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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.
A relatively simple semi-analytical model is developed for a well in a square with a fully penetrating fracture. Building upon this unit system with penetrating fracture. Building upon this unit system with uniform flux at the fracture face, the effects of storage, fracture length, fracture capacity, formation properties and reservoir boundaries are investigated.
New type curves which simultaneously incorporate concepts of uniform flux and finite fracture capacity are presented for both the constant flow rate and constant terminal pressure cases. The results of this study compare favorably with appropriate published data.
The practical application of the theoretical results to pressure transient testing of fractured wells is emphasized. The dominant flow regimes and their dimensionless time durations are identified for specification of appropriate data analysis methods. The result is establishment of a systematic method for evaluation of hydraulic fracture effectiveness. Furthermore, the technique can be readily applied for generating the corresponding well performance forecast. The utility of the method is demonstrated using real field data.
With the accelerating demand for domestic natural gas and oil, the use of large hydraulic fracture treatments for stimulation of low permeability reservoirs has become a common practice. Improved diagnostic techniques are needed for evaluating the effectiveness of these treatments. More precise post-stimulation assessment will allow for better control of design for future treatments. Also, accurate characterization of the fractured well is needed for performance forecasting and optimal reservoir development. Conventional fracture analysis methods are inadequate for evaluation of large propped fractures in tight formations. The application of these methods in such situations has often produced incorrect results.
The concept of the finite capacity fracture has done much to improve understanding of the influence of large hydraulic fractures upon well performance and pressure transient behavior. The characteristic flow pressure transient behavior. The characteristic flow behavior of the finite capacity fractured well has been discussed in the literature and type curves have been published for the evaluation of fracture effectiveness. published for the evaluation of fracture effectiveness. Practical considerations, however, have limited Practical considerations, however, have limited the use of available type curves and theoretical predictive methods. The finite capacity type curves predictive methods. The finite capacity type curves suffer from a lack of uniqueness. Indiscriminate use of these curves will produce erroneous results. While it has been demonstrated that a reliable analysis may be performed through a judicious application of type curves and analytical methods it is recognized that a better understanding of the characteristic fractured well flow behavior is needed. This knowledge can be acquired through development of relatively simple system models. With very sophisticated models, insight into the problem is often lost through the complexity of the solution.
In the present study, a tractable model is developed to describe the physics of the fractured well system. The model is based upon the drawdown behavior of a well intercepted by a finite capacity fracture where a uniform flux is maintained at the fracture face. Effects of storage, formation properties, fracture properties and reservoir boundaries are considered. Emphasis is placed upon the practical application of the Uniform Flux Finite Capacity Fracture (UFFCF) model to pressure transient testing and analysis.
The basic unit in the UFFCF model is a well in a closed square reservoir intersected by a finite capacity vertical fracture. The fracture fully penetrates the reservoir height and width. This unit system is shown schematically in Figure 1.
This study presents the solution for the pressure response during a slug test in a horizontal well.
The physical model consists of a fluid of small and constant compressibility flowing through an infinitely large anisotropic reservoir, with upper and lower impermeable boundaries. Wellbore storage and skin are included.
The solution is obtained by applying the concepts of instantaneous sources and Green's functions.
Both the uniform flux and the infinite conductivity models are discussed. Conclusions obtained by other authors using numerical simulation are reached analytically in this work, by employing a now method of treating the infinite conductivity model. These conclusions, presented in published studies on vertically fractured and on horizontal wells, include the existence of a steady-state flux distribution for long periods of time, as well as the fact that the pressure distribution during the stabilization period is independent of the production history.
This study presents the application of a new analytical model for analyzing pressure transient data for wells intercepted by a finite-conductivity vertical fracture in a closed square multilayered reservoir using the concept of a dual-porosity system. The solution with the single-layer solution; secondly, to verify the numerical solution with the analytical solution. A set of asymptotic analytical (early-, intermediate-and late-time) solutions is also presented.
In addition, we developed constant rate drawdown type curves applicable for analyzing transient pressure data. These type curves are used to match drawdown and buildup curves to determine fracture half-length, reservoir permeability and fracture conductivity. The results obtained from our type curves indicate that if the reservoir conductivity is known or can be calculated from given data, the estimated values of the aforementioned parameters compared favorably with the value of the designed parameter.
Although this study presents the solution to a two-layered system, it can be extended to a n-layered reservoir system.
The double-porosity models originated by Barenblatt et al and extended by many other are generally used to represent naturally fractured porous systems by superimposing two continua, one for fractured system and another for the porous matrix. In a previous study a double porosity model was used to previous study a double porosity model was used to simulate the behavior of tight multi-layered reservoirs intercepted by hydraulically induced vertical fracture of finite conductivity. Although several studies have been conducted on either wells producing double porosity reservoirs or fractured producing double porosity reservoirs or fractured wells in homogeneous reservoirs there are very few studies that include fractures in double porosity multi-layered reservoirs. Using the work of Barenblatt et al Warren and Root extended the transient pressure analysis of fractured water reservoirs to oil reservoirs and introduced the concept of "two porosity" system. While Warren and Root assumed pseudo-steady state interporosity flow Kazemi's and deSwaan's model allowed for transient interporosity flow. The concepts of wellbore storage and skin were later incorporated by Mavor and Cinco-ley.
