Journal of Computational and Applied Mathematics. A perturbation solution for bacterial growth and bioremediation in a porous medium with bio-clogging

Transcription

1 Journ of Computtion nd Appied Mthemtics 234 (2) Contents ists vibe t ScienceDirect Journ of Computtion nd Appied Mthemtics journ homepge: A perturbtion soution for bcteri growth nd bioremedition in porous medium with bio-cogging Miche Chpwny,, Stephen O Brien b, J.F. Wiims Deprtment of Mthemtics, Simon Frser University, Burnby, British Coumbi, Cnd b MACSI, Deprtment of Mthemtics & Sttistics, University of Limerick, Limerick, Irend rtice info bstrct Artice history: Received 25 Februry 29 Received in revised form 2 October 29 Keywords: Porous fow Bioremedition Asymptotic nysis Bio-cogging Free boundry We investigte fow probem of reevnce in bioremedition nd deveop mthemtic mode for trnsport of contmintion by groundwter nd the spreding, confinement, nd remedition of chemic wste. The mode is bsed on the fuid mss nd momentum bnce equtions nd simutneous trnsport nd consumption of the poutnt (hydrocrbon) nd nutrient (oxygen). Prticur emphsis is pced on the study of processes invoving the fu couping of rection, trnsport nd mechnic effects. Dimension nysis nd symptotic reduction re used to simpify the governing equtions, which re then soved numericy. 2 Esevier B.V. A rights reserved.. Introduction Over the pst few decdes, numerous studies on groundwter fow hve been deveoped with the im of predicting the rriv times nd spti distribution of poutnts in the subsurfce. The current study ims to coupe groundwter fow to the mode describing the degrdtion of poutnts in the subsurfce. We consider the trnsport of poutnts in sturted phretic quifer beow ground surfce in the form of dissoved compounds or prticutes in wter. These poutnts percote downwrd nd horizonty into the quifer due to grvity forces [ 3] nd dispersion [4,5]. At this stge, there is itte tht cn be done to contro their spred nd bioremedition is the chepest remedi procedure to cen the quifer utiizing indigenous bcteri [6]. In prticur, some bioremedition procedures invove ddition of nutrients to support the existing microbi ctivity. Exmpes re oxygen [7,4] nd nitrte [8], which re typic fctors tht imit the ctivity of bcteri. In the quifers, bcteri poputions grow in queous environments whie dhering to ech other nd to surfces or interfces s biofim [9,] or coonies []. Sometimes the biofim cn cuse pugging in the regions where nutrient nd poutnt concentrtions re high, [8,2]. As resut, bortory tests re sometimes crried out to determine the optim mounts of nutrients to dd. Too itte nutrient wi resut in ineffective growth rtes whie too much cn ed to the cogging of the quifer. For exmpe, [3] designed n experiment nutrient ddition strtegy tht cn dmpen excessive growth of bcteri t the injection point whie t the sme time minimising the concentrtion of poutnt t the end of given period of time. Most existing mthemtic modes consider cogging under bortory setups [8,4], or no bio-cogging under uniform fow veocity [5,7]. Here we wish to incorporte the processes of dvection, dispersion, sorption nd Monod kinetics [4,5,7,6], in the context of groundwter fow with bio-cogging under fied conditions. Pore cogging is introduced by expressing biomss concentrtion s noniner function of medium porosity [8,7]. Corresponding ddress: Deprtment of Mthemtics nd Sttistics, Consortium for Science nd Industry, University of Limerick, Limerick, Irend. E-mi ddress: mchpwn@mth.sfu.c (M. Chpwny) /$ see front mtter 2 Esevier B.V. A rights reserved. doi:.6/j.cm.2..24

2 27 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) ground surfce y pouted wter B phretic surfce x H phretic quifer seepge fce A impermebe bed L C D Fig.. Porous medium boundries nd the moving boundry of the mode probem. To obtin the poutnt dvective nd dispersive fuxes, we must know the fuid (i.e., the crrier) veocity. Thus we coupe the biodegrdtion mode to the necessry fow mode. The fow domin, often referred to s the dm probem, hs received considerbe ttention in the scientific engineering iterture (cf. [8, Ch. 4], [9], [2, Ch. 3], [2]). We remrk tht of these studies were imed t finding ccurte stedy stte soutions for the free boundry. In the present study, the free boundry is owed to move owing to the chnge in the physic properties of the porous medium. We begin in the next section by introducing the mthemtic mode under investigtion. The mode is rendered dimensioness in Section 3 nd in Section 4 we present reduced mode. An pproximte soution to the reduced mode is obtined in Section 4. using mtched symptotic expnsions nd ter (in Section 6) we present numeric soutions to both the reduced mode nd the correction terms. A compete two dimension mode which incudes the fu couping of rections nd fow is presented in Section The mthemtic mode To formute fesibe mode, we ssume tht. the site hs n dequte suppy of nutrients, is t suitbe temperture nd contins no chemics tht inhibit the biodegrdtion process, 2. there is no grzing of bcteri nd there re no toxic substnces present, 3. there is one species of bcterium, singe contminnt nd singe nutrient, 4. the biofim is treted s continuum nd vribes re described by verging quntities such s concentrtions nd voume frctions, 5. the quifer is ssumed to be very ong compred to the reservoir height H, H L, we define δ = H/L, where L is the horizont ength sce of the quifer. See Fig., 6. the quifer is of sufficient extent (in the z direction in Fig. ) tht two dimension mode is pproprite. 2.. The fow We consider two dimension trnsport of poutnts nd nutrients in homogeneous porous medium from pouted reservoir, Fig.. The domin is n quifer with moving boundry (phretic surfce) serving s the upper boundry nd impermebe surfce s the ower boundry. We ssume the zone beow the moving boundry is sturted porous medium with no moisture bove it. In the sturted domin, we mode fow using Drcy s w which retes the iquid veocity u (u,v ) to the pressure p in the foowing wy, u = k (θ) p µ θ x, v = k (θ) p µ θ y + ρ g, () where u nd v re the veocity components in the x nd y directions, θ is the voumetric iquid frction, µ is the iquid dynmic viscosity, ρ is the iquid density, g is the cceertion due to grvity nd k (θ) is the biomss ffected permebiity. The sterisk represents vribes with dimensions. Conservtion of the iquid requires tht θ + (θu ) =, ssuming there re no sources nd sinks within the sturted medium nd tht the chnge in density of the iquid (cused by presence of dissoved soutes) is negigibe. (2)

