SMOOTHED PARTICLE HYDRODYNAMICS MODEL FOR REACTIVE TRANSPORT AND MINERAL PRECIPITATION

Transcription

1 CMWRXVI SMOOTHED PARTICLE HYDRODYNAMICS MODEL FOR REACTIVE TRANSPORT AND MINERAL PRECIPITATION A. TARTAKOVSKY 1, T. SCHEIBE 1, G. REDDEN 2, P. MEAKIN 2, Y. FANG 1. 1 Pacfc Northwest Natonal Laboratory, P.O. Box 999, Rchland, WA 99352; 2 Idaho Natonal Laboratory, P.O. Box 1625, MS 2025, Idaho Falls, Idaho, ABSTRACT A new Lagrangan partcle model based on smoothed partcle hydrodynamcs was used to smulate pore scale precptaton reactons. The sde-by-sde njecton of reactng solutons nto two halves of a two-dmensonal granular porous medum was smulated. Precptaton on gran surfaces occurred along a narrow zone n the mddle of the doman, where the reactng solutes mxed to generate a supersaturated reacton product. The numercal smulatons qualtatvely reproduced the behavor observed n related laboratory experments. 1. INTRODUCTION It s generally recognzed that flud movement, mass transport and bogeochemcal transformatons that are observed at the feld scale are related to molecular-, cellular- and pore-scale processes for whch there s a growng body of scentfc knowledge. However, hgh-resoluton smulatons of such detaled nformaton cannot be carred out for complex, large-scale systems. The development of realstc and practcal feld-scale models wll depend on a better understandng of how ndvdual component processes are coupled, and better methods for passng nformaton from mcro-scale to contnuum models. Physcally based, broadly applcable computer models for mcro-scale multphase and mult-component flud flow, solute transport and the growth/dssoluton of sold phases n complex porous and fractured porous meda are needed to develop a better understandng of a wde range of bogeochemcal processes. In the past, grd-based (fnte elements or fnte dfference) and lattce Boltzmann methods have been used to smulate small-scale (pore scale or fracture aperture scale) reactve flow n porous meda. However, these methods are frequently plagued by artfcal numercal dffuson, volaton of mass conservaton or lack of Gallean nvarance, and they may suffer from numercal nstabltes for large dffuson coeffcents. The applcaton of grd-based models s also complcated by geometrcally complex boundares that change dynamcally as a result of precptaton, and the non-lnearty of the flow and reactve transport equatons. In ths paper, a Smoothed Partcle Hydrodynamcs (SPH) method s used to smulate pore-scale flow and reactve transport. The Lagrangan partcle nature of SPH allows physcal and chemcal effects to be modeled wth relatvely lttle code-development effort. In addton, geometrcally complex and/or dynamc boundares and nterfaces can be handled wthout undue dffculty. SPH was frst ntroduced by Lucy (1977) and Gngold and Monaghan (1977) to smulate flud dynamcs n the context of astrophyscal applcatons. SPH models have been successfully used to smulate a varety of subsurface process ncludng mcro-scale unsaturated (Tartakovsky and Meakn, 2005 a,b), saturated (Morrs et al., 1997, Zhu et al., 1

