Numerical Methods¶

This section describes the specification of numerical methods used by solvers.

Name	Type	Default	Description
FiniteElements	node	unique	Element: FiniteElements
FiniteVolume	node	unique	Element: FiniteVolume

Finite Element Discretization¶

We are currently refactoring the finite element infrastructure, and will update the documentation soon to reflect the new structure.

Finite Volume Discretization¶

Two different finite-volume discretizations are available to simulate single-phase flow in GEOSX, namely, a standard cell-centered TPFA approach, and a hybrid finite-volume scheme relying on both cell-centered and face-centered degrees of freedom. The key difference between these two approaches is the computation of the flux, as detailed below.

Standard cell-centered TPFA FVM¶

This is the standard scheme implemented in the SinglePhaseFVM flow solver. It only uses cell-centered degrees of freedom and implements a Two-Point Flux Approximation (TPFA) for the computation of the flux. The numerical flux is obtained using the following expression for the mass flux between cells $K$ and $L$ :

$F_{KL} = \Upsilon_{KL} \frac{\rho^{upw}}{\mu^{upw}} \big( p_K - p_L - \rho^{avg} g ( d_K - d_L ) \big),$

where $p_K$ is the pressure of cell $K$ , $d_K$ is the depth of cell $K$ , and $\Upsilon_{KL}$ is the standard TPFA transmissibility coefficient at the interface. The fluid density, $\rho^{upw}$ , and the fluid viscosity, $\mu^{upw}$ , are upwinded using the sign of the potential difference at the interface.

This is currently the only available discretization in the Compositional Multiphase Flow Solver.

Hybrid FVM¶

This discretization scheme overcomes the limitations of the standard TPFA on non K-orthogonal meshes. The hybrid finite-volume scheme–equivalent to the well-known hybrid Mimetic Finite Difference (MFD) scheme–remains consistent with the pressure equation even when the mesh does not satisfy the K-orthogonality condition. This numerical scheme is currently implemented in the SinglePhaseHybridFVM solver.

The hybrid FVM scheme uses both cell-centered and face-centered pressure degrees of freedom. The one-sided face flux, $F_{K,f}$ , at face $f$ of cell $K$ is computed as:

$F_{K,f} = \frac{\rho^{upw}}{\mu^{upw}} \widetilde{F}_{K,f},$

where $\widetilde{F}_{K,f}$ reads:

$\widetilde{F}_{K,f} = \sum_{f'} \Upsilon_{ff'} \big( p_K - \pi_f - \rho_K g ( d_K - d_f ) \big).$

In the previous equation, $p_K$ is the cell-centered pressure, $\pi_f$ is the face-centered pressure, $d_K$ is the depth of cell $K$ , and $d_f$ is the depth of face $f$ . The fluid density, $\rho^{upw}$ , and the fluid viscosity, $\mu^{upw}$ , are upwinded using the sign of $\widetilde{F}_{K,f}$ . The local transmissibility $\Upsilon$ of size $n_{\textit{local faces}} \times n_{\textit{local faces}}$ satisfies:

$N K = \Upsilon C$

Above, $N$ is a matrix of size $n_{\textit{local faces}} \times 3$ storing the normal vectors to each face in this cell, $C$ is a matrix of size $n_{\textit{local faces}} \times 3$ storing the vectors from the cell center to the face centers, and $K$ is the permeability tensor. The local transmissibility matrix, $\Upsilon$ , is currently computed using the quasi-TPFA approach described in Chapter 6 of this book. The scheme reduces to the TPFA discretization on K-orthogonal meshes but remains consistent when the mesh does not satisfy this property. The mass flux $F_{K,f}$ written above is then added to the mass conservation equation of cell $K$ .

In addition to the mass conservation equations, the hybrid FVM involves algebraic constraints at each mesh face to enforce mass conservation. For a given interior face $f$ between two neighboring cells $K$ and $L$ , the algebraic constraint reads:

$\widetilde{F}_{K,f} + \widetilde{F}_{L,f} = 0.$

We obtain a numerical scheme with $n_{\textit{cells}}$ cell-centered degrees of freedom and $n_{\textit{faces}}$ face-centered pressure degrees of freedom. The system involves $n_{\textit{cells}}$ mass conservation equations and $n_{\textit{faces}}$ face-based constraints. The linear systems can be efficiently solved using the MultiGrid Reduction (MGR) preconditioner implemented in the Hypre linear algebra package.

The implementation of the hybrid FVM scheme for Compositional Multiphase Flow Solver is in progress.