A Mathematical Model for the Analysis of the Pressure Transient Response of Fluid Flow in Fractal Reservoir

Abstract

This study uses similar construction method of solution (SCMS) to solve mathematical models of fluid spherical flow in a fractal reservoir which can avoid the complicated mathematical deduction. The models are presented in three kinds of outer boundary conditions (infinite, constant pressure, and closed). The influence of wellbore storage effect, skin factor, and variable flow rate production is also involved in the inner boundary conditions. The analytical solutions are constructed in the Laplace space and presented in a pattern with one continued fraction—the similar structure of solution. The pattern can bring convenience to well test analysis programming. The mathematical beauty of fractal is that the infinite complexity is formed with relatively simple equations. So the relation of reservoir parameters (wellbore storage effect, the skin factor, fractal dimension, and conductivity index), the formation pressure, and the wellbore pressure can be learnt easily. Type curves of the wellbore pressure and pressure derivative are plotted and analyzed in real domain using the Stehfest numerical invention algorithm. The SCMS and type curves can interpret intuitively transient pressure response of fractal spherical flow reservoir. The results obtained in this study have both theoretical and practical significance in evaluating fluid flow in such a fractal reservoir and embody the convenience of the SCMS.