The WIPP Source Sink Process Model¶

The WIPP Source Sink Process Model (WSSPM) determines gas and brine generation in the waste areas. It has been developed by Jennifer Frederick (Sandia National Laboratories - SNL), under the guidance of Brad Day (SNL), Todd Zeitler (SNL), and Glenn Hammond (SNL).

General Algorithmic Design¶

The WSSPM consists of several defined waste panel and inventory objects. Each waste panel object consists of a region object that defines the location of the waste panel, and a pointer to an inventory object. The waste panel object then creates its own inventory where it keeps track of several chemical species and chemical equation rate constants. Each waste panel determines gas and brine generation rates according to a function of the chemical equation rate constants and stoichiometric constants in each chemical equation involving gas or brine.

The process model is called at each Newton iteration, using the current iterate’s unperturbed and perturbed values for the liquid saturation to calculate the rate constants which determine gas and brine generation rates. When the process model is called, the rate constants are determined according to the remaining amounts of limiting chemical species in the relevant chemical equation. The gas and brine generation rates (which are a function of the rate constants and the stoichiometric constants) are used in the current iteration and are assigned in the governing equations for fluid mass conservation.

At the next time step, the chemical equation rate constants are used to update the inventory of remaining chemical species. Rate smoothing and tapering performed in the previous time step ensures that chemical species amounts do not go negative (e.g. overstep the update). The algorithm then starts over again and calculates the rate constants according to the updated inventory.

The following sections document the WSSPM algorithms in detail:

Effective Brine Saturation¶

The effective brine saturation, \(s_{eff}\), is obtained from the actual brine saturation, \(s\), according to:

(1)¶\[ \begin{align}\begin{aligned}s_{eff} = s - s_{min} + \left({s_{wick}\left({1.0 - e^{\alpha B}}\right)}\right)\\B = 200\left({max(s-s_{min},0)}\right)^{2}\end{aligned}\end{align} \]

where \(s_{min}\) is a chemistry cutoff brine saturation parameter below which chemistry does not occur (specified by SOCMIN in the input deck), \(s_{wick}\) is a wicking saturation for brine (specified by SAT_WICK in the input deck), and \(\alpha\) is a unitless rate smoothing parameter (specified by ALPHARXN in the input deck).

Furthermore, the value of \(s_{eff}\) is constrained to a value between 0 and 1 with the following conditionals:

(2)¶\[ \begin{align}\begin{aligned}s_{eff} = min(s_{eff},1)\\s_{eff} = max(s_{eff},0)\end{aligned}\end{align} \]

The conditionals are important because when \(s_{wick} > 0\) it is possible for \(s_{eff}\) to be larger than unity, which is not physical.

Anoxic Iron Corrosion Reaction¶

Iron corrosion is modeled by

(3)¶\[Fe + 2H_{2}O = Fe(OH)_{2} + H_{2}\]

For every mole of \(Fe\) consumed, 2 moles of brine are consumed, one mole of \(Fe(OH)_{2}\) is produced, and one mole of \(H_{2}\) is produced. The reaction rate constant for anoxic iron corrosion, \(K_{c}\), is calculated according to:

(4)¶\[K_{c} = \left({R_{CI}s_{eff} + R_{CH}\left({1-s_{eff}}\right)}\right)\]

(5)¶\[R_{CI} = \frac {r_{CI} D_{s} \rho_{Fe}} {W_{Fe}}\]

(6)¶\[R_{CH} = \frac {r_{CH} D_{s} \rho_{Fe}} {W_{Fe}}\]

where \(r_{CI}\) is the inundated corrosion rate in [m/s] (specified by CORRMCO2 in the input deck), \(r_{CH}\) is the humid corrosion rate in [m/s] (specified by HUMCORR in the input deck), \(D_s\) is the available iron surface area concentration in [m2/m3], \(\rho_{Fe}\) is the density of iron in [kg/m3], and \(W_{Fe}\) is the molecular weight of iron in [kg/mol]. The resulting units for the reaction rate \(K_{c}\) are [mol-Fe/m3/s]. The value for \(D_s\) is calculated as

(7)¶\[D_s = D_{sa} D_{conc}\]

where \(D_{sa}\) is the iron drum surface area in [m2] (specified by ASDRUM in the input deck), and \(D_{conc}\) is the iron drum concentration in the waste panel in [1/m3] (specied by DRMCONC in the input deck). The parameter DRMCONC should be equivalent to BRAGFLO’s ratio DRROOM/VROOM.

