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Abstract As reservoir pressure decreases with production, the conductivity of hydraulic fractures will also change. These changes in fracture conductivity may impact well productivity significantly, so it is important to investigate the behavior of dynamic fractures. Also, characterzing and forecasting dynamic fracture properties can help with well-spacing design and adjustments in development stategy. This paper proposes a semianalytical model to characterize the behavior of dynamically-closed fractures through pressure buildup test analysis. Time-dependent and space-varying fractures are considered in the modeling process. We note that, although time-dependent fractures are investigated in this study, calculation of variable-rate pressure buildup response is still based on the solution of linear differential equations governing fluid flow in a reservoir. Hence, the convolution theorem can be utilized to deal with the transient pressure buildup data. We recommend a modified exponential function to represent time-normalized fracture conductivity. The representation of dynamic fracture conductivity was handled with a stair-step function, and the fracture-flow model was solved numerically. Pressure buildup responses were obtained by coupling with an analytical matrix-flow model. When fracture conductivity is time-dependent, the pressure drop during the bilinear flow regime will increase. We discuss the effects of fracture conductivity changes and production history on the resulting pressure buildup response. We present a synthetic case study which demonstrates that pressure build-up tests conducted in different years allow us to diagnose changes in fracture conductivity during production history. The results of our interpretations can generate an equation that describes the relationship between fracture conductivity and time. Using the results of multiple pressure buildup test analyses, this method can be applied to forecast fracture characteristics in specific formations of interest.
A semianalytical solution was developed for the transient flow behavior of a reservoir with a well intersecting a partially-penetrating vertical fracture of finite conductivity. The transient pressure behavior of a well in this kind of system consists mainly of three flow periods: 1) the early time, 2) the infinite acting and 3) the pseudoradial flow periods.
The results of this study show that the flow behavior of the partially-penetrating fracture during the early time period is equivalent to that of a totally- penetrating fracture. This period consists of a penetrating fracture. This period consists of a bilinear flow period for low conductivity fractures, and of a linear reservoir flow period for moderate to highly-conductive fractures.
The onset of the infinite-acting flow period is directly proportional to the square of the dimensionless fracture height, which is defined as the ratio between the fracture height and the fracture half length. The results show that as the value of this ratio becomes small, the infinite-acting flow period starts at very early times, such that the bilinear and the linear reservoir flow periods might not appear in the well response for practical values of time.
The approximate start of the pseudo-radial flow period does not depend significantly on the fracture period does not depend significantly on the fracture conductivity, an the fracture penetration ratio, or on the dimensionless fracture height, for moderate to highly conductive fractures, and for fracture penetration ratios larger than about 0.2. The effect penetration ratios larger than about 0.2. The effect of the dimensionless fracture height on the pressure response of a partially-penetrating fractured well becomes negligible for penetration ratios larger than about 0.8.
The vertical location of the fracture affects the behavior of the well only after the upper and/or lower boundaries of the reservoir become noticeable in the pressure response of the well. The same solutions are pressure response of the well. The same solutions are found for the early time and for the infinite-acting flow periods, until the boundary effects become evident.
The effectiveness of hydraulic fracturing in increasing the productivity of damaged wells and wells located in low-permeability reservoirs has been recognized for many years.
It has been known for some time that data obtained from tests of fractured wells reflects the characteristics of the fractured well-reservoir system. Hence many studies have been undertaken to provide the means to evaluate the benefits of fracturing operations.
The effect of the conductivity of the fracture on the behavior of a fractured well-reservoir system was recognized early, and is reported in the works by van Poollen et al., Dyes et al:, McGuire and Sikora and Poollen et al., Dyes et al:, McGuire and Sikora and Prats. Steady-state results concerning the increase Prats. Steady-state results concerning the increase in productivity that a well would experience after fracturing were obtained by van Poolen et al., using a potentiometric model, by McGuire and Sikora, using an electric analog, and later by Prats through analytical procedures. Dyes et al. were mainly interested in the effect of fracturing on waterflooding operations. Their study was conducted using an electric analog.
