Flow in a Porous Medium with an Impervious “Hole” (COMSOL) Here, we solve the problem of two-dimensional flow of water of viscosity μ (at a temperature T = 293.15 K) in the rectangular region ABCD of the porous medium shown in Fig. 14.1(a), in which there is a central circular region E of essential

Question

Flow in a Porous Medium

with an Impervious "Hole" (COMSOL)

Here, we solve the problem of two-dimensional flow of water of viscosity [latex]\mu[/latex] (at a temperature [latex]T=293.15 \mathrm{~K}[/latex] ) in the rectangular region ABCD of the porous medium shown in Fig. 14.1(a), in which there is a central circular region E of essentially zero permeability, which allows no flow through it. The boundaries AB and CD are also impervious to flow.

The pressure along the two ends, AD and BC, are [latex]p=10[/latex] and [latex]p=0 \mathrm{psig}[/latex], respectively. The dimensions of the rectangle are length [latex]L=2 \mathrm{~m}[/latex] and height [latex]H=1 \mathrm{~m}[/latex], and the radius [latex]R[/latex] of the circle is [latex]0.3 \mathrm{~m}[/latex]. Both the rectangle and circle are centered on the origin [latex](0,0)[/latex]. The porosity of the medium is [latex]\varepsilon=0.175[/latex], and its constant permeability is [latex]\kappa=93[/latex] darcies, or [latex]0.917 imes 10^{-10} \mathrm{~m}^{2}[/latex] (see Eqn. (4.32) for the conversion factor).

1 darcy [latex]=1 \frac{\mathrm{cm} / \mathrm{s} \mathrm{cP}}{\mathrm{atm} / \mathrm{cm}} \doteq 0.986 imes 10^{-8} \mathrm{~cm}^{2}=1.06 imes 10^{-11} \mathrm{ft}^{2}.[/latex] (4.32)

Let [latex]p[/latex] denote the pressure, [latex]x[/latex] and [latex]y[/latex] the coordinates, and [latex]\mathbf{v}[/latex] the velocity vector. From Darcy's law (pages 207-208) and continuity (Eqn. (5.51)), we have:

[latex] abla \cdot \mathbf{v}=0.[/latex] (5.51)

[latex]\mathbf{v}=-\frac{\kappa}{\mu} abla p, \quad abla \cdot \mathbf{v}=0,[/latex] (14.1)

Substitution of [latex]\mathbf{v}[/latex] from Darcy's law into the continuity equation gives us (for constant [latex]\kappa[/latex] and [latex]\mu[/latex], which can then be eliminated) Laplace's equation:

[latex] abla^{2} p=\frac{\partial^{2} p}{\partial x^{2}}+\frac{\partial^{2} p}{\partial y^{2}}=0.[/latex] (14.2)

Thus, we shall use COMSOL to solve Laplace's equation for the pressure.

Accepted Answer

Select the Physics and Examine the Windows

Note: In the following, Select and left-click (L-click) mean the same thing. Right- or R-click (the same as Control-click on a Mac) is different and will give you more information about the item you are investigating.

Upon opening COMSOL 5.2a (or another recent version), you will see two icons in the upper left corner of the screen, as shown in Fig. 14.1(b).

Model Wizard. You should select it for any new problem, as in the present example. Note: A previous model can be opened using the File menu.
Blank Model. Use this option to open a model that is completely empty. The Model Wizard is a great method for starting models with a standard layout, but there are times when we need to build models from scratch. Thus, you would manually add only the needed components. One example use would be to construct a frequently accessed geometry.

Here, choose the Model Wizard icon, which will then give us the choice of dimensions, as in Fig. 14.2. Clearly, we select (L-click or simply click) the 2D choice.

There then appears a list of the various categories of problems that can be tackled by COMSOL, as in Fig. 14.3. Initially, there is just a plain list, with no subsets appearing (they will soon be developed). Now click in succession on the glyphs (the little rotating triangles) for Fluid Flow, Porous Media and Subsurface Flow, and Darcy's Law; the display should look exactly as in Fig. 14.3. Note that the line with Darcy's Law is highlighted, so now click successively on the Add and Study icons (Figs. [latex]14.3[/latex] and [latex]14.4(\mathrm{a})[/latex] ) to indicate that this is what we wish to investigate.

At the top left-hand corner of the screen, you will see the Select Study window, as in Fig. 14.4(b), so select the Stationary option. Finally, to complete the selection of the physics, click Done, as in Fig. 14.4(a), and you will see the complete set of the four principal user interfaces, as in Fig. 14.5.

