MODELING OF UBURFACE BIOBARRIER FORMATION B. Chen-Charpentier and 2 H.V. Kojouharov Departent o Matheatics, University o Wyoing, P.O. Box 3036, Laraie, WY 8207-3036; Phone: (307)766-422; Fax: (307)766-6838. 2 Departent o Matheatics, Arizona tate University, P.O. Box 804, Tepe, AZ 85287-804l; Phone: (480)965-3745; Fax:(480)965-89. ABTRACT Bioil-oring icrobes can or biobarriers to inhibit containant igration in groundwater. Also subsurace bioils have the potential or biotransoration o organic containants to less harul ors, thereby providing an in situ ethod or treatent o containated groundwater supplies. We present a atheatical and nuerical odel to describe the population distribution and growth o bacteria in porous edia. The odel is based on the convection-dispersion equation with nonlinear reaction ters. Accurate nuerical siulations are crucial to the developent o containant reediation strategies. We use the nonstandard nuerical approach that is based on nonlocal treatent o nonlinear reactions and odiied characteristic derivatives. It leads to signiicant, qualitative iproveents in the behavior o the nuerical solution. Nuerical results or a siple biobarrier oration odel are presented to deonstrate the perorance o the proposed new ethod. We show coparisons with experiental results obtained ro Montana tate s Center or Bioil Engineering. Key words: biobarriers, odels, bioils, siulations INTRODUCTION Controlling pollution in underground water is a very iportant and diicult proble. There are bacteria that will destroy any organic containants in subsurace regions. But or ost pollutants, including heavy etals, a ore proising concept is the creation o biobarriers or containent and reediation o containated soil and groundwater. Biobarriers are in situ barriers that are ored by stiulating the growth o bioil-oring icrobes that are already present or are introduced into the aquier. As the icrobial bioass increases it plugs the ree pore space low paths through porous edia, thereby reducing the hydraulic conductivity and ass transport properties (Cunningha et al., 99). By adequately choosing where to plug the porous ediu, it is possible to prevent the igration o groundwater containants ro hazardous waste sites. An even better scenario is to have biobarriers that will not only contain the containant plue but will also degrade it. Matheatical odels are needed to copleent experiental work into the use o bioils to or biobarriers. Matheatical odels help to explain the echaniss or low, solute transport, biological and cheical reactions, bioil accuulation, and natural biodegradation in porous edia. These generally lead to strongly coupled systes o nonlinear partial dierential equations that are diicult to solve. Analytical solutions or the ull, coupled probles are nonexistent, and nuerical ethods have probles such as instabilities and artiicial diusion. Here we use new ethods that are reliable, accurate, and eicient or the given odels. We apply the ethods to subsurace 228 Proceedings o the 2000 Conerence on Hazardous Waste Research
biobarrier oration. Without these ethods, results o nuerical siulations are o doubtul value. In this article, we use a new Eulerian- Lagrangian nuerical schee that eiciently handles the nuerically diicult convectiondoinated transport probles with nonlinear reaction ters. The convection-reaction part o a transport equation is approxiated using an exact tie-stepping schee. oe o the iportant eatures o the schee are the nonlocal odeling o nonlinear reactions and the ore sophisticated discretization o tie derivatives. It enables us to track sharp concentration ronts uch ore accurately than with standard nuerical schees. Having dealt with the ost diicult part o the proble, standard inite dierences are well suited or solving the reaining dispersion ter. This ethod was presented by Kojouharov and Chen (999). We copare the results obtained ro using our nuerical siulator with soe o the experiental results or short cores done by Cunningha et al. (99). The results copare very well, which is a very good validation o the odel. The siulator can now be used as a predictive tool to deterine values o paraeters which are diicult or ipossible to easure, and to help design experients, ield studies, and actual biobarriers. The outline o the paper is as ollows. In the next section, the governing syste o partial dierential equations is orulated or a twophase, two-species ixture. In section three, the non-standard nuerical ethod or solving the reactive solute transport proble in porous edia is given. To deonstrate the