Table 1 Key reservoir equations

\(\frac{\text{d}}{\text{d}t}\int _{V_{n}} M^{\kappa } \text{d}V_{n} = \int _{\Gamma _{n}} F^{\kappa } \cdot n \text{d}\Gamma _{n} + \int _{V_{n}} q^{\kappa } \text{d}V_{n}\)

(1)

\(M^{\text{m}} = \phi \left(S_{\text{L}}\rho _{\text{L}}X^{\kappa }_{\text{L}}+S_{\text{G}}\rho _{\text{G}}X^{\kappa }_{\text{G}}\right)\)

(2)

\(F^{\text{m}} = X^{\kappa }_{\text{L}}\rho _{\text{L}}u_{\text{L}}+X^{\kappa }_{\text{G}}\rho _{\text{G}}u_{\text{G}}\)

(3)

\(M^{\text{E}} = (1-\phi )\rho _{\text{R}}c_{\text{R}}T + \phi (\rho _{\text{L}}S_{\text{L}}U_{\text{L}}+\rho _{\text{G}}S_{\text{G}}U_{\text{G}})\)

(4)

\(F^{\text{E}} = -\lambda \nabla T + h_{\text{L}}\rho _{\text{L}}u_{\text{L}}+h_{G}\rho _{\text{G}}u_{\text{G}}\)

(5)

\(u_{\text{ph}} = -\frac{k_{a} k_{r,\text{ph}}}{\mu_{\text{ph}}}(\nabla {P}_{\text{ph}} - \rho_{\text{ph}}g)\)

(6)