One-dimensional non-Darcy flow in a semi-infinite porous media:a multiphase implicit Stefan problem with phases divided by hydraulic gradients
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摘要
One-dimensional non-Darcy flow in a semiinfinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established.Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
One-dimensional non-Darcy flow in a semiinfinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established.Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
One-dimensional non-Darcy flow in a semiinfinite porous media is investigated. We indicate that the non-Darcy relation which is usually determined from experimental results can always be described by a piecewise linear function, and the problem can be equivalently transformed to a multiphase implicit Stefan problem. The novel feature of this Stefan problem is that the phases of the porous media are divided by hydraulic gradients, not the excess pore water pressures. Using the similarity transformation technique, an exact solution for the situation that the external load increases in proportion to the square root of time is developed. The study on the existence and uniqueness of the solution leads to the requirement of a group of inequalities. A similar Stefan problem considering constant surface seepage velocity is also investigated, and the solution, which we indicate to be uniquely existent under all conditions, is established.Meanwhile, the relation between our Stefan problem and the traditional multiphase Stefan problem is demonstrated. In the end, computational examples of the solution are presented and discussed. The solution provides a useful benchmark for verifying the accuracy of general approximate algorithms of Stefan problems, and it is also attractive in the context of inverse problem analysis.
引文
[1].Rubinstein,L.:The Stefan Problem,vol.27.American Mathematical Society,Providence(1971)
2.Alexiades,V.,Solomon,A.D.:Mathematical Modeling of Melting and Freezing Processes.Hemisphere,Washington(1983)
3.Crank,J.:Free and Moving Boundary Problems.Clarendon Press,Oxford(1984)
4.Lunardini,V.J.:Heat Transfer with Freezing and Thawing.Elsevier,Amsterdam(1991)
5.Gupta,S.C.:The Classical Stefan Problem,Basic Concepts,Modelling and Analysis.Elsevier,Amsterdam(2003)
6.Voller,V.R.:A similarity solution for solidification of an undercooled binary alloy.Int.J.Heat Mass Transf.49,1981-1985(2006)
7.Storti,M.:Numerical modeling of ablation phenomena as twophase Stefan problems.Int.J.Heat Mass Transf.38,2843-2854(1995)
8.Chung,H.,Das,S.:Numerical modeling of scanning laser-induced melting,vaporization and resolidification in metals subjected to step heat flux input.Int.J.Heat Mass Transf.47,4153-4164(2004)
9.Rajeev,K.,Rai,N.,Das,S.:Numerical solution of a movingboundary problem with variable latent heat.Int.J.Heat Mass Transf.52,1913-1917(2009)
10.Lorenzo-Trueba,J.,Voller,V.R.:Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation.J.Math.Anal.Appl.366,538-549(2010)
11.Marcus,E.A.S.,Tarzia,D.A.:Explicit solution for freezing of humid porous half-space with a heat flux condition.Int.J.Eng.Sci.38,1651-1665(2000)
12.Lombardi,A.L.,Tarzia,D.A.:Similarity solutions for thawing processes with a heat flux condition at the fixed boundary.Meccanica36,251-264(2001)
13.Nicolin,D.J.,Jorge,R.M.M.,Jorge,L.M.M.:Stefan problem approach applied to the diffusion process in grain hydration.Transp.Porous Media 102,387-402(2014)
14.Nicolin,D.J.,Jorge,R.M.M.,Jorge,L.M.M.:Moving boundary modeling of conventional and transgenic soybean hydration:moisture profile and moving front experimental validation.Int.J.Heat Mass Transf.90,568-577(2015)
15.Pascal,F.,Pascal,H.,Murray,D.W.:Consolidation with threshold gradients.Int.J.Numer.Anal.Methods Geomech.5,247-261(1981)
16.Chen,Y.M.,Tang,X.W.,Wang,J.:An analytical solution of onedimensional consolidation for soft sensitive soil ground.Int.J.Numer.Anal.Methods 28,919-930(2004)
17.Tao,L.N.:The exact solutions of some Stefan problems with prescribed heat flux.J.Appl.Mech.48,732-736(1981)
18.Rogers,C.:Application of a reciprocal transformation to a twophase Stefan problem.J.Phys.A Math.Gen.18,L105-L109(1985)
