The Green Element Method for Numerical Well Test Analysis

To use pressure transient test data in computerised methods for integrated reservoir charcterisation numerical simulations of the well tests are typically required. Numerical artifacts occurring in the simulation must be avoided as much as possible so that they do not adversely affect the reservoir characterisation. This work explores the advantages of a hybrid boundary element method known as the Green element method for modeling pressure transient tests. Boundary element methods are a natural choice for the problem because they are based on Green’s functions, which are an established part of well test analysis. The classical boundary element method is limited to single phase flow in homogeneous media. This works 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 Green element approach to standard finite difference simulation show that both methods are able to model the pressure change in the well over time. When pressure derivative is considered however the finite difference method produces very poor results which would give misleading interpretations. The Green element method in conjunction with singularity programming reproduces the derivative curve very accurately. Introduction The boundary element method (BEM) was applied by Numbere and Tiab to generate steady-state streamlines in sectionally homogenous two-dimensional reservoirs. Masukawa and Horne considered immiscible displacement problems using BEM. Kikani and Horne applied BEM to generate pressure transients in arbitrarily shaped homogeneous reservoirs. The problem of flow in heterogeneous reservoirs was addressed by Sato and Horne who developed a perturbation based approach. This approach became increasingly computationally intensive as the reservoir hetereogeneity became more pronounced. The standard form of BEM is not applicable to flow in heterogeneous media so hybrid boundary element based schemes were considered. This study focused on the application of the Green Element method (GEM). Taigbenu first presented GEM in 1990 and described it as an element-by-element implementation of the boundary element method. Taigbenu considered the Laplace, diffusion, nonlinear Boussinesq and convection-diffusion equations. Taigbenu and Onyejekwe applied GEM to groundwater flow in the unsaturated zone. Archer and Horne discussed the application of the GEM and the Dual Reciprocity Boundary Element Method to one-dimensional pressure diffusion and tracer flow in heterogeneous media. Archer and Horne and Archer extended the analysis to two dimensions. Singularity programming was introduced to study well tests. This work compares the performance of the proposed GEM/singularity programming approach to finite difference simulation of well tests. Theory Treatment of Heterogeneity To use the classical boundary element method the differential equation being considered must include a ∇2 operator. The single-phase flow equation: ∇ · (k∇p) = φμct ∂p ∂t (1) is therefore not in a form suitable for solution by a boundary element method. 2 THE GREEN ELEMENT METHOD FOR NUMERICAL WELL TEST ANALYSIS SPE 62916 Taigbenu and Onyejekwe demonstrated how the singlephase flow equation could be solved by rewriting it as: ∇2p = −∇lnk · ∇p + φμct k ∂p ∂t (2) Incorporation of Singularity Programming Masukawa and Horne and Sato applied singularity programming in conjunction with boundary element methods to compute pressure transients. Unlike the current study Sato’s solution was performed in Laplace space. Singularity programming decomposes the solution into singular and nonsingular components: pD = p D + p s D (3) The singular solution used in this work is based on the pressure response of a well flowing at a given rate in an infinite homogeneous reservoir with uniform permeability, k0 (Dake): p = pi + qμ 4πk0h Ei (−φμctr2 4k0t ) (4) where Ei(x) is the exponential integral: Ei(x) = − ∫ ∞ −x e s ds (5) Out of regard for numerical stability dimensionless variables were used in the implementation of the singularity programming. The choice of dimensionless variables in this work follows that of Sato: pD = pi − p pi (6)