transport of bacteria in porous media: ii. a model for convective transport and growth

Transport of Bacteria in Porous Media: II. A Model for Convective Transport and Growth

A. K. Sarkar,'-* George Georgiou?t and Mukul M. Sharma',*t 'Departments of Petroleum Engineering and 2Chemical Engineering, The University of Texas at Austin, Austin, TX 78712

Received December 13, 1993iAccepted March 10, 1994

A model is presented for the coupled processes of bacterial growth and convective transport in porous media. The retention and transport of bacteria has been modeled using a fractional flow approach. The various mechanisms of bacteria retention can be incorporated into the model through selection of an appropriate shape of the fractional flow curve. Permeability reduction due to pore plugging by bacteria was simulated using the effective medium theory. In porous media, the rates of transport and growth of bacteria, the generation of metabolic products, and the consumption of nutrients are strongly coupled processes. Consequently, the set of governing conservation equations form a set of coupled, nonlinear partial differential equations that were solved numeri- cally. Reasonably good agreement between the model and experimental data has been obtained indicating that the physical processes incorporated in the model are ad- equate. The model has been used to predict the in situ transport and growth of bacteria, nutrient consumption, and metabolite production. It can be particularly useful in simulating laboratory experiments and in scaling microbial-enhanced oil recovery or bioremediation processes to the field. 0 1994 John Wiley & Sons, Inc. Key words: bacterial transport porous media

INTRODUCTION

The transport and growth of bacteria in porous media are essential to the implementation of bacterial/biochemical processes such as microbial enhanced oil recovery (MEOR) and bioremediation of contaminants in groundwater. The objective of this work is to develop a model for the transport and growth of bacteria in porous media that may be used for the analysis of laboratory results and for scaling these results to the field.

The transport of bacteria depends on the ratio of the cell size to the average pore throat size, the shape of the microorganisms, and the production of extracellular polymers on the bacterial In the accompanying article, Sarkar et al. l 5 have demonstrated experimentally that the flow velocity, injected concentration, presence of a residual phase, and the state of dispersion of the cell suspension (which is controlled by the concentration of dispersant, ionic strength, and temperature) are major factors affecting

* Present address: NIPER, Bartlesville, OK. t To whom all correspondence should be addressed.

the convective transport of bacteria in porous media. The retention of bacteria and permeability reduction were observed to be significantly higher in the first few centimeters (-2.5 cm) of the porous medium as compared with the downstream sections. Based on heuristic arguments, it was concluded that size exclusion, surface adhesion, bridging, and hydrodynamic exclusion are the major mechanisms for cell retention and permeability reduction. l7

In addition to transport, the distribution of bacteria within a porous reservoir is also determined by in situ cell growth, diffusion, and chemotaxis. The increase in cell mass in situ depends on the intrinsic kinetics of cell multiplication, which can be inferred, or at least approximated, by fermentation studies, as well as concentration gradients which de- termine the local concentration of nutrients and inhibitory metabolites. Thus, a model for transport and growth of bacteria must account for the mechanisms of bacteria retention, describe the kinetics of growth, and take into account the distribution of nutrients and metabolic products in porous media.

In earlier work, transport of bacteria has been modeled as a rate process, using expressions for rates of clogging and declogging or rates of deposition and release that do not account for the various mechanisms listed above.577 In a review of problems with using existing transport models to describe microbial propagation in porous media, Brown et al.4 pointed out the need for treating microorganisms as colloids rather than as molecular species in solution. The lack of sufficient data to validate existing models was also noted. Depending on the application, appropriate bioremediation/biorestoration mechanisms or microbial-enhanced oil recovery mechanism^'^ need to be included in the model for simulating a particular bacterial process.

The model described here has been used for simulating laboratory experiments on the transport and growth of bacteria in sandpacks. Experiments showing the effects of velocity, injected concentration, and the presence of two phases on transport of bacteria have been simulated, and the results on effluent bacteria concentration histories and permeability histories have been compared with experimental results. Reasonable agreement was obtained by adjusting the transport and effective radius parameters. The growth of bacteria in the sandpack was also simulated and the results of effluent glucose concentration histories and permeability

Biotechnology and Bioengineering, Vol. 44, Pp. 499-508 (1994) 0 1994 John Wiley & Sons, Inc. CCC 0006-3592/94/040499-10

profiles after static growth were successfully matched with experimental results by adjusting the growth parameters.

MODEL FORMULATION

The basic equations governing the transport of water, bacteria, nutrients, and metabolites are species mass conservation equations. In our formulation (n, - l) , component mass balance equations and one overall mass conservation equation are used. In addition, specific models for bacterial transport, growth, and permeability reduction need to be developed. Constitutive relations for rock properties, as well as initial and boundary conditions have to be specified to complete the mathematical description.

