降雨条件下非饱和带水—气二相流模拟研究
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摘要
降雨条件下非饱和带水-气二相流问题是目前国内外研究备受关注的问题。多数研究仅限于非饱和带中的水相运动,而对于气相的研究甚少,或者直接忽略气相的作用。实际上,水在非饱和带中的流动是水、气两相流体在土壤孔隙通道中相互驱替的一个复杂过程,尤其在干旱-半干旱地区大强度降雨、有压入渗等问题上,土壤中气相压力在入渗水流作用下,由于瞬间不可能完全消散,气压的急剧增加将对水流的运移产生不可忽略的影响。
本文在总结和分析前人成果的基础上,以非饱和带水-气二相流为研究主线,通过室内物理模拟实验、数值模拟以及理论分析等手段的结合,重点对降雨入渗条件下非饱和带水气二相压力变化、水气动力场特征以及影响因素等进行机理揭示。利用实测资料对非饱和带水、气二相耦合模型进行识别与校正,并应用该模型进一步揭示非饱和带水、气二相流的运移机理。获得了以下成果和认识：
(1)根据已有的实测资料对非饱和带水气耦合模型进行识别与验证,无论是梯度场.还是降速场的拟合,其整体形态的宏观效果较好,基本上反映了水气二相流的运移规律。
(2)通过水、气饱和度以及气相压力的变化,进一步验证了气相与水相之间的关系：气压与含水率或水相饱和度呈负相关性,与负压或气相饱和度呈正相关性。
(3)通过模型参数的敏感度分析可知,绝对渗透率、相对渗透率和毛细压力对模拟计算气压值的敏感度较强；介质的密度和孔隙度对计算结果几乎没有影响,敏感度较低。
(4)降雨条件下,水流基本以垂向运动为主,其垂向入渗速率远大于横向入渗速率；气体由于入渗水流的压缩作用将会冲破上覆水层由地表逃逸,主要以垂向运动为主。但气体随着湿润峰的推移逐渐运移至潜水面处,由于受到潜水面的阻滞作用,气体存在着明显的水平运移,此外,不同的侧边界条件对气体的排泄方式有重要的影响。
(5)非饱和带气压值的主要与降雨强度有关。不同的降雨强度对气压值的影响不同,降雨强度越大,水流的入渗速率越快,气压值的变化越为明显。降雨强度的变化导致水流入渗速率的改变,其是影响非饱和带气压值大小的最直接因素。
(6)根据剖面气压的模拟结果与物理实验结果对比分析,将降雨条件下剖面气压的变化特征主要划分为三个区：气压稳定区,分布在近地表处,气压值基本不变；气压变动区,分布在埋深约20cm-60cm处,气压在该区发生压缩和剧增且气压呈波动状态,该区是水-气两相发生相互驱替的主要区域；气压缓增区,该区分布在潜水面以上,气压值增加比较缓慢。
(7)通过水气耦合模型与单相流模型对非饱和带剖面水相压力的计算结果可得,耦合模型的计算值更接近实测值,拟合误差小于5%,而单相流模拟结果则偏大；通过两者对潜水面补给通量的计算可得,当降雨强度为120cm/h时,两种模型计算结果相差约240cm3/min；当降雨强度为0.24cm/h时,其结果相差仅为20cm3/min。降雨强度越大,两者的计算结果偏差越大,由此表明大强度降雨条件下气相的影响作用不容忽视。当降雨强度小于0.24cm/h时,耦合模型对潜水面补给通量的计算结果基本相等且耦合模型与单相流模型计算结果偏差较小,相对误差约10%。界定降雨强度小于0.24cm/h时,可忽略气相的影响作用。
The study of two phases flow(water and air)is a more concerned problem in unsaturated zone under infiltration condition. Most studies are only about water phase flow,but the air phase flow studies are less, or neglect the influential action of air phase directly. In fact, the water phase flow of unsaturated zone is a mutual displacement and complex process in soil pore, especially under high intensity precipitation and pressure underground infiltration in arid and semi-arid area, because the air phase can't dissipate fully under the influential action of infiltration, the air pressure sharp increases, which can't be neglected for water flow.
