Straight-line shale segment in uniform flow

Example 3-1. Straight-line shale segment in uniform flow. 

The arc tan solution. While Equation 3-1 is correct, it is not the best way to develop our ideas mathematically. To bring out the basic ideas naturally, we transform Equation 3-1 into radial polar coordinates, first setting 

If we substitute Equations 3-8 and 3-9 in Equation 3-7, and simplify the result using Equation 3-1, we obtain Laplace s equation in cylindrical coordinates, 

The elementary vortex solution. The basic elementary singularities used in this book can be developed from Equation 3-10. To show how the logarithmic solution arises, one might argue that the pressure about a circular well concentrically situated in a circular reservoir should not depend on 0. This being the case, we set the 9 derivative term in Equation 3-10 to zero, to obtain + (1/r) = 0, whose fundamental solution takes the well-known log r form. 

On the other hand, one might ask, What is the flow corresponding to vanishing r derivatives This limit leads to Eee = 0, which has the solution 

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