# Machine-Learning-Assisted Investigation of the Diffusion of Hydrogen in
Brine by Performing Molecular Dynamics Simulation

Sree Harsha Bhimineni  [  Tianhang Zhou  [  [<EMAIL_ADDRESS>Saeed
Mahmoodpour  [  Mrityunjay Singh  [  Wei Li  [  Saientan Bag  [  Ingo Sass  [
Florian Müller-Plathe  [

###### Abstract

Deep saline aquifers are one of the best options for large-scale and long-term
hydrogen storage. Predicting the diffusion coefficient of hydrogen molecules
at the conditions of saline aquifers is critical for the modelling of hydrogen
storage. The diffusion coefficient of hydrogen molecules in chloride brine
with different cations ($\mathrm{Na}^{+}$, $\mathrm{K}^{+}$,
$\mathrm{Ca}^{2+}$) containing up to 5 $\mathrm{mol/kg_{H_{2}O}}$
concentration is numerically investigated using molecular dynamics (MD)
simulation. A wide range of pressure (1-218 atm) and temperature (298-648 K)
conditions is applied to cover the realistic operational conditions of the
aquifers. We find that the temperature, pressure and properties of ions
(compositions and concentrations) affect the hydrogen diffusion coefficient.
An Arrhenius behavior of the effect of temperature on the diffusion
coefficient has been observed with the temperature independent parameters
fitted using the ion concentration under constant pressure. However, it is
noted that the pressure strongly affects the diffusive behavior of hydrogen at
the high temperature ($\geq$ 400 K) regime, indicating the inaccuracy of the
Arrhenius model. Hence, we combine the obtained MD results with four models of
machine learning (ML), including linear regression (LR), random forest (RF),
extra tree (ET) and gradient boosting (GB) to provide effective predictions on
the hydrogen diffusion. The resultant combination of GB model with MD data
predicts the diffusion of hydrogen more effectively as compared to the
Arrhenius model and other ML models. Moreover, a post hoc analysis (feature
importance rank) has been performed to extract the correlation between
physical descriptors and simulation results from ML models. Our work provides
a promising route for a quick and cost-effective diffusion coefficient
determination for multiple and complex brine solutions with a wide range of
temperature, pressure and ion concentration by the combination of MD
simulations and ML techniques.

###### keywords:

American Chemical Society, LaTeX

NIT Warangal] Chemical Engineering Department, National Institute of
Technology, Warangal, Telangana, India Contributed equally to this work
CUPB]State Key Laboratory of Heavy Oil Processing, College of Carbon
Neutrality Future Technology, China University of Petroleum (Beijing), Beijing
102249, China TU-Darmstadt]Eduard-Zintl-Institut für Anorganische und
Physikalische Chemie, Technische Universität Darmstadt, Alarich-Weiss-Str. 8,
64287 Darmstadt, Germany Contributed equally to this work TU-
Darmstadt]Technische Universität Darmstadt, Institute of Applied Geosciences,
Group of Geothermal Science and Technology, Schnittspahnstrasse 9, 64287
Darmstadt, Germany TU-Darmstadt]Technische Universität Darmstadt, Institute of
Applied Geosciences, Group of Geothermal Science and Technology,
Schnittspahnstrasse 9, 64287 Darmstadt, Germany TU-Darmstadt]Department of
Materials and Earth Sciences, Technische Universität Darmstadt, Darmstadt
64287, Germany TU-Darmstadt]Eduard-Zintl-Institut für Anorganische und
Physikalische Chemie, Technische Universität Darmstadt, Alarich-Weiss-Str. 8,
64287 Darmstadt, Germany TU-Darmstadt]Technische Universität Darmstadt,
Institute of Applied Geosciences, Group of Geothermal Science and Technology,
Schnittspahnstrasse 9, 64287 Darmstadt, Germany TU-Darmstadt] Eduard-Zintl-
Institut für Anorganische und Physikalische Chemie, Technische Universität
Darmstadt, Alarich-Weiss-Str. 8, 64287 Darmstadt, Germany IR,NMR,UV

##  1 Introduction

The development of wind and solar energy is the most promising of all types of
renewable energy. However, wind and solar power are difficult to predict
accurately due to their intermittency, which inevitably leads to an imbalance
between peak demand and peak production 1. On the other hand, these
fluctuations also cause technical challenges such as congestion in
transmission lines and disturbances in the power markets. The key to solving
intermittent problems is to establish a large-scale energy storage system
across seasonal time scales.

Hydrogen, as an alternative energy carrier, has characteristics of high
specific energy capacity (120 MJ/kg) 2 and high conversion efficiency, and has
proven to be one of the lowest cost power storage options 3. A recent report
for the International Hydrogen Energy Council has stated that by 2050,
hydrogen energy demand will account for more than 18 $\%$ of global final
energy consumption 4. Hydrogen can be produced using the concept of “Power to
Gas” 5, 6, 7 by means of electrolysis, utilizing a large amount of surplus
renewable electricity generated by renewable solar and wind energy, which
helps alleviate the main shortcomings of renewable energy generation:
intermittency, seasonality and geographic limitations 8, 9, 10. The stored
hydrogen can be used as fuel for electricity generation to make up for the
demand in the low power generation period (“Gas to Power”) by means of fuel
cells 11, 12. It can also be converted into liquid fuel or used as a substance
in various industries13. However, the large-scale hydrogen storage is very
challenging because of its very low density (0.08988 g/L) 14 at standard
conditions. Furthermore, there are some challenges associated with the storage
of materials.