This paper summarized and analyzed previous achievements, mainly studied the water and air flow in unsaturated zone, according to physical simulation experiment、numerical simulation and theoretical analysis,the author revealed the change of water and air pressure、dynamic fields characteristics and influencing factors, and employed measured data to identify and correct two phases coupled model of water and air and used this coupled model further revealed the transport mechanism of water and air Gaining the following results and conclusions:
(1) Using measured data to identify and correct two phases coupled model of water and air, from the fitting of gradient fields and deceleration fields we could see that the macro-effects of whole morphology were better, which basically reflected the migration rule of two phases flow.
(2)According to the change of water and air saturation and air pressure,we further verified the relationship between air phase and water phase:there was a negative correlation between air pressure and water content(water saturation),a positive correlation between air phase and negative head (air saturation).
(3)According to the analysis of model's sensitivity, the sensitivities of intrinsic permeability、relative permeability and capillary pressure were higher for calculating value of air pressure, the sensitivities of density and porosity were lower.
(4)Water flow was main vertical movement under ifiltration, but the vertical flow infiltration rate was higher than horizontal; Air phase would breakthrough overlying saturated zone and escaped from soil surface when air was compressed by infiltration, air was main about vertical movement.However, air phase would transport to water level by the pushing of wetting front, because of the blocking effect of water level, air was main horizontal movement, besides, different lateral boundary conditions had a very important effect on draining of air
(5)Air pressure values was main related to precipitation intensity in unsaturated zone. Different precipitation intensities had a different effect on air pressure, the precipitation intensity was higher, the water flow infiltration rate was faster, so the variation of air pressure was obvious. The variation of precipitation intensity resulted in the changes of water flow infiltration rate and it was the most direct factor affected the air pressure in unsaturated zone.
(6)According to comparative analysis between the simulation results of profile air pressure and physical simulation experiment results, the variation characteristics were divided into there areas:air pressure stable area, it was near the soil surface, air pressure was basically steady; Air pressure variation area:it distributed the depth between 20cm and 60cm, air was compressed and air pressure sharp increased in a state of fluctuation, this was the main area which mutual displacement of two phases occurred; Air pressure low-growth area:it distributed above the water level, the air pressure was slowly increased.
(7)According to the results of water pressure between water and air coupled model and single water flow, the results of coupled model were close to measured values and the fitting errors were less than 5%, but the results of single flow model were too large. According to the computation of water level recharge flux, when the precipitation reached 120cm per hour, the results deviation of two kinds of models was almost 240cm3 per minute; When the precipitation was 0.24cm per hour, the results deviation was about 20 cm3 per minute.The higher precipitation intensity, the larger deviation of both computation results. It showed that the influential action couldn't be neglected under high intensity precipitation. When the precipitation was less than 0.24cm per hour,the coupled model results of bottom flux were nearly equal, and the results deviation of bottom flux between coupled model and single model were too small, and the relative error was about 10%. So it could be neglected for the influential action of air phase when the precipitation intensity was less than 0.24cm per hour.
