Modeling Tools for Environmental Engineers and Scientists

functions, introduced in Section 6.22 in Chapter 6, in visualizing flow patternsin groundwater flow management.Consider a situation in which a contamination plume has been detected inan aquifer where a uniform flow exists in the x-direction. It is desired tolocate pumping wells on the y-axis symmetrically about the x-axis. The goalhere is to determine a combination of pumping rates and well spacing toensure that no contamination will pass between the wells. This can beachieved analytically in the case of two wells as follows: by superposing thepotential function for the uniform flow field with that for the two wells,the composite potential function can be obtained as follows: = –ux 4Qπ ln x 2 y – L 2 2 4Qπ ln x 2 y L 2 2 where Q is the pumping rate at each well, and L is the spacing between thewells. The condition of no flow between the wells implies that there is a singlestagnation point between the wells, or in terms of the potential andstream functions,oru(x,0) = 0 ∂ ∂xx,0= 0The task of differentiating the expression for φ, setting the resultingexpression to zero, and then solving it for x to check for stagnation conditions,although straightforward, can be tedious. However, Mathematica ® canbe used readily as shown in Figure 8.39 to differentiate the potential function,set the result to zero, and solve for x, to give the answer for x to check for thelocation of the stagnation point. In line In[1] in Figure 8.39, the generalexpression for φ is entered, and Mathematica ® is asked to substitute y = 0 inthe expression (indicated by the symbol /. {y→ 0}), to find φ(x,0). The resultis returned in line Out[1]. Then, in line In[2], Mathematica ® is asked to takethe last result (indicated by the % symbol), differentiate it with respect to x(indicated by the symbol D), set it to zero, and solve the result to find x.Mathematica ® performs this sequence of operations in symbolic form andreturns the result in line Out[2] as two possible roots for x. This example illustratesthe ability of Mathematica ® to present equations in two-dimensionalform and to perform standard mathematical calculi in pure symbolic form.To ensure only one root for the quadratic equation, the terms within thesquare root sign should cancel one another, or translating this mathematical© 2002 by CRC Press LLC