Influence of miscible viscous fingering of finite slices on an adsorbed solute dynamics

Transcription

1 PHYSICS OF FLUIDS, 83 9 Influence of miscible viscous fingering of finie slices on an adsorbed solue dynamics M. Mishra, M. Marin, and A. De Wi Nonlinear Physical Chemisry Uni and Cener for Nonlinear Phenomena and Complex Sysems, Service de Chimie Physique e Biologie Théorique, Universié Libre de Bruxelles (ULB), CP 3, 5 Brussels, Belgium Ecole Supérieure de Physique e de Chimie Indusrielles, Laboraoire de Physique e Mécanique des Milieux Héérogènes (PMMH), UMR 7636 CNRS, Universié Paris 6, Universié Paris 7, Rue Vauquelin, 753 Paris Cedex 5, France Received 7 April 9; acceped 5 June 9; published online 5 Augus 9 Viscous fingering VF beween miscible fluids of differen viscosiies can affec he dispersion of finie widh samples in porous media. We invesigae here he influence of such VF due o a difference beween he viscosiy of he displacing fluid and ha of he sample solven on he spaioemporal dynamics of he concenraion of a passive solue iniially dissolved in he injeced sample and undergoing adsorpion on he porous marix. Such a hree componen sysem is modeled using Darcy s law for he fluid velociy coupled o mass-balance equaions for he sample solven and solue concenraions. Depending on he condiions of adsorpion, he spaial disribuion of he solue concenraion can eiher be deformed by VF of he sample solven concenraion profiles or disenangle from he fingering zone. In he case of deformaion by fingering, a parameric sudy is performed o analyze he influence of parameers such as he log-mobiliy raio, he raio of dispersion coefficiens, he sample lengh, and he adsorpion reenion parameer k on he widening of he solue concenraion peak. The resuls highligh experimenal evidences obained recenly in reversed-phase liquid chromaography. 9 American Insiue of Physics. DOI:.63/.387 I. INTRODUCTION Miscible viscous fingering VF is an inerfacial fluid flow insabiliy ha occurs when a less viscous fluid displaces anoher more viscous miscible one in a porous medium, leading o he formaion of fingerlike paerns a he inerface of boh fluids. VF of one single inerface impacs a variey of pracical applicaions such as oil recovery, filraion, or hydrology for insance and has been he subjec of numerous invesigaions over he years since he pioneering work of Hill and Saffman and Taylor. 3 VF has also been observed in liquid chromaography, which is used o separae he chemical componens of a given sample by passing i hrough a column of porous medium. 4 VF also influences he spreading of finie widh samples in localized polluion zones in aquifers.,3 In he case of such viscous slices of finie exen, fingering is a ransien phenomenon as he mixing of boh sample and carrier fluids leads o an effecive decrease in ime of he logmobiliy raio. Even hough he spreading of he sample may look like Gaussian a long imes when dispersion becomes again dominan in he ranspor phenomena, he variance of he peak is larger han expeced because of fingering a earlier ime. A quaniaive characerizaion of he conribuion of such VF in he variance of he peaks has been conduced in he frame work of chromaographic, or polluion applicaions. This widening has been shown o be of larger exen when he sample is less viscous han he carrier fluid han when i is more viscous. 4 In chromaography and for problems of ranspor in aquifers, he finie size of he viscous sample and he solue adsorpion on he porous marix can be of imporance.,5,6 Recenly, a heoreical model aking ino accoun such adsorpion of he species ruling he viscosiy of he soluion during miscible VF of finie widh samples was developed by Mishra e al. 7 The growh raes of he unsable wave numbers of he VF insabiliy have been discussed by means of a linear sabiliy analysis limied o he iniial sage of developmen of he insabiliy. On laer sages, nonlinear dynamics have been analyzed by means of numerical simulaions when he viscosiy-modifying componen in he sample is reained or no. When his componen is reained, is effec is similar o ha of an unreained solue wih a +k smaller viscosiy, where k is he reenion facor. The ypical widh of he fingers is also decreased by a facor +k. In chromaography, his siuaion can occur for polymeric solues of relaively large molar mass as he inrinsic viscosiy of polymers increases wih heir molar mass. In mos liquid chromaographic analyses, he sample solues have however a relaively low molar mass, hence a oo low inrinsic viscosiy o induce a VF phenomenon affecing dispersion characerisics. However, in order o increase he solubiliy of he componens o be analyzed, he solues are ofen dilued in a solven whose chemical naure and hence viscosiy differ from hose of he displacing fluid. Then, VF may develop a he unsable sample solven/eluen inerface and affec he reained solue zones. Experimenal evidences ha he chromaographic peak shapes of solues elued under reversed-phase liquid chromaographic condiions are sig /9/ 8 /83//$5., 83-9 American Insiue of Physics Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

