Modeling Tools for Environmental Engineers and Scientists

flow is along the x-direction, for example, with U = u, the potential andstream functions simplify to:and = 0 – ux = 0 – uyNow, consider a well injecting or extracting a flow of ±Q located at the originof the coordinate system. By continuity, it can be seen that the value ofQ = (2π r) u r , where u r is the radial flow velocity. Substituting this into thedefinitions of the potential and stream yields in cylindrical coordinates:and = 0 ± 2Qπ (ln r) = 0 ± 2Qπ (θ)which can be transformed to the more familiar Cartesian coordinate system asand = 0 ± 4Qπ [ln (x 2 + y 2 )] = 0 ± 2Qπ tan –1 y x The results confirm that the stream lines are a family of straight lines emanatingradially from the well, and the potential lines are circles with the wellat the center, as expected.Because the stream functions and the potential functions are linear, byapplying the principle of superposition, the stream lines for the combinedflow field consisting of a production well in a uniform flow field can now bedescribed by the following general expression: = 0 + u(y cos – x sin ) ± 2Qπ tan –1 y x This result is difficult to comprehend in the above abstract form; however,a contour plot of the stream function can greatly aid in understanding the flowpattern. Here, the Mathematica ® equation solver package is used to model© 2002 by CRC Press LLC