#### Abstract

In this communication, we review recent studies by these authors for modeling the -*H*
equilibrium. With the aim of estimating the solubility of pure hydrocarbon hydrate former in pure water in equilibrium with gas hydrates, a thermodynamic model is introduced based on equality of water fugacity in the liquid water and hydrate phases. The solid solution theory of Van der Waals-Platteeuw is employed for calculating the fugacity of water in the hydrate phase. The Henry's law approach and the activity coefficient method are used to
calculate the fugacities of the hydrocarbon hydrate former and water in the liquid water phase, respectively. The results of this model are successfully compared with some selected experimental data from the literature. A mathematical model based on feed-forward artificial neural network algorithm is then introduced to estimate the solubility of pure hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates. Independent experimental data (not employed in training and testing steps) are used to examine the reliability of this algorithm successfully.

#### 1. Introduction

Gas hydrates are ice-like structures in which water
molecules, under pressure, form structures composed of polyhedral cages
surrounding gas molecule βguestsβ such as methane and ethane [1β4]. The most common gas hydrate structures are those of
structure I (I) and structure II (II), where each structure is composed
of a certain number of large and small cavities formed by water molecules [1, 2]. For a molecule to
enter a cavity, its size should be smaller than a certain value. Large molecule
guests which can enter only a limited number of large cavities require smaller
βhelp gasβ molecules to mainly fill some smaller cavities sufficiently to
stabilize hydrate crystals [1, 2]. It has been proved that gas hydrates occur in staggering
abundance in cold subsea, sea floor, and permafrost environments where
temperature and pressure conditions ensure their stability [1β5]. It is believed the amount of natural gas trapped in these deposits is much higher than the amount of natural gas existing in classical reserves [1, 2, 5]. Hydrate
technology has also been proposed as a means for separation from
industrial flue gases and sequestration in the deep ocean for reducing the
emission of greenhouse gases [1β5]. Gas hydrates can form as well in undersea piping and
above-ground pipelines, where they pose a major and expensive problem for the
petroleum industry [1, 2]. Hydrates are also being regarded as an alternate means of
gas transportation and storage [1, 2, 4]. It is believed that
the potential storage of gas in hydrate is comparable to gas storage in the
form of liquefied natural gas (*LNG*)
and compressed natural gas (*CNG*) [2, 6].

The liquid water-hydrate (-) equilibrium knowledge is necessary in the design of gas transportation and storage process and should be destined for proposed sequestration schemes [1β4, 6]. These factors, and the potential widespread abundance of gas hydrates in the cold subsea, sea floor, and permafrost environments [1β4, 6], warrant an understanding of - equilibrium.

Figure 1 shows typical solubility-temperature diagram for water-pure hydrate former (limiting reactant) system [2]. As can be seen, the temperature and pressure dependencies of the pure hydrate former solubility in pure water being in the liquid water-vapor (-) equilibrium region are different from the corresponding dependency in the - equilibrium region [1β4, 6β19]. The - equilibrium is strong function of temperature and pressure while the - equilibrium is strong function of temperature but very weak function of pressure [1β4, 6β19]. On the other hand, the pure hydrate former solubility in pure water being in the - equilibrium region increases with decreasing the temperature at a given pressure, while, the corresponding solubility in pure water being in the - equilibrium region decreases with decreasing the temperature at the same pressure [1β4, 6β19]. Furthermore, the metastable liquid water-vapor (-) equilibrium may extend well into the gas hydrate formation zone [1, 2, 6β19].

The experimental works done to describe the - equilibrium are limited mainly due to two factors: the possible extension of the metastable - equilibrium into the gas hydrate region and the experimental restraint that the existing analysis methods require modifications [1, 2, 6β19]. Literature survey reveals the availability of only few sets of experimental data for the - equilibrium [1, 2, 7, 11, 16β19]. Consequently, few reliable models are available in the literature for calculating the - equilibrium [1β4, 6β8, 11, 14β16].