The following assumptions have to be made in order to make the analysis tractable:

1. Only two incompressible fluid phases are present. 2. Precipitation/dissolution, cation exchange, or other

chemical reactions are not taken into consideration. 3. Bacteria, nutrient, and metabolites do not partition into

the nonaqueous fluid phase. 4. Pressure, volume, and temperature changes from chem-

ical reactions, bacterial growth, and mixing are negligi- bly small.

Material Conservation Equations

The differential form of the material balance equations for isothermal, multicomponent , multiphase transport of chemical/biochemical species in permeable media can be written as9

where W,, Nk, and R, are the overall concentration, net flux, and source terms, respectively, of a component k .

The overall concentration of component k ( w k ) may be expressed as:

np

1= 1

where n, is the total number of components and np is the total number of phases. Ckl refers to the volumetric concentration of component k in phase I , and t k refers to the volumetric concentration of component k adsorbed on the rock surfaces based on the original pore volume. $ is the porosity of the medium, S1 is the saturation of phase 1 , and Pk denotes the density of component k.

The flux term (N,) consists of convective, dispersive, and coupled dispersive fluxes. It may be expressed as:

(3 )

where Dk, and D ' , are hydrodynamic and coupled hydrodynamic dispersion coefficients, respectively. Subscript k' refers to the indices for the component with which component k is coupled. For example, in MEOR processes, there exist diffusive fluxes in addition to the molecular diffusive flux. Bacteria have a chemotactic diffusive flux driven by a nutrient concentration gradient (from a lower nutrient concentration toward a higher nutrient concentration) and a tumbling diffusive flux driven by the bacteria concentration itself (from a higher concentration of bacteria toward a lower concentration of bacteria).

The source terms are a combination of all rate terms for a particular component, e.g., injection, production, growth, and lysis (of bacteria), and may be expressed as:

where Qk indicates rate of injection or production and V, is the bulk volume. &I, Rlkl are the specific rate constant and the coupled specific rate constant of component k in phase one. First-order rate expressions have been used here for reaction rate terms. For example, in the case of the growth- limiting nutrient, the coupled rate coefficient indicates that the rate of consumption is dependent upon the concentration of live bacteria and the concentration of inhibitory metabolite and, in the case of metabolites, the rate of generation depends on the concentration of bacteria and the nutrient. These specific rate terms are discussed in detail later.

If the component density ( P k ) is assumed to be constant, then the final form of the material balance equation is:

The continuity equation describing the conservation of total mass can be obtained by summing the above material conservation equations over all components. Combining the continuity equation, Darcy's law, and the capillary pressure relation, and by neglecting gravity, the final form of the pressure equation in terms of the aqueous phase pressure ( P l ) , can be obtained as:

500 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 44, NO. 4, AUGUST 5, 1994

Details of this derivation are provided by Sarkar.17 Here K is the single phase permeability and h, is the relative mobility of phase I . Other terms are defined in the Nomencla- ture section.

Transport and Retention of Bacteria

The modeling of bacterial retention by the porous medium is crucial to the simulation of bacteria transport processes. Previously, several approaches have been employed to model the retention of colloidal particles: (i) empirical filtration coefficient models; (ii) trajectory analysis’*; and (iii) network models.’* In general, it is desirable to simulate this process as a transient one in which the retention of bacteria causes a change in the pore structure and local flow field resulting in a continuously changing (in time and space coordinates) local filtration coefficient. This, however, is tedious and impractical, especially when coupled with the problems of cell growth and metabolism.

We propose a fractional flow approach to modeling bacterial transport that considerably simplifies the conservation equations. A fractional flow function is defined that relates the local flowing cell concentration to the total (flowing + entrapped) cell concentration. It is assumed that local equi- librium is established and that there is a unique relationship between the concentration of flowing and trapped cells. The latter assumption implies that the rate of cell capture is much larger than the rate of convection. This is a reasonable approximation18 and it provides tremendous savings in mathematical and computational complexity of describing cell transport and capture. The shape of the fractional flow curve will depend on the mechanisms of bacterial retention. For example, a particular form of the fractional flow function can be derived for cases in which size exclusion is the only retention mechanism, as is the case for the flow of dilute suspensions when the particle size is comparable with the pore throat size. For the case of retention by size exclusion, the relation between the flowing bacteria concentration and the total bacteria concentration is linear above a threshold total bacteria concentration, C* . When the total bacteria concentration is lower than C* then all cells are retained and thus the concentration of flowing bacteria is zero. All cells in excess of C* flow because no more small pores are available to entrap them. This is, of course, an overly simplistic model for cell retention. In general, cells will partially plug pores that are both smaller and larger than cells and they will continue to do so until bridging causes an internal or external filter cake to form. In such situations, the fractional flow function will deviate from the above behavior. The higher the extent of bridging the lower the concentration of flowing bacteria. Eventually, the total bacteria concentration will be so high as to cause the flowing concentration to go to zero resulting in complete trapping. Similar nonlinear effects are observed in traffic flow the- ~ r y ’ ~ where the flux of automobiles increases and then de- creases as the traffic density increases.

