Primary variables

In a miscible multiphase simulation, where phases can appear/disappear depending on the thermodynamical conditions, the appropriate primary nonlinear variables can depend on the phase state of the system. For example, consider a two-phase, two-component model. If both fluid phases are present, a suitable set of primary variables are pressure of one phase, saturation of one phase, and temperature. If one phase disappears (for instance, due to dissolution of the gas fluid component into the liquid fluid phase), then phase saturation is no longer present in the governing equations, and is therefore not an appropriate primary variable. In this case, the mass fraction of a fluid component in the remaining phase is a suitable choice of primary variables. This is summarised in Table 1:

Table 1: Primary variables

Phase state	Variable 1	Variable 2	Variable 3
Single phase $(\alpha)$	$P_{\alpha}$	$T$	$x_{\alpha}^{\kappa}$
Two phases	$P_{\alpha}$	$T$	$S_{\alpha}$

There are two popular strategies to overcome this issue: primary variable switching, and using a persistent set of primary variables.

Primary variable switch

In this approach, the primary variables are switched depending on the phase state of the model. For example, if only a single fluid phase is present, the primary variables used might be the pressure, temperature and mass fraction of a component in the phase, see Table 1. If the phase state changes to a two-phase model, then the mass fraction variable is switched to now represent the saturation of one of the fluid phases.

This approach has been used in several flow simulators, for example TOUGH2 (Pruess et al., 1999)

Persistent set of primary variables

An alternative approach is to use a set of primary variables that remain independent in all phase states. This persistent primary variable approach has been implemented in PorousFlow for miscible two-phase flow. In this approach, the primary variables are pressure of one phase $P_{\alpha}$ , temperature $T$ and total mass fraction of a fluid component summed over all phases $ $z^{\kappa} = \frac{\sum_{\alpha} S_{\alpha} \rho_{\alpha} x_{\alpha}^{\kappa}}{\sum_{\alpha} S_{\alpha} \rho_{\alpha}}.$

References

K. Pruess, C. Oldenburg, and G. Moridis. TOUGH2 User's Guide, Version 2.0. Technical Report LBNL-43134, Lawrence Berkeley National Laboratory, Berkeley CA, USA, 1999.

@techreport{pruess1999,
    author = "Pruess, K. and Oldenburg, C. and Moridis, G.",
    title = "{TOUGH2 User's Guide, Version 2.0}",
    institution = "Lawrence Berkeley National Laboratory",
    address = "Berkeley CA, USA",
    number = "LBNL-43134",
    year = "1999"
}

Primary variable switch
Persistent set of primary variables
References