The objective of this work is to review
recent studies by these authors for modeling the - equilibrium [2β4]. A thermodynamic model based on
equality of water fugacity in the liquid water and hydrate phases for
determining solubility of pure hydrocarbon hydrate former in pure water being
in equilibrium with gas hydrates is first introduced [2]. The fugacity of water in the hydrate phase is calculated
using the solid solution theory of van der Waals-Platteeuw [20]. The hydrocarbon
hydrate former fugacity and water fugacity in the liquid water phase are
calculated using Henryβs law approach and the activity coefficient method,
respectively. The capability of this model is investigated by comparing its
predictions with some selected experimental data from the literature. It is demonstrated
that the results are in acceptable agreement demonstrating the capability of
the thermodynamic model developed in this work for estimating the solubility of
pure hydrocarbon hydrate former in pure water being in equilibrium with gas
hydrates. It is also shown that by assuming an average value equal to 1.2 for
the water activity coefficient, acceptable results can be obtained for
estimating the solubility of carbon dioxide in water at equilibrium with its
gas hydrate [3]. Finally, the capability of artificial neural network (*ANN*) algorithm, as an alternative tool,
for modeling the - equilibrium is successfully demonstrated [4].

#### 2. Liquid Water-Hydrate Equilibrium

The liquid water-hydrate equilibrium of a system is calculated, by equating the fugacities of water in the liquid water phase, , and in the hydrate phase, , as follows [1β3]:

The fugacity of water in the hydrate phase, , is related to the chemical potential difference of water in the filled and empty hydrate cages by the following expression [1, 2]: where is the fugacity of water in the hypothetical empty hydrate phase and represents the chemical potential difference of water in the filled () and empty () hydrates. and stand for universal gas constant and temperature, respectively.

The solid solution theory of van der Waals-Platteeuw [20] can be employed for calculating as follows [1, 2]: where is the number of cavities of type per water molecule in a unit hydrate cell [1, 2], stands for the Langmuir constant for hydrocarbon hydrate formerβs interaction with each type cavity, and is the fugacity of hydrate former [2].

The fugacity of water in the empty lattice can be expressed as follows [1, 2]: where , , , and are the vapor pressure of the empty hydrate lattice, the correction for the deviation of the saturated vapor of the pure (hypothetical) lattice from ideal behavior, the partial molar volume of water in the empty hydrate [1, 2, 21], and pressure, respectively. The exponential term is a Poynting type correction.

Equation (4) may be simplified by two assumptions: (1) that the hydrate partial molar volume equals to the molar volume and is independent of pressure and (2) that is relatively small (on the order of βMPa), so that [1, 2]. Therefore [1, 2],

Using the previous expressions, the following equation is obtained for the fugacity of water in the hydrate phase [2]: where is the fugacity of hydrocarbon hydrate former in the liquid water phase.

The Poynting correction term can be ignored up to intermediate pressures and therefore, the following equation can be obtained for calculating fugacity of water in hydrate phase [2, 3]: The fugacity of water in the liquid water phase can be expressed by [2, 3, 22]: where and are, respectively, the water mole fraction and the activity coefficient of water in liquid water phase being in equilibrium with gas hydrates. In the intermediate pressure range, the liquid water is an incompressible fluid, hydrocarbon hydrate former solubility is very small comparing to unity, and activity coefficient of water can be approximated to unity [22] (however it is necessary to be careful, as it is not the case at high pressures [22] where the nonideality of the liquid water phase and solubility become important). Therefore, (8) can be satisfactorily written as follows [2]: Using (7) and (9), the following expression is obtained [2]: or where the fugacity of hydrocarbon hydrate former in the liquid water phase up to intermediate pressures can be calculated using the following equation [2, 22]: where represents Henryβs constant for hydrocarbon hydrate former-water system. Therefore, the following final expression is obtained for estimating the solubility of pure hydrocarbon hydrate former in liquid water phase being in equilibrium with gas hydrates [2]:

