# Multiple Kinetic Parameterization in a Reactive Transport Model Using the Exchange Monte Carlo Method

^{1}

^{2}

^{*}

## Abstract

**:**

_{2}O system. We tested the robustness and accuracy of our method over a wide range of noise intensities. This methodology can be widely applied to kinetic analyses of various kinds of water–rock interactions.

## 1. Introduction

## 2. Methodology

#### 2.1. General Framework

_{i}, J

_{i}, and R

_{chem}are the porosity (-), concentration of i (mol/cm

^{3}solution), total transport flux of aqueous species i, and the rate of gain or loss of chemical species i by surface reactions (mol/cm

^{3}solution/s), respectively.

_{1}, P

_{2}, …, P

_{N}}. By running the reactive transport model with Θ, spatial variations in minerals and ionic concentrations can be forward-calculated. Given that the pressure–temperature (P–T) condition and timescales are constrained, Θ is inversely determined from maximum-likelihood estimation by minimizing the difference between the calculated and observable data (E(Θ)) as follows:

_{i,j}, y

_{i,j}(Θ), and σ

_{i}are the abundance of i at the j-th distance obtained from observable data, the abundance of i at the j-th distance calculated from Θ, and the standard deviation of the observable data for i, respectively. N

_{obs}is the number of observations in each dataset. Therefore, the value of E(Θ) denotes the average weighted square error per data point.

_{r}= [‒1,1]) to one of the unknown parameters, and by using log

_{10}(P

_{i,}

_{can}) = log

_{10}(P

_{i}) + n

_{r}, where P

_{i}and P

_{i,}

_{can}are the current unknown parameter and the candidate unknown parameter, respectively. Therefore, the candidate parameter vector is Θ

_{can}= {P

_{1,can}, P

_{2}, …, P

_{N}} (Figure 1a). E(Θ

_{can}) is then calculated from the candidate values for the parameters, and ΔE is obtained from ΔE = E(Θ) − E(Θ

_{can}), where E(Θ) and E(Θ

_{can}) are the values of E calculated from Θ (={P

_{1}, P

_{2}, …, P

_{N}}) and Θ

_{can}(={P

_{1,can}, P

_{2}, …, P

_{N}}), respectively. The candidate vector is accepted if the probability is min(1, exp(–ΔEβ

_{n})). β

_{n}is the “inverse temperature” on replica n, which controls the acceptance or rejection updates of the candidate values. At low β

_{n}, the candidate value is easily updated when ΔE > 0, but this is not the case at high β

_{n}. The parameter vector after this candidate update attempt is {P

_{1,new}, P

_{2}, …, P

_{N}}, where P

_{1,new}= P

_{1,can}when the candidate value is accepted. However, P

_{1,new}= P

_{1}is when the candidate value is rejected. After one local update, the next unknown parameter is changed in a similar way.

_{n}, were exchanged with probability min(1, exp(–ΔEΔβ

_{n})) (Figure 1b). This allows the local minima to be overcome, because a system at low β

_{n}can provide new local optimizers for a system at high β

_{n}, thereby allowing tunneling between the local minima and improving convergence to a global optimum.

_{n}and number of replicas are important since these values have a close relation to the exchange ratio (i.e., the acceptance ratio of the exchange process) [38], which are directly related to the efficiency and computational cost of parameter search. Since a large Δβ

_{n}(i.e., a small number of replicas) would result in a low average exchange ratio, Δβ

_{n}should not be too large. On the other hand, in order to make the average exchange ratio high, Δβ

_{n}has to be small (i.e., a large number of replicas is required), although the associated computational cost is huge. The number of replicas to be used in the EMC method is unclear and completely problem-dependent; they are chosen by trial and error. In a verification of the method described later, we used 20 replicas since >20 replicas are typically used [39,40]. The range of β was set from 20 to 350, and Δβ

_{n}between each replica was constructed as a geometrical progression, as it would improve sampling efficiency [28]. As discussed later, we show that the EMC method can find the true values, given the setting of Δβ

_{n}and number of replicas, as described above. Note that the exchange between the replicas occurred frequently during optimization, and the EMC method provided the true answer of multiple unknown parameters.

## 3. Synthetic Data: Silica Metasomatism in an Olivine–Quartz–H_{2}O System

#### 3.1. Model System

_{2}O system (Figure 2). This system was chosen for the following reasons: (1) The diffusional metasomatic zoning of talc and serpentine zones between quartz and olivine was classically modeled [41,42]. This system is relatively simple, but it involves the dissolution/precipitation processes of several minerals as well as element diffusion; accordingly, it serves as a good example of reactive transport processes in water–rock interactions. (2) The progress of the metasomatic reaction in the olivine–quartz–H

_{2}O system was experimentally investigated [43], which is a useful when considering a realistic model. (3) The reaction rate is critical to understanding the various processes in subduction zones, including metamorphism, earthquakes, and volcanism [44,45].