In 2019, the International Energy Agency reported that deep saline reservoirs
located at 500 to 2000 m below the Earth’s surface are one of the best options
for large-scale and long-term hydrogen energy storage 15. Saline aquifers are
porous media with substantial potential storage capacity (hundreds of
$km^{3}$) and are found all over the world 16. In a saline aquifer, compressed
hydrogen replaces the saline water in the micron-sized pores17, 18, which is
very beneficial for continuous flow injection and extraction of hydrogen. The
storage capacity of hydrogen is affected by many factors, such as the
reservoir’s porosity, temperature, and pressure. In addition, loss of hydrogen
is also possible because of the reactions with microorganisms, rocks and
fluids. The hydrogen loss caused by the diffusion of hydrogen in rocks and
fluids can be estimated by the mass transfer across the boundary layer 19,
which is a function of the molecular diffusion coefficient and the
concentration gradient of components 20, 21. Hence, it is necessary to study
the diffusion coefficient of hydrogen at the conditions of the aquifer.

As the conditions in the aquifer are extreme, the experimental study of
diffusion coefficient is challenging and expensive. Molecular dynamics (MD)
simulations are an attractive alternative. The different diffusion
coefficients (Self-, Maxwell-Stefan (MS), and Fick) have been intensively
evaluated for pure, binary, or multicomponent systems 22, 23, 24, 25 at
different conditions by MD simulation. Diffusion coefficients of various gases
such as $\mathrm{CH_{4}}$, $\mathrm{CO_{2}}$, and $\mathrm{N_{2}}$ in
water/brine have been measured by MD simulations 26, 27, 28, 29. A study by
Moultos et al. in 2014 examined various combinations of $\mathrm{CO_{2}}$ and
water force fields to predict $\mathrm{CO_{2}}$ diffusion in water for a wide
range of pressures and temperatures 27. More recently, Zhao et al. proposed a
correlation for predicting the diffusion of $\mathrm{H_{2}}$,
$\mathrm{CH_{4}}$, $\mathrm{CO}$, $\mathrm{O_{2}}$, and $\mathrm{CO_{2}}$ in
water near the critical point, using MD simulations 19. Omrani et al. 28 used
MD simulations to predict $\mathrm{CO_{2}}$ diffusion coefficient in
water/brine for a wide spectrum of pressure and temperature. They also
considered multicomponent systems of $\mathrm{CO_{2}}$-$\mathrm{SO_{2}}$-water
and $\mathrm{CO_{2}}$-$\mathrm{N_{2}}$-water to demonstrate the effects of
impurities on $\mathrm{CO_{2}}$ diffusion coefficient in water with
implication to $\mathrm{CO_{2}}$ sequestration 30. Recently, Bartolomeu and
Franco studied the thermophysical properties of supercritical $\mathrm{H_{2}}$
including the self-diffusion coefficient by MD simulation 31. Overall, MD
simulation is a valuable and accurate method for calculating diffusion
coefficients under different conditions, especially when experiments cannot be
carried out.

On the other hand, many studies have indicated the possibility of coupling
machine learning (ML) with MD simulation. Allers et al. 32 have integrated ML
models with the MD database on the prediction of the self diffusion
coefficients of Lennard-Jones fluids. They found that the artificial neural
net regression models provided superior prediction of diffusion in comparison
to the existing empirical relationships. Liang et al. 33 have obtained the
accurate thermophysical properties of $\mathrm{MgCl_{2}}$ \- KCl eutectics
such as density, viscosity, diffusion coefficients, and heat capacity from the
MD simulation by training the ML-based deep potential. Kirch et al. 34 have
combined MD results of the brine-oil interfacial tension with ML algorithms to
obtain a predictive model and estimate the effect of different parameters.
Recently, we have combined MD simulations and a genetic algorithm to design
the sequence-specific block copolymers with a high thermal conductivity35.

In this work, we have performed MD simulations to qualitatively study and
generate a compatible data set of diffusion coefficients of hydrogen into
brine. The results obtained from the MD simulations have been validated by
comparing them with the experimental data. The quantitative estimation of the
diffusion coefficient has been studied by using two formulations. Firstly, we
have investigated the influence of temperature on the diffusion coefficient by
the Arrhenius equation. The effect of ion concentration has been incorporated
into the temperature independent parameters of the equation. Secondly, we have
adapted different ML models trained by the MD data set to investigate the
complex effect of temperature, pressure and ion concentration in this work.
Moreover, the gradient boosting model has been used to estimate the importance
rank of different parameters on the diffusion coefficient of hydrogen
molecules.

