Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false > Class Template Reference

Specialization of the CCTpfaForchheimersLaw grids where dim=dimWorld. More...

#include <dumux/flux/cctpfa/forchheimerslaw.hh>

Description

template<class ScalarType, class GridGeometry>
class Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >

Specialization of the CCTpfaForchheimersLaw grids where dim=dimWorld.

Public Types
using	Scalar = ScalarType
	state the scalar type of the law More...
using	Cache = TpfaForchheimersLawCache< ThisType, GridGeometry >
	state the type for the corresponding cache More...

Static Public Member Functions
template<class Problem , class ElementVolumeVariables , class ElementFluxVarsCache >
static Scalar	flux (const Problem &problem, const Element &element, const FVElementGeometry &fvGeometry, const ElementVolumeVariables &elemVolVars, const SubControlVolumeFace &scvf, int phaseIdx, const ElementFluxVarsCache &elemFluxVarsCache)
	Compute the advective flux using the Forchheimer equation. More...
template<class Problem , class ElementVolumeVariables >
static Scalar	calculateTransmissibility (const Problem &problem, const Element &element, const FVElementGeometry &fvGeometry, const ElementVolumeVariables &elemVolVars, const SubControlVolumeFace &scvf)
template<class Problem , class ElementVolumeVariables , bool scalarPerm = std::is_same<typename Problem::SpatialParams::PermeabilityType, Scalar>::value, std::enable_if_t< scalarPerm, int > = 0>
static DimWorldMatrix	calculateHarmonicMeanSqrtPermeability (const Problem &problem, const ElementVolumeVariables &elemVolVars, const SubControlVolumeFace &scvf)
	Returns the harmonic mean of $\sqrt{K_0}$ and $\sqrt{K_1}$. More...
template<class Problem , class ElementVolumeVariables , bool scalarPerm = std::is_same<typename Problem::SpatialParams::PermeabilityType, Scalar>::value, std::enable_if_t<!scalarPerm, int > = 0>
static DimWorldMatrix	calculateHarmonicMeanSqrtPermeability (const Problem &problem, const ElementVolumeVariables &elemVolVars, const SubControlVolumeFace &scvf)
	Returns the harmonic mean of $\sqrt{\mathbf{K_0}}$ and $\sqrt{\mathbf{K_1}}$. More...

Static Public Attributes
static const DiscretizationMethod	discMethod = DiscretizationMethod::cctpfa
	state the discretization method this implementation belongs to More...

Member Typedef Documentation

Cache

template<class ScalarType , class GridGeometry >

using Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::Cache = TpfaForchheimersLawCache<ThisType, GridGeometry>

state the type for the corresponding cache

Scalar

template<class ScalarType , class GridGeometry >

using Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::Scalar = ScalarType

state the scalar type of the law

Member Function Documentation

calculateHarmonicMeanSqrtPermeability() [1/2]

template<class ScalarType , class GridGeometry >

template<class Problem , class ElementVolumeVariables , bool scalarPerm = std::is_same<typename Problem::SpatialParams::PermeabilityType, Scalar>::value, std::enable_if_t< scalarPerm, int > = 0>

static DimWorldMatrix Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::calculateHarmonicMeanSqrtPermeability ( const Problem &  problem, const ElementVolumeVariables &  elemVolVars, const SubControlVolumeFace &  scvf  )

inlinestatic

Returns the harmonic mean of $\sqrt{K_0}$ and $\sqrt{K_1}$.

This is a specialization for scalar-valued permeabilities which returns a tensor with identical diagonal entries.

calculateHarmonicMeanSqrtPermeability() [2/2]

template<class ScalarType , class GridGeometry >

template<class Problem , class ElementVolumeVariables , bool scalarPerm = std::is_same<typename Problem::SpatialParams::PermeabilityType, Scalar>::value, std::enable_if_t<!scalarPerm, int > = 0>

static DimWorldMatrix Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::calculateHarmonicMeanSqrtPermeability ( const Problem &  problem, const ElementVolumeVariables &  elemVolVars, const SubControlVolumeFace &  scvf  )

inlinestatic

Returns the harmonic mean of $\sqrt{\mathbf{K_0}}$ and $\sqrt{\mathbf{K_1}}$.