_{sat}) of 8.58 MPa in a porous medium. The porous medium was initially composed of 63 vol.% olivine (forsterite) grains and 37 vol.% fluid-saturated porosity (Figure 1). Porosity was relatively higher than that of metamorphic rocks, although the high porosity used in this study could be found at surface and subsurface water–rock hydrothermal environments, as well as in hydrothermal experiments with mineral powder. At distance (x) = 0, a source of aqueous silica (i.e., quartz) was present, and the aqueous silica in the solution was transported by diffusion into the porous olivine zone (Figure 2). Therefore, silica concentration changed in time and space due to diffusion and surface reactions. In response to the local silica concentration, several surface reactions could take place simultaneously.

#### 3.2. Kinetic Forward Model

_{2(aq)}, Mg

^{2+}, and H

^{+}) are expected to be involved in surface reactions in the system. Of course, these species could be implemented in the reactive transport model and they would normally be considered. However, several numerical models suggest that the serpentinization system could be modeled with an Mg-fixed frame [46,47]. Therefore, in this study, we simplified the reactions and assumed that the surface reactions in the olivine zone could be expressed by the following overall reactions:

_{2}SiO

_{4}+ 5SiO

_{2(aq)}+ 2 H

_{2}O → 2Mg

_{3}Si

_{4}O

_{10}(OH)

_{2}

_{2}SiO

_{4}+ SiO

_{2(aq)}+ 4H

_{2}O → 2Mg

_{3}Si

_{2}O

_{5}(OH)

_{4}

_{3}Si

_{2}O

_{5}(OH)

_{4}+ 2SiO

_{2(aq)}↔ Mg

_{3}Si

_{4}O

_{10}(OH)

_{2}+ H

_{2}O

_{2}SiO

_{4}+ 2H

_{2}O ↔ 2Mg(OH)

_{2}+ SiO

_{2(aq)}

_{2}+ 2SiO

_{2(aq)}↔ Mg

_{3}Si

_{2}O

_{5}(OH)

_{4}+ H

_{2}O

_{n}, A

_{i}, and C

^{n}

_{SiO2(aq),eq}are the reaction-rate constant (mol SiO

_{2(aq)}/cm

^{2}mineral/s) for the overall reaction n, the bulk surface area (cm

^{2}mineral/cm

^{3}rock) of mineral i, and the equilibrium silica concentration (mol SiO

_{2(aq)}/cm

^{3}solution) for reaction n, respectively. For the reactions where both the forward and backward reactions were considered (i.e., R3 and R5), the rate constant for reaction n was expressed as k

_{n}

_{+}for the forward reaction and k

_{n}

_{–}for the backward reaction (Figure 2). The C

^{n}

_{SiO2(aq),eq}values for R1–R5 were obtained as follows: (1) the equilibrium silica molarities (mol/kg solution) for the overall reactions R1–R5 are calculated from logK obtained from SUPCRTBL [49], assuming that the activities of water and minerals were one; and (2) the molarities are converted to volumetric concentrations (mol SiO

_{2(aq)}/cm

^{3}solution) by extrapolation of the partial molar volume of SiO

_{2(aq)}(–9.1 cm

^{3}/mol at 300 °C and P

_{sat}; [50]), as in Watson et al. [16]. Log K and C

^{n}

_{SiO2(aq),eq}values obtained for R1–R5 are summarized in Table 1.

_{i}represents the surface area of the reactant or product for each overall reaction. In the present study, we introduce effective rate constant k′

_{n}[34]:

_{n}has the unit of s

^{–1}, and k′

_{n}values for R1–R5 (k′

_{R1}, k′

_{R2}, k′

_{R3+}, k′

_{R3–}, k′

_{R4}, k′

_{R5+}, and k′

_{R5–}) are the fitting parameters in these calculations.

_{i}and ${\overline{V}}_{i}$ indicate the amount of mineral (mol/cm

^{3}rock) and the molar volume of minerals (cm

^{3}mineral/mol), respectively, which are sourced from Reference [51]. N

_{i}is the number of minerals in the model. In our calculations, forsterite, talc, lizardite, and brucite were considered; thus, N

_{i}= 4.

_{2(aq)}in a porous medium is given as follows [48,52]:

_{SiO2(aq)}is the diffusion coefficient of SiO

_{2(aq)}in pure solution. Formation factor (F) is defined as in Reference [52], where

_{SiO2(aq)}has only been determined for limited P–T conditions [16,54,55,56], and not for the P–T conditions of the present experiment, D

_{SiO2(aq)}is also one of the fitting parameters in our calculations.