2 A. TARTAKOVSKY, T. SCHEIBE, G. REDDEN, P. MEAKIN, Y. FANG 1999) and multphase (Tartakovsky and Meakn, 2004, 2006) flows and non-reactve and reactve transport (Zhu and Fox, 2001, 2002, Tartakovsky and Meakn, 2005c, Tartakovsky et al., a, b) n fractured and porous meda. We extended the Tartakovsky et al. (a, b) model for precptaton from a supersaturated soluton to smulate flow, mxng and reacton of two solutons, each contanng sngle solutes (A and B) that react to form a product (C). The smulatons results are compared wth results of laboratory experment n whch Na 2 CO 3 and CaCl 2 solutons were njected separately at the bottom of the dfferent halves of a vertcal 2-dmensonal flow cell (Fgure 2). Both numercal smulatons and laboratory results show that precptaton occurs n a narrow zone along the nterface between the two solutons. 2. SPH TRANSPORT EQUATIONS There are a number of ways to represent the processes nvolved n precptaton that vary n terms of the level of detal and degree to whch they correspond to known molecular and partcle nteractons. In ths paper we have chosen to represent precptaton as heterogeneous growth of the reacton product (C) from a supersaturated soluton. Ths approach s based on the dea that, provdng the supersaturaton ndex does not become too large, homogeneous nucleaton wll not occur, and precptaton wll occur as overgrowth on pre-exstng mneral surfaces. Future work wll ncorporate and test more explct and detaled expressons for the knetcs of nucleaton and growth. Flud flow and solutes mxng and reacton of the type A+B=C and precptaton of C can be descrbed by a combnaton of the contnuty equaton, dρ / dt = ρ v, (1) the lnear momentum conservaton equaton, 2 ext dv/ dt = 1/ ρ P+ μ/ ρ v+ 1/ ρf, (2) and the dffuson equatons for speces A, B, and C (Chopard et al., 1994), A A 2 A AB A B dc / dt = D C k C C, (3) B B 2 B AB A B dc / dt = D C k C C, (4) C C 2 C AB A B dc / dt = D C + k C C, (5) where v s the flud velocty, P s the flud pressure, F ext represents body forces (such as gravty actng on the flud denstes), μ s the flud vscosty, ρ s the flud densty, C A, C B and C C are the concentratons of a solutes A, B and C (the concentraton s defned as the mass dssolved n a unt volume of flud), D A, D B and D C are the molecular dffuson coeffcents of the correspondng speces n the solvent and k AB s a rate coeffcent for A + B = C, normalzed wth respect to the equlbrum solublty product K sp. For smplcty t s assumed that precptaton/dssoluton of solute C s descrbed by a frst-order knetc-reacton model at the flud/sold nterface: D C C C n = k( C C C eq ), (6) where C eq s the concentraton of C n equlbrum wth sold C, n s the unt vector n the drecton normal to the nterface pontng toward the flud and k s the local precptaton rate coeffcent. The normal velocty, v n, at whch the sold surface advances at poston x s on the flud-sold nterface nto the lqud s gven by 2

3 CMWRXVI C C v ( x ) = D C ( x ) n / ρ, (7) n s s s where ρ s s the densty of the precptated sold phase. The SPH method s based on the dea that contnuous feld A(r) can be approxmated as AS() r = A / nw ( r r ) where W s the SPH weghtng functon, r s the poston of partcle, n = ρ /m s the partcle number densty, ρ and m are the flud densty and mass of partcle. The SPH approxmaton of contnuous felds allows the mass and momentum conservaton equatons to be wrtten n a form of a system of ordnary dfferental equatons (ODEs) (Tartakovsky et al., 2005c), n = W( rj r ), j flud + sold partcles (8) and dv dt j P P = + W r r m n n 1 j 2 2 j flud+ j sold ( ) j ( μ + μ )( ) ( r 2 rj) W( r rj, h) nn j( r rj) ext 1 j v v j F + +. flud partcles (9) m m j flud+ sold Both moble fluds and sold boundares are represented by partcles. Partcles representng solds are frozen n space but they enter nto the calculaton of forces actng on the flud partcles (eq. (9)). The veloctes of the partcles representng sold flled regons are set to zero and the number densty of the flud at the partcle locatons s found from equaton (8). Followng Tartakovsky et al. (a, b) the system of dffuson/reacton equatons (3)-(6) can be cast n the form of a system of ODEs: A A A A dc D ( n + nj)( C Cj ) AB A B = 2 ( j) W ( j, h) k C C dt r r r r (10) j flud nn r r dc dt and B dc dt C j flud j( j) ( + )( ) 2 ( nn j( r rj) ) (, ) B B B D n nj C Cj AB A B = r r j W r r j h k C C (11) j flud ( + j)( j ) ( r 2 rj) W( r rj, h) nn j( r rj) ( ) C C C D n n C C = + k C C R C C δ AB A B C eq k k sold, (12) where C I s the concentraton of the solute I at flud partcle, R s the effectve partcle flud-sold reacton rate constant, ndcates summaton over all the flud partcles, k sold j flud ndcates summaton over all the sold partcles and δ k = 1 f r r k d and zero otherwse. The effectve partcle flud-sold reacton rate s gven by k nt RdN, where N nt, 3