A simple fractional flow curve was chosen to simulate the

experiments and illustrate the use of this method. The equation used is,

(7)

where

Cf c, = - CT

The trapped bacterial concentration (tk) is

Cf is the flowing bacterial concentration and CT is the total bacteria concentration, whereas Cm and C,, are the respec- tive dimensionless concentrations. The retention parameters, A, B, and C*, need to be determined experimentally.

The use of a fractional flow function is clearly an approximation. There are many situations, such as deep bed filtration, where this may not be a good assumption. In these cases, particle retention occurs by surface adhesion and bridging. The concentration of trapped particles increases in a complex manner with time and it is not possible to relate the flowing concentration of colloids with the trapped concentration. In the case of bacteria, additional terms for rates of bacteria trapping and entrainment need to be used in the conservation equations.

The transport of bacteria is very sensitive to the cell size. Thus, the shape of the fractional flow curve must depend on the “effective” bacterial size in suspension. The effective size does not necessarily correspond to the dimensions of a single cell, because a variety of aggregation processes can result in cell clustering. There are two approaches that can be adopted for obtaining the effective coagulated size of the cells. The state of coagulation of the suspension can be obtained by computing the potential energy of interaction between the individual bacteria, and hence, the collision efficiencies. However, this requires an involved computa- tion. A simpler empirical approach was used instead, where the bacterial floc size for a given fluid medium is assumed to depend on the suspended bacterial concentration and the fluid velocity. The effective radius of bacteria in the flowing suspension (rs) is represented as a function of the size of a single cell (rsi), using an empirical relation of the form,

where

rsD = r,/rsi (9)

CD = log(cf/cCC) (10)

where C,, is the critical coagulation concentration; V is the

SARKAR, GEORGIOU, AND SHARMA: BACTERIAL TRANSPORT IN POROUS MEDIA. I1 50 1

fluid velocity; and e, f, and F are empirical constants. The critical coagulation concentration is expected to be a function of ionic strength, temperature, and bacterial type. The empiricism introduced through the relations above is far from ideal and a more exact relationship based on the potential energy of coagulation needs to be developed in the future.

Permeability Reduction Model

The permeable medium is represented as a three-dimensional network of pore throats and pore bodies. l8 As bacteria plug flow channels, the conductance distribution, i.e., the pore size distribution of the network, changes. The effective medium theory (EMT) can be used to estimate the permeability reduction. EMT provides an approximate method for calculating the effective hydraulic conductance (g,) if the conductance distribution Lf(g)], i.e., the pore throat size distribution in a network is known.

where

Z a = - - l 2

g is the individual pore throat conductance and z is the coordination number of the network, i.e., the number of pore throats that join at each interior pore body. The conductance (g) may be expressed as a function of pore throat radius (r) as

gar" 3 c n c 4 (1 1)

The above equations can be combined to obtain"

whereflr) is the size distribution of open pores. This size distribution changes as bacteria are trapped and pores are plugged. As the local pore size distribution evolves in time so does the local permeability. The overall permeability is computed as the harmonic mean of the local permeabilities.

To illustrate the model a Rayleigh pore size distribution was chosen:

fir) = 20!re@ r 2 0, (13)

where a is a parameter in the distribution and the mean (M) is given by,

The ratio of the permeability of the damaged porous me-

dium (KO) to the initial permeability (KJ can be expressed as a function of the ratio of the corresponding mean con- ductances.

(15) KD gmD

Ki gmi Permeability ratio = - = -

where gmj and g, are the mean initial and damaged con- ductances for the porous medium, respectively.

