A two-dimensional implementation of the proposed method in a heterogeneous aquifer 611
is given in this section. The objective of this example is to illustrate the application of 612
32 the presented random walk approach in a more realistic setting. In doing this, we 613
analyze the need of fully describing non-linear chemical kinetics in heterogeneous 614
porous media. 615
We study a reactive transport problem in a 2D rectangular confined aquifer with 616
dimensions of 100×50 m2. The aquifer is characterized by a randomly generated log- 617
normally distributed hydraulic conductivity field 𝑌𝑌 = ln 𝐾𝐾 with a mean of 〈𝑌𝑌〉 = 3 and 618
a variance of 𝜎𝜎𝑌𝑌2 = 1. The 𝑌𝑌 field follows an isotropic exponential covariance function 619
model with integral scale of 𝑀𝑀𝑌𝑌 = 5 m. The other aquifer properties are assumed 620
homogeneous. Groundwater flow is considered at steady-state and subject to constant 621
head conditions at the lateral boundaries and impermeable conditions otherwise. As a 622
result, the mean flow direction is oriented in the x-direction and characterized by a 623
mean hydraulic gradient of 0.00622. The flow problem is solved with a finite-difference 624
code, MODFLOW-2000 [Harbaugh et al., 2000], with a domain discretized into regular 625
cells of size 0.5 m. The resulting cell face flows were used to compute the random walk 626
particle velocities according to the hybrid interpolation method suggested by LaBolle et 627
al. [1996]. 628
The reactive transport problem is similar to the one defined in example 2 but considers a 629
two-dimensional heterogeneous porous media. A schematic representation of the system 630
is shown in Figure 11. A plume of dissolved organic matter, retarded with respect to 631
groundwater by linear sorption, passes after some time through an oxygen plume. The 632
chemical reaction follows the Michaelis-Menten kinetic model with the same 633
formulation and parameter values as given in example 2. Table 2 shows the values of 634
the parameters. The concentrations of the reactants are initially uniform in two separate 635
rectangular areas depicted in Figure 11 and zero everywhere else in the domain. The 636
concentration of all compounds in the inflow water is zero at all times. 637
33 The fast method of Botev et al. [2010] was used to determine the kernel bandwidth with 638
a slight modification: the anisotropic kernel bandwidth matrix 𝑯𝑯 obtained by this 639
method was transformed into an isotropic bandwidth by matching the determinants, i.e., 640
ℎ2 = det (𝑯𝑯). As explained in section 3, the use of isotropic kernels facilitates the
641
computation of the compensation function 𝑔𝑔 at the 𝐗𝐗𝐴𝐴𝐴𝐴𝑖𝑖𝑗𝑗 location by linear interpolation. 642
However, the kernel obtained by this approach is suboptimal compared to the original 643
method by Botev et al. [2010], and presumably produced a slower convergence with 644
respect to the number of particles. 645
The convergence of the random walk solution was controlled by choosing a small 646
enough time step and by performing a sensitivity analysis with respect to the number of 647
particles. As expected, the convergence occurred for a higher number of particles 648
compared to the 1D examples. Nevertheless, by using only 32,768 particles of each 649