The instantaneous rates for each chemical species for anoxic iron corrosion are shown in the table below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown.

species	rate	STCO_##	units
Fe	\(-K_c\)	STCO_13 = -1	mol-Fe/m3/s
H2O	\(-2K_c\)	STCO_12 = -2	mol-H2O/m3/s
Fe(OH)2	\(+K_c\)	STCO_15 = +1	mol-Fe(OH)2/m3/s
H2	\(+K_c\)	STCO_11 = +1	mol-H2/m3/s

The initial amount of iron in an inventory is specified under the INVENTORY block with the following parameters: IRONCHW, IRONRHW, IRNCCHW, IRNCRHW. The total amount of iron in the inventory is the sum of these parameters.

Biodegradation Reactions of Cellulosics, Plastics, and Rubbers¶

Biodegradation reactions are modeled using a lumped approach:

(8)¶\[\frac {1}{6} C_{6}H_{10}O_{5} = X_m(H_2|C) H_{2} + X_m(H_2O|C) H_{2}O\]

This lumped model is derived from the two dominant reactions:

(9)¶\[C_{6}H_{10}O_{5} + 4.8H^{+} + 4.8NO_{3}^{-} = 7.4H_{2}O + 6CO_{2} + 2.4N_{2}\]

(10)¶\[C_{6}H_{10}O_{5} + 6H^{+} + 3SO_{4}^{2-} = 5H_{2}O + 6CO_{2} + 3H_{2}S\]

where Eq. (9) represents denitrification, and Eq. (10) represents sulfate reduction. Methanogenesis is not included. The formula \(C_{6}H_{10}O_{5}\) represents biodegradable materials, which can consist of cellulosics, rubbers, and plastics.

The reaction rates for biodegradation is calculated as

(11)¶\[K_{b} = \left({R_{BI}s_{eff} + R_{BH}\left({1-s_{eff}}\right)}\right)\]

(12)¶\[R_{BI} = \chi_b r_{BI} D_{c} P_{bio}\]

(13)¶\[R_{BH} = \chi_b r_{BH} D_{c} P_{bio}\]

where \(r_{BI}\) is the inundated biodegradation rate in [mol-cellulosics/kg-cellulosics/s] (specified by GRATMICI in the input deck), \(r_{BH}\) is the humid biodegradation rate in [mol-cellulosics/kg-cellulosics/s] (specified by GRATMICH in the input deck), \(D_{c}\) is the initial mass concentration of biodegradables in the waste panel in [kg-cellulosics/m3], \(P_{bio}\) is a unitless probability parameter of attaining sampled microbial gas generation rates (specified by BIOGENFC in the input deck), and \(\chi_b\) is a flag which is set to 0 or 1 depending on whether biodegradation is included in the simulation (as controlled by PROBDEG in the input deck). The resulting units for the reaction rate \(K_{b}\) are [mol-cellulosics/m3/s]. The value for \(D_c\) is calculated as

(14)¶\[D_{c} = \frac {M_{C_{6}H_{10}O_{5}}} {V}\]

where \(V\) is the waste panel volume (internally calculated according to the region object), and \(M_{C_{6}H_{10}O_{5}}\) is the total initial mass of biodegradables in the waste panel. The total initial mass of biodegradables is calculated as a function of input deck parameters according to

(15)¶\[M_{C_{6}H_{10}O_{5}} = M_{cellulosics} + \chi_{rp} \left(M_{rubbers} + \beta M_{plastics}\right)\]

where \(\beta\) is the unitless mass ratio of plastics to equivalent carbon in the waste panel (specified by PLASFAC in the input deck), \(\chi_{rp}\) is a flag which is set to 0 or 1 depending on whether rubbers and plastics are included in the simulation (as controlled by PROBDEG in the input deck), \(M_{cellulosics}\) is the sum of input deck parameters CELLCHW, CELLRHW, CELCCHW, CELCRHW, CELECHW, CELERHW, \(M_{rubbers}\) is the sum of input deck parameters RUBBCHW, RUBBRHW, RUBCCHW, RUBCRHW, RUBECHW, RUBERHW, and \(M_{plastics}\) is the sum of input deck parameters PLASCHW, PLASRHW, PLSCCHW, PLSCRHW, PLSECHW, and PLSERHW.