Derivation of Specific Rate Terms

The kinetics of bacterial growth, consumption of nutrient, and generation of metabolites needs to be specified in the material conservation equations. For bacteria, the specific rate constant (Rbl), considering growth and lysis processes, can be written as2

Rkl = Rbl = P - klys, (16)

For Bacillus licheniformis JF-2, the microorganism used in the experimental part of this set of papers, it was found that the dependence of the specific growth rate, p,, on the concentration of the limiting nutrient does not follow the well known Monod equation.6 Instead, the growth data can be fit to the semi-empirical Contois equation2:

(17) Pmax

1 + D - cbl

Cn

P =

where cbl and C, are concentrations of bacteria and nutrient, respectively. pmax and D are empirical constants that were determined experimentally. In this case, we used the growth rate equation for anaerobic conditions because the cells within the porous medium are deprived of oxygen. A typical value for the specific lysis rate for bacteria under nutrient limited conditions is klys = 0.01 h-'.5

The specific growth rate can be affected by inhibitory metabolites, such as alcohols or acids, which become important toward the end of the growth process when their concentration become significant.' In the case of B. licheniformis JF-2, grown under anaerobic conditions, lactic acid and 2,3-butanedibl accumulate in the fermentation broth, particularly during the late-exponential phase. '' High concentrations of either metabolic product can interfere with growth. This phenomenon is likely to be accentuated in the stagnant conditions within the pores of the sand bed. In the model, growth inhibition due to metabolite accumulation is mathematically expressed by saturation kinetia2

kmax kin

where kin is an experimentally determined constant and C j is the concentration of inhibitory metabolite.

The rate of consumption of the limiting nutrient (glucose) can be written as:

502 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 44, NO. 4, AUGUST 5,1994

1 Rlkl = R', = - p + rn

Y B

where Y, and rn are the yield and maintenance coefficients of bacteria, respectively.

The generation of metabolic products has been tradition- ally modeled using the Leudeking-Piret equation,* and the specific rate for metabolite production can be written as

The above equation can, in principle, be used to obtain the rate of production of several metabolic products, such as alcohols and organic acids, either of which can be growth inhibitory. Also, the production of carbohydrate polymers, which affect cell adhesion, and of surfactants that lower the capillary number can be modeled in the same way (Table I). In most cases, to simplify the problem, only the primary metabolic product of interest needs to be modeled. With the exception of the growth equation, which is particularly suit- able for anaerobic cultures of B. licheniforrnis JF-2, the above crcpressions provide a framework within which the growth kinetics of any microorganisms can be incorporated for use with our model. An implicit assumption in this approach is that the kinetics of microbial growth in suspended cultures are a good approximation of in situ microbial processes.

Constitutive Relations and Phase Behavior

Modeling the petrophysical properties of the rock and the fluid phase behavior is a very important aspect of this model. The relative permeability for each phase and the capillary pressure need to be specified as a function of the wetting phase saturation. This is done using Corey-type relations.' In addition to the constitutive relations, con- straints are placed on the saturations, phase concentrations, and total concentrations (to ensure that their sum is equal to unity ). We have taken a simple approach to phase behavior by assuming that the oleic phase consists of only the oil component, and that the bacteria, the nutrient, and the pro- duced metabolites remain in the aqueous phase. This assumption is valid for nearly all bacteria species and for polar metabolites.

Initial and Boundary Conditions

It is necessary to specify initial conditions for (n, - 1) overall component concentrations for the conservation equations and an aqueous phase pressure value for the pressure equation. The boundary conditions for no flow, inflow, and outflow boundaries have been treated as described earlier by Lake et al." and are listed in Sarkar.I7 The set of equations developed above completely describes the multicomponent, multiphase flow of bacteria and the associated components in porous media.

Numerical Solution Procedure

The mathematical model developed above is solved numer- ically using a finite difference technique. The finite difference formulation of the species conservation equation and overall material balance equation (pressure equation), treat- ment of initial and boundary conditions, and handling of different rate terms are discussed in detail by Sarkar17, and Pope et al.13 The model problem has been solved for a one-dimensional medium using a block-centered type of grid. An IMPES (implicit pressure and explicit saturation) technique is used to solve one pressure equation and (n, - 1) species conservation equations. A bandsolver has been used to solve the tridiagonal pressure matrix.

The model can be used for simulating injection of different quanti and compositions of fluids in the presence of one or more phases. Profiles (concentrations as a function of distance) and histories (concentration as a function of time) can be obtained for any component or phase.

RESULTS AND DISCUSSION

Up to 10 components in two phases have been considered in our simulations. The model can be easily adapted for additional components and phases, as needed. Sensitivity study of the model parameters showed that the shape of the fractional flow curve was critical in determining the location and concentration of cells in situ. The kinetic parameters for the microorganisms were found to be the next most important factor. As discussed later, the measurement of kinetic parameters in situ may be critical in adequately modeling bacterial processes in porous media.

