EXPLICIT SOLUTION TO GREEN AND AMPT INFILTRATION EQUATION

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1 EXPLICIT SOLUTION TO GREEN AND AMPT INFILTRATION EQUATION By Srgio E. Srrno ABSTRACT: An xlicit solution of th Grn nd Amt infiltrtion qution is rsntd by constructing dcomosition sris. Siml xrssions for th cumultiv infiltrtion dth nd th infiltrtion rt r roosd. Ths xrssions r vlid for d homognous soils undr onding conditions rsulting from intns rinfll vnts. Th solution ws comrd with th xct imlicit solution, nd with th Lmbrt W solution of th Grn nd Amt qution. It ws found tht with thr trms in th sris, th dcomosition solution hs mximum rror of bout 0.5%. Th inclusion of dditionl trms in th sris rducs th rror. With four trms, th dcomosition sris yilds n rror of bout 0.0%. Th ccurcy of thr-trm dcomosition solution sms dqut for most rcticl clcultions. Furthrmor, its simlicity of imlmnttion mks it suitbl for fst hnd clcultions. INTRODUCTION Th Grn nd Amt qution (Grn nd Amt 9) continus to b widly usd modl of tim volution of th cumultiv infiltrtion dth nd th infiltrtion rt in d homognous soils undr onding conditions tht dvlo during intns rinfll vnts. It is rltivly sy to imlmnt; it is hysiclly bsd, sinc it riss from finit-diffrnc liction of Drcy s lw; nd it lnds itslf to th following lictions: trnsint rinfll conditions (Chu 978), timvrying dth of onding (Frybrg t l. 980), nd soils in which th hydrulic conductivity chngs with dth (Bvn 984). It hs bn usd s bsic roch to comrhnsiv modls (Schmid 990). Among th difficultis in th imlmnttion of th Grn nd Amt modl r th ccurt stimtion of th tim to onding conditions (Min nd Lrson 97), nd th fct tht th Grn nd Amt qution givs th cumultiv infiltrtion imlicitly in trms of tim. To us th qution, th hydrologist must us itrtion to clcult th tim, t, for slctd vlu of th cumultiv infiltrtion, F. Onc th cumultiv infiltrtion vrsus tim is constructd in this mnnr, th infiltrtion rt vrsus tim curv my b infrrd. Brry t l. (99) drivd n xlicit solution tht mks us of th Lmbrt W function, which is rltivly unknown scil function tht nds to b vlutd ovr its sris xnsion, or bsd on som roximtions. In th nxt sction, nothr xlicit solution bsd on th liction of th mthod of dcomosition (Adomin 994) is rsntd. Dcomosition gnrts sris, much lik th Fourir sris, tht convrgs fst to th xct solution. Dcomosition hs bn lid to th nlyticl solution of nonlinr infiltrtion nd trnsort qutions (Srrno 996, 998; Srrno nd Adomin 996) without discrtiztion or linriztion. In th vrifiction sction, comrison with th xct imlicit solution nd with th Lmbrt W solution is rsntd for both th cumultiv infiltrtion nd th infiltrtion rt. DECOMPOSITION OF GREEN AND AMPT EQUATION Considr th infiltrtion of wtr into d homognous soil undr n intns constnt rinfll rt, (mm/h), such tht onding lyr of wtr of thicknss H (mm) hs dvlod Prof. of Hydrologic Engrg., Dt. of Civ. Engrg., Univ. of Kntucky, Lxington, KY E-mil: srgio@ngr.uky.du Not. Discussion on until Jnury, 00. To xtnd th closing dt on month, writtn rqust must b fild with th ASCE Mngr of Journls. Th mnuscrit for this