Asymptotic Solutions of the Diffusivity Equation and Their Applications

Abstract

Abstract Understanding how pressure fronts propagate (diffuse) in a reservoir formation is fundamental to welltest analysis and reservoir drainage volume estimation. We have developed an alternative approach to the solution of the 3-D diffusivity equation by directly solving the propagation equation for the "pressure front" defined as the maximum pressure response for an impulse source. The pressure front equation is a form of the Eikonal equation, which is a high frequency asymptotic solution of the diffusivity equation in heterogeneous reservoirs and whose properties are well developed in the literature. Most importantly, the Eikonal equation can be solved very efficiently by a class of solutions called the Fast Marching Methods (FMM) for a "diffusive time of flight" (DTOF) that governs the propagation of the "pressure front" in the reservoir. The "diffusive time of flight" can be used as a spatial coordinate to reduce the 3-D diffusivity equation into an equivalent 1-D formulation, leading to a simplified method for rapid reservoir modeling. Recent papers have explored the utility of this 1-D approach for performance prediction using finite difference numerical simulation. The method is especially well suited to the interpretation of the drainage volume, which is of great help in well spacing calculations and in the context of unconventional reservoirs, multi-stage fracture spacing optimization. In this paper we introduce an analytic solution technique for the diffusivity equation, which provides a direct relationship between production data and the reservoir drainage volume. The analytic formulation provides for the direct calculation and extension of many simple well test, rate transient and well performance concepts such as depth of investigation, welltest derivative, drainage volume, flow regimes and well productivity. As with other analytic approaches, these solutions allow superposition in space and in time, which allows for the solution for multiple wells, multiple flow rates, and bounded and composite reservoirs. We validate our approach against well-known solutions in pressure and rate transient analysis usually solved in Laplace space, including pressure transients with wellbore storage and skin. Our study demonstrates that the new approach yields results very close to the known solutions calculated via numerical inversion of the Laplace transform, and indicates how to extend these solutions to problems with heterogeneity and complex fractured well geometry.