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Comparison of Numerical Formulations for Two-phase Flow in Porous Media

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Abstract

Numerical approximation based on different forms of the governing partial differential equation can lead to significantly different results for two-phase flow in porous media. Selecting the proper primary variables is a critical step in efficiently modeling the highly nonlinear problem of multiphase subsurface flow. A comparison of various forms of numerical approximations for two-phase flow equations is performed in this work. Three forms of equations including the pressure-based, mixed pressure–saturation and modified pressure–saturation are examined. Each of these three highly nonlinear formulations is approximated using finite difference method and is linearized using both Picard and Newton–Raphson linearization approaches. Model simulations for several test cases demonstrate that pressure based form provides better results compared to the pressure–saturation approach in terms of CPU_time and the number of iterations. The modification of pressure–saturation approach improves accuracy of the results. Also it is shown that the Newton–Raphson linearization approach performed better in comparison to the Picard iteration linearization approach with the exception for in the pressure–saturation form.

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Authors and Affiliations

  1. Department of Civil Engineering, Sharif University of Technology, P.O. Box 11365-9313, Tehran, Iran

    B. Ataie-Ashtiani & D. Raeesi-Ardekani

Authors
  1. B. Ataie-Ashtiani
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  2. D. Raeesi-Ardekani
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Correspondence to B. Ataie-Ashtiani.

Appendices

Appendix A: Derivation of Coefficients of Two-fluid-phase Equations for Pressure–Saturation Formulations

1.1 PSP Elemental Matrices

Using iterative increment equations it is possible to put Eq. 15 in the form of Eq. 10 where:

$$ B_{{w_{j} }}^{n + 1,m} = - {\frac{{\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}};\quad C_{{w_{j} }}^{n + 1,m} = {\frac{\varphi }{\Updelta t}} $$
(A1,2)
$$ D_{{w_{j} }}^{n + 1,m} = {\frac{{\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} + \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}};\quad F_{{w_{j} }}^{n + 1,m} = - {\frac{{\lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}} $$
(A3,4)
$$ R_{{w_{j} }}^{n + 1,m} = - f_{w} \left( {S_{w}^{n + 1} ,P_{w}^{n + 1} } \right)^{m} $$
(A5)
$$ B_{{nw_{j} }}^{n + 1,m} = - {\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}};\quad C_{{nw_{j} }}^{n + 1,m} = - {\frac{\varphi }{\Updelta t}} $$
(A6,7)
$$ D_{{nw_{j} }}^{n + 1,m} = {\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} + \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}};\quad F_{{nw_{j} }}^{n + 1,m} = - {\frac{{\lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}}; $$
(A8,9)
$$ R_{{nw_{j} }}^{n + 1,m} = - f_{nw} (S_{w}^{n + 1} ,P_{w}^{n + 1} )^{m} $$
(A10)

The other coefficients of Eq. 10 are zero.