2 83- Mishra, Marin, and De Wi Phys. Fluids, 83 9 L y U µ µ y c a = c = nificanly dependen on he naure and/or composiion of he sample solven have been given recenly. This phenomenon has been clearly associaed wih he difference beween he eluen and sample solven viscosiies. To our knowledge, here is no quaniaive sudy or any modeling of he influence of VF on he shape of he chromaographic peaks of solues in his siuaion which ypically involves hree componens: he eluen or displacing fluid, he sample solven, and he reained solue. In his conex, he purpose of he presen sudy is o undersand how a VF insabiliy occurring beween he displacing fluid and he unreained sample solven affecs he spaioemporal disribuion of a reained solue iniially dissolved in he solven and o sudy he influence of he exen of reenion on he solue concenraion dynamics. We consruc a model for miscible VF of a hree componen sysem, considering he dynamics of he carrier fluid and of boh a solue and a solven in he sample, incorporaing a linear adsorpion isoherm for he solue. This model couples Darcy s law for he evoluion of he flow velociy wih an equaion describing he evoluion of he concenraion of he sample solven and wih a mass-balance equaion for he solue concenraion. This solue undergoes adsorpion-desorpion processes on he porous marix. Numerical simulaions are performed wih a viscosiymodifying sample solven and reained solues. The resuls are discussed in erms of a reenion parameer k quanifying he adsorpion phenomena. Theoreical explanaions of he disengagemen of solue and sample solven peak shapes are also presened for a pure dispersive case. The aricle is organized as follows: in Sec. II, we inroduce our hree componen model while in Sec. III, we describe our numerical inegraion mehod. Resuls are described in Sec. IV before conclusions are drawn in Sec. V. II. THREE COMPONENT MODEL x The sysem considered see Fig. is a wo dimensional D porous medium of lengh L x and widh L y in which a sample consising of a solue or analye in concenraion c a, dissolved in a sample solven of concenraion c and of viscosiy is injeced a an iniial ime =. The iniial lengh of he sample is W. This sample is displaced by anoher miscible fluid, he carrier fluid or eluen, differen from he sample solven which has a viscosiy and in which he sample solven concenraion c = and he solue concenraion is c a, =. The displacing fluid is injeced uniformly wih a mean velociy U along he x direcion. We assume ha he iniial solue or analye concenraion c a, in he sample solven is small and does no influence he W c a, c = c FIG.. Skech of he sysem a iniial ime. µ c a = c = L x viscosiy of he fluids. This viscosiy is however assumed o depend on he concenraion of he sample solven c hrough an exponenial funcion, c = e R c/c, where he log-mobiliy raio R is defined as R=ln /.If, R, and he rear inerface of he sample where he less viscous displacing fluid pushes he more viscous sample is unsable wih regard o VF. The fronal inerface is on he conrary sable. As he solue does no impac he viscosiy of he soluions, i will behave as a passive scalar in he flow. The spaioemporal dynamics of his solue s concenraion depends however on adsorpion on he porous marix. Once he sample is injeced in he porous marix he solue can adsorb ono he porous marix following he reversible adsorpion-desorpion reacion: k a c a,m ca,s. k d Here c a,m and c a,s are he concenraion of he solue in he mobile and saionary phases, respecively, while k a and k d are he adsorpion and desorpion kineic consans. Assuming ha he fluid is incompressible and he flow inside he porous medium is governed by Darcy s law 4, he governing equaions for he sysem are as follows: u =, p = c K p u, c + u c c = D x x + D c y y, c a,m + F c a,s + u c a,m = D ax c a,m x D ay c a,m y, 6 where u = u,v is he D fluid velociy wih u and v he velociy componens in he x and y direcions, respecively, K p is he permeabiliy of he porous medium, and p is he pressure. Equaion 5 is he convecion-diffusion equaion for he concenraion c of he sample solven ruling he viscosiy of he soluion. Equaion 6 is he mass-balance equaion for he solue concenraion c a, where F=V s /V m = o / o is he phase raio of he volume V s and V m of he saionary and mobile phases, where o is he oal porosiy or void volume fracion of he porous medium. 8 D x and D y are he dispersion coefficiens of he sample solven in he displacing fluid along he x and y direcions, respecively, while D ax and D ay are hose of he solue. Influence of Koreweg sresses 9 are here assumed negligible. Assuming a linear isoherm adsorpion dependence beween he concenraion c a,s and c a,m as c a,s = Kc a,m, where K=k a /k d is he equilibrium consan of he adsorpiondesorpion equilibrium, Eq. 6 becomes 7 Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

