Compositional flash calculations

The miscible flow models available in PorousFlow use a compositional flash to determine the amount of each fluid phase present for the given set of persistent primary variables using the Rachford-Rice equation (Rachford and Rice, 1952)

$$\sum_i \frac{z_i (K_i - 1)}{1 + V (K_i - 1)} = 0,$$(1) where $z_i$ is the total mass fraction of fluid component $i$ summed over all fluid phases $$z_i = \frac{\sum_{\alpha} S_{\alpha} \rho_{\alpha} x_{\alpha}^{i}}{\sum_{\alpha} S_{\alpha} \rho_{\alpha}},$$ $K_i$ is the equilibrium constant $$K_i = \frac{y_i}{x_i}$$ that relates the mass fraction of fluid component $i$ in the gas phase ($y_i$) to the mass fraction in the liquid phase ($x_i$), and $V$ is the mass fraction of fluid in the gas phase, which for a two-phase model is $$V = \frac{S_g \rho_g}{S_g \rho_g + S_l \rho_l}$$ where $S_g$ and $S_l$ are the saturations of the gas and liquid phases, respectively, and $\rho_g$ and $\rho_l$ are the densities of the gas and liquid phases, respectively.

The Rachford-Rice equation, Eq. (1), is solved for $V$, after which the unknown gas saturation can be calculated. The Rachford-Rice equation can be solved analytically for the case where there are two fluid components, whereby $V$ is $$V = \frac{z_i (K_1 - K_0) - (K_1 - 1)}{(K_1 - 1)(K_0 - 1)}.$$

For cases where there are more than two fluid components, however, Eq. (1) must be solved numerically. Fortunately, the Rachford-Rice equation has the nice numerical property that Eq. (1) is monotonically varying with $V$, so that numerical solution requires only a few iterations, and is therefore numerically inexpensive.

Once the vapor mass fraction $V$ has been calculated, the mass fractions of fluid component $i$ in each phase can be calculated $$x_i = \frac{z_i}{(K_i - 1) V + 1}, \quad y_i = \frac{z_i K_i}{(K_i - 1) V + 1}.$$