本文在总结和分析前人成果的基础上,以非饱和带水-气二相流为研究主线,通过室内物理模拟实验、数值模拟以及理论分析等手段的结合,重点对降雨入渗条件下非饱和带水气二相压力变化、水气动力场特征以及影响因素等进行机理揭示。利用实测资料对非饱和带水、气二相耦合模型进行识别与校正,并应用该模型进一步揭示非饱和带水、气二相流的运移机理。获得了以下成果和认识：
(1)根据已有的实测资料对非饱和带水气耦合模型进行识别与验证,无论是梯度场.还是降速场的拟合,其整体形态的宏观效果较好,基本上反映了水气二相流的运移规律。
(2)通过水、气饱和度以及气相压力的变化,进一步验证了气相与水相之间的关系：气压与含水率或水相饱和度呈负相关性,与负压或气相饱和度呈正相关性。
(3)通过模型参数的敏感度分析可知,绝对渗透率、相对渗透率和毛细压力对模拟计算气压值的敏感度较强；介质的密度和孔隙度对计算结果几乎没有影响,敏感度较低。
(4)降雨条件下,水流基本以垂向运动为主,其垂向入渗速率远大于横向入渗速率；气体由于入渗水流的压缩作用将会冲破上覆水层由地表逃逸,主要以垂向运动为主。但气体随着湿润峰的推移逐渐运移至潜水面处,由于受到潜水面的阻滞作用,气体存在着明显的水平运移,此外,不同的侧边界条件对气体的排泄方式有重要的影响。
(5)非饱和带气压值的主要与降雨强度有关。不同的降雨强度对气压值的影响不同,降雨强度越大,水流的入渗速率越快,气压值的变化越为明显。降雨强度的变化导致水流入渗速率的改变,其是影响非饱和带气压值大小的最直接因素。
(6)根据剖面气压的模拟结果与物理实验结果对比分析,将降雨条件下剖面气压的变化特征主要划分为三个区：气压稳定区,分布在近地表处,气压值基本不变；气压变动区,分布在埋深约20cm-60cm处,气压在该区发生压缩和剧增且气压呈波动状态,该区是水-气两相发生相互驱替的主要区域；气压缓增区,该区分布在潜水面以上,气压值增加比较缓慢。
(7)通过水气耦合模型与单相流模型对非饱和带剖面水相压力的计算结果可得,耦合模型的计算值更接近实测值,拟合误差小于5%,而单相流模拟结果则偏大；通过两者对潜水面补给通量的计算可得,当降雨强度为120cm/h时,两种模型计算结果相差约240cm3/min；当降雨强度为0.24cm/h时,其结果相差仅为20cm3/min。降雨强度越大,两者的计算结果偏差越大,由此表明大强度降雨条件下气相的影响作用不容忽视。当降雨强度小于0.24cm/h时,耦合模型对潜水面补给通量的计算结果基本相等且耦合模型与单相流模型计算结果偏差较小,相对误差约10%。界定降雨强度小于0.24cm/h时,可忽略气相的影响作用。
The study of two phases flow(water and air)is a more concerned problem in unsaturated zone under infiltration condition. Most studies are only about water phase flow,but the air phase flow studies are less, or neglect the influential action of air phase directly. In fact, the water phase flow of unsaturated zone is a mutual displacement and complex process in soil pore, especially under high intensity precipitation and pressure underground infiltration in arid and semi-arid area, because the air phase can't dissipate fully under the influential action of infiltration, the air pressure sharp increases, which can't be neglected for water flow.
This paper summarized and analyzed previous achievements, mainly studied the water and air flow in unsaturated zone, according to physical simulation experiment、numerical simulation and theoretical analysis,the author revealed the change of water and air pressure、dynamic fields characteristics and influencing factors, and employed measured data to identify and correct two phases coupled model of water and air and used this coupled model further revealed the transport mechanism of water and air Gaining the following results and conclusions:
(1) Using measured data to identify and correct two phases coupled model of water and air, from the fitting of gradient fields and deceleration fields we could see that the macro-effects of whole morphology were better, which basically reflected the migration rule of two phases flow.
(2)According to the change of water and air saturation and air pressure,we further verified the relationship between air phase and water phase:there was a negative correlation between air pressure and water content(water saturation),a positive correlation between air phase and negative head (air saturation).
(3)According to the analysis of model's sensitivity, the sensitivities of intrinsic permeability、relative permeability and capillary pressure were higher for calculating value of air pressure, the sensitivities of density and porosity were lower.
