Boundary Integral Equation Method for Unsteady Two Dimensional Flow in Unconfined Aquifer

Transcription

1 Pure ad Applied Mahemaics Joural 6; 5(): 5- Published olie February, 6 (hp: doi:.648j.pamj.65.3 ISS: (Pri); ISS: (Olie) Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer Azhari Ahmad Abdalla Deparme of Mahemaics, Prep-Year, Uiversiy of Hail, Hail, Saudi Arabia address: aalkhodor@homail.com To cie his aricle: Azhari Ahmad Abdalla. Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer. Pure ad Applied Mahemaics Joural. Vol. 5, o., 6, pp. 5-. doi:.648j.pamj.65.3 Absrac: I his paper, we employed he use of Boudary Iegral Equaio Mehod o obai umerical soluios of specific ucofied aquifer flow problems. Of he wo formulaios preseed i his paper, ha i which he piezomeric head ad is ormal derivaive are assumed o vary liearly wih ime over each ime sep has proved more accurae ha ha i which boh piezomeric head ad is ormal derivaive remai cosa a each ode hroughou each ime sep. Comparisos bewee Mehod of secio-3 ad he aalyical soluios have demosraed he superior accuracy of he iegral equaio formulaio. Keywords: BIEM, Ucofied Aquifer, Dupui Assumpio, Groudwaer Flow. Iroducio The sudy of groudwaer flow is of grea imporace due o he rapid icreasig for he demad of waer. This has ecessiaed he eed o eplore ew mehods of eploiig waer, especially waer i aquifers, oe of he mai asks i sudies cocerig aquifers flow is he deermiaio of he disribuio of he piezomeric head ad is ormal derivaive. I order o fid he disribuio of piezomeric head, φ, hroughou a aquifer i is ecessary o solve a parial differeial equaio subjec o specified iiial ad boudary codiios, [], [4]. I is desirable o calculae aalyical soluios o hese problems, bu his is seldom possible due, ofe, o irregulariies i he shape of aquifer boudaries. Cosequely, approimae soluios mus be calculaed eiher by aalogue mehods, which are based o he similariy bewee groudwaer flow ad oher physical sysems obeyig he same parial differeial equaio, or by umerical echiques [4], [7]. May umerical echiques bee employed o solve boh seady ad useady groudwaer flow problems. Probably he simples ad mos widely used of hese is he fiie eleme mehod ad he fiie differece mehod, [4], [3], [8]. Aoher more direc mehod of solvig ime-depede problems is wih iegral equaios, i which he ime derivaive is wrie i fiie-differece form, [], [], [3]. I his way, he soluio a oe ime ca be calculaed from he soluio a a previous ime i erms of a iegral equaio coaiig oly boudary daa ad a surface iegral over he eire soluio domai [], [6]. The pricipal advaage of iegral equaio echiques is ha if oly boudary daa is used i he calculaios he dimesioaliy of he problem is esseially reduced by oe. Thus, he amou of daa preparaio is cosiderably less ha ha required by eiher he fiie-differece or fiie-eleme mehods. Soluios wihi he flow regio ca readily be calculaed from a iegral equaio soluio, as i will be show, ad he advaage of reduced daa preparaio is sill sigifica because oly pois a which he soluio is required eed o be specified. The flow of groudwaer is esseially a hree dimesioal problem, bu i may real siuaios he horizoal dimesio of a aquifer are much greaer ha he verical dimesios. I such cases i is possible o accep he Dupui approimaio [5], [5], which assumes ha flow is horizoal. Cosequely, he problem is reduced o wo dimesios i he horizoal plae. The relaive scarciy of published work i groudwaer lieraure cocerig he use of he boudary eleme mehod (BEM) o solve he equaio ha describes he flow i ucofied aquifer, is maily due o he umerical difficulies ecouered i applyig his mehod o resolve o-lieariy. The accuracy of he BIEM depeds o he size of odes ad o he ype of he ierpolaio fucio [9].