Simulation of Convective Transport Followed by In Situ Growth of B. licheniformis JF-2

Simulations of bacteria transport at low Damkohler num- bers clearly demonstrated the influence of the fractional flow curve on the effluent and in situ cell concentrations. A sample result from the simulations for various values of A and B [parameters in the fractional flow relation, Eq. (7)] is shown in Figure 1. The bacteria concentration is made dimensionless with respect to the injected concentration. These results and other simulations show that the time re- quired for cells to first appear in the effluent depends largely on C* and, to a lesser extent, on A and B. The shape of the curve [effluent bacteria concentration vs. time or pore volumes (PV) injected] beyond this time is controlled entirely by A and B in Eq. (7), i.e., the shape of the fractional flow curve. Analytical solutions to some simple cases (NDA = 0, i.e., no cell growth or cell lysis) can be obtained using the method of characteristics in a manner similar to the traffic flow problem. l4

More complicated results are obtained when the rates of convection and cell metabolism are comparable. Another interesting limiting case is that of infinite Damkohler number, i.e., zero fluid velocity. The results of these "incuba-

SARKAR, GEORGIOU, AND SHARMA: BACTERIAL TRANSPORT IN POROUS MEDIA. II 503

1.0- 0 .I c1

d

h 2.04 Q U c

.......................................... C . : j 2.02 t

0 0 ,o 2.00

3

t

c)

i 2 1.98

a“ 1.96 - ’”\ I 5 1.94 I . . .

Q

.....- V W .-

.I c1

E

g

* 8 0.6 u

0.4 V Q L a u

.I

c1

; 0.2

0.0 0 2 4 6 8 1 0

Dimensionless time

Figure 1. Bacteria concentration ratio (bacteria concentration normalized with respect to the injected concentration) for different values of the retention parameters A and B in the fractional flow equation. Symbols: ( - ) , A = 1, B = 1, two-phase; (-),A = 1, B = 1, single phase; ( . . . . ) , A = 1, B = 4, single phase; ( - . - . - ) , A = 0.5, B = 4, single phase.

0.10 4 . c s

-0.08 0

0 0 2

2 : E 5

- 0.06 d - - 0.04

-0.02 v) a

a 0.00

tion” experiments depend entirely on the growth kinetics “in situ.” The ability of the simulator to model experimental results obtained for the two limiting cases of zero and infinite N D , are discussed in the next section.

An example of a simulation involving a sequence of injection and incubation steps is shown in Figure 2. In this example, a cell culture was injected into a porous medium containing a residual oil saturation for 0 =S to G 2. Nutrient alone was injected for 2 < rD =S 4. This was followed by an incubation period at the end of which brine was injected for 2 pore volumes. This is a typical injection sequence in an MEOR process. The results of the simulation during the incubation period are shown in Figure 2A and B. The primary metabolic product is a biosurfactant that reduces the interfacial tension between the oil and water phases. The rate of production of oil and water during the subsequent brine injection are shown in Figure 2C.

The results of the incubation period depend on the initial distribution of cells in the porous medium (i.e., the fractional flow curve and the injected concentration of cells) and the in situ growth kinetics. The results of the subsequent displacement with brine are controlled by the surface activity of the biosurfactant and the relative permeability and residual saturation constitutive relationships for the porous medium.

Comparison with Experimental Results Using B. licheniformis JF-2

To evaluate the ability of the model to simulate experimental results, a few laboratory data sets were chosen for which most of the model parameters had been measured. The sim-

ulated effluent bacteria concentration and permeability ratio histories of the upstream section were compared with experimental results reported by Sarkar et a1.15 and Sarkar” using B . licheniformis JF-2. Only the transport and growth parameters were adjusted to match those of the different bacteria used in the experiments.

A comparison between the simulated and experimentally measured bacteria effluent concentrations for a flow velocity of 25 ft/day and injected bacteria concentration of lo8

1.2 A

Time, hours


cells/mL is shown in Figure 3. The bacteria first appeared in the effluent at to = 1.3, whereas cell breakthrough (C/C, = 0.5) occurred at to = 3.7. The agreement between the measured and simulated cell concentrations is very good. Similarly, good agreement between the model’s predictions and the experimentally measured effluent bacteria concentration profiles was obtained for all injection velocities tested, which were in the range of 2 to 100 ft/day (data not shown).

Figure 4 shows a comparison of the predicted permeability history and experimental results at two flow velocities. The experimental data in both cases show a sudden decrease in permeability for small dimensionless times (to = 0.2). When the injection velocity was 25 ft/day the permeability did not change substantially for fD > 2, consistent with the results of the simulator. At the higher injection velocity the permeability ratio exhibited a slow decline, from 0.5 at rD = 2 to 0.36 at fD = 6. The physical mechanism for the slow decrease in permeability at the high flow velocity is not clear. Since the mechanism responsible for this phenomenon is not known, it is not taken into account in the simulator which predicts a constant permeability ratio of 0.42.