perorance o the proposed ethod o solution or the odel and the eectiveness o biobarriers or reducing the hydraulic conductivity, nuerical results and coparisons with experients are presented in section our. In the last section, a suary o results is presented. GOVERNING YTEM OF EQUATION Consider a three-phase ixture consisting o a liquid phase, a solid rock phase, and a bioil phase. Even though the bioil can be considered to be part o the solid phase, it is sipler to take it as a separate phase. The our olecular species present in the porous ediu are the bioil-oring icrobes, labeled M; the soluble containants or nutrients, labeled N; and the water and rock species. We assue that interactions in the syste occur only between the icrobial and nutrients species. Furtherore, we assue that the icrobes are iobile; i.e., they are attached to the rock as bioil. The undaental equation or transient groundwater low o constant density can be written in the ollowing or (Allen, 988): h h K =. (luid low) () t The single luid-low equation () arises ro the ass balance law h v + = t, (2) when we substitute or the speciic discharge vector v using the Darcy s law h v = K. (3) x h(x,t) denotes the hydraulic head; s is the Proceedings o the 2000 Conerence on Hazardous Waste Research 229
speciic storage; K is the saturated hydraulic conductivity; and (x,t) represent sources or sinks. The speciic discharge vector v(x,t), called Darcy velocity, represents the speed o the water. The transport and reaction o nutrients and the growth o icrobes are governed by a syste o partial dierential equations. ince the rock phase doesn t change, we assue that the solid rock atrix is stationary and that the diusion o icrobial and nutrient species in the solid phase is negligible. Thereore we can work only with the liquid and bioil phases as ollows: t ( φ ρ ) = r ( ρ, ρ ) M M M M N (icrobes) ρ N ( φnρn) + ( vρn) DN = rn( ρm, ρn). t (nutrients) (4) Here ρ i, i=m,n, represents the intrinsic ass density o icrobes and nutrients, respectively. For a single-luid low, the quantity N is the volue raction occupied by the liquid divided by the volue o the liquid plus the volue o the bioil; M is the volue o the bioil again divided by the volue o the liquid and bioil; D N (x,t) is the hydrodynaic dispersion coeicient or the nutrients; and r i represents the total rate at which species i is produced via reactions and sources. The growth rates are usually written in ters o concentrations and we will do that too. The icrobial death rate is assued to be proportional to the size o the bioil population. The rate o bioil growth is given by the ollowing Monod kinetics reactions: µ MA µ ( ) =, K + where µ MA is the axiu speciic growth rate; and K s is that value o the concentration o nutrients (ass o nutrients per unit o liquid volue) where the speciic growth rate µ () has hal its axiu value (Bailey and Ollis, 986). We assue that only the growth and accuulation o bioil in the pore spaces cause changes in the porous edia properties. Let be the current bioil concentration; then % = ρ M is the noralized bioil concentration. It ollows that the change in porosity, or sall initial bioil concentrations, is given by where φ 0 is the clean surace porosity. For the saturated hydraulic conductivity K we assue the ollowing or: K = K % nk 0 ( ) φ = φ, 0 % (5) (6) where K 0 is the initial hydraulic conductivity, and n k is an experientally deterined paraeter which takes values around 3 (Cleent et al., 996). For siplicity, ro now on we will drop the tilde ro the noralized bioil concentration. We assue there are no sources and sinks or the luid; thereore, =0. Also we are odeling very short cores with unior bioil distribution so we can take the velocity to be independent o x. Invoking all sipliying assuptions to equations (4) and using concentrations as the 230 Proceedings o the 2000 Conerence on Hazardous Waste Research
