This tutorial describes the 2D Equilibrium Sorption with 1st Order Decay applet, which computes 2D transient concentration plumes in a semi-infinite aquifer using the Cleary-Ungs (exact) and Domenico (approximate) analytical solutions. The result is displayed as a color-coded concentration field with optional cross-section plots.
For the mathematical background see the Solution Tutorial.
The applet is divided into three panels around a central concentration field canvas:
(click for cross section)
Click anywhere on the concentration field to draw a cross-section line (horizontal or vertical, depending on the selected Cross Sections mode). The bottom canvas shows the resulting concentration profile. Hover over the cross-section canvas to read values in the Slider Data table.
Source Parameters
| Parameter | Symbol | Units | Default | Description |
|---|---|---|---|---|
| Input Concentration | C₀ | M/L³ | 1.0 | Fixed source concentration at x = 0, along the source zone width Y |
| Source Width | Y | L | 1.0 | Width of the continuous line source in the transverse y-direction. The source extends from −Y/2 to +Y/2 centered at the origin. |
Model Parameters
| Parameter | Symbol | Units | Default | Description |
|---|---|---|---|---|
| Flow Velocity | v | L/T | 1.0 | Average pore-water velocity in the positive x-direction. Assumed spatially uniform. |
| Decay Rate | λ | 1/T | 0.0 | First-order aqueous-phase decay constant. Set to 0 for a conservative solute. |
| Retardation Factor | R | — | 1.0 | Linear equilibrium retardation factor: R = 1 + ρKd/θ. Must be ≥ 1; R = 1 means no sorption. |
| Long. Dispersivity | αx | L | 0.1 | Longitudinal dispersivity. The longitudinal dispersion coefficient is Dx = αx v. |
| Trans. Dispersivity | αy | L | 0.01 | Transverse dispersivity. The transverse dispersion coefficient is Dy = αy v. |
Output Time
| Parameter | Symbol | Units | Default | Description |
|---|---|---|---|---|
| Time | t | T | 1.0 | Time at which to evaluate the concentration field. The Domenico solution approaches steady state for large t; the Cleary-Ungs solution gives the full transient response. |
| Option | Description |
|---|---|
| Domenico | The Domenico (1987) closed-form approximate solution. Fast to compute; widely used in practice. Not valid at short times or near the source. |
| Cleary-Ungs | The exact Cleary-Ungs (1978) solution evaluated via Gauss-Legendre numerical quadrature. More accurate than Domenico at all times and distances. May exhibit oscillations at large Péclet numbers (x/αx); increase integration points if this occurs. |
| |Dom − Cleary-Ungs| | Displays the absolute pointwise difference between the two solutions. Use this to assess the accuracy of the Domenico approximation for a given parameter set. |
Integration Points (Cleary-Ungs)
The Integration Points dropdown sets the number of Gauss-Legendre quadrature points used to evaluate the Cleary-Ungs integral:
| Points | Description |
|---|---|
| 20 | Fastest; adequate for small Péclet numbers and smooth plumes |
| 60 | Good balance of speed and accuracy for most problems |
| 96 | Most accurate; recommended for large Pe or when oscillations appear with fewer points |
Coordinate Range
Set Min X, Max X, Min Y, Max Y in the right panel to define the spatial extent of the plotted domain. The concentration field is evaluated on a grid covering this region. Click Compute Conc. Field to apply a changed range.
Resolution
Controls the pixel block size (1×1 to 10×10 pixels per computed point). Smaller block size gives finer spatial detail but takes longer to compute. Start with a coarser resolution to check the plume shape, then refine.
Color Scheme
| Option | Description |
|---|---|
| Smooth Shading | Continuous color gradient from blue (low C/C₀) through cyan, green, yellow, to red (high C/C₀) |
| Contour by 5% Steps | Discrete color bands at every 5% increment of C/C₀ (20 bands) |
| Contour by 10% Steps | Discrete color bands at every 10% increment of C/C₀ (10 bands) |
| Custom No. of Contours | Enter any integer to set the number of equally spaced contour bands |
After computing the concentration field, click on the field canvas to draw cross-section lines. The Cross Sections radio buttons in the right panel control the type:
| Mode | Description |
|---|---|
| None | Clicking on the concentration field has no effect; no cross-section lines are drawn |
| Horizontal | Click to draw a horizontal line at a fixed y value; the bottom canvas shows C vs x along that transect |
| Vertical | Click to draw a vertical line at a fixed x value; the bottom canvas shows C vs y along that transect |
Multiple cross-section lines can be overlaid — each is drawn in a different color. The Active CS Lines list in the left panel lets you remove individual lines. Clear Cross Section Plots removes all lines at once.
Hover the mouse over the cross-section plot (bottom canvas) to read values along the profile. A vertical dashed line appears at the cursor position. The Slider Data Display table in the right panel updates live, showing:
- The position coordinate (x or y, depending on cross-section mode)
- The interpolated C/C₀ value for each active cross-section line at that position
Hovering over the concentration field (top canvas) shows the (x, y) coordinates and C/C₀ value at the cursor in the pointer info bar below the field.
Two export buttons are available in the right panel:
| Button | Description |
|---|---|
| Display Conc. Data | Opens a modal with the full x–y–C/C₀ grid for the computed concentration field. Downloadable as TXT or CSV. |
| Display CS Data | Opens a modal with the concentration profiles for all active cross-section lines. Downloadable as TXT or CSV. |