冻土水力传导系数的理论模型研究
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摘要
冻土水力传导系数多采用经验公式来描述,其结果缺少理论依据。该文从冰水界面水膜热力学理论出发,对克拉贝隆方程进行修正,得到孔隙水冻结温度与孔隙半径的关系式。基于此,结合毛细管束理论和土壤冻结特征曲线(SFCC),给出预测冻土水力传导系数的理论模型,并与前人的实测值和经验公式进行对比分析。结果表明:孔隙冻结温度随着孔隙半径的减小而下降,且温度下降速率也随之逐渐增大;考虑未冻孔隙水和未冻水膜作为水分的迁移通道,该模型计算值与试验结果具有很好吻合度,且优于经验公式,验证该模型的合理性;最后指出SFCC的拟合效果会影响该模型的预测结果。
The hydraulic conductivity of frozen soil is not only one of the main factors affecting the speed of moisture migration, but also an important model parameter in a large number of frost heave models. However, the study of the hydraulic conductivity for frozen soil is often described by empirical formulas. The calculation results lack a theoretical basis, and the predicted values of different empirical formulas often have enormous deviations. Therefore, it is debatable to use the empirical formula to predict the hydraulic conductivity coefficient of frozen soil. In order to reveal the process of moisture migration, according to the theory of water film thermodynamics at the ice-water interface, this paper points out that the water flow can be regarded as the Darcy flow under the equivalent pressure control in frozen soil. On this basis, the expression of pore water freezing temperature and pore radius is obtained. The capillary bundle theory is applied to the frozen soil and combined with the soil frozen characteristic curve to give a theoretical model for predicting the hydraulic conductivity of frozen soil, Meanwhile, the function of pore radius probability density is eliminated to make this proposed model convenience to calculate and use. The calculated values of this model are compared with the experimental data and empirical formulas of the predecessors. The results show that the pore freezing temperature decreases with the decrease of pore radius, and the temperature drop rate also increases. The freezing temperature reduction rate is significantly accelerated especially when the pore radius is less than 10-6 m. Considering the unfrozen pore water and the unfrozen water film as the migration channel of moisture, in general, the calculated values of this model agree well with the experimental results and are better than the empirical formula, which proves the rationality of the model. The predicted values of the hydraulic conductivity of frozen soil obtained by three empirical formulas are even different by four orders of magnitude for Qinghai-Tibet silty clay. Although the predicted value obtained by the Fowler formula is closer to the measured value, there is an empirical coefficient in the prediction formula. For the empirical coefficient is often arbitrarily determined in a certain range, and it is no clear physical meaning and calculation procedure, which leads to the reliability of the predicted value from the empirical formula is debatable. At the same time, the power function is used to fit the soil frozen characteristic curve(SFCC) in this model, which tends to infinity near 0 ℃. In order to ensure the continuity of the function of the SFCC, the piecewise function is used to describe the soil frozen characteristic curve, but the deviation of the calculated and measured values of the model within 0--0.035 7 ℃. Because the model is based on a quasi-steady state process, and the frozen soil is in the stage of intense phase change near 0 ℃. Thus, this paper pointed out that the fitting formula of the soil frozen characteristic curve is very important to this model for reducing this deviation near 0 ℃.
