基于分数阶导数的非饱和渗流问题研究
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摘要
非饱和渗流问题广泛存在于煤炭工程领域。现有实验表明,均方位移与时间成非线性关系,但经典的Richards方程未考虑这一情况。针对这一问题,本文通过引入时间分数阶导数,结合两种水力传导系数对新的Richards方程进行了数值求解。在此基础上,分析了时间分数阶Richards方程中的相关参数对含水量曲线的影响,并对已有论文的实验数据进行了拟合。结果表明,新的模型能够描述反常扩散条件下的非饱和渗流问题。
Unsaturated seepage phenomenon widely exists in coal engineering. The experimental showsthat there is a nonlinear relationship between mean square displacement and time,while the classicRichards equation fails to characterize. To solve this,we introduced time-fractional order into classic e-quation by involving two forms of hydraulic conductivity so as to get time-fractional Richards equationand numerical results were obtained. Furthermore,the effect of the parameters in the time-fractionalRichards equation on the water content has been discussed in this paper and the fitting results havebeen applied to the experimental results from former paper. The fitting results show that the new modelcan fit better to the seepage process in unsaturated porous media.
Unsaturated seepage phenomenon widely exists in coal engineering. The experimental showsthat there is a nonlinear relationship between mean square displacement and time,while the classicRichards equation fails to characterize. To solve this,we introduced time-fractional order into classic e-quation by involving two forms of hydraulic conductivity so as to get time-fractional Richards equationand numerical results were obtained. Furthermore,the effect of the parameters in the time-fractionalRichards equation on the water content has been discussed in this paper and the fitting results havebeen applied to the experimental results from former paper. The fitting results show that the new modelcan fit better to the seepage process in unsaturated porous media.
引文
[1]海国治,秦书玉,李宗翔.煤层注水湿润分布状态的数值模拟[J].煤炭学报,1997,22(5):73-76.Hai Guozhi,Qin Shuyu,Li Zongxiang.Numerical simulation of coal seam damped by water infusion[J].Journal of China Coal Society,1997,22(5):73-76.
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[4]杨晓杰,王嘉敏,张民,等.大强煤矿深部巷道围岩吸水软化规律试验研究[J].矿业科学学报,2017,2(5):432-438.Yang Xiaojie,Wang Jiamin,Zhang Min,et al.Study on strength softening of surrounding rock in deep roadway at Daqiang coal mine[J].Journal of Mining Science and Technology,2017,2(5):432-438.
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[10]El A A,Milczarek J.Neutron radiography study of water absorption in porous building materials:anomalous diffusion analysis[J].Journal of Physics D:Applied Physics,2004,37(16):2305.
[11]El Abd A,Czachor A,Milczarek J.Neutron radiography determination of water diffusivity in fired clay brick[J].Applied Radiation and Isotopes,2009,67(4):556-559.
[12]Zhou H W,Yang S,Zhang S Q.Conformable derivative approach to anomalous diffusion[J].Physica A:Statistical Mechanics and Its Applications,2018,491(1):1001-1013.
[13]Pachepsky Y,Timlin D,Rawls W.Generalized Richards'equation to simulate water transport in unsaturated soils[J].Journal of Hydrology,2003,272(1):3-13.
[14]Sun H G,Meerschaert M M,Zhang Y,et al.A fractal Richards'equation to capture the non-Boltzmann scaling of water transport in unsaturated media[J].Advances in Water Resources,2013,52(4):292-295.
[15]陈福来,任理.有限差分异质多尺度方法求解非饱和土壤水流问题的计算效率(Ⅱ):数值实验[J].水利学报,2010,41(7):771-777.Chen Fulai,Ren Li.Computational efficiency of the finite difference heterogeneous multiscale method for unsaturated flow problems in random porous mediaⅡ:numerical simulation[J].Journal of Hydraulic Engineering,2010,41(7):771-777.
[16]Hanert E.On the numerical solution of space-time fractional diffusion models[J].Computers&Fluids,2011,46(1):33-39.
[17]田富强,高龙,胡和平.基于CVODE求解器的饱和-非饱和土壤水分运动数值模型[J].中国科学:技术科学,2012,42(3):311-324.Tian Fuqiang,Gao Long,Hu Heping.A twodimensional Richards equation solver based on CVODEfor variably saturated soil water movement[J].Scientia Sinica(Technologica),2011,54(3):3251-3264.
[18]Podlubny I.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications[M].London:Academic Press,1998.
[19]Zhou H W,Wang C P,Han B B,et al.A creep constitutive model for salt rock based on fractional derivatives[J].International Journal of Rock Mechanics and Mining Sciences,2011,48(1):116-121.
[20]卢宁,William.非饱和土力学[M].北京:高等教育出版社,2012.
[21]Ozisik N.Finite difference methods in heat transfer[M].London:CRC Press,1994.
[22]Wei S,Chen W,Hon Y C.Characterizing time dependent anomalous diffusion process:a survey on fractional derivative and nonlinear models[J].Physica A:Statistical Mechanics and Its Applications,2016,462(5):1244-1251.
[23]Averjanov S F.About permeability of subsurface soils in case of incomplete saturation[J].English Collection,1950,7(1):19-21.
[24]Davidson J M,Stone L R,Nielsen D R,et al.Field measurement and use of soil-water properties[J].Water Resources Research,1969,5(6):1312-1321.
