Rate transient analysis of arbitrarily-oriented, hydraulically-fractured media

A semi-analytical mathematical solution to the Heat Equation has been derived to address the issue of rate forecasting from hydraulic fractures in unconventional oil reservoirs. The fracture is assumed to have a time-independent pressure profile along its length with a terminally declining flow rate. The analytical model makes use of the point solution to the heat equation for a constant rate in a bound system of a rectangular geometry. The constant pressure solution is constructed from the constant rate solution through convolution performed in the Laplace domain, then analytic inversion to the time domain. The line source solution is then acquired through analytic integration of the point source solution along a well trajectory. Results are generated using a simulator based on the mathematical model for arbitrarily oriented fractures in a 2D domain. Since the physical problem involves inverse modeling of flow rate data for reservoir characterization, diagnostics techniques for probing the fracture and drainage geometry are presented using derivative analysis of rate transient behavior of hydraulically-fractured media.