# Velocity field calculation for non-orthogonal numerical grids

## Abstract

Computational grids containing cell faces that do not align with an orthogonal (e.g. Cartesian, cylindrical) coordinate system are routinely encountered in porous-medium numerical simulations. Such grids are referred to in this study as non-orthogonal grids because some cell faces are not orthogonal to a coordinate system plane (e.g. xy, yz or xz plane in Cartesian coordinates). Non-orthogonal grids are routinely encountered at the Savannah River Site in porous-medium flow simulations for Performance Assessments and groundwater flow modeling. Examples include grid lines that conform to the sloping roof of a waste tank or disposal unit in a 2D Performance Assessment simulation, and grid surfaces that conform to undulating stratigraphic surfaces in a 3D groundwater flow model. Particle tracking is routinely performed after a porous-medium numerical flow simulation to better understand the dynamics of the flow field and/or as an approximate indication of the trajectory and timing of advective solute transport. Particle tracks are computed by integrating the velocity field from cell to cell starting from designated seed (starting) positions. An accurate velocity field is required to attain accurate particle tracks. However, many numerical simulation codes report only the volumetric flowrate (e.g. PORFLOW) and/or flux (flowrate divided by area) crossing cell faces.more »

- Authors:

- Savannah River Site (SRS), Aiken, SC (United States). Savannah River National Lab. (SRNL)

- Publication Date:

- Research Org.:
- Savannah River Site (SRS), Aiken, SC (United States)

- Sponsoring Org.:
- USDOE

- OSTI Identifier:
- 1172684

- Report Number(s):
- SRNL-STI-2015-00115

- DOE Contract Number:
- AC09-08SR22470

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING

### Citation Formats

```
Flach, G. P.
```*Velocity field calculation for non-orthogonal numerical grids*. United States: N. p., 2015.
Web. doi:10.2172/1172684.

```
Flach, G. P.
```*Velocity field calculation for non-orthogonal numerical grids*. United States. doi:10.2172/1172684.

```
Flach, G. P. Sun .
"Velocity field calculation for non-orthogonal numerical grids". United States.
doi:10.2172/1172684. https://www.osti.gov/servlets/purl/1172684.
```

```
@article{osti_1172684,
```

title = {Velocity field calculation for non-orthogonal numerical grids},

author = {Flach, G. P.},

abstractNote = {Computational grids containing cell faces that do not align with an orthogonal (e.g. Cartesian, cylindrical) coordinate system are routinely encountered in porous-medium numerical simulations. Such grids are referred to in this study as non-orthogonal grids because some cell faces are not orthogonal to a coordinate system plane (e.g. xy, yz or xz plane in Cartesian coordinates). Non-orthogonal grids are routinely encountered at the Savannah River Site in porous-medium flow simulations for Performance Assessments and groundwater flow modeling. Examples include grid lines that conform to the sloping roof of a waste tank or disposal unit in a 2D Performance Assessment simulation, and grid surfaces that conform to undulating stratigraphic surfaces in a 3D groundwater flow model. Particle tracking is routinely performed after a porous-medium numerical flow simulation to better understand the dynamics of the flow field and/or as an approximate indication of the trajectory and timing of advective solute transport. Particle tracks are computed by integrating the velocity field from cell to cell starting from designated seed (starting) positions. An accurate velocity field is required to attain accurate particle tracks. However, many numerical simulation codes report only the volumetric flowrate (e.g. PORFLOW) and/or flux (flowrate divided by area) crossing cell faces. For an orthogonal grid, the normal flux at a cell face is a component of the Darcy velocity vector in the coordinate system, and the pore velocity for particle tracking is attained by dividing by water content. For a non-orthogonal grid, the flux normal to a cell face that lies outside a coordinate plane is not a true component of velocity with respect to the coordinate system. Nonetheless, normal fluxes are often taken as Darcy velocity components, either naively or with accepted approximation. To enable accurate particle tracking or otherwise present an accurate depiction of the velocity field for a non-orthogonal grid, Darcy velocity components are rigorously derived in this study from normal fluxes to cell faces, which are assumed to be provided by or readily computed from porous-medium simulation code output. The normal fluxes are presumed to satisfy mass balances for every computational cell, and if so, the derived velocity fields are consistent with these mass balances. Derivations are provided for general two-dimensional quadrilateral and three-dimensional hexagonal systems, and for the commonly encountered special cases of perfectly vertical side faces in 2D and 3D and a rectangular footprint in 3D.},

doi = {10.2172/1172684},

journal = {},

number = ,

volume = ,

place = {United States},

year = {Sun Mar 01 00:00:00 EST 2015},

month = {Sun Mar 01 00:00:00 EST 2015}

}