3 83-3 Influence of miscible viscous fingering of finie slices Phys. Fluids, k c a,m c a,m c a,m + u c a,m = D ax x + D ay y, 8 where k =FK. Le us noe ha a he momen of injecion, all solue or analye molecules are in he mobile phase of volume V m so he concenraion of solue c a, =n/v m, where n is he oal number of analye moles injeced in he column. When he sample sars o move in he column a = and afer he adsorpion/desorpion equilibrium has been reached, he corresponding analye concenraion in he mobile c a,m and saionary c a,s phases are c a,m = n m /V m, c a,s = n s /V s, where n s and n m are he number of analye moles in he saionary and mobile phases, respecively. Hence, he conservaion of he oal number of solue moles n=n s +n m along wih c a,s =Kc a,m implies ha c a, V m = c a,m V m + c a,s V s = c a,m +k V m, which leads o c a,m = c a, / +k. Therefore, he mobile phase concenraion of he analye c a,m varies from in he pure eluen o c a,m =c a, / +k in he sample. To nondimensionalize he governing equaions, we choose he concenraion c and c a, / +k as he reference concenraion for he solven and solue concenraions, respecively, and U as he characerisic velociy. Defining a lengh scale L c =D x /U and a ime scale c =D x /U, he nondimensional quaniies are hen obained as xˆ = x L c, ŷ = y L c, ˆ = c, û = u U, vˆ = v U, p = p D x /K p, =, c = c, c c a,m = +k c a,m, c a, = D y D x, a = D a y D ax, = D a x D x. Inroducing a reference frame moving wih he injecion speed x = xˆ ˆ, y = ŷ, u = û, v = vˆ, = ˆ, he governing Eqs. 3 5 and 8 wih he concenraiondependen viscosiy Eq. become, afer dropping he superscrips, u =, p = c u + e x, c + u c = c x + c y, 9 +k c a,m + u k c a,m x + v c a,m y = c a,m c a,m x + a y, c = e Rc. 3 From he above equaions i is clear ha Eq. is decoupled from Eqs. 9. Hence, once he velociy field is deermined from Eqs. 9 and 3 for given R and, he ranspor of he solue concenraion c a,m can be analyzed easily for differen values of he analye parameers k,, a. Using his model, our goal is now o analyze he influence of he viscosiy dependence on he sample solven concenraion c on he spaioemporal disribuion of he solue s concenraion in he mobile phase c a,m. III. NONLINEAR SIMULATIONS Inroducing he sream funcion x,y, such ha u= / y and v= / x, and following Tan and Homsy, he momenum and concenraion equaions become = R c x x + c y y + y, c 4 c + c y x c x y = c x + c y, + k y c a,m x c a,m x y 5 +k c a,m = c a,m c a,m x + a y. 6 Equaion 4 is obained by eliminaing he pressure gradien by aking he curl of Darcy s law. Noe ha in absence of adsorpion k = and for a solue dispersing a he same rae as he sample solven = and a =, we recover he classical heoreical model of miscible VF sudied previously by many auhors.,,,3 The dynamics of he concenraion of he sample solven c and of he solue c a,m hen evolve exacly he same way and he dynamics of c a,m is merely ha of a passive scalar adveced by fingering beween he displacing fluid and he sample solven of he sample. The objecive here is o analyze he effec of k, i.e., see how he dynamics of he sample solven and of he solue can disenangle because of adsorpion. Equaions 4 6 are numerically solved using he pseudospecral mehod inroduced by Tan and Homsy. The code has been validaed here by reproducing he resuls of Tan and Homsy afer reducing he hree componen sysem Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

4 83-4 Mishra, Marin, and De Wi Phys. Fluids, 83 9 (d).8.6 c a,m.4 k =.5 k = k = k =5 R=, sample solven k = k =.. 3 FIG. 3. Transversely averaged concenraion profiles of he solue c a,m a = for differen values of k and simulaions of Fig.. x (e) IV. VISCOUS FINGERING IN THE THREE COMPONENT SYSTEM (c) FIG.. Color online Densiy plos of he mobile phase concenraion of he solue c a,m a successive imes in a frame moving a he injecion velociy wih L =5, =, =, a =, l=5, and R=; a k =, he dynamics corresponds o he VF dynamics of he sample solven concenraion, b k =., c k =.3, d k =.5, e k =, and f k =. From op o boom: =, 5,,, 4, 5, 7. o a wo componen one by choosing k =, =, and a =. Is convergence has also been esed in he case of finie sample fingering. The boundary condiions are periodic in boh x and y direcions. Along he axial coordinae x, his is no problem as long as he fingers exending o he lef do no inerac wih he righ of he sample by periodiciy and vice versa. Periodiciy along y is sandard for such paern formaion sudies., In dimensionless unis, he lengh and widh are L=UL x /D x and L =UL y /D x, respecively, while he dimensionless iniial lengh of he sample is l=uw/d x. The iniial condiions for boh c and c a,m correspond o a convecionless fluid