Figure 5A shows comparisons of the simulated and experimental effluent bacteria concentration ratio histories for two experiments in which the injected concentrations were lo7 and lo8 viable cells/mL (measured as colony forming units, cfu/mL). The simulated and experimental permeability ratio histories for the upstream section are compared in Figure 5B. At the low injected cell concentration the simulated results show no permeability reduction at all, which is completely consistent with the experimental data. In the case of a higher injected concentration the permeability ex- hibits a local minimum immediately after the onset of bacteria flow, similar to Figure 4B. For tD values greater than

0 1 2 3 4 5 Dimensionless Time

Figure 3. Comparison of the simulated and experimentally determined bacteria concentration in the absence of growth. The results are normalized with respect to the injected concentration and expressed as bacteria concentration ratio. The experimental data are from Sarkar et al.15 for an injected cell concentration of lo8 cfu/mL and velocity of 25 Wday. Solid line: simulation results; squares: experimental data from Sarkar et al.”

0 ..I CI

d

8 a“

r, Y .I I

a Q

L

0.8 -

0.6 -

0.4 -

o.2 1 o . o ! . , . , . , . , . , . I

0 1 2 3 4 5 6 Dimensionless Time

Figure 4. Comparison of simulated and experimental permeability ratio histories for the upstream section at two different fluid velocities: (-) simulation and (0) experimental results at 25/ft; (. . . . . .) simulation and (+) experimental results, 100 ft/day. Experimental data were from Sarkar et ai.15

1.0, the model simulates the change in bed permeability reasonably well. In fact, the difference between the predicted and experimentally determined permeability values is less than 2 10%.

Figure 6 shows a comparison of the simulated effluent bacteria concentration ratio and permeability history with experimental data for the case where a residual oil phase is present during the bacterial flood. Briefly, to establish a residual oil phase experimentally, a brine-saturated porous medium, in this case a sandpack, was used as a starting- point.” Decane was then used to displace the brine to an apparent residual saturation. Brine was subsequently used to displace the decane and obtain a residual oil saturation. This represents a depleted oil reservoir. It should be noted that in the two-phase flow experiments the permeability ratio is based on brine-effective permeability at residual oil saturation. The agreement between the simulation and the experimentally determined effluent bacteria concentration is good up to a fD of 6.0. However, at higher dimensionless times, the simulator overestimated the effluent bacteria concentration. A likely explanation for this discrepancy is that bacteria entrapment due to bridging becomes more extensive as a function of time leading to the formation of an external filter cake. Such time-dependent changes in the mechanism of bacteria retention cannot be accounted for by the fractional flow equation of the simulator. The simulator also predicts the overall profile of permeability decrease for fD > 1 .O (Fig. 6B). As in the single-phase case, the experimentally determined permeability shows a drastic decrease immediately after the onset of bacteria injection followed by a slight increase and a low monotonic decrease.

Simulation of Experiments with Other Bacteria

Stehmeier and Jackz0 conducted several single-phase bacteria transport experiments using sandpacks (-3 ft long).


0 .I c1

d 3 E .I U

U E aJ 0 E

u" Q k aJ V Q cp

.I

U

0 1 2 3 4 5 6

Dimensionless Time

I 9

1 B

u.tl

0.0 I . 1 - I . I . I .

0 1 2 3 4 5 6

Dimensionless Time

Figure 5. Comparison of the simulated and experimental (A) effluent bacteria concentration histories and (B) permeability histories at different injected cell concentrations. Symbols: (O), injected concentration 10' cfu/ mL; (+), injected concentration lo8 cfu/mL.

The flow velocities (87 and 35 ft/day) and the injected concentrations (- lo8 cfu/mL) for the two experiments that we simulated were similar to the conditions used in our experiments. Values for the growth and transport parameters were obtained from the experimental conditions specified by Stehmeier and Jack.*' Figure 7 shows a comparison between experimental and numerical results for bacteria effluent concentration histories for these experiments. Once again, the agreement is reasonably good at both fluid velocities. It should be noted that a higher value for parameter B in the fractional flow curve had to be used to account for the long delay in the appearance of bacteria in the effluent. This higher value is indicative of extensive surface adhesion of the bacteria, presumably mediated by the production of extracellular polymers. *O

CONCLUSIONS

We have developed a comprehensive model that accounts for bacterial propagation and growth in porous media in the presence of a nonaqueous phase. Although the model can be employed to simulate any process involving bacteria injec-

tion and in situ growth, its primary aim is to simulate microbial-enhanced oil recovery. In order to model the flow of the oleic phase as a function of microbially induced changes in permeability and interfacial tension we have used an approach similar to what had been employed earlier to simulate enhanced oil recovery." The retention of bacteria within the porous medium was modeled by a simple fractional flow equation using two experimentally determined parameters. Even though this formulation is empirical, we found that it leads to good agreement with experimental data obtained with two different bacteria. Once the fraction of retained bacteria has been estimated, then the permeability reduction can be calculated from the effective medium theory. l 8 Furthermore, component conservation equations were employed to model the coupled processes of cell growth, transport, and metabolite formation in porous media. Rate expressions for growth and metabolite formation were obtained by assuming that cellular processes follow