Equation (13a) allows easy
calculation of the solubility of pure hydrocarbon hydrate former in the liquid
water being in equilibrium with gas hydrates.
Its main advantages are *the availability of necessary input data and
the simplicity of the calculations*, as the calculations can be done in
Excel spread sheets [2]. Furthermore, as can be
seen in (13a), almost all terms are temperature dependent while not
pressure dependent, indicating the solubility of pure hydrocarbon hydrate former in liquid
water phase being in equilibrium with gas hydrates is strong function of
temperature and only very weak function of pressure. For the (and
other highly soluble gases like )-water system, the solubility of
in the water phase cannot be ignored and consequently, the water
activity cannot be set to unity. Taking into account the solubility in the water phase and also the water activity, the following
equation can be derived [3]:

##### 2.1. Model Parameters

In (13a) and (13b), the following values of for structure-I hydrates can be used [1, 2]: and for structure II:

The Langmuir constants (they are generally functions of temperature, pressure, composition and hydrate structure, while the effect of pressure is normally ignored [23]) accounting for the interaction between the hydrate former and water molecules in the cavities were reported by Parrish and Prausnitz [24] for a range of temperatures and hydrate formers [2]. However, the integration procedure was already followed in obtaining the Langmuir constants for wider temperatures using the Kihara [25] potential function with a spherical core according to the study by McKoy and SinanoΔlu [26]. In this work, the Langmuir constants for hydrate formerβs interaction with each type cavity have been determined using the equations of Parrish and Prausnitz [24]:

For pentagonal dodecahedra (small cavity): for tetrakaidecahedra (large cavity): where is in K and has units of reciprocal MPa. Constants are reported in Table 1.

The concept in (4) of universal empty hydrate vapor pressure for each structure prompted Dharmawardhana et al. [27] to calculate the from a number of simple hydrate three-phase ice-vapor-hydrate equilibrium. By equating the fugacity of water in the hydrate phase to that of pure ice at the three-phase line, Dharmawardhana et al. [27] obtained the following equation for the vapor pressure of the empty hydrate structure I [1, 2]: and for structure II: where is in MPa and is in K. It should be mentioned that the nonideality of water vapor pressure of the empty hydrate at saturation seems to be negligible due to the small quantity (typically, to βMPa).

The following values for Henryβs constant of hydrocarbon hydrate former-water can be used [2, 28]: where and are in K and MPa, respectively. Constants , , , and are given in Table 2.

The water vapor pressure can be obtained using the following expression [29]: where, and are, respectively, in K and MPa.

#### 3. Artificial Neural Network Algorithm

Artificial neural
networks have large numbers of computational units called neurons connected in
a massively parallel structure and do not need an explicit formulation of the
mathematical or physical relationships of the handled problem [4, 30β37]. The most
commonly used *ANN*s are the feed-forward
neural networks (FNN) [4, 36, 37], which are
designed with one input layer, one output layer and hidden layers [4, 33β35, 37]. The number of neurons in
the input and output layers equals to the number of inputs and outputs physical
quantities, respectively [4, 37]. The disadvantage
of *FNN*s is the determination of the
ideal number of neurons in the hidden layer(s); few neurons produce a network
with low precision and a higher number leads to overfitting and bad quality of
interpolation and extrapolation [4, 37]. The use of techniques such as Bayesian regularization, along with a Levenberg-Marquardt
algorithm [4, 38, 39], can help overcome this problem [4, 31, 32, 37].

In the *FNN* method, the input layer of the
network receives all the input data and introduces scaled data to the network [4, 37]. The data from the input
neurons are propagated through the network via weighted interconnections [4, 37]. Every neuron in
a layer is connected to every neuron in adjacent layers [4, 37]. The neuron within the hidden layer performs the following tasks: summation of the arriving weighted inputs (input vector )
and propagations of the resulting summation through an activation
function, ,
to the adjacent neurons of the next hidden layer or to the output neuron(s).
Three types of transfer functions are normally used: the exponential sigmoid,
tangent sigmoid and linear functions [4, 37]. In this work, the activation function is a linear
function [4]:
where stands for parameter of linear activation
function. A bias term, ,
is associated with each interconnection in order to introduce a supplementary
degree of freedom. The expression of the weighted sum, , to the th
neuron in the th layer () is [4, 37]:
where is the weight parameter between
each neuron-neuron interconnection. Using this feed-forward network with linear
activation function, the output, , of the neuron within the
hidden layer is [4]:

To achieve a better stability, the following scaling rule is applied to solubility of pure hydrocarbon hydrate former being in equilibrium with gas hydrates () before normalization [4]: where subscript exp represents experimental data.

