Modelling of Fractured Wells in Reservoir Simulation
Authors: Ding Y. 
Published: December 2007

Abstract
Fractured wells are frequently encountered in petroleum engineering. They can be hydraulically fractured wells to improve oil production or wells intersecting natural fractures. Due to the presence of the fractures, the flow behavior and the pressure distribution in the vicinity of the fractured well are complex and the numerical modelling for this kind of wells is complicated. In this paper, we will discuss the numerical methods for the fractured well modelling by reservoir simulators.

Introduction
The study of fractured wells has received more attention in well test analysis for a long time. Some analytical or semi-analytical formulas have been developed for several simple cases, such as a fully penetrating hydraulically fractured vertical well, a longitudinal fractured horizontal well or a transverse fractured horizontal well. But these formulas cannot be used for the flow simulation for a field study or for the study of the complex wells and fractures. To handle these cases, a reservoir simulator can be used.
During this decade, several numerical simulation studies for fractured wells have been published. In all these studies, fine gridblocks are used around the wells and the fractures. The requirement of grid refinement and the impacts of the grid discretization on the well performance is particularly discussed by Hegre. Although using grid refinement gives accurate simulation results, it is expensive and prohibitive for a full field simulation. In well test analysis, an equivalent wellbore radius rw, eq is introduced for the infinite conductivity fracture. However, this equivalent wellbore radius rw, eq can be used for a reservoir simulator only if the fracture is entirely included in one gridblock and rw, eq is smaller than the equivalent wellblock radius r0. So, it is necessary to study the techniques for the fractured well modelling with coarse grids. In this paper, we will propose a new method for the simulation of fractured wells. This method is based on the well model rather than on the gridding technique.
For the well modelling in reservoir simulation, a numerical productivity index (PI) is usually introduced to relate the wellbore pressure, the well block pressure and the well flow rate, since the singularity around the well causes different values between the wellbore pressure and the wellblock pressure. In most of the well models, the well trajectory and its position are not really considered. So these well models are limited for the complex well modelling, and are inappropriate for the fractured well modelling. Recently, a new kind of well model was proposed, which takes into account well configurations by using equivalent transmissibilities to improve the flow calculation in the well vicinity. This well model will be extended to model fractured wells in this paper.
In this paper, we will study particularly high conductivity fractures, and these fractures can be considered as a part of the well. To establish a well model, it is necessary to know a steady-state or pseudo-steady state pressure distribution around the well. In section 2, the integral formulae are introduced to represent the pressure distribution for a steady-state problem. This formula is general and is valid for any well configuration and any kind of fractures. In particular, we will present some applications to hydraulically fractured wells.

Modelling of Fractured Wells
A steady-state flow problem is considered to get the pressure solution around the wells and the fractures. This solution will be used for numerical PI and equivalent transmissibility calculations. The integral formulas are used for the pressure representation. These formulas were introduced for complex well modelling and are also suitable for fractured well modelling.
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