(4)Water flow was main vertical movement under ifiltration, but the vertical flow infiltration rate was higher than horizontal; Air phase would breakthrough overlying saturated zone and escaped from soil surface when air was compressed by infiltration, air was main about vertical movement.However, air phase would transport to water level by the pushing of wetting front, because of the blocking effect of water level, air was main horizontal movement, besides, different lateral boundary conditions had a very important effect on draining of air
(5)Air pressure values was main related to precipitation intensity in unsaturated zone. Different precipitation intensities had a different effect on air pressure, the precipitation intensity was higher, the water flow infiltration rate was faster, so the variation of air pressure was obvious. The variation of precipitation intensity resulted in the changes of water flow infiltration rate and it was the most direct factor affected the air pressure in unsaturated zone.
(6)According to comparative analysis between the simulation results of profile air pressure and physical simulation experiment results, the variation characteristics were divided into there areas:air pressure stable area, it was near the soil surface, air pressure was basically steady; Air pressure variation area:it distributed the depth between 20cm and 60cm, air was compressed and air pressure sharp increased in a state of fluctuation, this was the main area which mutual displacement of two phases occurred; Air pressure low-growth area:it distributed above the water level, the air pressure was slowly increased.
(7)According to the results of water pressure between water and air coupled model and single water flow, the results of coupled model were close to measured values and the fitting errors were less than 5%, but the results of single flow model were too large. According to the computation of water level recharge flux, when the precipitation reached 120cm per hour, the results deviation of two kinds of models was almost 240cm3 per minute; When the precipitation was 0.24cm per hour, the results deviation was about 20 cm3 per minute.The higher precipitation intensity, the larger deviation of both computation results. It showed that the influential action couldn't be neglected under high intensity precipitation. When the precipitation was less than 0.24cm per hour,the coupled model results of bottom flux were nearly equal, and the results deviation of bottom flux between coupled model and single model were too small, and the relative error was about 10%. So it could be neglected for the influential action of air phase when the precipitation intensity was less than 0.24cm per hour.
引文
[1]Faybishenko B A. Hydraulic behavior of quasi-saturated soils in the presence of entrapped air: laboratory experiments[J]. Water Resour Res,1995,31:2421-2435
[2]Touma J, Vachaud C, Parlange J Y. Air and water flow in a sealed,pounded vertical soil column: Experiment and model[J]. Soil Sci.,1984,137:181-187
[3]Bernadiner M G. A capillary microstructure of the wetting front [J]. Transport in porous media,1998, 30:251-265
[4]Lu T X, Biggar J E, Nielsen D R. Water movement in glass bead porous media 2.Experiments of infiltration and finger flow[J]. Water Resour Res,1994,30:3283-3290
[5]Weir G J, Kissling W M. The influence of airflow on the vertical infiltration of water into soil[J]. Water Resour Res,1992,28:2765-2772
[6]Peck A J. Moisture profile development and air compression during water uptake by bounded porous bodies:2. Horizontal columns[J].Soil Sci,1965a,100:333-340.