You will see four main windows and a small auxiliary one, from left to right:

(a) Model Builder, shown in Fig. 14.6, which, as you proceed through the various steps of solving a problem, will give you a tree that outlines all of the steps or "nodes."

(b) Settings, shown in Fig. 14.7, where you can assign values to items selected in the Model Builder.

(c) Graphics, shown in Fig. 14.8, which will eventually contain your geometry and the results.

(d) An Add Materials window to the right of the Graphics window.

(e) Message (below the Graphics window), not currently needed.

Take a few minutes to scan useful icons at the top, such as New, Open, Undo, Reset Desktop, Dynamic Help. Note that window widths can be adjusted by "grabbing" their boundaries and moving horizontally.

Note: Everything that follows is initiated in the Model Builder.

The small Add Materials window at the extreme right of the visual interface will be examined later.

Define the Parameters

Here, we establish values for the variables [latex]L, H, R[/latex], pleft, and pright, which will be used later. R-click Global Definitions and select Parameters. You can specify any units inside brackets following the value, and they will be converted to SI units. So, after entering [latex]\mathrm{L}=2[\mathrm{~m}], \mathrm{H}=1[\mathrm{~m}], \mathrm{R}=0.3[\mathrm{~m}][/latex], pleft [latex]=10[\mathrm{psi}][/latex], and pright [latex]=0[/latex] [psi] into the Name and Expression columns, the Settings window will appear as in Fig. 14.9.

Establish the Geometry

Here, we draw a rectangle and cut from it a central circle.

Under Component 1 in the Model Builder, R-click Geometry 1 and select Rectangle. In the Settings window, insert Width [latex]L[/latex], Height [latex]H[/latex], and Center at origin, as in Fig. 14.7 earlier. Click Build Selected in the upper left corner of the Settings window to display the rectangle in the Graphics window.

Then repeat the process for the circle---R-click Geometry 1 in the Model Builder, select Circle, and then go to the Settings window to enter Radius [latex]R[/latex] and Center at the origin. Build Selected then shows the circle as well as the rectangle, as in Fig. 14.10(a).

We now wish to remove the circle from the rectangle. In the Model Builder, R-click Geometry 1, Boolean and Partitions, Difference. Observe in the Settings window that Objects to Add is active (the box shows blue). Select the rectangle with a L-click, noting that it turns from red to blue and that [latex]\mathrm{r} 1[/latex] appears in the Add box. Activate the Objects to Subtract icon. Hover the mouse over the circle, if necessary scrolling with a mouse button, until the circle is red, then left-click so that it turns blue and c1 appears in the Subtract box. Note that "Keep interior boundaries" is checked --- unimportant here but would be needed if, for example, the circular area were filled with another porous medium of different permeability. Build Selected confirms the required geometry, as in Fig. 14.10(b).

The five icons shown in Fig. [latex]14.11[/latex] appear on the left-hand side just above the Graphics window and are useful when constructing the geometry and in viewing the graphical results. The first, second, and fourth---Zoom In, Zoom Out, and Zoom to Selection are self-explanatory. By clicking on the third icon, Zoom Box, and then outlining an area by dragging the mouse, you can magnify a selected area. A click on the fifth and very useful icon, Zoom Extents, will immediately fill the screen with as large an object as possible.

Define the Materials

To specify the properties of the fluid (water in our case), we have two options:

Manually enter the properties.
Select from a list of materials stored within COMSOL.

We shall choose the second option. Under Component 1 in the Model Builder, R-click Materials, Add Material, and note the Materials window to the right of the Graphics window. Click progressively on Liquids and Gases, Liquids, and Water, as in Fig. 4.12(a), then Add to Selection [latex](+)[/latex].

Note the new Settings window in Fig. 14.12(b), and follow the sequence Geometric Entity, Geometric Entry Level, Manual. Note the Selection icon is active. L-click the rectangle with the hole; note that it turns blue and that "1" appears in the Active window. The density and dynamic viscosity will be needed---both at a temperature of [latex]T=293.15 \mathrm{~K}[/latex] unless we specify otherwise. The porosity and permeability (note warnings in red) will soon be specified. Generally, in the Settings window, errors will be flagged in red, and inconsistent units will appear in brown.

