Pipe Pathways use a Laplace transform A method for solving certain types of differential equations that involves transforming the equations into an algebraic form that can be readily solved, and then using an inverse transform to retrieve the solutions of the original problem. approach to provide analytical solutions to a broad range of advectively-dominated transport processes involving one-dimensional advection, longitudinal dispersion (and diffusion), retardation, decay and ingrowth, and exchanges with immobile storage zones (e.g., matrix diffusion).
The geometry of the pathway is defined by specifying a length, a cross-sectional area, and a perimeter. Mass enters at one end of a Pipe (or along some specified length of the pipe), advects through (and disperses and diffuses within) the mobile zone of the Pipe, and then exits at the other end. Unlike Cell pathways, Pipe pathways contain only a single fluid medium (which is always, by definition, the Reference Fluid A special type of Fluid element that provides a basis for defining partition coefficients between media for the various species in the model (i.e., the ratio of the species’ concentration in the medium to its concentration in the Reference Fluid at equilibrium).). They can, however, contain solid media Materials (such as water, sand, clay, air) that constitute (are contained within) transport pathways. GoldSim provides two types of elements for defining media: Fluids and Solids. which can impact transport (e.g., by modifying the porosity of the pathway and/or acting to sorb and hence retard species The chemical (or non-chemical, such as bacterial or viral) constituents that are stored and transported through an environmental system in a contaminant transport model. In GoldSim, the Species element defines all of the contaminant species being simulated (and their properties).).
The boundary condition for the Pipe is as follows:
Concentration goes to 0 as x goes to ∞
Effectively, this allows dispersive/diffusive outfluxes, with a downstream concentration (immediately after the Pipe) that is actually quite close to the concentration in the Pipe (resulting in a low dispersive outflux).
Two types of simple contaminant retardation processes can be represented within a Pipe pathway A transport pathway element that is intended to represent a feature that essentially behaves as a fluid conduit, such as a fracture. Pipes are primarily used for simualting matrix diffusion in fractured groundwater pathways.:
- equilibrium partitioning between the fluid in the pathway and a user-specified infill medium; and
- equilibrium partitioning between the fluid in the pathway and a user-specified coating medium around the perimeter of the pathway (if, for example, the Pipe was being used to represent a fracture).
In addition to these linear retardation mechanisms, Pipe Pathways can also represent interchanges with two types of immobile storage zones along the length of the pathway (which would typically be applicable only if the Pipe was being used to simulate transport through a fracture):
- matrix diffusion zones, in which the transfer rate into and out of the zone is proportional to the concentration gradient and the diffusive properties of the zone; and
- a "stagnant" dispersive zone, in which the transfer rate into and out of the zone is proportional to the concentration difference and the flow rate in the pathway.
Finally, suspended Solids can be specified to be present in the Pipe. These Solids are assumed to be advected and dispersed along the Pipe, but are not subject to retardation processes or interactions with storage zones. Species which partition onto the suspended Solids are transported with them as they move through the Pipe.
Warning: Solubility constraints are not applied within Pipes. (They are only applied within Cells and Aquifers). Hence, if the concentration of a species entering a Pipe (e.g., via a boundary condition) exceeds the solubility limit, the concentration leaving a Pipe could exceed the limit.
The mathematical and computational details of how Pipe
pathways are implemented within GoldSim are provided in
- Advective-Dispersive Transport in Pipes
- Comparing Pipes and Aquifers
- Computing Pipe and Aquifer Pathway Concentrations Accounting for Transverse Dispersion
- Controlling the Pipe Solution Algorithm
- Defining a Sorptive Coating Material for a Pipe
- Defining Basic Pipe Properties
- Features and Capabilities of Pipes
- Flux Links to/from Pipes
- Pipe Pathway Outputs
- Saving Results for a Pipe
- Simulating Storage Zones in a Pipe
- Simulating Suspended Solids in a Pipe
- Simulating Time-Variable Pipe Properties
- Summary of Limitations on the Use of Pipe Pathways
- Viewing a Pipe in the Browser