[7]Wang Z, Feyen J, Nielsen D R, et al. Two-phase flow infiltration equations accounting for air entrapment effects[J]. Water Resour Res,1997,33:2759-2767
[8]Touma J, Vauclin M. Experimental and numerical analysis of two-phase infiltration in a partially saturated soil[J]. Transport in Porous Media.1986,1:27-55
[9]Grismer M E, Orang M N, Clausitzer V, et al. Effects of air compression and counterflow on infiltration into soils[J].J Irrig Drain Eng,1994,120:775-795
[10]Failow D J, Elrick D E. Field measurement of air-entry and water-entry soil water pressure heads[J]. Soil Sci Soc Am J,1996,60:1036-1039
[11]Massmann J W. Applying groundwater flow models in vapor extraction system design[J]. Journal of Environmental Engineering,1989,115 (1):129-149
[12]彭胜,陈家军等.非饱和带水气二相流国外研究综述.[J]水科学进展,2000,11(3)：333-338
[13]陈家军,彭胜等.非饱和带水气二相流参数确定实验研究.[J]水科学进展,2001,12(4)：467-471
[14]彭胜,陈家军等.非饱和带水气二相流实验研究.[J]土壤学报,2002,39(4)：505-511
[15]孙冬梅,朱岳明,张明进.非饱和带水.气二相流数值模拟研究[J].岩土工程学报,2007,29(4)：560-565
[16]邵龙潭,王助贫等.非饱和土中水流入渗和气体排出过程的求解.[J]水科学进展,2000,11(1)：8-13
[17]唐海行,苏逸深.考虑气压势影响的降雨入渗数值模拟研究.[J]水科学进展,1996,7(1)：8-13
[18]王锦山,尹伯悦等.水—气两相流在煤层中运移规律.[J]黑龙江科技学院学报,2005,15(1)：16-19
[19]孙元元.非恒定降雨入渗下砂土和壤土非饱和性状.[J]水利水运工程学报,2007, (1)：61-64
[20]王晓峰,刘光焰等.非饱和土体降雨入渗的全过程数值模拟.[J]水文,2007,27(1)：30-32
[21]薛强,冯夏庭等.水气两相系统K—S—P模型参数反演的最优估计.[J]水科学进展,2005,16(4)：488-493
[22]郑冰,陈家军,尉斌,等,多孔介质不溶混流体毛细压力与饱和度关系实验研究[J].水动力学研究与进展,2005,9(5)：665-672
[23]Johns S, Selker etc. Green and Ampt infiltration into soil of variable Pore Size with Depth.[J]Water Resour Res,1999,35(5):1685-1688
[24]Shlomo P, Neuman. Wetting Front Pressure Head in the Infiltration Model of Green and Ampt. [J] Water Resour Res,1976,12 (3):564-566
[25]Morel-seytoux H J, Khanji J. Derivation of equation of infiltration[J]. Water Resour Res,1974,10: 795-800
[26]Pinder Georage F, Linda M Abriola. On the simulation of nonaqueous phase organic compounds in the subsurface. [J] Water Resour Res,1986,22 (9):109S-119S
[27]Sleep B E, Sykes J F. Modeling the transport of volatile organics in variably saturated media [J] Water Resour Res,1989,25 (1):81-92
[28]Celia Michael A, Philip Binning. A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow [J] Water Resour Res,1992,28(10):2819-2828
[29]Kaluarachchi J J, Parker J C. An efficient finite element method for modeling multiphase flow[J].Water Resource Research,1989,25 (1):43-54
[30]Schrefler B A, Zhan Xiao Yong. A fully coupled model for water flow and air flow in deformable porous media [J].Water Resource Research.1993.29 (1):155-167.
[31]David Meiri. Two-phase flow simulation of air storage in an aquifer[J]. Water Resource Research. 1981.17 (5):1360-1366
[32]Huyakorn, P.S., and G.F.Pinder, Solution of two-phase flow using a new finite element technical, in Proceedings of the international Conference on Applied Numerical Modeling, University of south-ampton,1977.375-390.
[33]Lewis, R.W., E.A. Verner, and O.C.Zienkiewicz, A finite element approach to two-phase flow in porous media, in finite elements in fluids,1975.1:183-199
[34]陈月明编.油藏数值模拟基础[M].北京,石油大学出版社,1989
[35]陈家军,悉成刚,王金生.非饱和带水-气二相流数值模拟研究进展.[J]水科学进展,2000,11(2)：208-214.
[36]骆祖江,付延玲,王增辉.非饱和带水气二相渗流动力学模型[J].煤田地质与勘察,1999,27(5)：43-45.
[37]骆祖江,陈艺南,付延玲.水-气二相渗流耦合模型全隐士联立求解[J].煤田地质与勘察,2001,29(6)：36-38.
[38]悉成刚,陈家军等.水气二相流方程的一种离散数值解法[J].水科学进展,2002,13(3)：287-291.
[39]雅贝尔著,李竞生.多孔介质流体动力学[M].陈崇希译.北京：中国建筑出版社,1983.