3 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) The biodegrdtion We consider equtions modeing the interction of three quntities: the mss of substrte per unit voume of pore wter s (g/m 3 ), mss of nutrient per unit voume of the pore wter (g/m 3 ), nd mss of microbi ces per voume of the pore m (g/m 3 ), see for exmpe [7,8]. Foowing numerous bioremedition modes in the iterture (cf. [22,6,23]), the governing equtions re R s s (θs ) = div[θu s θd s ] k m φm K s + s K +, (3) s (θ ) = div[θu θd ] Xk m φm K s + s K +, (4) m = b(m m ) + Yk mm s K s + s K +, (5) where R s is the substrte retrdtion fctor, m (g/m3 ) is the indigenous microbi concentrtion, X (g/g) is the stoichiometric constnt for nutrient consumption, Y (g/g) is the yied, k m (/dy) is the mximum rte of substrte utiiztion, K (g/m 3 ) is the hf sturtion constnt for nutrient, K s (g/m 3 ) is the hf sturtion constnt for substrte s, b (/dy) is the microbi decy coefficient nd D (m 2 /dy) is digon dispersion-coefficient-mtrix with entries D nd D t corresponding to the ongitudin nd trnsverse hydrodynmic dispersion respectivey. Foowing [6], the retrdtion fctor R s = + ρ b K d /θ, where K d (m 3 /g) is the distribution coefficient nd ρ b (g/m 3 ) is the quifer buk density. The expression corresponds to iner equiibrium isotherm or dsorption eqution. Eqs. (2) (5) re couped by incuding the chnges in porosity due to the growing biomss. Note tht the rection terms in the eqution invove the pointwise concentrtions rther thn the verge concentrtion in Representtive Eementry Voume (REV), see [8]. In this study, we dopt the mcroscopic pproch [8], which mkes no ssumptions bout the microscopic biomss distribution. A microscopic mode for pore cogging wi be combintion of the biofim, microcoonies nd pugs ner throts. Consider representtive eementry voume U (m 3 /m 3 ). Then the voumetric frction of the iquid phse in the medium is, θ = U U = U U s U m = φ U m U U = φ φm = φ ρ m m ρ m, (6) where U, U s nd U m re the voumes occupied by iquid phse, the soid mtrix nd the biomss respectivey, ρ m is the constnt density of the biomss. We use φ = θ s to denote the constnt medium porosity negecting the presence of the biomss. Thus (6) retes the chnge in biomss concentrtion to medium porosity. The expression ssumes the soid phse cn be seprted into voume occupied by biomss nd voume occupied by the soi mtrix with the biomss responsibe for degrdtion ttched to the soi mtrix. We cose this system using n expression reting the iquid frction to the porous medi permebiity. Foowing [7], we write θ 9/6 k (θ) = k (φ), φ where k (φ) is the medium permebiity in the bsence of biomss. We define k (φ) = k = constnt, which is consistent with nondeformbe sturted porous medi in the bsence of biomss Boundry nd initi conditions We need boundry conditions to sove Eq. (2) in the sturted domin. On the soid impermebe surfce u ˆn =, i.e., the norm veocity is zero. At the Erth s surfce or the sturtion eve in the ground the pressure cn be tken s zero i.e., p = fter subtrcting off tmospheric pressure. A moving boundry occurs where we hve n interfce between iquid nd gs. We ignore fow in the unsturted region so tht the moving boundry position is given by the kinemtic condition nd wter fows between the impermebe surfce nd the moving surfce t y = h (x, t ). We therefore need to sove the fow equtions subject to the foowing boundry conditions v = on y =, Isotherms re dsorption retions between the iquid phse concentrtion, sy C nd sorbed phse concentrtion, sy C. The semi-empiric retion to represent the equiibrium sorption in [24], is given by C = Kd C n where K d is the distribution coefficient nd n is the Freundich exponent. If n =, the Freundich isotherm reduces to the iner isotherm.