unknowns gives the ollowing inal or o the governing syste o dierential equations: h K( ) = 0, µ ax (7) ( ) = kr, t Ks+ N µ ax + DN =, t x Y Ks ( v) + where k r is the irst-order endogenous decay rate, and Y is the yield rate coeicient. NUMERICAL METHOD FOR BIOFILM GROWTH Equation (7) represents a coupled syste o nonlinear, tie-dependent partial dierential equations that is very diicult to solve nuerically. A key objective o the nuerical siulation is to develop tie-stepping procedures that are accurate and coputationally stable. Dierent tie-stepping ideas can be applied to solve the governing syste o equations (Russell and Wheeler, 983). One possible tie-stepping approach is the sequential solution technique. The sequential ethod irst solves iplicitly or the Darcy velocity v at the current tie level by solving the luid low equation (). Then the species transport syste (4) is solved iplicitly or the concentrations and in a decoupled ashion (Ewing and Russell, 982). New values o porosity and pereability are then calculated and the cycle is repeated by calculating the new velocities. For the solution o the ordinary dierential equation odeling the low, we use a standard inite-dierence ethod to calculate h. Then we nuerically dierentiate using h v= K, (8) x to get the velocity ield. Consider the ollowing reaction-diusionadvection equation governing reactive species transport in porous edia: c c c + v D = R c t (). Here, c is the species concentration; v is the velocity; and D is the hydrodynaic dispersion tensor. ince v is independent o x, the nonlinear reaction ter R(c) has the expression () r(), c R c (9) = (0) where r(c) represents the total rate at which the species is produced via reactions and sources. Unortunately, there are only ew cases or which analytic solutions to the solute transport equation (9) exist. The or o equation (9) ranges ro parabolic to alost hyperbolic, depending on the ratio o convection to dispersion. While classical nuerical techniques, such as the standard inite-dierences or Galerkin inite-eleents, work well or probles o solute transport that are doinated by dispersive oveent, they suer ro severe nonphysical oscillations and excessive nuerical dispersion when convection doinates the dispersive eects. olutions o hyperbolic-type equations can be represented ro the initial data propagating over characteristic paths in the surace and can be viewed as dispersing away ro these paths, along which the concentration c is a sooth unction (Douglas and Russell, 982). Thereore, it is logical to design nuerical Proceedings o the 2000 Conerence on Hazardous Waste Research 23
procedures that recognize the hyperbolic nature o the convection-doinated solute transport probles, such as the Eulerian-Lagrangian ethods. In recent years, any such schees have been developed but still little has been done to iprove the nuerical solutions o probles in which nonlinear reactions are present. Nonlinear reaction ters play a signiicant role in applications involving bacterial growth and containant biodegradation in subsurace regions. Reerring to Kojouharov and Chen (999), we proposed a new Eulerian- Lagrangian nuerical ethod or solving the reactive solute transport equation (9). The nuerical solution o the convection-reaction part is deined using an exact tie-stepping schee. This enables us to ollow the transport and track sharp ronts uch ore accurately than with the standard nuerical schees. Having dealt with the ost diicult part o the transport proble (9), only the soothing property o the dispersion ter reains. Then, standard inite dierences or inite eleents are well suited or solving the dispersion part. We now apply the new ethod to the ollowing dispersion-ree syste o dierential equations: µ ax = k r t Ks + µ ax + v = t x Y K + s The icrobes equation is a linear irst order ordinary dierential equation whose exact solution is given by, (icrobes) (). (nutrients) where The exact tie-stepping schee or solving the nutrients transport equation ro syste () is given by the expression + x + x ln, (3) t t x + µ ax ( x ) where λ =, and the back- Y track point x x + = λ t e λ λ µ ( x) ax λ = ( Ks + x ) ( x ) k. K s = λ has the expression or constant in space, tie-dependent velocity ields: v(t)=p n- (t). We now add a diusion ter to the transport equation or the ollowing nutrients: µ ax + v DM =. t x Y Ks + (4) Applying the exact tie-stepping schee (3) to equation (4) yields the ollowing iplicit in nature, sei-discrete procedure: To coplete the construction o the new Eulerian-Lagrangian ethod or solving equation (4), we need to introduce an approxiation r, (2) x ( ) n x = x Pn + t P t, + + x x x + DM = t ( x) K + s λ ln. t ( x ) (5) 232 Proceedings o the 2000 Conerence on Hazardous Waste Research
