Green element method and singularity programming for numerical well test analysis

Abstract

One technique available to petroleum reservoir engineers to determine the properties (such as permeability and reservoir size) of oil and gas reservoirs is well test analysis. In a well test a well undergoes a step-change in its flowrate and the resulting variation in the well pressure is carefully measured. Traditionally these pressure responses are interpreted by comparing them to analytical solutions. However these solutions are limited to homogeneous reservoirs of regular shapes. An alternative is to compare the measured data to numerical simulations of the well test. This allows for more complex reservoir geometry and heterogeneity in the reservoir permeability to be included. Traditionally reservoir engineers use finite difference methods for these fluid flow calculations. These are prone to some numerical artifacts that make well test responses difficult to compute accurately. This work explores the advantages of a hybrid boundary element method (BEM) known as the Green element method (GEM) for modeling well tests. BEMs are a natural choice for the problem because they are based on Green's functions, which are an established part of well test analysis [Soc Petrol Engr J (1973) 285]. The classical BEM is limited to single phase flow in homogeneous media. This work presents formulations, which give computationally efficient means to handle heterogeneity. The accuracy of the scheme is further enhanced by incorporating singularity programming. Comparisons of the proposed GEM approach to conventional finite difference simulation, using the same gridding and timestepping, show that finite difference simulations of well test responses do not accurately reproduce the corresponding analytical solutions. GEM can accurately reproduce analytical solutions for the pressure and its derivative even using coarse gridding. It can also efficiently handle heterogeneity.