Streamline tracing

The classic streamline tracing problem. As if our new concepts were not complicated or confusing enough, the streamfunction T also has log r and 0 solutions. What do these mean physically, and how do these relate to their pressure counterparts Since our ultimate objective is to determine the streamfunction distribution once the pressure field is available numerically, we would like to determine the streamfunction expression that is complementary, first, to the source (or sink) solution... 

Discussion 4-4. Streamline tracing in the presence of multiple wells. 

A simple scheme. Now suppose that a numerical solution for the pressure field is available, for example, the finite difference solutions presented later in Chapter 7. The solution, for instance, may contain the effects of arbitrary aquifier and solid wall no-flow boundary conditions we also suppose that this pressure solution contains the effects of multiple production and injection wells. How do we pose the streamline tracing problem using T without dealing with multivalued functions The solution is obvious subtract out multivalued effects and treat the remaining single-valued formulation using standard methods. Let us assume that there exist N wells located at the coordinates (Xn,Yn), having... 

Comparison of Equations 4-66 and 4-69 shows that 9p(x,y)/9x = 9 F(x,y)/9y, while Equations 4-65 and 4-68 show that 9p(x,y)/9y = -9 F(x,y)/9x. Hence, the Cauchy-Riemann conditions are identically satisfied, so that the streamfunction in Equation 4-67 is complementary to the vortex pressure in Equation 4-64. These results provide expressions for streamline tracing in the presence of distributed singularities. In general, as we will show in Chapter 5, the flow past (or from) an arbitrary entity can be represented by superpositions of line sources and vortexes, respectively, having strengths f(x) and g(x). The net pressure field... 

So far, we have introduced different types of streamfunctions and explained their applications to streamline tracing. Unfortunately, these methods cannot be generalized to three-dimensional problems. But for such flows, a another type of 3D streamfunction or vector potential does exist, at least formally. This follows from the vector identity div curl T = 0, for any vector V. 

Discussion 4-7. Circle Theorem Exact solutions to Laplace s equation, 66 Discussion 4-8. Generalized streamline tracing and volume flow rate computations, 68... 

Discussion 4-9. Streamline tracing in 3D flows, 70 Discussion 4-10. Tracer movement in 3D reservoirs, 73 Fluid flow instabilities, 76 Problems and exercises, 78... 

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