Water Resources of the United States

# Visualizing the SUTRA dispersion model in 3D

## Background

The SUTRA dispersion model in three dimensions (3D) is a generalization of the original two-dimensional (2D) model. The 2D and 3D SUTRA dispersion models are described in detail in Section 2.5 of the SUTRA documentation. This page supplements the formal documentation by providing interactive visualizations of the 3D dispersion model. A VRML browser is required to view the visualizations interactively.

In the 3D SUTRA dispersion model, the longitudinal and transverse dispersivities, aL, aT1, and aT2, can depend on the direction of groundwater flow, i.e., the dispersion model can be anisotropic. SUTRA computes the value of the longitudinal dispersivity, aL, from the radius of an ellipsoid measured along the flow direction. Thus, aL is greatest when flow is along the longest axis of the ellipsoid. The transverse dispersivities, aT1 and aT2, are computed from radii of a second ellipsoid. These transverse radii are measured along two directions that are perpendicular to the flow direction and to each other.

The user controls the behavior of the 3D dispersion model by setting SUTRA input parameters that determine the dimensions and orientation of the two ellipsoids in space:

• For simplicity, both dispersivity ellipsoids are oriented with their principal axes aligned with the principal permeability directions, i.e., with the maximum (max), middle (mid), and minimum (min) permeability directions defined by the angles ANGLE1, ANGLE2, and ANGLE3 in dataset 15.
• The dimensions of the longitudinal dispersivity ellipsoid are determined by the dispersivities ALMAX, ALMID, and ALMIN in dataset 15. The principal radii in the max, mid, and min directions are (aLmax)1/2, (aLmid)1/2, and (aLmin)1/2, respectively, where aLmax, aLmid, and aLmin represent ALMAX, ALMID, and ALMIN, respectively.
• The dimensions of the transverse dispersivity ellipsoid are determined by the dispersivities ATMAX, ATMID, and ATMIN in dataset 15. The principal radii in the max, mid, and min directions are (aTmax)1/2, (aTmid)1/2, and (aTmin)1/2, respectively, where aTmax, aTmid, and aTmin represent ATMAX, ATMID, and ATMIN, respectively.

## 3D visualizations

The visualizations below are best viewed interactively, which will allow you to rotate, zoom, and pan the image. This requires a VRML browser.

The way in which SUTRA computes the longitudinal dispersivity, aL, as a function of flow direction is illustrated in the figure below, which corresponds to Figure 2.4b in the SUTRA documentation:

 To view a larger image, click on the figure. (37 KB GIF file)   To view an interactive "3D" visualization, CLICK HERE, then click and drag to rotate the image. (809 KB VRML file plus 9 KB of PNG files; requires a VRML browser)

Figure. How SUTRA calculates the longitudinal dispersivity, aL, as a function of flow direction. Here, v is the flow (velocity) vector; xmax, xmid, and xmin are coordinates aligned with the max, mid, and min directions, respectively; and aL is the squared radius measured along the flow direction. The principal radii of the ellipsoid (not labeled) have squared lengths of aLmax, aLmid, and aLmin, and are aligned with the max, mid, and min directions, respectively. Note that aL=aLmax for flow in the max direction, aL=aLmid for flow in the mid direction, and aL=aLmin for flow in the min direction.

The way in which SUTRA computes the transverse dispersivities, aT1 and aT2, as a functions of flow direction is illustrated in the figure below, which corresponds to Figure 2.4c in the SUTRA documentation:

 To view a larger image, click on the figure. (41 KB GIF file)   To view an interactive "3D" visualization, CLICK HERE, then click and drag to rotate the image. (905 KB VRML file plus 12 KB of PNG files; requires a VRML browser)

Figure. How SUTRA computes the transverse dispersivities, aT1 and aT2, as a functions of flow direction. Here, v is the flow (velocity) vector and xmax, xmid, and xmin are coordinates aligned with the max, mid, and min directions, respectively. The principal radii of the ellipsoid (not labeled) have squared lengths of aLmax, aLmid, and aLmin, and are aligned with the max, mid, and min directions, respectively; and aT1 and aT2 are squared radii measured in two directions perpendicular to the flow direction. These two directions, which are the transverse dispersion directions, correspond to the principal axes of the ellipse (called the slicing ellipse) formed by the intersection of the ellipsoid with the plane that passes through the origin and is perpendicular to the flow direction. Note that when groundwater flow is in one of the three principal permeability directions (max, mid, or min), aT1 and aT2 take on the values associated with the other two directions. Thus, the transverse dispersivities are aTmid and aTmin for flow in the max direction; aTmax and aTmin for flow in the mid direction; aTmax and aTmid for flow in the min direction.

U.S. Department of the Interior | U.S. Geological Survey
URL: https://water.usgs.gov/nrp/gwsoftware/sour/special/dispersion3D/dispersion3D.htm
Page Contact Information: Alden Provost <aprovost@usgs.gov>