A New Approximate Analytic Solution for Finite-Conductivity Vertical Fractures

Abstract

Summary new analytic solution, based on an approximate trilinear flow model, is developed to study the transient behavior of a well intercepted by a finite-conductivity vertical fracture. The solution accounts for the effects of skin, wellbore storage, and fracture storage. Both constant-pressure and constant-rate cases are considered. The solution is simple and reliable for short-time analysis. Combining this solution with a semilog asymptotic solution provides a reliable tool for analysis and formation evaluation of fractured wells. We also demonstrate that the optimization technique is a convenient means of formation parameter estimation. A set of early-time asymptotic solutions is also presented. These solutions provide qualitative and quantitative relations of the simultaneous influences of wellbore storage, fracture storage, and skin damage on early-time wellbore pressure behavior. Introduction The increased activities in exploiting tight reservoirs by means of hydraulic fracturing techniques have generated considerable interest in the development of pressure-testing procedures for evaluating fracture performance. procedures for evaluating fracture performance. Pressure or production analyses of fractured wells have been investigated with numerical as well as analytic methods. Cinco-Ley surveyed published numerical and analytic reservoir flow models for fractured wells. A numerical approach with a reservoir simulator can rigorously treat nonlinear fluid/rock properties, as well as formation heterogeneity and geometry. From the standpoint of ease of analysis, however, we prefer analytic models, if applicable, over numerical models. For pressure testing of wells, analytic models can be grouped into two types according to their solution methods: semianalytic and asymptotic analytic models. The semianalytic model was first developed by Gringarten et al. for infinite-conductivity fractured wells and was later extended to finite-conductivity fractured wells by Cinco-Ley et al. In these approaches, the governing linear partial differential equations were transformed first to a set of integral equations. Then, these integral equations were discretized in time and space to find the unknown variables of pressure and flow in the fracture. Adapting this algorithm for routine well testing purposes would require considerable computer coding and storage comparable to the requirement of a purely numerical approach. In asymptotic analytic solutions, a square-root-of-time solution and recently developed asymptotic bilinear solutions are used for formation evaluation of fractured wells. The square-root-of-time solution is applicable only to short and high-conductivity fractures. The bilinear model is applied when the influence of the flow from fracture tip is not felt within the fracture. Also, while the slope of pressure vs. the fourth root of time will provide an estimate of fracture conductivity, the fracture length cannot be obtained directly. Therefore, we conclude that no simple, suitable analytic model for formation evaluation of fractured wells, capable of providing both fracture length and conductivity, has been developed. We present a new analytic mathematical model for flow to a fractured well. On the basis of physical and mathematical reasoning, we approximate the flow between the formation and the fracture as having a trilinear behavior. This model, called the trilinear model, considers the effects of skin, wellbore storage, fracture storage, and constant-pressure and constant-rate cases. The solutions are simple and reliable for short-time analysis (the time before semilog straight-line behavior is reached) of a well intercepted by a vertical fracture. Combining our short-time solution with semilog asymptotic solutions provides a reliable tool for pressure testing of fractured provides a reliable tool for pressure testing of fractured wells. We also present a method of formation parameter estimation by means of an optimization technique. This procedure requires an optimization (or error minimization) procedure requires an optimization (or error minimization) subroutine. We demonstrate that the fracture parameters can be determined conveniently with the optimization technique and the trilinear model. Finally, we present early-time asymptotic solutions for both constant-pressure and constant-rate cases to illustrate the simultaneous influences of skin, wellbore storage, and fracture storage at early testing times. SPEFE P. 75