A summary of the logic about PROBDEG is summarized in the follow table:

PROBDEG value	Meaning	Flag value
0	No biodegradation occurs	\(\chi_{b} = 0\); \(\chi_{rp} = 0\)
1	Biodegradation occurs for cellulosics only	\(\chi_{b} = 1\); \(\chi_{rp} = 0\)
2	Biodegradation occurs for all materials	\(\chi_{b} = 1\); \(\chi_{rp} = 1\)

The instantaneous rates for each chemical species for biodegradation in Eq. (8) are calculated by assuming an average (and lumped) stoichiometry for the denitrification (Eq. (9)) and sulfate reduction ((10)) reactions.

\(N_{2}\) and \(H_{2}S\) are lumped and treated as \(H_{2}\). \(CO_{2}\) is not tracked because it is assumed to be entirely consumed in carbonation processes with magnesium materials in the repository. Nitrate and sulfate are also not tracked since they are assumed to be plentiful, however, the initial amount of nitrate and sulfate are used to calculate the stoichiometry coefficient for gas and brine production as follows:

(16)¶\[ \begin{align}\begin{aligned}X_m(H_2|C) = \frac{2.4}{6}F_{NO3} + \frac{3}{6}F_{SO4}\\X_m(H_2O|C) = \frac{7.4}{6}F_{NO3} + \frac{5}{6}F_{SO4}\end{aligned}\end{align} \]

where \(F_{NO3}\) is the fraction of carbon consumed through the denitrification reaction and \(F_{SO4}\) is the fraction of carbon consumed by sulfate reduction. The calculation of \(F_{NO3}\) and \(F_{SO4}\) is shown below:

A1 = TOTMOLBIO
A2 = GRATMICI * TOTKGBIO * SECPERYER * 10000.d0
MAX_C = min(A1,A2)
F_NO3 = MOL_NO3 * (6.d0/4.8d0) / MAX_C
F_NO3 = min(F_NO3,1.d0)
F_SO4 = 1.d0 - F_NO3

where TOTMOLBIO is the total moles of biodegradables in the waste panel, TOTKGBIO is the total mass of biodegradables in the waste panel, SECPERYEAR is a conversion factor that converts from seconds to years, and the value of 10,000 represents the WIPP safety period (years).

The instantaneous rates for each chemical species for biodegradation are shown in the table below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown. Note that \(H_2\) gas is produced during this reaction even though it is not a reactant in the chemical equations. This is due to the treatment of \(N_2\) and \(H_2S\) gases as \(H_2\) gas.

species	rate	STCO_##	units	notes
\(C_{6}H_{10}O_{5}\)	\(-K_b\)	STCO_24 = -1	mol-biodegradables/m3/s
\(H^{+}\)	n/a	n/a	n/a	not tracked
\(NO_3^-\)	n/a	n/a	n/a	not tracked
\(SO_4^{2-}\)	n/a	n/a	n/a	not tracked
\(H_{2}O\)	\(K_bX_m(H_2O\|C)\)	\(X_m(H_2O\|C)\)	mol-H2O/m3/s
\(CO_{2}\)	n/a	n/a	n/a	not tracked
\(N_{2}\)	n/a	n/a	n/a	lumped as H2
\(H_{2}S\)	n/a	n/a	n/a	lumped as H2
\(H_2\)	\(K_bX_m(H_2\|C)\)	\(X_m(H_2\|C)\)	mol-H2/m3/s

The initial amount of nitrate and sulfate in an inventory is specified under the INVENTORY block with the following parameters: NITRATE, SULFATE.

Iron Sulfidation Reaction¶

Iron sulfidation reactions are modeled as

(17)¶\[Fe(OH)_{2} + H_{2}S = FeS + 2H_{2}0\]

and

(18)¶\[Fe + H_{2}S = FeS + H_{2}\]

where Eq. (17) represents sulfidation of iron hydroxide (a corrosion product), and Eq. (18) represents sulfidation of iron.