Among subsequent works that contemplated the transient behavior of a vertically-fractured well were the work by Scott who used heat flow analogy, and a later work by Russell and Truitt who used a finite-difference formulation of the problem. Totally-penetrating fractures of infinite conductivity were considered in these studies.
Gringarten et al. obtained and analytical solution to the problem of transient flow of fluids towards fractured wells. Results were presented for the cases of wells with infinite conductivity and uniform flux vertical fractures. The method applied to solve these problems was based on the use of Green's and Source problems was based on the use of Green's and Source Functions whose usefulness in solving transient reservoir flow problems had been documented in a previous study. Type curves of the transient pressure previous study. Type curves of the transient pressure behavior of fractured wells were provided for use in type-curve matching procedures of well test data.
The purpose of this study is to develop exact analytic solutions for the pressure response of a finite conductivity fracture. This model should be able to verify the existence of various flow regimes found in earlier studies. It is hoped that this solution could be modified to give simplified expressions for well pressures for all times and all fracture conductivity ranges.
The present work poses and solves the problem of a vertical finite conductivity fracture of elliptical cross-section. The flow within the fracture is assumed to be incompressible and the reservoir is assumed to be infinite. The elliptical fracture geometry was chosen to facilitate the expression of fracture and reservoir pressures as eigenfunction expansions.
The solution is obtained by expressing the reservoir presure as a series of Mathieu functions, and the fracture pressure presure as a series of Mathieu functions, and the fracture pressure as a series of cosines. The coefficients in these series satisfy an infinite set of linear relations, termed Fredholm sum equations. Exact solutions to these sum equations are obtained in forms which resemble continued fractions of summations, or equivalently, which require iteration of rational forms. A great deal of effort has been expended to speed the calculation of the solutions, however, only partial success has been achieved.
The solutions become increasingly difficult to compute as time decreases. So, approximate solutions for well pressures are given for extremely low values of time. These solutions indicate that behavior of an elliptical fracture is essentially the same as that of a rectangular fracture. Indeed, the well pressures calculated in this work are quite dose to those for a rectangular fracture. Generally applicable simplified well solutions have not been found.
The topic of the pressure response of fractured wells is not new. Many models have been investigated which consider various aspects of the problem. However, these models either consider only a part of the problem, or they allow only approximate solution of the governing equations.
The most comprehensive model that has been investigated is the rectangular finite conductivity fracture model developed by Cinco and coauthors in a number of papers. The two most important of these are Cinco et al., where the model is proposed and the governing equations solved, and Cinco and proposed and the governing equations solved, and Cinco and Samaniego, where the behavior of the solution is investigated.
The solution procedure used in the first of these papers was a numerical solution of an integral equation. The method is computationally intensive, and, since it is numerical, yields only approximate results. The pressures computed using this method appear to be accurate, although it is difficult to say just how accurate they are.
In Cinco and Samaniego a simplified model was analyzed and used to identify the bilinear flow regime. The regimes of fracture linear flow and reservoir linear flow were also examined. The fracture linear flow regime results from expansion of fluid in the fracture. This regime is of too short a duration to be of practical use and so little is lost by assuming the fracture flow to be incompressible.
The intent of the present work is to present a model which depicts the flow of fluids into and through a finite conductivity vertical fracture. We seek an exact solution which can be used to find pressures anywhere in the fracture/reservoir system.
