Application of the Boundary Element Method and Dual Reciprocity Method to the Modeling of Well Testing

Fluid flow mathematical models often involve advection and/or diffusion equations, which is also the case with well testing and displacement process analyses in the petroleum industry. These models are either diffusion dominated, as described by a pressure diffusivity equation, or advection dominated (i.e. modeled by a saturation equation with a high Peclet number). Due to non-linearity in the flow functions and a sharp build up of fronts, difficulties will arise when standard numerical approximations are adopted.;functions and a more comprehensive time integration scheme. In general, previous researchers consider multi-phase fluid flow to be composed of two distinct stages: the determination of the velocity field and pressure response which gives rise to an elliptic pressure equation and the advection-diffusion process which gives rise to a hyperbolic saturation equation. In this study, the Boundary Element Method and Dual Reciprocity Boundary Element Method are applied to the pressure diffusivity equation; while the saturation is updated during each time step to compensate for the mobility change due to the multi-phase fluid flow characteristics. The saturation equation was recast into a suitable form using boundary integration by taking into account capillary pressure. In addition, the Finite Analytical Method is suggested to handle the advection dominated saturation equation when capillary effect is negligible. All the governing equations are solved in the Laplace domain to alleviate numerical error caused by a time derivative. The Stehfest algorithm was proposed with respect to the inversion of the solutions into the real domain.;Both the BEM and DRBEM can reproduce analytical solutions to certain degree with respect to the pressure and its derivative which are crucial to reservoir parameter estimation, the saturation profile which is essential relative to the prediction of oil recovery performance was updated using different scheme based on the nature of its equation.;In this study, previous research is reviewed and summarized, and a recently developed scheme is introduced which attempts to overcome several existing methodological shortcomings. At the same time, this work explores the advantage of the Conventional Boundary Element Method and a Hybrid Boundary Element Method known as the Dual Reciprocity Boundary Element Method which analyzes a pressure transient test and multi-phase flow performance. BEM is a natural choice for these problems because of its rigorous analytical base of Green functions, which have been extensively investigated as an established part of well test analysis in reservoir engineering. However, the classical BEM has been somewhat limited to single phase flow in homogeneous media. Recently, the DRBEM has been established as an effective alternate numerical tool for modeling various engineering problems. This study presents a derivation of the DRBEM, which provides a computationally efficient means to handle a well test and multi-phase flow in a homogeneous medium. The accuracy of the scheme can be further enhanced by incorporating singularity programming, global interpolation.