Source Functions in Early Time Pressure Transient Analysis

Abstract

The pressure responses of a reservoir can be obtained by convolving source functions and flow rates. Although the literature reports on deriving source functions analytically from the diffusivity equation, there is no study on deriving source functions using flow rates and pressure responses obtained from pressure transient tests. We therefore wanted to develop a methodology for obtaining the formation source functions using pressure data when pressure and flow-rate data are known. In addition, we wanted to study the characteristics of some source functions both from simulated data and from analytical methods in the literature.

We demonstrate that the pressure functions (solutions of the diffusivity equation) of a test well can be calculated by convolving flow rates with source functions, and that the source functions can be derived by deconvoluting the pressure drop and flow rates available from the pressure-test data. Pressure function, flow rate, or source function can be obtained when two of these three functions are known. A source function (in time domain) with the wellbore storage effect is a horizontal line in a very early time and coincides with the infinite line source function or the infinite surface cylinder source function after the end of the wellbore storage effect. The source functions are almost the same for different positive skin factors. For a negative skin factor, a source function is initially a horizontal line and subsequently coincides with the infinite line source or infinite surface cylinder source functions.

Keywords:

• boundary condition
• convolution
• deconvolution
• skin effect
• solution of diffusivity equation
• wellbore storage