The reaction rate is calculated as

(19)¶\[K_{s} = X_m(H_2S|C) K_b\]

where \(K_{b}\) is the biodegradation rate (the rate-limiting step which generates H2S), and \(X_m(H_2S|C)\) is the ratio of moles H2S produced per mole of cellulosics consumed. This is the parameter RXH2S in the BRAGFLO v6.02 User’s Manual. Currently the value of \(X_m(H_2|C)\) (SMIC_H2) used for \(X_m(H_2S|C)\) in PA calculations.

The rate constant \(K_{s}\) is split into \(K_{s}^{c}\), representing sulfidation of the corrosion products of iron (Fe(OH)2) (rate for Eq. (17)), and \(K_{s}^{i}\), representing sulfidation of metallic iron (rate for Eq. (18)). Fe(OH)2 sulfidation kinetically dominates Fe sulfidation. This is modeled by first determining the available Fe(OH)2 . If Fe(OH)2 is available in excess of H2S generation during a given time step, then \(K_{s}\) is entirely portioned to \(-K_s^c\), and \(-K_s^i\) is zero. If there is not enough Fe(OH)2 available to react with all of the H2S generated during a timestep, then \(K_{s}\) is first portioned to \(-K_s^c\) according to available Fe(OH)2, then the remaining \(K_{s}\) is portioned to \(-K_s^i\).

The instantaneous rates for each chemical species for iron sulfidation are shown in the table below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown. Note that \(H_2S\) is not tracked.

species	rate	STCO_##	units	notes
Fe	\(-K_s^i\)	STCO_43 = -1	mol-Fe/m3/s
Fe(OH)2	\(-K_s^c\)	STCO_35 = -1	mol-Fe(OH)2/m3/s
H2S	n/a	n/a	n/a	not tracked
FeS	\(+K_s^i\) \(+K_s^c\)	STCO_36 = +1 STCO_46 = +1	mol-H2/m3/s
H2	\(+K_s^i\) \(+K_s^c\)	STCO_31 = -1 STCO_41 = 0	mol-H2/m3/s mol-H2/m3/s
H2O	\(+2K_s^c\)	STCO_32 = +2	mol-H2O/m3/s

MgO Hydration Reaction¶

MgO hydration to brucite is modeled by

(20)¶\[MgO + H_{2}O = Mg(OH)_{2}\]

For every mole of MgO consumed, one mole of brine is consumed and one mole of brucite is produced. The reaction rate constant for for MgO hydration to brucite, \(K_{m}\), is calculated according to:

(21)¶\[K_{m} = \left({R_{MI}s_{eff} + R_{MH}\left({1-s_{eff}}\right)}\right)\]

(22)¶\[R_{MI} = max(r_{MI},r_{MH}) D_{m}\]

(23)¶\[R_{MH} = r_{MH} D_{m}\]

where \(r_{MI}\) is the inundated brucite rate in [mol-brucite/kg-MgO/s] (specified by BRUCITES or BRUCITEC in the input deck depending on deep brine intrusion), \(r_{MH}\) is the humid brucite rate in [mol-brucite/kg-MgO/s] (specified by BRUCITEH in the input deck), and \(D_{m}\) is the initial mass concentration of MgO. The value for \(D_{m}\) is calculated according to

(24)¶\[D_{m} = \frac {M_{biodegradables} W_{MgO} X} {V W_{biodegradables}}\]

where \(V\) is the volume of the waste panel in [m3], \(M_{biodegradables}\) is the total initial mass of biodegradables in [kg], \(W_{biodegradables}\) is the average molecular weight of the biodegradables in [kg/mol], \(W_{MgO}\) is the molecular weight of MgO, and \(X\) is a unitless MgO excess factor which is the ratio of moles of MgO to moles of organic carbon in the waste panel (specified by MGO_EF in the input deck). This amount of MgO is chosen because it should be a sufficient amount of MgO to remove CO2 produced during biodegradation reactions via the product brucite.