4 272 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) p =, v = h h + u x on y = h (x, t ), p = ρ g(h y ) on x =, p = on x = L. The st condition ssumes no ponding (i.e., no body of wter exists) t the seepge fce so tht the boundry is t tmospheric pressure. We consider bioremedition mode where the porous medium is ssumed to hve n verge nutrient infow concentrtion t x = nd constnt bckground concentrtion of the biomss m. The substrte eks through the sturted porous medi from the eft reservoir to the right t n verge concentrtion s, (see Fig. ). Thus ong the side AB, there is continuity of the poutnt nd nutrient fux. The boundry is ssumed to be sttionry so tht [u (c c ) + D c ] ˆn =, where c denotes the concentrtion of either the nutrient or the substrte. Aong AD, the surfce is impervious to fow of both the poutnt nd the nutrient so tht the norm veocity is zero. The boundry condition for substrte nd nutrient is ( c ) ˆn =. Aong CD, there is seepge fce. Pouted wter emerges from the porous medium into the environment nd no porous medium exists on the extern side. At this outfow boundry, we hve continuity of the diffusive fux cross the boundry so tht ( c ) ˆn = for both the substrte nd the nutrient. Aong the phretic surfce BC, i.e., y = h (x, t ), there is n interfce between two fuids. Beow the interfce there is sturted porous medium nd the region beow is ssumed to be unsturted (occupied by ir). The boundry is sttionry nd there is no diffusive fux norm to the surfce. The boundry condition ong this surfce is (D c ) ˆn =. The initi conditions ssume poutnt nd nutrient free (fter subtrcting off the bckground concentrtion) porous medi with bckground biomss concentrtion m. 3. Non-dimensionistion The system of equtions is rendered dimensioness by introducing the foowing sced vribes x = x L, y = y H, p = p ρ gh, t = t k ρ gh, k = k, µ L 2 k s = s, s u = u µ L k ρ gh, =, v = v µ k ρ g, m = m. (7) m In ddition, we so sce θ with φ. The dimensioness eqution governing the iquid fow is δ θ 2 = δ2 k(θ) p + p k(θ) x x y y +, (8) with the foowing boundry conditions p = on y =, y p =, δ 2 h = p = ( y) on x =, p = on x =, p k(θ) y + + δ 2 k(θ) p h x x on y = h, where k(θ) = ( mχ) 9/6 nd χ = m /ρ m. The rtio χ is mesure of the effects of biomss growth on medium porosity. A sm vue of χ impies tht the initi biomss mss per unit voume of the porous medium is very sm so tht it hs no effect on medium porosity nd permebiity. The dimensioness biodegrdtion equtions re R s (θs) = uθs + div(θd s) λ m s κ s + s κ +, (9) (θ) = uθ + div(θd ) λ λ 2 m s κ s + s κ +, () m = λ λ 4 (m ) + λ λ 3 m s κ s + s κ +, ()

5 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) where u = (u, v/δ 2 ) is the veocity, R s = + A/θ is the substrte retrdtion fctor (A is constnt), D = dig(ε,ε t /δ 2 ) is digon mtrix, κ s = K s /s nd κ = K /. The dimensioness prmeters re λ = m µ L 2 s k ρ gh k m, λ 2 = X, λ 3 = s Y m, λ 4 = s b m k m, nd we wi write ε = /Pe nd ε t = /Pe t. The corresponding boundry conditions re s foows: ong x = we hve s ε = u(s ), x ε = u( ), x ong the moving boundry y = h(x, t) we hve h s ε x x + ε t s δ 2 y =, ong the seepge fce x =, we hve ε h x x + ε t δ 2 y =, s x =, x =, nd ong the impermebe surfce y =, we hve Pe = k ρ gh µ D, Pe t = k ρ gh µ D t (2) s x =, x =. In Section 5, we consider some soutions of these two dimension equtions. In the next section we present both numeric nd pproximte nytic soutions to the reduced one dimension equtions in the imit of χ, δ 2 nd ε. 4. The reduced mode To mke some nytic progress, nd s one issue of reevnce is n estimte of the time tken for the poutnts to trve through the quifer, we wi first seek pproximte one dimension soutions in the spirit of show wter wves [25] (with c c(x, t)) bsed on the smness of δ. These soutions woud be pproprite for the cse of sm spect rtio, where the boundry condition t x = is pproximtey constnt or where the vertic Pecet number is rge enough to mke the concentrtion pproximtey one dimension [26]. We thus ssume tht c c(x, t), (x, t) nd m m(x, t) which stisfies the boundry conditions. 4.. Approximte one dimension equtions In the first instnce we wi negect the effect of pore cogging nd tke χ. Thus we wi ssume tht the medium permebiity k =, the retrdtion fctor R s ˆRs = + A = constnt nd the porosity θ =. The fow is governed by the eqution δ 2 2 ˆp x ˆp y 2 =, with the foowing boundry conditions ˆp = on y =, y ˆp =, δ 2 ĥ ˆp = y + ˆp = ( y) on x =, ˆp = on x =. + δ 2 ˆp ĥ x x on y = ĥ(x, t), The corresponding reduced bioremedition equtions (fter ppying the mss conservtion eqution) re (3) (4) ˆR s = û x + ε 2 ŝ x λ 2 ˆm ŝ â κ s + ŝ κ + â, (5) â = û â x + ε 2 â x λ λ 2 2 ˆm ŝ â κ s + ŝ κ + â, (6) ˆm = λ λ 4 ( ˆm ) + λ λ 3 ˆm ŝ â κ s + ŝ κ + â. (7)

