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Single-Phase Fracture Flow

The modelling of realistic jointed rock can be computationally expensive due to the presence of fractures that normally require highly refined mesh. The mesh size can be prohibitively fine when compared with the scale of the model domain. In cases where macroscopic behaviour of the jointed rock is of primary interest, the adoption of embedded fracture approach can be beneficial. 

This example uses ParaGeoInv to recover the effective permeability based on a target solution of a discrete model . Two homogenisation methods are used, namely the ubiquitous embedded fracture (UEF) and spatially embedded fracture (SEF).

Mesh model of target solution showing distribution of discrete fracture imported from a FracMan (*.fab) data file.

Evolution of pore pressure of target solution in matrix and fracture

Pore pressure distribution in homogenised models

Homogenised permeability of different homogenised models

Ubiquitous Embedded Fracture

In the UEF approach, the effect of fracture properties is accounted for in the compliance matrix of every element in the fracture media. Here we have five homogenised models with different number of divisions on all sides. The effect of mesh size is reflected in the change of effective permeability in both directions. Smaller mesh size accounts for higher pressure gradient, and thus lower permeability in both directions to maintain mass flux. 

Ubiquitous Embedded Fracture

In the UEF approach, the effect of fracture properties is accounted for in the compliance matrix of every element in the fracture media. Here we have five homogenised models with different number of divisions on all sides. The effect of mesh size is reflected in the change of effective permeability in both directions. Smaller mesh size accounts for higher pressure gradient, and thus lower permeability in both directions to maintain mass flux. 

Pore pressure distribution in homogenised models

Homogenised permeability of different homogenised models

Reduction of misfit as the algorithm approaches the optimal solution

Convergence of fracture permeability multiplier

Spatial Embedded Fracture

In the SEF approach, the continuum representation of fracture is positioned according to the distribution of discrete fractures. In this example, the optimisation variable is the fracture permeability multiplier, which is multiplied by the algorithm with the matrix permeability to obtain fracture permeability. The optimal fracture permeability multiplier converges to 68.01, indicating the fracture permeability is around 68 times higher than the matrix value. 

Comparison of pore pressure contours between discrete and homogenised models

Comparison of pore pressure contours between discrete and homogenised models

Spatial Embedded Fracture

In the SEF approach, the continuum representation of fracture is positioned according to the distribution of discrete fractures. In this example, the optimisation variable is the fracture permeability multiplier, which is multiplied by the algorithm with the matrix permeability to obtain fracture permeability. In this example, the optimal fracture permeability multiplier converges to 68.01, indicating the fracture permeability is around 68 times higher than the matrix value. 

Pore pressure distribution in homogenised model

Convergence of fracture permeability multiplier

Pore pressure distribution in discrete model

Reduction of misfit as the algorithm approaches the optimal solution

Mesh model showing continuum fracture representation

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