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The pressure transient behavior of a finite conductivity vertically fractured well is important in reservoir engineering because it provides valuable information concerning the well completion, and provides estimates of the in-situ reservoir properties and the propped fracture dimensions and conductivity. The pressure transient behavior of finite-conductivity wells has typically been analyzed using mathematical models that assume relatively simple fracture properties. Most of these models assume uniform fracture height, width, and proppant permeability distributions with respect to space, and many of these models also assume that fracture storage effects are negligible.
This paper presents the pressure transient solution of a well intersected by a finite-conductivity vertical fracture with a lengthwise variation in the fracture geometric and material properties. The general solution of the fracture fluid-flow problem, and an approximate solution of the problem in which the effect of fracture storage has been assumed to be negligible, are presented. Comparisons of the pressure transient behavior predicted with the two finite-conductivity fracture solutions are given. The effects of spatially varying fracture height, width, and material properties are investigated with these pressure transient solutions. The new solutions provide more realistic pressure transient analysis models which may be used to better design, implement, and evaluate hydraulic fracture treatments. They will also provide better agreement between the prediction of the fracture geometry and its post-treatment evaluation.
The pressure transient behavior of finite-conductivity vertical fractures has been studied extensively in recent years in an effort to obtain more reliable estimates of the geometry and the conductivity of the vertical fractures. Many of these studies have used semianalytic solutions to describe the pressure distributions in the reservoir and in the due to the production from the well. The semianalytic models that have been used in these studies have been commonly developed by the application of the Boundary Element Method (BEM).
Gringarten et al. introduced the use of the Boundary Element Method for the development of pressure transient solutions of vertically fractured wells by an adaptation of the analogous solutions of heat conduction in solids. Gringarten and Ramey demonstrated that Source and Green's functions could be used to construct useful solutions for pressure transient problems in oil and gas reservoirs. These techniques have subsequently been used extensively in the analyses of the pressure transient behavior of vertically fractured wells.
The application of the Boundary Element Method to the study of the pressure transient behavior of finite-conductivity vertical fractures was first made by Cinco-Ley et al. They developed a general semianalytic model in which the fracture geometry and material properties were assumed to be uniform. Their model also considered the fracture height to be equal to the reservoir thickness. This model has been commonly used for the analysis of the pressure transient behavior of wells with finite-conductivity fractures. It has also been a reference with which most of the later approximate models have been compared.
This paper presents a summary of the development of a semianalytic reservoir model for gas reservoirs with various reservoir complexities. The types of reservoir complexities that have been considered in this study include: multiple reservoir layers in which the reservoir layers may be infinite or finite in extent, dual or single porosity systems, and wells that have been fractured. The reservoir model that has been developed and used in this study was constructed with the appropriate Laplace domain pressure and rate-transient solutions that are available in well testing literature.
This reservoir model was developed to provide a more detailed and accurate means of analyzing or predicting the performance of a gas reservoir than is possible with a simple "type curve" approach, yet would be more computationally efficient than a finite difference reservoir model. The results of the semianalytic reservoir model have been validated with the reference solutions generated with a commercial finite difference reservoir simulator. The results of the semi-analytic reservoir model are comparable to the results of the finite difference simulator, yet require substantially less computation time and computer memory.
Reservoir simulation has been used for many years to predict and analyze the performance of wells completed in hydrocarbon producing reservoirs. Reservoir simulation is used to predict and analyze the performance of wells and reservoirs for many different purposes, including both primary and enhanced recovery projects. Simulation can also be a cost-effective way of predicting the future performance of a reservoir for a variety of production or injection scenarios.
A numerical solution method that has found widespread acceptance in reservoir simulation is the finite difference numerical scheme. Both fully implicit and IMPES (Implicit Pressure-Explicit Saturation) formulations are commonly used in reservoir simulation. There are also any number of special grid generation techniques that can be used with finite difference numerical reservoir simulators.
To analyze the production data of a reservoir with numerous complexities (such as multi-phase flow, layered systems, or fractured wells) may require weeks or months to complete using a finite difference simulation model. A significant amount of reservoir research using finite difference simulation models has been conducted.
