Inversion for the Input History of a Dye Tracing Experiment

Abstract

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.

From cave to spring: Understanding transport of suspended sediment particles in a fully phreatic karst conduit using particle analysis and geochemical methods

“…However, this is not always the only effect present and particles can as well exhibit both a higher flow velocity and a slightly higher dispersion than solute tracers (Goeppert & Goldscheider, 2008; Toran & Palumbo, 1992), which goes in line with the slightly higher dispersion of particles during the described experiment, whereas dispersivity is similar both for solute tracers and particles. As the ADM is a simplified approach for comparison of transport parameters that has been used in previous literature (Field & Li, 2011), differences in dispersion can only serve as a hint, but further experiments would be required to investigate these processes more in depth. In other studies, higher flow velocities of particles in contrast to solutes are also explained as pore exclusion processes, whereas particles are too large to enter smaller pores and therefore transported faster (Schiperski et al, 2016), however these processes are likely more relevant for the vadose zone and are only of minor relevance for this tracer test in a fully phreatic conduit.…”

Karst water resources in a changing world: Review of solute transport modelling approaches

“…To simulate karst transport, the two-region non-equilibrium (2RNE) approach is frequently used for simulating transport dynamics in karst aquifers (e.g., Assunção et al, 2023;Barberá et al, 2018;Birk et al, 2005;Ender et al, 2018;Geyer et al, 2007;Goeppert et al, 2020;Göppert & Goldscheider, 2008, among many others). Here, 2RNE only considers the flow along the karst conduit in which flow velocity is assumed to be constant whilst no water manuscript submitted to Review of Geophysics exchanges are considered in between the matrix and conduit components (Field & Li, 2011;Li & Liu, 2014). For the simulation of reactive processes in karst aquifers, 2RNE was extended by the inclusion of the retardation factor by Geyer et al (2007).…”

“…Yet, as there is not a unique solution for an inverse problem (Hartmann, 2018;Sivelle et al, 2020;Valota et al, 2002), it is often difficult to link the karst transport parameters with the actual transport dynamics. For that reason, karst transport models are more prone to suffering from the issue of non-uniqueness as compared to the flow models, particularly in describing and (physically) interpreting transport parameters (i.e., flow velocity, dispersivity) (e.g., Adinehvand et al, 2017;Chang et al, 2017;Johnston et al, 2009;Field & Li, 2011;Massmann et al, 2005, among many others). This often results in overparameterization and equifinality as exemplified in Figure 13.…”

A simple diagnostic model to estimate total sink recharge based on dye tracing experiments

“…This was shown by theory on partial differential equations in mathematics (Strauss 1992), and more generally by theory on linear and shift-invariant systems in signal processing (Lathi 2000). Nonetheless, Field & Li (2011) showed that it is impossible to invert for the history of contaminants injected at sinks directly from BTCs at springs, and therefore foreword modeling and fitting method have to be used.…”

“…The CxTFIT program is based on a two-regions non-equilibrium model (Toride et al 1993) to describe the case where strong solute transfer prevails between mobile-and immobile-water regions. It can reproduce strong skewness and long tailing in the predicted BTCs (Birk et al 2005;Geyer et al 2007;Göppert & Goldscheider 2008;Field & Li 2011), but conceptually has two major disadvantages: (1) flow velocity was assumed to be a constant along the conduit; and (2) dilution by dye-free waters from other tributary conduits, any fractures and pores of the rock was ignored. Another disadvantage in the application of the CxTFIT is that it requires specification of many parameters that could lead to mathematical complications and possible errors in parameterization.…”