#1: Huron-Vidal Mixing Rule. A Unique Combination of a Cubic Equation and a GE model.
Cubic equations have been an industrial standard in oil industry since the nineteen eighties, but in the original form those equations are unable to handle the mutual solubility between hydrocarbons and aqueous components. Formation water is often produced together with petroleum reservoir fluids and hydrate inhibitors may have to be added to prevent hydrate formation when unprocessed well streams are transported in subsea pipelines.
The chemical industry has long used the so-called GE (or Excess Gibbs Energy) models to handle mixtures of water, alcohols and glycols. Examples are UNIFAC, UNIQUAC, and NRTL. Water and hydrate inhibitors are polar compounds. At a molecular level these components form clusters when dissolved into a hydrocarbon phase. This is inconsistent with the classical mixing rules, which assumes a random distribution of molecules. The GE models have never obtained widespread use in the oil industry because the models are inapplicable at pressure higher than around 10 bar/1470 psi.
In 1978 Huron and Vidal (HV) proposed to combine the “local composition capabilities” of the GE models with the “high pressure capabilities” of a cubic equation of state. They rewrote the a-parameter of the equation of state to be a function of GE at infinite pressure. That was a purely mathematical operation, which in itself would not change the results of the equation of state. They further proposed to use a modified NRTL expression for GE at infinite pressure. NRTL stands for Non-Random-Two-Liquids and includes a “non-randomness” parameter, which will have a high value for binaries of molecules with a preference for its own kind of molecules as neighbors. In a binary mixture of H2O and C1, a H2O molecule would for example rather be next to other H2O molecules than next to C1 molecules.
Huron and Vidal’s work stand out from other similar models because their mixing rule reduces to the classical mixing rule of Soave or Peng & Robinson in the absence of aqueous components. That gives two advantages. One is that an EoS model developed for a water free reservoir fluid will still be valid if water and possibly hydrate inhibitors are later on to be added. The other advantage is that the number of model parameters to be estimated from experimental data is lower than for other GE models. Three parameters are needed for each binary, one for the interaction energy between a molecule of type 1 surrounded by molecules of type 2, one for the interaction energy between a molecule of type 2 surrounded by molecules of type 1, and a 3rd parameter expressing the non-randomness, i.e. how much preference a molecule has for being surrounded by other molecules of its own type. The interaction parameters are linear in temperature, so in reality five parameters are required for each binary. Not considering pairs of identical components, an N-component mixture has N*(N-1)/2 different component pairs. With five model parameters for each binary the total number of parameters is 2.5*N*(N-1). The Huron and Vidal model only needs interaction parameters for binaries with at least one aqueous components. The remaining parameters are found from the classical equation of state parameters. A detailed description of the Huron-Vidal model can be found in section 16.1.3 of Pedersen et al. (2014)
Calsep started work on the Huron and Vidal model in the early nineteen nineties and published its first paper about the model in 1993 (Kristensen et al.). Focus was on a satisfactory description of the loss of methanol hydrate inhibitor to the hydrocarbon phases. Over the years the application area has been extended to also cover mutual solubility of hydrocarbons and water/brine at HPHT conditions (Pedersen et al., 2001 and 2004), and hydrate depression of all commonly used hydrate inhibitors including MEG (Hemmingsen et al., 2011). PVTsim supports the HV mixing rule with both the SRK and the PR equations.
New experimental data has become available since Calsep started work on the Huron-Vidal model and in late 2014 Calsep decided to revisit the HV-parameters used in PVTsim for systems with water. An improved match was seen of the water content in gas especially at high pressure. Figures 1 and 2 show examples of HV model predictions with new old parameters. Figure 1 shows that the match of the water solubility in C1 gas with both SRK and PR is improved at pressures above 600 bar. Figure 2 shows that the match of the water solubility in a CO2 rich gas is much improved with the new parameters and a good match is seen at a temperature as high as 200 oC. The new HV parameters will be default in PVTsim Nova 2.0 to be released in the summer of 2015.
Calsep has worked out a validation report documenting the match of the mutual solubility of water and gas seen with new and old HV parameters. The report covers both SRK and PR. The data material comprises binary, ternary, five-component and multi component systems and temperatures from 0 to 250 oC and pressures from 1 to 2000 bara. Please contact Calsep if you are interested in a copy of the validation report.
Figure 1. Water (H2O) in methane (C1) simulation results with new and the old Huron-Vidal (HV) parameters 138°C, 150°C and 171°C. The figure at the left hand side show the SRK results the right hand figure the PR results.
Figure 2. Water (H2O) in 30 mole% methane (C1) / 70 mole% carbon dioxide (CO2) at 93°C and 200°C. The figure at the left hand side show the SRK results the right hand figure the PR results.
Hemmingsen, P.V., Burgass, R., Pedersen, K. S., Kinnari, K. and Sørensen, H., “Hydrate temperature depression of MEG solutions at concentration up to 60 wt%. Experimental data and simulation results”, Fluid Phase Equilibria 307, 2011, pp. 175-179.
Huron, M.J. and Vidal, J., “New mixing rules in simple equations of state for representing vapour-liquid equilibria of strongly non ideal mixtures”, Fluid Phase Equilibria 3, 255-271, 1979.
Kristensen, J.N., Christensen, P.L., Skovborg P., Pedersen, K.S., ”A Combined Soave-Redlich-Kwong and NRTL Equation of State for Calculating the Distribution of Methanol between Water and Hydrocarbon Phases”, Fluid Phase Equilibria 82, 1993, pp. 199-206.
Pedersen, K.S., Milter, J., Rasmussen, C.P., ”Mutual solubility of water and a reservoir fluid at high temperatures and pressures. Experimental and simulated data”, Fluid Phase Equilibria 189, 85-97, 2001.
Pedersen, K.S., Milter, J., ”Phase equilibrium between gas condensate and brine at HT/HP conditions”, SPE 90309, SPE ATCE, Houston, TX, September 26-29, 2004.
Pedersen, K.S., Christensen, P.L and Azeem, J., “Phase behavior of petroleum reservoir fluids”, 2nd edition, Taylor & Francis, Boca Raton, 2014.