是没有冻结的土壤，i a 是热扩散系数，0r 为半径，0τ为一个特定的时间，T 是井轴的温度，是当前井的半径，为世间。

在第一个阶段，冻土被加热时，初始条件和边界条件的形式是：

1)1(222=-=??r r

，θαθ (2.2) 0)1

0,22===M M t t ，（；θθθ (2.3) ∞→→r M ;2θθ (2.4) 在这里，0ττM M =t 对应的是冻土开始融化的时间，T T =M M θ为冻土初始温度分布，可以用无量纲形式表示。

在第二个阶段，当有两个区域：解冻区和冻结区。因此我们有了方程2.1为辅助条件。

)()()(1)(02t s r r t r t S ?==M M ；，；θθ (2.5) )(01)1(21111t S r r r

====-=??；；；θθθαθ (2.6) )(2211t S r dt

ds r r ==??+??-；θλθλ (2.7) ∞→→M r ；θθ2 (2.8)

在冻结区（3=i 低于多年冻土区）中，我们有：

01)1(0333===-=??t r r

；；；θθθαθ (2.9) ∞=→t ；03θθ (2.10)

00000200

)(3,2,1(r t S i qp r kr L i i i i τξθθτλλλα=T T =T T ==T ==**）（，，），， 在这里，k 是传热系数，*

i λ是导热系数，q 是水冰相得比热，L p 是大规模含

冰的土，)(0ζT 是在M T =T 时土壤的温度分布,)(0z T 是初始的土壤解冻，)(τζ

是

冻土的解冻半径。

（2.4）（2.8）（2.10）的成立是有条件的，在这里我们将介绍热影响半径R （t ）满足的条件：

)(032022t R r r

r ==??=??==M ，，，θθθθθθ (2.11) 现在，我们将使用积分法，在i=2的情况下对等式2.1积分，分别以2.2和2.11式为积分边沿。

122212212)1(=--=??=???r R R x r

r x rdr r θαθθ (2.12) 我们利用2.11的条件来化简2.12式，结果是：

r =-R (2.13)

把2.13式代入2.12，我们得到了可以代替2.1式（i=2）的近似的积分方程。 M -=????θθθdt

dR R dr r dt d rdr r R t R 12)(12 (2.13) 我们可以用下面的 的形式代入方程2.14中：

r t a t a r t a t r )( ln )()(32122++==）

（，θθ (2.15) 其中 是未知的时间，可以从相应的边界条件中求出来。

从2.14式中我们得到了R(t)方程的第一个阶段，其连同关系式2.15就可以确定出温度在冻土中的分布。

为了解决第二个阶段的问题，我们需要冻土区的温度分布，并且通过与等式

2.15类比的方法，得到下面的形式：

r a a r a ***++=3212 ln θ (2.16) 是从相应的边界条件中确定出来的。

我们用以下形式来表示解冻区的温度分布：

r

s ln )1r s ln (1112ααθ-+= (2.17)

我们把关系是2.16和式2.17代入解冻边界式2.7中，我们得到了关于解冻半径S(t)的方程式：

为了从R(t)中得到方程式，我们给等式2.1两边同时乘以r,并从S 积分到R 。我们把关系式2.6和2.11，用和上面相似的方法处理后，就会得到下面的方程式： s r R t

S x dt dR R dr r dt d =M ??-=?2212θθθ (2.18) 把2.16式代入2.18式中，并展开一系列的变换，我们就能得到一个常微分方程，连同方程S(t)，给出了一个系统的两个常微分方程：解冻半径的方程S(t)和热影响半径方程R(t)（第二阶段）。初始条件是1)(=M t S 和 M M =R t R )(。

在解冻区(h z H)用的是和冻结区相同的方法解决的，但是要用3=i 替换

2=i and 0θ for M θ。

描述在t=井喷发生的时刻，确定井喷的（最大可能）流量式，用的最多的是式1.8、1.10和1.11。

3、经过计算后，我们发现了合适的计算方法和个人计算机程序包。测试计算结果和已知的实验数据。

下面我们举一个关于亚马尔半岛的井喷计算结果。

深度以及高度分别为H=1500和h=250米的永冻土层。预计M ％的流体如下：4CH - 96.37, 62H C – 2.89, 83H C - 0 .05, 104H C - 0 .03, 5C - 0.01,