2 6 Azhari Ahmad Abdalla: Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer I his paper, BIEM wih he cosa ierpolaio fucio ad BIEM wih he liear ierpolaio fucio were formulaed ad used for he soluio of wo dimesioal, useady homogeous, ucofied aquifer flow (o icludig well). The resuls obaied by he wo mehods were ivesigaed ad a compariso bewee hem ad wih he eac soluio was held.. Iegral Equaio Techiques The horizoal dimesios of may aquifer are orders of magiude larger ha verical dimesios. I such cases, i is possible o make use of he Dupui approimaio, which assumes ha velociies are horizoal ad ha piezomeric heads are cosa alog a verical lie. Uder hese codiios, ucofied flows saisfy followig parial differeial equaio; [], [5] h.( T φ) = S + Qδ ( ) δ ( y y ) + R i i i () i i which S is he sorage coefficie or effecive porosiy, Q. is he flow rae i well i locaed a ( i, y i ), ad he rae of recharge, R I, is posiive for evaporaio ad egaive for repleishme, φ is he piezomeric head. The rasmissiviy, T, is give by []; z + B T(, y, ) = K(, y, z) dz () Z i which Z O (, y) is he elevaio of he boom boudary of he aquifer ad B (, y, ) is he sauraed aquifer hickess (which depeds o φ for ucofied flows). If he Dupui approimaio is applied, cofied flows saisfy h.( T h) = S + Qδ ( ) δ ( y y ) + R i i i (3) i Here S, is a fucio of he elasiciy of he aquifer. The rasmissiviy, T, is give by Eq.(), bu he aquifer hickess, B, is cosa a ay poi. Cosequely, Eq. () is oliear i φ, while Eq. (3) is liear. However, if Eq. () is liearized by assumig ha T is idepede of chages i φ, [6], [], he boh cofied ad ucofied flows will obey a parial differeial equaio of he same form. Furhermore, if he aquifer is homogeeous ad isoropic, boh cofied ad ucofied flows are govered by: T φ = S + Qδ ( ) δ ( y y ) + R (4) i i i i Where T ad S are cosas. Iegral equaio echiques was used i [] o aalyze problems wih combiaios of leaky, layered, cofied, ucofied, ad o isoropic aquifers uder seady sae codiios. To illusrae he basic echique for useady flows, cosider ucofied flow hrough a homogeeous, aisoropic, o-leaky regio, R, ha is bouded by a coour, Γ, ad i which he pricipal aes of seepage are parallel o he ad y aes. The permeabiliies for direcios parallel o he ad y aes are give by k ad k y respecively [6], ad if here is o recharge, Eq. () ca be wrie as φ φ S + = φ y K * * k I which * = ad y* = y* If a backward-differece k approimaio is adoped for he ime derivaive as i [5], Eq. (5) ca be wrie as s *, y [ φ( + )] φ( + ) = h( ) * k k I which values of φ a ime are kow ad a ime + are ukow I order o solve Eq. (6) by iegral equaio echiques, i mus be liearized [8], [4]. The followig was suggesed i []: [ φ s S ( ) ] [ ( ) ] ( ) X*, Y* k ( ) φ + + = φ + K φ (7) I which, φ ( + ) ) is a average value of φ over he regio. From Eq. (6), i, ca be show ha; y (5) (6) U ( P, Q ( Q, + ) S φ( p, + ) = φ ( Q, + ) U ( P, Q) ds( Q) (, ) (, ) Γ ( ) ( ) U P Q φ Q da (8) Q Q k R Where: S U( P, Q) = K r( P, Q) Kh( + ) Ad r (P, Q) is he disace from a fied poi, P, o he boudary o aoher poi, Q. The variable,θ, is he agle bewee he boudary ages a P. The soluio is obaied ieraively by assumig a value of φ ( + ) ad he solvig (9) Eq. (7) as i [7], []. The esimae of φ ( + ) is he revised ad he process repeaed uil here is esseially o chage i successive soluios. The obvious disadvaage of such a formulaio is ha because he fial erm o he righ side of Eq. (7) is o-zero, i geeral, iegraios mus be performed hroughou he soluio domai a each ime sep. Thus, alhough he mehod ca be eeded o solve problems i leaky aquifers wih recharge [], i does o have he advaage of reducig he dimesioaliy of he problem. The use of ime-depede fudameal soluios o