technique or discretizing the spatial derivatives involved in the dispersion ter. Let us consider the ollowing centered, weighted second dierence approxiation (Huyakorn and Pinder, 983): where is the hydrodynaic dispersion coeicient located at the center o a space increent, and x is the spatial grid size. Cobining the sei-discrete procedure (5) with the above spatial approxiation o the dispersion ter yields the non-standard dierence ethod or solving equation (4): where ( x) + + + + DM δx( DN δx ) = i (6) D D ( + ) ( ) + + + + + + M i i M i i i+ i 2 2, 2 x + xi + xi+ DM = D, M + t i+ 2 2 + i xi + + δx( DN δx ) = t i K + s i λi ln, t ( xi ) λ i µ = ax and the backtrack point Y + i x i (7) (8) has the expression ( ) xi = xi Pn + t Pn t. Reark.- In general, the backtrack point x i does not lie at a grid point. I the approxiate solution is being deterined by a initedierence procedure, the convective concentra- tion x i ust be evaluated by an interpolation o the approxiate solution values { i }at the grid points x i. NUMERICAL REULT We now turn to a set o nuerical experients to deonstrate the perorance o the proposed new ethod and the eectiveness o icrobial barriers or reducing the hydraulic conductivity property o porous edia. The governing syste o equations exained here has the ollowing or: where h h K = 0, v= K µ ax = G ( ) k r t Ks + 2 + v DN = 2 t µ ax G ( ), YK + s ( ) G =, + γ with γ typically sall, is introduced to restrict the growth o the icrobes as the pores are being plugged. h is the hydraulic head; is the noralized bioil concentration; and is the nutrients concentration. The non-diensional spatial doain considered here is Ω= [0,]. Assuptions ade in the above atheatical odel (20) are that all bacteria are attached to the solid rock surace as a part o the bioil structure and that the concentration o nutrients present in the solid phase is negligible. Changes in the hydraulic conductivity K, (luid low) (icrobes) (nutrients) (9) Proceedings o the 2000 Conerence on Hazardous Waste Research 233
are caused by the accuulation o solid-phase bioass in the pore spaces. We assue a piecewise steady-state luid low, due to the relatively slow changes in the porous edia properties (Cunningha et al., 99). The bioil concentration-porosity relation used is ( ) φ = φ. 0 (20) This orula is valid or sall initial bioil concentrations which is the case in the exaples that we do. The conductivity-reduction relationship exained here is given by the ollowing expression: 3 K = K. 0 (2) We siulate two o the experients done by Cunningha et al. (99): the ones or.70 and.54 sands. The experiental values are as ollows: For.70 sand: pereability k =3.9x0-6 c 2, i.e., hydraulic conductivity K =.2404 c/s or water at 5 C. For.54 sand: k =2.7x0-6 c 2, i.e., K=.635 c/s. For both sands, the initial porosity is φ 0 =0.35 and the reactor s length is 5 c, which was scaled to or the calculations. The boundary and initial conditions considered in the odel in agreeent with Cunningha et al. (99) are as ollows: (0,t) = 0 = 25 g/l (the nuerical results were scaled by a actor o 50, s.t. 0 =0.5 or calculation and graphing purposes), h(0,t)=0.5 c, h(,t)=0 c, i.e., head gradient =0.5 c/c, (x,0)= 0.5, and (x,0) = 0.02. The ollowing reaction paraeters are taken ro Taylor and Jae (990): µ ax = 0.00004/s, K s =0.799 g/l and Y=0.0975. Other paraeters used are as ollows: k r =0.000076/s, which was obtained ro calibrating the odel, and D N =0.0005 c 2 /s (The odel has a very low sensitivity to this paraeter.). The igures present the results o our calculation, together with soe o the experiental values shown in Figures 5 and 8 ro Cunningha et al. (99). We use concentrations instead o bioil thickness since we cannot calculate the thickness without aking assuptions on the distribution o icrobes. But, it is reasonable to assue that there is a linear relation between bioil thickness and icrobial concentration. Figure shows the variation o the noralized porosity with the noralized bioil concentration. ax is the axiu value o the icrobial concentration, and the and * sybols represent soe experiental results. Figure 2 is a plot o the pereability decrease and the increase in the icrobial concentration with tie. In our results,the noralized bioass goes to in about 2 days, the sae tie it takes the noralized pereability to decrease to about 0.. In Figure 5 (Cunningha et al., 99), the pereability also decays to 0. in about 2 days, but the noralized bioil thickness takes about 6 days to tend to. The dierence is due to the averag- 234 Proceedings o the 2000 Conerence on Hazardous Waste Research