2CO -0 .22,and 2N - 0.43 .。

井喷流量的计算有两种方式：

第一种是井喷发生在油井运转的启动过程中，其计算不包括解冻的情况。经过计算得出他的井喷流量率是m G =35.3千克/秒，井口的压力和温度分别为

p(0)=0.84兆帕和 T(0)=247科尔文。压力P(z)和温度T(z)在沿井方向的变化分别表示在图1中。

第二种情况是在井开始运营了352天以后，发生的井喷事故。其中有=35.1千克/秒、p(0)=1 .03兆帕和T(0) =257科尔文。压力P(z)的初始分布和混合物的温度T(z)变化分别表示在图2中。图3显示了解冻半径ξ(t)随时间的变化。

从这个结果中很明显的能看出，外部温度场的变化对对土壤的加热和解冻等的影响，对井喷事故的发生影响很小。但是对井口压力和温度参数的影响是非常大的。其压力和温度会影响我们选择抑制井喷事故的方法。

通过上面的计算，我们可以得出ξ(t)是随着时间的变化单调递增的，特别是在井运营的初期，ξ(t)的增减量式非常明显的。P(z)和T(z)随时间的推移变化不大。计算还表明，我们初步估计的井中和土壤中的情况特点是正确的。

从专业的角度来看，我们做了关于马亚尔半岛油井一系列相似的计算。这些结果证实了上面所说的观点。

REFERENCES

1 .V. D . Malevanskii and E . V . Sheberstov, Hydrodynamic Calculation of the Conditions of Suppression of Oil and Gas Well Gushers [in Russian], Nedra, Moscow (1990) .

2 .G. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York (1969) .

3 .R. I . Nigmatulin, Dynamics of Multiphase Media, Pt . 1 [in Russian], Nauka, Moscow (1987) .

4 . P. R . Gimer, V . I . Isaev, Yu . D . Raiskii, and G . D . Rozenberg, "Model and methods of calculating the steady nonisothermal two-phaseflow of multicomponent mixtures in vertical tubes,° in : International Conference on the Development of

Gas-Condensate Fields,Krasnodar, 1990 : Proceedings, Sec . 5 [in Russian], Krasnodar (1990), p . 112 .

5 .V. A . Mamaev, G . E. Odishariya, O . V . Klapchuk, et al ., Motion of Gas-Liquid Mixtures in Pipes [in Russian], Nedra, Moscow(1978) .

6 .M. M . Dubina and B . A . Krasovitskii, Heat Transfer Conditions and Mechanics of the Interaction of Pipelines and Wells with Soils[in Russian], Nauka, Novosibirsk (1983) .

7 .T . Goodman, "The heat-balance integral and its application to problems involving

a change of phase," Trans. ASME, 80, 335 (1958) .

8 .G. S . Gryaznov, Gas Well Design in Permafrost Regions [in Russian], Nedra, Moscos (1978) .

9.I . A . Charnyi, Underground Hydrogasdynamics [in Russian], Gostoptekhizdat, Moscow (1963) .

SIMULATION OF GAS-CONDENSATE WELL BLOWOUT UNDER

PERMAFRORT CONDITIONS

I. M. Astrakhan, S . A. Egorushkin, V . I . Isaev,

G. D. Rozenberg, and F . A . Slobodkina UDC

532 .529 .5 :622 .276 .5

The blowout problem for a gas-conjugate condensate well is considered in a new formulation . The effect of the fluid flow in the well shaft on the temperature of the surrounding rock and the reaction of the surrounding permanently frozen rock on the flow regime in the shaft during the thawing process are simulated and studied under ordinary operating and blowout conditions .

Blowouts in gas-condensate fields, especially in the Far North, may lead to serious environmental consequences . In order to choose a means of shutting down a gushing well it is necessary to know the flow rate and the distribution of pressure and

temperature along the well shaft . Various authors have made hydrodynamic

calculations for gushing gas [1] or gas-condensate wells, but without taking into account the chemical composition of the gas-condensate mixtures and their phase transitions [1] .

