The obviously small parameter that is arrived at when describing fractured wells (the ratio of width to length) has been employed for some time. Although the high conductivity of fractures has also been used to arrive at the steady state solution for the infinitely conductive fractures, the transient problem for the finite conductivity fracture has to be solved numerically. Still, high permeability within the fracture (as compared to that of the surrounding rock) offers additional possibilities for simplifying the transient solution of the problem. This opportunity is based on asymptotic analysis utilizing different time scales for flows within the fracture and in the rock.
Another difficulty related to describing fractured well behavior arises when there is interaction with other wells or the reservoir boundary. Within a finite reservoir, the fluxes on the fracture panels must be included in the solution of the problem as a whole. This can lead to very short time steps and make the problem 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, which are also compared to known numerical results employing detailed (non-asymptotic) solutions for the problem. It further suggests how this approach may be of significant practical utility to the industry as a means of making much faster and more accurate analysis of completion methods, well placements and spacing patterns, and other typical reservoir development decisions.
Progress in applying the Boundary Element Method (BEM) to reservoir engineering problems substantially enhances our understanding of reservoir performance. Great attention has been paid, in particular, to modeling flow in vertically fractured wells1–9.
Stationary solutions for infinitely conducting fractures were historically the first ones used in reservoir engineering10. Transient solutions for infinitely conducting fractures constituted the next stage in theory development1, and publications1–3 have boosted research on finite conductivity fractures. Most of this research was performed within the BEM. This is understandable because BEM provides adequate means for handling singular sources, which is not true for conventional methods (finite difference and finite element methods).
After publications Refs.2,3, improvements in accuracy were made. This was achieved through higher order approximations on the fracture4 and by applying different kinds of analytical approaches5,7. In developing a practical application for reservoir engineering11, we wanted to get high computational speed for problems with the fractures within arbitrary shaped reservoirs. This was achieved through utilization of different small parameters present in the problem.
A detailed description of the BEM and the mathematics involved can be found elsewhere9, 12–18. So, we will present only the features specific for this study.