Abstract
It follows from the previous papers that we can derive a S-function as long as the complex velocity potential is provided and S-function gives us a good approximation of pressure drops and derivatives. In this paper, S-function analysis will be expanded to free-form curves. For this purpose, we have to derive a complex velocity potential for the given boundary shape. The solution for this kind of problem is found in vulgate textbooks for complex variables, which is known as“ Schwarz-Christoffel conformal mapping”. This theory was discovered soon afterward, independently by Christoffel in 1867 and Schwarz in 1869. In decades, the development of computer enables us to calculate the conformal mapping actually. The typical Schwarz-Christoffel conformal mapping is a formula that will transform the real axis of the w-plane into a polygon in the z-plane; however, here we will mainly discuss about“ strip type transform”. It is rarely mentioned in public, although, this theory can be easily extended to the problem that sinks and /or sources( singular points) exist. Thus, we can obtain the complex velocity potential and equidistance surfaces on physical space of which boundary shapes are complicated. At last we can calculate S-functions for any boundary shapes and it enables us to analyze well testing.
