Table 1 Basic relationships for mass and momentum balance equations

Liquid advective flux (Darcy’s law)

\({\mathbf{q}}_{l}=\frac{-k}{{\mu }_{l}}\left(\nabla {p}_{l}+{\rho }_{l}g\nabla z\right)\)

Heat conduction (Fourier’s law)

\({{\varvec{i}}}_{{\varvec{c}}}=-{\lambda }_{b}\nabla T\)

Heat dispersion (Fourier’s law)

\({{\varvec{i}}}_{{\varvec{d}}}=-{\lambda }_{d}\nabla T\)

Liquid density

\({\rho }_{l}= {\rho }_{l0} \mathrm{exp} \left({\alpha }_{lp}\left({p}_{l}-{p}_{l0}\right)+{\alpha }_{lT}\left(T-{T}_{0}\right)\right)\)

Solid density

\({\rho }_{s}= {\rho }_{s0}\mathrm{ exp}\left({\alpha }_{sp}\left(p-{p}_{s0}\right)+{\alpha }_{sT}\left(T-{T}_{0}\right)\right)\)

Thermal retardation coefficient

\(R=\frac{\phi {c}_{l}{\rho }_{l}+(1-\phi ){c}_{s}{\rho }_{s}}{\phi {c}_{l}{\rho }_{l}}\)

Bulk thermal conductivity

\({\lambda }_{b} ={(1-\phi )}^{n}{\lambda }_{s}+{\phi }^{n}{\lambda }_{l}\)

Dispersive conductivity

\({\lambda }_{d}\) = \({c}_{l}{\rho }_{l}d\left|{\mathbf{q}}_{l}\right|\)