Appendix B — Appendix B — Model Toolkit

This appendix collects the mathematical machinery behind the reaction–transport models discussed in Chapters 8 and 9. Readers who skipped the Deep Dive sidebars can find the full framework here; readers who followed every sidebar can use this as a concise reference.

B.1 B.1 The diagenetic equation

The conservation (balance) equation for any species in one-dimensional sediment is Berner’s diagenetic equation (Berner 1980):

\[ \frac{\partial \hat{C}}{\partial t} = -\frac{\partial F}{\partial x} + \sum R \]

where \(\hat{C}\) is the bulk concentration (averaged over a volume larger than several grain diameters but smaller than the macroscopic gradient), \(F\) is the flux, and \(\sum R\) is the net source or sink from all reactions.

For a solute in porewater:

\[ \hat{C} = \varphi\, C \]

where \(\varphi\) is porosity and \(C\) is concentration per unit volume of porewater.

For a solid species:

\[ \hat{C} = (1 - \varphi)\, B \]

where \(B\) is the amount per unit volume of solids (Boudreau 1997).

B.1.1 Steady state

At steady state relative to the sediment–water interface (\(\partial \hat{C}/\partial t = 0\)), changes are observed only when following a layer downward:

\[ \frac{D\hat{C}}{Dt} = w\,\frac{\partial \hat{C}}{\partial x} \]

where \(w\) is the burial velocity of solids.

B.2 B.2 Transport terms

B.2.1 Advection

Solute advective flux:

\[ F_A = \varphi\, u\, C \]

Solid advective flux:

\[ F_A = (1 - \varphi)\, w\, B \]

where \(u\) is porewater velocity and \(w\) is solid burial velocity.

B.2.2 Compaction

During compaction, porewater moves upward relative to sediment grains but usually still moves downward in the interface-anchored reference frame — appearing slower than the solids.

B.2.3 Externally impressed flow (Darcy)

\[ u_x = -\frac{k}{\varphi\,\mu}\,\frac{\partial p'}{\partial x} \]

where \(k\) is permeability, \(\mu\) is dynamic viscosity, and \(p'\) is the reduced pressure (Boudreau 1997).

B.3 B.3 Reaction rate expressions

B.3.1 Respiration (dual-Monod)

\[ R_{\text{resp}} = k_{\text{resp}} \cdot [B] \cdot \frac{[\text{TED}]}{[\text{TED}] + K_m^{\text{TED}}} \cdot \frac{[\text{TEA}]}{[\text{TEA}] + K_m^{\text{TEA}}} \]

where TED = terminal electron donor, TEA = terminal electron acceptor (Jin and Bethke 2005; Thullner, Regnier, and Van Cappellen 2007).

B.3.2 Hydrolysis (Michaelis–Menten)

\[ R_{\text{hydr}} = k_{\text{cat}} \cdot [E] \cdot \frac{[\text{POM}]}{[\text{POM}] + K_m^{\text{POM}}} \]

where \(k_{\text{cat}}\) is the turnover number, \([E]\) is enzyme concentration, and POM is particulate organic matter (A. W. Dale, Regnier, and Cappellen 2006; Thullner, Regnier, and Van Cappellen 2007).

B.3.3 Thermodynamic factor

\[ F_T = \frac{1}{\exp\!\left(\frac{\Delta G_r + F\,\Delta\Psi}{RT}\right) + 1} \]

This factor smoothly transitions from 1 (far from equilibrium) to 0 (at equilibrium), preventing reactions from proceeding past their thermodynamic limit (Jin and Bethke 2005; Regnier et al. 2011).

B.3.4 Temperature dependence (Arrhenius)

\[ k = k^\circ \exp\!\left[-\frac{E_a}{R}\left(\frac{1}{T} - \frac{1}{298.15}\right)\right] \]

Note: The Arrhenius equation is semi-empirical, derived for elementary reactions. Apparent \(E_a\) values are generally calculated from rate measurements (Arndt et al. 2013; Leal, Blunt, and LaForce 2015).

B.4 B.4 Growth, yield, and decay

Standard biomass model (Thullner, Regnier, and Van Cappellen 2007; A. W. Dale et al. 2010):

\[ \frac{\partial [B]}{\partial t} = Y \cdot R_{\text{resp}} - \mu_{\text{dec}} \cdot [B] \]

Yield coefficient approaches:

Approach	Reference	Notes
Theoretical (energy-based)	Rittmann and McCarty (2001)	\(R^2 = 0.9\)
Theoretical (Gibbs dissipation)	Heijnen and Van Dijken (1992)	\(R^2 = 0.9\)
Empirical	Roden and Jin (2011)	\(Y = 0.28 + 0.0211 \cdot (-\Delta G')\)

Monod growth kinetics:

\[ r_X = \mu_{\max}\,\frac{S}{K + S} \cdot X \]

B.5 B.5 Partial equilibrium

When aqueous reactions are much faster than mineral reactions, the partial equilibrium assumption replaces stiff differential equations with algebraic constraints for the fast reactions (Leal, Blunt, and LaForce 2015):

\[ r = k \cdot a_i \left(1 - \frac{Q}{K}\right) \]

With possible adjustments near equilibrium:

\[ r = k \cdot a_i \left[\left(1 - \frac{Q}{K}\right)^\xi\right]^\nu \]

B.6 B.6 Software and tools

Several geochemical modeling codes implement the frameworks described above:

• PHREEQC (Parkhurst and Appelo, 1999, 2013)
• The Geochemist’s Workbench (Bethke, 2007)
• EQ6 (Wolery and Daveler, 1992)
• CHESS (van der Lee and Windt, 2002)

For readers who want to build and run their own reaction–transport simulations of the processes described in this book, the open-source PorousMediaLab provides a Python-based framework for modeling biogeochemical reactions in porous media. It implements many of the rate expressions and transport equations presented here, and can serve as a hands-on companion to the mathematical theory.