A Purely Numerical Approach For Analyzing Fluid Flow To A Well Intercepting A Vertical Fracture
Authors: Narasimhan T.N. , Palen W.A. 
Published: December 2007

Abstract
A numerical method, based on an Integral Finite Difference approach, is presented to investigate wells intercepting fractures in general and vertical fractures in particular. Such features as finite conductivity, wellbore storage, damage, and fracture deformability and its influence as permeability are easily handled. The advantage of the numerical approach is that it is based on fewer assumptions than analytic solutions and hence has greater generality. Illustrative examples are given to validate the method against known solutions. New results are presented to demonstrate the applicability of the method to problems not apparently considered in the literature so far.

Introduction
The problem of fluid flow to a well intercepting a single vertical fracture H of considerable interest in the field of petroleum engineering. The transient pressure response of such a well to fluid production has been investigated by previous workers production has been investigated by previous workers using analytical solutions, some of which are evaluated numerically. Despite their power, analytical methods have certain limitations. To overcome these limitations, numerical methods in conjunction with fast computing devices can be utilized with great advantage. The purpose of this paper is to describe and demonstrate the power of a fairly general numerical model in studying fluid flow to a well intercepting a vertical fracture.

The Problem
The problem of interest is schematically represented in Fig. 1. The reservoir is assumed to be homogeneous, horizontal, a really infinite and of thickness h. A vertical fracture of width w, length 2Xf and extending, throughout the thickness of the reservoir and is fully penetrated by a well of radius rw. It is assumed that the well produces fluid at a prescribed (usually constant) rate. The problem is to predict the pressure transient problem is to predict the pressure transient behavior of the system. in general and the well in particular. particular. Previous Work Previous Work Although interest in wells intercepting vertical fractures dates back to 1960, the first comprehensive study of the pressure transient response of such a well was performed by Gringarten et all who studied the case of flow to an infinitely conducting vertical fracture, with a hypothetical zero-radius well at its center. This and a few subsequent studies considered not only an infinite reservoir but also various types of bounded reservoirs. Later, Cinco-Ley et al extended the Green's functions approach of Gringarten et all to consider the mare realistic cases of finite conductivity vertical fractures. In their studies, Cinco-Ley et al coupled the Green's fraction solution of the earlier work with a one dimensional, linear solution for fluid flow within the fracture, the coupling procedure being subject to continuity of fluxes at the procedure being subject to continuity of fluxes at the fracture surface. To evaluate their solutions, Cinco-Ley et al used a numerical method. In subsequent studies, Cinco-Ley and Samaniego extended the solutions to include effects of wellbore storage and formation damage around the fracture.
Although they do discretize the fracture into several segments in their numerical evaluation, Cinco-Ley et al's method is basically analytic or at best semi-analytic in nature.

Limitations of Analytic Approach
Use analytic approach has formed the backbone of well test interpretation and reservoir analysis since the 30's and has provided innumerable valuable insights in understanding reservoir response. Yet, tractable analytic solutions can be obtained only if the system under consideration is simple.