reactant, the estimated error in the total amount of product generated was below 1% as 650
compared to the solution obtained with 131,072 particles. Figure 12 shows the three 651
particle plumes at different moments of the simulation (for a better visualization, only a 652
random subsample of 5,000 particles is shown), and the corresponding KDE 653
reconstruction of the product concentrations. 654
655
The actual impact of the reaction kinetics on the problem solution depends on whether 656
mixing or chemical kinetics is the limiting process. In order to illustrate this, we 657
compare the evolution of the CO2 production with the following equivalent second- 658 order reaction, 659 𝑟𝑟(𝐱𝐱, 𝑡𝑡) =𝑘𝑘 𝑘𝑘𝑓𝑓 CH2O 𝑘𝑘O2 𝑐𝑐CH2O 𝑐𝑐O2,
34 for three different values of 𝑘𝑘𝑓𝑓 ranging from five times smaller to five times higher than 660
the value given in Table 2. Figure 13 shows that for a very fast reaction rate the process 661
is mixing-limited (in this case mixing is driven by the difference in the retardation 662
coefficients), and therefore the reaction kinetics do not have a significant effect on the 663
results. These kind of reactions can be modeled as instantaneous (ref. xavi), as long as 664
the mixing process is well represented by the transport model. On the other hand, in 665
slow reactions, the reaction kinetics can make a very important difference in the results. 666
7. Conclusions 667
We have presented a new random walk particle tracking method to simulate reactions 668
with complex kinetics involving two reactants. Reactive transport is solved in two 669
stages: the first one corresponding to the chemical reactions, and the second one to the 670
standard advection-dispersion equation. The method is based on the representation of 671
particles by optimal kernel functions. This way, we derived the probability that a given 672
particle reacts with any particle associated with other reactants. In the proposed 673
methodology, complex kinetic reactions require linearizing a function of the local 674
concentrations at the location of highest probability density of encounter between 675
potentially reactive particle pairs. The implementation of the probability of reaction in 676
random walk models has been achieved in this paper by particle annihilation, but other 677
approaches such as particle mass variations can easily be incorporated. 678
679
In addition, two simple relationships should be satisfied to fulfill stoichiometry: one 680
relating the mass of interacting particles with the stoichiometric coefficients, and 681
another one relating the mass of particles produced from reactions with the 682
stoichiometric coefficients. In practice, the first relationship requires a careful choice of 683
35 the mass of the particles injected. The second relationship determines the mass of 684
particles produced from the chemical reaction. 685
686
Several synthetic examples demonstrate the potential applicability of the method in a 687
wide range of applications, ranging from reaction-rate laws with fractional exponents to 688
acidic dissolution and precipitation systems with nontrivial activity coefficients. Results 689
have shown that a good match with a finite difference solution is obtained with 690
relatively few particles. The method has been demonstrated to converge to the solution 691