This is a specialization for tensor-valued permeabilities.

calculateTransmissibility()

template<class ScalarType , class GridGeometry >

template<class Problem , class ElementVolumeVariables >

static Scalar Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::calculateTransmissibility ( const Problem &  problem, const Element &  element, const FVElementGeometry &  fvGeometry, const ElementVolumeVariables &  elemVolVars, const SubControlVolumeFace &  scvf  )

inlinestatic

The flux variables cache has to be bound to an element prior to flux calculations During the binding, the transmissibility will be computed and stored using the method below.

flux()

template<class ScalarType , class GridGeometry >

template<class Problem , class ElementVolumeVariables , class ElementFluxVarsCache >

static Scalar Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::flux ( const Problem &  problem, const Element &  element, const FVElementGeometry &  fvGeometry, const ElementVolumeVariables &  elemVolVars, const SubControlVolumeFace &  scvf, int  phaseIdx, const ElementFluxVarsCache &  elemFluxVarsCache  )

inlinestatic

Compute the advective flux using the Forchheimer equation.

see e.g. Nield & Bejan: Convection in Porous Media [41]

The relative passability $\eta_r$ is the "Forchheimer-equivalent" to the relative permeability $k_r$. We use the same function as for $k_r$ (VG, BC, linear) other authors use a simple power law e.g.: $\eta_{rw} = S_w^3$

Some rearrangements have been made to the original Forchheimer relation:

$$\mathbf{v_\alpha} + c_F \sqrt{\mathbf{K}} \frac{\rho_\alpha}{\mu_\alpha } |\mathbf{v_\alpha}| \mathbf{v_\alpha} + \frac{k_{r \alpha}}{\mu_\alpha} \mathbf{K} \nabla \left(p_\alpha + \rho_\alpha g z \right)= 0$$

This already includes the assumption $k_r(S_w) = \eta_r(S_w)$:

• $\eta_{rw} = S_w^x$ looks very similar to e.g. Van Genuchten relative permeabilities
• Fichot, et al. (2006), Nuclear Engineering and Design, state that several authors claim that $k_r(S_w), \eta_r(S_w)$ can be chosen equal
• It leads to the equation not degenerating for the case of $S_w=1$, because I do not need to multiply with two different functions, and therefore there are terms not being zero.
• If this assumption is not to be made: Some regularization needs to be introduced ensuring that not all terms become zero for $S_w=1$.

This non-linear equations is solved for $\mathbf{v_\alpha}$ using Newton's method and an analytical derivative w.r.t. $\mathbf{v_\alpha}$.

The gradient of the Forchheimer relations looks as follows (mind that $\sqrt{\mathbf{K}}$ is a tensor):

$$f\left(\mathbf{v_\alpha}\right) = \left( \begin{array}{ccc} 1 & 0 &0 \\ 0 & 1 &0 \\ 0 & 0 &1 \\ \end{array} \right) + c_F \frac{\rho_\alpha}{\mu_\alpha} |\mathbf{v}_\alpha| \sqrt{\mathbf{K}} + c_F \frac{\rho_\alpha}{\mu_\alpha}\frac{1}{|\mathbf{v}_\alpha|} \sqrt{\mathbf{K}} \left( \begin{array}{ccc} v_x^2 & v_xv_y & v_xv_z \\ v_yv_x & v_{y}^2 & v_yv_z \\ v_zv_x & v_zv_y &v_{z}^2 \\ \end{array} \right)$$

Note

We restrict the use of Forchheimer's law to diagonal permeability tensors so far. This might be changed to general tensors using eigenvalue decomposition to get $\sqrt{\mathbf{K}}$

Member Data Documentation

discMethod

template<class ScalarType , class GridGeometry >

const DiscretizationMethod Dumux::CCTpfaForchheimersLaw< ScalarType, GridGeometry, false >::discMethod = DiscretizationMethod::cctpfa

static

state the discretization method this implementation belongs to

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The documentation for this class was generated from the following file:

• cctpfa/forchheimerslaw.hh