embedding a recangular sample of concenraion c=, c a,m = and of size L l in a c=, c a,m = background. The middle of he sample is iniially locaed a x=4l/5. The laeral sides of he sample correspond o wo back o back sep funcions beween c= and c= or beween c a,m = and c a,m = wih an inermediae poin where c=c a,m =/+Ar, where r is a random number beween and and A is he ampliude of he noise of he order 3. This noise is used o rigger he fingering insabiliy on a perinen compuing ime. (f) Densiy plos of c a,m, he concenraion of he solue in he mobile phase are ploed a successive imes in Fig. for L =5, l=5, =, =, a =, R= and differen values of he reenion parameer k on a gray scale from black o whie corresponding o c a,m = and c a,m =, respecively on color, c a,m = o correspond o blue and red. The sysem is shown in a frame of reference moving wih he injecion speed of he eluen. The figures are ploed wih a consan aspec raio. The VF dynamics due o he unfavorable conras beween he viscosiy of he displacing fluid and ha of he sample solven is shown in Fig. a. Such a VF of finie slices in wo componen sysems has already been sudied quaniaively in deail before.,,4,7,4 In absence of any adsorpion, i.e., for k = and c a,s = he solue remains in he mobile phase and follows he dynamics of he flow as a passive scalar. Is fingering is hen he same as ha of he sample solven Fig. a. Figures b f show he cases when he solue in he injeced sample adsorbs ono he porous marix i.e., k. I is observed ha he VF dynamics of he solven acs upon he evoluion of he solue concenraion, c a,m. In he presence of adsorpion he reained solue zone develops fingers when eiher one inerface or boh rear and fronal inerfaces of i come in conac wih he unsable inerface of he sample solven zone Figs. b f. This is in agreemen wih experimenal observaions in chromaography columns. This reained solue zone disengages from he sample solven zone wihou any disorion a any of he wo inerfaces for larger values of k. These above resuls can be quaniaively analyzed using he ransversely averaged profiles of he mobile phase concenraion of he solue defined as c a,m x, = L L ca,m x,y, dy. 7 c a,m x, is ploed in Fig. 3 for differen values of he reenion parameer k a a fixed ime = for he nonlinear dynamics of Fig.. The reference x= posiion along he x-axis is se such ha he cener of an unreained analye Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

5 83-5 Influence of miscible viscous fingering of finie slices Phys. Fluids, 83 9 zone would occupy a ha posiion in absence of VF. If R =, no fingering occurs and c a,m evolves only via dispersion. If R= bu k = he curve corresponding o his nonadsorpive case shows he characerisic bumps and he broadening due o he VF dynamics induced by he difference in viscosiy of he sample solven and of he displacing fluid.,4,7 When adsorpion is occurring k i is observed ha as k increases, he solue zone is rearded wih regard o he sample solven zone. For some k, boh he rear and fronal inerfaces ge disored see he doed curve for k =. since boh rear and fronal inerfaces of he solue concenraion field are affeced by VF of he sample solven zone given by he curve wih k =. For larger values of k boh rear and fronal inerfaces of he solue plug remain nondeformed. This occurs when he solue plug ges disengaged from he solven plug before VF has had ime o disperse he sample solven. The solue ransversely averaged profiles feaure hen he sandard error funcion characerisic of simple dispersion. To undersand he condiions for a disengagemen wihou disorion of he reained solue zone from he solven zone, le us discuss in Sec. IV A a simplified model for hree componen sysems. A. Dispersive model of disengagemen of solue zone from sample solven zone The hree componen sysem is here considered in he case of isoeluoropic sample solvens for which he reenion facor k of he solue is he same in boh he sample solven and he eluen. I is assumed ha a recangular plug of a given sample solven conaining a dissolved solue occupies iniially a lengh L inj of he porous medium. Because he solue is reained on he porous marix, i moves along he column more slowly han he zone of he unreained sample solven concenraion disribuion. From Eqs. 4 6, i is clear ha, for he case of a sable displacemen obained when R= and =, he cener of mass of he solue plug propagaes oward he lef of he sample plug wih a speed k / +k in he nondimensional moving frame of reference and wih a speed / +k in he nondimensional seady frame of reference which corresponds o he dimensional speed U/ +k. I is of ineres o find he criical ime, cri, a which he solven zone and he solue zone will cease o overlap. This ime can be obained analyically in he pure dispersive case, in he absence of VF. In ha case indeed, he solue and solven concenraion disribuions will cease o overlap when he posiion of he rear