Moreover, it is usual to consider only the isothermal flow regime . Blowouts may occur both when a well is started up and sometime after it begins to be operated . It should be noted that during operation the mass fluid flow rate is kept constant and its magnitude is known .

In the fields of the Far North, a characteristic feature of which is the presence of permafrost, the consequences of a blowout are aggrevated by the deep thawing of the soil in the process of operating the well . In order to calculate the thawing it is

necessary to know the temperature distribution in the well shaft and its variation with time while the well is being operated .Therefore the investigation of the motion of the fluid along a well shaft drilled in permafrost is a problem that combines the

calculation of both the actual flow of fluid (gas, gas-condensate mixture, oil) in the well and the temperature field in the surrounding rock . It is assumed that the flow in the well may be regarded as steady, since according to preliminary estimates the characteristic times (the time of passage of the fluid along the well shaft and the time taken by the rock temperature on the well wall to change by 1 0 ) differ by 2-3 orders .

1 . In constructing a mathematical model of the motion of a gas-condensate, i .e ., two-phase multicomponent mixture along a well shaft we will make the following assumptions : the motion in the well is steady and one-dimensional ; the pressures and temperatures of the phases are the same and constant over the cross section of the well ; in each cross section the conditions of local thermodynamic equilibrium are satisfied for the volume of mixture passing through that section in unit time ; the diffusion approximation of a multivelocity continuum is used for describing the motion of the multicomponent mixture .

When these assumptions are taken into account, the system of equations

describing the motion of a two-phase multicomponent mixture along the shaft of a vertical well in the operating regime has the form [2-4] :

coast G G G g m =+=1 (1.1)

]})1([1{])1([21

2221222ρ??ρρ??ρλρ-+--++=l s g l s g m m G G F dz d G G DF g dz dp (1.2) )()]2

()2([22T T DK g W i G W i G dz d c m l l l g g g --=+++πρ (1.3) where l G 、m G and g G are the mass flow rates of the mixture and gas and

liquid phases, respectively ; p and Tare the pressure and absolute temperature of the mixture ; D and F are the diameter and area of the well channel cross section ; ?is the true gas content ; m ρ、 l ρ and s ρ are the densities of the mixture and the gas and liquid phases ; m λ is the hydraulic resistance coefficientin the Darcy-Weisbach formula ; g i and l i are the enthalpies of the gas and liquid phases ; l w and g w are

their velocities ; g isthe acceleration of gravity ; k is the heat transfer coefficient ;

c T =)(?c T is the rock temperature at the well wall ; an

d ? is acoordinat

e measured downwards along the well axis .

In order to close the system (1 .1)-(l .3) we will use the equations of state of the

two phases and the empirical relations

Moscow. Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika

Zhidkosti i Gaza, No .3, pp . 101-108, May-June, 1994 . Original article submitted January 22, 1993 . 0015-462819412903-0380$12 .

1994 Plenum Publishing Corporation

given in [5] : ),(T P g g ρρ=;),(T P l l ρρ= (1.4)

),Re ,,,,(m m m l g b l We Fr ρρμμβ??= ，),Re ,,,,,(m m m l

g b l m m We Fr ερρμμ?λλ= (1.5)

gD w Fr m m 2= ，σρρg l m m Dw We -=22，l

b g l m m Dw μββμ?βρ?βρ)1()1()1(Re 1212-++--=-- The empirical relations (1 .5) have a different form for different flow regimes .

Depending on the velocity of the mixture, which is determined, inter alia, by the

volume concentration of the gas phase, four regimes can be distinguished : bubble, annular, dispersed-annular, and dispersed . The specific forms of relation (1 .5) for these regimes are given [5] .

We will also use the well-known relations that follow from the definition of the

parameters of two-phase flow :

1)(,)1(,--=-==l

l g g g g l l l g g g G G G w F G w F G ρρρβρ??ρ (1.6)

)(1,)1(l

l g g m l g m G G F w ρρρβ?ρρ+=-+= Here, is the flow gas content ;

l μμ，b and g μare the viscosities of water and the liquid and gas phases ; is the relative roughness of the pipe walls ; is the liquid-gas interfacial tension ; and m w is the volume-average velocity of the

mixture .

