mslumped-parameter (zero-dimensions!) groundwater model of Bangladesh

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1 mslumed-pmete (zeo-dimensions!) goundwte model of Bngldesh You gol in this oblem set is to develo bucket model fo the hydology of ou study site in Bngldesh. Using this model, you will investigte how the develoment of dy-seson iigted gicultue in Bngldesh my hve ffected its goundwte hydology. Figue 1 is concetul eesenttion of ou study site. We e inteested in modeling the hydology of the sndy quife, which undelies ou entie study site nd is confined by eltively imemeble cly lye. The fction of the quife e coveed by ech of the onds (f ), ice fields (f g ), nd ives (f ) is: f =0.1 f =0.02 f g =0.65 H, H, H efe to the heds of the quife, ond, nd ive esectively. In this model, you cn ssume tht tees t ou site hve shllow oots tht do not extend ll the wy into the quife. So, by efeence to Figue 1, ET tee, which is the flux out of the quife due tee tnsition, is equl to zeo. The couled equtions fo the tes of chnge of H nd H e: dh = k * ( H H ) E Eqution 1 dh S * = f * k * ( H H ) + f * k * ( H H ) qi Eqution 2 In Eqution 1, E is the te of evotion fom the ond e unit ond e. In Eqution 2, q i is the net flux out of the quife esulting fom the iigtion of ice fields, e unit quife e. The following questions will hel you develo models fo E nd q i. The sediments between the ond nd quife hve conductnce of k =0.01 dy -1. The sediments ccumulted t the ive bottom hve highe conductnce of k =0.1 dy -1. The stotivity (S) of the quife is S=0.01. The units of [Length/Time]. dh nd dh e Figue 1 The heds in the ives t ou site e contolled by the wte levels in the Gnges ive, nd e not ffected by the quife hed. So, the evolution of H duing the dy seson is indeendent of H. Field dt fo H s function of time e lotted in ed in Figue 2. 1

2 Figue 2: Dt on Aquife (geen nd bown), Rive (ed), nd Pond (blue) heds t ou study site. Assume tht the only significnt flows into nd out of the quife occu duing the dy seson, since duing the iny seson (lso efeed to s the monsoon seson o the flooding seson), the heds of the quife, ives nd onds e equl so tht no flows occu between them. You gol is to model the evolution of H nd H ove the dution of the dy seson. Bsed on Figue 2, the dy seson extends oximtely between Novembe 1 st nd My 31 st. At its beginning, following the flooding seson, the heds in the ives, onds, nd quife e ll t 4m. The following execises nd questions will guide you though develoing you model nd undestnding you esults. ) The ive hed, H, ffects the quife hed H (see Eqution 2). Howeve, H itself isn t ffected by H since the Gnges Rive contols it. We will wok with H s known inut into Eqution 2. This cn be done by secifying the vlues of H t diffeent times in the dy seson bsed on collected field dt (Figue 2). Altentively, we cn use simlified eesenttion of the H dt shown in Figue 2 by modeling H (t) s Sine function of time t, so tht the esulting function esembles the collected dt. t H = H m * Sin * π + π + 4 Eqution 3 (in metes) tsn H m is the mlitude of H (t), H m = 2.5 m tsn is the dution of the dy seson. 1. Check tht this eqution fo H (t) gives oite vlues of H fo t=0, t=tsn, nd t=0.5*(tsn). t 2. Show tht nothe equivlent eqution fo H (t) is H = H m * Sin * π + 4 (in metes). tsn b) Evotion t ou study site cn be modeled with efeence to the evotion t the neest meteoologicl sttion in Munshigonj (with the cystl bll). Figue 3 shows the estimted evotion t Munshigonj, which will be efeed to s the efeence evotion (Eo). Eo is usully 2

3 exessed s flow e unit e of the evoting sufce with units of [Length/Time]. In Figue 3, the units of Eo e mm/month. Figue 3 Eo dt is collected egully nd elibly, so it is useful to estimte evotion t ou site in eltion to Eo. The ctul evotion te of the onds t ou site (e unit ond e) cn be modeled s E = C *E o C is coefficient estimted emiiclly, by collecting field dt on the evotion tes of the onds nd elting them to E o. C ws found to be 1.4. Similly, the ctul tnsition te of ice lnts in the gicultul fields t ou site (e unit field e) cn be modeled s E g = C g *E o. C g ws found to be 1.1. Using dt fom Figue 3, estimte the men vlue of E o duing the dy seson. You will need to convet you esult into units of metes/dy, since it will be n inut into you equtions fo dh / nd dh /. c) Assume tht the fmes um out wte fom the quife to iigte thei ice fields t te. The ice co tnsies some of the wte lied to the fields, nd the est ecoltes bck though the cly lye into the quife. 1- If you ignoe the lg between the times when wte is lied to the fields nd when it eches the quife, wht is the net flux out of the quife tht esults fom iigtion (lbeled q i in the eqution fo dh /)? Use the eqution fo E g in t (b) to exess q i in tems of known quntities. Mke sue you exess it e unit quife e, since it will be one of the fluxes in Eqution 2 tht detemine dh /. 2- Figue 2 shows tht the fmes do not iigte thei cos duing the entie dy seson. Wite simle model fo q i (t), using you nswe to t (c) nd infomtion on the stt nd end of the iigting eiod fom Figue 2. 3

