One-dimensional radial flow equation, of which permeability and porosity are in inversely proportional to the radius, is equivalent to the equation of linear flow. In other words, if we take the radial variations of permeability and porosity into consideration, we can represent the linear flow by one-dimensional radial flow equation. To expand this idea and to apply it for various types of inner and outer boundary shapes, we defined the S-function as "The functions of reservoir properties in radial direction which should be used for a radial composite model in order to represent the pressure transient behaviors at the well location for given inner- and/or outer- boundary shapes". We found that the approximate S-function can be easily estimated from complex velocity potentials for simple boundary shapes(SPE100174).<br>In this paper, the S-functional analyses are expanded to free-form boundary shapes. For this purpose, we have to derive the complex velocity potential for arbitrary boundary shapes. The solutions for these kinds of problems are known as “Schwarz-Christoffel conformal mapping”. In decades, the development of the computer technologies enables us to calculate the conformal mapping easily. <br>The typical Schwarz-Christoffel conformal mapping is a formula that will transform the complex velocity potential in a canonical domain into a polygon of the physical domain. This theory can be extended to the problem that sinks and /or sources (singular points) exist. Thus, we can obtain the complex velocity potential and at last we can calculate the pressure behavior for arbitrary shaped boundary problems. Here we will mainly discuss a “strip type transform” for the bent channel connected to fan system and about a “disk type transform” for the several hydraulic plane fractures which have different directions each other.


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