CE 504 Computational Hydrology Infiltration Equations Fritz R. Fiedler

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1 CE 504 Compuaional Hdrolog Infilraion Equaion Fri R. Fiedler 1 Ricard Equaion 2 Green-Amp equaion 3 Oer Hdrologi are concerned wi e paial and emporal pariioning of rainfall ino urface and uburface flow, and e ubequen movemen in and beween ee wo general componen of e drologic ccle. A ime we ma neglec one componen; for eample, if e focu i on even-cale urface runoff in emi-arid environmen, once waer i infilraed i ma be conidered lo from e em. However, allow laeral uburface flow i imporan o deermining orm runoff in mo umid environmen. Ricard Equaion Ricard equaion i e fundamenal governing equaion for uburface flow. I i derived b combining a ma balance on an elemenal volume of porou media wi Darc law. In careian coordinae, wi d, d, d defining e conrol volume, e ma balance reul in q q q 0 1 were i e volumeric moiure conen volume of waer / oal volume, and q, q, and q are dicarge per uni area in e,, and direcion, repecivel. Te fir erm repreen e cange in waer orage wiin e elemenal volume, and e remaining ree erm are e cange in pecific dicarge in eac of e ree direcion. In e direcion, Darc law i wrien q 2 were i e draulic conducivi in e direcion, and i e draulic ead, given b 3 were i e oil ucion ead and i e elevaion above ome arbirar daum. Soil ucion ead i equal o p/γ, were p i e oil waer preure le an amoperic for unauraed oil and γ i e pecific weig of waer. In unauraed oil, bo and are a funcion of oil moiure conen, i.e.,,,. Subiuing ee and Equaion 3 ino Equaion 2 reul in q 4 and inroducing e oil waer diffuivi d D 5 d Equaion 4 become

2 D q 6 Equaion 6 a limied uili in laered oil, a i diconinuou a laer inerface. Te draulic conducivi funcion can be wrien a wen i aumed o be a unique funcion of. Combining Darc law and e ma balance equaion in ree dimenion reul in 7 Defining e pecific moiure capaci o be C d/d, and ubiuing i and Equaion 3 ino Equaion 7 1 C 8 Te oluion of i equaion require, C, and o be known. For onedimenional flow in e verical direcion, i.e., infilraion, e fir wo erm on e rig ide of Equaion 8 are no required. For auraed condiion, C i ero, and i conan and equal o e auraed draulic conducivi. In i cae, Equaion 8 reduce o Infilraion during rainfall can be imulaed uing Ricard equaion b pecifing appropriae boundar condiion. A e ground urface, eier a flu equal o e rainfall rae i pecified prior o ponding, or a ead equal o e dep of ponded waer i pecified. Before ponding, e oil a e ground urface i no auraed. Tee condiion can be wrien p r 0 0, 0, 1 10 p > 0, 0, 0 11 Noe a, for eample, e noaion 0, refer o e ground urface 0 a ime, and would be negaive below e ground urface in e andard careian coordinae em. A auraion i ero, below auraion i i negaive, and 0 refer o a mall ponded dep of waer. Te lower boundar condiion i locaed a e waer able wen e dep o auraion i mall, or b a uni gradien flu wen e dep o auraion i large. Tee are wrien 0 0, > L 12 0 > q 13 Of coure, iniial condiion mu be pecified a well. Numerical oluion o a mied form of Ricard equaion are beneficial in a ma conervaion i eaier o acieve. In e verical dimenion, i form i wrien 1 14

3 Parameer Soluion of Ricard equaion require knowledge of e and relaionip. For eample, van Genucen 1980 propoed e following wo relaionip r r < 0 n m 1 α 0 15 [ ] 1/ m m 1 1 S 2 e l S 16 e were m 1-1/n for n > 1, S e - r / - r, r i e reidual volumeric moiure conen, and l, α and n are van Genucen model parameer. Wile oer form of e and relaionip ue ome differen parameer,,, and r, repreening auraed draulic conducivi, moiure conen a auraion, and reidual moiure conen, repecivel, are common oil model parameer. Tere are variou mean o obain ee required relaionip, including direc meauremen, ue of value pical for a given oil pe, and regreion beween oil eural properie and model parameer. Te laer are omeime referred o a pedoranfer funcion. Te compuer program Roea, developed b Marcel G. Scaap 1999 of e Soil Salini Laboraor, Agriculural Reearc Service, U.S. Deparmen of Agriculure, provide mean o eimae e van Genucen 1980 model parameer baed on Soil eural clae; Sand, il and cla percenage; Sand, il and cla percenage and bulk deni; Sand, il and cla percenage, bulk deni and e value of a 330 cm 33 kpa; and Sand, il and cla percenage, bulk deni and e value of a 330 and 15,000 cm 33 and 1500 kpa. Green-Amp Equaion Te Green-Amp meod Green and Amp, 1911 i baed on e pical approimaion a waer infilrae a a arp weing fron wi auraed condiion beind e weing fron. Tu e cumulaive infilraion, F, i given b F L η i L 17 were L i e dep of e weing fron below e ground urface, η i e poroi, and i i e iniial moiure conen. Te infilraion rae, f, i deermined wi Darc law 0 L L f L L 18 were 0 i e dep of ponded waer, wic i picall aumed o be negligible. Equaion 16 and 17 are combined o obain

