Asymptotic Description of Vertically Fractured Wells Within the Boundary Element Method
Authors: Kryuchkov S. , Sanger S. 
Published: March 2010

Abstract
The typical parameter that is used when describing fractured wells (the ratio of width to length) has been employed for some time. The high conductivity of fractures has allowed the use of a steady-state solution for the infinitely conductive fractures; however, the transient flow modelling problem for fractures of finite conductivity has to be solved numerically. Still, the high permeability within the fracture (as compared to that of the surrounding matrix) offers additional possibilities for simplifying the transient solution for modelling the flow in the fractures. This opportunity is based on asymptotic analysis using different time scales for the flow within the fracture and in the lower permeability surrounding rock matrix.
Another difficulty related to describing fractured well behaviour arises when there is interaction with other adjacent wells or the reservoir drainage boundaries. For a finite reservoir, the fluxes on the fracture faces (panels) must be included in the solution of the problem as a whole. This can lead to very short time steps and make the numerical solution of the problem very computationally intensive. However, asymptotic analysis, based on the proximity of fractures to other objects in the reservoir, makes it possible to effectively de-couple the solution for the fracture from the solution of the reservoir as a whole, which increases computational efficiency substantially.
This paper presents the conceptual grounds and mathematical details of asymptotic analysis and gives examples of calculations based on this approach. These results are compared to known numerical results employing detailed (non-asymptotic) solutions for the problem. This work illustrates how this approach may be of significant practical use to the industry as a means of conducting faster and more accurate analysis of completion methods, well placement and spacing patterns, and other reservoir development decisions.

Introduction
Progress in applying the Boundary Element Method (BEM) to reservoir engineering problems has substantially enhanced our understanding of reservoir performance. Great attention has been paid in the past, in particular, to modelling flow in vertically fractured wells(1–9).
Stationary solutions for infinitely conducting fractures were historically the first ones used in reservoir engineering(10). Transient solutions for infinitely conducting fractures constituted the next stage in theory development(1), and a number of publications(1–3) have detailed research on finite conductivity fractures. Most of this research was performed using the BEM. This is understandable because the BEM provides a stable method for handling singular sources, which is not true for other conventional numerical solution methods (such as finite difference and finite element).
Cinco-Ley et al.(2, 3) made improvements in the accuracy of the numerical solution generated using BEM methods. This was achieved through higher order approximations on the fracture(4) and by applying different kinds of analytical approaches(5, 7). In developing a practical application for reservoir engineering(11), the objective of this model was to obtain a high level of computational speed for problems with the fractures within arbitrarily-shaped reservoirs. This was accomplished through the use of different parameters to define in the flow modelling problem.