r ws submittd for rviw nd ossibl ubliction on Fbrury 8, 000; rvisd August 8, 000. This r is rt of th Journl of Hydrologic Enginring, Vol. 6, No. 4, July/August, 00. ASCE, ISSN /0/ /$8.00 $.50 r g. Pr No. 9. on th ground surfc t tim t (h) ftr rinfll bgn. At ny tim t (h) ftr th strt of rinfll, th wtting front is loctd vrticl distnc L (mm) from th ground surfc. Th rssur hd t th wtting front, f (mm), is ngtiv if w tk th mbint tmoshric rssur s rfrnc. Nglcting ir ntrmnt, th ortion of th soil btwn th ground surfc nd th wtting front is sturtd nd hs volumtric wtr contnt of = n, whr n is th soil orosity. For th soil dr thn th wtting front t tim t, th wtr contnt is = i, whr i is th initil wtr contnt tht is, th wtr contnt rior to th storm. Alying Drcy s lw, nd roximting th hydrulic grdint s finit diffrnc btwn th ground surfc nd th wtting front, th infiltrtion rt t th ground surfc, f (mm/h), is givn by f L H f(t) =K () L whr K = sturtd hydrulic conductivity (mm/h). Nglcting th onding hd, H, nd introducing th cumultiv infiltrtion dth in millimtrs s F(t) = 0 f() d =(n i )L, t () bcoms f = K ; = f(n i) () F whr. dnots th bsolut vlu. Exrssing f(t) =df(t)/dt, () bcoms th ordinry diffrntil qution df K K =0; t t ; F(t )=F () dt F whr F = cumultiv infiltrtion t th tim of onding (mm). Eq. () dscribs th cumultiv infiltrtion dth t ny tim t t. Multilying by dt; srting vribls; intgrting t btwn t nd t, nd F btwn F nd F; nd rrrnging, w obtin th trditionl Grn nd Amt solution F F(t ) F(t ) t = ln t ; t t (4) K K F(t) W cll (4) th xct solution of th roximt infiltrtion modl [()], lthough F is not givn xlicitly in trms of t. To obtin th cumultiv infiltrtion function vrsus tim, th hydrologist must slct tril vlus of F nd substitut thm in (4) to obtin th corrsonding tim of occurrnc t. Using scil form of th soil-wtr hysicl rltionshis (i.., th wtr contnt vrsus rssur-hd rltionshi, nd th hydrulic conductivity vrsus rssur-hd rltionshi), Brry t l. (99) obtind rticulr solution of th Richrds qution nd showd tht, undr th bov ssumtions, th cumultiv infiltrtion t th ground surfc rducs to th Grn nd Amt qution [(4)]. Thus, () my b xrssd s 6 / JOURNAL OF HYDROLOGIC ENGINEERING / JULY/AUGUST 00 J. Hydrol. Eng., 00, 6(4): 6-40

2 F ln(f ) B(t) =0; B(t) =K(t t ) F ln(f ) (5) By mking th substitution y = (F ), (5) rducs to (B(t))/ w y w = x; x = ; w = (6) Th solution to (6) is simly w = W(, x), whr W(,) dnots th Lmbrt W function. Exnding (B(t))/ F(t) = W, ; t t (7) Th Lmbrt W function is rltivly unknown scil function tht solvs (6), which hs n infinit numbr of solutions for ch nonzro vlu of x (Corlss t l. 996). As rsult, Lmbrt W hs n infinit numbr of brnchs, scilly in th comlx ln. Exctly on of ths brnchs is nlytic t zro. This brnch is rfrrd to s th rincil brnch of Lmbrt W, nd is dnotd by W(0, x). Th othr brnchs ll hv brnch oint t zro, nd ths brnchs r dnotd by W(k, x), whr k is ny nonzro intgr. Th rl brnch tht corrsonds to th solution of th infiltrtion qution [()] occurs whn k =, s xrssd in (7). Brry t l. (99) showd tht th brnch whn k = 0 corrsonds to cillry ris t th wtr tbl. Th Lmbrt W