6 274 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) The boundry conditions re s foows: t x = we hve ε (, t) = û(, t)[ŝ(, t) ], x ε nd t x = we hve â (, t) = nd (, t) =. x x The initi conditions re â (, t) = û(, t)[â(, t) ], (8) x (9) ŝ(x, ) =, â(x, ) =, ˆm(x, ) =. (2) It wi be shown ter how, t eding order, the fow mode is reduced to finding the moving boundry position ony. The reduced perturbtion mode cn be soved in two stges: soving for the stedy stte fow veocity nd then using the resut in the biodegrdtion equtions. In fct, the eding order pproximtion reduces the mode from moving boundry probem to free boundry probem Soution to the reduced fow probem Considering the equtions bove, we choose distinguished imit δ 2 = ε nd seek soutions of the form ˆp = ψ + ε ψ + O( ε 2 ), nd ĥ = f + ε f + O( ε 2 ). (2) Substituting Eqs. (2) into (3), t eding order we hve 2 ψ y 2 =, which, together with the boundry conditions, cn be soved to give ψ = f y. (22) At the next order we obtin: 2 ψ = 2 f y 2 x, for ψ = on y =, 2 y which reduces to ψ y = y 2 f x 2. Note tht Eqs. (22) (23) cn be combined to give the Dupuit Forchheimer eqution, i.e., f = f f, x x (23) (24) with eft side boundry condition f (, t) =. It is not obvious wht boundry condition to use on the seepge fce since from Eq. (22) we require tht f (, t) = y. This wi be overcome by obtining numeric soution to the fu probem. However, from expressions (22) nd (23) it cn be deduced tht the fow veocity is given by û f x, ˆv ε y 2 f x 2, which re the expressions tht rete fow veocity to the free surfce. From numeric point of view we ony need to sove the fow probem once in order to obtin soution to the reduced mode. In prticur, we hve to find the position of the free boundry y = f (x). The fow probem cn be soved by the Dupuit ssumptions bsed on the pproximtion tht the quifer is ong nd thin s outined in [2], in which cse the fow is essentiy in the horizont direction. In the cse of phretic fow, the phretic surfce wys termintes t point bove the wter tbe of the body of open wter present outside the fow domin. This region (CD in Fig. ), is known s the seepge fce. The pproximtion in [2] does not ppy in regions where vertic fow cnnot be negected such s t the seepge fce. Using this pproximtion, the free surfce cn be found to be f = ( x) /2. In fct the veocity profie wi be found to be infinite t x = using Eq. (25). However, it cn be shown tht f = νx, (25)

7 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) b Fig. 2. () Pressure profie computed for ε =.5. (b) Grid used to resove the seepge fce. Note the mesh refinement towrds the seepge fce Computing the exct position of the free boundry The fow equtions re discretized in spce t node (i, j) using centered finite difference pproximtion. The discretized finite difference pproximtions re substituted into the governing equtions to give set of gebric equtions tht re soved using the SOR (Successive Over Rextion) method. In prticur, we discretize the medium so tht the nodes wys terminte on the free boundry (see [27]). A vribe grid is used in directions Numeric strtegy. Choose n initi guess for the surfce y = ĥ i (x), nd sove (8) in the region Ω encosed by x nd the free surfce y = ĥ i (x). The boundry conditions re given in (4). 2. Find the pressure ˆp Γ = ˆp(x, y = ĥ i (x)), i.e., pressure on the grid points ong the free boundry. 3. If the pressure ˆp Γ =, stop: then the phretic surfce is given by y = ĥ i (x), otherwise go to step (4). 4. If ˆp Γ (x, ĥ i (x)) > Λ, where Λ is sm positive number, move the points on the free boundry ccording to ĥ i+ = ĥ i + C ˆp Γ (x, ĥ i (x)), where C is positive constnt; then go to (2) nd repet with new nd better pproximtion y = ĥ i+ (x). The pressure is positive inside the sturted region nd negtive outside it. 2 Thus steps (3) nd (4) in the numeric gorithm depend on whether the point is inside or outside the sturted region. At the free boundry we hve two boundry conditions, see Eqs. (4). Thus our numeric gorithm hs to stisfy the two conditions simutneousy. The itertive process invoves soving for ˆp in region with fixed boundries t every stge. To resove the seepge fce ccurtey nd efficienty we hve used non-uniform grid. Given tht the height of the fce is not known in dvnce, we hve constructed grid bsed on the interfce when there is no seepge fce, i.e., h(x) = x. This is the most singur sitution nd so provides suitbe grid in the gener cse. We simpy tke uniform prtition in the y direction nd mp tht into the x direction, tht is, we uniformy discretize y nd then sove for x vi y = x. This procedure gives x i = ( (i/n ) 2 ), where there re N grid points on the interv [, ], nd defines geometric grid with exceent resoution ner the seepge fce, see Fig. 2. The resuting phretic surfce is given in Fig. 3(). In prticur, we fitted the pproximte soution ĥ = νx to the numeric soution. The best fit gve ν =.992. In the figure, we so give comprison of the pproximte soution ĥ = ( x) /2,[2, Ch. 3] (no seepge fce) nd the fitted numeric resut. Fig. 2() gives the pressure profie for the cse with seepge fce, nd Fig. 3(b) gives the retionship between horizont distnce nd veocity in the x direction. If we hd used the boundry condition ĥ(, t) =, then the free surfce woud be perpendicur to the x-xis t x =. In fct using (25), the horizont veocity using (25) is û 2 ( x) /2 which is infinite t x = Approximte soution to the reduced biodegrdtion equtions In this section we wi consider the method of mtched symptotic expnsions to obtin n pproximte soution to the biodegrdtion equtions. The initi conditions ssume poutnt nd nutrient free porous medium nd constnt indigenous biomss concentrtion ˆm =. The boundry nd initi conditions re given in (8) (2). 2 In prtiy sturted regions, wter is hed under surfce tension dhering to the porous medi grins nd ny fow occurs (between grins) through micropores. The surfce tension provides pressure ess thn tmospheric. But tmospheric pressure is used s dtum to mesure soi wter pressure, thus vues ess thn tmospheric re t negtive pressure (or tension).