19.Cherniha,R.M.,Cherniha,N.D.:Exact solutions of a class of nonlinear boundary value problems with moving boundaries.J.Phys.A Math.Gen.26,L935-L940(1993)
20.Broadbridge,P.,Pincombe,B.M.:The Stefan solidification problem with nonmonotonic nonlinear heat diffusivity.Math.Comput.Model.23,87-98(1996)
21.Briozzo,A.C.,Tarzia,D.A.:An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients.IMA J.Appl.Math.67,249-261(2002)
22.Voller,V.R.,Swenson,J.B.,Paola,C.:An analytical solution for a Stefan problem with variable latent heat.Int.J.Heat Mass Transf.47,5387-5390(2004)
23.Cherniha,R.,Kovalenko,S.:Exact solutions of nonlinear boundary value problems of the Stefan type.J.Phys.A Math.Theor.42,355202(2009)
24.Cherniha,R.,Kovalenko,S.:Lie symmetries of nonlinear boundary value problems.Commun.Nonlinear Sci.Numer.Simul.17,71-84(2012)
25.Zhou,Y.,Wang,Y.J.,Bu,W.K.:Exact solution for a Stefan problem with latent heat a power function of position.Int.J.Heat Mass Transf.69,451-454(2014)
26.Zhou,Y.,Xia,L.J.:Exact solution for Stefan problem with general power-type latent heat using Kummer function.Int.J.Heat Mass Transf.84,114-118(2015)
27.Zhou,Y.,Zhou,G.Q.,Bu,W.K.:Approximate analytical solution for nonlinear multiphase Stefan problem.J.Thermophys.Heat Transf.29,417-422(2015)
28.Xie,K.H.,Wang,K.,Wang,Y.L.,et al.:Analytical solution for onedimensional consolidation of clayey soils with a threshold gradient.Comput.Geotech.37,487-493(2010)
29.Zhou,Y.,Bu,W.K.,Lu,M.M.:One-dimensional consolidation with a threshold gradient:a Stefan problem with rate-dependent latent heat.Int.J.Numer.Anal.Methods Geomech.37,2825-2832(2013)
30.Liu,W.C.,Yao,J.,Wang,Y.Y.:Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient.Int.J.Heat Mass Transf.55,6017-6022(2012)
31.Liu,W.C.,Yao,J.,Chen,Z.X.:Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability.Acta.Mech.Sin.30,50-58(2014)
32.Mitchell,J.K.,Younger,J.S.:Abnormalities in Hydraulic Flow Through Fine-Grained Soils.ASTM Special Publication,American Society for Testing and Materials,Philadelphia(1967)
33.Hansbo,S.:Consolidation of clay with special reference to influence of vertical sand drains.Swed.Geotech.Inst.Proc.18,45-50(1960)
34.Tang,X.W.,Onitsuka,K.:Consolidation by vertical drains under time-dependent loading.Int.J.Numer.Anal.Methods Geomech.24,739-751(2000)
35.Qin,A.F.,Sun,D.A.,Tan,Y.W.:Analytical solution to onedimensional consolidation in unsaturated soils under loading varying exponentially with time.Comput.Geotech.37,233-238(2010)
36.Olver,F.W.J.,Lozier,D.W.,Boisvert,R.F.,et al.:NIST Handbook of Mathematical Functions.Cambridge University Press,New York(2010)
37.Wilson,D.G.:Existence and uniqueness for similarity solutions of one dimensional multi-phase Stefan problems.SIAM J.Appl.Math.35,135-147(1978)
38.Tao,L.N.:The heat conduction problem with temperature dependent material properties.Int.J.Heat Mass Transf.32,487-491(1989)
2.Alexiades,V.,Solomon,A.D.:Mathematical Modeling of Melting and Freezing Processes.Hemisphere,Washington(1983)
3.Crank,J.:Free and Moving Boundary Problems.Clarendon Press,Oxford(1984)
4.Lunardini,V.J.:Heat Transfer with Freezing and Thawing.Elsevier,Amsterdam(1991)
5.Gupta,S.C.:The Classical Stefan Problem,Basic Concepts,Modelling and Analysis.Elsevier,Amsterdam(2003)
6.Voller,V.R.:A similarity solution for solidification of an undercooled binary alloy.Int.J.Heat Mass Transf.49,1981-1985(2006)
7.Storti,M.:Numerical modeling of ablation phenomena as twophase Stefan problems.Int.J.Heat Mass Transf.38,2843-2854(1995)
8.Chung,H.,Das,S.:Numerical modeling of scanning laser-induced melting,vaporization and resolidification in metals subjected to step heat flux input.Int.J.Heat Mass Transf.47,4153-4164(2004)
9.Rajeev,K.,Rai,N.,Das,S.:Numerical solution of a movingboundary problem with variable latent heat.Int.J.Heat Mass Transf.52,1913-1917(2009)
10.Lorenzo-Trueba,J.,Voller,V.R.:Analytical and numerical solution of a generalized Stefan problem exhibiting two moving boundaries with application to ocean delta formation.J.Math.Anal.Appl.366,538-549(2010)
11.Marcus,E.A.S.,Tarzia,D.A.:Explicit solution for freezing of humid porous half-space with a heat flux condition.Int.J.Eng.Sci.38,1651-1665(2000)