inerface of he sample solven, x s,r, will be ahead of he fronal inerface of he solue zone, x a,f. In a pure dispersive regime, he widhs of he solue as well as of he sample solven zones change in ime from heir iniial dimensional widh W o heir corresponding dispersive widhs. By assuming ha all he values of he normal disribuion in he pure dispersive displacemen lie wihin wo sandard deviaions of he mean, he upsream end of he rear inerface of he sample solven zone is lagging, by a disance s, behind he mean posiion of his rear inerface. The downsream end of he fronal inerface of he solue zone is cri l = k FIG. 4. Dimensionless criical ime cri a which he solue concenraion disribuion ceases o overlap wih he sample solven one in absence of fingering as a funcion of k for differen sample lenghs l wih =. ahead, by a disance a, of he mean posiion of he fronal inerface. Here s and a are he spaial sandard deviaions ha he sample solven zone and he analye zone, respecively, would have a ime for vanishing W Dirac injecion pulse. Assuming ha he rear of he sample, a ime =,isa posiion x=, one has in he seady frame of reference wih dimensional quaniies, x s,r = U s, 8 x a,f = W + U +k + a, 9 where is he ime elapsed since sample injecion and U is he displacing fluid and sample solven velociy. Noing ha s = Dx and a = +k D x, hen cri is he ime a which x s,r =x a,f, i.e., U cri Dx cri = W + U +k cri + +k D x cri. Using L c =D x /U and c =D x /U as a characerisic lengh and ime as defined before, he dimensionless form of Eq. becomes k +k cri + +k cri l =, where l=w/l c as before. Equaion gives wo roos for cri. I can easily be shown ha Eq. has a unique soluion for all he values of only by considering he roo wih posiive sign of square roo of discriminan of Eq. and hence he corresponding criical ime is cri = +k k + +k + + +k + k. In Fig. 4 ploing cri as a funcion of k shows ha he larger l +k Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

6 83-6 Mishra, Marin, and De Wi Phys. Fluids, 83 9 he reenion parameer k, he smaller cri. Moreover, cri sauraes o an asympoic value equal o + +l a larger values of k. When increasing he iniial widh of he sample l his asympoic cri increases. For k =, =, and l=5 in his pure dispersive case, one can obain from Eq. ha he disengagemen of he reained solue and solven zones occurs a =383. However, in he presence of VF, his disengagemen occurs a a ime close o =5 see Fig. e, which is much larger han for he pure dispersive displacemen. The presence of VF induces hus a delay in he disengagemen of he solue plug from he solven plug. I means ha he unaffeced solue zone can be obained a a laer ime in he presence of VF han for pure dispersion of sample solven. B. Variance of he ransverse averaged profile To quanify he influence of VF on he disorion of he concenraion peak of he solue plug, we nex compue he variance a of he ransversely averaged profiles of concenraion c a,m x, as,4,7 a = L c a,m x, x m dx L, 3 c a,m x, dx where m = L xf x, dx is he firs momen of f x, =c a,m x, / L c a,m x, dx which is he probabiliy densiy funcion of he coninuous disribuion c a,m x,. Similarly a variance can be calculaed for he solven plug for ransversely averaged profile c x, defined along he line above. From Eqs. 5 and 6 he variance o of a sable R= solven peak and he variance a,o of he mobile phase solue peak can be calculaed o be o =l /+ and a,o =l / + / +k, respecively. The erm l / corresponds o he conribuion due o he iniial widh l of he sample, while he erm linear in is due o a dispersive mixing. As done in previous sudies of VF of finie slices, when fingering akes place, we can exrac he quaniy f = o and a,f = a a,o, which gives he conribuion o he oal variance and a due o VF. The effecs of he adsorpion parameer k on he oal variance a are ploed in Fig. 5 a for he simulaions presened in Fig.. Similar o he case of VF in wo componen sysems, he variance a increases more in ime han a,o because of he widening of he peak due o fingering conribuions. However, in he presence of adsorpion, he variance reduces as k increases even if, for small values of k.3, he variance does no have a significan differen value as he effecs of he sample solven VF are large and he peaks are more disored. I is clearly observed from Fig. 5 b ha he variance for k = or increases only due o dispersion afer he ime when he solue plug has disengaged from he sample solven one. Indeed, afer a ransien increase due o VF he shape of a is again he same as for pure dispersion, i.e., direcly proporional o a,o =l /+ / +k. The numerical simulaion for R=, k =, and he heoreical observaion exacly coincide Fig. 5 b which can be considered as a furher validaion of he presen numerical mehod. 