3 Pure ad Applied Mahemaics Joural 6; 5(): 5-7 solve problems havig he same form as Eq. (4) was firs preseed ad he rasie hea coducio hrough a aisoropic medium was aalyzed i [4]. The mehod has sice bee used by may, [], [5] where he soluios o he followig form of he hea coducio equaio were preseed; V C V = () V (, y, = ) = V (, y) for (, y) R () V (, y, ) g (, y, = ) for (, y) oγ Γ (3) i which V is he emperaure, C is he hermal diffusiviy, ad Eq. () is a special form of Eq. (4), Eq. () is he iiial codiio which mus be saisfied hroughou he soluio domai, R, ad Eqs. () ad (3) are he boudary codiios which mus be saisfied o he respecive porios of he boudary coour, Γ = Γ Γ. A fudameal soluio, u, of Eq. () is give i [4] as: V (, y, ) = f (, y, ) for (, ) y oγ Γ () [ (, ) ] 4 C( τ ) r P Q u[ ( Q) ( P), y( Q) y( P), τ ] = u[ r( P, Q), τ ] =, e 4 C ( τ ) (4) i which r (P, Q) is he disace from a fied poi, P, wihi R, o aoher poi, Q, ad τ [5], so (9) ca be wrie as: V ( Q, τ ) u[ r( P, Q), τ ] V ( P, ) = C u[ r( P, Q), τ ] V ( Q, τ ) ds( Q) dτ + [ [ (, ), ] (, = ] ( ) ( ) u r P Q V Q τ da (5) Q Q R Γ +Γ If P is a poi o he boudary coour; θ V ( Q, τ ) u[ r( P, Q), τ ] V ( P, ) = C u[ r( P, Q), τ ] V ( Q, τ ) ds( Q) dτ + [ u[ r( P, Q), V ] ( Q, τ = ) ] da R ( ) ( ) Γ +Γ Q Q (6) I which θ is he agle bewee he boudary ages a p. The soluio of equaio (6) was calculaed by assumig ha V V ad remaied cosa over a sufficiely small ime sep ad by performig he ime iegraios sepwise as i [4]. [6]. Usig his assumpio i Eq. (5) ad chagig he order of iegraio gives; θ V ( Q, ) u[ r( P, Q), τ ] V ( P, ) = [ (, ), ] (, ) ( ) C ( ) u r P Q τ dτ V Q dτ ds Q ( ) Γ + Γ Q Q [ u[ r( P, Q), ] V ( Q, τ ) ] + = da R (7) The ime iegrals o he righ side of Eq. (7) ca be evaluaed aalyically, ad i ca be show ha; [ r ( p, Q)] u[ r( p, Q),, ς ] cos( r, ) 4 C( ) dτ = e (8) ( Q) Cr( p, Q) I which (r, ) is he agle bewee r (P, Q) ad he ui ormal a poi Q. The obvious disadvaage of usig Eq. (7) is ha, as wih he formulaio as meioed i [], a surface iegral mus be calculaed a he sar of each ime sep, ad hus egaig, o some ee, he advaage of iegral equaios over he more radiioal echiques. However, he mehods do reai heir accuracy, i was claimed i [] ad [6], ha he grid spacig used o calculae surface iegrals ca be greaer ha ha used i a fiie-eleme soluio [4]. I [4], boh Time depede fudameal soluio, liear ime ierpolaio fucios i Eq. (7) were used o solve aisymmeric problems, i a effor o improve accuracy. A aleraive formulaio for he soluio of Eq. (6) was preseed i [] V by calculaig he coour iegrals, assumig ha V ad varied liearly alog a sraigh lie joiig adjace odes ad iegraig he epressio aalyically aroud he boudary. The rapezium rule was used o calculae iegrals i he ime domai. I was cocluded i [], ha he iiial porios of he ime iegraios were criical ad had o be calculaed accuraely o avoid he iroducio of a cosise error io he soluio. This formulaio, also, required he evaluaio of a iegral over he soluio domai. A soluio of Eq. (6) may be obaied by dividig he ime ierval τ io ime seps ad reversig he order of iegraio o give;