##  2 Methodology

###  2.1 MD Simulation

We use the TIP4P/2005 model36 for $\mathrm{H_{2}O}$ because of its
applicability and accuracy in wide temperature and pressure ranges37. A two-
site model38 fitted by the experimental $PVT$ curve of bulk $\mathrm{H_{2}}$
and validated by the diffusion data is then used for $\mathrm{H_{2}}$. For the
chloride ($\mathrm{Cl^{-}}$) ions, we use the OPLS model39. For cations, the
OPLS model39 ($\mathrm{Na^{+}}$ and $\mathrm{K^{+}}$) and Aqvist model40
($\mathrm{Ca^{2+}}$) are used since they have been validated against densities
and osmotic pressures of aqueous solutions over a large temperature and
salinity range 41, 42. The potential function, $U_{ij}$, used to describe the
non-bonded interactions between two particle $i$ and $j$ with a Lenard-Jones
potential, and an electrostatic term (expressed by a Coulomb potential) is:

$U_{ij}=4\varepsilon_{ij}\left[\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{12}-\left(\frac{\sigma_{ij}}{r_{ij}}\right)^{6}\right]+\frac{q_{i}q_{j}}{4\pi\varepsilon_{0}r_{ij}}$
(1)

where $\varepsilon_{ij}$ and $\sigma_{ij}$ are the Lennard-Jones interaction
parameters between the atoms $i$ and $j$; $r_{ij}$ is the distance between
them; $q_{i}$ and $q_{j}$ are their charges, and $\varepsilon_{0}$ is the
vacuum permittivity. The force field parameters of the models used in this
study are tabulated in Table 1.

Table 1: Force field parameters used for $\mathrm{H_{2}O}$ with TIP4P/2005 model36, $\mathrm{H_{2}}$ with Yang’s model38, $\mathrm{Cl^{-}}$ with OPLS model39, $\mathrm{Na^{+}}$ and $\mathrm{K^{+}}$ with OPLS model39 and $\mathrm{Ca^{2+}}$ with Aqvist model40 in this work. Molecule | Force center or charge |  $\mathrm{\varepsilon}$ (kcal/mol) |  $\mathrm{\sigma}$ (Å) | q (e)
---|---|---|---|---
$\mathrm{H_{2}O}$ | O | 0.185200 | 3.1589 | -1.1128
$\mathrm{H1}$ | 0 | 0 | 0.5564
$\mathrm{H2}$ | 0 | 0 | 0.5564
$\mathrm{H_{2}}$ | $\mathrm{H1}$ | 0.019890 | 2.7200 | 0
$\mathrm{H2}$ | 0.019890 | 2.7200 | 0
$\mathrm{K^{+}}$ | $\mathrm{K}$ | 0.000328 | 4.9300 | 1
$\mathrm{Na^{+}}$ | $\mathrm{Na}$ | 1.607040 | 1.8974 | 1
$\mathrm{Ca^{2+}}$ | $\mathrm{Ca}$ | 0.449702 | 2.4120 | 2
$\mathrm{Cl^{-}}$ | $\mathrm{Cl}$ | 0.117840 | 4.4172 | -1

The Lennard-Jones interaction parameters between unlike atoms $(i\neq j)$ are
calculated using the Lorentz-Berthelot mixing rule43:

$\varepsilon_{ij}=\sqrt{\varepsilon_{i}\varepsilon_{j}}$ (2)
$\sigma_{ij}=\frac{\sigma_{i}+\sigma_{j}}{2}$ (3)

MD simulations are performed in the Large-scale Atomic/Molecular Massively
Parallel Simulator (LAMMPS). 44 Three types of systems, namely single cation
systems ($\mathrm{K^{+}}$, $\mathrm{Na^{+}}$, and $\mathrm{Ca^{2+}}$), binary
cation systems ($\mathrm{K^{+}}$-$\mathrm{Na^{+}}$,
$\mathrm{K^{+}}$-$\mathrm{Ca^{2+}}$ and $\mathrm{Na^{+}}$-$\mathrm{Ca^{2+}}$),
and ternary cation systems
($\mathrm{Na^{+}}$-$\mathrm{K^{+}}$-$\mathrm{Ca^{2+}}$) are simulated.
Examples of the compositions are summarized in Table 2. The number of ions is
chosen according to the desired molality. For example, 216 particles of
$\mathrm{Na^{+}}$ and $\mathrm{Cl^{-}}$ represent a molality of 3
$\mathrm{mol/kg_{H_{2}O}}$ in 4000 $\mathrm{H_{2}O}$ molecules. To cover a
wide range of the brine composition, literature is examined and 5
$\mathrm{mol/kg_{H_{2}O}}$ is selected as the upper limit for the salinity in
this study 45, 46, 47, 48, 49, 50, 51, 52, 53, 54. The total charge is zero in
all systems. The cut-off distance for the Lennard-Jones potential and the
real-space Coulombic cut-off distance have been used as 12 Å and 8.5 Å,
respectively. It should be noted that the long-range part of the electrostatic
forces is evaluated using the particle-particle particle-mesh (PPPM) solver55.
The bonds and angles of $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$ molecules are
fixed using the SHAKE algorithm56. For all the simulations, a 0.4 ns
preliminary run with $T_{0}$ = 300 K and $P_{0}$ = 1 atm have been performed,
followed by a 1.1 ns equilibration in the isothermal-isobaric (NPT) ensemble
using the Nosé-Hoover thermostat and barostat with a temperature damping
parameter of 100 fs and pressure damping parameter of 1000 fs. Three
production runs with each of 2 ns are run in the microcanonical (NVE)
ensemble, and the simulations are sampled every 10 ps to perform the analysis.
The time step used for all the runs is 1 fs.

Table 2: Clarification of brine compositions of the systems simulated in this
work.