[40]雷志栋,杨诗秀,谢森传.土壤水动力学[M].北京：清华大学出版社,1988.
[41]Oak M J, Baker L E, Thomas D, etal.Three-Phase Relative Permeability of Bereas and stone[J].Journal of Petroleum Technology,1990,42(8):1054-1061
[42]VAN GENUCHTEN M T. A closed form equation predicting the hydraulic conductivity unsaturated soils [J]. Soil Sci Soc of Amer J,1980,44:892-898
[43]MUALEM Y. A new model for predicting the hydraulic conductivity of unsaturated porous media [J]. Water Resour Res,1976,12 (3):513-522
[44]Buckingham, E. Contributions to our knowledge of the aeration of soils[J]. Soils Bull,1904,25:5-52
[45]Joel Massmann, Daniel Farrier. Effects of Atmospheric Pressure on Gas transport in the Vadose Zone[J]. Water Resour Res,1992,28 (3):777-791
[2]Touma J, Vachaud C, Parlange J Y. Air and water flow in a sealed,pounded vertical soil column: Experiment and model[J]. Soil Sci.,1984,137:181-187
[3]Bernadiner M G. A capillary microstructure of the wetting front [J]. Transport in porous media,1998, 30:251-265
[4]Lu T X, Biggar J E, Nielsen D R. Water movement in glass bead porous media 2.Experiments of infiltration and finger flow[J]. Water Resour Res,1994,30:3283-3290
[5]Weir G J, Kissling W M. The influence of airflow on the vertical infiltration of water into soil[J]. Water Resour Res,1992,28:2765-2772
[6]Peck A J. Moisture profile development and air compression during water uptake by bounded porous bodies:2. Horizontal columns[J].Soil Sci,1965a,100:333-340.
[7]Wang Z, Feyen J, Nielsen D R, et al. Two-phase flow infiltration equations accounting for air entrapment effects[J]. Water Resour Res,1997,33:2759-2767
[8]Touma J, Vauclin M. Experimental and numerical analysis of two-phase infiltration in a partially saturated soil[J]. Transport in Porous Media.1986,1:27-55
[9]Grismer M E, Orang M N, Clausitzer V, et al. Effects of air compression and counterflow on infiltration into soils[J].J Irrig Drain Eng,1994,120:775-795
[10]Failow D J, Elrick D E. Field measurement of air-entry and water-entry soil water pressure heads[J]. Soil Sci Soc Am J,1996,60:1036-1039
[11]Massmann J W. Applying groundwater flow models in vapor extraction system design[J]. Journal of Environmental Engineering,1989,115 (1):129-149
[12]彭胜,陈家军等.非饱和带水气二相流国外研究综述.[J]水科学进展,2000,11(3)：333-338
[13]陈家军,彭胜等.非饱和带水气二相流参数确定实验研究.[J]水科学进展,2001,12(4)：467-471
[14]彭胜,陈家军等.非饱和带水气二相流实验研究.[J]土壤学报,2002,39(4)：505-511
[15]孙冬梅,朱岳明,张明进.非饱和带水.气二相流数值模拟研究[J].岩土工程学报,2007,29(4)：560-565
[16]邵龙潭,王助贫等.非饱和土中水流入渗和气体排出过程的求解.[J]水科学进展,2000,11(1)：8-13
[17]唐海行,苏逸深.考虑气压势影响的降雨入渗数值模拟研究.[J]水科学进展,1996,7(1)：8-13
[18]王锦山,尹伯悦等.水—气两相流在煤层中运移规律.[J]黑龙江科技学院学报,2005,15(1)：16-19
[19]孙元元.非恒定降雨入渗下砂土和壤土非饱和性状.[J]水利水运工程学报,2007, (1)：61-64
[20]王晓峰,刘光焰等.非饱和土体降雨入渗的全过程数值模拟.[J]水文,2007,27(1)：30-32
[21]薛强,冯夏庭等.水气两相系统K—S—P模型参数反演的最优估计.[J]水科学进展,2005,16(4)：488-493