4 8 Azhari Ahmad Abdalla: Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer θ V ( Q, τ ) u[ r( P, Q), τ V ( P, ) = C [ u[ r( P, Q), τ ] V ( Q, τ ) ] dτds( Q) + [ u[ r( P, Q), V ] ( Q, τ = ] da, ( ) Γ +Γ k = Q R (9) I which l = ad =. A firs glace, i may appear ha Eq. (9), is o beer ha ay of he previous formulaios because of he surface iegral ha appears o he righ side. However, because he goverig parial differeial equaio, Eq. (), is liear wih cosa coefficies, he pricipal of superposiio ca be applied. I which V o is he iiial codiio a =, give by Eq. (), he; If V = V V () θ V ( Q, τ ) u[ r( P, Q), τ ] V ( Q, ) = C [ u[ r( P, Q), τ ] V ( Q, τ ) ] τds( Q) k = ( Q) ( Q) Γ +Γ () Which mus be solved subjec o he followig boudary codiios V i V (, y, ) f (, y, ) V(, y) =, for (, y) o V V (, y, ) = g (, y, ) (, y ), for (, y) o Γ Γ () Γ Γ (3) Alhough he amou of daa sorage ad compuaioal effor required is greaer for his mehod ha for he ohers i he field, he advaage of reduced daa preparaio is very sigifica. Eq. () was used i [6] o aalyze problems of hermal shock by assumig ha V ad remai cosa hroughou each ime sep. Such a soluio ca be used as a sarig poi for he soluio of he liearized groudwaer flow equaio (4), for φ whe R =. A secod formulaio, i which φ ad vary liearly wih ime, will be used i a equaio aalogous o Eq. (9) ad he resuls compared wih hose obaied whe φ ad are assumed cosa over each ime sep. 3. Soluio of Homogeous Useady Ucofied Aquifer Flow Equaio wihou Well Homogeous regios wihou well ca be modeled by [5]: θ u φ y a u φ dsdτ (,, ) = ( ) Γ +Γ (4) i which θ is he ierior agle bewee he boudary ages a (, y) ad a is he raio of rasmissiviy o sorage coefficie i.e. k S S = sorage coefficie, φ (, y, ) piezomeric head, Q = flow rae. = ime. A algebraic approimaio o Eq. (4), ca be wrie of M odes, which gives M equaios coaiig m ukows ( φ ad a each ode). The remaiig M equaios required for a complee soluio are obaied from he boudary codiios. To solve equaio (4) wo mehods was used. 3.. Mehod () - BEIM wih Cosa Ierpolaio The simples way o iegrae Eq. (4) umerically is o divide he ime ierval (, ) io ime seps ad o assume ha boh h a remai cosa a each ode hroughou each ime sep. The order of iegraio i Eq (4) ca be chaged o give; r cos(, ) φ r 4a φ ( p)cos( r, ) γ r r e e ds E [ ] + ( p)l[ ] ds + r r 4a 4a Γ +Γ Γ +Γ (5) γ k e h k + = + φ 4a k = Γ +Γ Γ +Γ ( p) lrds E l[ G r ds I which G is as defied as

5 Pure ad Applied Mahemaics Joural 6; 5(): 5-9 r r G = E E 4 a ( k ) 4 a ( k ) Ad F is as defied as r r cos( r, ) 4 a ( ) k 4 a ( k ) F = e e r (6) (7) The righ had side of Eq. (5) coais oly kow iformaio from previous ime seps, while he lef had side coais he ukow values of φ ad a ime =. Eq. (5) is applied successively o each boudary ode, ad his ogeher wih he boudary codiios, gives a se of simulaeous equaios which ca be solved by Gauss Elimiaio o give he soluio for φ ad a each boudary ode. The firs iegral o he lef had side of Eq. (5) is calculaed umerically as i [5], ad he secod iegral is umerically approimaed by Eqs. (9) ad (). The fial erm o he lef had side of Eq. (4) was approimaed as i [3], while he iegral o he righ had side ca be calculaed by he umerical schemes usig he iegral formulaios give i [5]. 