Courses: The GoldSim Contaminant Transport Module:

Unit 8 - Modeling Spatially Continuous Processes: Diffusive Transport

Lesson 1 - Unit 8 Overview

These motions were such as to satisfy me, after frequently repeated observation, that they arose neither from currents in the fluid, nor from its gradual evaporation, but belonged to the particle itself.

Robert Brown

In previous Units we have been discussing representing parts of our system using well-mixed compartments.  This is appropriate in many instances (e.g., when dealing with a system of well-mixed tanks, a stratified lake, or well-mixed soil compartments).  We saw that mathematically these kinds of systems can be represented using a set of coupled ordinary differential equations. In these systems, there were two key assumptions:

  • Each compartment could be assumed to be well-mixed (i.e., concentrations were the same throughout the compartment); and
  • The mass transport from one location to the other was only a function of the mass (or concentration) in the “upstream” compartment.

However, in many cases, it will not be appropriate to represent parts of the system in this way.  In some parts of the system, both of these assumptions will be difficult to defend.  In particular,

  • Concentrations will vary spatially in a continuous manner so that assuming a relatively large well-mixed region would be inappropriate; and
  • The mass transport rate at any point cannot be described in terms of a single concentration or mass, but must be described in terms of a concentration gradient (a concentration difference over some distance).

Another way to think about this is that whereas when representing parts of your system as well-mixed compartments there is an implicit spatial component to the system (since the compartments are in fact spatially separate), in other cases there is an explicit spatial component to the system that directly impacts mass transport rates.

These parts of a system (with an explicit spatial component) require a different approach to representing and solving the mass transport equations. Mathematically, we need to represent these systems not as ordinary differential equations, but as partial differential equations, in that concentration is a function of both time and space. This is the first of two Units that describes how to model such systems in GoldSim.

The kinds of systems of this type that are typically the most important in most mass transport models are advectively-dominated.  Examples include mass transport through an aquifer or a surface water body such as a river, pond or lake. In these models, mass transport is described by the advection-dispersion equation (which we discussed briefly in Unit 3). Unit 9 will discuss how GoldSim represents systems described by this equation.

In this Unit, however, we are going to discuss a simpler partial differential equation: the diffusion equation (Fick’s Second Law). Diffusion is the physical process by which a substance tends to spread steadily from regions of high concentration to regions of lower concentration due to the random thermal motions of suspended atoms, molecules and particles referred to as Brownian motion (named after the Mr. Brown who provided the quote at the beginning of this Lesson).

Diffusion is a relatively slow process (in water), and in many systems will be of limited importance (since transport will be advectively-dominated).  However, for parts of some systems (e.g., engineered waste disposal facilities) diffusion can be a critical process. These systems typically have time frames of interest that are very long (tens, hundreds or many thousands of years). Diffusion can also play a critical role in many other (shorter-term) processes that you may not be aware of (e.g., transport through very small boundary layers). And, of course, if your fluid of interest is air rather than water, diffusion rates are orders of magnitude faster.

Even if you think you would never need to model a diffusive process, you should still work through this Unit as it will improve your understanding of the Contaminant Transport Module and provide a necessary introduction for topics discussed in the next Unit (advective-dispersive transport).

This Unit has two Exercises, and numerous Examples that we will walk through together. It has a total of 12 Lessons (including this overview and a summary at the end).