To develop the *ANN*, the datasets are subdivided into
3 classes: training, testing and validation [4, 37]. After partitioning the datasets, the training
set is used to adjust the parameters. All synaptic weights and biases are first
initialized randomly. The network is then trained; its synaptic weights are
adjusted by optimization algorithm, until it correctly emulates the
input/output mapping, by minimizing the average root-mean-square error [4, 37]. The optimization method
chosen in this work was the Levenberg-Marquardt algorithm [38, 39], as mentioned earlier. The testing set is used during the
adjustment of the networkβs synaptic weights to evaluate the algorithms
performance on the data not used for adjustment and stop the adjusting if the
error on the testing set increases. Finally, the validation set measures the
generalization ability of the model after the fitting process [4, 37].

#### 4. Results and Discussion

##### 4.1. Thermodynamic Model

Among the - equilibrium
data reported in the literature for the solubility of methane in pure water
being in equilibrium with gas hydrates, those reported by Yang [16], Servio and Englezos [10] and Kim et al. [11] seem to be the most reliable. These data are reported in Tables 3, 4, and 5, respectively. As can be seen, the temperature range is from
274.15 to 281.7βK, and the pressures are up to 143.62βMPa. Other
experimental data reported in the literature for methane solubility in pure
water being in equilibrium with gas hydrates have not been considered in this
work, as they are not consistent with other literature data. Tables 3β5 also show the predictions of the models developed in this work and the absolute deviations (*ADs*). As can be seen, (13a) with no adjustable parameter shows
encouraging results. The predictions of this thermodynamic model for the
solubility of methane in pure water being in equilibrium with gas hydrates show
less than 18% absolute deviation and the average absolute deviation (*AAD*) among all the experimental and
predicted data is 7.3% [4].

Limited information is available
for the - equilibrium
of ethane + water system. Yang [16] and
Kim et al. [11] measured ethane solubility in pure
water being in equilibrium with gas hydrates. These experimental data along
with the predictions of (13a) and *AD*s are shown in
Tables 6 and 7. As can be observed, the
temperature range is from 277.3 to 278.5βK, and the pressures are up to 151βMPa. This table also shows that the temperature change has no effect on experimental gas solubility
data [11, 16] indicating the experimental data reported by Yang [16] and Kim et al. [11] for ethane solubility in pure
water being in equilibrium with gas hydrates are not reliable. The maximum *AD* between the experimental and predicted data for the ethane + water system is less than 32% and the *AAD* is less than 22% [4].

Limited information is also available for the propane
solubility in pure water being in equilibrium with gas hydrates. Gaudette
and Servio [19] measured these data, which are reported in
Table 8. As can be seen, the temperature range is from 274.16 to 276.16βK, and
the pressures are up to 0.358βMPa. The *AAD* is equal to 8% and the maximum *AD* between
the experimental and predicted data is less than 17%. For
this system, the vapor pressure of the empty hydrate
lattice in the thermodynamic model was slightly
tuned, because it can be regarded as one of the deviations sources in the
thermodynamic model [1, 4].

Among the - equilibrium
data reported in the literature for the solubility in pure water at
equilibrium with its gas hydrate, those reported by Servio and Englezos [9] and Yang et al. [7] seem to be the most reliable. These data are reported in Tables 9 and 10, respectively. As can be seen, the temperature range is from 273.95 to 282.95βK, and the pressures are up to 14.2βMPa. Tables 9 and 10 also show the predictions of the thermodynamic
model used in this work and the absolute deviations. In this table, the
experimental data reported at 275.95βK and 2βMPa was used to adjust the
activity coefficient of water in (13b). An average value equal to 1.2 yielded very good results (*AD* = 0%). Using this value for the activity
coefficient of water, (13b) was applied to predict the solubility in pure
water at equilibrium with its gas hydrate. As demonstrated in Tables 9 and 10, (13b) shows encouraging results. The predictions show less than
14% absolute deviation and the average absolute deviation among all the experimental and predicted data is less than 7%. *The average value equal to 1.2 eliminates
any need to use a model for the activity coefficient of water* reconfirming
the simplicity of the model for predicting solubility of carbon dioxide in pure
water at equilibrium with its gas hydrate [3].

