Solved with COMSOL Multiphysics 4.0a. COPYRIGHT 2010 COMSOL AB.

Transcription

1 Rock Fracture Flow Introduction A potential flow model describing fluid movement in a rock fracture uses the Reynolds equation, also known as the cubic law equation which involves the following variables: ρg a 3 H 12μ = 0 Fluid density, ρ The acceleration of gravity, g The fluid s dynamic viscosity, μ The fracture s aperture or width, a(x, y) The scaled pressure, H = H(x, y), also called hydraulic head This model uses interpolation of aperture data defined in a text file. Model Definition The definition of the last variable, hydraulic head is H = p z ρg where z equals the height and p represents fluid pressure. After eliminating the constant factor in the diffusion coefficient, you end up with the equation ( a 3 H) = 0 Notice that you cannot eliminate the aperture expression a 3 because this example assumes that a is a nonconstant function of x and y, and thus it falls under the influence of the divergence operator. ROCK FRACTURE FLOW 1

2 Notes About the COMSOL Implementation COMSOL Multiphysics does not include a physics interface for potential flow, but you can use the Transport of Diluted Species interface available under Chemical Species Transport for this model; the PDE solved here is identical to the Reynolds equation. You must make one minor mental adjustment here: In the user interface the hydraulic head is referred to as the concentration. You can, however, rename the dependent variable as H while adding the Physics Interface. The computational domain is rectangular and well inside the sampled aperture data matrix. Set a hydraulic head of 20 mm at the upper boundary and 0 mm at the lower boundary. This creates a pressure difference of 20 mm that drives the fluid flow. Both the left and right boundaries have symmetry boundary conditions. The COMSOL installation includes a text file, aperture_data.txt, containing the sample aperture data for this model in the form of a 100-by-100 matrix. This synthetically generated data set corresponds to an aperture with a fractal dimension of 2.6. You import the aperture data into the COMSOL Multiphysics user interface by defining an interpolation function, which you then use as the aperture a in the cubic law equation. 2 ROCK FRACTURE FLOW

3 Results The plot in Figure 1 shows the flux using colored surface data and the hydraulic head as the z-coordinate (height). Figure 1: The flux and the hydraulic head. The plot in Figure 2 provides a visualization of the aperture data. ROCK FRACTURE FLOW 3

4 Figure 2: The interpolated aperture data shown as a combined surface and height plot. Model Library path: COMSOL_Multiphysics/Geophysics/rock_fracture_flow Modeling Instructions MODEL WIZARD 1 Go to the Model Wizard window. 2 Click the 2D button. 3 Click Next. 4 In the Add Physics tree, select Chemical Species Transport>Transport of Diluted Species (chds). 5 Click Add Selected. 6 In the Concentration edit field, type H. 7 Click Next. 4 ROCK FRACTURE FLOW

5 8 In the Studies tree, select Preset Studies>Stationary. 9 Click Finish. GLOBAL DEFINITIONS Interpolation 1 1 In the Model Builder window, right-click Global Definitions and select Functions>Interpolation. Define an interpolation function using the aperture data available in a file. 2 Go to the Settings window for Interpolation. 3 Locate the Parameters section. From the Data source list, select File. 4 Click the Browse button. 5 Browse the course folder \Complete Exercises\Hands on #2 - Rock Fracture Flow and select the file rock_fracture_flow_aperture_data.txt. 6 Click the Import button. 7 Find the Functions subsection. In the Functions table, enter the following settings: FUNCTION POSITION IN FILE aperture 1 GEOMETRY 1 The model geometry is simply a rectangle. Rectangle 1 1 In the Model Builder window, right-click Model 1>Geometry 1 and select Rectangle. 2 Go to the Settings window for Rectangle. 3 Locate the Size and Shape section. In the Width edit field, type In the Height edit field, type Locate the Position section. In the x edit field, type In the y edit field, type In the Model Builder window, right-click Rectangle 1 and select Build Selected. TRANSPORT OF DILUTED SPECIES 1 In the Model Builder window, click Model 1>Transport of Diluted Species. 2 Go to the Settings window for Transport of Diluted Species. ROCK FRACTURE FLOW 5

