We present a set of new analytical solutions to the single layer reservoir problem, both in real time and Laplace space. The solutions are derived assuming a cuboid shaped reservoir using a method of integral transforms. The method can be applied to calculate the pressure as a function of position and time when using any continuous function to describe the production rate of a point source. Successive integration of the point source solution can be performed to calculate the average bottom hole pressure of a well.

These equations are applicable to partially penetrating vertical, horizontal and fractured wells and take into account superposition effects in multi-well and multi-rate scenarios. Notably, regarding fractured wells, we are able to accurately model the case of a finite conductivity fracture with non-Darcy flow as well as those of infinite conductivity. The generality of our method allows any continuous function of position and time to be used to treat either pressures or fluid fluxes on the boundaries.

Also, using solutions in Laplace space we are able to model naturally fractured reservoirs, wellbore storage, non-Darcy D-factors as well as constant well pressure production, also all within a full field multi-well scenario. Our method, therefore, provides a powerful alternative to simulation in terms of reservoir modeling.

We present a comparison of our solutions with that generated using a commercial finite difference simulator for a variety of problems in terms of accuracy and speed. We find amazing accuracy with massive gains (factors>300) in CPU times for fracture problems in particular.

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