[22]郑冰,陈家军,尉斌,等,多孔介质不溶混流体毛细压力与饱和度关系实验研究[J].水动力学研究与进展,2005,9(5)：665-672
[23]Johns S, Selker etc. Green and Ampt infiltration into soil of variable Pore Size with Depth.[J]Water Resour Res,1999,35(5):1685-1688
[24]Shlomo P, Neuman. Wetting Front Pressure Head in the Infiltration Model of Green and Ampt. [J] Water Resour Res,1976,12 (3):564-566
[25]Morel-seytoux H J, Khanji J. Derivation of equation of infiltration[J]. Water Resour Res,1974,10: 795-800
[26]Pinder Georage F, Linda M Abriola. On the simulation of nonaqueous phase organic compounds in the subsurface. [J] Water Resour Res,1986,22 (9):109S-119S
[27]Sleep B E, Sykes J F. Modeling the transport of volatile organics in variably saturated media [J] Water Resour Res,1989,25 (1):81-92
[28]Celia Michael A, Philip Binning. A mass conservative numerical solution for two-phase flow in porous media with application to unsaturated flow [J] Water Resour Res,1992,28(10):2819-2828
[29]Kaluarachchi J J, Parker J C. An efficient finite element method for modeling multiphase flow[J].Water Resource Research,1989,25 (1):43-54
[30]Schrefler B A, Zhan Xiao Yong. A fully coupled model for water flow and air flow in deformable porous media [J].Water Resource Research.1993.29 (1):155-167.
[31]David Meiri. Two-phase flow simulation of air storage in an aquifer[J]. Water Resource Research. 1981.17 (5):1360-1366
[32]Huyakorn, P.S., and G.F.Pinder, Solution of two-phase flow using a new finite element technical, in Proceedings of the international Conference on Applied Numerical Modeling, University of south-ampton,1977.375-390.
[33]Lewis, R.W., E.A. Verner, and O.C.Zienkiewicz, A finite element approach to two-phase flow in porous media, in finite elements in fluids,1975.1:183-199
[34]陈月明编.油藏数值模拟基础[M].北京,石油大学出版社,1989
[35]陈家军,悉成刚,王金生.非饱和带水-气二相流数值模拟研究进展.[J]水科学进展,2000,11(2)：208-214.
[36]骆祖江,付延玲,王增辉.非饱和带水气二相渗流动力学模型[J].煤田地质与勘察,1999,27(5)：43-45.
[37]骆祖江,陈艺南,付延玲.水-气二相渗流耦合模型全隐士联立求解[J].煤田地质与勘察,2001,29(6)：36-38.
[38]悉成刚,陈家军等.水气二相流方程的一种离散数值解法[J].水科学进展,2002,13(3)：287-291.
[39]雅贝尔著,李竞生.多孔介质流体动力学[M].陈崇希译.北京：中国建筑出版社,1983.
[40]雷志栋,杨诗秀,谢森传.土壤水动力学[M].北京：清华大学出版社,1988.
[41]Oak M J, Baker L E, Thomas D, etal.Three-Phase Relative Permeability of Bereas and stone[J].Journal of Petroleum Technology,1990,42(8):1054-1061
[42]VAN GENUCHTEN M T. A closed form equation predicting the hydraulic conductivity unsaturated soils [J]. Soil Sci Soc of Amer J,1980,44:892-898
[43]MUALEM Y. A new model for predicting the hydraulic conductivity of unsaturated porous media [J]. Water Resour Res,1976,12 (3):513-522
[44]Buckingham, E. Contributions to our knowledge of the aeration of soils[J]. Soils Bull,1904,25:5-52
[45]Joel Massmann, Daniel Farrier. Effects of Atmospheric Pressure on Gas transport in the Vadose Zone[J]. Water Resour Res,1992,28 (3):777-791