Figure. Variation in siulated noralized edia porosity with noralized bioil concentration. The triangles and the stars represent the experiental values or.70 and.54 sands, respectively, ro Cunningha et al., 99. ing o the bioil thickness done by Cunningha et al. (99), where the doinant coponent is or glass spheres (which we are not odeling). Figure 3 shows the growth o bioass together with the decrease in nutrients. The aount o bioass reaches a axiu steady state at about 2 days, which coincides with the tie it takes or the nutrients to reach their iniu. Figure 2. Noralized porous edia pereability decrease corresponding to increased noralized icrobial concentration versus tie. The icrobial concentration curve is the average or both types o sand. The triangles and the stars represent experiental pereability values or.70 sands and.54 sands, respectively, ro Cunningha et al., 99. CONCLUION A new class o nuerical ethods has been developed or solving one-diensional, transient convective-dispersive transport equations with nonlinear reactions. Large tie steps can be taken without aecting the accuracy o the nuerical solution. The appropriate tie-step size or a particular odel proble can be deterined by physical considerations, rather than stability, convergence, or consistency reasons. The proposed new ethods have been successully applied to biobarrier oration odels incorporating Monod kinetics reactions. Nuerical results conired the theoretical and experiental predictions that icrobial barriers are eective or anipulating the porous edia properties in general, and or reducing the hydraulic conductivity in particular. The agreeent is very good and shows that the odel can reproduce experiental results and in the uture be used as a predictive tool. However, the curves in Figure are closer together than the corresponding experiental ones. One reason is that we are plotting bioass concentrations instead o bioil thickness. Another possible reason is that we took all the bacteria to be in bioil or with no signiicant detachent, so all the bioass reduces the porosity and pere- Proceedings o the 2000 Conerence on Hazardous Waste Research 235
REFERENCE Allen, M. B., 988. Basic Mechanics o Oil Reservoir Flows, Multiphase Flow in Porous Media, M. B. Allen III, G. A. Behie, and J. A. Trangenstein, Lecture Notes in Engineering 34, C. A. Brebia and. A. Orszag, Eds., pringer- Verlag, New York, pp. -8. Bailey, J. E., and D. F. Ollis, 986. Biocheical Engineering Fundaentals, McGraw-Hill, Inc., NY. Figure 3. Increase in icrobial concentration and decrease o noralized nutrient concentration with tie. ability. In practice there is detachent and the ree- loating icrobes will not change the physical properties o the ediu. Also, or the.54 sand, the pore channels are saller and the velocities higher, which would increase the detachent in this case and add to the separation o the curves. ACKNOWLEDGMENT The Departent o Matheatics, University o Wyoing, and the Great Plains/Rocky Mountain Hazardous ubstance Research Center are acknowledged or inancial support. Although this article has been unded in part by the U.. Environental Protection Agency under assistance agreeent R-89653, through the Great Plains/Rocky Mountain Hazardous ubstance Research Center, it has not been subjected to the agency s peer and adinistrative review and, thereore, ay not necessarily relect the views o the agency. No oicial endorseent should be inerred. Characklis, W. G., and K. C. Marshall, 990. Bioils, John Wiley and ons, Inc., NY. Cleent, T. P., B.. Hooker, and R.. keen, 996. Microscopic Models or Predicting Changes in aturated Porous Media Properties Caused by Microbial Growth, Ground Water, 34:5, pp. 934-942. Cunningha, A. B., W. G. Characklis, F. Abedeen, and D. Craword, 99. Inluence o the Bioil Accuulation on Porous Media Hydrodynaics, Environ. ci. Technol., 25:7, pp. 305-3. Douglas, J. Jr., and T. F. Russell, 982. Nuerical Methods or Convection-Doinated Diusion Probles Based on Cobining the Method o Characteristics with Finite Eleent or Finite Dierence Procedures, IAM J. Nuer. Anal., 9, pp. 87-885. Ewing, R. E., and T. F. Russell, 982. Eicient Tie-tepping Methods or Miscible Displaceent Probles in Porous Media, IAM J. Nuer. Anal., 9, pp. -66. 236 Proceedings o the 2000 Conerence on Hazardous Waste Research
Huyakorn, P.., and G. F. Pinder, 983. Coputational Methods in ubsurace Flow, Acadeic Press, NY. Kojouharov, H. V., and B. M. Chen, 999. Non-tandard Methods or the Convective-Dispersive Transport Equation with Nonlinear Reactions, Nuer. Methods Partial Dierential Equations, 5, pp. 67-624. Russell, T. F., and M. F. Wheeler, 983. Finite Eleent and Finite Dierence Methods or Continuous Flows in Porous Media, Frontiers in Applied Matheatics, Vol. : The Matheatics o Reservoir iulation, R.E. Ewing, Ed., IAM, Philadelphia, pp. 35-06. Taylor,.W., and P.R. Jae, 990. ubstrate and Bioass Transport in a Porous Mediu, Water Resources Research, 26, pp. 28-294. Proceedings o the 2000 Conerence on Hazardous Waste Research 237