In addition to the quality of the experimental data reported in the literature, the deviations may be attributed to other factors, as well: the Langmuir constants and the vapor pressures of the empty hydrate lattice are consistent with initial data on the liquid water/ice-vapor-hydrate equilibrium. Since the experimental conditions go far below the initial hydrate formation conditions, the assumptions that are evidently valid at the initial hydrate formation conditions may be invalid elsewhere [1]. Accurate determination of fugacities has also important effect on the predictions. Henryβs law approach for calculating fugacities of pure and heavy hydrocarbon hydrate formers may not be very accurate, as Henryβs constants are normally developed based on experimental data and measuring these data for heavy hydrocarbon hydrate formers may not be easy [2β4].

##### 4.2. Artificial Neural Network Algorithm

The *ANN* algorithm detailed in
Table 11 with one hidden layer was used to calculate/predict the logarithm of
solubility of pure hydrocarbon hydrate former in pure water being in
equilibrium with gas hydrates as a function of temperature and carbon number of hydrocarbon hydrate former
(which represents the type of hydrocarbon), because the logarithm of solubility is
approximately linear function of temperature for a given pure hydrocarbon
hydrate former. It should be
mentioned that plenty of data should generally be used for developing *AAN* algorithms, especially for highly
nonlinear systems. In our case, where the logarithm of solubility of pure
hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates
is approximately linear function of temperature, few sets of
data for training (data reported in Tables 3β8) can be used to develop the *ANN* algorithm. However, the more data
for training, the more reliable *ANN* results. Having this in mind, one neuron in the hidden layer yielded acceptable results
according to both the accuracy of the fit (minimum value of the objective
function) and the predictive power of the neural network [4].

Tables 3β8 show the predicted/calculated
solubility values using the *ANN* algorithm
developed in this work along with the absolute deviations. As can be seen in
Tables 3β5, the results obtained using the *ANN* algorithm for methane solubility in pure water being in
equilibrium with gas hydrates show less than 9% absolute deviation and the
average absolute deviation among all
the experimental and predicted data is 3.4%. The maximum *AD* and the *AAD* between the
experimental data and predicted data using the *ANN* algorithm for ethane solubility in pure water being in
equilibrium with gas hydrates are about 31% and 22%, respectively, as mentioned in
Tables 6 and 7. The deviations are attributed to experimental data [11, 16], which are not temperature dependent and cannot
be considered reliable.
Because of unreliability of the experimental data [11, 16] for the ethane + water system, the pseudo-experimental data generated from the thermodynamic
model were used for training the *ANN* algorithm. Table 8
shows that using the *ANN*, the maximum *AD* between the experimental and predicted data for propane
solubility in pure water being in equilibrium with gas hydrates is below 5%
and the *ADD* equals to 1.6% [4].

Using the *ANN* algorithm, the solubility of -butane (the heaviest pure hydrocarbon hydrate former) in pure
water being in equilibrium with gas hydrates, for which there is no
experimental data in the literature, was predicted. These data along with the
predictions of the *ANN* algorithm for
methane, ethane, and propane are shown in Figure 2. As can be observed, the *ANN* predicts the true behavior of
solubilities of pure hydrocarbons hydrate formers in pure water being in
equilibrium with gas hydrates (the solubility decreases by increasing the
carbon number of pure hydrocarbon hydrate former and increases by increasing
the temperature), demonstrating the capability of the *ANN* algorithm for modeling the - equilibrium of water + hydrocarbon systems [4].