function is rltd to th tr gnrting function, T(x), common in th nlysis of lgorithms discilin. Whthr or not (7) is considrd closd-form solution is subjct to intrrttion. Th Lmbrt W function hs th sris xnsion (Corlss t l. 996) i i () i i 8 4 W(x) = x = x x x x (i )! i= x x 4 5 (8) which convrgs for x </. A usful roximtion is givn in (9) nd (9b) W(x) 0.665[ ln(x )]ln(x ) 0.04; 0 x 500 (9) W(x) ln(x 4) ln[ln(x)]; x > 500 (9b) ln(x) Th roximtion corrsonds to th rl brnch (/ < x, W > ); tht is, k = 0. Corlss t l. (996, 997) rovid n xcllnt rviw of roximtions, xmls, lictions, nd rfrncs to this function tht rs in mny lictions of nginring nd scinc. Symbolic lgbr softwr, such s Ml (Wtrloo Ml Inc., Wtrloo, Ontrio, Cnd), now rovid roximtions to th Lmbrt W function s intrinsic functions. As of this writing, most clcultors do not offr it s stndrd function. Th infiltrtion rt is obtind by diffrntiting (7) with rsct to t s KW, (B(t))/ f(t) = ; t t (0) (B(t))/ W, Now ttmtd is n xlicit solution of () using th mthod of dcomosition (Adomin 994), which llows th solution of nonlinr qutions. A solution to F is sought in trms of mor commonly known functions, such s th nturl logrithm. From th mny dcomosition otions vilbl, on could strt from th diffrntil qution [()] itslf nd obtin sris solution. Howvr, it sms simlr to strt from th Grn nd Amt solution [(4)]. Rwriting (4) F F = K(t t ) F ln () F which my b writtn s F F = K(t t ) F NF; NF = ln () F Exnding th nonlinr ortor NF F = K(t t ) F A () i i=0 nd th Adomin olynomils r dfind s dnf 0 A 0 = NF 0 ; A = F df 0 (4,b) dnf0 F d NF0 A = F df! df (4c) 0 0 dnf0 d NF0 F d NF0 df0 df 0! df 0 A = F FF (4d) Th olynomils A n r gnrtd for ch nonlinrity so tht A 0 dnds only on F 0, A dnds only on F 0 nd F, A dnds only on F 0, F, F, nd so on. All of th F n comonnts r nlytic nd clculbl. Th trm n=0 F n constituts gnrlizd Tylor sris bout th function F 0. Somtims th mgnitud of th soil rmtrs is such tht th sris n convrgs ridly nd th n-trm rtil sum n = j=0 F j, th roximnt, srvs s n ccurt nough nd rcticl solution. For furthr discussion on th convrgnc roblm of dcomosition sris, th rdr is rfrrd to Chrruult t l. (99), Chrruult (989), nd Abboui nd Chrruult (994). It is lso imortnt to mntion th rigorous mthmticl frmwork for th convrgnc of dcomosition sris dvlod by Gbt (99, 99, 994). H connctd th mthod of dcomosition to wll-known formultions whr clssicl thorms (.g., fixd oint thorm, substitutd sris, tc.) could b usd. For discussion on th convrgnc of dcomosition sris with scil rfrnc to infiltrtion qutions, s Srrno (998). Th roosd dcomosition xnsion diffrs from th Tylor sris xnsion roosd by Schmid (990). Th conct bhind dcomosition constructs ch trm in th sris bsd on th (nonlinr) ortor of th qution nd th nlyticl form of th rvious trms in th sris. From () nd (4) F = A = ln F = A = ln F 0 = K(t t ) F K(t t ) F F = A 0 = ln F (5) (5b) K(t t ) F F K(t t ) F (5c) K(t t ) F F [K(t t ) F ] K(t t ) F ln F [K(t t ) F ] (5d) JOURNAL OF HYDROLOGIC ENGINEERING / JULY/AUGUST 00 / 7 J. Hydrol. Eng., 00, 6(4): 6-40