The hydraulic conductivity of frozen soil is not only one of the main factors affecting the speed of moisture migration, but also an important model parameter in a large number of frost heave models. However, the study of the hydraulic conductivity for frozen soil is often described by empirical formulas. The calculation results lack a theoretical basis, and the predicted values of different empirical formulas often have enormous deviations. Therefore, it is debatable to use the empirical formula to predict the hydraulic conductivity coefficient of frozen soil. In order to reveal the process of moisture migration, according to the theory of water film thermodynamics at the ice-water interface, this paper points out that the water flow can be regarded as the Darcy flow under the equivalent pressure control in frozen soil. On this basis, the expression of pore water freezing temperature and pore radius is obtained. The capillary bundle theory is applied to the frozen soil and combined with the soil frozen characteristic curve to give a theoretical model for predicting the hydraulic conductivity of frozen soil, Meanwhile, the function of pore radius probability density is eliminated to make this proposed model convenience to calculate and use. The calculated values of this model are compared with the experimental data and empirical formulas of the predecessors. The results show that the pore freezing temperature decreases with the decrease of pore radius, and the temperature drop rate also increases. The freezing temperature reduction rate is significantly accelerated especially when the pore radius is less than 10-6 m. Considering the unfrozen pore water and the unfrozen water film as the migration channel of moisture, in general, the calculated values of this model agree well with the experimental results and are better than the empirical formula, which proves the rationality of the model. The predicted values of the hydraulic conductivity of frozen soil obtained by three empirical formulas are even different by four orders of magnitude for Qinghai-Tibet silty clay. Although the predicted value obtained by the Fowler formula is closer to the measured value, there is an empirical coefficient in the prediction formula. For the empirical coefficient is often arbitrarily determined in a certain range, and it is no clear physical meaning and calculation procedure, which leads to the reliability of the predicted value from the empirical formula is debatable. At the same time, the power function is used to fit the soil frozen characteristic curve(SFCC) in this model, which tends to infinity near 0 ℃. In order to ensure the continuity of the function of the SFCC, the piecewise function is used to describe the soil frozen characteristic curve, but the deviation of the calculated and measured values of the model within 0--0.035 7 ℃. Because the model is based on a quasi-steady state process, and the frozen soil is in the stage of intense phase change near 0 ℃. Thus, this paper pointed out that the fitting formula of the soil frozen characteristic curve is very important to this model for reducing this deviation near 0 ℃.
引文
[1]周幼吾,郭东信,邱国庆,等.中国冻土[M].北京:科学出版社,2000.
[2]Harlan R L.Analysis of coupled heat-fluid transport in partially frozen soil[J].Water Resource Research,1973,9(5):1314-1323.
[3]汪仁和,李栋伟.正冻土中水热耦合数学模型及有限元数值模拟[J].煤炭学报,2006(6):757-760.Wang Renhe,Li Dongwei.Moisture-temperature coupling mathematical model in freezing soil and finite element numerical simulation[J].Journal of China Coal Society,2006(6):757-760.(in Chinese with English abstract)
[4]Wu Daoyong,Lai Yuanming,Zhang Mingyi.Heat and mass transfer effects of ice growth mechanisms in a fully saturated soil[J].International Journal of Heat and Mass transfer,201586(1):699-709.
[5]黄兴法,曾德超,练国平.土壤水热盐运动模型的建立与初步验证[J].农业工程学报,1997,13(3):37-41.Huang Xingfa,Zeng Dechao,Lian Guoping.A numerical model for coupling movement of water heat and salt in soil under the conditions of frozen,unfrozen,saturated and unsaturated[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),1997,13(3):37-41.(in Chinese with English abstract)
[6]Padilla F,Villeneuve J P.Modeling and experimental studies of frost heave including solute effects[J].Cold Regions Science and Technology,1992,20(2):183-194.
[7]刘月,王正中,王羿,等.考虑水分迁移及相变对温度场影响的渠道冻胀模型[J].农业工程学报,2016,32(17):83-88.Liu Yue,Wang Zhengzhong,Wang Yi,et al.Frost heave model of canal considering influence of moisture migration and phase transformation on temperature field[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2016,32(17):83-88.(in Chinese with English abstract)
[8]Liu Zhen,Yu Xiong.Coupled thermo-hydro-mechanical model for porous materials under frost action:Theory and implementation[J].Acta Geotechnica,2011,6(2):51-65.
[9]Zhou Jiazuo,Li Dongqing.Numerical analysis of coupled water,heat and stress in saturated freezing soil[J].Cold Regions Science and Technology,2012,72(3):43-47.
[10]Wu Daoyong,Lai Yuanming,Zhang Mingyi.Thermohydro-salt-mechanical coupled model for saturated porous media based on crystallization kinetics[J].Cold Regions Science and Technology,2017,133(1):94-107.