Finally,
it should be mentioned that the *AD*s obtained using the thermodynamic
model are generally higher than the *AD*s
obtained using the *ANN* algorithm.
This was expected, as none of the parameters of this thermodynamic model were
adjusted [4].

#### 5. Conclusions

We made a review on the recently developed tools by these authors for predicting the liquid water-hydrate equilibrium of the water-hydrocarbon systems. A thermodynamic model based on equality of fugacity concept was introduced for estimating the pure hydrocarbon hydrate former solubility in pure water being in equilibrium with gas hydrates. The model employed the solid solution theory of van der Waals-Platteeuw [20] for calculating the fugacity of water in the hydrate phase while Henryβs law approach and the activity coefficient method were used to calculate the pure hydrocarbon hydrate former fugacity and water fugacity, respectively, in the liquid water phase. The main advantages of this model are the availability of input data and the simplicity of the calculations by ignoring the effect of pressure on the liquid water-hydrate equilibrium. The reliability of model was tested for three hydrocarbon hydrate formers: methane, ethane, and propane. Acceptable agreements were achieved between the results of this model and some selected experimental data from the literature. The study showed a need for generating reliable experimental data for the liquid water-hydrate equilibrium, especially for the systems containing heavy hydrocarbon hydrate former. The deviations of the model predictions were attributed to different factors, especially the quality of the experimental data, correlations employed for the Langmuir constants and the vapor pressures of the empty hydrate lattice, and finally expressions for the components fugacities.

Furthermore,
it was shown that the model can be extended to the + water
system by using an *average value equal to
1.2 for the activity coefficient of water in the + water system*,
which *eliminates any need to use a model
for the activity coefficient of water*. This value was already assumed equal
to unity for the single hydrocarbon hydrate former + water system.

A feed-forward artificial neural network algorithm with one hidden layer, which uses a modified Levenberg-Marquardt optimization algorithm [38, 39], was developed for estimating the solubility of single hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates. This algorithm has one output neuron (logarithm of solubility of single hydrocarbon hydrate former in pure water being in equilibrium with gas hydrates), two input neurons (temperature and carbon number of hydrocarbon hydrate former, which represents the type of hydrocarbon) and one neuron in the hidden layer taking advantage of a linear activation function. The model was trained and developed using more recent and reliable data reported in the literature and its predictions were successfully compared with some independent experimental data (not used in developing the artificial neural network algorithm) for three hydrate formers: methane, ethane, and propane.

*Nomenclature*

AAD: | Average absolute deviation |

AD: | Absolute deviation () |

ANN: | Artificial neural network |

CNG: | Compressed natural gas |

FNN: | Feed-forward (back propagation) neural network |

LNG: | Liquefied natural gas |

: | Parameter of Henryβs constant correlation |

: | Parameter of Henryβs constant correlation |

: | Langmuir constant |

: | Parameter of Henryβs constant correlation |

: | Parameter of Henryβs constant correlation |

: | Hydrate |

: | Henryβs constant |

: | Input vector |

: | Liquid |

: | Number of experimental data |

: | Number of feed inputs |

: | Output |

: | Pressure |

: | Universal gas constant |

: | Weighted sum |

: | Temperature |

: | Vapor |

: | Parameter of Langmuir constant correlation |

: | Parameter of Langmuir constant correlation |

: | Bias |

: | Parameter of Langmuir constant correlation |

: | Parameter of Langmuir constant correlation |

: | Fugacity |

: | Activation function |

: | Layer |

: | Molar volume |

: | Number of cavities of type per water molecule in a unit hydrate cell |

: | Weight parameter between each neuron-neuron interconnection |

: | Mole fraction in the liquid phase |

: | Parameter of activation function |

*Greek Letters*

: | Chemical potential |

: | Activity coefficient |

Correction parameter |

*Superscripts*

: | Hypothetical empty hydrate |

: | Hydrate |

: | Liquid state |

Sat: | Saturation condition |

*Subscripts*

: | Hydrocarbon hydrate former |

exp: | Experimental data |

max: | Maximum value |

min: | Minimum value |

g: | Gas |

Water |