1.2 PSN Elemental Matrices

Applying the Newton–Raphson method to Eq. 15 yields

$$ \begin{aligned} A_{{w_{j} }}^{{n + 1,m}} & = {\frac{{\partial R_{{w_{j} }}^{{n + 1,m}} }}{{\partial S_{{w_{{j - 1}} }} }}} = {\frac{{\partial R_{{w_{j} }}^{{n + 1,m}} }}{{\partial \lambda _{{w_{{j - {\frac{1}{2}}}} }}^{{n + 1,m}} }}}\,{\frac{{d\lambda _{{w_{{j - {\frac{1}{2}}}} }}^{{n + 1,m}} }}{{dS_{{_{{w_{{j - 1}} }} }}^{{n + 1,m}} }}} \hfill \\ & = {\frac{1}{2}}\,{\frac{{P_{{w_{j} }}^{{n + 1,m + 1}} - P_{{w_{{j - 1}} }}^{{n + 1,m + 1}} - \rho _{w} g\Updelta z}}{{\Updelta z^{2} }}}\,{\frac{{d\lambda _{{w_{{j - 1}} }}^{{n + 1,m}} }}{{dS_{{_{{w_{{j - 1}} }} }}^{{n + 1,m}} }}} \hfill \\ \end{aligned} $$
(A11)
$$ \begin{gathered} C_{{w_{j} }}^{n + 1,m} = {\frac{{\partial R_{{w_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j} }} }}} = {\frac{{dR_{{w_{j} }}^{n + 1,m} }}{{dS_{{w_{j} }}^{n + 1,m} }}} + {\frac{{\partial R_{{w_{j} }}^{n + 1,m} }}{{\partial \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{w_{j} }}^{n + 1,m} }}{{\partial \lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}\,{{dS_{{_{{w_{j} }} }}^{n + 1,m} }}} \hfill \\ = {\frac{\varphi }{\Updelta t}} + {\frac{1}{2}}\,{\frac{{\partial \lambda_{{w_{j} }}^{n + 1,m} }}{{\partial S_{{_{{w_{j} }} }}^{n + 1,m} }}}\,{\frac{{ - P_{{w_{j + 1} }}^{n + 1,m + 1} + 2P_{{w_{j} }}^{n + 1,m + 1} - P_{{w_{j - 1} }}^{n + 1,m + 1} }}{{\Updelta z^{2} }}} - {\frac{{\rho_{w} g}}{\Updelta z}} \hfill \\ \end{gathered} $$
(A12)
$$ \begin{aligned} E_{{w_{j} }}^{{n + 1,m}} &= {\frac{{\partial R_{{w_{j} }}^{{n + 1,m}} }}{{\partial S_{{w_{{j + 1}} }} }}} = {\frac{{\partial R_{{w_{j} }}^{{n + 1,m}} }}{{\partial \lambda _{{w_{{j + {\frac{1}{2}}}} }}^{{n + 1,m}} }}}\,{\frac{{d\lambda _{{w_{{j + {\frac{1}{2}}}} }}^{{n + 1,m}} }}{{dS_{{_{{w_{{j + 1}} }} }}^{{n + 1,m}} }}} \hfill \\ & = {\frac{1}{2}}\,{\frac{{P_{{w_{j} }}^{{n + 1,m + 1}} - P_{{w_{{j + 1}} }}^{{n + 1,m + 1}} - \rho _{{`w}} g\Updelta z}}{{\Updelta z^{2} }}}\,{\frac{{d\lambda _{{w_{{j + 1}} }}^{{n + 1,m}} }}{{dS_{{_{{w_{{j + 1}} }} }}^{{n + 1,m}} }}} \hfill \\ \end{aligned} $$
(A13)
$$ A_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j - 1} }} }}} = {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial P_{{c_{j - 1} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j - 1} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} $$
(A14)
$$ \begin{gathered} C_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j} }} }}} = - {\frac{\varphi }{\Updelta t}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j} }} }}^{n + 1,m} }}} \hfill \\ + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial P_{{c_{j} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dS_{{_{{w_{j} }} }}^{n + 1,m} }}} \hfill \\ \end{gathered} $$
(A15)
$$ E_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j + 1} }} }}} = {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j + 1} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial P_{{c_{j + 1} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j + 1} }}^{n + 1,m} }}{{dS_{{_{{w_{j + 1} }} }}^{n + 1,m} }}} $$
(A16)

where the capillary capacity \( c_{w} = {\frac{{dP_{c}^{{}} }}{{dS_{{_{w} }}^{{}} }}} \) (Touma and Vauclin 1986) and \( {\frac{{d\lambda_{nw}^{{}} }}{{dS_{{_{w} }}^{{}} }}} \) is negative, and \( {\frac{{d\lambda_{w}^{{}} }}{{dS_{{_{w} }}^{{}} }}} \) is positive. The other coefficients of Eq. 10 are the same as those of equation PSP elemental matrix.