[2]周哲,卢义玉,葛兆龙,等.水-瓦斯-煤三相耦合作用下煤岩强度特性及实验研究[J].煤炭学报,2014,39(12):2418-2424.Zhou Zhe,Lu Yiyu,Ge Zhaolong,et al.Theoretical and experimental study on strength characteristics of coal under coupling effect of water and gas[J].Journal of China Coal Society,2014,39(12):2418-2424.
[3]常远,张晓虎,何满潮.不饱和十字交叉型裂隙流动的尺寸效应[J].矿业科学学报,2018,3(4):342-348.Chang Yuan,Zhang Xiaohu,He Manchao.The size effect of the unsaturated cross intersection fracture flow[J].Journal of Mining Science and Technology,2018,3(4):342-348.
[4]杨晓杰,王嘉敏,张民,等.大强煤矿深部巷道围岩吸水软化规律试验研究[J].矿业科学学报,2017,2(5):432-438.Yang Xiaojie,Wang Jiamin,Zhang Min,et al.Study on strength softening of surrounding rock in deep roadway at Daqiang coal mine[J].Journal of Mining Science and Technology,2017,2(5):432-438.
[5]刘永茜,张浪,李浩荡,等.含水率对煤层气渗流的影响[J].煤炭学报,2014,39(9):1840-1844.Liu Yongqian,Zhang Lang,Li Haodang,et al.Effect of moisture content on CBM seepage[J].Journal of China Coal Society,2014,39(9):1840-1844.
[6]陈曦,于玉贞,程勇刚.非饱和渗流Richards方程数值求解的欠松弛方法[J].岩土力学,2012,33(增1):237-243.Chen Xi,Yu Yuzhen,Cheng Yugang.Under-relaxation methods for numerical solution of Richards equation of variably saturated flow[J].Rock and Soil Mechanics,2012,33(S1):238-243.
[7]Richards L A.Capillary conduction of liquids through porous mediums[J].Physics,1931,1(5):318-333.
[8]孔祥言.高等渗流力学[M].合肥:中国科学技术大学出版社,1999.
[9]Gerolymatou E,Vardoulakis I,Hilfer R.Modelling infiltration by means of a nonlinear fractional diffusion model[J].Journal of Physics D:Applied Physics,2006,39(18):4104.
[10]El A A,Milczarek J.Neutron radiography study of water absorption in porous building materials:anomalous diffusion analysis[J].Journal of Physics D:Applied Physics,2004,37(16):2305.
[11]El Abd A,Czachor A,Milczarek J.Neutron radiography determination of water diffusivity in fired clay brick[J].Applied Radiation and Isotopes,2009,67(4):556-559.
[12]Zhou H W,Yang S,Zhang S Q.Conformable derivative approach to anomalous diffusion[J].Physica A:Statistical Mechanics and Its Applications,2018,491(1):1001-1013.
[13]Pachepsky Y,Timlin D,Rawls W.Generalized Richards'equation to simulate water transport in unsaturated soils[J].Journal of Hydrology,2003,272(1):3-13.
[14]Sun H G,Meerschaert M M,Zhang Y,et al.A fractal Richards'equation to capture the non-Boltzmann scaling of water transport in unsaturated media[J].Advances in Water Resources,2013,52(4):292-295.
[15]陈福来,任理.有限差分异质多尺度方法求解非饱和土壤水流问题的计算效率(Ⅱ):数值实验[J].水利学报,2010,41(7):771-777.Chen Fulai,Ren Li.Computational efficiency of the finite difference heterogeneous multiscale method for unsaturated flow problems in random porous mediaⅡ:numerical simulation[J].Journal of Hydraulic Engineering,2010,41(7):771-777.
[16]Hanert E.On the numerical solution of space-time fractional diffusion models[J].Computers&Fluids,2011,46(1):33-39.
[17]田富强,高龙,胡和平.基于CVODE求解器的饱和-非饱和土壤水分运动数值模型[J].中国科学:技术科学,2012,42(3):311-324.Tian Fuqiang,Gao Long,Hu Heping.A twodimensional Richards equation solver based on CVODEfor variably saturated soil water movement[J].Scientia Sinica(Technologica),2011,54(3):3251-3264.
[18]Podlubny I.Fractional differential equations:an introduction to fractional derivatives,fractional differential equations,to methods of their solution and some of their applications[M].London:Academic Press,1998.
[19]Zhou H W,Wang C P,Han B B,et al.A creep constitutive model for salt rock based on fractional derivatives[J].International Journal of Rock Mechanics and Mining Sciences,2011,48(1):116-121.
[20]卢宁,William.非饱和土力学[M].北京:高等教育出版社,2012.
[21]Ozisik N.Finite difference methods in heat transfer[M].London:CRC Press,1994.
[22]Wei S,Chen W,Hon Y C.Characterizing time dependent anomalous diffusion process:a survey on fractional derivative and nonlinear models[J].Physica A:Statistical Mechanics and Its Applications,2016,462(5):1244-1251.
[23]Averjanov S F.About permeability of subsurface soils in case of incomplete saturation[J].English Collection,1950,7(1):19-21.
[24]Davidson J M,Stone L R,Nielsen D R,et al.Field measurement and use of soil-water properties[J].Water Resources Research,1969,5(6):1312-1321.