3 K(t t ) F F [K(t t ) F ] K(t t ) F ln F [K(t t ) F ] K(t t ) F ln F [K(t t ) F ] F = A = ln 4 (5) Mor trms my b sily drivd. Fctorizing, on obtins logrithmic sris whos cofficints r tim sris with closd-form rrsnttion F(t) =F (t) ln 0 F 0(t) F 0(t) F F (t) F 0(t) F 0(t) F 0(t) ln ln F F (t) F [F 0(t) ]...; t t F 0 (t) (6) FIG.. Comrison btwn Lmbrt W nd Dcomosition Solutions FIG.. Rltiv Error in Prcntg 8 / JOURNAL OF HYDROLOGIC ENGINEERING / JULY/AUGUST 00 J. Hydrol. Eng., 00, 6(4): 6-40

4 Th infiltrtion rt is obtind ftr diffrntiting (6) with rsct to t K F0 F0 f(t) =K ln F F F ; t t F 0 (7) VERIFICATION To gin n id on th ccurcy of (6), ssum K = mm/ h, F = = mm, nd t = h. Fig. illustrts comrison btwn th Lmbrt W solution [(7)] nd th dcomosition solution with thr trms on th right sid of (6). Th grmnt btwn th two solutions is xcllnt, nd cn b imrovd by clculting mor trms in th dcomosition sris. Fig. shows th rltiv rror in rcntg btwn th two solutions for th cumultiv infiltrtion. Th mximum rltiv rror is bout 0.5%. Th mximum rltiv rror is rducd to bout 0.0% if on includs th fourth trm on th right sid of (6). If th hydrologist uss th first two trms in (6) only, th rror incrss to bout 0.9%, which might b cctbl for rliminry hnd-clcultor stimtions. Now conductd is mor dtild comrison, using th xct solution [(4)] s lid by Dingmn (994,. 4), FIG.. Comrison mong Lmbrt W, Dcomosition, nd Exct Solutions FIG. 4. Infiltrtion Rts ccording to Diffrnt Solutions JOURNAL OF HYDROLOGIC ENGINEERING / JULY/AUGUST 00 / 9 J. Hydrol. Eng., 00, 6(4): 6-40

5 which consistd of th stimtion of th cumultiv infiltrtion nd th infiltrtion rt undr constnt rciittion intnsity. Th following r th rmtr vlus ftr som dimnsionl trnsformtion (Dingmn 994): = 5 mm/min, K =.08 mm/min, n = 0.49, i = 0.5, nd f = 66 mm. Th tim to onding ws stimtd s (Dingmn 994) Kf(n i) K t = = =.6 min (8) ( K) ( K) Accordingly, th cumultiv infiltrtion t th tim of onding is F = t = 8.66 mm. Th rmtr = f (n i )=.454 mm. Fig. shows comrison mong th xct solution [(4)], th Lmbrt W solution [(7)], nd th dcomosition solution with thr trms on th right sid of (6). Agin th grmnt mong th thr solutions is xcllnt. Th dcomosition solution slightly ovrstimts th vlus of cumultiv infiltrtion. This discrncy could b rducd by including mor trms of th sris in th clcultions. Fig. 4 shows th corrsonding volution of th infiltrtion rt ccording to () nd (4) for th xct solution, (0) for th Lmbrt W solution, nd (7) for th dcomosition solution, rsctivly. Not tht f =, whn t < t. As xctd, th ccurcy of th dcomosition solution dcrss whn diffrntiting F. For rly tims ftr onding, nd for rolongd tims ftr wtting, th dcomosition solution sms dqut. By fr th most imortnt dvntg of th dcomosition solution is its simlicity of imlmnttion. SUMMARY AND CONCLUSIONS An xlicit solution of th Grn nd Amt infiltrtion qution ws drivd by constructing nd truncting dcomosition sris. Siml sris xrssions for th cumultiv infiltrtion dth nd th infiltrtion rt wr roosd. Th solution ws comrd with th xct imlicit solution, nd with th Lmbrt W solution. It ws found tht with thr trms in th sris, th dcomosition solution hs