Appendix B: Derivation of Coefficients of Two-fluid-phase Equations for Pressure–Saturation-modified Formulations

2.1 PSMP Elemental Matrices

Using finite difference solution of the modified pressure–saturation equation with a Picard iteration scheme for the nonlinear coefficients for Eq. 20 yields

$$ \begin{aligned} - & \varphi {\frac{{S_{{w_{j} }}^{n + 1} - S_{{w_{j} }}^{n} }}{\Updelta t}} - {\frac{{(\lambda_{nw} .{\frac{{\partial P_{w} }}{\partial z}})_{{j + {\frac{1}{2}}}}^{n + 1} - (\lambda_{nw} .{\frac{{\partial P_{w} }}{\partial z}})_{{j - {\frac{1}{2}}}}^{n + 1} + (\lambda_{nw} cc_{w} {\frac{{\partial S_{w} }}{\partial z}})_{{j + {\frac{1}{2}}}}^{n + 1} - (\lambda_{nw} cc_{w} .{\frac{{\partial S_{w} }}{\partial z}})_{{j - {\frac{1}{2}}}}^{n + 1} }}{\Updelta z}} \\ + & \rho_{nw} g{\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1} - \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1} }}{\Updelta z}} = 0 \\ \end{aligned} $$
(B1)
$$ A_{{nw_{j} }}^{n + 1,m} = \left( {\lambda_{nw} cc_{w} } \right)_{{j - {\frac{1}{2}}}}^{n + 1,m} ;\quad C_{{nw_{j} }}^{n + 1,m} = - {\frac{\varphi }{\Updelta t}} - \left( {\lambda_{nw} cc_{w} } \right)_{{j - {\frac{1}{2}}}}^{n + 1,m} - \left( {\lambda_{nw} cc_{w} } \right)_{{j + {\frac{1}{2}}}}^{n + 1,m} ; $$
(B2,3)
$$ E_{{nw_{j} }}^{n + 1,m} = \left( {\lambda_{nw} cc_{w} } \right)_{{j + {\frac{1}{2}}}}^{n + 1,m} $$
(B4)

Where the \( cc_{w} = {\frac{{dS_{w} }}{{dP_{c} }}} \) is the inverse of capillary capacity c w . The other coefficients of Eq. 10 are the same as those of equation PSP elemental matrix.

2.2 PSMN Elemental Matrices

Applying the Newton–Raphson method to Eq. 20 yields

$$ A_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j - 1} }} }}} = {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial cc_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{dcc_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} $$
(B5)
$$ C_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j} }} }}} = {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial cc_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{dcc_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} $$
(B6)
$$ E_{{nw_{j} }}^{n + 1,m} = {\frac{{\partial R_{{nw_{j} }}^{n + 1,m} }}{{\partial S_{{w_{j + 1} }} }}} = {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial \lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{d\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j + 1} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{wn_{j} }}^{n + 1,m} }}{{\partial cc_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{dcc_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{dS_{{_{{w_{j + 1} }} }}^{n + 1,m} }}} $$
(B7)

where \( {\frac{{dcc_{w} }}{{dS_{w} }}} = {\frac{{d^{2} P_{c} }}{{dS^{2}_{w} }}} \) (Eq. 24a,b). The other coefficients of Eq. 10 are the same as those of equation PSP elemental matrix.

Appendix C: Derivation of Coefficients of Two-fluid-phase Equations for Pressure–Pressure Formulations

