Analytical solution of the one‐dimensional time‐dependent transport equation

Authors:
  • HBH. A. Basha
    H. A. Basha
  • FEF. S. El‐Habel
    F. S. El‐Habel
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Abstract
An analytical model which provides an approximate description of scale‐dependent transport is presented. The model is based on the advection‐dispersion equation but with the dispersion coefficient dependent on the travel time of the solute from a single input source. The time dependence of the dispersion coefficient can assume arbitrary functional forms. The governing equation, which includes a time‐varying dispersion coefficient, linear equilibrium adsorption, and first‐order reaction, is reduced to the heat diffusion equation after a series of transformations. Analytical solutions pertaining to a time‐varying mass injection in an infinite medium with arbitrary initial distribution of concentration and functional dependence of the dispersion coefficient can be easily derived from the model. Since the analytical solution is in an integral form, different temporal variation of the dispersion coefficient over different time intervals can also be incorporated. It is shown that the solution reduces to the well‐known result for the case of constant dispersion and mass injection. In this study, particular solutions for various dispersion functions and mass injection scenarios are presented. These include linear, exponential, and asymptotic variation of the dispersion functions and instantaneous as well as continuous mass injection. The analytical results could model the transport of solute in a hydrogeologic system characterized by a dispersion coefficient that varies as a function of travel time from the input source. It could provide a modeling solution to solute transport problems in heterogeneous media and be used as a suitable model for the inversion problem, especially since more than one fitting parameter is available to fit the field tracer data which exhibit a scale effect.

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