Systems |  $\mathrm{c_{Na^{+}}}$* |  $\mathrm{c_{Ca^{2+}}}$* |  $\mathrm{c_{K^{+}}}$* |  $\mathrm{c_{Cl^{-}}}$* | c*
---|---|---|---|---|---
Single | 3.0 | 0.0 | 0.0 | 3.0 | 3.0
Single | 0.0 | 3.0 | 0.0 | 6.0 | 3.0
Single | 0.0 | 0.0 | 3.0 | 3.0 | 3.0
Binary | 1.5 | 1.5 | 0.0 | 4.5 | 3.0
Binary | 0.0 | 1.5 | 1.5 | 4.5 | 3.0
Binary | 1.5 | 0.0 | 1.5 | 3.0 | 3.0
Ternary | 1.0 | 1.0 | 1.0 | 4.0 | 3.0

  * •

*Concentrations are in $\mathrm{mol/{kg_{H_{2}O}}}$.

  * •

$\mathrm{c_{Na^{+}}}$: Concentration of $\mathrm{Na^{+}}$ ions;
$\mathrm{c_{Ca^{2+}}}$: Concentration of $\mathrm{Ca^{2+}}$ ions;
$\mathrm{c_{K^{+}}}$: Concentration of $\mathrm{K^{+}}$ ions and $\mathrm{c}$:
Total cation concentration.

The diffusion coefficient is obtained from the Einstein relation of the mean
squared displacement (MSD) with diffusion time57 as follows:

$D=\lim_{t\to\infty}\frac{\langle[r_{i}(t)-r_{i}(0)]^{2}\rangle}{6t}$ (4)

where $r_{i}(t)$ is the position vector of $\mathrm{H_{2}}$ at time $t$, and
$\langle$ $\rangle$ indicates the average over all $\mathrm{H_{2}}$ molecules
and time origins. The viscosity is determined according to the Green-Kubo
formula 58, 59:

$\eta=\frac{V}{k_{B}T}\int_{0}^{\infty}\langle
P_{\alpha\beta}(t_{0})P_{\alpha\beta}(t_{0}+t)\rangle dt$ (5)

where $k_{B}$ is the Boltzmann constant, $V$ is the volume of the simulation
box, and the $P_{\alpha\beta}$ are the off-diagonal components of the stress
tensor.

###  2.2 Machine Learning

By the use of the scikit-learn package 60, four different machine learning
models have been implemented on the simulated MD data to give predictions of
the diffusion coefficient (Figure 1). They are (i) linear regression (LR),
(ii) random forest (RF), (iii) extra trees (ET) and (iv) gradient boosting
(GB). Hyperparameters which are tunable parameters used to control the
learning process are tuned to optimize the models. The data set contains 5
features: temperature (T), pressure (P), and three cation concentrations
($\mathrm{c_{Na^{+}}}$, $\mathrm{c_{Ca^{2+}}}$ and $\mathrm{c_{K^{+}}}$) which
are used to predict the diffusion coefficient. Details of these models have
been summarized in the Supporting Information.

Figure 1: The diffusion coefficient of hydrogen is predicted through
Arrhenius-fitting and machine learning methods in this work. The temperature,
pressure, concentration and composition of ions are input features.

The mean absolute error (MAE) and root mean squared error (RMSE) are evaluated
to compare the accuracy of the models as follows:

$e_{MAE}=\frac{\sum_{i=1}^{m}|y_{i}-y_{pi}|}{m}$ (6)

$e_{RMSE}=\sqrt{\frac{\sum_{i=1}^{m}(y_{i}-y_{pi})^{2}}{m}}$ (7)

where $y_{i}$ and ypi represent the actual and the predicted values of the
diffusion coefficient, respectively. The coefficient of determination ($R^{2}$
score), another metric to compare the accuracy, is given by:

$R^{2}=1-\frac{SS_{res}}{SS_{tot}}$ (8)

where SSres is the sum of squared residuals given by
$\sum_{i=1}^{m}(y_{i}-y_{pi})^{2}$ and SStot is the total sum of squares given
by $\sum_{i=1}^{m}(y_{i}-y_{av})^{2}$ where ${y_{av}}$ is the average of the
actual values of diffusion coefficients. The best possible score of R2 is 1.0,
under the circumstance that all data is predicted correctly. A model that gets
a zero score of R2 predicts the expected value of the data set without
considering the input features. Besides the $R^{2}$ score calculated using the
testing data for the comparison of the accuracy of ML models, the mean cross-
validated $R^{2}$ score (MCRS)61 is calculated over 10 cross-validation splits
of the data with Eq. 8 to tune the hyperparameters of the ML models.

Moreover, the GB model has been used to acquire the feature importance rank
(FIR) in this work. Feature importance is the normalized measure of the
frequency of an individual feature to be used in the training process. For a
higher FIR, the specific feature is used more frequently to split the nodes.
As a result, it provides the information of the correlation of different
features with the diffusion coefficient of hydrogen molecules.