3.. Mehod - BIEM wih Liear Ierpolaio A secod, way o iegrae Eq. (4), is o divide he ime ierval (, ) io ime seps as before, bu o assume ha boh φ ad vary liearly wih ime hrough ou each ime sep. Followig he iegraio we obai: G + p G p + F + p + p ds p ds γ r e ( ) ( ) φ φ φ ( ) ( ) l 4 a Γ +Γ Γ +Γ = G + ( p) G ( P) + φ ( p) F ds?? Γ +Γ k K k k k k k k k k k k [ G + G + ( p) G ( p) + ( p) G ( p) + φ F + κ F ] ds k = Γ +Γ (8) By applyig his equaio o each ode successively ad usig a quadraure formula such as he rapezium rule, a se of simulaeous equaios is obaied which ca be solved accordig o he boudary codiios for φ ad a each ode for r=, r,. The soluio for φ a ay poi (, y) coaied iside he regio R, a ime = is give by Γ +Γ u Q r φ(, y, ) = a ( u φ ) dsdτ E 4T 4a for, y i R (9) variables; φ φ = L = L y y = l = L Q Q = L (3) Oce φ ad have bee deermied a all boudary odes by he mehods give i. above, φ a ay ieral poi ca be calculaed direcly by umerically iegraig he righ had side of Eq. (9). The iegraio is grealy simplified because here are o sigulariies i he iegrad. The assumpio ha φ ad vary liearly bewee successive ime seps improve accuracy oo much. 4. Comparisos Bewee Mehod- ad Mehod- The resuls obaied i he above secio will be compared wih kow aalyical soluios. Resuls ca be epressed mos simply by iroducig he followig dimesioless I which L is a characerisic legh i he problem. Alhough soluios ca be obaied for a umber of boudary coour shapes, oe of he mos commoly used will be he square because aalyical soluios for flows i his regio is readily calculaed. As he boudary codiios o eiher side of a sharp corer may be differe ad because here is a discoiuiy i geomery here, odes are placed adjace o, raher ha eacly a he corers, as show i he figure, ha he disace bewee a sharp corer ad he adjace odes should be approimaely oe quarer of he usual odal spacig. The coribuio ha hese corer odes make o he umerical iegrals are calculaed by assumig ha values of h ad o he boudary coour bewee such a ode ad he sharp corer are cosa ad equal o he values a he ode.

6 Azhari Ahmad Abdalla: Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer Fig.. odal Arrageme for Square Regio. Suppose ha hree sides of he square regio show i fig. above are impermeable ad ha φ = everywhere wihi he regio a =, as give i [9] he eac soluio is; Ad α ( ) e φ (, ) = cos α α (3) (., ) α = = e (3) i which α = ( + ). umerically calculaed values of φ, boh o he boudary coour ad a seleced pois wihi he flow regio iself, are compared wih he values obaied from he aalyical soluio of Eq. (3). Because a =l. is ifiie a = I (Eq. (3).), here are ihere problems i usig Mehod of secio 3., i which h ad are assumed o vary liearly wih ime over each ime sep. Whe calculaig values of ad φ, a he ed of he firs ime sep, values of h ad a = are required, as described by Eq. (4). Thus, icreasig he umber of seps, he compuaioal cos rises rapidly. I should be oed ha he compuaioal ime required by Mehod is approimaely eighy perce of ha red by Mehod. whe same ime seps are used for boh ses of calculaios. Cosequely, he large isabiliy i a X =, ad he mior isabiliy i φ, ha occur ear =l., are o be epeced iiially. However a ecelle feaure of his mehod is ha despie he iiial isabiliy, he umerical soluio closely approimaes he aalyical soluio afer oly a few ime seps. Also, umerically calculaed values of φ wihi he flow regio are i good agreeme wih Eq. (3), eve a he ed of he firs ime sep. The