Validation and Extension of Asymptotic Solutions of Diffusivity Equation and Their Applications to Synthetic Cases
Authors: Wang Zhenzhen , Li Chen , King Michael 
Published: February 2017

Abstract
We have previously introduced a transformation based upon the Fast Marching Methods (FMM) to describe the multi-dimensional diffusivity equation with heterogeneity as an effective one dimensional diffusivity equation in a streamtube. In the current study we develop and validate new asymptotic analytic approximations to this problem, which provide for a number of novel applications including rapid numerical simulation, reservoir and well characterization, sensitivity-based inversion using production data, and dynamic upscaling and downscaling. The novel semi-analytic asymptotic pressure approximation for the solution of an equivalent 1-D diffusivity equation is able to approximate the 3-D solution with heterogeneity. Earlier approaches have relied upon the numerical solution of the 1-D equation, and provide all the flexibility expected of a numerical approach. However, analytic solutions provide for the derivation of explicit relationships between the geometry of a propagating pressure "front" within a reservoir and pressure and rate measured at wells. In the current study, we extend the analytic treatment beyond simple fixed rate draw-down and test the predictions against analytic and numerical synthetic cases.
In this paper we provide a systemic validation of our semi-analytic solution technique and extend its utility to more realistic cases, including large changes in reservoir properties, pressure transient analysis with wellbore storage, and rate transient analysis in bounded reservoirs with fixed rate or fixed BHP production. This technique provides us with the ability to describe pressure propagation from fractured wells into the surrounding formations, which provides for a better drainage volume characterization, which is beneficial for both well spacing calculation and multi-stage fracture spacing optimization in unconventional reservoirs. It not only provides for the direct calculation of various welltest, rate transient and well performance concepts such as depth of investigation, welltest derivative, flow regimes and well productivity, but it can also predict pressure and flux distribution maps at any time of interest. Our study verifies that the new approach yields results very close to those generated by commercial simulators, indicating its promising application to rapid field production data analysis. As with other analytic approaches, the derived asymptotic solutions satisfy superposition in space and time, which allows for further application to cases with multiple wells and varying flow rates.
We show the validation of novel semi-analytic asymptotic pressure solutions to the diffusivity equation, which extend the calculation of rate and pressure transient from homogeneous reservoirs with regular well geometries to a series of reservoir problems with hydraulic fractured wells and reservoir heterogeneity. The treatment we present in this paper is faster than numerical finite difference simulation and allows for the development of fundamental relationships between reservoir performance and reservoir and well characteristics.