4 A. TARTAKOVSKY, T. SCHEIBE, G. REDDEN, P. MEAKIN, Y. FANG the average number of flud partcles that nteract wth each sold partcle, depends on the partcle number densty, d and, n general, the partcle radal dstrbuton functon, g(r). Precptaton and dssoluton are modeled by trackng the masses, m, of the sold partcles, whch change accordng to: dm 1 = R ( C j C eq) δj s. (13) dt j fn j Once the mass of a sold partcle reaches twce the orgnal mass, m 0, the nearest flud partcle precptates, becomng a new sold partcle, and the masses of both the new and old sold partcles are set to m 0. Smlarly, f the mass of a sold partcle reaches zero, the sold partcle becomes a new flud partcle. Snce the new flud partcles wll be very close to the sold boundary, where the flud velocty s very small, the ntal velocty of a new flud partcle s set to zero. 2.1 Implementaton of the SPH model. At each tme step n a smulaton, the partcle number denstes, n, at each of the partcles are calculated usng equaton (8) and the pressure at each partcle s obtaned usng the equaton of state P = P eq n /n eq. Then the rght hand sde of equaton (9) s evaluated and new concentratons are calculated. Partcle acceleratons, a = dv /dt, are found from equaton (9), and flud partcle veloctes, postons and concentratons are found by ntegratng the SPH equaton of moton usng the explct velocty Verlet algorthm (Allen and Tldesley, 2001). An M 6 splne functon (Schoenberg, 1946) was used for the SPH smoothng functon. FIGURE 1. Flow and precptaton for two Peclet numbers. 4

5 CMWRXVI 3. NUMERICAL RESULTS AND LABORATORY EXPERIMENT Fgure 1 shows smulatons of the mxng of two reactve fluds and mneral precptaton n the porous medum for two dfferent Peclet numbers. The solutes A (red partcles) and B (blue partcles) were njected at the same rate nto the rght and left parts of the computatonal doman. As the solutons mx, A and B react accordng to equatons (3)-(5) and produce C, whch precptates on mneral surfaces accordng to equaton (6). The black partcles represent the dstrbuton of dssolved C, the gray partcles represent mneral grans and the green partcles are precptated sold C. To ntalze the smulatons, partcles were placed randomly nto a box (n unts of h), and the SPH equaton (9), wth F ext = 0, and perodc boundary condtons n all drectons were used to brng the system nto an equlbrum state. The equlbrum partcle densty was n eq = 16 h -2 (16 partcles n an area of h 2 ). The coeffcent P eq n the equaton of state was P eq = 20 and the vscosty was μ = 1. After equlbrum was reached, the partcles at postons x covered by the dscs used to represent sold grans were frozen to form mpermeable boundares to the flow. After the flud partcles were dentfed, each flud partcle was assgned A and B concentraton of zero and a C concentraton of Ceq. A body force was then appled n the y drecton. No flow boundary condtons were mposed at the sol gran boundares and at the boundares of the computatonal doman n the x drecton, and perodc flow boundary condtons were used n the y drecton. Partcles extng the flow doman at y = 64 were returned nto the flow doman at x = 0. The concentratons C A =1, C B = 0, C C = C eq (C eq = 0.15) were assgned to the partcles enterng at y = 0 n the left part of doman and C A =0, C B = 1, C C = C eq n the rght part of doman. Mxng of the solutes A and B and subsequent formaton of the speces C occurs only n the narrow zone n the mddle of the doman. The wdth of ths zone s practcally ndependent of the Peclet number, Pe, but the concentraton of C ncreases wth ncreasng Pe. The mxng zone decreases wth tme as precptate reduces contact between solutes A and B. The flud velocty feld s not affected sgnfcantly by precptaton as precptaton occurs manly n a narrow zone n the drecton of flow. The numercal results qualtatvely agree wth results of laboratory experments. Two solutes (Na 2 CO 3 and CaCl 2 ) were njected at the bottom of the vertcal 2-dmensonal flow cell as shown on the Fgure 2. The flud cell was approxmately 60 cm hgh and 60 cm wde. FIGURE 2. Na 2 CO 3 and CaCl 2 solutons njected separately at the bottom of the dfferent halves of a vertcal 60cm x 60 cm flow cell. The whte lne n the mddle of flow cell s a zone of calcum carbonate precptaton. 5

