Biodegradation Reaction

1. Description

Simulation of biodegradation reaction in TOUGH4 is based on the TMVOCBio, (Aquater, 2002; Battistelli, 2003; 2004), which was included in the TMVOC module of TOUGH3. TOUGH4 extends this functionality to any EOS module. The modeling capabilities include simulating multiple biodegradation reactions mediated by different microbial populations or based on different redox reactions, thus involving different electron acceptors. It uses the modified Monod model to represent biodegradation reactions, and this general form effectively accounts for the substrate uptake as well as various limiting factors. The following is the key methodology for implementing biodegradation reaction simulation.

Assumptions

To implement biodegradation reactions within the TOUGH4, a number of simplifying assumptions have been considered. The assumptions underlying the implementation of biodegradation reactions and the mathematical formulation are fully described in Aquater (2002) and Battistelli (2003, 2004). The main assumptions are summarized as:

(1) Bacteria transport is negligible.

(2) All biomass is considered active in the biodegradation process.

(3) Diffusion processes into and out of the biophase are neglected, so biodegradation rates depend directly on aqueous phase solute concentrations.

(4) Predation of microbes by other microorganisms is neglected.

(5) Bioreactions are not affected by chemical equilibria.

(6) Each microbial population can be involved in several degradation processes, each one acting on a single organic substrate.

(7) The time needed for acclimation of microbial populations to new substrate and electron acceptors concentration levels is neglected as it is generally much smaller than the overall time scale for biodegradation in subsurface media.

(8) Changes of porous medium porosity due to biomass growth are neglected as well as the related effects on medium permeability (clogging).

(9) A minimum biomass concentration, specified by the user, is maintained in the absence of any organic contaminants degradation.

Implementation of Multiple Biodegradation Reactions

TOUGH4 follows the numerical formulation used by the T2LBM EOS module (Oldenburg, 2001) of the TOUGH2 reservoir simulator (Pruess et al., 2012), developed to model the aerobic/anaerobic biodegradation of solid waste in municipal landfills. The original T2LBM formulation has been modified to model a number of different degradation processes defined by the user, following the BIOMOC general formulation of biodegradation reactions (Essaid and Bekins, 1997), and to take into consideration the changes of aqueous phase density and saturation in the updating of microbial concentration in the aqueous phase, to account for the changes occurring in unsaturated or multiphase flow conditions.

The degradation rate of substrates and related consumption of electron acceptors and nutrients, as well as the generation rate of by-products and heat, are computed for assembling the balance equations for the mass components and thermal energy during each simulated time step. The mass and energy balance equations are solved by fully coupled with reactive transport of mass components. The total uptake of each substrate within a simulated time step is considered as a sink/source term in the mass balance equation. Since the transport of microorganisms is assumed to be negligible, no mass balance equations for the microbial populations are set up. Biomass growth and decay are tracked during the simulation by updating local biomass concentration in the aqueous phase after each converged time step, accounting also for the changes in aqueous phase saturation and density.

This numerical approach of treating biodegradation reactions as internal sink/sources computed for each Newton- Raphson iteration introduces an additional source of numerical errors, known as splitting error, generated by the decoupling of governing equations (Valocchi and Malmstead, 1992). This error depends on the time step length and is also related to the magnitude of reaction rates (Kanney et al., 2003; University of Texas, 2000). Thus, the numerical solution of biodegradation reactions is sensitive to time step length, depending on the rate of reactions of the modeled process, and may suffer from numerical instability if the time step is too long to accurately model fast biodegradation rates. To avoid this issue, users may need to properly assign a maximum time step value, reduce the convergence criterion of mass balance solution, or allow time step increase only when convergence is achieved with a small number of iterations.

Mathematical Formulations

The degradation rate is expressed using modified forms of the Monod kinetic rate equation, which can account for various limiting factors, such as the limitation by the substrate, the electron acceptor and nutrients, on the uptake rate of substrate. Two approaches are included: (1) a multiple Monod kinetic rate equation, which assumes all the limiting factors simultaneously affect the substrate uptake rate (Borden and Bedient, 1986; Waddill and Widdowson, 1998), and (2) a minimum Monod model, which assumes that the substrate uptake rate is controlled by the most limiting factor among those acting for the specific substrate (Kindred and Celia, 1989).

Below the degradation rate dS/dt of a primary substrate by a microbial population B is shown for the case in which only the limitation of the substrate and the electron acceptor availability is considered, even though limitations due to any other solute dissolved in the aqueous phase, including nutrients, can be modeled by invoking the appropriate Monod term. Note that biomass growth inhibition, toxicity effects, as well as competitive and non-competitive inhibition effects are also included as the limiting factors.

$\dfrac{dS}{dt}=\mu_{0B}B$ (7-34)

where S is substrate concentration, in kg substrate/kg aqueous phase; B is biomass concentration, in kg biomass/kg aqueous phase. The specific substrate utilization rate $\mu_{0B}$ is given by

$\mu_{0B}=\dfrac{\mu_{max,B}}{I_{NC}}f_Tf_{SW}f_{Monod}\dfrac{1}{I_B}$ (7-35)

where $\mu_{max,B}$is maximum substrate utilization rate by biomass, in kg substrate/(kg biomass's); $f_T$is inhibitive function of temperature (Oldenburg, 2001); $f_{SW}$is inhibitive function of water saturation (El-Kadi, 2001); $I_B$ is biomass inhibition factor; $I_{NC}$is non-competitive inhibition factor. The Monod kinetic rate term $f_{Monod}$ is different for the multiple Monod kinetic model and minimum Monod model. Detailed definitions can be found from the literature.

The rate of biomass change with the effect of death rate $\delta$is

$\dfrac{dB}{dt}=Y\mu_{0B}B-\delta B=-Y\dfrac{dS}{dt}-\delta B$ (7-36)

where Y is biomass yield coefficient, in kg biomass/kg substrate.

Following the BIOMOC approach (Essaid and Bekins, 1997), the formulation of the degradation rate is extended for the case where a generic substrate is simultaneously degraded by different microbial populations involving the same or different electron acceptors. When a substrate S(j) is degraded in $N_{proc}(j)$simultaneous degradation reactions mediated by different microbial populations B(ip), ip = 1, $N_{proc}(j)$, the total degradation rate for the substrate S(j) is

$\dfrac{dS(j)}{dt}=\displaystyle\sum_{ip=1}^{N_{proc}(j)} -\mu_{0B}(ip)B(ip)$ (7-37)

The same approach can be extended to simultaneous degradation processes of multiple substrates, providing the capability to model a number of different degradation processes defined by the user, following the general formulation of biodegradation reactions described above.

For more detailed discussions of the mathematical models, users may refer to the mentioned literatures and the user manual of TMVOCBio.

1. Input requirements

The simulation of biodegradation reactions can be evoked by the keyword "BIODG". If "BIODG" appears in the main file and parameters for biodegradation and definition for the bioprocesses are correctly inputted, the model will include the simulation of biodegradation reactions.