Both models have been established to exhibit three well defined flow periods. These periods include: early time, which is dominated by storativity of the natural fractures, the intermediate time, when the fluid transfer from matrix to fractures becomes the dominating factor and the pressure in the network of the fractures tends towards stabilization and lead to transition period, and long time, when the flow is controlled by the storage capacity of the entire system. The behavior of a double porosity system is normally correlated by the fracture storage capacity parameter w and the interporosity flow parameter nf or parameter w and the interporosity flow parameter nf or dimension less matrix hydraulic diffusivity nmaD.
In the absence of fractures and layers, the studies by Cinco-Ley et al and Cinco-Ley and Samaniego-V. provide a complete analysis of the behavior of a well provide a complete analysis of the behavior of a well intersecting a finite conductivity fracture. For double porosity reservoirs Houze et al developed a model to study the behavior of wells intersected by infinite conductivity and uniform flux vertical fractures.
The effect of the partial penetration of an infinite conductivity fracture on the transient pressure behavior of a vertically fractured well is investigated.
Analysis of results shows that the pressure behavior of a well intersected by a partially-penetrating infinite conductivity vertical fracture can be divided into three flow periods: 1) the early time flow period which is characterized by a formation linear flow as in the case of a fully-penetrating infinite-conductivity vertical fracture, 2) the infinite-acting flow period and 3) the pseudoradial flow period which develops after the effects of the vertical boundaries of the reservoir are felt in the pressure behavior of the well.
A log-log graph of log(hf/h)pWD versus log t shows a slope of one half during the early time, flow period of a well with an infinite-conductivity period of a well with an infinite-conductivity partially penetrating fracture. The time for the end of the partially penetrating fracture. The time for the end of the early time flow period is directly related to the square of the dimensionless height of the fracture, hfd which is defined as the ratio between the height of the fracture and its half length. This time becomes so small for small values of hfD that the linear formation flow period will not show up for practical purposes.
The solutions during the infinite-acting flow period are, for the different values of penetration period are, for the different values of penetration ratio and a given value of characterized by a single envelope curve in the log-log graph previously mentioned.
The time for the start of the pseudoradial flow period shows no dependence on the penetration ratio and period shows no dependence on the penetration ratio and on the dimensionless fracture height except for fractures with dimensionless heights larger than about 1 and penetration ratios smaller than 1/5.
Even though solutions were mainly obtained for the case of a fracture located at the middle of the formation, the effect of offsetting the fracture was investigated for some particular cases. As expected, the solutions are the same up to the end of the infinite acting flow period which slightly varies as a function of the fracture location.
The solutions are presented in the form of type curves suitable for transient pressure analysis procedures. procedures
Most of the work done on the behavior of fractured wells have considered a totally-penetrating fracture and only a few studies have dealt with the effects of a partially-penetrating fracture. partially-penetrating fracture. Tinsley et al. used an electrolytic model to deter mine the effect of the height of the fracture on the production of a well intersected by a finite- production of a well intersected by a finite- conductivity vertical fracture under steady-state conditions.
Cinco derived an analytical solution for the unsteady-state distribution created by a well intersected by a partially-penetrating uniform-flux inclined fracture in an infinite slab reservoir. An expression for the pseudo-skin factor was derived as well, based on a late time approximation to the solution. No further analysis was however presented.
Raghavan et al. investigated the effect of the partial penetration of a vertical fracture on the partial penetration of a vertical fracture on the transient pressure behavior of a fractured well. A uniform flux across the fracture was assumed, for which case an analytical solution was possible. An approximate solution for the transient pressure behavior well with a partially-penetrating infinite- conductivity fracture was obtained by evaluating the uniform flux solution at a point in the fracture which was assumed to yield the infinite-conductivity solution. This approximation was an extension of the relations between the infinite and uniform-flux solutions previously found by Gringarten et al., in the case of a previously found by Gringarten et al., in the case of a totally-penetrating vertical fracture, and by Muskat in the case of steady-state flow towards a partially-penetrating well. The solution to the partially- partially-penetrating well. The solution to the partially- penetrating infinite-conductivity problem would have required penetrating infinite-conductivity problem would have required however of a semi-analytical type of solution.
Mao studied the productivity of vertically fractured wells using a potentiometric model. Finite conductivity and partial penetration were considered.
It is the purpose of this paper to formulate and to solve properly the unsteady-state flow problem to a well with an infinite-conductivity partially- penetrating fracture and to present the results in a suitable penetrating fracture and to present the results in a suitable form for fracture design and evaluation procedures.