8 276 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) b Fig. 3. () Comprison of the pproximte soution nd the numeric soution. The broken ine is the pproximte soution nd the soid ine is the fitted numeric soution. The pproximte soution is given by ĥ = ( x) /2 nd ε =.5. Height of seepge fce is.92. (b) Longitudin veocity with dimensioness distnce Outer soution Consider the dimensioness governing equtions given in (5) (7) in the imit of rge Pecet number, ε. We choose distinguished imit λ = α ε, α = O(), so the governing equtions reduce to ˆR s = û x α ε ˆm ŝ â κ s + ŝ κ + â, â = û â x αλ 2 ε ˆm ŝ κ s + ŝ â K + â, ˆm = αλ 4 ε ( ˆm ) + αλ 3 ε ˆm ŝ â κ s + ŝ κ + â, (28) with boundry conditions ŝ(, t) =, â(, t) =, nd initi conditions â (, t) =, x ŝ(x, ) =, â(x, ) =, ˆm(x, ) =. (, t) =, x In Eq. (26), the initi nd boundry conditions introduce discontinuity or shock moving t speed s (t) stisfying ξ (t) = [ûŝ]+ s, [ˆRs ŝ] + where the squre brckets represents jump in the encosed quntity between its vue on the eft side ( ) nd the right side (+) of the shock. The shock position is given by ξ s (t) = tu stisfying the initi condition ξ s () =. For simpicity we ˆR s wi use û = û = constnt in the pproximte soution. Computing the outer soution up to O( ε ) terms, we obtin αx ε ŝ(x, t) = û κ + κ s +, x û t/ˆrs ;, x > û t/ˆrs, αλ 2 x ε û κ + κ s +, x û t/ˆrs ; â(x, t) =, û t/ˆrs x û t;, x > û t, + αλ 3 ε ˆm(x, t) = κ + κ s + t, x û t/ˆrs ;, x > û t/ˆrs, for λ = α ε. Here we observe tht the effects of the rection terms occur in the O( ε ) terms. (26) (27) (29)

9 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) Inner soution Consider Eq. (5) governing the substrte concentrtion, i.e., ˆR s = û x + ε 2 ŝ x α ε 2 ˆm ŝ â κ s + ŝ κ + â. (3) We introduce the interior-yer coordintes x =[x ξ s (t)]/ε n nd t = τ where ξ s (τ) is singe (smooth) curve cross which the outer soution is discontinuous. We so use ŝ(x, t) = c( x,τ), â(x, t) = g( x,τ)nd ˆm(x, t) = η( x,τ)to denote the soution in this yer. Bncing terms requires n = /2, so we hve c ˆR s τ ξ (τ) s c c ˆRs ε x = û ε x + 2 c x α ε 2 η c φκ s + c g φκ + g, (3) on x (, ). The boundry conditions come from mtching with the outer soution. Tht is c A s, s x nd c, s x +, (32) where A s = αx ε û κ + κ. We ssume tht c( x,τ) = c s + ( x,τ) + O( ε ), g( x,τ) = g ( x,τ) + O( ε ) nd η( x,τ)= η ( x,τ)+ O( ε ) so tht the O() eqution (in the inner yer) for the poutnt trnsport gives [û ˆRs ξ s (τ)] c x = nd since c = x, we hve ξ û s (τ) = which determines the shock position, i.e., ξ s (τ) = û ˆR τ/ˆrs stisfying ξ() =. This s soution grees with the shock position obtined from the outer soution, see (29). Anysis of the first order terms gives the soution in the inner yer. Appying the boundry conditions we hve ŝ(x, t) = A s 2 A s 2 erf x û t/ˆrs ε ˆR s, 4t which is the eding order pproximte soution for the substrte eqution. A simir nysis of the eqution governing the cceptor concentrtion gives the shock oction t ξ (τ) = û τ. Simiry, for the nutrient we hve â(x, t) = A 2 A 2 erf x û t, ε 4t where A = αλ ε 2 x û κ + κ. s + Consider the eqution governing the decy nd production of biomss. The soution ˆm( x,τ)foows from the pproximte soutions ŝ( x,τ)nd â( x,τ). In the inner region we hve: η τ ξ m(τ) η ε x = αλ 4 ε (η ) + αλ 3 ε ˆm c κ s + c g κ + g. (33) We seek soution to (33) in the form η( x,τ)= η ( x,τ)+ ε η ( x,τ)+ for c( x,τ) c ( x,τ)nd g( x,τ) g ( x,τ). The O() expnsion gives ξ m (τ) =, since η = x, stisfying the initi condition ξ m() =. The O( ε ) expnsion gives η τ =. Using the initi condition η ( x, ) = we hve, t eding order, η ( x,τ)=. The O(ε ) terms re η τ = αλ c g 4(η ) + αλ 3 η. κ s + c κ + g But η ( x,τ)= nd c ( x,τ)nd g ( x,τ)re given bove, so we hve η τ = αλ c g 3η, κ s + c κ + g where x (, ). Integrting gives: i.e., η = τ αλ 3 η c κ s + c g κ + g dτ, ˆm = + t ŝ â ε αλ 3 κ s + ŝ κ + â dt. To verify these resuts we use the numeric method of ines. This invoves discretizing the equtions in spce to obtin set of ordinry differenti equtions which re then integrted using MATLAB routine ode5s. For the simutions we set both the retive nd bsoute error toernce to 6.