##  3 Results and Discussions

###  3.1 Validations

In order to check the effect of $\mathrm{H_{2}}$ concentration on the
diffusion coefficient, we have examined systems with 2, 10, 20, 30, 40, 50,
100, 150 and 200 $\mathrm{H_{2}}$ molecules with 4000 water molecules
(concentration of $\mathrm{H_{2}}$ $\sim$ 0.0005 $\mathrm{mol/mol_{H_{2}O}}$
\- 0.05 $\mathrm{mol/mol_{H_{2}O}}$) at 298 K and 1 atm. The diffusion
coefficient of $\mathrm{H_{2}}$ fluctuates with large errors in case of the
low solubility (Figure 2 (a)). This can be explained by the significant
deviation of MSD from the linear relationship with poor statistics, as shown
in Figure 2 (b). The deviation is reduced and the diffusion coefficient of
$\mathrm{H_{2}}$ is practically constant within its error bars as the number
of $\mathrm{H_{2}}$ molecules increases beyond 30 (Figure 2 (c)). Wilke et
al.62 proposed an empirical equation for the diffusion coefficients of gas in
water as:

$D\propto\frac{T}{\mu}$ (9)

where T is the temperature and $\mu$ is the viscosity. We have inserted
experimental results63 into Eq. 9 and interpolated the diffusion coefficient
as 0.593$\cdot\mathrm{10^{-8}}$ $\mathrm{m^{2}/s}$ for $\mathrm{H_{2}}$
molecules at $T$ = 298 K and $P$ = 1 atm. Details are summarized in the
Supporting Information Table S1. On the other hand, Kallikragas et al.64 have
examined MD simulations with SPC/E water potential65 and the potential devised
by Cracknell66 for $\mathrm{H_{2}}$ to obtain the diffusion coefficient as
0.522$\cdot\mathrm{10^{-8}}$ $\mathrm{m^{2}/s}$ for $\mathrm{H_{2}}$ molecules
at $T$ = 298 K and $P$ = 1 atm. The diffusion coefficient of 40
$\mathrm{H_{2}}$ molecules ($D$ = 0.524$\cdot\mathrm{10^{-8}}$
$\mathrm{m^{2}/s}$) in this work shows good stability (Figure 2 (a)) and is
comparable to both the above experimental and simulation results. Therefore,
the number of 40 $\mathrm{H_{2}}$ molecules (0.01 $\mathrm{mol/mol_{H_{2}O}}$)
has been used in all simulations in the upcoming discussions. We have also
examined the variation of MSD curves of 40 $\mathrm{H_{2}}$ in pure water at
(288, 298, 308, 318, 328, 338, 348, 358 and 368) K and 1 atm in Figure 3. The
MSD curves of different temperatures show a comprehensible difference at the
end of the simulation. The linearity of all MSD plots versus time proves that
the diffusion coefficients of $\mathrm{H_{2}}$ can be calculated accurately
and credibly. Moreover, Wang67 has reported the experimental value of self
diffusion coefficient of $\mathrm{H_{2}O}$ as 0.257$\cdot\mathrm{10^{-8}}$
$\mathrm{m^{2}/s}$ and Holz et al.68 have reported it as
0.230$\cdot\mathrm{10^{-8}}$ $\mathrm{m^{2}/s}$ at $T$ = 298 K and $P$ = 1
atm, which are in close agreement with the value obtained in this work which
is 0.233$\cdot\mathrm{10^{-8}}$ $\mathrm{m^{2}/s}$.

Figure 2: (a) The variation of the diffusion coefficient of $\mathrm{H_{2}}$
in pure $\mathrm{H_{2}O}$ versus the concentration of $\mathrm{H_{2}}$ at T =
298 K and P = 1 atm. The dashed red line is obtained from Eq. 9 using the data
provided by Wise et al.63 and the dashed green line is the result from other
molecular dynamics (MD) work by Kallikragas et al. 64. Mean squared
displacement (MSD) versus time for systems with (b) 2 $\mathrm{H_{2}}$
molecules and (c) 30 $\mathrm{H_{2}}$ molecules at $T$ = 298 K and $P$ = 1
atm. Figure 3: The variation of mean squared displacement (MSD) of
$\mathrm{H_{2}}$ in pure $\mathrm{H_{2}O}$ versus time with respect to
different temperatures (288, 298, 308, 318, 328, 338, 348, 358 and 368 K) at 1
atm.

We further compare simulated and experimental KCl and NaCl solutions 69 to
validate brine systems (Figure 4). Due to the lack in diffusion coefficients
for brine systems in the literature, the values of density and viscosity are
selected for the comparison. The increasing trend of our simulations are in
agreement with the experimental data regarding density (Figure 4 (a)) and
viscosity (Figure 4 (b)) as the ion concentrations increase.69 The magnitude
of the deviations in viscosity ((Figure 4 (b))) could be affected by the
complexity of the measurement in the experiment. These observations and
discussions confirm that our systems are well-parameterized compared to
previous literature.

Figure 4: The variation of (a) the density and (b) viscosity of KCl and NaCl
solutions versus the ion concentrations at 298 K and 1 atm.69

