Table 1 Constitutive relationships used in the established model

Density, \(\rho\)

\(\rho = {\rho_l}{s_l} + {\rho_v}(1 - {s_l})\)

Darcy velocity, \(\vec u\)

\(\vec u = \varepsilon \vec V\), \(\vec V\) is the physical velocity

Relative permeability of liquid phase and vapor phase, \({k_{rl}}\), \({k_{rv}}\)

\({k_{rl}} = {s_l}^3\), \({k_{rv}} = {(1 - {s_l})^3}\)

Kinematic viscosity, \(\nu\)

\(\nu = 1/(\frac{{{k_{rl}}}}{{\nu_l}} + \frac{{{k_{rv}}}}{{\nu_v}})\)

Relative mobility of liquid phase and vapor phase, \({\lambda_l}\), \({\lambda_v}\)

\({\lambda_l} = \frac{{\nu {k_{rl}}}}{{\nu_l}}\), \({\lambda_v} = \frac{{\nu {k_{rv}}}}{{\nu_v}}\)

Mixture pressure, p

\(\nabla p = {\lambda_l}\nabla {p_l} + {\lambda_v}\nabla {p_v}\)

Permeability, \(K\)

\(K = \frac{{d_p^2{\varepsilon^3}}}{{150{{(1 - \varepsilon )}^2}}}\)

Kinematic density, \({\rho_k}\)

\({\rho_k} = {\rho_l}{\lambda_l} + {\rho_v}{\lambda_v}\)

Specific enthalpy of fluid, \({h_f}\)

\({h_f} = [{\rho_l}{s_l}{h_l} + {\rho_v}(1 - {s_l}){h_v}]/\rho\)

Kinematic enthalpy of fluid, \({h_k}\)

\({h_k} = {\lambda_l}{h_l} + {\lambda_v}{h_v}\)

Total mass flux, \(\vec j\)

\(\vec j = - D({s_l})\nabla {s_l} + {k_{rv}}{\lambda_l}\frac{{K({\rho_l} - {\rho_v})}}{{\nu_v}}\vec a\)

Capillary diffusion coefficient, \(D({s_l})\)

\(D({s_l}) = \frac{{\sqrt {K\varepsilon } }}{\nu }{\lambda_l}(1 - {\lambda_l})\sigma [ - J({s_l})^\prime]\)

Leverett correlation [35], \(J({s_l})\)

\(J({s_l}) = 1.417(1 - {s_l}) - 2.120{(1 - {s_l})^2} + 1.263{(1 - {s_l})^3}\)

Effective heat transfer coefficient of fluid, \({k_{feff}}\)

\({k_{feff}} = \varepsilon {k_v}(1 - {s_l}) + \varepsilon {k_l}{s_l}\)

Effective heat transfer coefficient of solid, \({k_{seff}}\)

\({k_{seff}} = (1 - \varepsilon ){k_s}\)