5. Flow Assessment - BjornNyberg/NetworkGT GitHub Wiki

Suggested workflow within NetworkGT for assessing flow potential within a network and producing flow simulations.

Percolation and Permeability

Aperture

The user can also vary the coefficient C, which is related to the material properties of the host rock and generally ranges between 10-2 and 10-5. Alternatively, the user can define a constant aperture for all fractures within the network. In addition, the aperture is combined with a user specified fracture porosity (ranging from 0-1) to calculate the intrinsic permeability (millidarcy's; mD) and transmisivity (mD.m) of the fractures.

  1. Aperture Field for Transmisivity - Defines the maximum and average aperture of a fracture trace, based on a relationship between maximum aperture (A) and fracture length (L) whereby A = C.L^0.5.
  2. Group Fracture Length By - The user can supply a field to sum the fracture length based on specified fracture ID. This is recommended if the fracture network input are branches and it is desired to calculate the aperture based on the entire fracture line length.
  3. C Value - Coefficient C is related to the material properties of the host rock and generally ranges between 10-2 and 10-5.
  4. Fracture Porosity - User specified fracture porosity (ranging from 0-1).
  5. Apply a constant aperture (optional) - Apply a constant aperture if value is greater than 0.

Percolation

Defines the percolation threshold for a given sample area as a critical dimensionless intensity (B22C). Expected values for B22C can be derived for any network topology and are defined for four fracture orientation cases: 1) randomly orientated fractures; 2) orthognal fractures; 3) conjugate (60 degree) fractures; 4) oblique (30 degree) fractures. The user has the option to adjust the values of B22C and B22 based on the branch length variation, defined as the coefficient of variance (CV).

  1. Topology Parameters - Topology parameters polygon/contour grid dataset.
  2. Branch Length Variance (CV) - The user has the option to adjust the values of B22C and B22 based on the branch length variation, defined as the coefficient of variance (CV).
  3. Plot - Plot the calculated percolation measurements.

Permeability Tensor

Calculates an analytical estimate of effective permeability (fracture network plus matrix) for a specified sample area/contour grid. The user has the option to correct the effective permeability using a topology based hydraulic connectivity measure for the fracture network. The output adds fields to the attribute table of the topology parameters polygon/contour grid defining an effective permeability tensor (Kxx, Kxy, Kyy) and values for the principle permeability axis (K1, K2, K1 azimuth). Permeablity is given in millidarcy's (mD).

  1. Network - Linestring geometry of branch network with asscoiated I-I, I-C and C-C data.
  2. Topology Parameters - Topology parameters polygon/contour grid dataset.
  3. Rotation - Rotate the permeability tensor in degrees.
  4. Matrix Permeability - Permeability of the matrix given in millidarcy's (mD).
  5. Matrix Permeability Field - Specify the matrix permeability by a field avilable in the topology parameters dataset.
  6. Transmisivity Field - Specify the transmisivity (mD.m) by a field avilable in the topology parameters dataset. If not specified, the value is autoamtically taken from the value calculated in the Aperture tool.
  7. Hydraulic Conductivity - Calculate effective permeability using a topology based hydraulic conductivity measure of a fracture network.

Permeability Ellipse

Plots the effective permeability tensor for a given sample area as an ellipse. The principle axes of the ellipse are defined as the square root of the maximum permeability (K1) and minimum peremability (K2), which are given in millidarcy's (mD).

  1. Permeability Tensors - Grid defining an effective permeability tensor (Kxx, Kxy, Kyy) and values for the principle permeability axis (K1, K2, K1 azimuth).
  2. Normalise Equal Area - Normalise the permeability ellipse to K1 to create an equal area plot.
  3. Combine - Combine all graphs on one plot.

Solving and Simulating Flow

1D Flow

Solves for 1D flow across a fracture network with user defined boundary pressure conditions. The output creates a 1D flow linestring of the fracture network with additional attribute fields including Pressure (Pa), Flow Velocity (m/s) and the Azimuth of the flow direction. N.B. The larger the fracture network the more computational time the simulation takes. It is, therefore, recommended that the user applies the Simplify Network tool to help reduce computational time.

  1. Fracture Network - Linestring geometry of the fracture network with calculated fracture transmissivities (mD.m).
  2. Select Flow Direction - Select the flow direction as left to right or bottom to top and vice versa.
  3. Number of step - Number of steps to calculate for the flow tracer.
  4. End Time - End time in the flow tracer in seconds. Time between each step is taken End Time / Number of steps.
  5. Viscosity (Pa.s) - Viscosity of the fluid in Pa.s.
  6. Low Pressure (Pa) - Low pressure boundary value in Pa.
  7. High Pressure (Pa) - High pressure boundary value in Pa.
  8. Mesh Size (m) - Size of the mesh created for the fracture network.

2D Flow

Solves for 2D flow across a rectangular contour grid with user defined boundary pressure conditions. The user has the option to define the flow direction and pressure conditions between two open boundaries. The required input is a topology parameters contour grid with a calculated effective permeability tensor (Kxx, Kxy, Kyy). The output creates a 2D flow contour grid with additional attribute fields including Pressure (Pa), Flow Velocity (m/s) and the Azimuth of the flow direction. Additionally the user can simulate the flow of a tracer through the contour grid by defining an end time (s) and a number of time-steps. This will output a Timeseries contour grid containing tracer concentrations (ranging from 0-1) for each time-step.

