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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.
This paper presents analytic solutions of the pressure transient behavior of a well intersected by a finite-conductivity fracture in an infinite-acting, or in cylindrically or rectangularly bounded finite reservoirs. These solutions include the practical effects of reservoir permeability anisotropy and dual porosity behavior. Those solutions are analytic, and thus do not require discretization in space.
The analytical solutions of the finite-conductivity fracture transient behavior presented in this paper eliminate the numerical difficulties associated with other mathematically rigorous finite-conductivity fracture solutions that have been reported in the literature. Both the pressure and rats transient responses can be accurately evaluated using the finite conductivity fracture solutions presented in this paper. This is especially important for low-conductivity fractures, for which the pressure and rate transient behavior is often difficult to evaluate accurately using the solutions available in the literature.
A model recently presented by Cinco et al. for the transient pressure behavior of wells with finite conductivity vertical fractures was modified to include the effects of wellbore storage and fracture damage. An infinitesimal skin was considered around the fracture, and it was handled as a dimensionless factor defined as (pi/2)(wd/xf)[(k/kd) - 1].
It was found that the well behavior is importantly affected by the fracture damage. When plotted as a function of log pwD vs lot tD for short plotted as a function of log pwD vs lot tD for short times, results show flat, almost horizontal lines that later become concave upward curves asymptotically approaching the curve for undamaged fractures. This behavior is shown even by slightly damaged fractures. It also was found that important information about the fracture characteristics may not be determined when wellbore storage effects are present. present
It has been shown that the increase in the productivity of a well created by hydraulic productivity of a well created by hydraulic fracturing depends on fracture characteristics, such as fracture conductivity, length, penetration, and also on a possible damage to the penetration, and also on a possible damage to the formation immediately surrounding the fracture. During the last few years, there has been a continuously increasing interest in the determination of the characteristics and orientation of fractures by means of transient pressure analysis. Most of these methods consider the fracture to be of infinite conductivity or of uniform flux; others consider finite conductivity fractures. Generally, these methods assume that there is no skin damage around the fracture. Evans proposed a pressure analysis technique considering fracture skin damage. He assumed the flow from the formation to the fracture to be linear, passing through two porous media in series, one being the damaged zone around the fracture and the other the undamaged formation. Ramey and Gringarten discussed the transient well behavior of vertically fractured wells with large wellbore storage, and suggested a matching technique for analyzing pressure data. Recently, Raghavan discussed pressure analysis techniques for vertically fractured wells, including the effects of wellbore storage and skin. He assumed the fracture to be of uniform flux, and presented general characteristics of the pressure transient behavior for these systems.
The purpose of this study is to present solutions for the transient wellbore pressure behavior of a well crossed by a finite conductivity vertical fracture, considering the effect of a damaged zone around the fracture and wellbore storage. It is also intended to show the general flow characteristics of these fractured systems.
MATHEMATICAL MODELS AND METHODS OF SOLUTION
The transient flow toward a well with a finite conductivity vertical fracture surrounded by a damaged zone was studied by using a modified version of the model presented by Cinco et al. The following assumptions were considered.
1. An infinite, homogeneous, isotropic reservoir of permeability k, porosity phi, and thickness h.
2. The formation is produced through a vertically fractured well. The wellbore is intersected by a fully penetrating vertical fracture of permeability kf, porosity cf, width w, and permeability kf, porosity cf, width w, and half-length xf. All production of fluid is via the fracture.
3. There is a zone of reduced permeability caused by fracturing fluid loss around the fracture. This region has a permeability ks and width ws.
4. The porous medium contains a slightly compressible fluid of viscosity mu and compressibility c.
5. All formation, fracture and fluid properties are independent of pressure.
6. Gravity effects are negligible and pressure gradients are small everywhere.
During the last few years, there has been an explostion of information in the field of well-test analysis. Because of increased physical understanding of transient fluid flow, it is possible to analyze the entire pressure history of a well test, not just long-time data as in conventional analysis.1 It is now often possible to specify the time of beginning of the correct semilog straight line and determine whether the correct straight lie has been properly identified. It is also possible to identify wellbore storage effects, and the nature of wellbore stimulation as to permeability improvement, or fracturing, and to quantitatively analyze those effects.
Such accomplishments have been augmented by attempts to understand the short-time pressure data from well testing - data that were often classified as too complex for analysis. One recent study of short-time pressure behavior2 showed that it was important to specify the physical nature of the stimulation in considering the behavior of a stimulated well. That is, stating that the van Everdingen-Hurst infinitesimal skin effect was negative was not sufficient to define short-time well behavior. For instance, acidized (but not acid-fractured) and hydraulically fractured wells might not necessarily exhibit the same behavior at early times, even though they could possess the same value of negative skin effect.
In the same manner, hydraulic fracturing leading to horizontal or vertical fractures could produce the same skin effect, but with possibly different short-time pressure data. This could then provide a way to determine the orientation of fractures created by this type of well stimulation. In fact, it is generally agreed that hydraulic fracturing usually results in one vertical fracture, the plane of which includes the wellbore. Most studies of the flow behavior for a fractured well consider vertical fractures only.3-11
Yet it is also agreed that horizontal fractures could occur in shallow formations. Furthermore, it would appear that notch-fracturing would lead to horizontal fractures. Surprisingly, no detailed study of the horizontal fracture case had been performed until recently.12 A solution to this problem was presented by Gringarten and Ramey.13 In the course of their study, it was found that a large variety of new transient pressure behavior solutions useful in well and reservoir analysis could be constructed from instantaneous Green's functions.14 Possibilities included a well with a single vertical fracture in an infinite reservoir, or at any location in a rectangle.
Abstract During hydraulic fracturing process, different hydraulic loading and stress status of formations result in hydraulic fractures with various geometries and properties. Several propagation models including PKN and KGD have been widely applied in fracturing design and implementation. However, in the process of post-stimulation modeling, fractures are usually simplified with uniform geometries and conductivity distribution; therefore the effects of actual fracture geometry and proppant properties on the well transient pressure and production performance remain unclear. This study intends to comprehensively study the fractured wells with 2-D and 3-D non-uniform geometry and conductivity distribution. In the development of shale gas reservoirs and tight oil formations, horizontal well multistage fracturing is the key technology. The modeling results presented in this paper can help offer valuable information of reservoir properties, evaluate the conductivity distribution of propped fractures, simulate more realistic fracture configurations, and help optimize fracture treatment process and fractured wells’ performances with improved accuracy. A semi-analytical approach coupling fluid flow in reservoir and fractures existed in more realistic shape with non-uniform conductivity distribution has been developed to obtain well transient pressure and production responses. Source and sink function method is utilized to solve unsteady state flow problems of fluid flowing from reservoir to non-uniform fractures with geometries that are well defined in PKN, KGD and other generally ideal models. The effect of fracture conductivity with linear and stepwise distribution, and elliptic fracture shape variations has been investigated. Comparison study has been highlighted to illustrate effects of fracture geometry and conductivity distributions. Realistic hydraulic fractured wells with non-uniform fracture geometry and conductivity have been studied to showcase a consistent workflow of entering fracture properties from hydraulic fracturing models and outputting fractured well performance prediction in post-stimulation reservoirs. Instead of assuming pseudo-steady state flow status between reservoir and fracture, unsteady state flow problems related to non-uniform fracture geometric have been solved in a semi-analytical manner with solution of near analytical accuracy. More realistic fracture geometries estimated from fracture propagation models can be entered into post-stimulation models without idealized simplification; thus the gap between fracture propagation and post-stimulation modeling has been fulfilled.