Apparent Macrodispersion at the Forced Gradient Tracer Test

A forced gradient tracer test is carried out by artificially generating a forced flow field, often through a single or multiple pumping or injecting wells, then subsequently observing the tracer concentration at a fixed point. Three different forced gradient tracer tests that are commonly employed by hydrogeologists are diverging, converging, and two-well tests. The forced gradient tracer tests have the advantage of accelerating the process of transport, thus can obtain the result timely. In addition, they are often less expensive because they use only a few wells or boreholes. Their disadvantage is that the non-uniform flow pathways have been introduced because of the usage of pumping or injecting wells, which make breakthrough curves much more difficult to interpret. As far as we know, there is no exact theory and closed-form analytical solutions that can describe the breakthrough curve in a forced gradient test because of the difficulty of incorporating the non-uniform flow into the transport equation. Theories based on small perturbations have been developed and broadly employed to interpret the forced gradient tests. This article will provide exact closed-form analytical solutions for two different kinds of forced gradient flow by neglecting the local-scale dispersion. The first is the two-well test with equal pumping and injecting rates, the second is the converging test with regional flow. The purpose is to assess the ¡°apparent¡± macrodispersion caused exclusively by the non-uniformity of the flow. The breakthrough curves due to the ¡°apparent¡± macrodispersion will be obtained. These solutions can be used as the first cut calculations of concentrations in non-uniform flow fields. Because the solutions are exact, they can be used to test previous approximation theories such as the small perturbation method by neglecting the local-scale dispersion.