3.1 PPP Elemental Matrices

Applying the Picard method to Eq. 25a,b yields

$$ \left\{ \begin{gathered} \varphi c_{{w_{j} }}^{n + 1} {\frac{{P_{{nw_{j} }}^{n + 1} - P_{{nw_{j} }}^{n} - P_{{w_{j} }}^{n + 1} - P_{{w_{j} }}^{n} }}{\Updelta t}} \hfill \\ - {\frac{{\left( {\lambda_{w} .{\frac{{\partial P_{w} }}{\partial z}}} \right)_{{j + {\frac{1}{2}}}}^{n + 1} - (\lambda_{w} .{\frac{{\partial P_{w} }}{\partial z}})_{{j - {\frac{1}{2}}}}^{n + 1} }}{\Updelta z}} + \rho_{w} g{\frac{{\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1} - \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1} }}{\Updelta z}} = 0 \hfill \\ - \varphi c_{{w_{j} }}^{n + 1} {\frac{{P_{{nw_{j} }}^{n + 1} - P_{{nw_{j} }}^{n} - P_{{w_{j} }}^{n + 1} - P_{{w_{j} }}^{n} }}{\Updelta t}} \hfill \\ - {\frac{{(\lambda_{nw} .{\frac{{\partial P_{nw} }}{\partial z}})_{{j + {\frac{1}{2}}}}^{n + 1} - (\lambda_{nw} .{\frac{{\partial P_{nw} }}{\partial z}})_{{j - {\frac{1}{2}}}}^{n + 1} }}{\Updelta z}} + \rho_{nw} g{\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1} - \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1} }}{\Updelta z}} = 0 \hfill \\ \end{gathered} \right. $$
(C1)
$$ \left\{ \begin{gathered} y_{w} \left( {P_{nw}^{n + 1} ,P_{w}^{n + 1} } \right)^{m + 1} = \varphi c_{{w_{j} }}^{n + 1,m} {\frac{{P_{{nw_{j} }}^{n + 1,m + 1} - P_{{nw_{j} }}^{n} - P_{{w_{j} }}^{n + 1,m + 1} - P_{{w_{j} }}^{n} }}{\Updelta t}} \hfill \\ - {\frac{{\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} {\frac{{P_{{w_{j + 1} }}^{n + 1,m + 1} - P_{{w_{j} }}^{n + 1,m + 1} }}{\Updelta z}} - \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} {\frac{{P_{{w_{j} }}^{n + 1,m + 1} - P_{{w_{j - 1} }}^{n + 1,m + 1} }}{\Updelta z}}}}{\Updelta z}} + \rho_{w} g{\frac{{\lambda_{{w_{{g + {\frac{1}{2}}}} }}^{n + 1,m} - \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{\Updelta z}} = 0 \hfill \\ y_{nw} \left( {P_{nw}^{n + 1} ,P_{w}^{n + 1} } \right)^{m + 1} = - \varphi c_{{w_{j} }}^{n + 1,m} {\frac{{P_{{nw_{j} }}^{n + 1,m + 1} - P_{{nw_{j} }}^{n} - P_{{w_{j} }}^{n + 1,m + 1} - P_{{w_{j} }}^{n} }}{\Updelta t}} \hfill \\ - {\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} {\frac{{P_{{nw_{j + 1} }}^{n + 1,m + 1} - P_{{nw_{j} }}^{n + 1,m + 1} }}{\Updelta z}} - \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} {\frac{{P_{{nw_{j} }}^{n + 1,m + 1} - P_{{nw_{j - 1} }}^{n + 1,m1 + 1} }}{\Updelta z}}}}{\Updelta z}} + \rho_{nw} g{\frac{{\lambda_{{nw_{{g + {\frac{1}{2}}}} }}^{n + 1,m} - \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1} }}{\Updelta z}} = 0 \hfill \\ \end{gathered} \right.\, $$
(C2)
$$ C_{{w_{j} }}^{n + 1,m} = {\frac{{\varphi c_{{w_{j} }}^{n + 1,m} }}{\Updelta t}};\quad D_{{w_{j} }}^{n + 1,m} = - {\frac{{\varphi c_{{w_{j} }}^{n + 1,m} }}{\Updelta t}} + {\frac{{\lambda_{{w_{{j + {\frac{1}{2}}}} }}^{n + 1,m} + \lambda_{{w_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}}; $$
(C3,4)
$$ R_{{w_{j} }}^{n + 1,m} = - y_{w} (S_{w}^{n + 1} ,P_{w}^{n + 1} )^{m} $$
(C5)
$$ C_{{nw_{j} }}^{n + 1,m} =- {\frac{{\varphi c_{{w_{j} }}^{n + 1,m} }}{\Updelta t}} + {\frac{{\lambda_{{nw_{{j + {\frac{1}{2}}}} }}^{n + 1,m} + \lambda_{{nw_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\Updelta z^{2} }}};\quad D_{{nw_{j} }}^{n + 1,m} = {\frac{{\varphi c_{{w_{j} }}^{n + 1,m} }}{\Updelta t}}; $$
(C6,7)
$$ R_{{nw_{j} }}^{n + 1,m} = - f_{nw} (S_{w}^{n + 1} ,P_{w}^{n + 1} )^{m} $$
(C8)