Abstract Pressure-transient models are presented for evaluation of the behavior of vertical, vertically fractured, and horizontal wells in radial and linear (three-region) composite reservoirs with moving fluid fronts. The Laplace Transform Finite Difference (LTFD) numerical solution methodology combined with the well- known Buckley-Leverett (BL) frontal-advance equation have been used to develop solutions for the moving boundary problem. Hybrid semi-analytic and numerical solutions have been constructed for finite-conductivity vertical fractures and infinite-conductivity horizontal wellbores. Complete descriptions of the mathematical models are presented; the pressure-transient solutions, frontal position and velocity, and saturation distributions. Indications are that monitoring of the transient behavior permits detection of water encroachment to the producing well, prior to breakthrough. This enables modifications in the production operations to be taken proactively to delay breakthrough. The results of the pressure-transient models reported in this paper have been compared with the available moving boundary radial composite solutions in the literature and the results of the vertically fractured and horizontal well solutions have been validated using analytic and numerical reservoir simulation. Six well and reservoir model combinations have been considered in this investigation for which oilfield applications exist for each composite system considered. These include solutions of the pressure-transient behavior of an unfractured vertical well in a radial composite reservoir, a vertically fractured well in linear and radial composite systems, a horizontal well in radial and linear composite systems, and a vertical fracture intersected by a horizontal well in a linear composite system. Extension of the solution methodology used in this study for evaluating the pressure-transient behavior of a selectively completed horizontal wellbore in a cylindrical composite reservoir has also been considered in this study. Each of these solutions include moving fluid fronts, whose position and velocity are determined from the frontal advance model and fractional flow theory. General fractional flow solutions have been implemented that utilize conventional laboratory relative permeability measurements.
Abstract This paper presents the results of an investigation concerning the development of more accurate predictive and interpretive models of the boundary-dominated flow performance of vertically fractured wells located in closed rectangularly bounded reservoirs. In particular, improvements in the characterization of the dimensionless productivity index of vertically fractured wells in closed rectangularly bounded reservoirs during boundary-dominated flow have been made using a mathematically rigorous model for pseudosteady state flow. This model has been used to develop predictive and analysis graphical design charts of the dimensionless productivity index for improved fracture stimulation design and evaluation. Other issues investigated in this work were the development of a general relationship for evaluating the pseudosteady state shape factor of a vertically fractured well in a closed rectangularly bounded reservoir, along with a review of appropriate applications of the apparent wellbore radius concept to vertically fractured wells in finite reservoirs. Example applications of the dimensionless productivity index and pseudosteady state shape factor solutions developed in this work are provided for fracture stimulation design. Fracture Design Using Dimensionless Productivity Index There have been a number of technical articles that have appeared in the literature since beginning in at least 1998 concerning the use of the dimensionless productivity index as a measure for improved fracture stimulation design under boundary-dominated flow conditions. One of the first investigations that specifically pertains to this subject is one reported by Valko and Economides. Since that time, a number of researchers have developed interpretation models and analyses that employ the dimensionless productivity index as the basis for improved fracture stimulation design. Direct field applications of this fracture stimulation design technique have been reported in the literature for moderate and high permeability reservoirs. Extension of the single phase analyses that have been commonly employed in previous studies were later extended to consider multiphase flow, in an investigation that also involved the development of a dimensionless productivity index solution for vertically fractured wells in closed cylindrical and rectangularly bounded reservoirs. The extension of the dimensionless productivity analyses for multiphase flow applications reported in Ref. 2 also verified the relationship between the dimensionless productivity index (expressed in terms of the pressure drawdown between the average reservoir pressure and the sandface flowing pressure,) and the classic transient testing dimensionless wellbore solutions for pseudosteady state flow, expressed in terms of the pressure drawdown of the system between the initial pore pressure and the sandface flowing pressure (Pi-Pwf) first reported by Ramey and Cobb. A thorough investigation on the use of the dimensionless productivity index as a design criteria for fracture stimulation design was performed by Meyer and Jacot. In that investigation, the authors developed a general solution from resistivity theory for computing the dimensionless productivity index of a vertically fractured well with an arbitrary fracture conductivity distribution, as well as providing a more fundamental and theoretical basis for the apparent wellbore radius of very low conductivity fractured wells than had existed previously. At this point, the general concept of fracture stimulation design using the dimensionless productivity index as a basis has been reasonably well established. Principally what remains to be done at this point in fracture stimulation design using the productivity index solution are further improvements in the accuracy of the design relationships, and development of an analytic and more theoretical basis for some of the design relationships that have been developed numerically or empirically from field experimentation.
This paper was prepared for presentation at the 1999 SPE Annual Technical Conference and Exhibition held in Houston, Texas, 3–6 October 1999.