[11]冯瑞玲,蔡晓宇,吴立坚,等.硫酸盐渍土水-盐-热-力四场耦合理论模型[J].中国公路学报,2017,30(2):1-10,40.Feng Ruiling,Cai Xiaoyu,Wu Lijian,et al.Theoretical model on coupling process of moisture-salt-heat-stress field in sulfate salty soil[J].China Journal of Highway and Transport,2017,30(2):1-10,40.(in Chinese with English abstract)
[12]牛玺荣.硫酸盐渍土地区路基水、热、盐、力四场耦合机理及数值模拟研究[D].西安:长安大学,2006.Niu Xirong.The Study and Numerical Simulation on Moisture-Heat-Salt-Stress Coupled Mechanism in the Sulphate Saline Soil Subgrade[D].Xi'an:Chang'an University,2006.(in Chinese with English abstract)
[13]Burt T P,Williams P J.Measurement of hydraulic conductivity of frozen soils[J].Canadian Geotechnical Journal,1974,11(4):647-650.
[14]Burt T P,Williams P J.Hydraulic conductivity in frozen soils[J].Earth Surface Processes,1976,1(4):349-360.
[15]Horiguchi K,Miller R D.Hydraulic conductivity functions of frozen materials[C]//Proc.4th Int.Conf.Permafrost,National Academy Press,Washington D C,1983:504-508.
[16]Watanabe Kunio,Osada Yurie.Simultaneous measurement of unfrozen water content and hydraulic conductivity of partially frozen soil near 0°C[J].Cold Regions Science and Technology,2017,142(10):79-84.
[17]Watanabe Kunio,Osada Yurie.Comparison of hydraulic conductivity in frozen saturated and unfrozen unsaturated soils[J].Vadose Zone Journal,2016,15(5):1-7.
[18]Nixon J F D.Discrete ice lens theory for frost heave in soils[J].Canadian Geotechnical Journal,1991,28(6):843-859.
[19]Taylor G S,Luthin J N.A model for coupled heat and moisture transfer during soil freezing[J].Canadian Geotechnical Journal,1978,15(4):548-555.
[20]Weigert A,Schmidt J.Water transport under winter conditions[J].Catena,2005,64(2):193-208.
[21]周扬,周国庆,周金生,等.饱和土冻结透镜体生长过程水热耦合分析[J].岩土工程学报,2010,32(4):578-585.Zhou Yang,Zhou Guoqing,Zhou Jinsheng,et al.Ice lens growth process involving coupled moisture and heat transfer during freezing of saturated soil[J].Chinese Journal of Geotechnical Engineering,2010,32(4):578-585.(in Chinese with English abstract)
[22]胡坤.冻土水热耦合分离冰冻胀模型的发展[D].徐州:中国矿业大学,2011.Hu Kun.Development of Separated Ice Model Coupled Heat and Moisture Transfer in Freezing Soils[D].Xuzhou:China University of Mining and Technology,2011.(in Chinese with English abstract)
[23]Scherer G W.Crystallization in pores[J].Cement and Concrete Research,1999,29(8):1347-1358.
[24]Israelachvili J N.Intermolecular and Surface Forces-intermolecular and Surface Forces[M].London:Academic Press,1991.
[25]Dash J G,Fu H,Wettlaufer J S.The premelting of ice and its environmental consequences[J].Report on Progress in Physics,1995,58(1):115-167.
[26]St?hli M,Jansson P E,Lundin L C.Soil moisture redistribution and infiltration in frozen sandy soils[J].Water Resources Research,1999,35(1):95-103.
[27]李广信.高等土力学[M].北京:清华大学出版社,2011.
[28]Tarnawski V R,Wagner B.On the prediction of hydraulic conductivity of frozen soils[J].Canadian Geotechnical Journal,1996,33(1):176-180.
[29]Fowler A C,Krantz W B.A generalized secondary frost heave model[J].SIAM Journal on Applied Mathematics.1994,54(6):1650-1675.
[30]O’Neill K.The physics of mathematical frost heave models:A review[J].Cold Regions Science and Technology,19836(3):275-291.