0 1.0

a A .I U

0 .- c) E 5 Y

0

u Q L aJ 0

.I

U

2

0.6 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Dimensionless Time

1 . o y

0.8

0.6

0.4

o'2 1 o.o ! . , . 1 . , . , . 1 . , . , . 1

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0

Dimensionless Time Figure 6. Comparison of the simulated and experimental results for the two phase case. (A) Effluent bacteria concentration histories. (B) Perme- ability histories. Solid lines: simulation results; open squares: experimental data from Sarkar et aL15 Parameter values for the simulation were as follows:C,, = 0.164E-3;n = 0.12E-05;FT = 5.0:e = l .OS , f= 0.285; A = 3.0, B = 10.0.

506 BIOTECHNOLOGY AND BIOENGINEERING, VOL. 44, NO. 4, AUGUST 5,1994

0.8

0.6

0.4

0.2

0.0 0 5 10 15 20 25 30

Dimensionless Time

Figure 7. Comparison of simulated and experimental effluent bacteria concentration histories for a different microorganism. Experimental data were obtained from Stehmeier and Jackz0 (runs #20 and 22) corresponding to injection flow velocities of 87 fVday (A) and 35 ft/day (0). Parameter values for the simulation were as follows: Cc, = 0.164 X IOW3; n = 0.12 X lo-'; F , = 5.0; e = 1.08,f = 0.285; A = 5.5 , B = 75.0.

the same kinetics in porous media as they do in suspended cultures under laboratory conditions. This is clearly a necessary first approximation, because there is so little known about the kinetics of bacterial growth in natural systems.

The model we have developed represents a first attempt to simulate the complex processes that take place upon injection and subsequent growth of bacteria in porous media. It presents a methodology for coupling the quantitative description of microscopic phenomena relating to bacterial growth and migration to macroscopic, directly measurable parameters such as the permeability ratio and the fractional flow of the aqueous and oleic phases. Admittedly, the model is quite complicated and some degree of empiricism was involved in the fractional flow curve formulation and in the definition of the effective radius employed in the effective medium theory approach. Although the use of such semi-empirical expressions is undesirable in general, it was necessary in this case in order to keep the model mathematically tractable. The reasonable agreement between our model and the experimental data indicates that the overall approach is satisfactory. We have demonstrated very good agreement between the simulated bacteria effluent concentration and data obtained with two different systems. Also, the permeability change prediction matched reasonably well the experimental data although some discrepancies were evident for small values of the dimensionless time.

The authors acknowledge the financial support provided by the Department of Energy, Grant DE-FG07-89BC14445 (to G.G. and M.M.S.); the Texas Advanced Research Program, Grant 003658-0458 (to M.M.S.); and the corporate sponsors of the Enhanced Oil and Gas Recovery Research Program (to G.G.)

NOMENCLATURE a A , B

minimum fraction of pore throats open to flow retention parameters in fractional flow equation

parameters in the specific rate constant for metabolite production threshold concentration below which all the particles will be

critical coagulation concentration flowing particle concentration trapped and adsorbed particle concentration concentration of a component k in phase 1 total particle concentration coupled dispersion coefficient for a component k in phase I dispersion coefficient for a component k in phase 1 effective radius parameters for concentration and velocity effects conductance distribution for conductive pores pore throat radius distribution function damaged pore throat radius distribution function empirical constant in effective radius equation individual pore throat conductance effectivehean hydraulic conductance in a network mean conductance for a damaged medium mean conductance for an undamaged medium absolute permeability in the flow direction permeability of the damaged medium permeability of the undamaged medium relative permeability of phase I cell lysis rate constant maintenance coefficient for cells original number of pore throats per unit bulk volume exponent in Fq. (1 1) number of components Damkohler number net flux of component k [mass/(area time)] net flux of component k in phase 1 number of phases pressure in aqueous phase capillary pressure between phase 1 and 1' pressure in phase 1 pore volumes injected rate of injection (positive sign) or production (negative sign) of component k coupled specific rate constant for component k in phase 1 specific rate constant for component k in phase 1 pore throat radius effective radius of cells experimentally determined cell radius saturation of a phase 1 time dimensionless time (pore volumes injected/pore volume of bed) superficial velocity of phase I bulk volume of sandpack overall mass concentration (masshulk volume) yield coefficient (mass of cells per mass of limiting nutrient) fraction of nonconducting pores coordination number of the network

trapped

Subscripts

i , j . k correspond to x, y , z of the coordinate system k index for a component k' index for the component that is coupled with k 1 index for a phase

Greek symbols

V gradient operator pE I$ porosity of the medium p,,,,=

number-averaged density of solid particles

maximum specific growth rate of the bacteria


pI density of phase 1 pI viscosity of phase 1 X, X, total relative mobility

relative mobility of phase 1 ( =K&,)

References

1.