_{2(aq)}involving diffusion and surface reactions in the olivine region, as follows:

_{SiO2(aq)}at t = 0 (initial conditions) was uniformly set to 7.12 × 10

^{–8}mol/cm

^{3}, similar to the value of C

_{SiO2(aq)}obtained after a 24 h run in hydrothermal serpentinization experiments on olivine and NaCl–NaHCO

_{3}fluid [20]. The value of C

_{SiO2(aq)}at x = 0 (boundary conditions) was set to 7.41 × 10

^{–6}mol/cm

^{3}, which is a quartz-saturated SiO

_{2(aq)}concentration at 300 °C and vapor-saturated pressure. Given that our model does not consider the nucleation of secondary minerals, we added a small amount (1.0 × 10

^{–7}vol.%) of seed lizardite, talc, and brucite at t = 0.

#### 3.3. Synthetic Datasets

_{obs}= 70), and the distributions were obtained from the reactive transport model (Equation (9)) after 180 h of reaction time when using the following randomly generated parameters: log

_{10}(D

_{SiO2(aq)}) = –3.0; log

_{10}(k′

_{R1}) = –4.0; log

_{10}(k′

_{R2}) = –3.0; log

_{10}(k′

_{R3+}) = –6.0; log

_{10}(k′

_{R3−}) = –8.0; log

_{10}(k′

_{R4}) = –1.0; log

_{10}(k′

_{R5+}) = –2.0; log

_{10}(k′

_{R5–}) = –3.0; and log

_{10}(m) = –0.47. To improve the efficiency of the parameter sampling using the EMC method, the parameter search range was limited from 0 to –10 (in log units) for D

_{SiO2(aq)}and k′

_{n}, and 0 to 0.7 (log units) for m. The observation noise of σ

_{y}= 10

^{–1.5}, which equates to a 6.3% relative deviation of the maximum amount of each mineral, was added to the mineral and porosity data (Figure 3a–e). The noisy datasets (i.e., red data points in Figure 3a–e) were then analyzed using our method to observe its effectiveness in extracting the known reaction-rate constants.

## 4. Results and Discussion

#### 4.1. Extraction of Reaction Kinetic Parameters from Noisy Datasets with the Reactive Transport Model

_{SiO2(aq)}and k′

_{n}were uniformly set to 0.05 log units, whereas the ranges for m were set to 0.01 log units.

_{SiO2(aq)}, k′

_{R1}, k′

_{R2}, k′

_{R4}, k′

_{R5+}, and m appeared to be normally distributed, whereas the histograms of the other parameters (k′

_{R3+}, k′

_{R3–}, and k′

_{R5–}) had a uniform distribution with single peaks at around −10 and/or 0 (Figure 4). These single peaks would be an artificial effect due to a boundary condition of the parameter search ranges, which were limited to 0 to −10 in log units.

_{SiO2(aq)}, k′

_{R1}, k′

_{R2}, k′

_{R4}, and k′

_{R5+}; Figure 4), each true value was within the average value (±1σ) of the sampled parameters, indicating that the true parameters were successfully estimated despite errors from the noisy datasets. In particular, the reaction rate constant for R2 (k′

_{R2}), which controls the amount of lizardite (Figure 1), is estimated with low error (log

_{10}(k′

_{2}) = –2.99 ± 0.05; Figure 4). This is due to the fact that lizardite (0–10 mmol/cm

^{3}rock; Figure 3c) was the most abundant secondary mineral (cf. 0–0.15 mmol/cm

^{3}rock for talc (Figure 3b); 0.013 mmol/cm

^{3}rock for brucite (Figure 3d)) and that this reaction also controlled the amount of observed olivine and porosity (Figure 3a,e). This suggests that the main reaction controlling the observed mineral distribution can be qualitatively constrained using its estimated parameter errors.

_{R3+}, k′

_{R3–}, and k′

_{R5–}; Figure 4), the proposed method was unable to estimate the true values. This is because the parameters have a broad minimum, and therefore, any value could result in a good fit. This reflects the fact that these reactions (i.e., forward and backward reactions for R3, and backward reaction for R5) do not affect the amounts of different minerals present, and these reactions could be excluded from the model for simplicity. In fact, the backward reaction of R5 with a reaction-rate constant of k′

_{R5–}only occurs when C

_{SiO2(aq)}< C

^{R5}

_{SiO2(aq),eq}, which was never the case in our calculations (Figure 5). Therefore, the proposed method could also exclude this reaction path (i.e., k′

_{R3+}, k′

_{R3–}, and k′

_{R5–}could be set to zero), as it is not important in reconstructing the observed mineral distributions.

#### 4.2. Dependence of Estimation Accuracy on Observation Noise

_{y}= 10

^{–2.0}and 10

^{–1.2}, which corresponded to 2.0% and 12.6% deviations, respectively, relative to the maximum amount of each mineral. The fitted noisy datasets are shown in Supplementary Figure S1.