[31]Lundin L C.Hydraulic properties in an operational model of frozen soil[J].Journal of Hydrology,1990,118(4):289-310.
[32]张虎,张建明,张致龙,等.冻结状态青藏粉质黏土的渗透系数测量研究[J].岩土工程学报,2016,38(6):1030-1035.Zhang Hu,Zhang Jianming,Zhang Zhilong,et al Measurement of hydraulic conductivity of Qinghai-Tibet Plateau silty clay under subfreezing temperatures[J].Chinese Journal of Geotechnical Engineering,2016,38(6):1030-1035.(in Chinese with English abstract)
[33]Brooks R H,Corey A T.Properties of porous media affecting fluid flow[J].Journal of the Irrigation and Drainage Division,1966,92(2):61-90.
[2]Harlan R L.Analysis of coupled heat-fluid transport in partially frozen soil[J].Water Resource Research,1973,9(5):1314-1323.
[3]汪仁和,李栋伟.正冻土中水热耦合数学模型及有限元数值模拟[J].煤炭学报,2006(6):757-760.Wang Renhe,Li Dongwei.Moisture-temperature coupling mathematical model in freezing soil and finite element numerical simulation[J].Journal of China Coal Society,2006(6):757-760.(in Chinese with English abstract)
[4]Wu Daoyong,Lai Yuanming,Zhang Mingyi.Heat and mass transfer effects of ice growth mechanisms in a fully saturated soil[J].International Journal of Heat and Mass transfer,201586(1):699-709.
[5]黄兴法,曾德超,练国平.土壤水热盐运动模型的建立与初步验证[J].农业工程学报,1997,13(3):37-41.Huang Xingfa,Zeng Dechao,Lian Guoping.A numerical model for coupling movement of water heat and salt in soil under the conditions of frozen,unfrozen,saturated and unsaturated[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),1997,13(3):37-41.(in Chinese with English abstract)
[6]Padilla F,Villeneuve J P.Modeling and experimental studies of frost heave including solute effects[J].Cold Regions Science and Technology,1992,20(2):183-194.
[7]刘月,王正中,王羿,等.考虑水分迁移及相变对温度场影响的渠道冻胀模型[J].农业工程学报,2016,32(17):83-88.Liu Yue,Wang Zhengzhong,Wang Yi,et al.Frost heave model of canal considering influence of moisture migration and phase transformation on temperature field[J].Transactions of the Chinese Society of Agricultural Engineering(Transactions of the CSAE),2016,32(17):83-88.(in Chinese with English abstract)
[8]Liu Zhen,Yu Xiong.Coupled thermo-hydro-mechanical model for porous materials under frost action:Theory and implementation[J].Acta Geotechnica,2011,6(2):51-65.
[9]Zhou Jiazuo,Li Dongqing.Numerical analysis of coupled water,heat and stress in saturated freezing soil[J].Cold Regions Science and Technology,2012,72(3):43-47.
[10]Wu Daoyong,Lai Yuanming,Zhang Mingyi.Thermohydro-salt-mechanical coupled model for saturated porous media based on crystallization kinetics[J].Cold Regions Science and Technology,2017,133(1):94-107.
[11]冯瑞玲,蔡晓宇,吴立坚,等.硫酸盐渍土水-盐-热-力四场耦合理论模型[J].中国公路学报,2017,30(2):1-10,40.Feng Ruiling,Cai Xiaoyu,Wu Lijian,et al.Theoretical model on coupling process of moisture-salt-heat-stress field in sulfate salty soil[J].China Journal of Highway and Transport,2017,30(2):1-10,40.(in Chinese with English abstract)
[12]牛玺荣.硫酸盐渍土地区路基水、热、盐、力四场耦合机理及数值模拟研究[D].西安:长安大学,2006.Niu Xirong.The Study and Numerical Simulation on Moisture-Heat-Salt-Stress Coupled Mechanism in the Sulphate Saline Soil Subgrade[D].Xi'an:Chang'an University,2006.(in Chinese with English abstract)
[13]Burt T P,Williams P J.Measurement of hydraulic conductivity of frozen soils[J].Canadian Geotechnical Journal,1974,11(4):647-650.