2.

3.

4.

5.

6 .

7.

8.

9.

Aiba, S., Shoda, M., Nagatani, M. 1968. Kinetics of product inhibition. Biotechnol. Bioeng. 10: 845. Bailey, J. E., Ollis, D. F. 1986. Biochemical engineering fundamen- tals. 2nd edition. McGraw-Hill, New York. Bhuyan, D. 1989. Development of an alkaline-surfactant-polymer compositional reservoir simulator, Ph.D. dissertation, Department of Petroleum Engineering, The University of Texas at Austin, Austin, TX, USA. Brown, M. J. , Moses, V., Robinson, J . P., Springham, D. G. 1986. Microbial enhanced oil recovery: progress and prospects. CRC Crit. Rev. Biotechnol. 3: 159. Corapcioglu, M. Y., Haridas, A. 1985. Microbial transport in soils and groundwater: a numerical model. Adv. Water Res. 5: 189. Goursaud, J. C. 1989. Production of a biosurfactant by Bacillus licheniformis JF-2, M.S. thesis, University of Texas at Austin, Austin, TX, USA. Jang, L. K., Sharma, M. M., Findley, J. E., Chang, P. W., Yen, T. F. 1982. An investigation of the transport of bacteria through porous media. Proceedings of 1982 International Conference on MEOR. Shangri-La, Afton, OK, USA (CONF-8205140, Dist. Cat. UC-92a). Kalish, P. J., Stewart, J. A, , Rogers, W. F. , Bennett, E. 0. 1964. The effect of bacteria on sandstone permeability. J. Petrol. Tech. 37: 805. Lake, L. W. 1989. Enhanced oil recovery. Prentice-Hall, Englewood Cliffs, NJ.

10. Lake, L. W., Pope, G. A,, Carey, G. F., Sepehmoori, K. 1984. Isothermal, multiphase, multicomponent fluid-flow in permeable Me- dia. Part I: Description and mathematical formulation. In Situ 8: 1.

11. Lin, S. , Minton, M. A, , Sharma, M. M., Georgiou, G. 1994. Struc- tured and immunological characterization of the biosurfactant pro- duced by Bacillus licheniformis JF-2. Appl. Env. Microbiol. 60 (1)31-38.

12. Payatakes, A. C., Tien, C. , Turian, R. M. 1973. A new model for granular porous media. AIChE. J . 19: 58.

13. Pope, G. A, , Lake, L. W., Carey, G. F., Sepehmoori, K. 1984. Isothermal, multiphase, multicomponent fluid-flow in permeable media. Part 11: Numerical techniques and solution. In Situ 8: 17.

14. Rhee, H.-K., Rutherford, A, , Amundson, N. R. 1986. First-order partial differential equations, vol. I. Theory and application of single equations. Prentice-Hall, Englewood Cliffs, NJ.

15. Sarkar, A. K., Georgiou, G., Sharma, M. M. 1994. Transport of bacteria in porous media: I. An experimental investigation. Biotech- nol. Bioeng. 44: 00-00.

16. Sarkar, A. K., Goursaud, J. C., Sharma, M. M., Georgiou, G. 1989. A critical evaluation of MEOR processes. In Situ 43: 207.

17. Sarkar, A. K. 1992. Simulation of microbial enhanced oil recovery processes, Ph.D. dissertation, The University of Texas at Austin, Austin, Texas.

18. Sharma, M. M., Yortsos, Y. C. 1987. Transport of particulate suspensions in porous media: model formulation. AIChE. J . 33: 1363.

19. Shaw, J . C., Bramhill, B., Wardlaw, B., Costerton, J . W. 1985. Bacterial fouling in a model core system. Appl. Environ. Microbiol. 49: 693.

20. Stehmeier, L. G. , Jack, T. R. 1987. Evaluation of sacrificial agents in bacterial transport in laboratory cores. Report No. 00315, Nova/ Husky Research Corp. Ltd.

21. Wu, G., Sharma, M. M. 1989. A model for precipitation and dissolution processes with precipitate migration. AIChE. J. 35: 1385.


transport of bacteria in porous media: ii. a model for convective transport and growth

Documents