DIFFERENTIAL EQUATIONS AND THEIRS APPLICATION TO THE SOIL MOISTURE STUDY

Transcription

1 Bllein UASVM oriclre 652/2008 pissn ; eissn DIFFERENTIAL EQUATIONS AND TEIRS APPLICATION TO TE SOIL MOISTURE STUDY Florica MATEI Viorel BUDIU Maei DIRJA Ioana POP Maria MICULA USAMV Clj-Napoca 3-5 Manasr Sree Clj-Napoca ewords: Darc s eaion Ricard s eaion diffsion eaion soil moisre Absrac: In is noe we make a connecion beween diffsion eaion and eaions of Darc and Ricard sed in sd e soil drodnamics. INTRODUCTION Mos psical penomena e.g. flid dnamics can be modeled sing parial differenial eaion. A parial differenial eaion is an eaion a conains parial derivaives of nknown fncion. Two of e mos common eamples are speeded of milk in a cp of coffee or warming of a rod wen a e end of i ere is one or wo sorce of ea. Te firs one sd e diffsion of milk in a cp of coffee; e milk concenraion depends pon ime elapsed b e ime wen someone p e milk in a coffee and b e place were e por e milk. Te second one sd e dissipaion of e ea ino e rod a depends b e ime and b e posiion of considered poin ino e rod. Also we can imagine ow e waer disribes ino e soil laers is penomenon depend also b e ime and space and a e end of is paper we sow a i governed b e diffsion eaion. YDRODYNAMICS OF SOIL WATER AND DIFFUSION EQUATION Te drodnamics of e waer ino soil is a ver comple penomenon. In order o ndersand is penomenon soil scieniss ave made some models for e flow of waer ino soil. Depending b e parameers a will be considered and e wa a we modeling e soil waer flow ino e soil we can obain differen pe of eaions. I is imporan o ave eilibrim beween e assmpions and e parameers involved in a model. Taking ino consideraion previos affirmaion we ave differen pe of eaions a models e waer flow ino soil. 1. Darc s eaion Te assmpions in is case are: soil is saraed wi waer; waer is flowing in all pores nder a posiive pressre ead. Usall e soil is asi-saraed wi waer b for is case of saraed flow e impac of pores filled wi air is no considered. Te soil is paced in a orional clinder conneced in bo sides wi vessel filled wi waer mainaining in bo side a consan level of waer. If e waer level in e lef side is V iger a on e rig side e waer will flow from lef o rig. Te fl densi A were V is e volmeric overflow is ime and A is e area of a cross secion of e 594

2 clinder perpendiclar o e direcion of flow. Saring from is relaion e Darc ave obained e following eperimenal law: s 1 were: represens e oal poenial ead and represens e erm de o elevaion in case wen e soil is p in a verical clinder; s represens saraed dralic condcivi of e soil. Oer limiaion of Darc s eaion is given b e low gradien. 2. Unsaraed flow in rigid soil A rigid soil is a do no cange eir blk volme wi cange of waer conen [3]. Te assmpion in is case is a pores a are filled wi air cold resarae or drain; e capillari effec is no ake ino consideraion. Te eperimen in is case considers an nsaraed soil colmn and we can make an analog wi a flow in a spon wi an insalled resisance. Te fl densi depends pon e dralic gradien and is governed b a eaion similar o 1: 2 L were L is e soil leng and is nsaraed soil condcivi [LT -1 ]. Becase e soil is no saraed and flow occrs primaril in pores filled wi waer is smaller an s for e same b saraed soil. From is reason will be a fncion of e soil waer poenial ead and eaion 2 became d 3 d If we work in 2 or 3 dimensions we ave: 4 Bckingam was e firs a ad described e dependence of nsaraed flow pon e poenial gradien so eaion 3 and 4 are named Darc-Bckingam eaions. Tese pe of eaions are adeae for describing nsaraed flow onl if e soil waer conen is no canging in ime b is case is ver seldom one. Wen and are canging in ime eaion 3 or 4 ms be combined wi e conini eaion. Conini eaion relaes e rae of cange in ime for wi e spaial rae of cange for in a smaller elemenar soil volme. Te resl is a non linear eaion and even for simpl condiions e solion is difficl o find. Te fl densi is described b Darc-Bckingam eaion and e rae of filling or emping of e pores of e soil is described b e eaion of conini. Consider e volme elemen aving e edges of leng and. Te difference beween e volme of waer flowing ino e volme elemen and volme of waer a flowing o is eal o e difference of waer conen in e elemen of ime. Te rae of inflow in direcion is. We assme a e rae of cange for is coninos so e rae of oflow is. Te inflow volme is and e oflow volme is. Te difference beween inflow and oflow in direcion is: 5 Te differences beween inflow and oflow in and direcion are: 595