Define the Fluid Properties

We now specify the porosity and permeability of the porous medium, as in Fig. 14.13(a). In the Model Builder, R-click Darcy’s Law and select Fluid and Matrix Properties 1. Under Settings, with Manual Selections, choose Domain 1 by a left-click, noting that it turns from red to blue and that "1" appears in the Active window. (Alternatively, choose All Domains in the selection option). Note that Density and Dynamic viscosity are from water, as already noted. Then, under User Defined, insert Porosity [latex]=\varepsilon_{p}=0.175[/latex] and Permeability [latex]=\kappa=93[/latex] darcies [latex]=[/latex] [latex]100 imes 0.917 imes 10^{-8} \mathrm{~cm}^{2}=0.917 imes 10^{-10} \mathrm{~m}^{2}[/latex], which we enter as [latex]0.917^{*} 10 ^{\wedge}-10[/latex].

Note that the default temperature [latex]T=293.15 \mathrm{~K}[/latex] can be changed at this stage if desired. You can also look at the equations being solved (continuity and Darcy's law), as in Fig. 14.13(b), by clicking on the glyph just before Equations.

Define the Boundary Conditions

L-click No-Flow 1 and note that all eight boundaries (four segments are needed for the circle) are by default set to No-Flow. To override for the left and right boundaries, R-click Darcy's Law in the Model Builder, select Pressure, and then under Settings select the left-hand boundary [latex](\# 1)[/latex], and insert pleft, as in Fig. 14.14. Then R-click Darcy's Law and Pressure again, and under Settings select the right-hand boundary (#4), insert pright, Add. L-click No-Flow 1 and observe the overriding pressure conditions for the left- and right-hand boundaries.

Create the Mesh

As discussed in Chapter 13, the finite-element method employed by COMSOL subdivides the solution region into many elements and then solves a large set of simultaneous equations for the solution values at the nodes (where the elements join one another). Generally, a finer mesh will generate a more accurate solution, but at the expense of increased computing time. COMSOL automatically generates the mesh and chooses a shape for the elements depending on the problem at hand. In our fairly simple problem, all the elements are triangles --- see Fig. 14.15.

To generate the mesh, Select Mesh 1 in the Model Builder, and under Element Size in the Settings window, select Element Size Finer, followed by Build All. Right-click Mesh 1, select Statistics to see the number of elements---or click the Messages tab under the graphics window (which will eventually also show the computing time taken, as in Fig. 14.16.

Solve the Problem

To solve the simultaneous equations, return to the Model Builder and R-click Study. Select Compute, and watch the solution development by selecting the Progress tab under the Graphics screen. If all goes well, the solution will first be displayed as a Surface Plot of the pressure, as in Fig. 14.17.

Display the Results

As we progress from left to right, the color changes from deep red (high pressure, approaching nearly [latex]69,000 \mathrm{~Pa}[/latex] ) to deep blue (low pressure, approaching zero [latex]\mathrm{Pa})[/latex], and values for the pressure can be obtained by consulting the color bar at the right on which you can navigate to see coordinates and values. Alternatively, a click on any point will give the exact coordinates and interpolated pressure, appearing in the Message area.

We next obtain a plot of the isobars, as in Fig. 14.18. In the Model Builder, R-click Results and select 2D Plot Group. R-click 2D Plot Group 2, and select Contour. In Settings, note 20 levels. Also note Expression shows the pressure [latex]p[/latex]. Select Plot. Now change to 40 levels and click Plot again. To facilitate reproduction here, we changed the isobar coloring to uniformly black. The Zoom Box icon above the window enables more detail in a prescribed area.

Finally, velocity vectors can be added to the contours, as in Fig. 14.19. In the Model Builder (we always initiate an action there), R-click 2D Plot Group 2 and select Arrow Surface (which gives isobars and velocities). In the Settings window, we see Darcy's velocity field. Plot. Note from Fig. 14.20 that the qualities of the arrows may be modified --- for example, we adjusted the length of the arrows by using the slider. In any event, the arrows are always normal to the isobars, because the velocity vector is proportional to the gradient of the pressure [latex]\mathbf{v}=-(\kappa / \mu) abla p[/latex].

In this example, we have given much detail, but we have by no means covered all the capabilities of COMSOL. There are many other details to be learned, and several of these are covered in the other 10 COMSOL examples in Chapters 6,7 , [latex]8,9,11[/latex], and 12 .

Question 14.1: Flow in a Porous Medium with an Impervious “Hole” (COMSOL) H...

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