Matrix/DFE Node Splitting and Dual Node Connectors

DFEs can have a permeability either larger or smaller than that of the neighbouring matrix elements (standard finite-elements in FEFLOW), but with the classical Common Node approach they are only active if they are more permeable than the matrix elements. There is therefore no distinction in process variable value (hydraulic-head, concentration, temperature, etc) between matrix node and discrete feature node, for they are identical.

In FEFLOW 7.1 we have introduced the concept of the Dual Node and Dual-Node Connector (DNC) that corresponds to splitting existing common-nodes into 2 distinct nodes (the originally shared Common Node at the matrix element level, and the newly created Dual Node at the DFE level) and adding an extra one-dimensional connection (the DNC) between Common Node and Dual Node.

From a geometrical point of view, these two nodes share the same (X,Y,Z) coordinates. However a connection is established and formulated using the definition of a coupling-length  (a characteristic length defining the extension of the Dual-Node Connector) and of conductive and capacitive properties.

Schematic representation of the concept of dual-nodes and dual-node connectors. An example with four quadrangular finite elements and two 2D discrete feature elements (in blue)

 

Flow through the DNC is thus one-dimensional and is expressed using a Darcy-type formulation. Cross-sectional area, storage, conductivity and coupling-length are parameters required for the connector in order to solve the flow equation. The parameters conductivity and coupling-length are combined to formulate a conductance (or exchange) coefficient X for the DNC, X = KDNC / Lc. The exchanges can be further increased or decreased by using a modulation function.

 

With the dual-node approach, DFEs less permeable than the matrix can still interact with matrix elements. This does not mean that dual-node DFEs can be used to act as a barrier to flow. They simply can have their own parameterization (possibly less permeable than the matrix) and sets of boundary conditions, and they can be connected to the matrix through the usage of DNC elements.

The concept of Dual Nodes extends the FEFLOW capabilities to control exchanges between matrix and DFE networks. This is particularly relevant for mass and heat transport processes for which the standard, common-node approach is suffering from artificial diffusion/dispersion invading matrix elements. With the concept of Dual Nodes applications covering cases of fracture and karstic flows, coupling with surface water processes, or fluid/mass/heat/age exchange in dual-porosity models, can be handled in a more physical and robust manner.

In variable-saturation scenarios (Richards' equation), the DNCs are handled the same way as standard DFES or Multilayer Wells are. FEFLOW uses a smooth transition of relative conductivity depending on the saturated length of the connector. Further details are found in the section Other Settings of the Problem Settings dialog.

 

Creating dual-nodes from a nodal selection

A given selection of nodes can be converted to dual-nodes. The action correspond to duplicating the nodes and adding 1D connections (DNCs) between parent and child nodes. The option can be used to simulated dual-porosity effects.

 

Creating dual-nodes from a selection of DFEs

A given selection of DFEs can be converted to a dual-node representation. The action corresponds to (i) creating the dual-nodes (together with the DNCs) and (ii) assigning the dual-nodes to the DFEs.

 

 

The table below shows an example of a 1D fracture network embedded in a 2D mesh, in which a conservative mass transport is solved. Matrix hydraulic conductivity is Kmatrix= 1 m/d and DFE hydraulic conductivity is KDFE = 500 m/d. Hydraulic gradient is constant and defined from left to right. A pumping-well is locally perturbating the flow. Mass concentration is constantly injected at four nodes of the domain left border, where DFEs are in contact with the boundary (injection thus occurs directly in the DFE network). The concentration breakthroughs are recorded at the matrix node and its dual node (i.e., in the DFE) for the position indicated by the flag on the pictures. The scenarios presented in the table modify the exchange coefficient by either changing the DNC hydraulic conductivity KDNC or the coupling length Lc:

  • Case 0: KDNC = 0.001 Kmatrix, Lc = 1 m

  • Case 1: KDNC = 0.01 Kmatrix, Lc = 1 m

  • Case 2: KDNC = 0.1 Kmatrix, Lc = 1 m

  • Case 3: KDNC = 1.0 Kmatrix, Lc = 1 m

  • Case 4: KDNC = 1.0 Kmatrix, Lc = 0.01 m

 

Case

Concentration Distribution at final time

Concentration Breakthrough

Remarks

0

DFEs and matrix elements are strongly disconnected. The breakthrough curves are very distinct. The signal in the matrix is around 1% of the signal observed in the DFEs. Matrix invasion is very limited.

1

DFEs and matrix elements 10 times more connected than in case 0. The breakthrough curves are still very distinct, but matrix invasion is increased. The signal in the matrix is above 10% of the signal observed in the DFEs.

2

DFEs and matrix elements 100 times more connected than in case 0. The breakthrough curves are still distinct but are getting closer to each other. The signal in the matrix is above 50% of the signal observed in the DFEs. Matrix invasion is getting stronger.

3

DFEs and matrix elements 1000 times more connected than in case 0. The breakthrough curves are getting close to each other. The signal in the matrix is around 80% of the signal observed in the DFEs. Matrix invasion is getting stronger.

4

The combination of hydraulic conductivity and coupling length yields an exchange coefficient that is big enough to generate a behaviour similar to the one resulting from the common-node approach. Both breakthrough curves are nearly identical. Matrix is deeply invaded by contaminant mass.

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