###  3.2 Variation of the $\mathrm{\bf H_{2}}$ diffusion coefficient

The diffusion coefficients of $\mathrm{H_{2}}$ in different chloride brines
have been calculated at 298 K and 1 atm (Figure 4 (a)) and at 648 K and 218
atm (Figure 4 (b)). The concentration of chloride brines is varied from 0.5
$\mathrm{mol/kg_{H_{2}O}}$ to 5 $\mathrm{mol/kg_{H_{2}O}}$. First, it is
observed that the diffusion coefficients of $\mathrm{H_{2}}$ at high
temperature and pressure (Figure 4 (b)) are much larger than that at low
temperature and pressure (Figure 4 (a)) irrespective of the compositions and
molalities of ions. There are previous studies which investigated the effect
of temperature and pressure on the diffusion coefficients of gases in pure
water and brine 70, 71, 72. The trends of our results are in agreement with
the previous studies. Second, there is a tendency for the diffusion
coefficient to decrease with an increase in the molality for most systems in
Figure 4. However, systems containing single cation $\mathrm{K^{+}}$ show an
increase and then a decreasing trend of the diffusion coefficient versus the
molality at high temperature and pressure (Figure 4 (b)). To understand the
mechanism underlying the increased diffusion coefficient, we calculate the
intermolecular radial distribution functions (Supporting Information Figure
S2). We find that the interaction between $\mathrm{H_{2}}$ and
$\mathrm{K^{+}}$ becomes weaker as the molality increases, resulting in a
lower aberration to the movement of $\mathrm{H_{2}}$ from $\mathrm{K^{+}}$
(Supporting Information Figure S2). Then, the the disturbance in the motion of
hydrogen molecules could be explained by two compensatory effects, namely, the
interaction between $\mathrm{H_{2}}$ and $\mathrm{K^{+}}$ and the interaction
between $\mathrm{H_{2}}$ and $\mathrm{H_{2}O}$. A similar trend has also been
observed for water molecules in brine systems containing $\mathrm{K^{+}}$ 73.
Kim et al. found that the diffusion coefficient of water molecules with
structure-breaking ions (e.g., $\mathrm{K^{+}}$) increases as the ion
concentration increases, whereas the diffusion coefficient of water molecules
with structure-making ions (e.g., $\mathrm{Na^{+}}$ and $\mathrm{Ca^{2+}}$)
decreases.73 However, it is noted that the effect of $\mathrm{Na^{+}}$ and
$\mathrm{Ca^{2+}}$ are more stronger than $\mathrm{K^{+}}$ in binary and
ternary systems, and an overall reduction in diffusion coefficient is observed
(Figure 4 (b)). Hence, we focus on the ternary cations systems to evaluate the
effect of temperature on the diffusion coefficient of $\mathrm{H_{2}}$ in the
next section.

Figure 5: The variation of the diffusion coefficient of $\mathrm{H_{2}}$ in
saline brine versus the total cation concentration varying from 0.5
$\mathrm{mol/kg_{H_{2}O}}$ to 5 $\mathrm{mol/kg_{H_{2}O}}$ at (a) $T$ = 298 K
and $P$ = 1 atm and (b) $T$ = 648 K and $P$ = 218 atm.

###  3.3 Dependence of the $\mathrm{H_{2}}$ diffusion coefficient on
temperature, pressure and cation concentration

The diffusion coefficient of gases in liquids strongly depends on the
temperature.63, 70, 71 As a first approximation, the Arrhenius equation74 was
proposed for the prediction of the temperature dependence of the diffusion
coefficient as:

$D=D_{0}exp\left(\frac{-E_{d}}{RT}\right)$ (10)

where $D_{0}$ is the diffusion coefficient at infinite temperature, $E_{d}$ is
the diffusion activation energy, $T$ is temperature and $R=8.314$
$\mathrm{J/(mol\cdot K)}$ is the universal gas constant. This formula has been
used to describe the diffusion behavior of different gases in water at the
constant pressure75, 76, 77, 78. Recently, Chen et al. 72 found that at
constant pressure, the temperature dependence of the diffusion coefficient of
$\mathrm{CH_{4}}$ in $\mathrm{NaCl}$, $\mathrm{Ca{Cl_{2}}}$, $\mathrm{NaBr}$
and $\mathrm{NaI}$ brine can also be predicted by Eq. 10. Similar exponential
behavior of the diffusion coefficients of hydrogen in ternary cation systems
has been observed in this work (Figure 6).

A straight line fit has been found for the $lnD$ versus $\frac{1}{T}$ (Eq. 11)
with different ion concentrations (Figure 6) and the $D_{0}$ and $E_{d}$ have
been extracted.

$lnD=\left(\frac{-E_{d}}{R}\right)\frac{1}{T}+(lnD_{0})$ (11)

Parameters $E_{d}$ (in kJ/mol) and $lnD_{0}$ ($D_{0}$ in $\mathrm{m^{2}/s}$)
are then linearly fitted to the total cation concentrations (Figure 7 (a) and
(b)) as :

${E_{d}}=0.1143c+15.6728$ (12) $ln(D_{0})=-0.1300c-12.9297$ (13)

Figure 6: Diffusion coefficients and the corresponding Arrhenius fits as a
function of temperature for ternary cation systems
($\mathrm{Na^{+}}$-$\mathrm{K^{+}}$-$\mathrm{Ca^{2+}}$) with total cation
concentrations varying from 0.5 $\mathrm{mol/kg_{H_{2}O}}$ to 5
$\mathrm{mol/kg_{H_{2}O}}$ at $P$ = 218 atm.

(a)

(b)

Figure 7: Concentration dependence and linear fits of (a)
${E_{d}}=0.1143c+15.6728$ and (b) $lnD_{0}=-0.1300c-12.9297$ at $P$ = 218 atm,
where $c$ is the total cation concentration.