6 A. TARTAKOVSKY, T. SCHEIBE, G. REDDEN, P. MEAKIN, Y. FANG The whte lne n the mddle of the flow cell s a zone of calcum carbonate precptaton produced by reacton of Na 2 CO 3 and CaCl 2. Despte the fact that equatons (3) (6) provde only a carcature of the calcte precptaton process (there s no ntermedate reacton product, C, n the expermental system), precptaton occurs only n the narrow zone along the nterface between the two solutons n both the experments and smulatons. On the other hand, Darcy scale smulatons of ths experment usng contnuum model code such as HYDROGEOCMEM (Yeh et al., 2004) predcted a sgnfcantly wder precptaton zone. The reason for the falure of the contnuum model s that mxng and precptaton n the experment s very localzed and produces hgh concentraton gradents n a regon that s only several sol grans wde. Ths may lead to the falure of advectve dsperson models based on the local averages. FIGURE 3. Rate of change of <C C > due to reacton between solutes A and B versus the product of the average concentratons <C A ><C B > for two Peclet numbers. FIGURE 4. Rate of change of <C C > due to precptaton as a functon of <C C > - C eq for two Peclet numbers. C Fgure 3 shows the rate of change of the average solute C concentraton, C due to reacton between solutes A and B versus the product of the average concentratons <C A ><C B > as a functon of Pe. The concentraton of -th solute averaged over the entre pore volume of I I I the computatonal doman, V, s gven by C = C ( x) dx / V C / N V f. It can be flud partcles seen that the reacton producng solute C s not lnear wth respect to <C A ><C B >. Ths fgure also shows that averagng of the concentratons over a volume (even one as small as that used n the SPH smulatons) may lead to erroneous results. At late tmes durng njecton, when the formaton of precptates sgnfcantly lmts drect contact and mxng between solutes A and B, the rate of producton of <C C > decreases wth ncreasng <C A ><C B >, whle the dc / dt lnearly ncreases wth ncreasng C A C B at the poston of partcle accordng to C equaton (5). Fgure 4 shows the rate of change of <C C > due to precptaton as a functon of <C C > - C eq for two Peclet numbers. Ths fgure ndcates that the effectve precptaton rate (slope of the curve) s defned n two dstnct branches. The lower branch corresponds to the frst part of the smulaton durng whch the solute concentraton ncreases as the mxng zone of solutes A and B develops. The upper branch corresponds to the second part of the smulatons durng whch pore fllng due to precptate formaton reduces mxng between A 6

7 CMWRXVI and B. The complex behavor ndcates that the effectve reacton rate coeffcent depends on the local hstory of the reactve transport process. It can be also seen that effectve precptaton rate ncreases wth ncreasng Peclet number. 4. CONCLUSIONS Laboratory experments were performed to nvestgate the njecton of reactve solutons nto dfferent halves of a quas two-dmensonal porous medum, and the experment was smulated usng a two-dmensonal model. Mxng of the solutes A and B and subsequent precptaton of the solute C was observed n a narrow zone n the mddle of doman. The wdth of ths zone was found to be practcally ndependent of the Peclet number, but the precptaton rate ncreased wth ncreasng Pe. The mxng zone decreased wth tme as precptated mnerals reduced contact between solutes A and B. The smulatons and experments lead to smlar results because n both precptaton occurs most rapdly n regons n whch the solute concentraton product C A C B s largest and the removal of A and B due to reacton or precptaton s essentally rreversble. The precptaton greatly effects mxng of the two solutes, and ths llustrates why the smple contnuum-scale model does not properly predct the wdth of the reacton zone. It s possble that, by usng a very hghly-resolved numercal grd n the mxng zone and ncorporatng the reactons n the contnuum model (ncludng changes n dffuson rates assocated wth precptaton), the contnuum model mght adequately predct the macroscopc precptaton features shown n Fgure 2. We are performng further numercal studes to test ths hypothess. As an alternatve, we are also explorng the ntegraton of poreand contnuum-scale models drectly usng a hybrd modelng approach. In ths approach, the pore-scale SPH model wll be used to smulate flow, transport, and reacton n the small porton of the expermental doman correspondng to the mxng zone, and the contnuumscale reactve transport model wll be used for the remander of the expermental doman (where local mxng and reactons are nsgnfcant). Ths approach has been successfully appled to other problems n whch hghly localzed processes manfest themselves n macroscopc phenomena n complex ways (for example, mcrofracture propagaton n materals scence). We beleve that, for porous meda problems that nvolve strong couplng between flow, transport, and reacton processes, a hybrd model approach wll not only have computatonal advantages, but wll also provde a more fundamentally sound representaton of the underlyng physcs and chemstry of the system. ACKNOWLEDGEMENTS Ths research was supported by the Laboratory Drected Research and Development program at the Pacfc Northwest Natonal Laboratory and the Envronmental Management Scence Program of the Offce of Scence, U.S. Department of Energy. The Pacfc Northwest Natonal Laboratory s operated for the U.S. Department of Energy by Battelle under Contract DE-AC06-76RL01830, and the Idaho Natonal Laboratory s operated for the U.S. Department of Energy by the Battelle Energy Allance under Contract DE-AC07-05ID

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