4 d) Code the system of equtions fo dh / nd dh / into Mtlb function M-file, which defines you diffeentil equtions nd is n inut file fo the ode45 function. The quife hed nd ond hed should be the only unknowns in you equtions, in ddition to you indeendent vible t. Mke sue you define ll othe vibles nd metes you include in you equtions. Solve fo you quife nd ond heds using the ode45 function. ode45 oututs the solutions (H (t) nd H (t)), s well s the vecto of time oints (t) fo which H nd H e solved. Click on the button Woksce on the left side of you Mtlb window to see these oututs (Figue 4). Figue 4 In the Woksce window, double click on the vecto of time oints nd on the mtix of heds to see thei vlues. Wht e the two columns of the hed mtix? 1- Plot the ode45 solutions fo H (t) nd H (t). Figue out how to lot H (t) setely: to etieve the vlues fo H (t) setely fom the hed mtix outut by ode45, you need to secify which ows nd which column of the hed mtix you e inteested in. 2- Ovely on you lot of H (t) nd H (t) the lot fo ive hed H (t). To do this, you will need to e-define the eqution fo H (t) t the commnd omt, even though you ve defined it in you initil function M-file. This is becuse ny vible o mete you define in you function M-file is intenl to tht function. The only oututs into you Woksce e the ode45 solutions, i.e. you quife nd ond heds nd the vecto of time oints t. Confim tht once you define H t the commnd omt, it shows u in you Woksce. How mny ows does the H vecto hve? Why? To ovely the H (t) cuve on to of you lot fo H (t) nd H (t), tye hold on t the commnd omt befoe enteing the lot (t,h) commnd. e) Inteet you esulting lot. How does the onset of iigtion ffect H (t), H (t) nd H (t)? Come you lot to the dt esented in Figue 2. 4

5 f) Think bout how you might imove you model so tht it bette eflects the conditions t ou study site. As you elbote you model, un it nd lot the solutions fo H (t) nd H (t), then ovely the cuve fo H (t). See if you modifictions to the model oduce esults tht gee bette with the dt in Figue 2. Below e coule of suggestions fo imoving you model. The fist suggested modifiction is equied, the second is otionl, nd you cn come u with othe ossible modifictions. If you hve n ide fo imoving you model, Hnn cn hel you with the coding. 1- In t b, you secified E o to be constnt vlue equl to the vege vlue of E o ove the dy seson. Bsed on the dt in Figue 3, fomulte bette model fo E o (t) nd code it into you function M-file. 2- (OPTIONAL, but oduces nice esults) The q i flux you descibed in (c-1) is the mximum flux when ll the ice fields e being iigted. Howeve, this mximum flux isn t mintined duing the whole iigting eiod, since mny fields e iigted fo only otion of this eiod. To ccount fo this, include fcto i in you eqution fo dh /, which multilies q i nd secifies the fction of gicultul e tht is ctully iigted t ny time t. Define i (t) s Sine function of t, vying fom 0.5 t the stt of the iigting eiod, to 1 in the middle of this eiod, nd bck to 0.5 t its end. This model fo i (t) imlies tht hlf the ice fields e ctully iigted initilly, nd this fction inceses with time until ll the fields e being iigted, nd then oceeds to decline bck to 0.5 t the end of the iigting eiod. i (t) must be secified s zeo outside the iigting eiod. Refe to (c-2) fo infomtion on the extent of the iigting eiod, nd to t () to emind youself how to model vible s Sine function of time. g) So f, you ve been looking t the solutions fo the quife nd ond heds. These e evolving in time due to the fluxes between the diffeent wte bodies. In the next few questions you will be investigting these fluxes. 1- Define the eqution fo FluxP(t), the flux between the quife nd the ond e unit e of quife. In you eqution, when is the esult fo FluxP ositive, nd would tht imly echge to o dischge fom the quife? If needed, djust the signs in you eqution so tht ositive FluxP indictes echge to the quife, i.e. flow fom the ond to the quife. 2- Similly, define the eqution fo FluxR(t), the flux between the quife nd the ive e unit e of quife. In you eqution, when is FluxR ositive, nd would tht imly echge to o dischge fom the quife? If needed, djust the signs in you eqution so tht ositive FluxR indictes echge to the quife. 3- Cete new Mtlb M-file, in which you code the equtions fo FluxP(t) nd FluxR(t). Mke sue you nme the vibles defined by these equtions: FluxP nd FluxR (you ll soon see why this is necessy). You will be unning this scit fte you un ode45, so it will use the solutions fo H (t) nd H (t) tht ode45 oduces. As in t (d-1), to etieve the solutions fo H (t) setely, you need to secify which ows nd which column of you outut hed mtix you e inteested in. Similly fo H (t). You scit will lso use the vecto of t vlues (time oints) oduced by ode45. Remembe tht you need to define ny othe metes o vibles used in you equtions fo FluxP(t) nd FluxR(t) (such s H, k, k, f nd f ), even though you ve defined them in you initil function M-file. This is becuse these vibles nd metes e intenl to you function, nd so en t sved by Mtlb into you Woksce when you un ode45 (this is exlined in t d). 5