Above discussions on the Arrhenius model of diffusion coefficient is under the
constant pressure. We further investigate the effect of pressure on the
diffusion coefficient of $\mathrm{H_{2}}$. At room temperature (298 K), the
effect of pressure on the diffusion coefficient is negligible for both low and
high ion concentrations (Figure 8 (a) and (b)), which has also been observed
for the diffusion of $\mathrm{CO_{2}}$ and $\mathrm{N_{2}}$ in water70 at room
temperature. This implies that the Arrhenius fit Eq.11 can be applied
regardless of the pressure at room temperature over a pressure range of 1 to
218 atm. However, the diffusion coefficient of $\mathrm{H_{2}}$ shows a strong
dependence on the pressure and cation concentration as the temperature
increases from 298 K to 648 K in Figure 8. At low ion concentration (Figure 8
(c)), diffusion coefficient of $\mathrm{H_{2}}$ decreases with the pressure.
The reduction of diffusion coefficient can be explained by the fact that the
viscosity of the solution increases as pressure increases79, 80, 81, which in
turn increases the resistance to the motion of the $\mathrm{H_{2}}$ molecules
significantly. At high ion concentrations (Figure 8 (d)), the diffusion
coefficient remains constant with respect to pressure. This non-decreasing
trend of diffusion coefficient results from the large number of ions in the
system which already confines the motion of the hydrogen molecule due to
increased collisions at low pressure82. These observations imply that the
development of a fitting function which considers the pressure, cation
compositions and cation concentrations could be complicated. Zhao et al.19
have developed a more exact model for hydrogen in pure water. However, it
requires additional information on the density and viscosity of the solution,
which require additional efforts in experimenting or simulating for the data.
Therefore, new robust models need to be considered.

Figure 8: The variation of diffusion coefficient of $\mathrm{H_{2}}$ in the
ternary cation systems ($\mathrm{Na^{+}}$-$\mathrm{K^{+}}$-$\mathrm{Ca^{2+}}$)
versus pressure at (a) $T$ = 298 K and low ion concentration $c$ = 0.5
mol/kg$\mathrm{H_{2}O}$; (b) $T$ = 298 K and high ion concentration $c$ = 5
mol/kg$\mathrm{H_{2}O}$; (c) $T$ = 648 K and low ion concentration $c$ = 0.5
mol/kg$\mathrm{H_{2}O}$ and (d) $T$ = 648 K and high ion concentration $c$ = 5
mol/kg$\mathrm{H_{2}O}$.

###  3.4 Prediction of the diffusion coefficient of $\mathrm{H_{2}}$ based on
machine learning models

Here, we use four ML methods to predict the diffusion coefficient concerning
temperature, pressure, and ion concentrations. We first identify the proper
size of training data, that is, how much data is needed for training to
achieve the proper accuracy. Investigation of learning curve has been
performed and summarized in the Supporting Information. The prediction
accuracy plateaued at about 197 training samples where the data set contained
a total of 282 samples. Hence, 70 $\%$ of the data set was selected as
training samples. Then, except the simple LR model, the hyperparameters of the
other three implemented models were further optimized by the MCRS calculated
over ten cross-validation splits to determine the optimal model configuration.
The number of estimators (nest), the depth (maxdep), and the number of
features (maxfeat) are hyperparameters for the RF, ET, and GB models.
Additionally, the influence of the learning rate on the prediction accuracy
has also been investigated for the GB model. The detailed optimisation process
is summarized in the Supporting Information and obtained hyperparameters are
listed in Table 3.

Table 3: Hyperparemeters of the machine learning methods in this work. Model | Hyperparameter | Value
---|---|---
RF | nest | 800
maxdep | 13
maxfeat | 5
ET | nest | 100
maxdep | 10
maxfeat | 5
GB | nest | 600
learning rate | 0.1
maxdep | 2
maxfeat | 5

To compare the accuracy of ML models, the RMSE, MAE, and $\mathrm{R^{2}}$
score of ML models and the Arrhenius model have been calculated. It is clear
to deduce that all ML models have fewer errors compared with the Arrhenius
model reference (Figure 9). This implies that ML models can accurately predict
the diffusion coefficient and hence the introduction of ML provides an
economical and time-saving route for predicting the diffusion coefficient of
new systems rather than going through the whole process of conducting new MD
simulations to obtain data, which requires heavy computational resources and
efforts.

Figure 9: Metrics used to compare the accuracy of gradient boosting (GB),
extra trees (ET), random forest (RF), linear regression (LR), elastic net (EN)
and naive baseline (NB) methods which are calculated using the testing data
set containing 85 entries. (a) Mean absolute error (MAE); (b) Root mean
squared error (RMSE) and (c) $\mathrm{R^{2}}$ score.

Furthermore, we have compared $lnD$ from simulations with predicted $lnD$ for
the GB and Arrhenius models in Figure 10. It is observed that almost all
points fall near the straight line with slope 1 from the GB model (Figure 10
(b)), whereas there are more divisions in the Arrhenius model (Figure 10 (b)).
We therefore conclude that the machine learning based GB model accurately
captures the fluctuation of diffusion coefficient of $\mathrm{H_{2}}$ with
known parameters, which are generally considered in the realistic conditions.
The GB model is then used to produce the feature importance rank (FIR), which
is an additional measure of feature relevance. As a starting point, we
calculate FIR that covers the entire temperature range (Table 4 (a)). The
temperature ($\sim$ 0.945) shows the strongest correlation with the diffusion
coefficient of $\mathrm{H_{2}}$ among all features. The ranking of the
correlation to the diffusion coefficient of $\mathrm{H_{2}}$ is
$\mathrm{Ca^{+2}}$ ($\sim$ 0.032) $>$ $\mathrm{Na^{+}}$ ($\sim$ 0.019) $>$
$\mathrm{K^{+}}$ ($\sim$ 0.001). The value of $\mathrm{K^{+}}$ is so low that
its effect on the diffusion coefficient becomes insignificant when combined
with other ions as demonstrated in the previous observation (Figure 4 (b)).
The effect of pressure is insignificant ($\sim 0.002$). After that, we divided
the data set into two parts in order to compute the FIR within low temperature
range (298 K $\leq$ T $<$ 400 K, Table 4 (b)) and high temperature range (400
K $\leq$ T $\leq$ 648 K, Table 4 (c)). It is observed that the diffusion
coefficient of $\mathrm{H_{2}}$ shows a strong dependence on the pressure
(from $\sim$ 0.0005 to $\sim$ 0.007) as the temperature increases. This lends
credence to our previous discussions in Figure 8 regarding the pressure effect
on the diffusion coefficient. Consequently, it is evident that the conclusions
deduced in the previous discussions are in agreement with the results of FIR.

