Determination Of The Optimal Mixture Distribution

The proportions of groundwater contributing to Ash Meadows National Wildlife Refuge from the Spring Mountains, Pahranagat Valley, and the Nevada Test Site are determined using the concentrations of the rare earth elements in samples collected from wells and springs within each of these areas. In a previous study, the mixing potions were deterined using a trial and error method where different proportions of water from each of the three sites were tested to find the best match to the Ash Meadows waters. This method results in a solution that may not be optimal. In this study, the determination of the optimal mixing proportion is formulated as a restricted minimization problem, which is then solved by the method ofLagrange multipliers. Introduction Understanding the flow of groundwater is of crucial importance when evaluating the health risks associated with underground contamination. Information on the origin of groundwater as well as the direction of flow is provided through the use of conservative groundwater tracers. In the case of multiple sources of groundwater, these tracers can also be used to determine the proportions that each acquifer system contributes to a given area. Johannesson et al. (1979) investigated the use of rare earth elements (REE) as tracers of groundwater flow into the Ash Meadows National Wildlife Refuge (see Fig. 1). The rare earth elements (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb and Lu) were shown to behave relatively conservatively (travel along with the groundwater) in this environment. Earlier studies using deuterium (Winograd and Friedman, 1972) and uranium isotopes (Cowart, 1979; Osmond and Cowart, 1982) Transactions on Modelling and Simulation vol 22, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X 430 Boundary Element Technology Figure 1. Site map showing Pahranagat Valley, Spring Mountains, NTS, and Ash Meadows. suggested that the groundwater in Ash Meadows is a combination of water recharged in the Spring Mountains and groundwater flow from Pahranagat Valley (see map in Figure 1). Recharge from the Spring Mountains was identified as the primary source (60-70%) of the Ash Meadows water. Winograd and Thordarson, (1975) argued that a small portion of Na-enriched waters characteristic of the groundwaters from the NTS must also contribute to the mixture in order to explain the higher Na concentrations of the Ash Meadows groundwaters when compared to those of the Spring Mountains and Pahranagat Valley. In their investigation, Johannesson et al (1997) collected six groundwater samples from Ash Meadows, three from Pahranagat Valley, three from the Spring Mountains, and two from the Nevada Test Site, The concentrations of REEs in these samples were determined by inductively coupled plasma mass spectrometry (ICP-MS). A trial and error method was then used to determine the proportion of groundwater from the Spring Mountains, Pahranagat Valley, and NTS that produces REE concentrations similar to those in Ash Meadows groundwaters. In this paper we formulate the problem of determination of the optimal mixing proportions as a restricted minimization problem, and then solve it by the method of Lagrange multipliers. Transactions on Modelling and Simulation vol 22, © 1999 WIT Press, www.witpress.com, ISSN 1743-355X Boundary Element Technology 431 Mathematical Formulation of the Problem The problem of finding the mixture of groundwaters of Pahranagat Valley (PV), Spring Mountains (SM), and the Nevada Test Site (NTS) that resembles the groundwaters of Ash Meadows (AM) can be formulated as a restricted multiple linear regression problem, and then solved using a restricted minimization method: Find the relative proportions of groundwaters of PV, SM, and NTS (p, p, p^ such that