_{y}= 10

^{–1.2}. The errors were low when the observable noise was low. For example, log

_{10}(D

_{SiO2(aq)}) values were estimated to be –2.96 ± 0.06 at σ

_{y}= 10

^{–2.0}, –2.96 ± 0.24 at σ

_{y}= 10

^{–1.5}, and –2.80 ± 0.26 at σ

_{y}= 10

^{–1.2}(Figure 6).

## 5. Summary and Outlook

_{obs}) decreases, the error on parameters estimated by the proposed methodology would increase, as suggested by previous studies on parameter estimation [57,58]. Although the effects of sparseness in the dataset on the estimation error were not tested, in this study’s example at least, the effect of the amount of data was not large since the spatial patterns of individual minerals are rather smooth.

## Supplementary Materials

_{y}= 10

^{–2.0}and (b) σ

_{y}= 10

^{–1.2}used in the accuracy tests. Gray lines show the values predicted using the reactive transport model (Equation (9), with the parameters described in Section 3) and the red data points are the noisy data.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Schematic flow chart of the exchange Monte Carlo method for four replicate systems. (

**a**) Flow chart of local update. (

**b**) The local update was repeated after the exchange process. See Section 2.1 for details. MCS: Monte Carlo step.

**Figure 2.**Schematic illustration of the reactive transport model considered in this study. At x = 0, SiO

_{2(aq)}was transported by diffusion at a rate of D

_{SiO2(aq)}through the porous media of powdered forsterite, and secondary-mineral (talc, lizardite, and brucite) was produced. The gray bold line shows the overall reaction between two minerals. See Section 2.1 for details.

**Figure 3.**Synthetic datasets with a noise level of σ

_{y}= 10

^{–1.5}used in the accuracy tests. (

**a**–

**e**) Plots with gray lines show true values predicted using the reactive transport model (Equation (9), with the parameters given in Section 3) for (

**a**) forsterite, (

**b**) talc, (

**c**) lizardite, (

**d**) brucite, and (

**e**) pores. Red data points are synthetic noisy data.

**Figure 4.**Histograms showing estimated candidate values sampled using the exchange Monte Carlo method from the noisy datasets (σ

_{y}= 10

^{–1.5}; Figure 3a–e). Each gray region represents the average ± one standard deviation of the parameters sampled using the EMC method. Dotted red lines show the true values of each parameter described in Section 3. To allow comparison of each parameter, the bin ranges for D

_{SiO2(aq)}and k′

_{n}were uniformly set to 0.05 log units, whereas the range for m was set to 0.01 log units.

**Figure 5.**Spatiotemporal evolution of silica activity (a

_{SiO2(aq)}) predicted using the reactive transport model (Equation (9), with the parameters given in Section 3). a

_{SiO2(aq)}(and thus C

_{SiO2(aq)}) is always larger than C

^{R5}

_{SiO2(aq),eq}.

**Figure 6.**Noise effects on the accuracy of the estimated parameters. The dotted red lines show the true values of each parameter described in Section 3. The parameter errors are low when the observable noise (σ

_{y}) is low.

Reaction Number | Log K ^{§} | C^{n}_{SiO}_{2}_{(aq),eq} ^{ξ} (mol/cm^{3} Solution) |
---|---|---|

R1 | 17.148 | 2.649 × 10^{–7} |

R2 | 6.263 | 3.887 × 10^{–10} |

R3 | 5.442 | 1.354 × 10^{–6} |

R4 | –4.395 | 2.868 × 10^{–8} |

R5 | 9.724 | 9.789 × 10^{–9} |

^{§}Value at 300 °C and P

_{sat}obtained from SUPCRTBL [49].

**Calculated from log K at 300 °C and P**

^{ξ}_{sat}(see main text for details).

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Oyanagi, R.; Okamoto, A.; Tsuchiya, N.
Multiple Kinetic Parameterization in a Reactive Transport Model Using the Exchange Monte Carlo Method. *Minerals* **2018**, *8*, 579.
https://doi.org/10.3390/min8120579

**AMA Style**

Oyanagi R, Okamoto A, Tsuchiya N.
Multiple Kinetic Parameterization in a Reactive Transport Model Using the Exchange Monte Carlo Method. *Minerals*. 2018; 8(12):579.
https://doi.org/10.3390/min8120579

**Chicago/Turabian Style**

Oyanagi, Ryosuke, Atsushi Okamoto, and Noriyoshi Tsuchiya.
2018. "Multiple Kinetic Parameterization in a Reactive Transport Model Using the Exchange Monte Carlo Method" *Minerals* 8, no. 12: 579.
https://doi.org/10.3390/min8120579