  1. Fracture Network - Grid defining an effective permeability tensor (Kxx, Kxy, Kyy) in millidarcy's (mD).
  2. xx permeability - Define the field that conains kxx permeability in millidarcy's (mD).
  3. xy pereambility - Define the field that contains kxy permeability in millidarcy's (mD).
  4. xy pereambility - Define the field that contains kyy permeability in millidarcy's (mD).
  5. Select Flow Direction - Select the flow direction as left to right or bottom to top and vice versa.
  6. Number of step - Number of steps to calculate for the flow tracer.
  7. End Time - End time in the flow tracer in seconds. Time between each step is taken End Time / Number of steps.
  8. Viscosity (Pa.s) - Viscosity of the fluid in Pa.s.
  9. Low Pressure (Pa) - Low pressure boundary value in Pa.
  10. High Pressure (Pa) - High pressure boundary value in Pa.>

Theory

Darcy Law

We consider a saturated porous media with one liquid phase (e.g., water), also we assume that the porous media is inert and at rest. The Darcy law models the velocity in [m/s] and pressure in [Pa], and is given by

In the previous model, we have further assumed that the gravitational effects are not present as well as that the density of the liquid phase and porosity do not change in time and space. The porous media is the domain is represented in Figure 1.

The data in equation 1a are the permeability in [mD] and the dynamic viscosity in [Pa s] of the liquid phase. Coupled to equation 1a and considering the nomenclature shown in Figure 1 we have the following boundary conditions

where is the outward unit normal of the outer boundary . The value of and , both in [Pa], are constant pressure boundary conditions, with the assumption that . The operator tr indicates the evaluation of the argument on the portion of the corresponding boundary.

The set of equations 1 defines the Darcy problem.

Tracer transport

We consider that a passive tracer is injected into the porous media and is transported from the advective field , computed by equation 1. The equation that describes the motion of its concentration is

where we have supposed that the porosity is constant in time and we have considered a null initial condition. Coupled to equation 2a we consider the following boundary conditions on the inflow part of , indicated with , as

where is the constant concentration inflow value.

The set of equations 2 defines the tracer problem.

Fracture Darcy law

We consider now the Darcy law on a network of fractures by discarding the effect of the porous media. We apply the model reduction approach to introduce the equations for the fractures thought as mono-dimensional objects. Following this approach some of the variables considered before will be implicitly scaled by the fracture aperture , thus their unit of measures will be different. To fix the ideas consider the fracture network given in Figure 2.

The fracture Darcy law models the velocity (flux) in [m2/s] and pressure in [Pa] for each fracture branch as

where in [mD m] is the effective permeability or transmissibility, i.e. already multiplied by the fracture aperture, and the dynamic viscosity as introduced in equation 1. The differential operators in 3a are defined tangentially with respect to the fracture branch . At each intersection , with the intersection index, we prescribe conservation of mass for all the fracture branches involved in the intersection. Denoting by the set of fracture branch indexes that are involved in the intersection , we have

where the vector indicate the outward unit vector that points outside from and is tangential to it. Finally, to complete the model we need to impose suitable boundary conditions for all the floating parts of the fracture branches. In the case a fracture branch touches the external bounding box, still denoted with , we impose problem specific boundary conditions. Otherwise, the so-called tip condition is considered being null flux. We have

Where we have considered and prescribed constant pressure boundary data in [Pa], with the assumption that . The last condition in equation 3c is the tip-condition.

The set of equations 3 defines the fracture Darcy problem.

Fracture tracer transport

We consider now that a passive tracer is injected into the fracture network and is transported by the advective field for each fracture branch, computed by 3. The equation that describes the motion of its concentration , on each fracture branch, is

where is the effective porosity in [m], i.e. already multiplied by the fracture aperture. Being a purely advective problem, at each intersection we consider an upstream model for the fracture branch coupling. To correctly write the model, we need to introduce the tracer concentration at the intersection and the conservation now becomes

The value represents the upstream concentration for the fracture branch . To complete the model, we introduce the boundary conditions on the inflow part of the external bounding box , indicated with

Where is the constant concentration inflow value.

The set of equations 4 defines the fracture tracer problem.

Citation

If you use the 1D or 2D Flow tools in your research, we ask you to cite NetworkGT and PorePy. In addition, 1D fracture meshing is provided by Gmsh.

Nyberg, B., Nixon, C.W., Sanderson, D.J., 2018. NetworkGT : A GIS tool for geometric and topological analysis of two-dimensional fracture networks. Geosphere 14(4), pp. 1618-1634.

Keilegavlen, E., Berge, R., Fumagalli, A., Starnoni, M., Stefansson, I., Varela, J., Berre, I. PorePy: An Open-Source Software for Simulation of Multiphysics Processes in Fractured Porous Media. arXiv:1908.09869

Geuzaine, C., and Remacle, J.-F., 2009. Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), pp. 1309-1331.