5 4 σa 3 6 x5 σa k = x4 In order o undersand he influence of VF of he sample solven on he broadening of he solue plug, Fig. 6 shows he conribuion o he sandard deviaion due o fingering, a,f, as a funcion of ime for differen values of he reenion parameer k. We know from Ref. ha for he solven plug f sars o deviae from zero a he onse ime of VF and nex increases wih ime before sauraing o an asympoic value afer some ime when dispersion akes over again. Similarly he variance of he solue plug deviaes from he same iniial consan f of he solven plug when he solue disribuion is deformed by fingering. I sauraes o an asympoic value afer some ime when he effecs of he VF are weakening because of he complee disengagemen of he solue and solven plugs. I is seen from Fig. 6 a ha he ime inerval beween he onse ime of fingering and he ime of sauraion is decreasing wih increasing k. For k =,.,.5 sauraion occurs a a ime larger han whereas for k =,,3 sauraion occurs a a ime close o 3, 5, and, respecively. I shows ha he larger k, he quicker he disenanglemen beween solue and solven plugs. The onse ime of he broadening of he solue plug in presence of VF increases and reaches a maximum for a criical value of k here close o k =.5, see Fig. 6 b hen sars decreasing for furher increasing of k. I can how k = k = FIG. 5. a Variance a of c a,m x, as a funcion of ime for differen values of k and simulaions of Fig.. b Enlargemen of a for k =, and R= solid lines wih comparison of he pure dispersion value a,o =l /+ / +k obained when R= dashed lines. Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

7 83-7 Influence of miscible viscous fingering of finie slices Phys. Fluids, k = k = 3..5 L d x k δ L d k =,.,.5,, k = k = k = FIG. 6. a Conribuion of fingering o he sandard deviaion a,f as a funcion of ime for differen values of k wih L =5, =, =, a =, l=5, and R=; b zoom a early imes. ever no be smaller han he onse ime of he solven VF or, in oher words, han he onse ime for he non adsorpive solue displacemen. I is observed ha for some cases, for example, Fig. e a 4, a disorion a he fronal inerface, which can be due o he ineracion of ha inerface wih VF a an earlier ime, can remain for quie a long ime before being smoohed ou by dispersion. This phenomenon can also be seen in Fig. 6 a for k = where he sauraion of he variance o an asympoic value is already achieved a = 4, i.e., he broadening is only due o he dispersion. C. Spreading lengh of solue zone FIG. 7. a Spreading lengh L d of he solue as a funcion of ime for he simulaions of Fig. and differen values of k. b Corresponding log-log plos for he pure dispersive case R=; solid lines and comparison wih he R= dashed lines cases wih reenion. The curve wih label is o show he characerisic slope corresponding o he pure diffusion. The mixing lengh which is nohing else bu he lengh of he fingering zone is an imporan measure in he sudy of VF as i gives informaion on he lengh of he zone where he wo miscible fluids mix wih each oher.,7 Similarly, in order o undersand he lengh of he displacemen of he solue zone due o VF, we quanify he spreading of solue fingers by he lengh l d of he inerval in which c a,m x, 3. The evoluion of he spreading lengh of he solue defined as L d =l d l is ploed for differen reenion parameers k in Fig. 7 a. I shows ha a lower k, he lengh of he solue displacemen zone have similar rends whereas a large k, he lengh L d decreases for increasing k. Figure 7 b depics he evoluion of L d normalized by / +k for differen k wihou he effecs of VF i.e., for pure dispersion when R= hrough a log-log plo. I shows ha in a pure dispersion regime, he displacemen lengh is he same for all k. The normalized lengh / +k is he sandard deviaion of he solue concenraion profile which can be found by solving he linear par of Eq.. I is seen ha he displacemen lengh evolves hen proporionally o. Figure 7 b shows ha for R=, k = and, afer he solue band ineraced wih VF and compleely disenangled from he VF zone he displacemen of he solue is dominaed again by dispersion. However his dispersion of he solue band is no direcly proporional o like in a pure dispersive case, raher i moves wih a power of which is less han.5. I is ineresing o noe ha he effecs of VF reduce he power-law relaionship of he dispersion lengh of he solue wih ime. D. Parameric sudy I is well known ha he larger he value of he logmobiliy R, he more inense VF and he faser he fingers ravel.,4, From Fig. 8 a we find ha VF wih R=3 affecs he broadening of solue plug for larger values of he reenion parameer k han for R= shown in Fig. 6, i.e., he ime needed for a disengagemen process reduces for larger values of k. So, a fixed R, here is a criical k above which he broadening of he solue plug is no affeced by Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