Figure 10: (a) Fitting data from gradient boosting; (b) fitting data from
Arrhenius equation (Eq. 11) versus data set from molecular dynamics (MD)
simulations of ln(D ($\mathrm{m^{2}/s}$)) utilizing 197 training samples and
85 testing data samples.

Feature | Importance value
---|---
1/T | 0.9454
$\mathrm{c_{Ca^{2+}}}$ | 0.0322
$\mathrm{c_{Na{+}}}$ | 0.0192
P | 0.0021
$\mathrm{c_{K^{+}}}$ | 0.0011
(a) Overall data set containing samples with 298 K $\leq$ T $\leq$ 648 K.

Feature | Importance value
---|---
1/T | 0.7713
$\mathrm{c_{Ca^{2+}}}$ | 0.1711
$\mathrm{c_{Na{+}}}$ | 0.0550
$\mathrm{c_{K^{+}}}$ | 0.0021
P | 0.0005
(b) Data set containing samples with 298 K $\leq$ T $<$ 400 K.

Feature | Importance value
---|---
1/T | 0.8167
$\mathrm{c_{Ca^{2+}}}$ | 0.0916
$\mathrm{c_{Na{+}}}$ | 0.0812
P | 0.0073
$\mathrm{c_{K^{+}}}$ | 0.0032
(c) Data set containing samples with 400 K $\leq$ T $\leq$ 648 K.

Table 4: Feature Importance Rank (FIR) of five parameters estimated using
gradient boosted method using the data set obtained from MD simulations.

##  4 Conclusion

We employ a computational framework of the combination of molecular dynamics
(MD) simulation and machine learning (ML) models to determine the diffusion of
hydrogen molecules in the extreme conditions present in saline aquifers. To
the best of our knowledge, feeding the MD data into the ML-based prediction of
diffusion coefficient of $\mathrm{H_{2}}$ by considering temperature,
pressure, and ion concentrations ($\mathrm{NaCl}$, $\mathrm{Ca{Cl_{2}}}$, and
$\mathrm{KCl}$) is reported for the first time. Initially, we have verified
that there is no observable effect of the concentration of hydrogen on its
diffusion coefficient in water. The diffusion coefficients of hydrogen and
water molecules have been compared with the experimental data to confirm the
validity of our simulation settings. A decrease in the diffusion coefficient
of $\mathrm{H_{2}}$ with an increase in the cation concentration is evident.
This implies that high salinity reservoirs are more suitable for the hydrogen
storage due to the lower diffusion coefficients.

Furthermore, our results demonstrate that the temperature of systems is the
major determinant of the diffusion coefficients of the $\mathrm{H_{2}}$,
whereas the effect of pressure on the diffusion coefficient is inconsequential
for low temperature conditions (298 K). The traditional quantitative
description, namely, the Arrhenius formula, can well predict the temperature
dependence of the diffusion coefficient of $\mathrm{H_{2}}$ in brine at the
constant pressure. However, we find that the diffusion coefficient of
$\mathrm{H_{2}}$ in brine with low ion concentrations reduces significantly
with the pressure at 648 K, implying the inaccuracy of the Arrhenius formula
considering the effect of pressure and compositions of individual cations. The
models which can incorporate the versatile behaviour of the diffusion
coefficient, such as Zhao’s model19, requires the information of viscosity and
density of the solution, which are indirect parameters requiring further
experiments or simulations.

Therefore, we have applied four ML algorithms (linear regression (LR), random
forest (RF), extra trees (ET), and gradient boosting (GB)) to obtain the best
model considering the accuracy, completeness, and simplicity. All the models
show higher accuracy than the Arrhenius model from the evaluation of root mean
squared error (RMSE), mean absolute error (MAE), and $\mathrm{R^{2}}$ score.
When compared to other models, the GB model is the one that provides the best
accurate predictions regarding the diffusion coefficients of hydrogen. In
addition, the feature importance rank deduced using the GB algorithm is in
agreement with the conclusions obtained about the effect of different
parameters on the diffusion coefficient.

Overall, our work demonstrates that the Arrhenius fitting model can predict
the diffusion coefficient for the low temperature regime (T$\leq$400 K).
However, for a wide range of temperature and pressure range in saline
aquifers, using MD simulations to feed GB model is a promising approach to
define a model which predicts the diffusion behavior of $\mathrm{H_{2}}$ and
we believe that this computational framework has a significant potential for
advancing the development of the large-scale hydrogen storage over seasonal
time scales.

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