Finite-state models of transport phenomena in hydrologic systems

Abstract

Transport phenomena in hydrologic systems are simulated with finite-state models (FSMs), which are similar to mixing cell models in that they utilize a mixing cell as their basic subdivision, yet are more flexible, capable of modeling more complex systems, and easier to manipulate than previous mixing cell models. The basic FSM equations are discrete, recursive forms of the continuity equation for mass transport and the storage equation for fluid transport. Different types of mixing and flow can be simulated by specifying appropriate algorithms for use in the basic equations. Finite-state models thus have a physical basis, although they avoid the use of differential equations. The FSM digital computer model can simulate systems in one, two, or three spatial dimensions with relative facility. In many important cases, transit number and age number distributions can be calculated. These distributions, and especially their means, are useful in determining fluid residence times in hydrologic systems. Two aquifer systems are modeled using finite-state models. In a portion of the Tucson Basin Aquifer of southern Arizona a three-dimensional, steady flow FSM is used to account for the observed carbon-14 age distribution in the aquifer without assuming piston flow in the aquifer and without evaluating dispersion parameters. This model provides a first approximation of the three-dimensional flow distribution, an estimate of the long-term average annual recharge, and fluid residence times in the aquifer. The second FSM, two-dimensional and non-steady flow, accounts for the transient distribution of tritium in the Edwards Limestone of south-central Texas. This aquifer is a highly anisotropic, nonhomogeneous karst aquifer that is difficult to model by traditional methods. In both models, first guesses for cell volumes and flow distributions were made on the basis of available hydrogeological data. Saturated, unsaturated, and open-channel flow also are examined. Flow algorithms for the basic FSM storage equation follow the theory of linear systems, although in certain regimes, especially those involving unsaturated flow, it may be necessary to develop nonlinear flow algorithms. This was not attempted. It is also shown that the finite-state model can simultaneously model the transport of mass and fluid in a hydrologic system. The FSM also has the potential for modeling heat transport, which may prove useful in simulating geothermal reservoirs as well as other systems involving heat transport.