problem associaed wih he ifiie value of a = does o arise wih Mehod of secio 3. because values of h ad a a = are o required for he soluio a he ed of he firs ime sep, as saed i Eq. (4). However, because Mehod assumes ha φ ad are cosa hroughou each ime sep, ime seps mus be closely spaced o esure accurae resuls. Immediaely followig he isaaeous rise i h ime seps are closely spaced, bu as ime proceeds ad he sep legh icreases, here is a loss i accuracy. Alhough he resuls ca be improved by decreasig he ime sep legh, he compuaioal cos becomes prohibiive. I should be oed ha he compuaioal ime required by Mehod is approimaely eighy perce of ha required by Mehod. The isabiliy ha occurs wih Mehod followig isaaeous chages i hi is o really cosidered a problem for groudwaer flows because chages i φ are ever isaaeous i real siuaios. If he isaaeous rise a =l. is replaced wih; φ ( =., ) = e > (33) I which E is a posiive cosa, very rapid chages ca be modelled by choosig a sufficiely large value of E. More gradual chages i φ, are modelled by subsiuig a smaller value of E i he righ side of Eq. (33) The eac soluio o his problem, calculaed from Eq. (3) usig he Duhamel super posiio iegral, [4] ( ) cos α φ (, ) = e e e α (34) α ( α ) α (., ) e e = = (35) α i which α = ( + ). The righ side of Eq (35) is equal o zero a =. for all values of ecep for = α. Cosequely here is o discoiuiy i 3h ad he problems eperieced wih Mehod a = are elimiaed. A isaaeous rise i φ, a =l. is closely approimaed by subsiuig a values of i he righ side of Eq. (33). The umerical ad aalyical values of φ o he boudary

7 Pure ad Applied Mahemaics Joural 6; 5(): 5- coour, ad he values a pois wihi he flow regio are compared wih hose from he aalyical soluio. The soluios for a =l. are compared also wih he aalyical soluio values. There is ow ecelle agreeme bewee he aalyical soluios ad hose obaied usig Mehod i secio 3., ad here is o evidece of ay umerical isabiliy. There is virually o differece bewee he soluios obaied usig Mehod for a 5 isaaeous rise i h ad for h ( =.) = e This is o be epeced because he wo aalyical soluios are almos ideical ecep for differe values of a =l. a =, which do o appear i he umerical calculaios. The accuracy is limied by he legh of he ime sep as oulied previously. More gradual chages i h a =l. ca be modeled by subsiuig a value of = i he righ side of Eq. (33). because φ, varies more slowly ha i he Ad previous eamples, larger ime seps were chose. Soluios for h aroud he boudary coour ad a seleced pois wihi he flow regio are ad soluios for a =l. were compared wih he aalyical soluios. The soluios obaied usig Mehod are i ecelle agreeme wih he aalyical soluios, while hose obaied usig Mehod are somewha less accurae due o he large ime seps used. The hree cases eamied so far all have seady-sae soluios which are approached as, becomes ifiie. However, if he boudary codiio a =l. is a periodic fucio of ime, flow wihi he square regio is oscillaory. For eample, whe he boudary codiio a X=. is; φ ( =., ) = si, > he eac soluio calculaed from Eq. (3) as i [9] is; ( ) φ (, ) = si cos α 4 α (cos α e ) + si (36) α 4 α + ( =., ) = 4 α \ (cos α e ) + si ) + si (37) 4 α + i which α = ( + ). Because of he relaively large ime seps seleced, umerical calculaios have bee aemped oly wih Mehod, he more accurae of he schemes preseed i secio. Also, sice o discoiuiy i φ or occurs a = (see Eqs. (37 ad 38)), Mehod will be sable. Values of h o he boudary coour ad a seleced pois wihi he flow regio, ad values of a =, are compared wih hose from he aalyical soluio. Oce agai all he resuls are i ecelle agreeme. 