mximum rror of bout 0.5%. Th inclusion of dditionl trms in th sris rducs th rror. With four trms, th dcomosition sris yilds n rror of bout 0.0%. Th ccurcy of thrtrm dcomosition solution sms dqut for most rcticl clcultions. Furthrmor, its simlicity of imlmnttion mks it suitbl for fst hnd-clcultor stimtions. Th ccurcy of th dcomosition solution dcrss for th infiltrtion rt, bing dqut t th tims immditly ftr onding, nd t rolongd infiltrtion. ACKNOWLEDGMENTS Th suort nd ncourgmnt of th Ntionl Scinc Foundtion, grnt numbr BES , r grtly rcitd. REFERENCES Abboui, K., nd Chrruult, Y. (994). Convrgnc of Adomin s mthod lid to diffrntil qutions. Com. & Mthmtics with Alictions, 8(5), Adomin, G. (994). Solving frontir roblms in hysics Th dcomosition mthod, Kluwr Acdmic, Boston. Brry, D. A., Prlng, J-Y., Sndr, G. C., nd Sivln, M. (99). A clss of xct solutions for Richrds qutions. J. Hydro., 4, Bvn, K. (984). Infiltrtion into clss of vrticlly non-uniform soils. Hydrologicl Sci. J., 9, Chrruult, Y. (989). Convrgnc of Adomin s mthod. Kybrnts, 8(), 8. Chrruult, Y., Sccomrdi, G., nd Som, B. (99). Nw Rsults for convrgnc of Adomin s mthod lid to intgrl qutions. Mth. nd Com. Modlling, 6(), Chu, S. T. (978). Infiltrtion during n unstdy rin. Wtr Rsour. Rs., 4, Corlss, R. M., Gonnt, G. H., Hr, D. E. G., Jffry, D. J., nd Knuth, D. E. (996). On th Lmbrt W function. Adv. in Com. Mth., 5, Corlss, R. M., Jffry, D. J., nd Kuth, D. E. (997). A squnc of sris for th Lmbrt function. Proc., ISSAC 97, W. W. Kuchlin, d., ACM, Nw York, Dingmn, S. L. (994). Physicl hydrology, Mcmilln, Nw York. Frybrg, D. L., Rdr, J. W., Fnzini, J. B., nd Rmson, I. (980). Aliction of th Grn nd Amt modl to infiltrtion undr timdndnt surfc wtr dths. Wtr Rsour. Rs., 6, Gbt, L. (99). Equiss d un théori décomositionnll t liction ux qutions ux dérivés rtills. Doctorl dissrttion, Ecol Cntrl d Pris, Frnc (in Frnch). Gbt, L. (99). Th dcomosition mthod nd linr rtil diffrntil qutions. Mth. nd Com. Modlling, 7(6),. Gbt, L. (994). Th dcomosition mthod nd distributions. Com. & Mthmtics with Alictions, 7(), Grn, W. H., nd Amt, G. A. (9). Studis on soil hysics. : Th flow of ir nd wtr through soils. J. Agric. Sci., 4(), 4. Min, R. G., nd Lrson, C. L. (97). Modling infiltrtion during stdy rin. Wtr Rsour. Rs., 9(), Schmid, B. (990). Drivtion of n xlicit qution for infiltrtion on th bsis of th Min-Lrson modl. Hydrologicl Sci. J., Oxford, Englnd, 5, Srrno, S. E. (996). Hydrologic thory of disrsion in htrognous quifrs. J. Hydrologic Engrg., ASCE, (4), Srrno, S. E. (998). Anlyticl dcomosition of th non-linr unsturtd flow qution. Wtr Rsour. Rs., 4(), Srrno, S. E., nd Adomin, G. (996). Nw Contributions to th solution of trnsort qutions in orous mdi. Mth. nd Com. Modlling, 4(4), / JOURNAL OF HYDROLOGIC ENGINEERING / JULY/AUGUST 00 J. Hydrol. Eng., 00, 6(4): 6-40

Chapter 11 Calculation of

Chapter 11 Calculation of Chtr 11 Clcultion of th Flow Fild OUTLINE 11-1 Nd for Scil Procdur 11-2 Som Rltd Difficultis 11-3 A Rmdy : Th stggrd Grid 11-4 Th Momntum Equtions 11-5 Th Prssur nd Vlocity Corrctions 11-6 Th Prssur-Corrction