Inversion for the Input History of a Dye Tracing Experiment

Abstract

2 * Abstract: The advection-dispersion model (ADM) is a good tool for simulating transport of dye or solutes in a solution conduit. Because the general problem of transport can be decomposed into two problems, a boundary-value problem and an initial-value problem, the complete solution is a superposition of the solutions for these two problems. In this paper, the solution for the general problem is explained. A direct application of the solution for the boundary-value problem is dye-tracing experiments. The purpose is inclusion of the input history of a solute dye into the ADM. The measured breakthrough curve of a dye-tracing experiment is used to invert for the release history of the dye at the input point through the ADM. It is mathematically shown that the breakthrough curve can not be directly used to invert for the boundary condition at a tracer release point. Therefore, a conductance-fitting method is employed to obtain the input history. The inverted history for a simple example is then shown to be a step function with amplitude of 420 mg/L and a duration of 10 minutes. Simulations illustrate that the breakthrough curves at downstream springs provide a means for understanding the migration of dye. A discussion of the implication of the solution for an initial-value problem (e.g., simulating transport of preexisting solutes such as dissolved calcium carbonate in solution conduits) is also included.