6 3 Locate the Transport Mechanisms section. Clear the Convection check box. Now define the domain settings including reaction rate expressions. Diffusion 1 In the Model Builder window, click Diffusion. 2 Go to the Settings window for Diffusion. 3 Locate the Diffusion section. In the D H edit field, type aperture(x,y)^3. Reactions 1 1 In the Model Builder window, right-click Transport of Diluted Species and select Reactions. 2 Select Domain 1 only. Next, define the boundary conditions. Symmetry 1 1 In the Model Builder window, right-click Transport of Diluted Species and select Symmetry. 2 Select Boundaries 1 and 4 only. Concentration 1 1 In the Model Builder window, right-click Transport of Diluted Species and select Concentration. 2 Select Boundary 2 only. 3 Go to the Settings window for Concentration. 4 Locate the Concentration section. Select the Species H check box. Concentration 2 1 In the Model Builder window, right-click Transport of Diluted Species and select Concentration. 2 Select Boundary 3 only. 3 Go to the Settings window for Concentration. 4 Locate the Concentration section. Select the Species H check box. 5 In the c 0,H edit field, type 20. STUDY 1 1 From the Model Builder window s View Menu, choose Show More Options. 6 ROCK FRACTURE FLOW

7 Solver 1 1 In the Model Builder window, right-click Study 1 and select Show Default Solver. 2 In the Model Builder window, right-click Solver Configurations>Solver 1>Stationary Solver 1 and select Adaptive Mesh Refinement. 3 Go to the Settings window for Adaptive Mesh Refinement. 4 Locate the General section. In the Maximum number of refinements edit field, type In the Maximum number of elements edit field, type In the Model Builder window, right-click Study 1 and select Compute. RESULTS 2D Plot Group 2 1 In the Model Builder window, right-click Results and select 2D Plot Group. 2 In the Model Builder window, right-click 2D Plot Group 2 and select Surface. 3 Go to the Settings window for Surface. 4 In the upper-right corner of the Expression section, click Replace Expression. 5 From the menu, choose Diffusive flux magnitude (chds.dfluxmag_h). 6 Locate the Coloring and Style section. Clear the Color legend check box. 7 In the Model Builder window, right-click Surface 1 and select Height Expression. 8 Go to the Settings window for Height Expression. 9 In the upper-right corner of the Expression section, click Replace Expression. 10 From the menu, choose Concentration (H). 11 In the Model Builder window, click 2D Plot Group Go to the Settings window for 2D Plot Group. 13 Locate the Plot Settings section. Select the Title check box. 14 In the associated edit field, type Surface: Diffusive flux, Height: Hydraulic head. 15 Click the Plot button. 16 Click the Zoom Extents button on the Graphics toolbar. 2D Plot Group 3 1 In the Model Builder window, right-click Results and select 2D Plot Group. 2 In the Model Builder window, right-click 2D Plot Group 3 and select Surface. 3 Go to the Settings window for Surface. ROCK FRACTURE FLOW 7

8 4 Locate the Expression section. In the Expression edit field, type aperture(x,y). 5 In the Model Builder window, right-click Surface 1 and select Height Expression. 6 Go to the Settings window for Height Expression. 7 Locate the Expression section. From the Height data list, select Expression. 8 In the Expression edit field, type aperture(x,y). 9 In the Model Builder window, click 2D Plot Group Go to the Settings window for 2D Plot Group. 11 Locate the Plot Settings section. Select the Title check box. 12 In the associated edit field, type Surface and height: Aperture (mm). 13 Click the Plot button. 14 Click the Zoom Extents button on the Graphics toolbar. 8 ROCK FRACTURE FLOW