B41oa oil and Gas Processing Section a flow Assurance Heriot-Watt University

n
i
i
i
H
Y
RT
1
1
ln
υ
µ
µ
β
………………
..
……
..(1.20)
In which 
i
υ
is the number of cavities of type 
i
and 
i
Y
is the probability that a 
cavity of type 
i
is occupied by a gas molecule: 
P
c
P
c
Y
i
i
i
+
=
1
…………………………………………
.(1.21)
The term 
i
c
is a function of the gas molecule and the cage type (as well as 
temperature) and 
P
is the pressure. The vdW+P approach assumes: 
1. The contribution of the water framework to the Gibbs energy of the 
hydrate lattice is INDEPENDENT of the gas molecule type and the 
number of cages that are occupied. 
2. No cavity contains more than one gas molecule 
3. The interactions between gas molecules within the hydrate are 
negligible. 
4. Quantum effects may be ignored. 
While this approach is theoretically sound it really only applied to a single gas 
species, so Parrish and Prausnitz (1972) extended the model to accommodate 
the multi-component gas mixtures that are relevant to the oil and gas industry 
∑
∑
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
−
=
−
n
i
j
i
j
i
H
Y
RT
1
,
1
ln
υ
µ
µ
β
……………
.
…
..(1.22) 
and 
j
ij
j
j
ij
i
K
P
c
P
c
Y
∑
+
=
1
,
………………………………
.
…
(1.23)

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TOPIC 1: Gas Hydrates 
 
 
 
56 
©H
ERIOT
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NIVERSITY B41OA December 2018 v3 
In which the sum in both equations (1.22) and (1.23) is over all the 
j
components in the gas mixture. The pressure 
j
P
is the partial pressure of the 
component 
j
in the gas mixture. 
All components are included because the gas molecules compete for cavities 
to occupy, so as a cavity becomes occupied the probability that a gas 
molecule can enter the hydrate is reduced. 
Parrish and Prausnitz also replaced the partial pressure with fugacity in order 
to deal with any non-ideal behaviour in the gas phase, so that equation (1.23) 
becomes 
j
ij
j
K
i
i
K
f
c
f
c
Y
ˆ
1
ˆ
,
∑
+
=
…………………………………
.(1.24)
Ng and Robinson (1977) extended the model further to include liquid phase 
hydrocarbons and calculated the fugacities from the Peng-Robinson equation 
of state. Any other relevant equation of state can be used with minor 
modifications to the equations. To do the calculations equation (1.25), shown 
below, must be solved. 
0
=
−
α
µ
µ
H
………………………………………
..(1.25)
This requires that the chemical potential of water in the aqueous phase must 
be equal to that of the water in the hydrate phase. This is iterative and for a 
single component gas is straightforward because the type of hydrate formed is 
known. 
For a gas mixture, where the hydrate structure formed is not necessarily 
known, the calculation must be done for EACH of the possible hydrate 
structures: 
•
If the results is a difference GREATER than zero for the first hydrate 
structure, this is the stable structure and the calculation ends. 
•
If the difference is LESS than zero, the calculation must be performed 
until the chemical potentials are identical for the other structure. 
Commercially available software is available to do these calculations. 
•
EQUI-PHASE Hydrate from D.B. Robinson and Associates. 
•
HWHYD from Heriot-Watt University. 
•
CSMHYD from the Sloan group at the Colorado School of Mines. 
•
The process modelling software Hysys, Aspen and Prosim all contain 
hydrate prediction tools. 

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TOPIC 1: Gas Hydrates 
 
 
 
57 
©H
ERIOT
-W
ATT
U
NIVERSITY B41OA December 2018 v3