FDPDEType: FDPDE_STOKESDARCY3FIELD

Stokes-Darcy equations (3-field formulation)

The three-field formulation is an extension of the Stokes equations for conservation of mass and momentum of a two-phase mixture to include terms related to Darcy flux

where $P$ pressure and $\textbf{u}$ velocity are the unknowns. However, they will have a different meaning than in single-phase Stokes equations (i.e., fluid or dynamic pressure, solid velocity). The coefficients $A, B, C$ are defined as in Stokes questions. The $D_1, D_2, D_3, D_4, D_C$ represent the Stokes-Darcy coupling coefficients.

The 3-field formulation comes from decomposing the liquid pressure into dynamic, static and compaction components: $P_\ell=p+P_{lith}+\mathcal{P}$.

They briefly represent: $A$ - effective viscosity, $B$ - right-hand side for the momentum equation (i.e., body forces), $C$ - right-hand side for the continuity equation (i.e., incompressibility), $D_1, D_C$ - compaction, $D_2, D_3, D_4$ - coefficients for Darcy flux.

The FDPDE_STOKESDARCY3FIELD coefficients and unknowns are located on the staggered grid in the following way

where:

• A is a scalar field but located in both centers A_center (dof=1) and corners A_corner (dof=0),
• B is vector field located on faces, B = [Bx (dof=0), Bz (dof=0)],
• C is scalar field located in centers (dof=0),
• D1 is scalar field located in centers (dof=2),
• D2 is scalar field located on faces, D2x (dof=1), D2z (dof=1),
• D3 is vector field located on faces, D3x (dof=2), D3z (dof=2),
• D4 is scalar field located on faces, D4x (dof=3), D4z (dof=3),
• DC is scalar field located in centers (dof=3),

See Tests for examples with FDPDE_STOKESDARCY3FIELD

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