The system of 12 equations (1 .1)-(1 .6) contains 18 unknowns : g G ，l G ，m w ,g w ,l w , m ρ，l ρ，g ρ，P ,T,m λ，?，g i ，l i ，g μ，l μ，βand σ. In solving this system the temperature () is assumed to known . At the initial instant it corresponds to the geothermal gradient, and subsequently is determined in calculating the external temperature field by the method described below . Since for a given mixture

composition g i ，l i ，g μ，l μ，βand σ can be determined from the corresponding thermodynamic calculations as functions of p and T, from (1 .1)-(1 .6) for a given mass flow rate g G it is possible to determine )(?P and )(?T . We introduce new functions and transform Eqs . (1 .2)-(1 .3) :

g D T P G G F T P m m l

l g g ρφλρ??ρ+=Φ=Φ-+=Φ=Φ122222112),(];)1([1),( (1.7)

)2

()2(),(2233l l l g

g g w i G w i G T P +++=Φ=Φ; )(),(44T T DK g T P c m --=Φ=Φπρ ])1([1)(132141432T

p dz dT T T dz dp ?Φ?-?Φ?+Φ?=?Φ?Φ-?Φ?Φ?=φ； (1.8) T p T p ?Φ??Φ?-?Φ??Φ?+=?1331)1( (1.9) In order to integrate (1 .8) for known m G initial conditions are assigned in the

form p(H)=*p , T(H)=*T , where H is the depth of the well, and *p and *T are the pressure and temperature at the well bottom .

In the case of blowout the quantities *p and m G are unknown and two

conditions must be specified in order to integrate the system (1 .8) .

One of the these is the empirical relation for the inflow of fluid into the well [9]

222)(m

m k G G B +A =P -P * (1.10) Here, is the known reservoir pressure ; and A and B are filter coefficients

determined from the results of an investigation of the wells in the steady-state regime In the presence of blowout critical conditions exist at the well mouth (z=0) ; from this there follows the second condition 0→∞→dz dT dz dp ，

when z=0 or, as follows from (1 .8) :

0=?； 0=z (1.11) 2 . In order to determine the external temperature on the well wall it is

necessary to solve the problem of heat transfer between the well and the surrounding rock .

As indicated above, the flow in the well is assumed to be steady, and the thermal field in the soil to be unsteady . Therefore the problem is solved as follows . At the initial instant the temperature is determined from the geothermal gradient and by

means of Eqs . (1 .8) p(z) and T(z) are found for given values of **T ，p and m G . Then, on the assumption that on a certain time interval T(z) is constant, the external temperature field is calculated from the equations presented below and a new value of T(z) is found . The calculations are continued until T(z) changes by a given amount T . The relations p(z) and ）

（z

c T in the well shaft are again foun

d from th

e calculated values o

f ）

（z c T and the previously assumed **T ，p and m G . after which the calculation of the external temperature field is continued . Then the p(z) and T(z) in the well shaft are found again, and so on .

In deriving the equations describing the temperature fields in the surrounding

rock, we will make the following assumptions[6] : the water in the rocks undergoes phase transition at T=0°C, the vertical heat fluxes in the well's thermal influence zone are small as compared with the radial fluxes, and the heat flux in the well due to conduction is small as compared with that due to convection . We will assume that initially for 0 z h the soil is permanently frozen, while for h

< z H it is unfrozen . In the permafrost zone we will distinguish two stages of the

process .

In the fast stage the soil is assumed to be frozen at a temperature T< 0°C . The beginning of the second stage corresponds to the point at which the temperature of the soil becomes zero at the well wall . In the second stage the soil can be divided into two zones : thawed and frozen, separated by a phase transition boundary that moves away from the center of the well with time .

In all these cases the temperature distribution in the soil is described by the heat conduction equation, which can be written in the dimensionless form :

20

00022)1(r a x r r t r r r x t i

i i i i i i i τζττθθθθ===T T =??+??=??，，，， (2.1) Here,1=i corresponds to the thawed soil, 2=i to the frozen soil, and 3=i to the unfrozen soil, i a is the thermal diffusivity, 0r is the radius of the well, 0τ is a characteristic time, T is the temperature in the well shaft, ζ is the current value of the radius, and τ is time .

