Derivation of equations describing solute transport in ground water

Abstract

A general equation describing the three-dimensional transport and dispersion of a reacting solute in flowing ground water is derived from the principle of conservation of mass. The derivation presented in this report is more detailed but less rigorous than derivations published previously. The general solute-transport equation relates concentration changes to hydrodyriamic dispersion, convective transport, fluid sources and sinks, and chemical reactions. Because both dispersion and convective transport depend on the velocity of ground-water flow, the solute-transport equation must be solved in conjunction with the ground-water flow equation. INTRODUCTION In recent years there has been an increased awareness of problems of ground-water contamination. Reliable predictions of contaminant movement can only be made if we understand and can quantitatively describe the physical and chemical processes that control solute transport in flowing ground water. Several reports have been published recently that develop and present solute-transport equations to compute the concentration of a dissolved chemical species in ground water as a function of space and time. Examples of these, reports include Reddell and Sunada (1970), Bear (1972), and Bredehoeft and Finder (1973) . The three main processes affecting solute transport, and consequently chemical concentrations, are convective transport, hydrodynamic dispersion (including diffusion and mechanical dispersion), and chemical reactions. Because convective transport and hydrodynamic dispersion depend on the velocity of ground-water flow, the solute-transport equation must be considered in conjunction with the ground-water flow equation. Aquifers generally have heterogeneous properties and complex boundary conditions. Therefore, the solution of the partial differential equations that describe the solute-transport processes generally require the use of a deterministic, distributed parameter, digital simulation model. Among the reports that describe or present numerical models to solve the solutetransport equations are Reddell and Sunada (1970), Bredehoeft and Finder (1973), Finder (1973), Ahlstrom and Baca (1974), Gupta and others (1975), Grove (1976), and Lantz and others (1976). Furthermore, several documented case histories show that where adequate hydrogeologic data are available, solute-transport models can be used to compute the rates and directions of spreading of contaminants from known or projected sources. Examples of model applications to field problems include Konikow and Bredehoeft (1974), Robertson (1974), Robson (1974), Konikow (1976), and Segol and Finder (1976). These models use either finite-difference methods, finite-element methods, or the method of characteristics. The selection of the numerical method depends largely on the nature of the specific field problem, but also depends to some extent on the mathematical background of the analyst. Although solute-transport models are best utilized when the analyst is thoroughly familiar both with the equations and with the numerical algorithm, the increasing availability of documented and published programs affords the opportunity for the use of a model by persons with only minimal familiarity with both. The basic purpose of this report is to derive a general form of the solute-transport equation from general principles in a more detailed, step-by-step, but less rigorous manner than has been done in previously published literature. The report is intended to serve as an introduction to quantitative modeling of solute-transport processes in ground water. It will also show how the general solute-transport equation can be modified or simplified for application to a variety of different types of field problems. Because of the interrelation between the flow equation and the solute-transport equation, the former will also be presented in some detail although not specifically derived. It is assumed that the mathematical background of the reader includes at least a familiarity with partial differential equations. GROUND-WATER FLOW General flow equation A quantitative description of ground-water flow is a prerequisite to accurately representing solute transport in aquifers. A general form of the equation describing the transient flow of a compressible fluid in a nonhomogeneous anisotropic aquifer may be derived by combining Darcy f s Law with the continuity equation. By following the developments of Cooper (1966) and of Bredehoeft and Finder (1973), the general flow equation may be written in cartesian tensor notation as: 3x. pki* y 3P J'8P 8P