Transient Pressure Behavior for a Well With a Finite-Conductivity Vertical Fracture
Authors: Cinco L. Heber , Samaniego V. F. , Dominguez A. N. 
Published: December 2007

Abstract
A mathematical model was developed to study the transient behavior of a well with a finite-conductivity vertical fracture in an infinite slab reservoir. For values of dimensionless time of interest, to &gt;10, the dimensionless wellbore pressure, p, can be correlated by the dimensionless group; wk / x k, where w, k, and x are the width, permeability, and half length of the fracture, respectively, and k represents the formation permeability.
Results when plotted as a function of P vs log to give, for large t, a 1.151-slope straight line; hence, semilogarithmic pressure analysis methods can be applied. When plotted in terms o/ log P vs log t, a family of curves of characteristic shape result. A type-curve matching procedure can be used to analyze early time transient procedure can be used to analyze early time transient pressure data to obtain the formation and fracture pressure data to obtain the formation and fracture characteristics.

Introduction
Hydraulic fracturing is an effective technique for increasing the productivity of damaged wells or wells producing from low permeability formations. Much research has been conducted to determine the effect of hydraulic fractures on well performance and transient pressure behavior. The results have been used to improve the design of hydraulic fractures. Many methods have been proposed to determine formation properties and fracture characteristics from transient pressure and flow rate data. These methods have been based on either analytical or numerical solutions of the transient flow of fluids toward fractured wells. Recently, Gringarten et al. made an important contribution to the analysis of transient pressure data of fractured wells. They presented a type-curve analysis and three basic presented a type-curve analysis and three basic solutions: the infinite-fracture conductivity solution (zero pressure drop along a vertical fracture the uniform flux solution for vertical fractures, and the uniform flux solution for horizontal fractures.
Although the assumption of an infinite fracture conductivity is adequate for some cases, we must consider a finite conductivity for large or very low flow capacity fractures. Sawyer and Locke studied the transient pressure behavior of finite-conductivity vertical fractures in gas wells. Their solutions cannot be used to analyze transient pressure data because only specific cases were presented.
In this study, we wanted to prepare general solutions for the transient pressure behavior of a well intersected by a finite-conductivity vertical fracture. The solutions sought should be useful for short-time or type-curve analysis. We also wanted to show whether conventional methods could be applied to analyze transient pressure data for these conditions. A combination of both methods, as pointed out by Gringarten to al., should permit an pointed out by Gringarten to al., should permit an extraordinary confidence level concerning the analysis of field data.

STATEMENT OF THE PROBLEM AND DEVELOPMENT OF FLOW MODELS
The transient pressure behavior for a fractured well can be studied by analyzing the solution of the differential equations that describe this phenomenon with proper initial and boundary conditions. To simplify the derivation of flow models, the following assumptions are made.An isotropic, homogeneous, horizontal, infinite, slab reservoir is bounded by an upper and a lower impermeable strata. The reservoir has uniform thickness, h, permeability, k, and porosity, which are independent of pressure.The reservoir contains a slightly compressible fluid of compressibility, c, and viscosity, mu, and both properties are constant.Fluid is produced through a vertically fractured well intersected by a fully penetrating, finite-conductivity fracture of half length, x, width, w, permeability, k, and porosity, phi . These fracture permeability, k, and porosity, phi . These fracture characteristics are constant. Fluid entering the wellbore comes only through the fracture.
A system with these assumptions is shown in Fig. 1. In addition, we assume that gravity effects are negligible and also that laminar flow occurs in the system.