| [TOC] | 3. Model Implementation | [Prev. Page] | [Next Page] |
This chapter discusses the development of the numerical model. The model simulates two-dimensional saturated steady-state flow with advection and dispersion of multiple reactive solutes. It includes the processes of Langmuir and Freundlich isotherms, as well as first order and higher reactions. It uses single, double, and competitive Monod kinetics to represent biological processes. The model is also capable of representing other processes such as intermediate toxicity and cometabolic transformation by combinations of these reactions.
The fundamental equation that represents the sum of these processes is:
|
(3.1) |
Where: d C/d t = time rate of change in concentration (M/L3·T) C = concentration (M/L3) D = dispersion coefficient (L2/T) V = pore water velocity (L/T) R = net rate of reaction (sink/source term) (M/L3·T)
There are several options available when solving equation (3.1). Finite difference and finite element methods are very popular. Operator-splitting methods break down equation (3.1) into sub-problems and solve each sub-problem sequentially or simultaneously using the most appropriate method. A literature review was conducted to determine which methods have been employed by other researchers. Tables 3.1.1 through 3.1.4 summarize the results of a literature search of numerical models involving biodegradation.
The tables below illustrate that no single model includes all of the desired processes outlined above. Construction of a new model is necessary. Construction is chosen over modification of an existing program for multiple reasons. Modifying an existing model would require time and effort to learn how the existing code works. It would also be easy to introduce programming errors due to unfamiliarity with the existing code. In addition, most existing programs are written in FORTRAN. Inclusion of multiple different types of reactions is far easier to implement in C++.
Table 3.1.1 Comparison of Numerical Methods, Dimensions and Flow Type
| Author(s) | Model | Dimen-sions | Flow Type & Kinetics | Advection & Dispersion | Number of Components |
|---|---|---|---|---|---|
| Chen & McTernan (1992) | MMGTM | Three | Steady-State Saturated | Transient | Two, Substrate and Oxygen |
| Chen et al. (1992) | One | Steady-State Saturated | Transient Coupled | Five | |
| Dhawn (1993) | -- | One | Steady-State Saturated | Diffusion Only‡ | Two, Substrate and Oxygen |
| Kindred (1989) | -- | One | Steady-State Saturated | Transient, Coupled | Three |
| Lindstrom (1992) | -- | One | Steady-State Saturated | Transient, Decoupled | Two |
| Mills (1991) | COMETº | Two | Steady-State Saturated | Transient, Coupled | One |
| Odencrantz (1992) | -- | Two | Steady-State Saturated | Transient, Decoupled | Two, Electron Donor & Acceptor |
| Rifai (1988) | Bioplume II | Two | Steady-State Saturated | Transient Decoupled | Two, Hydrocarbon and Oxygen |
| Semprini & McCarty (1991b) | -- | One | Steady-State Saturated | Transient, Coupled | Two, Electron Donor & Acceptor |
| Sleep, Sykes (1993) | -- | Three | Variably Saturated, Transient | Transient, Coupled | Arbitrary |
| Srinivasan, Mercer (1988) | Bio1D | One | Steady-State Saturated | Transient Coupled | 1º and 2º Substrates and Oxygen |
| Taylor & Jaffé (1990b) | -- | One | Transient, Saturated | Transient, Coupled | Two, Substrate & Biomass |
| Tim & Mostaghmi (1991) | VIROTRANSæ | One | Variably Saturated, Transient | Transient, Coupled | One, Virus |
| Zheng (1993) | -- | Three | Steady-State Saturated | Transient, Decoupled | One |
| What is Desired | At Least Two | Steady-State or Transient | Four or greater |
†This study was for biological stimulation
only, not a biodegradation study.
‡Macropores
ºColloids-Metal Transport Model incorporates
EPA’s CML model, movement of contaminants on moving
colloids.
æVirus Transport Model
Table 3.1.2 Comparison of Numerical Methods, Biological Reactions
| Author(s) | Model | Biological Kinetics | Cometab-olism | Competitive Inhibition | Contaminant Availability |
|---|---|---|---|---|---|
| Chen & McTernan (1992) | MMGTM | Single, Double Monod & First-Order | No | No | Soluble and Sorbed |
| Chen et al. (1992) | No | Yes | No | Soluble Only | |
| Dhawn (1993) | -- | Double Monod | -- | No | Soluble Only |
| Kindred (1989) | -- | Double Monod | Yes | No | Soluble Only |
| Lindstrom (1992) | -- | Double Monod | No | No | Soluble Only |
| Mills (1991) | COMET | -- | -- | -- | -- |
| Odencrantz (1992) | -- | Single, Double Monod & Biofilm | -- | -- | Soluble Only |
| Rifai (1988) | Bioplume II | Instantaneous | No | No | Soluble Only |
| Semprini & McCarty (1991b) | -- | Double Monod & Biofilm | -- | -- | Soluble Only |
| Sleep, Sykes (1993) | -- | -- | -- | -- | -- |
| Srinivasan, Mercer (1988) | Bio1D | Double Monod† | No | No | Soluble Only |
| Taylor & Jaffé (1990b) | -- | Biofilm, (effectiveness factor) | -- | -- | Soluble Only |
| Tim & Mostaghmi (1991) | VIROTRANS | First-Order Decay | -- | -- | -- |
| Zheng (1993) | -- | -- | -- | -- | -- |
| What is Desired | Monod and First Order | Yes | Yes | Soluble and Sorbed |
†Double Monod Kinetics were modified by including a term that accounts for the minimum substrate value below which nothing happens.
