∂(θC^k)/∂t = (∂/∂x_i)[θD_ij(∂C^k/∂x_j)] – (∂/∂x_i)[θv_iC^k] + q_sC_s^k + ∑R_n

where:
θ = porosity
C^k = concentration of species k
D_ij = dispersion coefficient tensor
v = linear pore water velocity
x = distance
t = time
q_s = flow from sources and sinks
R_n = chemical reactions

(Zheng et al. 2010)

According to the user’s manual, no numerical solution technique is completely satisfactory for this equation, because it contains both a “spatial first derivative term (advection) and the spatial second derivative term (mechanical dispersion and molecular diffusion)” within the same equation (Zheng et al. 2010). However, various techniques are used to solve it, each with strengths and limitations.

The standard finite-difference method is as follows:

Rθ(∂C/∂t) = – (∂/∂x)[θv_xC] – (∂/∂y)[θv_yC] – (∂/∂z)[θv_zC] + L(C)

where:
L(C) = non-advection terms including dispersion/diffusion, fluid sink/source, and chemical reactions
R = retardation factor

(Zheng et al. 2010)

This solution is only appropriate for model problems that are not dominated by advection.  All of the solutions methods rely on some form of a mass balance performed on concentration through the faces of a grid cell, and are thus conceptually equivalent to the groundwater flow equation in MODFLOW.

Zheng, C., Weaver, J., & Tonkin, M. (2010). MT3DMS, a modular three-dimensional multispecies transport model—user guide to the hydrocarbon spill source (HSS) package. Athens, Georgia: Prepared under contract to the US Environmental Protection Agency.

CUAHSI is hosting a Conference on Hydroinformatics and Modeling, this July 17-19 in Logan, Utah.  The conference will cover efficient data storing and sharing, earth systems modeling, cloud computing, and remote sensing.

CUAHSI is currently accepting abstracts for this conference, here.

• Bloomberg recently reported that Saudia Arabia has approved a plan to spend $220 million on water and sanitation projections.  Saudi Arabia is currently the world’s biggest producer of desalinated water.

• Meanwhile in a recent New York Times opinion piece, Thomas Friedman discussed the role of water in Syria’s current conflict.  The civil war is concurrent with the country’s worst drought in modern history.

• A recent piece about traveling in Libya points out that that gas is cheaper than water in the country.  Filling a car’s tank would cost about $5, while a single liter of water costs $0.80 (via CNN).

In the 2010 Proceedings of the Student World Water Forum, he wrote:

“A perched aquifer often develops in the soil zone during periods of heavy rainfall, delivering interflow to channels in a watershed. The soil zone has been described as the upper most region of the vadose zone where plant and soil processes enhance storage and permeability, providing a fast pathway for water and solutes to streams. The soil zone connects the watershed surface to the deeper unsaturated, and saturated zones. Modeling of soil zone processes has been used to gain understanding of watershed hydrologic processes. Currently MODFLOW-2005 does not simulate dynamic near-surface hydrologic processes such as, infiltration, hortonian runoff, dunnian runoff, and return flow. The Perched Soil-Zone package (PSZ) for MODFLOW-2005 is being developed to address these major components of hillslope hydrology for simulating watershed processes in the context of basin-scale groundwater flow modeling.”

In his 2011 dissertation, Henson compared a new version of this package, the MODFLOW Soil-Zone Processes Package, with Richard’s Equation for infiltration.  His research indicated that the MODFLOW-SZP was more efficient that Richard’s Equation in numerical modeling for his test cases.  Henson’s dissertation, A Model for Simulating Soil-Zone Processes at the Regional Scale, is available for purchase from Amazon.com.

Thunderstorm by NOAA Photo Library (via wikimedia commons)

The new Groundwater Vistas 6 supports MODFLOW-USG, a code that frees the MODFLOW numerical process from the need for a rectangular grid.  This allows the user to develop complex shapes within the grid, such as trangular meshes around wells.

• In reporting on the history measurement, the New York Times explained that the carbon dioxide concentration in earth’s atmosphere hasn’t been so high in at least three million years. Based on glacial ice records, scientists estimate that carbon dioxide has fluctuated between 180 and 280 ppm over the last 800,000 years.

• What does climate change mean right now? CNN has published several videos on some of the immediate affects of climate change. One of these changes, higher pollen count, is making me miserable at the moment. The average pollen count is expected to double in the next 30 years.

Commenter Eric_t on bitbucket.org wrote: “In online programming discussions, Fortran seems to have gotten an undeserved bad reputation. Most of the time, comments of the language are based on the Fortran 77 edition of the language, and fail to take into account that the language has seen four major revisions since then.”  Jon Be, commenter on the computer-programming-forum.com, suggested that people who program in Fortran are often not computer scientists, which means their code looks messy and disorganized to computer scientists.  I suppose I may be guilty of that myself.

A Michigan State University page explains: “Lower level languages such as Fortran and C have been around for a long time and usually get a bad reputation. Newer is better, right? Not always! There are some decisive advantages to lower level languages. In particular, they’re blazingly fast for numerical work.

As the video below explains, Fortran is a language for scientists, engineers, and mathematicians.  Modern languages are often put to very different uses, such as web design.  Computer scientists would prefer to use the right tool for their particular job, and, unless the job involves solving complex mathematical problems, the right tool generally isn’t Fortran.  But for numerical modelers, Fortran is the best, because it is the fastest.