8 83-8 Mishra, Marin, and De Wi Phys. Fluids, k = k =..5 FIG. 9. Same as Fig. 6 excep for = k = FIG. 8. a Same as Fig. 6 excep for R=3; b zoom a early imes. VF. This criical k increases wih increasing R. As a corollar, he ime during which he solven and solue plugs inerac increases for a fixed k as R increases. The onse of fingering on he solue plug occurs early for large R, which is seen from Fig. 8 b, asfork =.5 he onse ime of solue fingers for R= is close o =9 see Fig. 6 b and for R=3 i is nearer o =4. For R=3, he onse ime of fingering of he solue plug firs increases wih increasing k, hen reaches a maximum for a criical k close o. This criical k is larger han in he R= case see Fig. 6 b where i was found o be close o.5. Furhermore, i is observed ha for larger R, he fingering variance akes a longer ime o reach an asympoic value for a fixed k, which reveals ha fingers ravel faser for large R. On he oher hand, decreasing he raio of dispersion coefficiens =D y /D x for a fixed value of R has a desabilizing effec and he sample solven VF is more inense. As a consequence decreasing leads o a sronger influence of VF on he displacemen of he solue zone. Since small ransverse dispersion D y favors longiudinal growh of VF by allowing he fingers o survive for a longer ime, he solue zone needs a longer ime o disenangle from he solven plug and be unaffeced by VF. This can be seen from a comparison of Figs. 6 and 9 for differen values of k. For k = he criical ime cri 3 when =. as compared o cri =5 for = in Fig. 6. So, for small dispersion raio, he criical k above which he solue plug is unaffeced by VF a anyime increases. Similarly his criical k will increase for larger L as he VF is hen more inense oo. Figure shows he influence of he iniial sample lengh l on he broadening of he solue zone in presence of sample solven VF. The larger he iniial exen of he sample, he larger he mixing zone beween he sample solven and he displacing fluid. Hence he VF zone ineracs longer wih he solue zone hereby increasing he broadening of a,f. The smaller l for a fixed k, he faser he solue zone disenangles from he sample solven zone, hence he faser he emporal dependence of a,f reaches a plaeau, as seen in Fig.. The criical k for he disengagemen of boh zones increases wih increasing sample lengh l. Evenually, Fig. shows he influence on a,f of he raio of he longiudinal dispersion coefficiens =D ax /D x of he solue in he mobile phase and of he sample solven. For larger values of, VF of he solven affecs a,f during a longer ime. Sauraion of a,f occurs herefore a a laer ime for increasing. The onse of he broadening of he solue zone due o VF occurs quicker for increasing as seen in Fig. b. This is due o he fac ha a larger value of implies larger axial dispersion of he solue zone favoring he longiudinal growh of he solue concenraion disribuion. This allows he solue o inerac on longer disances and for a longer ime wih he VF of he sample solven. In spie of his augmened duraion of he ineracion beween he analye and sample zones, which allows more ime for he analye zone o be affeced by he VF process on l = l = 5 l = 56 FIG.. a,f as a funcion of ime for differen values of he iniial sample lengh l for L =5, =, =, a =, R=, and k =. Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

9 83-9 Influence of miscible viscous fingering of finie slices Phys. Fluids, σ a,f he upsream side of he sample solven zone, he sauraion value of a,f is observed o decrease on increasing dela in Fig. a. In fac, as he axial dispersion of he analye increases, is ransverse dispersion also increases since a is here kep equal o. This promoes a faser merging of he fingers of he analye zone and a resuling lower VF conribuion o he sandard deviaion of his zone. This finding illusraes he significan role played by ransverse dispersion for limiing he VF effec. V. CONCLUSION 6 4 δ=. 5 When a passive solue undergoing adsorpion on he porous marix is iniially dissolved in a solven which is more viscous han he displacing fluid, is ranspor properies can be affeced by VF phenomena aking place a he inerface where he carrier fluid displaces he sample solven. The dynamics of he solue concenraion has been characerized here in such condiions on he basis on numerical simulaions of a hree componen model consising in Darcy s law coupled o he evoluion equaion for he solue and solven concenraions, his laer one conrolling he viscosiy of he soluion. If he solue is unreained and diffuses a he same rae as he solven, he solue is merely a passive scalar following he fingering aking place beween he displacing fluid and he solven. If he solue is, on he conrary, reained on he porous marix, is dynamics can more or less disenangle from he fingering area depending on he srengh of δ=. FIG.. a Effecs of fingering on a,f as a funcion of ime for differen values of wih L =5, =, k=, a =, l=5, and R=; b zoom a early imes. he reenion facor k. The concenraion peak of he solue can, in ha case, feaure deformaions due o he fingering processes eiher on is back, on boh fronal and rear