3 Te cange in waer conen for e enire represenaive volme is e sm of 5 6 and 7 a means: 8 wen 0 eaion 8 became 9 Combining 9 wi 3 we ave: 10 If we work in one dimension and we assme a e soil is isoropic en 11 Eaions 10 and 11 are called Ricard s eaions. 3. Waer flow in non rigid soils. Wen a soil swells or srinks de o e waer conen e eaions menion above do no work. A eor for e ree dimensional case is an open problem. Tere are some resls for one dimensional case [4 5] and describe weing of arificiall repacked soil colmn in a laboraor. 4. Connecion o e diffsion eaion. Te diffsion eaion is a parabolic parial differenial eaion. Te general forms of is eaion are: 2 2 D one dimensional diffsion eaion 12 D n dimension diffsion eaion 13 D n dimension diffsion convecion eaion 14 f D 15 diffsion eaion wi ea sorce 15 In wa i follows we empasie e connecion beween diffsion eaion in forms menioned above and Darc s and Ricard s eaions. Te general case is e diffsion eaions; in order o sow e eivalence we ms o connec diffsion coefficien D wi dralic condcivi [6]. If we se e Ricard s eaion became: 16 Te connecion beween soil waer diffsivi D and is D 17

4 en eaion 16 became: D 18 and represens e Ricard s eaion in diffsivi form and is in e form of classical maemaical eaion 14. Wen e graviaional erm is negleced e above eaion became D 19 Te las eaion as e same form as eaion 12. Te diffsivi form for Darc- Bckingam eaion is D 20 Te main reason for e se of differenial eaion in diffsivi form is given b e redcion of e nmber of variable. CONCLUSIONS De o pecliar condiion of eac eperimen a we sd in mos of e cases e model se one of e eaions menion above. Tere is some pariclar pe of soil a works wi oer maemaical model e. g. [1]. Te soil scieniss prefer e Darc or Ricard eaions and maemaicians work wi diffsion eaion. For bo sides remains e re problem: finding e solion of ese eaions. In order o find e solion is mandaor a eaion saisf some iniial Diricle problem or bondar condiion Newman problem or bo condiions. We menion some iniial and bondar condiion: iniial condiions e vales for and for all. Wen e iniial condiion demands d 0 0 along e enire colmn. If e soil waer conen i is consan wi d dep a fl corresponding o e ni gradien of eis. If i is ver small e downward fl ma be ver small. bondar condiions. A bondar for e 1-dimensional problem is e opograpical srface; e oer bondar can be a waer able if e soil colmn as a finie leng or a defined waer conen or waer fl. If e colmn is semi-infinie e oer bondar is wen. Even for ose cases is no eas o solve is differenial eaions someimes is possible o find a analical solions in e oer i is possible o solve e problem nmericall [2]. Acknowledgmens: Tis paper as been spored b CNCSIS PNII projec ID_893. REFERENCES 1. Bojie F Yang Z. Wang Y. Zang Pingwen 2001 A maemaical model of soil moisre disribion on ill slopes of e Loess Plaea Science in Cina Vol. 44 No iel D Inrodcion o environmenal soil psics Springer Verlag. 3. ilek M. Donald R. Nielsen 1994 Soil drolog Caena Verlag. 4. Pilip J. R drosaics and drodnamics in selling soils Waer Resor. Res. 5 pp Smiles D. E. M. J. Rosenal 1968 Te movemen of waer in swelling maerials As. J. Soil Res. 6 pp Warrick A. W Soil waer dnamics Oford Univ. Press. 597

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