4 F f F 19 and ince f df/d, and equaion can be derived for F F F ln1 20 Tee equaion are valid wen e urface i ponded; in rainfall-runoff applicaion, i occur wen e rainfall rae i greaer an e infilraion capaci, f. Mein and Laron 1973 developed equaion o predic e ime o ponding under ead uniform rainfall baed on e aumpion a prior o ponding, all rainfall infilrae and infilraion occur a a arp weing fron. A e ponding ime, p, e cumulaive infilraion F p i p, wic can be ubiued ino Equaion 18 and olved for p p 21 i i So for < p, f i and F i, and for > p, Equaion 19 i ued o compue F, and f i compued uing Equaion 18. Noe a Equaion 19 i nonlinear and mu be olved wi an ieraive ecnique. Ti ecnique can be eended o unead rainfall b racking F and comparing i o f rougou e orm. A eograp compoed of dicree, ead pule of rainfall of duraion i ued o decribe i. F F i wile f < i. Wen e urface i ponded rougou e ime incremen F F F ln F 22 If f < i, en ponding occur during e ime inerval. Uing f i and F F p in Equaion 18 and olving for F p F p 23 i Ten p ' were Fp F ' 24 i Te cumulaive infilraion a e end of a ime period in wic ponding a occurred i compued uing Equaion 21, uing F F p and - '. Figure 1 illurae e algorim ued o compue f and F, and erefore rainfall ece, for unead rainfall.

5 Figure 1. Infilraion compuaion uing e Green-Amp meod for unead rainfall afer Cow e al., Parameer Te parameer and ued in e Green-Amp equaion are no e ame a oe ued in Ricard equaion; raer, e ould be conidered parameer pecific o i equaion. Te draulic conducivi ued in e Green-Amp meod i ofen aken o be ½ of e auraed value. Te weing fron ucion i generall conidered o be a conan for a given oil. Table 1 provide pical value. Noe a on i able, e effecive poroi e η - r.

6 Table 1. Tpical Green-Amp 1911 parameer afer Cow e al., Oer Infilraion Equaion oiakov Ti equaion i wrien α f k 25 were k and α are conan derived from infilraion daa. Tere i no good wa o obain ee conan from baic oil properie, u i equaion i no frequenl ued. Horon Horon empirical infilraion i β f f c f o f c e 26 were f o i e maimum infilraion rae, and f c i e ead ae infilraion rae. Te coefficien β i a rae conan. Te ree parameer of i equaion are picall deermined from infilraion daa, u uffer from e ame limiaion a an empirical model. Holan Holan empirical infilraion equaion i

7 1.4 f GI A S a f c 27 were f i in ince per our, GI i a crop grow inde a range from 0.1 o 1.0, A repreen macropore aociaed wi plan roo, and f c i e ead ae infilraion rae. Pillip Ti equaion i baed on a erie oluion o Ricard equaion in a omogeneou oil wi conan iniial moiure conen and conan ead mainained a e oil urface. I i wrien 5 f 0.5S were S i a parameer known a e oil orpivi. Smi-Parlange Ti equaion i wrien α f 1 ep αf / B 1 were α and B are parameer a can be relaed o oil pe, oil ucion, and oil moiure defici. 29 Reference Cow, V. T., D. R. Maidmen, and L. W. Ma, Applied Hdrolog, McGraw Hill, Inc., Green, W. H., and G. A. Amp, Sudie on oil pic: 1. Flow of air and waer roug oil, Journal of Agriculural Science, 4, 1-24, Mein, R. G., and C. L. Laron, Modeling infilraion during ead rain, Waer Reource Reearc, 92, , Scaap, M. G., Roea uer manual, Soil Salini Laboraor, Agriculural Reearc Service, U.S. Deparmen of Agriculure p:// acceed Ocober 27, 2003, publied Van Genucen, M. T., A cloed-form equaion for predicing draulic conducivi of unauraed oil, Soil Science Socie of America Journal, 44, , 1980.