4. Implemented in Code_Aster#

In this chapter, we specify how the relationships described in Chapter 3 are integrated.

Attention, finite volumes are currently only available only with the law of hydraulic coupling HYDR_VGMqui makes it possible to correctly manage phase apparation/disappearance by treating negative capillary pressures. The only hydraulic models available today are therefore of the Mualem Van-Genuchten type.

4.1. Description of the elements#

2 types of models exist depending on the scheme used:

Modeling	Corresponding Schema	Compatible Flow Law
D_ PLAN_HH2SUDA	VFDA	LIQU_AD_GAZ_VAPE, LIQU_AD_GAZ
3D_ HH2SUDA	VFDA	LIQU_AD_GAZ_VAPE, LIQU_AD_GAZ

Table 4.1-1 : Finished Volume Models

The main unknowns are capillary pressure (PRE1) and gas pressure (PRE2) and are located at the center of the cells as well as in the middle of the edges (cf.).

Illustration 2: Quadratic element

This therefore gives for a quadrangle a storage like this:

Support	DDL
Face \(\sigma 1\)	PRE1
	PRE2
Face \(\sigma 2\)	PRE1
	PRE2
Face \(\sigma 3\)	PRE1
	PRE2
Face \(\sigma 4\)	PRE1
	PRE2
Center \(K\)	PRE1
	PRE2

Table 4.1-2 : Storage of unknowns

The mesh elements are defined in 2D for triangles with 7 knots and quadrilaterals with 9 knots (« unused » vertices + middle of the edges + center) as well as in 3D for hexahedra with 27 knots.

There are no mesh elements that would exclude the vertex nodes but the latter are not taken into account (cf.).

4.2. Calculation of generalized stresses and deformations#

The finite volume diagrams benefit greatly from the structure defined for the finite elements in [R7.01.10] and [R7.01.11].

Thus, the generalized constraints at the center of the element are physically the same as in finite elements, namely:

\({m}_{l}^{w},{\mathrm{F}}_{\mathrm{l}}^{\mathrm{w}};{m}_{g}^{w},{\mathrm{F}}_{\mathrm{g}}^{\mathrm{w}};{m}_{g}^{h},{\mathrm{F}}_{\mathrm{g}}^{\mathrm{h}};{m}_{l}^{h},{\mathrm{F}}_{\mathrm{l}}^{\mathrm{h}}\) as well as the generalized deformations: \({p}_{c},\mathrm{\nabla }{p}_{c};{p}_{\text{g}},\mathrm{\nabla }{p}_{\text{g}}\).

At interfaces, on the other hand, flows actually contain what is needed for the continuity equation. The latter may be different depending on the scheme used (see section 3.2.4).

Aster component name	Content at the center of \(K\)	Content on the faces \(\sigma \epsilon \delta K\)
M11	\({\rho }_{l}^{w}\varphi {S}_{l}-{({\rho }_{l}^{w}\varphi {S}_{l})}^{-}\)	0
FH11 *	\(\mathrm{\sum }_{\delta K}^{}{F}_{l}^{w}\mathrm{.}n\)	\({F}_{l}^{w}\mathrm{.}{n}_{K,\sigma }+{F}_{g}^{w}\mathrm{.}{n}_{K,\sigma }\)
M12	\({\rho }_{g}^{w}\varphi {S}_{g}-{({\rho }_{g}^{w}\varphi {S}_{g})}^{-}\)	0
FH12 *	\(\mathrm{\sum }_{\delta K}^{}{F}_{g}^{w}\mathrm{.}n\)	\({F}_{l}^{h}\mathrm{.}{n}_{K,\sigma }+{F}_{g}^{h}\mathrm{.}{n}_{K,\sigma }\)
M21	\({\rho }_{g}^{h}\varphi {S}_{l}-{({\rho }_{g}^{h}\varphi {S}_{g})}^{-}\)	0
FH21 *	\(\sum _{\delta K}^{}{F}_{g}^{h}\mathrm{.}n\)	0
M22	\({\rho }_{l}^{h}\varphi {S}_{l}-{({\rho }_{l}^{h}\varphi {S}_{l})}^{-}\)	0
FH22 *	\(\sum _{\delta K}^{}{F}_{l}^{h}\mathrm{.}n\)	0

Table 4.2-1 : Generalized constraints for schemas VFDA (* HH2SUDA )

4.3. Integration#

As in finite elements, the main integration loop is done by element. On the other hand, within an element we will loop over the nodes (there is no longer any concept of integration points, but just approximation points which are the nodes here). Since this diagram has its nodes in the center as well as on each edge, this here implicitly allows you to loop the interfaces. Finally, the finite element structure is not modified.

In the end, we can summarize the integration of the various equations treated in the following table (written for a quadrangle but easily generalizable):

Support	Equation	Type
Face \(\sigma 1\)	\({R}_{\sigma 1}^{w}\)	Continuity water flow
Face \(\sigma 1\)	\({R}_{\sigma 1}^{h}\)	Continuity flow of \(h\)
Face \(\sigma 2\)	\({R}_{\sigma 2}^{w}\)	Continuity water flow
Face \(\sigma 2\)	\({R}_{\sigma 2}^{h}\)	Continuity flow of \(h\)
Face \(\sigma 3\)	\({R}_{\sigma 3}^{w}\)	Continuity water flow
Face \(\sigma 3\)	\({R}_{\sigma 3}^{h}\)	Continuity flow of \(h\)
Face \(\sigma 4\)	\({R}_{\sigma 4}^{w}\)	Continuity water flow
Face \(\sigma 4\)	\({R}_{\sigma 4}^{h}\)	Continuity flow of \(h\)
Center \(K\)	\({R}_{K}^{w}\)	Conservation of the mass of water
Center \(K\)	\({R}_{K}^{h}\)	Conservation of the mass of \(h\)

4.4. Internal variables#

The internal variables are here:

Number	Aster component name	Content
1	\(\mathit{V1}\)	\({\rho }_{\text{lq}}\text{-}{\rho }_{{\text{lq}}_{}}^{0}\)
2	\(\mathit{V2}\)	\(\phi \text{-}{\phi }^{0}\)
3	\(\mathit{V3}\)	\({p}_{\text{vp}}\mathrm{-}{p}_{\text{vp}}^{0}\)
4	\(\mathit{V4}\)	\({S}_{\text{lq}}\)
5	\(\mathit{V5}\)	\({P}_{\text{c}}\)
6	\(\mathit{V6}\)	\({P}_{\text{g}}\)

Note: remember that here the Gauss points in the Aster sense are the nodes of the element.

4.5. Validation#

The following table shows some examples of validation test cases for classical physical problems:

Test cases	Phenomenon	Tested models
wtnp117	Capillary rebalancing	D_ PLAN_HH2SUDA (c)
wtnp120	Appearance/disappearance of phase in a bar	D_ PLAN_HH2SUDA (a) 3D_ HH2SUDA (c)
wtnp121	Water-saturated bar subjected to pressure shock	D_ PLAN_HH2SUDA (a) 3D_ HH2SUDA (e)
wtnp122	Bar saturated with gas subjected to a pressure shock	D_ PLAN_HH2SUDA (a)
wtnp123	Gas injection around a gallery	D_ PLAN_HH2SUDA (a)
wtnp124	Liakopoulos test: gravity drainage of a water column	D_ PLAN_HH2SUDA (a)