with an increasing number of particles. This rate of convergence depends on the type of 692
chemical reaction, i.e., on the shape of the compensation function g. Finally, a 2D 693
example dealing with non-linear Michaelis-Menten biodegradation in a randomly 694
heterogeneous aquifer is provided to illustrate the capabilities of the method in a more 695 realistic setting. 696 697 698 Acknowledgements 699
Financial support was provided by the Spanish Government, through projects WE- 700
NEED, PCIN-2015-248; ACWAPUR, PCIN-2015-239, and INDEMNE, CGL2015- 701
69768-R (MINECO/FEDER). GS acknowledges financial support by AGAUR. The 702
paper provides all the information needed to replicate the results. The codes and the 703
output data from the simulations are freely available from the authors upon request. 704
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40 856 857 858 859 Tables 860
Table 1: Simulation parameter values used in example 1.* 861 𝐀𝐀 𝐁𝐁 𝐂𝐂 (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 (𝟒𝟒𝟒𝟒, 𝟔𝟔) (𝟓𝟓𝟒𝟒, 𝟔𝟔) − 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝟏𝟏 𝐦𝐦𝐦𝐦𝐦𝐦 𝟏𝟏 𝐦𝐦𝐦𝐦𝐦𝐦 𝟒𝟒 𝜶𝜶, 𝜷𝜷, 𝜸𝜸 𝟐𝟐. 𝟑𝟑 𝟏𝟏. 𝟑𝟑 𝟏𝟏 𝜽𝜽 (eq. 37) 𝟐𝟐. 𝟑𝟑 𝟏𝟏. 𝟑𝟑 − 𝑹𝑹 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝒌𝒌𝒇𝒇 𝟔𝟔 (𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑)−𝟐𝟐.𝟔𝟔 𝐝𝐝𝐝𝐝𝐲𝐲−𝟏𝟏 𝒒𝒒 𝟒𝟒. 𝟑𝟑 𝐦𝐦/𝐝𝐝𝐝𝐝𝐲𝐲 𝝓𝝓 𝟒𝟒. 𝟐𝟐𝟓𝟓 𝑫𝑫 𝟒𝟒. 𝟒𝟒 𝐦𝐦𝟐𝟐/𝐝𝐝𝐝𝐝𝐲𝐲 𝝉𝝉 𝟖𝟖𝟒𝟒 𝐝𝐝𝐝𝐝𝐲𝐲𝐝𝐝
* (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 are the mean and standard deviation defining the initial normal distribution of
862
solute particles in space, 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is the total amount of substance at the start of the simulation,
863
and 𝜏𝜏 is the total simulated time. The other variables are defined in the text. 864
Table 2: Simulation parameter values used in example 2.* 865 𝐂𝐂𝐇𝐇𝟐𝟐𝐎𝐎 𝐎𝐎𝟐𝟐 𝐂𝐂𝐎𝐎𝟐𝟐 (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 (𝟕𝟕𝟒𝟒, 𝟐𝟐) (𝟓𝟓𝟒𝟒, 𝟖𝟖) − 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝟏𝟏 𝐦𝐦𝐦𝐦𝐦𝐦 𝟏𝟏 𝐦𝐦𝐦𝐦𝐦𝐦 𝟒𝟒 𝜶𝜶, 𝜷𝜷, 𝜸𝜸 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝑲𝑲𝑪𝑪𝑯𝑯𝟐𝟐𝑶𝑶, 𝑲𝑲𝑶𝑶𝟐𝟐(eq.39) 𝟏𝟏. 𝟔𝟔𝟔𝟔𝟔𝟔𝟕𝟕 𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑 𝟒𝟒. 𝟒𝟒𝟏𝟏𝟓𝟓𝟔𝟔 𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑 − 𝑹𝑹 𝟑𝟑 𝟏𝟏 𝟏𝟏 𝒌𝒌𝒇𝒇 𝟒𝟒. 𝟏𝟏𝟓𝟓 (𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑) 𝐝𝐝𝐝𝐝𝐲𝐲−𝟏𝟏
41
𝒒𝒒 𝟒𝟒. 𝟑𝟑𝟐𝟐 𝐦𝐦/𝐝𝐝𝐝𝐝𝐲𝐲
𝝓𝝓 𝟒𝟒. 𝟐𝟐𝟓𝟓
𝑫𝑫 𝟒𝟒. 𝟓𝟓 𝐦𝐦𝟐𝟐/𝐝𝐝𝐝𝐝𝐲𝐲
𝝉𝝉 𝟔𝟔𝟓𝟓 𝐝𝐝𝐝𝐝𝐲𝐲𝐝𝐝
* (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 are the mean and standard deviation defining the initial normal distribution of
866
solute particles in space, 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is the total amount of substance at the start of the simulation,
867
and 𝜏𝜏 is the total simulated time. The other variables are defined in the text. 868
869
Table 3: Simulation parameter values used in example 3.* 870 𝐂𝐂𝐝𝐝𝟐𝟐+ 𝐂𝐂𝐎𝐎 𝟑𝟑 𝟐𝟐− 𝐂𝐂𝐝𝐝𝐂𝐂𝐎𝐎 𝟑𝟑 (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 (𝟐𝟐𝟓𝟓, 𝟓𝟓) (𝟑𝟑𝟓𝟓, 𝟖𝟖) − 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝟐𝟐. 𝟓𝟓 𝐦𝐦𝐦𝐦𝐦𝐦 𝟐𝟐. 𝟓𝟓 𝐦𝐦𝐦𝐦𝐦𝐦 𝟒𝟒 𝜶𝜶, 𝜷𝜷, 𝜸𝜸 𝟏𝟏 𝟏𝟏 𝟏𝟏 å𝑪𝑪𝒊𝒊𝟐𝟐+, å𝑪𝑪𝑶𝑶 𝟑𝟑 𝟐𝟐− (eq. 43) 𝟒𝟒. 𝟔𝟔 𝐧𝐧𝐦𝐦 𝟒𝟒. 𝟓𝟓 𝐧𝐧𝐦𝐦 − 𝑹𝑹 𝟏𝟏 𝟏𝟏 ∞ 𝒌𝒌𝒐𝒐𝒐𝒐𝒔𝒔 𝟒𝟒. 𝟒𝟒𝟒𝟒𝟐𝟐 (𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑)−𝟐𝟐.𝟔𝟔 𝐝𝐝𝐝𝐝𝐲𝐲−𝟏𝟏 𝒌𝒌𝒆𝒆𝒒𝒒 𝟏𝟏𝟒𝟒−𝟐𝟐.𝟑𝟑 (𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑)𝟐𝟐 𝒒𝒒 𝟒𝟒. 𝟏𝟏 𝐦𝐦/𝐝𝐝𝐝𝐝𝐲𝐲 𝝓𝝓 𝟒𝟒. 𝟐𝟐𝟓𝟓 𝑫𝑫 𝟒𝟒. 𝟏𝟏𝟓𝟓 𝐦𝐦𝟐𝟐/𝐝𝐝𝐝𝐝𝐲𝐲 𝝉𝝉 𝟏𝟏𝟒𝟒𝟒𝟒 𝐝𝐝𝐝𝐝𝐲𝐲𝐝𝐝
