Pressure-Transient Analysis for Fractured Wells Producing at Constant Pressure

Abstract

Summary. Past solutions to many well-test problems considered the wells to be produced at a constant rate, It has become apparent that there are many cases in which constant-pressure production is more common. Gas wells, geothermal wells, and even oil wells are operated at constant pressure during most of their producing life. Constant-pressure well-test data for fractured wells are commonly analyzed using type-curve-matching techniques that may result in nonunique solutions. By the use of a semianalytic model, a set of correlations has been developed that reduces the nonuniqueness problem, as well as accounts for well-test data in which the well is produced at high rates. When applied to high-flowrate wells, this technique uses the bilinear flow period to determine an apparent fracture conductivity for constant-pressure drawdown or buildup tests. These apparent values can be corrected to true conductivities by the use of correlations presented in this study. These relationships were derived through computer simulation of a wide range of possible situations. In summary, this paper presents for the first time a technique that analyzes buildup and drawdown data from wells produced at constant pressure with turbulent flow in the fracture. Two example applications, one for buildup and one for drawdown, are included to demonstrate the use of these techniques. Introduction Hydraulic fracturing has become a common means of well stimulation to increase productivity of low-permeability, oil and gas wells, as well as to counter any formation damage around the wellbore. Massive hydraulic fracturing (MHF) treatments for formations with permeabilities of 0.1 md or less have become more and more permeabilities of 0.1 md or less have become more and more applicable, thus making it possible to develop vast reserves locked in low-permeability sands throughout the world. Projects of such size are quite expensive, however: an MHF project may consume up to 500,000 gal [1900 m ] of treating fluid and up to I X 10 Ibm [0.45 X 10 kg] of proppant. Considering the amount of time, effort, and money expended, an proppant. Considering the amount of time, effort, and money expended, an accurate evaluation of the success of the fracture treatment is essential. Advances in technology have necessitated the development of techniques to analyze reservoir pressure data from fractured wells. Gringarten and Ramey and Gringarten el al. first generated solutions for three special cases of a fractured system: infinite-conductivity vertical fracture, uniform-flux vertical fracture, and horizontal fracture. Cinco-Ley et al. extended this analysis to the more general case of finite-conductivity vertical fractures and presented a graphic technique to estimate fracture conductivity. presented a graphic technique to estimate fracture conductivity. Agarwal et al. developed graphic solutions for the case of constant pressure at the wellbore. The type curves developed in Ref 5 are of great interest, but unique solutions are difficult to obtain. More recently, Guppy et al. presented semianalytic solutions for this problem that proved to be very accurate at early times and presented new type curves for the constant-pressure case. presented new type curves for the constant-pressure case. Most conventional constant-pressure postfracture analysis methods are not valid when producing at high flow rates because of possible turbulence in the fracture. Wattenbarger and Ramey possible turbulence in the fracture. Wattenbarger and Ramey investigated the effects of nondarcy flow in the formation and concluded that the effects of turbulent flow within the fracture are more sig-nificant than that in the formation, except for very small fractur lengths. Holditch and Morses used a computer simulator to model the effect of turbulence in the calculation of fracture characteris-tics. The results, however, were purely qualitative in nature and could not be applied to specific cases. Refs. 9 through 12 analyzed nonDarcy flow in finite-conductivity fractures and presented solutions for three cases: constant-pressure drawdown, constantrate drawdown, and constant-rate buildup. No solution has been presented so far for analyzing buildup data from wells produced at constant pressure. The constant-pressure method, relying on type-curve analysis, generally provides nonunique solutions and is quite cumbersome to apply. The purpose of this paper is to provide alternative solution techniques for both the buildup and the drawdown cases for wells producing at high flow rates and to avoid the use of type-curve analysis. Toward this end, this paper also addresses the solution to the constant-pressure buildup case. Dimensionless Variables Use of dimensionless groups allows general solutions to be obtained and to be related to quantitative parameter values in a more physically meaningful way. physically meaningful way. Reciprocal Dimensionless Rate. We define the reciprocal dimensionless rate, l/qd, for the case of liquid as follows:The flow rate varies with time because constant-pressure production is assumed. The equation for gas wells, in which the pressure drop is expressed in the real-gas pseudopressure, Pp, to account for the variation of viscosity and gas compressibility with pressure, isDimensionless Time. The dimensionless time is defined asThe viscosity-compressibility product, is expressed at initial reservoir conditions for gas wells. SPEFE P. 169