In the fast stage, when the frozen soil is being heated, the initial and boundary conditions have the form :

1)1(222=-=??r r

，θαθ (2.2) 0)1

0,22===M M t t ，（；θθθ (2.3) ∞→→r M ;2θθ (2.4) Here 0ττM M =t corresponds to the onset of thawing of the frozen soil, and

T =M M θ is the initial temperature distribution in the frozen soil, written in dimensionless form .

In the second stage, when there are two zones - thawed and frozen - present in the soil, the equation (2 .1) is supplemented by the conditions

)()()(1)(02t s r r t r t S ?==M M ；，；θθ (2.5) )(01)1(21111t S r r r

====-=??；；；θθθαθ (2.6) )(2211t S r dt

ds r r ==??+??-；θλθλ (2.7) ∞→→M r ；θθ2 (2.8)

In the unfrozen zone (below the permafrost zone) for 3=i we have 01)1(0333===-=??t r r

；；；θθθαθ (2.9) ∞=→t ；03θθ (2.10)

00000200

)(3,2,1(r t S i qp r kr L i i i i τξθθτλλλα=T T =T T ==T ==**）（，，），， Here, k is the heat transfer coefficient, *

i λ is the thermal conductivity coefficient,

q is the specific heat of water-ice phase transition, L p is the mass ice content of the soil, )(0ζT is the temperature distribution in the soil for M T =T , )(0z T is the initial temperature distribution in the unfrozen soil, and )(τζ is the moving phase transition boundary (thawing radius) .

Instead of the conditions at infinity (2 .4), (2 .8), and (2 .10), we introduce the thermal influence radius R(t) at which the following conditions are satisfied :

)(032022t R r r

r ==??=??==M ，，，θθθθθθ (2.11) For solving the problem in stage I we will use the integral method of [7] . From Eq . (2 .1) for 2=i and the conditions (2 .2)and (2 .11) it follows that

122212212)1(=--=??=???r R R x r

r x rdr r θαθθ (2.12) Using relation (2 .11), we reduce the left side of equality (2 .12) to the form :

M -=????θθθdt

dR R dr r dt d rdr r R t R 12)(12 (2.13) Substituting (2 .13) in (2 .12), we obtain instead of (2 .1) (with i=2) the approximate integral equation

122212)1(=M --=?r R x dt

dR R dr r dt d θαθθ （2.14） We will find the function 0 2 entering into Eq . (2 .14) in the form [6] : r t a t a r t a t r )( ln )()(32122++==）

（，θθ (2.15)

where a1(t) are unknown functions of time which are determined from the

corresponding boundary conditions .

From (2.14) we obtain an equation for R(t) in stage I which, together with

relation (2 .15), gives the temperature distribution in the frozen soil .

In order to solve the problem in stage II we take the temperature profile in the frozen zone, by analogy with equality(2 .15), in the form [8] :

r a a r a ***++=3212 ln θ (2.16) The functions a~ are determined from the corresponding boundary conditions . We take the temperature distribution in the thawed zone in the form [8] :

r

s

ln )1r s ln (1112ααθ-+= (2.17) Substituting relations (2 .16) and (2 .17) in the condition on the thawing

boundary (2 .7), we obtain the equation for the thawing radius S(t) .

In order to obtain the equation for R(t) we multiply all the terms of Eq . (2 .1) (with i=2) by r and integrate from S to R. After transformations similar to those carried out above, using relations (2 .6) and (2 .11), we obtain

s r R t

S x dt dR R dr r dt d =M ??-=?2212θθθ (2.18) Substituting (2 .16) in (2 .18) and carrying out a series of transformations, we obtain an ordinary differential equation which, together with the equation for S(t), gives a system of two ordinary differential equations for determining the thawing

radius S(t)and the thermal influence radius R(t) in stage II . The initial conditions are 1)(=M t S and M M =R t R )(. In the unfrozen zone (h z H) the solution is constructed in accordance with

the same procedure as in the frozen zone, but with the index 3=i substituted for 2=i and 0θ for M θ.