Table 3.1.3 Comparison of Numerical Methods, Biomass Transport, and Sorption Kinetics
| Author(s) | Model | Biomass Transport | Biomass Growth | Sorption Partitioning | Sorption Kinetics |
|---|---|---|---|---|---|
| Chen & McTernan (1992) | MMGTM | No | Yes, Decoupled† | Linear, Langmuir & Freundlich | Equilibrium & Non-Equilibrium |
| Chen et al. (1992) | -- | No | Yes | Linear | Equilibrium |
| Dhawn (1993) | -- | No | Yes, Decoupled† | Linear | Equilibrium |
| Kindred (1989) | -- | No | Yes | Linear | Equilibrium |
| Lindstrom (1992) | -- | No | Yes | Linear | Equilibrium |
| Mills (1991) | COMET | Colloid Transport | -- | Linear for Soil and Colloids | Equilibrium |
| Odencrantz (1992) | -- | No | Yes, Decoupled† | Linear | Equilibrium |
| Rifai (1988) | Bioplume II | No | No | Linear | Equilibrium (inferred) |
| Semprini & McCarty (1991b) | -- | No | Yes, Decoupled† | Linear | Equilibrium & Non-Equilibrium |
| Sleep, Sykes (1993) | -- | -- | -- | none | none |
| Srinivasan, Mercer (1988) | Bio1D | No | No | Linear, Langmuir & Freundlich | Equilibrium |
| Taylor & Jaffé (1990b) | -- | Yes | Yes | -- | -- |
| Tim & Mostaghmi (1991) | VIROTRANS | Yes | No | Linear for Virus | Equilibrium |
| Zheng (1993) | -- | -- | -- | Linear | Equilibrium |
| What is Desired | Desired for Completeness | Yes | Linear, Langmuir & Freundlich | Equilibrium and Non-Equilibrium |
†Decoupled from the fluid flow equation, does not effect fluid flow.
Table 3.1.4 Comparison of Numerical Methods, Numerical Techniques
| Author(s) | Model | Hard Coded for Acceptor | Numerical Technique | Abiotic Reactions |
|---|---|---|---|---|
| Chen & McTernan (1992) | MMGTM | Yes | Crank-Nicholson Finite-Difference, Newton-Raphson | No |
| Chen et al. (1992) | Yes | Galerkin F.E. Picard Iteration | No | |
| Dhawn (1993) | -- | Yes | ISML† | No |
| Kindred (1989) | ` | Yes | Optimal Test Function | No |
| Lindstrom (1992) | -- | Yes | Eulerian-Legrangian | No |
| Mills (1991) | COMET | -- | -- | -- |
| Odencrantz (1992) | -- | No | Operator-splitting | No |
| Rifai (1988) | Bioplume II | Yes | Method of Characteristics | No |
| Semprini & McCarty (1991b) | -- | No | Finite Difference, Runge-Kutta | No |
| Sleep, Sykes (1993) | -- | -- | Finite Difference, IMPESC | No |
| Srinivasan, Mercer (1988) | Bio1D | Yes | Finite-Difference Crank Nicholson Newton-Raphson | No |
| Taylor & Jaffé (1990b) | -- | -- | Galerkin Finite Element, Weighted Finite-Difference | No |
| Tim & Mostaghmi (1991) | VIROTRANS | -- | Galerkin Finite Element, Newton-Raphson, Picard Iteration | No |
| Zheng (1993) | -- | -- | MOC block-Centered, Finite-Difference, Foreword Particles | No |
| What is Desired | -- | No | -- | Desired for Completeness |
†The coupled set of ODE’s were integrated using the International Mathematical & Scientific Library subroutine LGEAR and DPDES.
| [Home] | [Table of Contents] | [Prev. Page] | [Next Page] |
| A Two Dimensional Numerical Model for Simulating the
Movement and Biodegradation of Contaminants in a Saturated Aquifer © Copyright 1996, Jason E. Fabritz. All Rights Reserved. |
|||