10 278 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) b Fig. 4. Comprison of the numeric soution nd the pproximte soution for () ε =.5 nd t =., (b) ε =.5 nd t =.7. Fig. 5. Inner nodes system Sensitivity to ε To demonstrte the ccurcy of the numeric soution, in Fig. 4 we pot the symptotic soution nd numeric soution for fixed vue of t nd different vues of ε. In gener, the two soutions re so cose tht their fronts re indistinguishbe for ε =.5. The shrpening of the shock yer s ε decreses is so evident. In Appendix, we show how the effects of sm pore cogging on the fow fied cn be investigted. 5. The fu mode Finite difference nd finite eements methods hve been used in mny probems invoving free or moving boundries [3,27]. Chpwny nd O Brien [3] used tri free boundry method to predict the position of free surfce. This method is bsed on mking n initi guess t the position of the free boundry, Γ (k) sy. Then n pproximtion for the pressure fied inside the sturted porous domin is computed using the given conditions for the fixed boundries together with one of the two conditions on the free boundry. The improved free boundry position Γ (k+) is found by demnding tht the remining free boundry condition is stisfied. The process is terminted when the Γ s gree to given degree of ccurcy. In the present study, both expicit nd impicit finite difference methods re used to simutneousy compute the position of the moving boundry nd the concentrtion profies. The discretized finite difference expressions wi ow for non-uniform grid spce in the directions. The corresponding nod discretiztion is shown in Fig. 5 which eds to modifiction of the derivtives in the governing equtions. Introducing the Lgrngin formution requires d dt = + ẏ y + ẋ x, to describes the chnge whie moving ong the spce coordinte. The dot denotes derivtive with respect to time. We discretize the equtions by repcing the spce derivtives s foows; for the non-uniform grid we hve θ s s i,j+ s i,j s i,j s i,j θ i,j+/2 θ i,j /2, (35) y y j /2 j j (34)

11 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) b Fig. 6. () The moving boundry position t t =.. (b) The moving boundry position t t =. Tbe Seected input dt for the simutions. Symbo Description Vue Units Reference K s Hydrocrbon sturtion constnt 2. g/m 3 [6] K Oxygen sturtion constnt. g/m 3 [6] m Bckground biomss concentrtion 2.8 g/m 3 [8] s Infuent hydrocrbon concentrtion 26. g/m 3 [6] Infuent oxygen concentrtion 6.5 g/m 3 [6] b Microbi decy constnt. /dy [22] k m Utiiztion constnt.2 /dy [6] D Longitudin diffusion constnt.25 m 2 /dy [6] D t Trnsverse diffusion constnt.25 m 2 /dy [6] ρ m Biomss density g/m 3 [8] ρ Liquid wter density. 6 g/m 3 [2] ρ b Buk quifer density.6 6 g/m 3 [8] µ Dynmic viscosity g/dy/m [2] k Permebiity 2. m 2 [2] Y Microbi yied coefficient.5 [6] φ Porosity.39 [8] X Stoichiometric constnt 2.4 [6] Dimensioness prmeters: A.28 λ.47 Eq. (2) λ Eq. (2) λ Eq. (2) λ Eq. (2) χ Biomss rtio.5 ε Longitudin diffusivity.5 Eq. (2) ε t Trnsverse diffusivity.66 Eq. (2) δ 2 Aspect rtio.5 Section 2 where j = y j y j, j /2 = ( j + j )/2, θ j+/2 = (θ j + θ j+ )/2 nd θ j /2 = (θ j + θ j )/2. For the first derivtives we ppy three-point bckwrd formu (uθs) x 3[uθs] i,j 4[uθs] i,j +[uθs] i 2,j b i + b i 2. (36) The other prti derivtives re discretized in the sme wy. The resuting semi-discretiztion is fuy second order ccurte in spce nd eds to system of ordinry differenti equtions for the discrete soution vues which we then integrte in time using Mtbs stiff sover ode5s. We set both retive nd bsoute error toernces for ode5s to Simutions Fig. 7 gives the profies for the veocity, poutnt concentrtion, nutrient concentrtion, biomss concentrtion nd the concentrtion history t point x =. nd y =.42. In the figures, we use the prmeter vues given in Tbe.