12.Lombardi,A.L.,Tarzia,D.A.:Similarity solutions for thawing processes with a heat flux condition at the fixed boundary.Meccanica36,251-264(2001)
13.Nicolin,D.J.,Jorge,R.M.M.,Jorge,L.M.M.:Stefan problem approach applied to the diffusion process in grain hydration.Transp.Porous Media 102,387-402(2014)
14.Nicolin,D.J.,Jorge,R.M.M.,Jorge,L.M.M.:Moving boundary modeling of conventional and transgenic soybean hydration:moisture profile and moving front experimental validation.Int.J.Heat Mass Transf.90,568-577(2015)
15.Pascal,F.,Pascal,H.,Murray,D.W.:Consolidation with threshold gradients.Int.J.Numer.Anal.Methods Geomech.5,247-261(1981)
16.Chen,Y.M.,Tang,X.W.,Wang,J.:An analytical solution of onedimensional consolidation for soft sensitive soil ground.Int.J.Numer.Anal.Methods 28,919-930(2004)
17.Tao,L.N.:The exact solutions of some Stefan problems with prescribed heat flux.J.Appl.Mech.48,732-736(1981)
18.Rogers,C.:Application of a reciprocal transformation to a twophase Stefan problem.J.Phys.A Math.Gen.18,L105-L109(1985)
19.Cherniha,R.M.,Cherniha,N.D.:Exact solutions of a class of nonlinear boundary value problems with moving boundaries.J.Phys.A Math.Gen.26,L935-L940(1993)
20.Broadbridge,P.,Pincombe,B.M.:The Stefan solidification problem with nonmonotonic nonlinear heat diffusivity.Math.Comput.Model.23,87-98(1996)
21.Briozzo,A.C.,Tarzia,D.A.:An explicit solution for an instantaneous two-phase Stefan problem with nonlinear thermal coefficients.IMA J.Appl.Math.67,249-261(2002)
22.Voller,V.R.,Swenson,J.B.,Paola,C.:An analytical solution for a Stefan problem with variable latent heat.Int.J.Heat Mass Transf.47,5387-5390(2004)
23.Cherniha,R.,Kovalenko,S.:Exact solutions of nonlinear boundary value problems of the Stefan type.J.Phys.A Math.Theor.42,355202(2009)
24.Cherniha,R.,Kovalenko,S.:Lie symmetries of nonlinear boundary value problems.Commun.Nonlinear Sci.Numer.Simul.17,71-84(2012)
25.Zhou,Y.,Wang,Y.J.,Bu,W.K.:Exact solution for a Stefan problem with latent heat a power function of position.Int.J.Heat Mass Transf.69,451-454(2014)
26.Zhou,Y.,Xia,L.J.:Exact solution for Stefan problem with general power-type latent heat using Kummer function.Int.J.Heat Mass Transf.84,114-118(2015)
27.Zhou,Y.,Zhou,G.Q.,Bu,W.K.:Approximate analytical solution for nonlinear multiphase Stefan problem.J.Thermophys.Heat Transf.29,417-422(2015)
28.Xie,K.H.,Wang,K.,Wang,Y.L.,et al.:Analytical solution for onedimensional consolidation of clayey soils with a threshold gradient.Comput.Geotech.37,487-493(2010)
29.Zhou,Y.,Bu,W.K.,Lu,M.M.:One-dimensional consolidation with a threshold gradient:a Stefan problem with rate-dependent latent heat.Int.J.Numer.Anal.Methods Geomech.37,2825-2832(2013)
30.Liu,W.C.,Yao,J.,Wang,Y.Y.:Exact analytical solutions of moving boundary problems of one-dimensional flow in semi-infinite long porous media with threshold pressure gradient.Int.J.Heat Mass Transf.55,6017-6022(2012)
31.Liu,W.C.,Yao,J.,Chen,Z.X.:Analytical solution of a double moving boundary problem for nonlinear flows in one-dimensional semi-infinite long porous media with low permeability.Acta.Mech.Sin.30,50-58(2014)
32.Mitchell,J.K.,Younger,J.S.:Abnormalities in Hydraulic Flow Through Fine-Grained Soils.ASTM Special Publication,American Society for Testing and Materials,Philadelphia(1967)
33.Hansbo,S.:Consolidation of clay with special reference to influence of vertical sand drains.Swed.Geotech.Inst.Proc.18,45-50(1960)
34.Tang,X.W.,Onitsuka,K.:Consolidation by vertical drains under time-dependent loading.Int.J.Numer.Anal.Methods Geomech.24,739-751(2000)
35.Qin,A.F.,Sun,D.A.,Tan,Y.W.:Analytical solution to onedimensional consolidation in unsaturated soils under loading varying exponentially with time.Comput.Geotech.37,233-238(2010)
36.Olver,F.W.J.,Lozier,D.W.,Boisvert,R.F.,et al.:NIST Handbook of Mathematical Functions.Cambridge University Press,New York(2010)
37.Wilson,D.G.:Existence and uniqueness for similarity solutions of one dimensional multi-phase Stefan problems.SIAM J.Appl.Math.35,135-147(1978)
38.Tao,L.N.:The heat conduction problem with temperature dependent material properties.Int.J.Heat Mass Transf.32,487-491(1989)