The instantaneous rates for each chemical species for MgO hydration to brucite are shown in the table below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown.

species	rate	STCO_##	units
MgO	\(-K_m\)	STCO_57 = -1	mol-MgO/m3/s
H2O	\(-K_m\)	STCO_52 = -1	mol-H2O/m3/s
Mg(OH)2	\(+K_m\)	STCO_58 = +1	mol-Mg(OH)2/m3/s

Brucite and MgO Carbonation¶

Brucite carbonation to hydromagnesite is modeled by

(25)¶\[5Mg(OH)_{2} + 4CO_{2} = Mg_{5}(CO_{3})_{4}:4H_{2}O\]

MgO carbonation to magnesite is modeled by

(26)¶\[MgO + CO_{2} = Mg(CO_{3})\]

These two reactions are modeled in a similar fashion to the iron hydroxide and iron sulfidation reactions. CO2, generated from the biodegradation reaction, is assumed to be the limiting reactant. The CO2 generation rate is given by

(27)¶\[K_{h} = X_m(CO_2|C) K_b\]

where \(X_m(CO_2|C) = 1\) is the ratio of moles of CO2 produced per mole of carbon generated by the biodegradation reactions, and \(K_{b}\) is the biodegradation rate.

The rate constant \(K_{h}\) is then split between the brucite carbonation reaction, \(K_{h, brucite}\) and the MgO carbonation reaction, \(K_{h, MgO}\). Brucite carbonation is assumed to kinetically dominate MgO carbonation. If the amount of brucite available to react during a timestep is in excess of the CO2 generated, then \(K_{h}\) is entirely portioned to \(K_{h, brucite}\), and \(K_{h, MgO}\) is zero. If the amount of brucite available to react during a timestep is less than the CO2 generated, then \(K_{h}\) is portioned to \(K_{h, brucite}\) such that all of the available brucite is consumed, then the remaining fraction of \(K_{h}\) is portioned to \(K_{h, MgO}\).

The instantaneous rates for each chemical species for brucite carbonation and MgO carbonation are shown in the tables below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown.

species	rate	STCO_##	units	notes
Mg(OH)2	\(K_{h, brucite}\)	STCO_68 = -1.25	mol-Mg(OH)2/m3/s
CO2	n/a	n/a	n/a	not tracked
Hydromagnesite	\(K_{h, brucite}\)	STCO_60 = +0.25	mol-hymag/m3/s

species	rate	STCO_##	units	notes
MgO	\(K_{h, MgO}\)	STCO_77 = -1	mol-MgO/m3/s
CO2	n/a	n/a	n/a	not tracked
Magnesite	\(K_{h, MgO}\)	STCO_79 = +1	mol-MgCO3/m3/s

Hydromagnesite Dehydration Reaction¶

Hydromagnesite is not considered thermodynamically stable under repository conditions, and is expected to dehydrate to form magnesite, producing brine, as modeled by

(28)¶\[Mg_{5}(CO_{3})_{4}:4H_{2}O = 4MgCO_{3} + Mg(OH)_{2} + 4H_{2}O\]

The reaction rate constant for hydromagnesite dehydration, \(K_{y}\), is calculated according to:

(29)¶\[K_{y} = R_{hymagcon} C_{hymag}\]

where \(R_{hymagcon}\) is the hydromagnesite conversion rate in [mol-hydromagnesite/kg-hydromagnesite/s] (specified by HYMAGCON in the input deck), and \(C_{hymag}\) is the current mass concentration of hydromagnesite in the waste panel (calculated internally). The units of \(K_{y}\) are [mol-hydromagnesite/m3/s].

The instantaneous rates for each chemical species for hydromagnesite dehydration are shown in the table below. Positive rates indicate a source while negative rates indicate a sink. The default value for the stoichiometric matrix is also shown.

species	rate	STCO_##	units
Hydromagensite	\(-K_y\)	STCO_80 = -1	mol-hymag/m3/s
MgCO3	\(+4K_y\)	STCO_89 = +4	mol-MgCO3/m3/s
Mg(OH)2	\(+K_y\)	STCO_88 = +1	mol-Mg(OH)2/m3/s
H2O	\(+4K_y\)	STCO_82 = +4	mol-H2O/m3/s

Gas Generation Rate¶

The gas generation rate is calculated by summing the hydrogen and nitrogen rates from each of the modeled reactions. The following reactions produce or consume H2 gas:

reaction	rate	STCO_##	units
anoxic iron corrosion	\(+K_c\)	STCO_11 = +1	mol-H2/m3/s
biodegradation	\(K_b X_m(H_2\|C)\)	\(X_m(H_2\|C)\)	mol-H2/m3/s
FeOH sulfidation	\(+K_s^c\)	STCO_41 = -1	mol-H2/m3/s

The hydrogen gas generation rate is the sum of the “rate” column in the table above,

(30)¶\[R_{gas} = K_c + K_bX_m(H_2|C) + K_s^c\]

This rate is assigned as a gas source term in the governing equations for fluid flow (see GENERAL Mode).