Although even a perfunctory survey of the literature suggests that considerable information is available on the response of finite-conductivity fractures in single-layer systems, the influence of the settling of propping agents and the effect of fracture height on the well response need to be examined. These topics are examined in this paper. We suggest methods to analyze well performance when the fracture paper. We suggest methods to analyze well performance when the fracture conductivity is a function of fracture height and fracture length. The performance of wells with fracture height greater than the formation performance of wells with fracture height greater than the formation thickness is documented. The consequences of being unable to contain the fracture within the pay zone are also examined. Although incidental to this study, we found that solutions presented by various authors are not in agreement for all time ranges. In this paper, we discuss a systematic procedure to obtain a grid (mesh) so that paper, we discuss a systematic procedure to obtain a grid (mesh) so that accurate results are obtained by a finite-difference model. This procedure can be used for both two-dimensional (2D) and three-dimensional (3D) problems. problems. Introduction
This paper examines the performance of wells intercepting finite-conductivity vertical fractures. Although much work has been presented in this area of pressure analysis, several aspects of well behavior have yet to be examined. We examine some of these topics. In this work we examine the influence of vertical variations in fracture conductivity on well performance. Concerns regarding the effect of the settling of propping agents on well productivity addressed in this paper complement our work on the influence of lateral variations in fracture conductivity. We also examine situations where the fracture extends below and/or above the productive interval. We consider two possibilities: (1) the fracture length is assumed to be fixed and the fracture height is variable (volume of fracture treatment is variable); and (2) the fracture volume is assumed to be fixed (the product of the fracture half-length and fracture height is assumed to be constant). The latter case is of interest if one is not able to contain the fracture within the pay zone of interest. In the former case, we show that this approach provides a means to increase the effective fracture conductivity. These topics have not previously been examined in the literature. Verification of the finite-difference model used in this study consumed a significant portion of the time spent in this study. Although some works have reported problems in obtaining accurate solutions, no guidelines for choosing a grid (mesh) for this problem are available. We give empirical guidelines for systematically choosing a grid to obtain accurate solutions. We believe that these guidelines will significantly reduce the time spent by researchers in developing their own models and consider it an important contribution. We also present methods to modify grids developed for a given set of conditions if the fracture and/or reservoir dimensions are changed.
SPE Members Abstract This paper presents an investigation of the pressure response on hydraulically fractured wells flowing at constant flow rate through an asymmetric vertical fracture. The pressure behavior of wells intercepting asymmetric fractures of both infinite and finite conductivity was investigated by solving numerically and analytically the mathematical model. New solutions of the dimensionless wellbore pressure under production at constant flow rate are developed and are presented in terms of an asymmetry factor. New curves for these systems were generated and the deviation from the classical solution was readily detected. Some qualitative criteria to interpret the intensity of this effect are provided. Results of our investigation demonstrate that the relative position of the well in the fracture, i.e. the asymmetry condition, is an important consideration for the fracture characterization. Simulated pressure well tests indicated that at early times for fractures of moderate conductivity (CD < 5) the characteristic slope of one fourth is present, except for those cases of intense asymmetry (0.85 < xxlt; 1) where no evidence of straight line having one fourth slope was observed. However, it was also detected that at intermediate fracture conductivities (5 < CD < 50), even the case of complete asymmetry shows the characteristic slope of one fourth. It was also observed that as the asymmetry factor increases the end of the bilinear flow occurs earlier. The tabular solutions presented in this paper describe quantitatively the pressure behavior of fractured wells producing from asymmetric fractures for a wide range of asymmetry conditions. Our results are relevant in improving the fracture characterization of fractured wells as well as in the design of fracturing operations. Introduction Virtually all previous theoretical analyses of fractured wells have used the same restrictive assumptions used by Gringarten et. al, Cinco et. al, Wong et. al and Tiab postulating a symmetrycally homogeneous fracture frame. A few studies of the fracture asymmetry for fractured wells are available, while many others are available for the behavior of the symmetric fractures. The relevance of fracture asymmetry to pressure analysis was first discussed by Crawford and Landrum forty two years ago, but its significance in fracturing design applications has not yet been fully recognized. Demonstration of asymmetry effects in transient pressure analysis have also been made recently by Rodriguez et. al and Resureicao y Rodriguez. Among other evidences of this phenomenon, we would like to cite the fact that perforating scheme in wells, some heterogeneities along with the stress field gradients of developed reservoirs may be responsible of the presence of asymmetric fractures. This work is concerned with the analysis of fracture asymmetry effect on the pressure response of fracture wells. Solutions for this case are presented in tabular form for various asymmetry factors. The analysis is based on the numerical and analytical solutions to this problem, assuming constant conductivity and Darcy flow conditions.