5. Coclusio The iegral equaio echiques preseed here have bee foud o be saisfacory for solvig useady, wo dimesioal ucofied aquifer flow problems. The use of ime-depede fudameal soluios has led o accurae iegral equaio formulaios for solvig he useady Dupui equaio i he homogeeous regios. Of he wo formulaios preseed i his paper, ha i which he piezomeric head ad is ormal derivaive are assumed o vary liearly wih ime over each ime sep has proved more accurae. Comparisos bewee Mehod of secio-3 ad he aalyical soluios have demosraed he superior accuracy of he iegral equaio formulaio. Refereces [] ALI A Ameli (4), "Semi-aalyical mehods for simulaig he groudwaer-surface waer ierface", i fulfillme of he hesis requireme for he degree of Docor of Philosophy i Civil Egieerig Waerloo, Oario, Caada. [] Albrech, J. ad Collay, L. (Eds.), (98) "umerical Treame of Iegral Equaios", Birkhauser Verlag, Basel, 98. [3] Bear. Jacob, (978). Dyamic of Fluid i Porous Media, Dover Books o Egieerig, ISB-3: [4] Brebbia, C. A., Telles, J. C. F. ad Wrobel, L. C., (984), Boudary Elemes Techiques. Spriger-Verlag, Berli, 984. [5] Dupui, J. (863). Esudes Théoriques e Praiques sur le mouveme des Eau das les caau découvers e à ravers les errais perméables (Secod ed.). Paris: Duod. [6] Dubois, M. ad Buysse, M., (98), "Trasie Hea Trasfer Aalysis by he Boudary Iegral Equaio Mehod", i ew Developmes i Boudary Eleme Mehods, Brebbia, Co (Ed.), C. M. L. Publicaios, Souhampo, 98. [7] Goldberg, M. A. (Ed.), (979), "Soluio Mehods for Iegral Equaios: Theory ad Applicaios, Pleum Press, ew York, 979.

8 Azhari Ahmad Abdalla: Boudary Iegral Equaio Mehod for Useady Two Dimesioal Flow i Ucofied Aquifer [8] James A, Lige, Philip L-F. Liu, (983) "The Boudary Iegral Equaio Mehod for Porous Media Flow", Lodo, George Alle ad UWI, 983. [9] Lafe, O. E., Ligge, J. A. ad Liu, P. L. F. (98) "BIEM Soluios o Combiaios of Leaky, Layered, Cofied, Ucofied, oisoropic Aquifers ll, Waer Resources Research, Vol. lj, o. 5, pp , Ocober, 98. [] Leo, G. P., Liu, P. L. F. ad Ligge, J. A. (979) "Boudary Iegral Equaio Soluio o Aisymmeric Poeial Flows;. Basic Formulaio;. Recharge ad Well Problems i Porous Media ll Waer Resources Research, Vol. 5, o. 5, pp. -5, Ocober, 979. [] Leo, G. P., Liu, L. F. ad Ligge, J. A. (98) "Boudary Iegral Soluios o Three-Dimesioal Ucofied Darcys Flow", Waer Resources Research, Vol. 6, o. 4, pp , Augus, 98. [] Massoe, C. E. (965) "umerical Use of Iegral Procedures i Sress Aalysis", Ziekiewicz, O. C. ad Holiser, G. S. (Eds.), Joh Wiley ad Sos, Ic., ew York. [3] M.. Vu a,, S. T. guye a, c, M. H. Vu a, b, (5)," Modelig of fluid flow hrough fracured porous media by a sigle boudary iegral equaio", Egieerig Aalysis wih Boudary Elemes 59 (5) [4] orrie, D. H. ad de Vries, G. (978) "A Iroducio o Fiie Eleme Aalysis ll, Academic Press, ew York, 978. [5] Phooledra K. Mishra ad Krisopher L. Kuhlma (3), Ucofied Aquifer Flow Theory - from Dupui o prese, physics. flu. dy [6] Wrobel, L. C. ad Brebbia, C. A. (98), "A Formulaio of he Boudary Eleme Mehod for Aisymmeric Trasie Hea Coducio", Ieraioal Joural of Hea ad Mass Trasfer, Vol. 4, o. 5, pp , 98.