Numerical Methods¶
This section describes the specification of numerical methods used by solvers.
Name 
Type 
Default 
Description 

FiniteElements 
node 
unique 

FiniteVolume 
node 
unique 
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 finitevolume discretizations are available to simulate singlephase flow in GEOSX, namely, a standard cellcentered TPFA approach, and a hybrid finitevolume scheme relying on both cellcentered and facecentered degrees of freedom. The key difference between these two approaches is the computation of the flux, as detailed below.
Standard cellcentered TPFA FVM¶
This is the standard scheme implemented in the SinglePhaseFVM flow solver. It only uses cellcentered degrees of freedom and implements a TwoPoint Flux Approximation (TPFA) for the computation of the flux. The numerical flux is obtained using the following expression for the mass flux between cells and :
where is the pressure of cell , is the depth of cell , and is the standard TPFA transmissibility coefficient at the interface. The fluid density, , and the fluid viscosity, , 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 Korthogonal meshes. The hybrid finitevolume scheme–equivalent to the wellknown hybrid Mimetic Finite Difference (MFD) scheme–remains consistent with the pressure equation even when the mesh does not satisfy the Korthogonality condition. This numerical scheme is currently implemented in the SinglePhaseHybridFVM solver.
The hybrid FVM scheme uses both cellcentered and facecentered pressure degrees of freedom. The onesided face flux, , at face of cell is computed as:
where reads:
In the previous equation, is the cellcentered pressure, is the facecentered pressure, is the depth of cell , and is the depth of face . The fluid density, , and the fluid viscosity, , are upwinded using the sign of . The local transmissibility of size satisfies:
Above, is a matrix of size storing the normal vectors to each face in this cell, is a matrix of size storing the vectors from the cell center to the face centers, and is the permeability tensor. The local transmissibility matrix, , is currently computed using the quasiTPFA approach described in Chapter 6 of this book. The scheme reduces to the TPFA discretization on Korthogonal meshes but remains consistent when the mesh does not satisfy this property. The mass flux written above is then added to the mass conservation equation of cell .
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 between two neighboring cells and , the algebraic constraint reads:
We obtain a numerical scheme with cellcentered degrees of freedom and facecentered pressure degrees of freedom. The system involves mass conservation equations and facebased 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.