Generalized Analytical Solution for Reservoir Problems with Multiple Wells and Boundary Conditions

Abstract

Abstract 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. Introduction Reservoir simulation is an essential tool for the management of oil and gas reservoirs. Prediction of pressure-production behaviour under various operating conditions allows, among other benefits, proper investment decisions to be made. In order to make such a prediction one must construct a reservoir model. History matching observed behaviour of the reservoir must validate the parameters of this model. Ideally, finite difference numerical simulators are used to construct reservoir models. However, in order to make full use of such a tool a large amount of reliable data is required. Also a full study, including a history-matching analysis, may take months to carry out. Therefore, there is a need for an alternative tool that honours the physics of fluid flow and at the same time offers a solution many orders quicker. Analytical solutions are fast and provide a broad understanding of the reservoir dynamics. The equations applicable to laminar flow of fluids in a porous medium were the results of Darcy's experimental study of the flow characteristics of sand filters. This combined with the equation of continuity and an equation of state for slightly compressible fluid gives the diffusivity equation, which is the equation for pressure diffusion in porous medium. Solution of the diffusivity equation under different boundary condition forms the basis for prediction of bottom hole pressure response of a producing well. These analytical solutions are generally applicable for a single well and used widely in the area of well testing. The efficiency of analytical models is generally judged by accuracy and speed.