* (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 are the mean and standard deviation defining the initial normal distribution of
871
solute particles in space, 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is the total amount of substance at the start of the simulation,
872
and 𝜏𝜏 is the total simulated time. The other variables are defined in the text. 873
874
Table 4: Simulation parameter values used in example 4.* 875
𝐇𝐇+ 𝐂𝐂𝐝𝐝𝟐𝟐+ 𝐇𝐇𝐅𝐅𝟒𝟒
(𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 (𝟗𝟗𝟒𝟒, 𝟓𝟓) (𝟗𝟗𝟒𝟒, 𝟐𝟐𝟒𝟒) −
𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 𝟐𝟐 𝐦𝐦𝐦𝐦𝐦𝐦 𝟒𝟒 𝐦𝐦𝐦𝐦𝐦𝐦 𝟒𝟒
42 𝜽𝜽 (eq. 45) 𝟏𝟏. 𝟔𝟔 −𝟒𝟒. 𝟖𝟖 − 𝑹𝑹 𝟏𝟏 𝟏𝟏 𝟏𝟏 𝒌𝒌𝒇𝒇 𝟒𝟒. 𝟕𝟕𝟐𝟐 (𝐦𝐦𝐦𝐦𝐦𝐦/𝐦𝐦𝟑𝟑)𝟒𝟒.𝟐𝟐 𝐝𝐝𝐝𝐝𝐲𝐲−𝟏𝟏 𝒒𝒒 𝟒𝟒. 𝟓𝟓 𝐦𝐦/𝐝𝐝𝐝𝐝𝐲𝐲 𝝓𝝓 𝟒𝟒. 𝟐𝟐𝟓𝟓 𝑫𝑫 𝟒𝟒. 𝟒𝟒 𝐦𝐦𝟐𝟐/𝐝𝐝𝐝𝐝𝐲𝐲 𝝉𝝉 𝟕𝟕 𝐝𝐝𝐝𝐝𝐲𝐲𝐝𝐝
* (𝝁𝝁𝒙𝒙 , 𝝈𝝈𝒙𝒙)𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 are the mean and standard deviation defining the initial normal distribution of
876
solute particles in space, 𝒎𝒎𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊𝒊 is the total amount of substance at the start of the simulation,
877
and 𝜏𝜏 is the total simulated time. The other variables are defined in the text. 878 879 880 Figures 881 882
Figure 1: Schematic example of a product between two Gaussian pdf’s in one 883
dimension. The product (yellow) is another Gaussian function centered in 𝑋𝑋𝐴𝐴𝐴𝐴 and with 884
43 a standard deviation ℎ𝑎𝑎𝑏𝑏 = �𝐻𝐻𝐴𝐴𝐴𝐴. Its integral over x is the probability of collocation of 885
the two particles. 886
887
888
Figure 2: Solute concentrations in example 1, at the start of the simulation and after 80 889
days. The error zone around the Particle Tracking curves indicates ±1 standard 890
deviation. 891
44 892
Figure 3: Evolution in time of the total amount of substance of each compound in 893
example 1. The error zone around the Particle Tracking curves indicates ±1 standard 894
deviation. 895
896
Figure 4: Solute concentrations in example 2, at the start of the simulation and after 65 897
days. The error zone around the Particle Tracking curves indicates ±1 standard 898
deviation. 899
45 900
Figure 5: Evolution in time of the total amount of substance of each compound in 901
example 2. The error zone around the Particle Tracking curves indicates ±1 standard 902
deviation. 903
904
Figure 6: Solute concentrations in example 3, at the start of the simulation and after 905
100 days. The error zone around the Particle Tracking curves indicates ±1 standard 906
deviation. 907
46 908
Figure 7: Evolution in time of the total amount of substance of each compound in 909
example 3. The error zone around the Particle Tracking curves indicates ±1 standard 910
deviation. 911
912
Figure 8: Solute concentrations in example 4, at the start of the simulation and after 7 913
days. The error zone around the Particle Tracking curves indicates ±1 standard 914
deviation. 915
47 916
Figure 9: Evolution in time of the total amount of substance of each compound in 917
example 4. The error zone around the Particle Tracking curves indicates ±1 standard 918
deviation. 919
920
Figure 10: The two measured error components for different initial number of particles 921
of the reactants. 922
48 923