User Interface Tutorial — 3D Equilibrium Sorption with 1st Order Decay [Domenico]

This tutorial describes the 3D Equilibrium Sorption with 1st Order Decay applet. It extends the 2D Domenico model to three dimensions, adding a vertical (z-direction) source extent and a vertical dispersivity. The result is displayed as a 2D color-mapped concentration field in the horizontal (x–y) plane at a user-specified depth z, with optional cross-section plots.

For the mathematical background see the Solution Tutorial.

1Interface Overview

The layout is identical to the 2D applet, with the addition of vertical source height and dispersivity parameters:

LEFT PANEL — Inputs

Source Parameters (C₀, Y, Z)

Model Parameters (v, λ, R, αₓ, αᵧ, α_z)

Output (z, t)

Compute / Solution Tutorial

Concentration Field Plot (x–y plane at depth z)

Color-mapped C/C₀ field
(click for cross section)

(x, y) = — C/C₀ = —

Cross Section Plot

C vs x or C vs y profile

RIGHT PANEL — Display

Coord. Range (min/max X, Y)

Resolution

Color Scheme

Cross Sections (mode)

Clear CS Plots

Display Conc. Data

Display CS Data

Slider Data (hover CS)

The concentration field always shows the x–y horizontal plane at the observation depth z. To visualize a different depth, change the z value and recompute.

2Source & Model Parameters

Source Parameters

Parameter	Symbol	Units	Default	Description
Input Concentration	C₀	M/L³	1.0	Fixed source concentration at x = 0 over the rectangular patch source
Source Width	Y	L	1.0	Width of the rectangular patch source in the horizontal transverse y-direction (source extends from −Y/2 to +Y/2)
Source Height	Z	L	1.0	Height of the rectangular patch source in the vertical z-direction (source extends from −Z/2 to +Z/2)

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	α_x	L	0.1	Longitudinal dispersivity. D_x = α_x v.
Trans. Dispersivity (y)	α_y	L	0.01	Horizontal transverse dispersivity. D_y = α_y v.
Vert. Dispersivity (z)	α_z	L	0.005	Vertical transverse dispersivity. D_z = α_z v. Controls spreading in the z-direction.

Typical dispersivity ratios: In practice, α_y ≈ 0.1 α_x and α_z ≈ 0.01–0.05 α_x. Using α_z ≈ α_y gives a 2D-like plume with symmetric transverse spreading.

3Output Parameters

Parameter	Symbol	Units	Default	Description
Vertical Coordinate	z	L	0.0	Observation depth at which the x–y concentration slice is evaluated. z = 0 is the horizontal centerline of the source. This is independent of the source height Z.
Time	t	T	1.0	Time at which to evaluate the solution. The Domenico solution approaches steady state for large t.

Note: Source Height Z sets the vertical extent of the source at x = 0, while Vertical Coordinate z is the depth of the observation plane — they are independent inputs. Setting z = 0 gives the centerline slice; setting z = Z/2 probes the top edge of the source.

Validity: Like the 2D case, the Domenico solution requires t > ~5 α_x/v and x/α_x > ~30 for results within 5% of the true value.

4Display Options

Coordinate Range

Set Min X, Max X, Min Y, Max Y in the right panel to define the spatial extent of the x–y plane shown. 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 is slower to compute.

Color Scheme

Option	Description
Smooth Shading	Continuous color gradient from blue (low C/C₀) to red (high C/C₀)
Contour by 5% Steps	Discrete color bands at every 5% of C/C₀
Contour by 10% Steps	Discrete color bands at every 10% of C/C₀
Custom No. of Contours	Enter any integer to set the number of equally spaced bands

5Cross Sections

After computing, click on the concentration field canvas to draw cross-section lines at the observation depth z. The Cross Sections radio buttons in the right panel control the type:

Mode	Description
None	Clicking on the field has no effect; no cross-section lines are drawn
Horizontal	Click at a y-value; the bottom canvas shows C vs x at that y (at depth z)
Vertical	Click at an x-value; the bottom canvas shows C vs y at that x (at depth z)

Multiple lines overlay in different colors. The Active CS Lines list allows removing individual lines. Clear Cross Section Plots removes all.

Tip: Use a horizontal cross section at y = 0 (the plume centerline) to plot the longitudinal concentration profile at depth z. Use vertical cross sections at several x values to trace the transverse width evolution of the 3D plume.

6Slider Data

Hover the mouse over the cross-section plot (bottom canvas) to read values. A vertical dashed line appears at the cursor position. The Slider Data Display table in the right panel shows:

The position coordinate (x or y, depending on cross-section mode)
The observation depth z (fixed, as set in Output Parameters)
The interpolated C/C₀ for each active cross-section line at the cursor 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.

7Export Data

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 horizontal plane at depth z. 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.

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