3.2 PPN Elemental Matrices

Applying the Newton–Raphson method to Eq. 25a,b yields

$$ A_{{\alpha_{j} }}^{n + 1,m} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{nw_{j - 1} }} }}} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j - 1} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j - 1} }}^{n + 1,m} }}{{dP_{{_{{nw_{j - 1} }} }}^{n + 1,m} }}} $$
(C9)
$$ B_{{\alpha_{j} }}^{n + 1,m} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{w_{j - 1} }} }}} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j - 1} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j - 1} }}^{n + 1,m} }}{{dP_{{_{{w_{j - 1} }} }}^{n + 1,m} }}} $$
(C10)
$$ \begin{aligned} C_{{\alpha_{j} }}^{n + 1,m} = & {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{nw_{j} }} }}} = {\frac{{dR_{{\alpha_{j} }}^{n + 1,m} }}{{dP_{{_{{nw_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{nw_{j} }} }}^{n + 1,m} }}} \\& + & {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{nw_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial c_{{\alpha_{j} }}^{n + 1,m} }}}\,{\frac{{\partial c_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{nw_{j} }} }}^{n + 1,m} }}} \\ \end{aligned} $$
(C11)
$$ \begin{aligned} D_{{\alpha_{j} }}^{n + 1,m} = & {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{w_{j} }} }}} = {\frac{{dR_{{\alpha_{j} }}^{n + 1,m} }}{{dP_{{_{{w_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j - {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{w_{j} }} }}^{n + 1,m} }}} \\ &+ & {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{w_{j} }} }}^{n + 1,m} }}} + {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial c_{{\alpha_{j} }}^{n + 1,m} }}}\,{\frac{{\partial c_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j} }}^{n + 1,m} }}{{dP_{{_{{w_{j} }} }}^{n + 1,m} }}} \\ \end{aligned} $$
(C12)
$$ E_{{\alpha_{j} }}^{n + 1,m} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{nw_{j + 1} }} }}} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j + 1} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j + 1} }}^{n + 1,m} }}{{dP_{{_{{nw_{j + 1} }} }}^{n + 1,m} }}} $$
(C13)
$$ F_{{\alpha_{j} }}^{n + 1,m} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial P_{{w_{j + 1} }} }}} = {\frac{{\partial R_{{\alpha_{j} }}^{n + 1,m} }}{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}}\,{\frac{{\partial \lambda_{{\alpha_{{j + {\frac{1}{2}}}} }}^{n + 1,m} }}{{\partial P_{{_{{c_{j + 1} }} }}^{n + 1,m} }}}\,{\frac{{dP_{{c_{j + 1} }}^{n + 1,m} }}{{dP_{{_{{w_{j + 1} }} }}^{n + 1,m} }}} $$
(C14)

These coefficients above are applied for wetting and nonwetting phase (α = w, nw).

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Ataie-Ashtiani, B., Raeesi-Ardekani, D. Comparison of Numerical Formulations for Two-phase Flow in Porous Media. Geotech Geol Eng 28, 373–389 (2010). https://doi.org/10.1007/s10706-009-9298-4

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