6 4- The net flux out of the quife due to iigtion, e unit quife e, is FluxI(t)= i (t)*q i (t) o simly q i (t), deending on whethe o not you comleted the otionl t (f-2). Add the eqution fo FluxI(t) to you new M-file. Nme the vible defined by this eqution: FluxI. Remembe you will need to define ny vible o mete you use in this eqution tht is not n outut of ode45. Use the models fo q i (t) nd E o (t) tht you ceted in ts (c-2) nd (f-1), esectively. Since q i (t) nd E o (t) hve diffeent vlues deending on the time eiod (fo exmle, q i (t) is zeo outside of the iigting eiod), you need fo loo tht secifies these vibles fo evey time oint in the t vecto outut by ode45. Instuctions to hel you wite the code fo FluxI(t) e vilble on stell unde Mteils, in the document FluxI.doc. Does you eqution oduce ositive o negtive vlues fo FluxI(t)? Does FluxI(t) constitute dischge fom o echge to the quife? i) Run you new scit by tying its filenme t the commnd omt, to obtin solutions fo FluxP(t), FluxR(t), nd FluxI(t). Remembe tht FluxP(t) nd FluxR(t) e functions of H (t) nd H (t), so you need to hve un ode45 fist to obtin the solutions fo the quife nd ond heds. Plot these fluxes nd inteet you esults. Note esecilly chnges in sign in FluxP(t) nd FluxR(t). Wht do they men? Ae they consistent with you lot fo H (t), H (t) nd H (t)? Also inteet chnges in the sloes of these flux cuves. j) Fom the Mteils section of the stell website, downlod the Mtlb scit TotlFluxes.m nd sve it into you cuent diectoy, whee you e sving you othe scits. Run it to obtin solutions fo the vibles: totldischgep, totlechgep, totldischger, totlechger, nd totldischgei. The TotlFluxes.m scit uses you esults fo FluxP(t), FluxR(t), nd FluxI(t). Tht is why you hd to give secific nmes to the vectos contining the solutions fo these fluxes. The TotlFluxes.m scit integtes these fluxes ove the dution of the dy seson to comute the vibles: totldischgep: The totl dischge fom the quife to the ond e unit quife e ove the dy seson. totlechgep: The totl echge fom the ond to the quife e unit quife e ove the dy seson. totldischger: The totl dischge fom the quife to the ive e unit quife e ove the dy seson. totlechger: The totl echge fom the ive to the quife e unit quife e ove the dy seson. totldischgei: The totl flow out of the quife due to iigtion e unit quife e ove the dy seson. Once you un the scit, Mtlb sves these vibles nd thei vlues into you Woksce. Confim tht the vibles eesenting flows out of the quife hve negtive vlues, while those eesenting flows into the quife hve ositive vlues. 1- Wht is the totl flow out of the quife e unit quife e ove the dy seson? Wht is the totl flow into the quife e unit quife e ove the dy seson? 2- Wht is the net chnge in quife hed ove the dy seson (efe to you lot fo H, H, nd H o lot H (t) gin)? Given tht S=0.01, wht chnge in volume of wte e unit quife e does this coesond to? 3- Bsed on you nswes to (j-1) nd (j-2): does you model fo the quife hed conseve mss? 6

7 k) The quife t ou study site hs deth D=100m, nd vege oosity Ө = 0.25 (25%). 1- Wht is the volume of wte contined in the quife? 2- Assume tht ove ye, the only flows into nd out of the quife t ou site occu duing the dy seson. Using you nswes to (j-1) nd (k-1), clculte the esidence time of wte in the quife. If you find tht the totl outflows fom nd inflows into the quife e not equivlent, tke thei vege. l) Use you model to figue out how the quife s wte budget ws chnged s esult of the develoment of iigted gicultue in Bngldesh. 1- Adjust you model so tht no wte is extcted fom the quife fo iigtion. You cn do this by chnging the vlue of f g to 0 in you initil function M-file tht defines you system of equtions fo dh / nd dh /. 2- Run ode45, nd lot H (t) nd H (t). Ovely the lot of H (t). Come you lot fo H (t), H (t) nd H (t) in the bsence iigted gicultue to the lot of these heds tht you oduced when you ccounted fo iigtion in you model. 3- Run the M-file scit you wote tht solves fo FluxP(t), FluxR(t), nd FluxI(t). Plot these fluxes. Agin, come these esults, in the bsence of iigtion, to the esults of the model in which you ccounted fo iigtion. 4- Run the scit you downloded, TotlFluxes.m. Comute the esidence time of wte in the quife in the bsence of uming fo iigtion. Come it to the esidence time tht you comuted in (k-2). 7