The calculations are carried out by the method described above up to the moment 1t t = at which blowout begins . For determining the blowout (maximum possible)

flow rate Eqs . (1 .8) and relations (1 .10) and (1 .11) are used .

3 . For carrying out the calculations we developed suitable algorithms and a package of personal computer programs . The results of the test calculations were compared with known experimental data [4, 6, 8] .

Below, we present the results of calculations carried out for a typical well on the Yamal peninsula .

The depth of the well H=1500 m, the thickness of the layer of permanent frozen soil h=250 m . [tape unclear] of the expected fluid in M% was as follows : 4CH - 96.37, 62H C – 2.89, 83H C - 0 .05, 104H C - 0 .03, 5C - 0.01, 2CO -0 .22,and 2N - 0.43 .

Blowout flow calculations were made for two variants .

In the fast variant the well blew out directly during startup, i .e ., the calculations were made without allowance for thawing ; it was found that the blowout flow =35.3 kg/sec, and that the pressure and temperature at the well mouth p(0)=0.84

MPa and T(0)=247°K . The temperature distribution on the well wall corresponded to the geothermal gradient .Figure 1 shows the variation of the pressure p(z) and the temperature T(z) in the well with depth .

In the second variant the well blew out 352 days after it started operating ; it was found that m G =35 .1 kg/sec ; p(0)=1 .03MPa; and T(0) =257K . The initial

distributions of the pressure p(z) and temperature T(z) of the mixture over the depth of the well under operating conditions are reproduced in Fig . 2 . Figure 3 shows the variation of the thawing radius ξ(t) with time in relation to depth (curves 1-3 correspond to z-200, 100, and 0 m, respectively) .

From these results it is clear that the change in the external temperature field as a result of the heating and thawing of the soil has practically no effect on the blowout flow, but has a significant influence on such parameters at the pressure and

temperature at the well mouth, which determine the choice of means of suppressing the flow .

It follows from the calculations that at all depths )(t increases monotonically with time, the increase being especially sharp during the initial period of operation of the well . The p(z) and T(z) relations show little variation with time . The calculations also showed that the preliminary estimates of the relationship between the characteristic times for the processes in the well and the soil were correct .

At the request of the industry, a series of similar calculations were made for a number of specific wells on the Yamal peninsula. The results of these calculations confirmed the conclusions presented above .

REFERENCES

1 .V. D . Malevanskii and E . V . Sheberstov, Hydrodynamic Calculation of the

Conditions of Suppression of Oil and Gas Well Gushers [in Russian], Nedra, Moscow

(1990) .

2 .G. Wallis, One-Dimensional Two-Phase Flow, McGraw-Hill, New York (1969) .

3 .R. I . Nigmatulin, Dynamics of Multiphase Media, Pt . 1 [in Russian], Nauka, Moscow (1987) .

4 . P. R . Gimer, V . I . Isaev, Yu . D . Raiskii, and G . D . Rozenberg, "Model and methods of calculating the steady nonisothermal two-phaseflow of multicomponent mixtures in vertical tubes,° in : International Conference on the Development of Gas-Condensate Fields,Krasnodar, 1990 : Proceedings, Sec .

5 [in Russian], Krasnodar (1990), p . 112 .

5 .V. A . Mamaev, G . E. Odishariya, O . V . Klapchuk, et al ., Motion of Gas-Liquid Mixtures in Pipes [in Russian], Nedra, Moscow(1978) .

6 .M. M . Dubina and B . A . Krasovitskii, Heat Transfer Conditions and Mechanics of the Interaction of Pipelines and Wells with Soils[in Russian], Nauka, Novosibirsk (1983) .

7 .T . Goodman, "The heat-balance integral and its application to problems involving

a change of phase," Trans. ASME, 80, 335 (1958) .

8 .G. S . Gryaznov, Gas Well Design in Permafrost Regions [in Russian], Nedra, Moscos (1978) .

9.I . A . Charnyi, Underground Hydrogasdynamics [in Russian], Gostoptekhizdat, Moscow (1963) .