12 272 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) Depth.6.4 Depth Width () Poutnt concentrtion Width (b) Nutrient concentrtion Depth.6.4 Concentrtion Width Time (c) Biomss concentrtion. (d) Concentrtions cose to boundry x =..8.8 Depth.6.4 Depth Width (e) Trnsverse veocity Width (f) Longitudin veocity. Fig. 7. A the profies were tken t t =. In (d), the concentrtions were tken t the point (x =., y =.5). In Fig. 6, we show simutions for the moving boundry position t two different time intervs. There is significnt increse in biomss concentrtion in the regions where the nutrient nd poutnt concentrtions overp, see Fig. 7. This hs the direct effect of decresing the height of the phretic surfce which is consistent with the predictions from the pproximte one dimension soution, see Appendix. 6. Concusions nd future work We hve considered mode for bioremedition ppied to the seepge of poutnts through n erth w. The mode invoves fu couping of the fuid veocity nd the degrdtion process. As opposed to ideized situtions in the iterture,

13 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) the current mode is soved for fied fow conditions. The numeric soution is vidted by obtining n pproximte soution in the imit of ow bckground biomss concentrtion. The bio-cogging phenomenon, s given by the correction terms, suggests owering of the moving boundry position nd decrese in the soute injection concentrtions. Studies of stedy stte fow probems of seepge through erth ws re vibe in the iterture, see for exmpe [2]. Such probems my be soved, to resonbe degree of ccurcy, by the Dupuit pproximtion. However, in the present formution the Dupuit pproximtion fis to predict the seepge fce height which is n integr prt of the physic ppiction of the mode. Thus we hve presented numeric strtegy which computes the seepge fce nd the position of the moving boundry. We note tht the correction terms predict no chnge in the seepge height, see boundry conditions to Eq. (48) nd the two dimension simutions. The equtions being soved re compex nd the pproximte soution invoved doube expnsion for χ nd ε in the distinguished imit, δ 2 = ε. This mode nd soution strtegy cn cery be extended to more gener systems incuding more nutrient nd biomss species s we both fuy two- nd three dimension quifers. Acknowedgements This work ws supported by grnts from the Ntur Sciences nd Engineering Reserch Counci of Cnd nd the MITACS Network of Centres of Exceence. SOB cknowedges the support of the Mthemtics Appictions Consortium for Science nd Industry ( funded by the Science Foundtion Irend mthemtics inititive grnt 6/MI/5. Appendix A.. The effect of nonzero O(χ) on the fow In Section 4. we computed the eding order soution for fow in the form ˆp = ĥ y, û ĥ x, ˆv ε y 2 ĥ x 2, where ĥ = ĥ(x), ŝ = ŝ(x, t), â = â(x, t) nd ˆm = ˆm(x, t) re known. Here we investigte the effect of sm pore cogging on the fow by seeking expnsions of the form c ĉ + χĉ 2. The O(χ) eqution governing the correction to the pressure fied is given by 2 ˆp ε x + 2 ˆp 2 y = ε 2 κ x with the foowing boundry conditions ˆm ĥ ˆm ε x, (37) ˆp =, on y =, y ĥ ˆp ĥ =, ε = ˆp y + ĥ ˆp ε x y + ĥ x ĥ x κ ˆm, on y = ĥ, (39) (38) ˆp =, on x =, ˆp =, on x =. The corresponding O(χ) biodegrdtion equtions re given by (4) (4) ˆR s = û x + ε 2 ŝ x + F s(x, t) λ 2 Γ m (x, t) (42) â = û â x + ε 2 â x + F (x, t) λ 2 λ 2 Γ m (x, t) (43) where ˆm = λ λ 4 ˆm + λ λ 3 Γ m (x, t), F s (x, t) = ˆRs ŝ ˆm ˆRs û x ε ˆm x x, F (x, t) = â ˆm û â x ε ˆm â x x, (44)

14 2722 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) b Fig. 8. () The corrected moving boundry position t t =.. (b) The corrected concentrtion profies t t =. for ε =.5. In both figures we djust χ to. to highight the effects of cogging. nd Γ m (x, t) = ˆmâŝ + ( ˆmâ + ˆm â)ŝ (κ + â)(κ s + ŝ) ˆmâŝâ (κ s + ŝ)(κ + â) 2 ˆmâŝŝ (κ s + ŝ) 2 (κ + â), nd ˆRs = A ˆm where ŝ(x, t), â(x, t) nd ˆm(x, t) re the known eding order soutions obtined for ε. The boundry conditions t x = re ε x = ûŝ + û (ŝ ), ε â x = ûâ + û (â ), with the initi conditions ŝ (x, ) =, â (x, ) =, ˆm (x, ) =. To compete the mode, the veocity components re û = ˆp x + κ ˆm ĥ x, v = ˆp y. where κ = 9/6. Simir to the pproch t eding order, we ony need to sove for the moving boundry correction to find correction to the fow veocity. Note tht the moving boundry correction is now time dependent. A.2. Soutions Using the distinguished imit δ 2 = ε we seek soution for the pressure in the form ˆp = ϕ + ε ϕ + O( ε ). From Eq. (37), we hve 2 ϕ y 2 =, which cn be integrted to give (45) ϕ = ĥ, (46) stisfying the boundry condition ϕ (x, h ) = ĥ nd ϕ (x, ) =. The O( ε y ) terms in (37) give 2 ϕ x + 2 ϕ = κ ˆm ĥ ˆm 2 y 2 x x, which cn be simpified to yied 2 ϕ = κ ˆm ĥ ˆm y 2 x x 2 ĥ x. 2 (47)