inerfaces, or only on he fronal par depending wheher k is small, of inermediae value or large. Nondeformed peaks of solue disribuion are obained above a criical value of k when he solue plug rapidly disenangles from he solven one before fingering has had ime o ac. The criical ime a which his disenanglemen akes place has been calculaed analyically in he pure dispersion case and has been shown o be smaller han he disenanglemen ime when fingering akes place. A parameric sudy furhermore shows ha his criical ime is an increasing funcion of he log-mobiliy raio R and of he iniial lengh l of he sample bu a decreasing funcion of he raio of dispersion coefficiens of he solven. Moreover if he raio beween he axial dispersion coefficien of he solue and ha of he solven is increased, he solue ineracs on longer disances wih he fingering which increases herefore he broadening of he solue peak. These resuls are in agreemen wih recen experimenal findings in chromaography columns. ACKNOWLEDGMENTS M.M. from ULB graefully acknowledges a posdocoral fellowship of he FRS-FNRS Fonds de la Recherche Scienifique, Belgium. A.D. acknowledges financial suppor from he Communaué Française de Belgique Acions de Recherches Concerées Programme, Prodex and FNRS. The collaboraion beween our ULB and ESPCI eams is financially suppored by a French Programme d Acions Inégrées under Conrac No. 8948YD and Belgian CGRI Tournesol gran. G. M. Homsy, Viscous fingering in porous media, Annu. Rev. Fluid Mech. 9, S. Hill, Channeling in packed columns, Chem. Eng. Sci., P. G. Saffman and G. I. Taylor, The peneraion of a fluid ino a porous medium or Hele-Shaw cell conaining a more viscous liquid, Proc. R. Soc. London, Ser. A 45, M. Czok, A. Kai, and G. Guiochon, Effec of sample viscosiy in highperformance size-exclusion chromaography and is conrol, J. Chromaogr. A 55, E. J. Fernandez, T. Tucker Noron, W. C. Jung, and J. G. Tsavalas, A column design for reducing viscous fingering in size exclusion chromaography, Bioechnol. Prog., D. Cherrak, E. Guerne, P. Cardo, C. Herrenknech, and M. Czok, Viscous fingering: A sysemaic sudy of viscosiy effecs in mehanolisopropanol sysems, Chromaographia 46, M. L. Dickson, T. T. Noron, and E. J. Fernandez, Chemical imaging of mulicomponen viscous fingering in chromaography, AIChE J. 43, B. S. Broyles, R. A. Shalliker, D. E. Cherrak, and G. Guiochon, Visualizaion of viscous fingering in chromaographic columns, J. Chromaogr. A 8, H. J. Cachpoole, R. A. Shalliker, G. R. Dennis, and G. Guiochon, Visualising he onse of viscous fingering in chromaography columns, J. Chromaogr. A 7, S. Keunchkarian, M. Rea, L. Romero, and C. Casells, Effec of sample solven on he chromaographic peak shape of solues elued under reversed-phase liquid chromaographic condiions, J. Chromaogr. A 9, 6. G. Rousseaux, A. De Wi, and M. Marin, Viscous fingering in packed chromaographic columns: Linear sabiliy analysis, J. Chromaogr. A 49, Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp

10 83- Mishra, Marin, and De Wi Phys. Fluids, 83 9 A. De Wi, Y. Berho, and M. Marin, Viscous fingering of miscible slices, Phys. Fluids 7, V. Krez, P. Beres, J.-P. Hulin, and D. Salin, An experimenal sudy of he effecs of densiy and viscosiy conrass on macrodispersion n porous media, Waer Resour. Res. 39, 3, DOI:.9/WR M. Mishra, M. Marin, and A. De Wi, Differences in miscible viscous fingering of finie widh slices wih posiive and negaive log-mobiliy raio, Phys. Rev. E 78, G. R. Johnson, Z. Zhang, and M. L. Brusseau, Characerizing and quanifying he impac of immiscible-liquid dissoluion and nonlinear, raelimied sorpion/desorpion on low-concenraion eluion ailing, Waer Resour. Res. 39,, DOI:.9/WR S. E. Serrano, Propagaion of nonlinear reacive conaminans in porous media, Waer Resour. Res. 39, 8, DOI:.9/WR M. Mishra, M. Marin, and A. De Wi, Miscible viscous fingering wih linear adsorpion on he porous marix, Phys. Fluids 9, G. Guiochon, A. Felinger, D. G. Shirazi, and A. M. Kai, Fundamenals of Preparaive and Nonlinear Chromaography Elsevier, San Diego, 6. 9 C.-Y. Chen and E. Meiburg, Miscible displacemen in capillary ubes: Influence of Koreweg sresses and divergence effecs, Phys. Fluids 4, 5. H. H. Hu and D. D. Joseph, Miscible displacemen in a Hele-Shaw cell, Z. Angew. Mah. Phys. 43, P. Peijeans and T. Maxworhy, Miscible displacemen in capillary ubes. Par. Experimens, J. Fluid Mech. 36, C. T. Tan and G. M. Homsy, Simulaion of nonlinear viscous fingering in miscible displacemen, Phys. Fluids 3, C. T. Tan and G. M. Homsy, Sabiliy of miscible displacemens in porous media: Recilinear flow, Phys. Fluids 9, C.-Y. Chen and S.-W. Wang, Miscible displacemen of a layer wih finie widh in porous media, In. J. Numer. Mehods Hea Fluid Flow, 76. Downloaded 4 Aug 9 o Redisribuion subjec o AIP license or copyrigh; see hp://pof.aip.org/pof/copyrigh.jsp