Brine Generation Rate¶

The brine generation rate is calculated by summing the H2O rates from each of the modeled reactions. The following reactions produce brine:

reaction	rate	STCO_##	units
anoxic iron corrosion	\(-2K_c\)	STCO_12 = -2	mol-H2O/m3/s
biodegradation	\(K_b X_m(H_2O\|C)\)	\(X_m(H_2O\|C)\)	mol-H2O/m3/s
iron sulfidation	\(+2K_s^c\)	STCO_32 = +2	mol-H2O/m3/s
MgO hydration	\(-K_m\)	STCO_52 = -1	mol-H2O/m3/s
Hymag dehydration	\(+4K_y\)	STCO_82 = +4	mol-H2O/m3/s

The brine generation rate is the sum of the “rate” column in the table above,

(31)¶\[R_{H2O} = -2 K_c + K_bX_m(H_2O|C) + 5 K_b^s + 2 K_s^c - K_m + 4 K_y\]

This rate is assigned as a liquid source term in the governing equations for fluid flow (see GENERAL Mode) after taking into account the weight of salt:

(32)¶\[R_{brine} = R_{H2O} / (1 - 0.01S)\]

where \(S\) is the salt weight percent as indicated by SALT_PERCENT.

Reaction Rate Smoothing and Tapering¶

Prior to using the calculated rate constants from each model reaction, the rates are smoothed and tapered. The main purpose of smoothing and tapering the reaction rate constants is to avoid running out of a reactant during the duration of the current timestep when the remaining inventory is updated. Rate smoothing is implemented according to

(33)¶\[K_{smoothed} = K \left({1 - e^{\left({\alpha \frac{C}{C_i}}\right)}}\right)\]

The smoothed rate is a function of the raw calculated rate, \(K\), and the ratio of the current concentration of a relevant species to its initial concentration in the waste panel. The parameter \(\alpha\) is specified by ALPHARXN in the input deck. When the relevant species concentration relative to it’s initial concentration falls low, the reaction rate constant is decreased so that it follows a smooth curve to zero. For each modeled reaction, the relevant species is typically the limiting species that appears on the left hand side of the chemical equation, as summarized in the following table:

reaction	species for C/Ci
anoxic iron corrosion	Fe
biodegradation	C6H10O5
iron sulfidation	Fe
MgO hydration	MgO
brucite carbonation	MgO
HM dehydration	MgO

Immediately after smoothing, the smoothed reaction rate constant is additionally tapered (e.g. the rate is limited to the amount of the limiting reactant divided by the current timestep size). While the concept is similar to smoothing, the relevant species for tapering can be different for each equation, especially when there are multiple equations per modeled reaction (as is the case for biodegradation and iron sulfidation), and it is possible for there to be more than one limiting species. The following table summarizes the species which are used to taper each reaction rate:

reaction	rate	species for tapering
anoxic iron corrosion	\(K_c\)	Fe
biodegradation	\(K_b\)	C6H10O5
Fe sulfidation	\(K_s^i\)	Fe
Fe(OH)2 sulfidation	\(K_s^c\)	Fe(OH)2
MgO hydration	\(K_m\)	MgO
brucite carbonation	\(K_{h, brucite}\)	Mg(OH)2
MgO carbonation	\(K_{h, MgO}\)	MgO
HM dehydration	\(K_y\)	hydromagnesite

Chemical Species Inventory Update¶

At the beginning of each time step, each tracked chemical species is updated using the reaction rate constant calculated at the end of the previous time step and the length of the previous time step. The update is calculated according to

(34)¶\[C_{t} = C_{t-1} + \left( K_{t-1} dt \right)\]

where \(C_{t}\) is the updated concentration for the current time step, \(C_{t-1}\) is the old concentration in the previous time step, \(K_{t-1}\) is the rate constant calculated in the previous time step, and \(dt\) is the length of the previous time step.