A Simple and Rigorous Mathematical Model for Estimating Inflow Performance of Wells Intersecting Long Fractures
Authors: Guo B. , Schechter D.S. 
Published: December 2007

Abstract
This paper presents a simple and more rigorous mathematical model for predicting performance of vertical and horizontal wells intersecting fractures fully penetrating reservoir sections. An important feature of the new model is that it allows rigorous coupling of flow in the matrix and flow in the fracture. A unique pressure distribution in the fracture was consistently used both for flow in the matrix and for flow in the fracture. Another feature of the new model is its simplicity for easy use because the equations in the model take very simple forms. The newly developed mathematical model has been used for productivity analyses of hydraulically fractured horizontal wells in the Daqing field, China, and vertical wells in naturally fractured reservoirs in the Spraberry Trend Area field, West Texas. A good agreement was observed between the actual production rate and model calculated production rate for the Daqing horizontal wells. The use of the model, combined with stress-sensitive permeability of the natural fractures, has captured the characteristics of rapid decline in productivity of the Spraberry vertical wells. This paper provides reservoir engineers with a practical tool for analyzing inflow performance of wells intersecting long fractures.

Introduction
A model study of a waterflooding pilot in Spraberry Trend Area indicated a NE-SW trend of the major fractures. A contrast of 144/1 was required for the major/minor fracture trend permeability to match the pilot response. This strong anisotropic effective permeability implies the existence of well inter-connected, long natural fractures in the Spraberry reservoir. A characteristic of flow in long natural fractures is the pressure variation along the fracture should be significantly higher compared to that in a hydraulic fracture or a short natural fracture. Unfortunately, method for analyzing flow behavior in reservoirs with long fractures is not readily available from the literature. The objective of this study was to develop a simple method for estimating inflow performance relationship (IPR) of vertical and horizontal wells intersecting long fractures.
Several analytical solutions have been presented for transient flow in fractured reservoirs. Numerical models have also been developed for simulating fluid flow in fractured reservoirs. However, it is still customary for reservoir engineers to use equations derived for steady flow conditions. This is not only because the analytical transient-flow solutions and numerical simulators are not convenient to use in construction of IPR, but also because steady or pseudosteady flow prevails as the dominating flow mechanism in the lifetime of most oil wells. Therefore, steady flow equations are more attractive than transient flow equations and numerical models for well productivity analysis.
Van Poollen investigated productivity versus permeability damage in hydraulically induced fractures by means of mathematical analyses of electrical-model studies. The maximum simulated fracture length was 0.6 time of drainage radius. Productivity of the fractured wells was graphically presented in terms of ratio of oil production rate with fracture to that without fracture. It was concluded that damage to the formation immediately surrounding the fracture has only a minor effect on the productivity of the well; the damage to the fracture flow capacity has a major effect on the productivity of the well; and most production would enter the fracture in that portion of the fracture furthest away from the wellbore
Dyes et al. reported the influence of fracture in well productivity based on experimental results obtained from their Carter Electric Analyzer. The maximum simulated fracture length was 0.75 time of the drainage distance between the well and the element boundary. McGuire and Sikora provided more experimental data obtained from the Carter Electric Analyzer. The maximum simulated fracture length was equal to the drainage distance between the well and the element boundary.
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