[14]Burt T P,Williams P J.Hydraulic conductivity in frozen soils[J].Earth Surface Processes,1976,1(4):349-360.
[15]Horiguchi K,Miller R D.Hydraulic conductivity functions of frozen materials[C]//Proc.4th Int.Conf.Permafrost,National Academy Press,Washington D C,1983:504-508.
[16]Watanabe Kunio,Osada Yurie.Simultaneous measurement of unfrozen water content and hydraulic conductivity of partially frozen soil near 0°C[J].Cold Regions Science and Technology,2017,142(10):79-84.
[17]Watanabe Kunio,Osada Yurie.Comparison of hydraulic conductivity in frozen saturated and unfrozen unsaturated soils[J].Vadose Zone Journal,2016,15(5):1-7.
[18]Nixon J F D.Discrete ice lens theory for frost heave in soils[J].Canadian Geotechnical Journal,1991,28(6):843-859.
[19]Taylor G S,Luthin J N.A model for coupled heat and moisture transfer during soil freezing[J].Canadian Geotechnical Journal,1978,15(4):548-555.
[20]Weigert A,Schmidt J.Water transport under winter conditions[J].Catena,2005,64(2):193-208.
[21]周扬,周国庆,周金生,等.饱和土冻结透镜体生长过程水热耦合分析[J].岩土工程学报,2010,32(4):578-585.Zhou Yang,Zhou Guoqing,Zhou Jinsheng,et al.Ice lens growth process involving coupled moisture and heat transfer during freezing of saturated soil[J].Chinese Journal of Geotechnical Engineering,2010,32(4):578-585.(in Chinese with English abstract)
[22]胡坤.冻土水热耦合分离冰冻胀模型的发展[D].徐州:中国矿业大学,2011.Hu Kun.Development of Separated Ice Model Coupled Heat and Moisture Transfer in Freezing Soils[D].Xuzhou:China University of Mining and Technology,2011.(in Chinese with English abstract)
[23]Scherer G W.Crystallization in pores[J].Cement and Concrete Research,1999,29(8):1347-1358.
[24]Israelachvili J N.Intermolecular and Surface Forces-intermolecular and Surface Forces[M].London:Academic Press,1991.
[25]Dash J G,Fu H,Wettlaufer J S.The premelting of ice and its environmental consequences[J].Report on Progress in Physics,1995,58(1):115-167.
[26]St?hli M,Jansson P E,Lundin L C.Soil moisture redistribution and infiltration in frozen sandy soils[J].Water Resources Research,1999,35(1):95-103.
[27]李广信.高等土力学[M].北京:清华大学出版社,2011.
[28]Tarnawski V R,Wagner B.On the prediction of hydraulic conductivity of frozen soils[J].Canadian Geotechnical Journal,1996,33(1):176-180.
[29]Fowler A C,Krantz W B.A generalized secondary frost heave model[J].SIAM Journal on Applied Mathematics.1994,54(6):1650-1675.
[30]O’Neill K.The physics of mathematical frost heave models:A review[J].Cold Regions Science and Technology,19836(3):275-291.
[31]Lundin L C.Hydraulic properties in an operational model of frozen soil[J].Journal of Hydrology,1990,118(4):289-310.
[32]张虎,张建明,张致龙,等.冻结状态青藏粉质黏土的渗透系数测量研究[J].岩土工程学报,2016,38(6):1030-1035.Zhang Hu,Zhang Jianming,Zhang Zhilong,et al Measurement of hydraulic conductivity of Qinghai-Tibet Plateau silty clay under subfreezing temperatures[J].Chinese Journal of Geotechnical Engineering,2016,38(6):1030-1035.(in Chinese with English abstract)
[33]Brooks R H,Corey A T.Properties of porous media affecting fluid flow[J].Journal of the Irrigation and Drainage Division,1966,92(2):61-90.