Numerical Methods

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

XML Element: NumericalMethods

Name	Type	Default	Description
FiniteElements	node	unique	XML Element: FiniteElements
FiniteVolume	node	unique	XML 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 $K$ , $\rho^{avg}$ is the average fluid density, $d_K$ 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.

For Compositional Multiphase Flow Solver there are two options to compute the average density, $\rho^{avg}$ . The desired option can be selected using the gravityDensityScheme parameter:

ArithmeticAverage: $\rho^{avg}$ is computed using simple arithmetic average: $\rho^{avg} = 0.5 \cdot (rho_K + rho_L)$ , where $rho_K$ and $rho_K$ are densities in the two cells.
PhasePresence: average phase density is computed using checking for phase presence:
- $\rho^{avg} = 0.5 \cdot (\rho_K + \rho_L)$ if phase is present in both cells $K$ and $L$
- $\rho^{avg} = \rho_K$ if phase is present only in cell $K$
- $\rho^{avg} = \rho_L$ if phase is present only in cell $L$

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.

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.