15 M. Chpwny et. / Journ of Computtion nd Appied Mthemtics 234 (2) Integrting (47) together with the boundry conditions eds to ĥ = ĥ κ ˆm ĥ ˆm x x 2 ĥ + ĥ ĥ x 2 x x ĥ x κ ˆm. (48) The boundry conditions to (48) come from (4) nd (4), i.e., ĥ (, t) = ĥ (, t) =. To compete the nysis, we find numeric soution to Eqs. (42), (43), (44) nd (48). In prticur, Eq. (48) gives correction to the position of the moving boundry in the presence of growing biomss popution. Eq. (46) competes the soution. The O(χ) equtions together with the eding order soution give the pproximte soution to (9) () in the form s(x, t) ŝ + χŝ, (x, t) â + χâ, nd m(x, t) ˆm + χ ˆm. Interestingy, this (crude) mode predicts decrese in the height of the phretic surfce s biomss concentrtion increses since ess wter fows into the domin. This phenomenon is evident in Fig. 8 where there is significnt decrese in poutnt nd nutrient concentrtion t the infow boundry. Note tht for χ =.5, the corrected boundry position nd the eding order soution re indistinguishbe. Thus, for iustrtive purposes, Fig. 8 is shown for χ =.5. References [] H.V. Nguyen, J.L. Nieber, P. Oduro, L.W. Dekker, T.S. Steenhuis, Modeing soute trnsport in wter repeent soi, J. Hydro. 25 (999) [2] S.M. Hssnizdeh, M.A. Cei, H.K. Dhe, Dynmic effect in the cpiry pressure-sturtion retionship nd its impct on unsturted fow, Vdose Zone J. (22) [3] M. Chpwny, S.B.G. O Brien, Corner nd strt-up singurities in porous fow, J. Comput. App. Mth. 76 () (25) [4] K.T.B. McQurrie, E.A. Sudicky, E.O. Frind, Simution of biodegrdbe orgnic contminnts in groundwter:. Numeric formution in princip directions, Wter Resour. Res. 26 (2) (99) [5] F.J. Moz, M.A. Widdowson, L.D. Benefied, Simution of microbi growth dynmics couped to nutrient nd oxygen trnsport in porous medi, Wter Resour. Res. 22 (8) (986) [6] W.S. Dockins, G.J. Oson, G.A. McFeters, S.C. Turbk, Dissimitory bcteri suphte reduction in Montn groundwters, Geomicrobio. J. 2 (98) [7] R.C. Borden, P.B. Bedient, Trnsport of dissoved hydrocrbons infuenced by oxygen-imited biodegrdtion. Theoretic deveopment, Wter Resour. Res. 22 (3) (986) [8] J. Kidsgrd, P. Engesgrd, Numeric nysis of bioogic cogging in two dimension snd box experiments, J. Contm. Hydro. 5 (2) [9] S.K. Tiwri, K.L. Bowers, Modeing biofim growth for porous medi ppictions, Mth. Comput. Modeing 33 (2) [] B. Chen, Y. Li, Simution of thick biofim growth t the microsce, App. Numer. Mth. 4 (22) [] H.J. Dupin, P.L. McCrty, Impct of coony morphoogies nd disinfection on bioogic cogging in porous medi, Environ. Sci. Techno. 34 (2) [2] M. Thuner, L. Mucire, M.H. Schroth, J. Zeyer, W. Kinzebch, Interction between wter fow nd sprti distribution of microbi growth in two-dimension fow fied in sturted porous medi, J. Contm. Hydro. 58 (22) [3] S. Chw, S.M. Lenhrt, Appiction of optim contro to bioremedition, J. Comput. App. Mth. 4 (2) 8 2. [4] M. Thuner, M.H. Schroth, J. Zeyer, W. Kinzebch, Modeing of microbi growth experiment with biocogging in two-dimension sturted porous medi fow fied, J. Contm. Hydro. 7 (24) [5] R. Murry, J.X. Xin, Existnce of trveing wves in biodegrdtion mode for orgnic contminnts, SIAM J. Mth. An. 3 () (998) [6] L. Semprini, P.L. McCthy, Comprison between simutions nd fied resuts for in-situ biorestortion of corinted iphtics: Prt. Biostimution of methnotrophic bcteri, Ground Wter 29 (3) (99) [7] T.P. Cement, B.S. Hooker, R.S. Skeen, Mcroscopic modes for predicting chnges in sturted porous medium properties cused by microbi growth, Ground Wter 34 (5) (996) [8] J. Ber, Hydruics of Groundwter, in: McGrw-Hi Series in Wter Resources nd Environment Engineering, New York, 979. [9] G. Kedy, A Dupuit pproximtion for the rectngur dm probem, IMA J. App. Mth. 44 (99) [2] A.C. Fower, Mthemtic Modes in the Appied Sciences, Cmbridge University Press, 997. [2] J.H. Knight, Improving the Dupuit Forchheimer groundwter free surfce pproximtion, Adv. Wter Res. 28 (25) [22] S. Oy, A.J. Vocchi, Anytic pproximtion of biodegrdtion rte for in situ bioremedition of groundwter under ide rdi fow conditions, J. Contm. Hydro. 3 (998) [23] S.W. Tyor, P.R. Jeffé, Substrte nd biomss trnsport in porous medium, Wter Resour. Res. 26 (9) (99) [24] S.N. Mury, A.K. Mitt, Appicbiity of equiibrium isotherm modes for the biosorptive uptkes in comprison to ctivted crbon-bsed dsorption, J. Environ. Eng. 32 (2) (26) [25] R.S. Johnson, A Morden Introduction to the Mthemtic Theory of Wter Wves, Cmbridge University Press, 997. [26] S.B.G. O Brien, M.A. Hyes, A mode for grvity driven fow of thin iquid soid suspension with evportion effects, Z. Mth. Phys. 56 (25) 22. [27] J. Crnk, Free nd Moving Boundry Probems, Oxford University Press Inc., New York, 984.