## 2.25.2. Governing Mass-Balance Relationship[¶](#governing-mass-balance-relationship "Permalink to this headline")
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The model ([Figure 2.25.1](#figure-methane-schematic)) accounts for CH4 production in the anaerobic fraction of soil (_P_, mol m\-3 s\-1), ebullition (_E_, mol m\-3 s\-1), aerenchyma transport (_A_, mol m\-3 s\-1), aqueous and gaseous diffusion (\\({F}\_{D}\\), mol m\-2 s\-1), and oxidation (_O_, mol m\-3 s\-1) via a transient reaction diffusion equation:

(2.25.1)[¶](#equation-24-1 "Permalink to this equation")\\\[\\frac{\\partial \\left(RC\\right)}{\\partial t} =\\frac{\\partial F\_{D} }{\\partial z} +P\\left(z,t\\right)-E\\left(z,t\\right)-A\\left(z,t\\right)-O\\left(z,t\\right)\\\]

Here _z_ (m) represents the vertical dimension, _t_ (s) is time, and _R_ accounts for gas in both the aqueous and gaseous phases:\\(R = \\epsilon \_{a} +K\_{H} \\epsilon \_{w}\\), with \\(\\epsilon \_{a}\\), \\(\\epsilon \_{w}\\), and \\(K\_{H}\\) (-) the air-filled porosity, water-filled porosity, and partitioning coefficient for the species of interest, respectively, and \\(C\\) represents CH4 or O2 concentration with respect to water volume (mol m\-3).

An analogous version of equation [(2.25.1)](#equation-24-1) is concurrently solved for O2, but with the following differences relative to CH4: _P_ = _E_ = 0 (i.e., no production or ebullition), and the oxidation sink includes the O2 demanded by methanotrophs, heterotroph decomposers, nitrifiers, and autotrophic root respiration.

As currently implemented, each gridcell contains an inundated and a non-inundated fraction. Therefore, equation [(2.25.1)](#equation-24-1) is solved four times for each gridcell and time step: in the inundated and non-inundated fractions, and for CH4 and O2. If desired, the CH4 and O2 mass balance equation is solved again for lakes (Chapter 9). For non-inundated areas, the water table interface is defined at the deepest transition from greater than 95% saturated to less than 95% saturated that occurs above frozen soil layers. The inundated fraction is allowed to change at each time step, and the total soil CH4 quantity is conserved by evolving CH4 to the atmosphere when the inundated fraction decreases, and averaging a portion of the non-inundated concentration into the inundated concentration when the inundated fraction increases.

![Image 1: ../../_images/image14.png](https://escomp.github.io/ctsm-docs/versions/master/html/_images/image14.png)

Figure 2.25.1 Schematic representation of biological and physical processes integrated in CLM that affect the net CH4 surface flux ([Riley et al. 2011a](https://escomp.github.io/ctsm-docs/versions/master/html/tech_note/References/CLM50_Tech_Note_References.html#rileyetal2011a)). (left) Fully inundated portion of a CLM gridcell and (right) variably saturated portion of a gridcell.[¶](#id13 "Permalink to this image")

